Fluidic Nozzle Throats in Solid Rocket Motors [1st ed.] 978-981-13-6438-9;978-981-13-6439-6

Table of contents :
Front Matter ....Pages i-xi
Introduction (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 1-19
Steady Characteristics of a Gas–Gas Aerodynamic Throat (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 21-51
The Characteristic Function and Nozzle Efficiency (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 53-67
The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 69-93
Secondary Flow TVC for Fluidic-Throat Nozzles (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 95-133
Gas–Liquid Fluidic Throat (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 135-169
Thrust Modulation Process of Fluidic Throat for Solid Rocket Motors (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 171-182
Erosion Characteristics of Fluidic Throat in Solid-Rocket Motors (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 183-195
Nozzle Damping of the Fluidic Nozzle (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 197-206
System Application Modes and Key Technologies (Kan Xie, Xinmin Chen, Junwei Li, Yu Liu)....Pages 207-237

Citation preview

Kan Xie · Xinmin Chen · Junwei Li · Yu Liu

Fluidic Nozzle Throats in Solid Rocket Motors

Fluidic Nozzle Throats in Solid Rocket Motors

Kan Xie Xinmin Chen Junwei Li Yu Liu •

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Fluidic Nozzle Throats in Solid Rocket Motors

123

Kan Xie Jet Propulsion Lab, School of Aerospace Engineering Beijing Institute of Technology Beijing, China Junwei Li School of Aerospace Engineering Beijing Institute of Technology Beijing, China

Xinmin Chen China Academy of Launch Vehicle Technology Beijing, China Yu Liu Beihang University Beijing, China

Funded by B&R Book Program ISBN 978-981-13-6438-9 ISBN 978-981-13-6439-6 https://doi.org/10.1007/978-981-13-6439-6

(eBook)

Jointly published with National Defense Industry Press, Beijing, China The print edition is not for sale in China Mainland. Customers from China Mainland please order the print book from National Defense Industry Press. Library of Congress Control Number: 2019932829 © Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 This work is subject to copyright. All rights are reserved by the Publishers, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

It is a worldwide trend to develop solid-rocket engines with the trust being able to adjustwithin a broad range in real time. This key technology is under great and urgent demands in many national defense technologies even for many developed countries during these years. For example, these technologies include missile defense systems, solid rocket-ramjet engines, avoidance maneuver of warheads, buffer systems for landers (reentry capsules), multi-mission missiles, energy management technology for wide flight profiles, etc. During the past few decades, researchers have done a great deal of theoretical and experimental work of the thrust-controlled rocket engine, and exploited many techniques and design methods. Practical ones have already been in use today, such as gel propellant engines, the attitude- and orbit-control system of solid rocket motors on the kinetic kill vehicle (KKV), lateral force engines of the “Asters” surface-to-air missile, etc. The new fluidic nozzle throat (FNT) technology is rapidly developed in the field of the aeroengine and space engine. The technology is technically novel and has a wide prospect. This is because it can not only be applied to realize the real-time adjustment of thrust scales and directions of the solid-rocket engines, but also controls flow in solid propellant gas generators and actively protects the nozzle-throat insert of solid-rock engines that work for long hours from thermal damage. Considering that there are very few resources for the public to directly refer to, the authors combined basic theories with their own experience and summarized these in this book. We also hope that this book can inspire people to be a member to participate in further research in this field. The book comes in ten chapters. Chapter 1 is co-written by Yu Liu and Kan Xie; Chaps. 2–4 are written by Kan Xie; Chaps. 5 and 6 are co-written by Xinmin Chen and Kan Xie; Chaps. 7 and 8 are co-written by Junwei Li and Kan Xie; Chaps. 9 and 10 are co-written by Yu Liu and Kan Xie; the book is finally compiled and edited by Kan Xie. The drafting of the book was supported by leaders and peers of the departments of weapons and aerospace industry of Beijing Institute of Technology and Beijing University of Aeronautics and Astronautics, who provided valuable comments and suggestions. Academician Zhusheng Liu from the Chinese Academy of Sciences is highly appreciated for his reviewing efforts and approval to v

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Preface

the first draft of the book. Professor Ningfei Wang from the Chinese Academy of Sciences also provided guidance to the book. We also would like to express our heartfelt thanks to the staffs in No. 101 laboratory of the Sixth Research Academy of China Aerospace Science and Technology Corporation and Xi’an Modern Chemistry Research Institute for assistance in the ground-based experiments and motor schemes of fluidic-throat nozzle of solid rocket motors. We also would like to appreciate Xiaolong Yi, Dongfeng Yan, Yang Cheng, Long Bai, Jiahui Song, Qimeng Xia, Yuanyuan Wei, Xiangrui Zou, Yuejie Li, Haixia Zhou, Haoxiang Yuan, and Mingjie Zhong for their great help in writing and organizing this book. Moreover, we sincerely appreciate the reviewers and editors for their work and constructive suggestions, which indeed greatly help to improve our work. Beijing, China

Kan Xie

Contents

1

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Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Application Background of Fluidic-Throat Nozzles . . . . . . . 1.2 Theory and Characteristics of Fluidic Throat for Solid Rocket Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Thrust Modification Theory . . . . . . . . . . . . . . . . . 1.2.2 Work Modes and Characteristics of Fluidic Throat for Solid Rocket Motors . . . . . . . . . . . . . . . . . . . . 1.3 General Study Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 General Introduction . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Study on TVC of Fluidic-Throat Nozzles Based on Secondary Flow Injection . . . . . . . . . . . . . . . . . 1.4 Design Procedure of Solid Rocket Motors with FNT . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Steady Characteristics of a Gas–Gas Aerodynamic Throat . . . . . 2.1 The Effective Throat Area of Fluidic Throat . . . . . . . . . . . . . 2.2 The Analytical Methods for Steady Aerodynamic Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Experimental Methods and Apparatus for Cold-Flow Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Computational Fluidic Dynamics . . . . . . . . . . . . . . . 2.3 Flow Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Loop-Slot Fluidic Throat . . . . . . . . . . . . . . . . . . . . 2.3.2 Fluidic Throat with the Rounded-Hole Injectors . . . . 2.3.3 Relationships Between Dimensionless Jet Position and Characteristics of the Fluidic Throat . . . . . . . . . 2.4 Relationship Between Secondary Flow Ratio and Total Pressure Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Secondary Flow Parameters and Effective Throat Area . . . . .

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2.5.1

Total Temperature of Secondary Flow and Equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 The Mass Flow Rate Ratio Between Secondary Flow and Main Flow . . . . . . . . . . . . . . . . . . . . . . . 2.5.3 Comparison of Annular-Tuyere, and Rounded-Hole, Injectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 The Number of Secondary Flow Injectors . . . . . . . . 2.5.5 Area Ratio of the Secondary Flow . . . . . . . . . . . . . . 2.5.6 Injection Angle of the Secondary Flow . . . . . . . . . . 2.5.7 Effect of the Expansion Ratio on the Secondary Flow Injector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Effect of the Main Nozzle Parameters . . . . . . . . . . . . . . . . . 2.6.1 Convergent Section of the Main Nozzle . . . . . . . . . . 2.6.2 Back Pressure Ratio . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Comparison of Secondary Flow Injection Schemes in the Fluidic Throat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Layout of Secondary Flow Injectors . . . . . . . . . . . . 2.7.2 Negative and Active Schemes . . . . . . . . . . . . . . . . . 2.7.3 Multi-injector Combination Scheme . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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The Characteristic Function and Nozzle Efficiency . . . . . . . . . . 3.1 Performance Characterization and Calculation Method for the Fluidic Throat in Solid Rocket Motors . . . . . . . . . . 3.1.1 Characteristic Function . . . . . . . . . . . . . . . . . . . . . 3.1.2 Thrust Efficiency of FNTs . . . . . . . . . . . . . . . . . . 3.2 Curves of Characteristic Function for FNTs . . . . . . . . . . . . 3.3 The Thrust Efficiency of the Fluidic-Throat Nozzle . . . . . . . 3.3.1 Modified Mass Flow Rate Ratio Versus Thrust Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Back Pressure Ratio NPR Versus Thrust Efficiency 3.3.3 Injector Area Ratio Versus Thrust Efficiency . . . . . 3.3.4 Injector Angles Versus Thrust Efficiency . . . . . . . . 3.3.5 Injector Numbers Versus Thrust Efficiency . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Disturbance of the Particle Phase on the Fluidic Throat . . 4.2 Two-Phase Flow Theory and the FLNT-V1.1 Analytical Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Particle–Gas Two-Phase Flow Models . . . . . . . . 4.2.2 Control Functions of Two-Phase Flow . . . . . . . . 4.2.3 Nozzle Loss and Standard Calculation Cases . . .

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4.3

The Influence of Particle Sizes . . . . . . . . . . . . . . . . . . . . 4.3.1 Cases of Axial-Gap Nozzles . . . . . . . . . . . . . . . 4.3.2 Cases of Round-Hole Nozzles . . . . . . . . . . . . . . 4.4 The Influences of Mass Fraction of Particle Phase . . . . . 4.4.1 Cases of Axial-Gap Nozzles . . . . . . . . . . . . . . . 4.4.2 Cases of Round-Hole Nozzles . . . . . . . . . . . . . . 4.5 Influences of Profiles of Particle-Size Distribution . . . . . . 4.5.1 Control Performance . . . . . . . . . . . . . . . . . . . . . 4.5.2 Two-Phase Flow Fields and Particle Tracks . . . . 4.5.3 Choosing Injector Positions in Two-Phase Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Efficiency of Fluidic-Throat Nozzle in Two-Phase Flow Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Secondary Flow TVC for Fluidic-Throat Nozzles . . . . . . . . . 5.1 Work Modes and TVC Manners . . . . . . . . . . . . . . . . . . 5.2 Performance Characterization of TVC . . . . . . . . . . . . . . 5.3 Six-Component Forces Tests of Secondary Flow TVC . . 5.3.1 Six-Component Force Test Methods and Thrust Stand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Mechanical Models of the Six-Component Force Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Measurement Error and Calibration . . . . . . . . . . 5.4 Injector Positions and TVC Characters . . . . . . . . . . . . . . 5.5 Influences of Fluidic-Throat Jet on SVC . . . . . . . . . . . . 5.5.1 Asymmetric Injection in 2D FNTs . . . . . . . . . . . 5.5.2 Actual Lateral Forces Modification . . . . . . . . . . 5.6 Influence Factors of TVC . . . . . . . . . . . . . . . . . . . . . . . 5.6.1 Influence of Back Pressure NPR . . . . . . . . . . . . 5.6.2 Influence of Injection Angle . . . . . . . . . . . . . . . 5.6.3 Influence of Injector Area Ratio . . . . . . . . . . . . 5.7 Arrangement Concepts of Round-Hole Injectors . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Gas–Liquid Fluidic Throat . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Flow Characters of Gas–Liquid Throat . . . . . . . . . . . . . 6.2 Liquid Injection Forms and Atomization . . . . . . . . . . . 6.2.1 Test Methods . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.2 Atomization in the Nozzle Plume . . . . . . . . . . 6.2.3 Atomization in the Throat . . . . . . . . . . . . . . . . 6.3 Comparison Between Gas and Liquid Secondary Flows

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Thrust Modulation Process of Fluidic Throat for Solid Rocket Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Calculation Method of Internal Ballistics . . . . . . . . . . . . . . 7.1.1 Description of Mass Sources . . . . . . . . . . . . . . . . . 7.1.2 Discrete Method for N-S Equations . . . . . . . . . . . . 7.2 Modulation Process at Fixed Mass Flow Rates . . . . . . . . . . 7.3 Unsteady Process for the Axial-Gap Fluidic Throat . . . . . . . 7.3.1 Typical Unsteady Work Process . . . . . . . . . . . . . . 7.3.2 Influence of Chamber Volume . . . . . . . . . . . . . . . 7.3.3 Influence of Injection Angle . . . . . . . . . . . . . . . . . 7.4 Thrust Modulation for the Active Hole-Type Fluidic Throat 7.4.1 Characteristics of Mass Flow Rate Modification . . . 7.4.2 Thrust Modification Characteristic . . . . . . . . . . . . . 7.4.3 Filling Process . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Erosion Characteristics of Fluidic Throat in Solid-Rocket Motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Material of Nozzle Lining . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Calculation Method for Predicting Erosion Rate of the Nozzle Throat in Solid Rocket Motors . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Boundary Conditions at Gas–Solid Boundary . . . . . . 8.2.2 Gas Composition . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Chemical Reaction System . . . . . . . . . . . . . . . . . . . 8.2.4 Calculation Methods of Physical Parameters of Gas . 8.3 Erosion of Traditional SRM Nozzles—The Borie Nozzle Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Erosion of FNT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Active Thermal Protection Method Based on FNT . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Nozzle Damping of the Fluidic Nozzle . . . . . . . . . . . . . . . . . . . . 9.1 Wave Attenuation Method for Predicting Nozzle Damping of the Fluidic Nozzle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 Validation of Numerical Prediction Method for Nozzle Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 Effect of the Secondary Flow Injection on Nozzle Damping 9.4 Effect of Temperature of the Secondary Flow on Nozzle Damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 System Application Modes and Key Technologies . . . . . . . . . 10.1 System Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.1 ACS System . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1.2 Main Propulsion System . . . . . . . . . . . . . . . . . . 10.1.3 System Types of Different Work Media . . . . . . . 10.2 Typical Engine Structure . . . . . . . . . . . . . . . . . . . . . . . . 10.3 Injector Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Solid Generator of Secondary Flow . . . . . . . . . . . . . . . . 10.4.1 Reliable Ignition Technology . . . . . . . . . . . . . . 10.4.2 Gas Filtering Technology . . . . . . . . . . . . . . . . . 10.4.3 Gas Cooling Technology . . . . . . . . . . . . . . . . . 10.4.4 Pressure Regulation Technology . . . . . . . . . . . . 10.5 Thrust Modulation and Performance . . . . . . . . . . . . . . . 10.5.1 Modulation Modes . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Typical Thermal Experimental Results . . . . . . . . 10.5.3 Modification Process with Gas-Catalysis Particle Two-Phase Flow Injection . . . . . . . . . . . . . . . . 10.5.4 Performance and Thrust Modification Arrange . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Chapter 1

Introduction

Abstract As the power unit of missiles, an engine is required to have the ability of thrust control, especially with the ability of random thrust control, to enhance the vehicle maneuverability and penetration ability of missiles. Thrust control technology is an important research field of solid rocket motors. Compared to the thrustpredetermined solid rocket motor (single-chamber dual-thrust motor, etc.), random thrust control can more reasonably distribute the energy of propellant according to real-time work needs, which is a development trend for solid rocket motors. Achieving random control of thrust will mean a major breakthrough in solid rocket motor technology. Randomly changing the nozzle-throat area is an effective method of adjusting the rocket motor thrust. In a fixed nozzle contour, there are mainly two types of methods to change the nozzle-throat area: mechanical methods and fluid injection. This chapter mainly introduces the research of fluid methods.

1.1 Application Background of Fluidic-Throat Nozzles As the power unit of missiles, an engine is required to have the ability of thrust control, especially with the ability of random thrust control, to enhance the vehicle maneuverability and penetration ability of missiles. Thrust control technology is an important research field of solid rocket motors. Compared to the thrust-predetermined solid rocket motor (single-chamber dual-thrust motor, etc.), random thrust control can more reasonably distribute the energy of propellant according to real-time work needs, which is a development trend for solid rocket motors. Achieving random control of thrust will mean a major breakthrough in solid rocket motor technology. Since the 1960s, scholars in other countries have done a lot of work on the theoretical and experimental research of thrust control on solid rocket motors. Many technical approaches and solutions have been explored, and some have practical use such as the booster motor of Trident missile, the gel-propellant engine program studied in the former Soviet Union, etc. As of 1993, the United States has conducted extensive research on small propulsion systems to meet the needs of the Theater Missile Defense (TMD) System. TMD system has to cover large combat airspace with a minimum number of launch units and missiles, and defend against target groups at close or long range. This requires a fully controllable orbit control and attitude © Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_1

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1 Introduction

control system to control the interceptor. The maneuvering flight predicts the interception error by correcting lateral thrust and guides the kinetic energy warhead to directly collide with the target for effective interception. The development process has demonstrated that the solid attitude control system is as flexible as the liquid attitude control system, while the former is safer and more reliable. Randomly changing the nozzle-throat area is an effective method of adjusting the rocket motor thrust. In a fixed nozzle contour, there are mainly two types of methods to change the nozzle-throat area: mechanical methods and fluid injection. An important mechanical method of using a moving pintle to change the throat area requires the installation of a drive mechanism system (in Fig. 1.1) [1–6]. The drawback of this method is that the drive servos that drive the pintle are generally large in size and mass so that they are primarily suitable for small solid rocket motors. Figure 1.1 shows a coaxial pintle solid rocket motor developed by Beijing Institute of Technology. The size and mass of the servomechanism of the pintle are comparable to those of the engine body. When the method is applied to a large solid rocket motor, it substantially increases the additional structural mass and the severe ablation of the pintle reduces the reliability of the method shown in Fig. 1.1b. A fluidic nozzle throat (FNT) is a method that changes the flow area and the shape of throat of the primary flow as a result of the interaction between the secondary and primary flows (fluidic injection). There are two main methods for injection of FNT of a solid rocket motor. One is the scheme based on vortex valves [7–16], that is, the secondary flow is injected tangentially at the inlet of the combustion chamber nozzle so that the mainstream produces a convolution to reduce circulation area (Fig. 1.2a). In this solution, a central splitting vehicle is required in the combustion chamber, which increases the structural mass, and the swirling tends to increase the deposition of particles in the combustion chamber. In addition, the convolution also increases the radial momentum of the jet at the outlet of the nozzle, resulting in a reduction in the thrust coefficient, and is therefore primarily suitable for flow regulation of solid gas generators. This book mainly introduces the scheme and technology of the other FNT.

(a) Structure of an engine

(b) Erosion of the pintle by copper infiltrated tungsten material

Fig. 1.1 A coaxial pintle solid motor system designed by Beijing Institute of Technology

1.1 Application Background of Fluidic-Throat Nozzles

3

(b) Scheme of reversed secondary flow injection (a) Scheme based on vortex valve tangential injection of secondary flow

Fig. 1.2 Two fluidic nozzle schemes in rocket engines

The second FNT scheme is to inject a secondary flow opposite to the primary flow in the vicinity of the throat, so that the flow area of the mainstream is reduced by the extrusion of the secondary flow and the increase of the flow resistance (see Fig. 1.2b). Control of the size of the main throat area is achieved by adjusting parameters of the secondary flow (gas or liquid), including flow, pressure, and working pulse width. At the earliest, National Aeronautics and Space Administration (NASA) launched a research project called Fluidic Injection Nozzle Technology (FLINT) [17–31] and conducted a lot of research on the application of this program in aeroengines. Besides, they listed this technology as the first choice for jet propulsion systems of future advanced drones and fighters. The research shows that the injection scheme of FNT has the following characteristics and advantages: (1) There is no moving part in the scheme and it is of high reliability; (2) The control systems for nozzle expansion ratio and vector can be integrated to simplify the engine system. If the secondary flow is injected symmetrically in the throat, it will act to control the throat area. If the asymmetric injection near the throat tilts the nozzle sonic surface, the primary flow will deflect in the subsonic zone, thereby changing the direction of the thrust; (3) The abovementioned sonic surface tilt-induced vector control over fluidic throat shows higher efficiency than the shock-induced vector control of the secondary flow injected into the expansion section, and the thrust loss is small; (4) Secondary flow itself provides additional thrust. Based on the above research, Europe and the United States have successively implemented a number of research plans on engine technology such as the integrated high-performance turbine engine technology (IHPTET) and versatile, affordable, advanced turbine engines (VAATEs) in recent years. They have developed and verified the engine prototype of the thrust vectoring nozzle of the FNT. In order to improve the fourth-generation turbofan engine with a small bypass ratio and a push-to-weight ratio of 10 and the newly developed fifth-generation turbofan engine

4

1 Introduction

with a small bypass ratio and a push-to-weight ratio of 15, it provides an important technical basis. The negative aerodynamic mass of a new aeroengine using an FNT (Fig. 1.3b) is greatly lower than that of a conventional mechanically adjustable nozzle (Fig. 1.3a), and the mechanical structure is simpler and more reliable. The successful application and verification of fluidic nozzle-throat technology in aeroengines provides a good foundation for the research and development of random thrust control in the application of this technology to solid rocket engines. Recently, people have begun to pay attention to this kind of FNT technology used in rocket engines. The University of New South Wales in 2012 conducted preliminary assessment and research on the characteristics of FNT and key technical solutions to solid rocket motors. It is believed that the solid FNT can be combined with the vector control system of secondary flow to adjust the thrust size and direction at the same time, so that a more economical, simple, and practical thrust control system can be developed for solid rocket engines. The results coincided with our research on the FNT system in solid rocket motor, which was conducted in 2009–2011.

(a) A mechanical vectoring nozzle in air breathing motor

(b) FNT Concepts in air breathing motor Fig. 1.3 Comparison of traditional mechanical nozzles and FNT for aeroengines

1.1 Application Background of Fluidic-Throat Nozzles

5

Although FNT technology is still a developing technology, it is foreseeable that the development and breakthrough of this technology will be the foundation of a new generation of vectoring nozzles for stealth drones, aeroengines with adjustable expansion ratio, flow-controllable solid gas generators, and solid rockets with thrust control. The technology will play an important role in the development. At the same time, it will provide a technical reserve for the above fields. The application prospects of the FNT technology are listed in Fig. 1.4. It is also noted that FNT may be easier to apply to solid rocket motors. Because the FNT is a completely new technology for the aeroengine, the secondary flow from the compressor will involve the change of the thermal cycle and control mode of the entire aeroengine, and the aeroengine should be redesigned and finalized. Therefore, the entire development cycle is long and the development work is heavy. Unlike aeroengines, secondary flow technology as a shock vector control (SVC) has long

Fig. 1.4 Application prospect of FNT technology

6

1 Introduction

been applied to many types of solid rocket engines [32, 33]. This has accumulated rich engineering experience and application basis for the realization and application of FNT to solid rocket engines. Much design scheme of the FNT in solid rocket motor can refer to the existing design such as the form, combination, thermal protection structure, system arrangement, etc. Therefore, if the performance of the FNT meets the requirements of engineering applications, the original SVC technology of the secondary flow can be engineered with fewer modifications. Therefore, the FNT technology in solid rocket motors has good inheritance and low development cost; some components of the injection system of the original secondary flow, such as tanks and valves of the secondary flow, can be retained. Figure 1.5 shows the sketch map of the third-stage engine of the Militia III. The tank of the secondary flow is installed at the tail of the nozzle. The liquid working fluid is pressurized by the high-pressure gas cylinder. The basic parameters of the entire injection system of the secondary flow are shown in Table 1.1.

Fig. 1.5 The third-stage solid rocket motor on Militia III Table 1.1 Injection parameters on liquid secondary flow system Project

Parameter

Project

Parameter

Working medium of injection

Strontium perchlorate solution

Injection pressure (on average)

4.7 MPa

Density of working medium

1.95 g/cm3

Material of tank

Ti-4Al-4V Titanium alloy

Mass of working medium

27.4 kg

Spray hole arrangement

Four groups separated by 90° and three orifices per group

Pressure of tank when injecting

5 MPa

Net mass of system structure

19.1 kg

1.2 Theory and Characteristics of Fluidic Throat for Solid Rocket Motors

7

1.2 Theory and Characteristics of Fluidic Throat for Solid Rocket Motors 1.2.1 Thrust Modification Theory Since the thrust of a solid rocket engine is restricted by the fuel-burning surface, it is difficult to change its size during work. According to the working pressure and thrust formula of the engine,  1  Pc = aρ p C ∗ A/At 1−n

(1.1)

F = C F Pc At

(1.2)

where Ab is the burning area of propellant, At is the geometric area of throat, a is the burning rate constant, C* is the characteristic velocity, F is the thrust, n is the burning rate exponent, Pc is the chamber pressure, and ρ p is the propellant density. It can be seen from the above thrust formula that the thrust F is to be changed only by changing the burning surface Ab or the geometrical spray area At . But the Ab cannot be changed because the medicine column has been designed in advance. So the only one can be changed is the At (such as the mechanical method or the FNT). For the selected propellant, its density ρ p , burning rate coefficient a, and characteristic speed C* are constant. If the grain is a constant surface burning, Ab is a constant and then Eq. (1.1) can be written as 1

Pc = K 1 Atn−1

(1.3)

  1 where K 1 = aρ p C ∗ A 1−n = const. The general thrust coefficient C F does not change much and can be regarded as a constant. So Eqs. (1.2) and (1.3) can be written as At = (F/K 2 ) 1

Pc = K 3 F n 1

n−1 n

(1.4) (1.5)

where K 2 = C F K 1 = const and K 3 = K 1 K 2n = const. From Eq. (1.5), if the thrust of the solid rocket engine is doubled, the combustion 1 chamber pressure needs to be changed by a factor of 2 n . For solid propellant, 0.3 < n < 1. If the burning rate exponent is n = 0.3, the pressure will need to be increased to 10 times the original pressure. Obviously, the shell cannot withstand or the thrust adjustment range is reduced. The pressure is only 2.37 times the original pressure if n = 0.8. Therefore, propellant with high-pressure burning rate exponent is used for

8

1 Introduction

adjusting the effective area of the throat to adjust the thrust of the solid rocket engine at present; in addition, n is generally between 0.6 and 0.8. This book only discusses positive pressure burning rate exponent.

1.2.2 Work Modes and Characteristics of Fluidic Throat for Solid Rocket Motors Compared to the working mode of the FNT of the aspirating aeroengine, the FNT of the solid rocket engine mainly presents the following differences in the working characteristics and methods: I. Adjustment mode Because the aeroengine requires an increase in the throat area of the nozzle during the afterburner operation, the geometric area of throat of the FNT is designed for maximum thrust requirements. During most of the working hours in cruise, it is always necessary to inject secondary flow to narrow the throat to meet the requirements for small flow under cruise work. When it is necessary to burn after the forced combustion, the injection of the secondary flow is turned off to enlarge the throat. Therefore, how to solve the secondary flow source and usage amount for the aeroengine is a prominent problem. If the secondary fluid is self-contained, the mass required will be great. The solid rocket motor is just the opposite. The narrowing of the throat of the nozzle will increase the pressure of the combustion chamber and the burning rate of the propellant, finally increasing the thrust of the solid rocket engine. Thus, for a solid rocket motor, there is no need to inject secondary flow to form a fluidic throat when no maneuvering or acceleration is required. When the maneuver or the ballistic change is to be performed, it is necessary to inject the secondary flow to reduce the area of the throat, so the secondary flow carried can be relatively small in mass. In addition, the working flow rate of primary flow on the aeroengine is independent on the area of throat of the fluidic nozzle, while the flow rate of the solid rocket motor is related to the effective area of throat of the primary flow. II. Working medium of the secondary flow In order to solve the above problem relating the use of the secondary flow of the FNT, the aeroengine generally does not have a secondary flow medium itself. The secondary flow is obtained by taking a small portion of the injection nozzle from the air drawn by the aeroengine compressor (Fig. 1.3b), so air is generally taken as the working medium of the secondary flow in the aeroengine FNT. For solid rocket motors, liquid working medium or other high-pressure gases (which can be produced by solid gas generators) are generally used to reduce the volume of the storage tank for the secondary flow. If it is a liquid working medium, the tank can be pressurized by a gas generator or a high-pressure gas cylinder. The liquid working medium is not unique, such as the solid rocket motor used on the second and third stages of the Militia III missile. Both engines use SVC of secondary flow, and the secondary

1.2 Theory and Characteristics of Fluidic Throat for Solid Rocket Motors

9

flow medium used in the second stage is freon [32, 33], while the third stage uses strontium perchlorate solution. Due to the flexibility of the choice of secondary injection working medium, FNT in the solid rocket motor can further improve the choking capability of the new nozzle by selecting an appropriate working medium. III. Inner flow field environment of the nozzle At present, metal powders are generally added to the propellant in solid rocket motors to increase the specific impulse and energy. When the solid propellant with metal powders is burned, gas–particle two-phase flow is formed in the nozzle [34]. In addition, for the case where the liquid working medium is used as the secondary flow, a gas–liquid two-phase flow is also formed in the vicinity of the fluidic throat of the solid rocket motor. The gas in an aeroengine can be considered as a pure gas phase. This condensed phase parameter under multiphase flow conditions not only affects engine performance and thrust [34–41] but also influences the performance of FNT. In addition, if the position of the orifice for the secondary flow is not properly selected, the particulate phases (such as Al2 O3 ) may block the nozzle of the secondary flow during operation, such that the FNT does not operate as designed. By comparing the working modes and conditions of the FNT in the above two applications of aeroengines and solid rocket motors, it can be seen that there is actually a significant difference between the two. This difference not only affects the choice of design parameters for fluidic throats but, more importantly, also allows the FNT to exhibit different performances and characteristics. So it is not possible to simply copy or replicate the design method and system of the aeroengine FNT. These different conditions, including operating parameters, protocols, and design guidelines, are also issues to be discussed and addressed in the chapters of the book.

1.3 General Study Conditions 1.3.1 General Introduction At present, the concept and research on the injection of secondary flow into the nozzle of a fixed geometry surface to form a fluidic throat are mostly directed at aeroengine systems, which are mainly taken the lead by the United States. There are few research materials on the fluidic-throat nozzle of solid rocket motors disclosed abroad. The shape and characteristics of the fluidic throat are directly related to the parameters of the secondary injection flow and the injection mode. The secondary injection method can be divided into two types: constant injection and pulse injection [42, 43]. Foreign research mainly focuses on the constant injection method [17], and recently pulse injection has become a research hotspot of secondary injection flow technology in fluidic-throat nozzles.

10

1 Introduction

I. Europe and the United States Martin A I proposed the concept of “pneumatic variable nozzle” in 1957 [17], which is the predecessor of the concept of “fluidic throat.” He used a one-dimensional isentropic compressible flow theory to analyze the interaction of two injections at the throat of a convergent nozzle. The theory uses a vortex model to describe the penetration between two fluids, while the mixing process assumes that the two fluid components are identical. The theory can be used to initially determine the characteristics of the fluidic throat and some design parameters. However, the predicted performance of the fluidic-throat nozzle is quite different from the experimental results, and the turbulence performance of the fluidic-throat nozzle is not good. In the 1960s, the United States proposed the concept of “fluidic throat—FNT” [17–22], which defined the effective control area factor of the fluidic throat (equivalent to the flow coefficient) to evaluate the turbulent performance of the fluidic throat. Some experimental researches have been done on the application of solid rocket engines. The results of the hot test show that the fluidic injection method is feasible for controlling the thrust of solid rocket engines. It is found that passive fluidic-throat solutions are not as efficient as that of active solutions. However, limited to the calculation conditions at that time, the fluidic-throat nozzle on the solid rocket engines was not parameterized, so the turbulence performance achieved by the experimental device was not as high as now. In 1995, NASA launched a research project called “Fluidic Injection Nozzle Technology (FLINT)” [26–31]. The purpose is to research and develop nozzle control technology based on secondary flow injection, which mainly involved the following three directions: (1) Vector control based on secondary flow at the throat; (2) control of variable fluidicthroat area; and (3) enhancement technique for the injection of mixing flows. The research program aims to replace the cumbersome mechanical vectoring nozzles on existing aeroengines with fluidic nozzle control systems to simplify and integrate aeroengine systems. The units involved in the research are the Air Force Science Research Office (AFOSR), McDonnell Douglas, Georgia Tech, Lockheed Aircraft Systems Research Laboratory (LMTAS), General Electric Fighter Engine Research Laboratory (GEAE), etc. A notable result of the program was the establishment of a verification test bench and principle machine for a full-size aeroengine fluidic-throat nozzle [25]. In 1997, FLINT plans to complete the numerical study of the constant secondary flow injection method and the experimental study of the cold flow of the reduction ratio engine. The experiments in this stage mainly focused on the secondary/primary total pressure ratio of 1:3, and the total combustion chamber pressure/back pressure ratio of 1.2:7 (including the under-expansion and over-expansion of the nozzle) and the mass flow rate of the secondary/primary flow of 0–30% of the fluidic throat in the parameter range. The influence of the injection parameters and structural form of the secondary flow on the effective area of the fluidic throat is obtained, thus establishing some key design parameters, such as (1) the position of the secondary injection flow. When injected at the geometric throat, the effective area of the throat varies the most; (2) The greater the mass flow rate of the secondary flow, the more obvious

1.3 General Study Conditions

11

Fig. 1.6 Vortex series of secondary flow in rocket engine nozzle through pulse injection

the turbulence performance of the fluidic throat increases; (3) Injection angle. The effective area of the throat varies greatly while injecting countercurrent with the primary flow; (4) The greater the ratio of the total pressure of the secondary flow to the primary flow, the better the turbulence performance of the fluidic throat; (5) In the case of the same injection parameters, the annular nozzle has better performance than the nozzle in the form of a circular hole. In addition, their research shows that the computational fluid dynamic (CFD) results are compared with the results of coldflow test, with the error only within 1%, so the simulation results are credible and can be used to assist in the design of fluidic-throat nozzles. At that time, the experimental research is able to achieve a 28% change in the effective area of the nozzle throat with 7% of the primary flow. However, for the actual requirements of the aeroengine, such consumption for secondary mass flow is still quite large, so researchers have been working to find ways to further reduce secondary mass flow. Subsequent research in the FLINT program shifted the focus to the pulse injection approach [24]. It is found that compared to constant injection, pulse injection has the following characteristics: the pressure pulsation of the secondary flow nozzle causes the injection flow to generate a vortex series (Fig. 1.6), and the generation of the vortex series enhances the interaction of the injection and the primary flow, which helps to increase the turbulence performance of the secondary flow. Meanwhile, the researchers attempted to reduce the secondary mass flow rate required to be injected into the fluidic-throat nozzle. Previous studies using pulse injection technology as a means of enhancing injection mixing have focused on the interactions of low-velocity jets and primary flow. With the need of supersonic combustion technology and the development of secondary flow injection in the field of jet noise reduction, flow separation control, and aircraft jet drag reduction, the research scope of pulse jet is gradually expanded to

12

1 Introduction

the transonic and supersonic flow pulsation structure [44–48]. These studies laid the theoretical and experimental foundation for the pulse injection of fluidic-throat nozzles. In recognition of the fact that the pulse injection structure not only enhances injection flow mixing but also increases resistance of primary flow, researchers began using pulse injection in 1999 as a primary means of further improving turbulence performance and reducing consumption of secondary mass flow. The representative work in this stage is that P. J. Vermeulen used the method of acoustic-vibration coupling to make the pulsation frequency of the secondary flow reach several kilohertz in 2000. This method can conveniently use the outside air and reduce the required mass of high-pressure air from the compressor [31]. In 2004, M. Dziuba and T. Rossman used a resonant tube to increase the pulsation frequency of the secondary flow to tens of kilohertz [49]. In 2007, D. Baruzzini applied the fluidic-throat nozzle concept to liquid rocket engines and combined active pulse injection with passive constant injection to enhance the turbulent performance of the fluidic throat on the rocket engine and to reduce secondary mass flow usage [42, 43]. The “passive constant injection” is to use a part of the high-temperature gas in the combustion chamber of the liquid rocket engine as secondary injection flow. This study evaluates the advantages and feasibility of fluidic-throat nozzle technology for adjusting the nozzle expansion ratio of ground-launched orbiting liquid rocket engines. In addition, in order to find a way to generate the abovementioned high-frequency pulsating secondary flow, the researchers also carried out a large number of theoretical and experimental research, the representative work of which is those carried out by Yagle [28] and Dziuba [49]. They conducted experiments and CFD simulation calculations on the method of generating high-frequency sonic pulse injection using active resonance tubes. In 2005, the resonant tubes designed by M. Dziuba and T. Rossman produced 15–45 kHz pulsations with little mass flow rate of high-pressure air or helium without moving parts. But until now, compared to constant injection, the research on fluidic throat based on pulse injection has only stayed in the exploration stage, and there are still differences in the actual contribution and gain of this injection scheme. The best turbulence performance obtained in experiments, numerical simulations, and theoretical studies did not increase much more than constant injection as the researchers expected. In summary, the fluidic-throat nozzle technology has been widely studied and its application range has been actively expanded. II. Other countries In the former Soviet Union, A. M. Venitzki also mentioned the concept of “aerodynamic control method” [50] and gave a schematic diagram of active and passive secondary injection systems. A. M. Venitzki believes that compared to the mechanical method of adjusting the throat of the nozzle, the advantage of the aerodynamic control method is that the additional mass of the structure is small, and the secondary flow can contribute a part of the thrust. However, due to the technical secrecy of the former Soviet Union, no more public information is available on the application of the technology in engineering.

1.3 General Study Conditions

(a) The SVC

13

(b) The oblique throat control

(c) The groove throat inclination method

Fig. 1.7 Several typical TVC methods based on secondary flow injection

At present, China mainly studies the thrust vector control (TVC) of the oblique throat of the aeroengine based on secondary flow injection. The representative work mainly includes the work of Nanjing University of Aeronautics and Astronautics and the Air Force Engineering University [51], but these works have not specifically studied the turbulence performance (or adjustment of expansion ratio) of the fluidicthroat nozzle. In the field of solid rocket engines, the previous study of secondary flow injection was also mainly focused on shock-induced TVC [51–59].

1.3.2 Study on TVC of Fluidic-Throat Nozzles Based on Secondary Flow Injection The fluidic-throat nozzle system can be combined with the secondary flow injection TVC system to greatly improve the practical application value of the fluidic-throat nozzle and the efficiency of the system. NASA has studied and compared the following injection systems of the secondary flow with potential application value: SVC method, oblique throat control method, reverse flow TVC, and co-current TVC [60–64]. The SVC of the secondary flow (Fig. 1.7a) and the oblique throat control (Fig. 1.7b) are considered to be the most suitable for combination with the fluidicthroat nozzle. NASA has conducted a large number of parametric studies and coldflow experiments on these two vector control techniques. Among them, the study of oblique throat is mainly for nozzles with a small expansion ratio used on aeroengines [65–68]. In general, the side force performance of the shock wave induction method is superior to the oblique throat method, but the nozzle efficiency of the former is lower. This is because when the oblique throat method is used, the primary flow starts to deflect in the subsonic region, and there is no strong shock wave in the flow field [67, 68]. The groove throat inclination method (Fig. 1.7c), also named doublethroat vector control, is developed on the basis of the throat inclination method. This method can obtain a multi-thrust vector angle that is larger than the oblique throat, but the double-throat nozzle configuration is not suitable for a solid rocket engine.

14

1 Introduction

Many studies have been conducted on the induced shock wave method applied to general solid rocket engines [69–72]. However, these studies are all directed at the absence of secondary flow at the throat. For fluidic-throat nozzles of solid rocket engines, the system needs to be able to adjust the throat area under certain conditions, to change the thrust amplitude and the direction of the thrust. At this time, nozzle at the throat and at divergent section will have injections at the same time. If the injection parameters are not designed properly, the two injections will have strong interference and disturbance, which will affect the control ability of the secondary flow thrust vector. Therefore, under the condition of injections interference in the abovementioned fluidic-throat nozzles of solid rocket engines, how to balance the control performance of the throat area and the performance of the SVC is a new engineering problem. The University of New South Wales conducted a preliminary study of the above vector control techniques based on secondary flow injection and interference problems of injections at fluidic throats in solid rocket engines [73]. This book will introduce the research work of the author in this area in Chaps. 5 and 6.

1.4 Design Procedure of Solid Rocket Motors with FNT Whether in the final engineering application or in the process of developing the technology, the primary problem to be solved is how to design a fluidic-throat nozzle that meets the use conditions of solid rocket engines and a high overall performance under certain overall indicators. Without proper design methods, guidelines, and theoretical guidance, any practical research may not be in place. Of course, the accumulation of theory or practice is primary about improving these design methods. The design flow and method required for a solid rocket engine with a fluidic-throat nozzle is illustrated in Fig. 1.8. The “rectangular frame” in the figure represents the flow and steps of this type of engine design; the “blank quad box” section represents traditional methods, guidelines, etc.; and the “gray quadrilateral frame” represents the new guidelines and design methods that need to be added to the traditional nozzle design of solid rocket engines. It is also the content and work highlighted in this book. For different mission requirements, the overall indicators of solid rocket engines are not identical, so the overall indicators in the figure are divided into two parts; the indicators in the dashed box are the vector control requirements for the fluidic-throat nozzles. For some system solutions, this part of the indicator is not required and can be omitted as appropriate. In the new guidelines and design methods that need to be supplemented, the “ratio of mass flow pressure to total pressure” relationship of the nozzle during steady-state operation of the fluidic throat, the “effect of injection parameters of secondary flow on the efficiency of the nozzle,” and the “flow coefficient ratio to mass flow rate ratio” of the fluidic-throat curve will be described in Chaps. 2, 3, and 6 (secondary flow including gases and liquids). “The effects and corrections of particles relative to the fluidic throat” are discussed in Chap. 4. As previously mentioned, the combination of fluidic-throat and secondary flow TVC of solid rocket engines will result in greater system performance. For the nozzle of the

1.4 Design Procedure of Solid Rocket Motors with FNT

Fig. 1.8 Design of a solid rocket engine with a fluidic-throat nozzle

15

16

1 Introduction

solid rocket engine with large expansion ratio, the most suitable and the most inherited technology is the SVC method of the secondary flow. The “SVC characteristics and performance of the fluidic-throat nozzle” is introduced in Chaps. 5 and 6; the “ablative characteristics of fluidic-throat nozzles,” “thrust adjustment characteristics of fluidic-throat nozzles,” and the key technologies and practices of fluidic-throat nozzles on solid rocket engines are described in Chaps. 7–10. The general design idea of a fluidic-throat nozzle on a solid rocket engine is that the geometry of the nozzle can be designed according to the traditional nozzle design method and criteria. When it is necessary to take into account the turbulence performance or thrust response time of the fluidic throat in the selection of some parameters, the geometric profile can be appropriately modified or the appropriate geometric parameters can be reselected according to the methods provided in this book. The grain design of solid rocket motors can still refer to existing methods. When designing the secondary flow injector on the divergent section, the initial parameters can be determined by referring to the design method and criteria of the traditional SVC system. Then according to the interference and performance loss of injection at the throat, the position and parameters of the injector are appropriately adjusted to take into account throat control performance and TVC performance.

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16. Yu, X.J., He, G.Q., Li, J., et al.: Numerical analysis of flow in variable thrust SRM. AIAA Paper 2007-5801 (2007) 17. Martin, A.I.: The aerodynamic variable nozzle. J. Aeronaut. Sci. 24(5), 357 (1957) 18. McArdle, J.G.: Internal characteristics and performance of an aerodynamically controlled, variable discharge convergent nozzle. NACA TN4312, July 1958 19. McAulay, J E.: Cold-air investigation of three variable-throat-area convergent-divergent nozzles. NASA TM X-42, September 1959 20. Blaszak, J.J., Fahrenholz, F.E.: Rocket Thrust Control by Gas Injection. Massachusetts Institute of Technology, Naval Supersonic Laboratory, Technical Report 430, November 1960 21. Gunter, F.L., Fahrenholz, F.E.: Final Report on a Study of Rocket Thrust Control by Gas Injection. Massachusetts Institute of Technology, Naval Supersonic Laboratory, Technical Report 448, May 1961 22. Zumwalt, G.W., Jackomis, W.N.: Aerodynamic throat nozzle for thrust magnitude control of solid fuel rockets. Am. Rocket Soc. J. (1962) 23. Catt, J., Miller, D.N.: A static investigation of fixed-geometry nozzles using fluidic injection for throat area controls. AIAA Paper 95-2604 24. Hawkes, T.: Lessons learned in the development of a national cooperative programe. AIAA Paper 97-3348 25. Weber, Y.S., Bower, D.L.: Advancements in exhaust system technology for the 21st century. AIAA Paper 98-3100 26. Miller, D.N., Yagle, P.J., Hamstra, J.W.: Fluidic throat skewing for thrust vectoring in fixedgeometry nozzles. AIAA Paper 99-0365 27. Vakili, A., Sauerwein, S., Miller, D.: Pulsed injection applied to nozzle internal flow control. AIAA Paper 99-1002 28. Yagle, P.J., Miller, D.N., Bender, E.E., et al.: A computational investigation of pulsed ejection. AIAA Paper 2002-3278 29. Miller, D.N., Yagle, P.J., Bender, E.E., et al.: A computational investigation of pulsed injection into a confined, expanding crossflow. AIAA Paper 2001-3026 30. Williams, R.G., Vittal, B.R.: Fluidic thrust vectoring and throat control exhaust nozzle. AIAA Paper 2002-4060 31. Vermeulen, P.J.: An experimental study of the mixing behaviour of an acoustically pulsed air jet with a confined crossflow. AIAA Paper 88-3296 32. Zhang, Z. (ed.): Minuteman Intercontinental Ballistic Missiles. China Astronautic Publishing House, Beijing (1997) 33. China Aerospace Corporation: An Encyclopaedia of World Missile and Space Engines. Military Science Publishing House, Beijing (1999) 34. Li, Y., Zhongqin, Z., Zhao, Y.: Theory of Solid-Propellant Rocket, pp. 111–112. National Defense Industry Press, Beijing (1985) 35. Sun, M., Fang, D., Zhang, C.: The experimental and theoretical studies of two-dimensional two-phase nozzle flows. Acta Aeronaut. Astronaut. Sin. 9(11), 572–576 (1988) 36. Dunn, B.M., Durbin, M.R., Jones, A.L., et al.: Short range attack missile (SRAM) propulsion, 3 decades history. AIAA Paper 94-3059, June 1994 37. Quilici, J.L.: Nozzle development for the proposed AGM-130 rocket motor. AIAA Paper 841415, June 1984 38. Xie, K., Liu, Yu., Ren, J., Liao, Y.: An ideal method for the two-phase ring plug nozzle design. J. Solid Rocket Technol. 30(3), 223–228 (2007) 39. Kan, X., Yu, L., Junxue, R., Yunfei, L.: Design methods of two-phase axial plug nozzle. ACTA Aeronaut Astronaut Sin 28(6), 1339–1344 (2007) 40. Li, Y., Chen, L., Jian, Z.: Numerical study of two-phase nozzle flow with classified particles. J. Solid Rocket Technol. 26(3), 32–34 (2003) 41. Kliegel, J.R.: Gas particle nozzle flows. In: Ninth International Symposium on Combustion, pp. 811–826. Academic Press, New York (1963) 42. Baruzzini, D., Domel, N., Miller, D.N., et al.: Pulsed injection flow control for throttling in supersonic nozzles—a computational fluid dynamics design study. AIAA Paper 2007-4215

18

1 Introduction

43. Domel, N.D., Baruzzini, D., Miller, D.N., et al.: Pulsed injection flow control for throttling in supersonic nozzles—a computational fluid dynamics based performance correlation. AIAA Paper 2007-4214 44. Rona, A.: Control of transonic cavity flow instability by streamwise air injection. AIAA Paper 2004-682 45. Deere, K.A., Berrier, B.L., Flamm, J.D.: A computational study of a new dual throat fluidic thrust vectoring nozzle concept. AIAA Paper 2005-3502 46. Gamba1, M., Clemens, N.T., Jones, I.: Strongly-forced turbulent non-premixed jet flames in cross-flow. AIAA Paper 2007-1418 47. Gamble, E., Haid, D., Cannon: Improving off-design nozzle performance using fluidic injection. AIAA Paper 2004-1206 48. Haid, D., Gamble, E.J.: Nozzle aft body drag reduction using fluidics. AIAA Paper 2004-3921 49. Dziuba, M., Rossmann, T.: Active control of a sonic transverse jet in supersonic cross-flow using a powered resonance tube. AIAA Paper 2005-897 50. Venitzki, A.M.: Solid Propellant Rocket Engine (trans Jinkang Y). National Defense Industry Press, Beijing (1981) 51. Yongjiu, L.: Fluidic thrust vectoring control technology. Aircr. Des. 28(2) (2008) 52. Zhi, C.: Design Rule of Dual-Throat Fluidic Thrust Vectoring Nozzles and Exploration of Rear Fuselage Integration. Nanjing University of Aeronautics and Astronautics, Nanjing (2007) 53. Yongsheng, Z., Yankui, W., Xiaowei, Y., Xueying, D.: Design of secondary-divergent vectoring nozzle based on secondary fluidic injection. J. Beijing Univ. Aeronaut. Astronaut. 33(3) (2007) 54. Wang, Q., Fu, Y., Eriqitai: Computation of three dimensional nozzle flow field with fluidic injection. J. Propul. Technol. 23(6), 441–444 (2002) 55. Zhang, Q., Lv, Z., Wang, G., Liu, Z., Jin, J.: Numerical simulation of an axisymmetric fluidic vectoring nozzle. J. Propul. Technol. 25(2), 139–143 (2004) 56. Deng, Y., Zhong, Z., Song, W.: Computational investigation of secondary flow thrust vector control technology used in a convergent-divergent nozzle. J. Solid Rocket Technol. 28(1), 29–32 (2004) 57. Qiao, W., Cai, Y.: A study on the two-dimensional thrust vectoring nozzle with secondary flow injection. J. Aerosp. Power 16(3), 273–278 (2001) 58. Lu, B., Xu, X., Zhou, M.: Numerical simulation on rectangular jet vector nozzle. Aeroengine 34(1), 16–18 (2008) 59. Dechuan, S.: Study on Supersonic Flow Field with Secondary Injection and Its Control Parameters. Northwestern Polytechnical University, Xi’an (2002) 60. Deer, K.A.: Summary of fluidic thrust vectoring research conducted at NASA langley research center. AIAA 2003-3800 (2003) 61. Mason, M.S., Crowther, W.J.: Fluidic thrust vectoring for low observable air vehicles. AIAA 2004-2210 (2004) 62. Deere, K.A.: Computational investigation of the aerodynamic effects on fluidic thrust vectoring. AIAA 2000-3598 (2000) 63. Wing, D.J.: Static investigation of two fluidic thrust-vectoring concepts on a two-dimensional convergent-divergent nozzle. NASA TM-4574 (1994) 64. Chiarelli, C., Johnsen, R.K., Shieh, C.F., et al.: Fluidic scale model multi-plane thrust vector control test results. AIAA 93-2433 (1993) 65. Miller, D.N., Yagle, P.J., Hamstra, J.W.: Fluidic throat skewing for thrust vectoring in fixed geometry nozzles. AIAA-99-0365 66. Zhang, X., Wang, R., Yang, F.: Influence of double gas flow on fluid control vectoring nozzle. J. Solid Rocket Technol. 30(4), 295–298 (2007) 67. Zhou, M., Wang, R., Zhang, X., Xu, X.: Effect of jet flow distribution on fluidic throat skewing nozzle. J. Propul. Technol. 29(1), 58–61 (2008) 68. Jing, L., Wang, Q., Eriqitai, : Computational analysis of two fluidic thrust-vectoring concepts on nozzle flow field. J. Beijing Univ. Aeronaut. Astronaut. 30(7), 597–601 (2004) 69. Richard, J.Z.: Thrust vector control by liquid injection for solid propellant rockets. AIAA 75-1225 (1975)

References

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70. Berdoyes, M.: Hot gas thrust vector control motor. In: 28th JPC, AIAA 92-3551 71. Green, C.J.: Liquid injection thrust vector control. AIAA J. 1(3), 573–578 (1963) 72. Huang, J., Fan, C.: Testing Technology of Solid-Propellant Rocket Engines. Astonautic Publishing House, Beijing (1989) 73. Ali, A., Rodriguez, C.G., Neely, A.J., Young, J.: Combination of fluidic thrust modulation and vectoring in a 2D Nozzle. In: 48th AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit, 30 July–01 August 2012, Atlanta, Georgia, AIAA 2012-3780

Chapter 2

Steady Characteristics of a Gas–Gas Aerodynamic Throat

Abstract This chapter mainly discusses the flow details and characterization performance of gas–gas fluidic throat, including the effective throat area of fluidic throat, the analytical methods for steady aerodynamic characteristics, flow characteristics of fluidic, relationship between secondary flow rate ratio and total pressure ratio, relationship between secondary flow parameters and effective throat area, effect of the main nozzle parameters, and comparison of secondary flow injection schemes in the fluidic throat.

2.1 The Effective Throat Area of Fluidic Throat On the premise that the propellant burn area is known, to predict the combustion chamber pressure and thrust variation in a solid rocket motor during the design phase (c.f. Sect. 1.2.1), it is necessary to know the primary flow area of the nozzle throat. Usually, the effective throat area is used to characterize the flow area of primary flow at the nozzle throat. The effective throat area A˜ t is defined as follows:  m˙ o RTc,o Pc,o o  2(γγ +1  −1) 2 √ = γ γ +1 A˜ t =

(2.1) (2.2)

where γ is the specific heat ratio, R is the ideal gas constant, m˙ o is the mass flow rate of the primary flow, and Pc,o and T c,o are the total pressure and temperature in the combustion chamber. Effective throat area can be easily used to analyze a nozzle throat, so as to obtain a measure of thrust variation therein. The mass flow rate of a convergent–divergent nozzle under choked conditions is given by m˙ =

Cd At Pc  √ RTc

© Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_2

(2.3)

21

22

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

where C d is a flow coefficient and At is the geometric nozzle throat area. The relationships between geometric nozzle throat area and effective throat area can be obtained with Eqs. (2.1) and (2.3). A˜ t = Cd At

(2.4)

The actual throat of the nozzle throat area is A˜ t C d At Cd = = Cdo At Cdo A˜ to

(2.5)

where C do is the flow coefficient before secondary flow injection and C d is the flow coefficient after secondary flow injection, both of which relate to primary flow. C do can be obtained by calibration in advance: to obtain C d and the corresponding real effective throat area, the ratio of flow coefficient at times before, and after, secondary flow injection is needed. So, when discussing the change in fluid effective area at the throat, it is usually expressed in terms of C d /C do . In addition, effective throat area is mainly associated with the ratio of secondary flow to primary flow: the modified flow rate is as follows:  w = m˙ Tc

(2.6)

The ratio of modified flow rates can be defined, when the absolute temperature of the secondary flow is equal to that of the primary flow, as follows: ws /wo = (m˙ s /m˙ o )Tc,o =Tc,s

(2.7)

With this modified flow rate ratio, the results of a cold-flow test can be generalized to conditions in which the secondary and primary flows are at different temperatures. For example, if the primary flow is a high-temperature gas, the secondary flow is a lower temperature inert gas, or primary and secondary flows are both gaseous, but occur at different temperatures. The modified flow rate ratio will be treated as an independent variable in later chapters, especially when the secondary flow has the same absolute temperature as the primary flow, which the modified flow rate ratio equal to the actual flow ratio. Mastery of the steady-state characteristics of the fluid nozzle throat is the key to its design according to the thrust-based specifications. The “ratio of flow coefficient−modified flow rate ratio” (C d /C do −ws /wo ) curve is one of the basic curves describing fluid nozzle throat choke performance. Obtaining the ratio of flow coefficient−modified flow rate ratio curve under different secondary flow parameters can lead to a quantitative characterization of the influence of various factors on the choke performance and provide a reference for actual fluid nozzle throat design.

2.2 The Analytical Methods for Steady Aerodynamic Characteristics

23

2.2 The Analytical Methods for Steady Aerodynamic Characteristics 2.2.1 Experimental Methods and Apparatus for Cold-Flow Testing I. Experimental principles and systems To obtain the general rule for sizing a fluid nozzle throat, plot the relationship between its choke performance and structural parameters, and find the operating parameters under secondary flow conditions, a set of cold-flow tests are usually conducted early in the development stage. After obtaining a correlation curve, thermal experiments focus on the key parameters, which can greatly reduce development costs and risk. A typical solid fat throat cold-fluid test system is shown in Fig. 2.1. High-pressure gas is used as the primary and secondary flow working medium in the testing of the gas–gas fluidic throat. To ensure even gas injection rates, primary and secondary flows are symmetrically injected into each chamber through two routes after the sonic nozzles, which are used to limit the mass flow rate of each flow (Fig. 2.2). Before testing, the sonic nozzles must be calibrated, and there are six pressures requiring measured during the experiment: the pressures before, and after, the sonic nozzles in the secondary flow P25 and P29; the pressures before, and after, the sonic nozzles in the primary flow P26 and P27; and the secondary flow chamber pressure P31; and the primary flow chamber pressure P30, so that the mass flow rate ratio (m˙ s /m˙ o ) can be obtained from P25 and P26 and the corresponding sonic nozzle calibration curve (Fig. 2.2) likewise. After the secondary flow is injected into the throat, the effective throat area decreases and the primary flow chamber pressure increases in the case of a constant primary mass flow rate. According to Eqs. (2.3) and (2.5), the ratio of flow coefficients (C d /C do ) can be obtained from the primary flow chamber pressure P30 before, and after, secondary flow injection: the variation of mass flow rate ratio (m˙ s /m˙ o ) can be obtained from P25/P26. In addition, with a constant ratio of secondary flow injection area and geometric throat area, the total pressure ratio of secondary flow and primary flow (Ps /Po ) changes with the mass flow rate ratio. Figure 2.2 shows a typical calibration curve for a mass flow limiting device. After the mass flow limiting device reaches the choking state, the nozzle mass flow rate and pressure values before the mass flow limiting device will have a strong linear relationship. The expansion of the mass flow limiting device is accompanied by a shock wave; airflow after the shock wave becomes subsonic and the refrigerant pressure adjusts within a certain range automatically and matches the downstream pressure. The mass flow limiting device can prevent some downstream pressure fluctuations from the upstream flow disturbance and keep the experimental flow rate constant. After linear fitting of the calibration data, according to the flow in Eq. (2.3), the flow coefficient C d0 of the mass flow limiting device can be calculated

24

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

Fig. 2.1 Schematic of cold-flow experimental system

2.2 The Analytical Methods for Steady Aerodynamic Characteristics

25

Fig. 2.2 Pressure-flow curve for the mass flow limiting device

(it is usually close to 1). In addition, it is necessary to pay attention to the pressure range in the experiment, and to ensure that the temperature in the experiment is close to the temperature used in calibration. II. General cold-flow experimental engine A general cold-flow experimental engine can be used to study the influence of various geometric design parameters and the performance of the type thereof on the fluid nozzle throat, while also taking into account its feasibility and economy. The experimental engine device has a conveniently changeable nozzle throat part and can guarantee the universality of other parts. A general cold-flow experimental engine structure, which can meet the above requirements, is shown in Fig. 2.3. The general structure of this kind of universal device can be seen in the convergence of the fluidic-throat nozzle of a circular hole, which can also be used to investigate the fluidic-throat nozzle of the ring, and the two nozzles are respectively shown in Fig. 2.3a, b. In Fig. 2.3, the main flow gas chamber 1 and the convergence Sect. 2.2 are common parts, and the nozzle assembly with its dual flow is seen at 3–12 which are replaceable/interchangeable parts. This kind of cold-flow experimental engine has the versatility, the secondary flow nozzle structure is changeable, and the total number of parts needing to be processed is lower: part replacement, in these experiments, is thus convenient. The study of the ring-type fluidic throat focuses on the value in the theoretical analysis. The flow loss of this fluidic throat is the smallest, and it can be used as an ideal, 2-d, model for the fluidic-throat nozzle: it is easy to obtain the general rule governing the throat area of the fluid, besides, it is suitable for a parametric study. From the point of view of subsequent experimental results and simulation, the choking rules governing a ring seam-type nozzle throat with a hole-type nozzle throat are consistent.

26

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

(a) Hole type fluidic nozzle throat

(b) Ring seam type fluidic nozzle throat 1—Gas collecting chamber 4—Hole type FNT shell 7—End sealing O-ring 10—Ring type FNT shell

2—Contraction 3—Hole type FNT main component 5—End sealing O-ring

6—Axial plane sealing O-ring

8—End sealing O-ring 9—Ring type FNT internal component 11—End sealing O-ring

12—Axial plane sealing O-ring

Fig. 2.3 General cold-flow test engine structure (schematic)

Figure 2.4 shows the cross section of the replacement nozzle part (corresponding to component 4 in Fig. 2.3), and the sign of all secondary flow nozzle structural parameters. Table 2.1 lists four structural parameters for two typical solid-fluid nozzles using this universal device (Fig. 2.5). The secondary-flow nozzle structure using in the solid rocket motor can be divided into three types: contraction–expansion, contraction, and straight-hole (see Fig. 2.6 and Chap. 9). The contraction type can be seen as a special case of a contraction–expansion (nozzle expansion ratio = 1).

2.2 The Analytical Methods for Steady Aerodynamic Characteristics

Cross-section

27

Cross-section A-A

Fig. 2.4 Replaceable FNT engine throat Table 2.1 Nozzle structural parameters: four experimental examples of secondary flow Nozzle label

Area ratio (%)

n

Do /Dt

α

Λ

Nozzle type

90°

30°

Convergence hole

1008

10

8

3.5

2008

20

8

2.5

2016

20

16

2

HF

20

–

4

1008

2008

Girth (loop slot)

2016

HF

Fig. 2.5 The four types of nozzle

The structure of the straight-hole type is simple, and it is often used in solid rocket motors; however, the nozzle used in a solid rocket motor, especially near the nozzle throat, differs slightly in the injector in the liquid rocket engine head, in the effects of the thickness of the thermal protective structure of the solid rocket motor, and in the depth of the larger engine nozzle which is longer and narrower [1–6], as shown in Fig. 2.4. Four examples in Table 2.1 of the secondary flow nozzle throat area and the main nozzle geometric throat area ratio As,t /At are 10 and 20%, and investigations of the number of nozzles are of two kinds (8 and 16). The secondary flow nozzle area and the number of nozzles are basic design parameters for the fluid nozzle throat. In actual design, after the ratios of secondary flow rate and mainstream flow rate are selected,

28

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

Fig. 2.6 Inject port configuration schematic

Converging-diverging

Converging

Straight

the area ratio of the injector directly determined the total pressure of the secondary flow. The value of the secondary flow pressure is related to the structural quality of the secondary flow injection system: the higher the pressure of the secondary flow, the greater the quality delivered to the tank. However, under conditions of a certain nozzle area ratio, the number of nozzles determines the processing difficulty, accuracy, and occurrence of ablative cases in the throat. If there are too many nozzles, the diameters of these holes will be too small for easy machining and they will be easily blocked during operation, e.g., impurities in the second flow (upstream) or the substances mixed in the main stream will soon cause melting. If there are too few nozzles, the diameters of these holes will be too big, which will lead to ablation and the partial deterioration of the thermal stress environment. At the same time, the number of nozzles is directly related to the choke performance in the FNT. III. Typical pressure stress curve Figure 2.7 shows a typical environmental pressure and cold-flow environmental timing curve which uses nozzle 2016. When secondary flow is injected into the nozzle throat (P25 and P29 will increase), the main flow will be compressed, and the FNT will take shape, so the nozzle throat effective area of the main flow will decrease, and the pressure P30, which in the main flow chamber starts to increase, stabilizes at another balance pressure. From the figure, it may be seen that, in the initial phase, the pressure responds quickly and increases rapidly; by the end phase, the pressure changes smoothly to another equilibrium value: there are no overshoots or shocking during the whole process. From the pressure curve in Fig. 2.7, different ratios of secondary flow to main flow (flow ratio C d /C do ) are found that demonstrate the change in effective nozzle throat area. In addition, to examine the influence exerted by the cavity volume of the plenum on the pressure regulating time of the fluidic-throat nozzle, we can insert corks of different lengths into the gas chamber in the cylindrical cavity of the device during testing. As shown on the left-hand side of Fig. 2.8, compared to Fig. 2.7, the cavity

2.2 The Analytical Methods for Steady Aerodynamic Characteristics

29

Fig. 2.7 Pressure–time curve, nozzle 2016

volume of the cylindrical section in the cold engine on the right-hand side of Fig. 2.8 is reduced by about 40%. Figure 2.8 shows that, with the second stream injected, although the pressure increasing processes in the plenum are similar to those in Fig. 2.7, the conditions, with a cavity of smaller volume, reach a new equilibrium within about 27% less time. This is because when the volume of the cavity is reduced, the gas-filling process in the cavity will be decreased, and the plenum pressure rise time will also be shortened. Numerical simulation of unsteady CFD also predicts a consistent phenomenon (see Chap. 7). Simulation under the actual conditions of a heat engine test has pressure increasing processes similar to those in the cold engine experiments, where the larger the volume of the combustion chamber cavity, the longer the time taken for the pressure to reequilibrate. This provides a reference for the design of actual fluidic-throat nozzles in engines. With the continuous consumption of propellant, the cavity volume inside the engine increases, which leads to the increase of the time for adjusting fluidic nozzle throat pressure. For solid rocket motors that have strict requirements on thrust response or mass flow response, it requires careful design of the cavity volume of the engine throughout. Selecting the appropriate injection parameters, in order to ensure that response times meet index requirements, is essential while such fluidic-throat nozzles are used in solid rocket motors to adjust their thrust. In fact, techniques for controlling the throat area to control solid rocket motor thrust (such as pintles and eddy valve technology mentioned in Chap. 1) will generally run into these problems. When the effective throat area decreases, gas combustion products cannot be completely discharged from the combustion chamber at its current pressure: it then begins to fill the cavity of the combustion chamber; as a result, the combustion chamber pressure is increased; and for the propellant whose pressure index is positive,

30

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

Fig. 2.8 Pressure–time curve with the cork in, nozzle 2016

the chamber pressure rises will further cause the burning rate to increase, and at the same time, the gas flow rate increases. With more gas filling the chamber, the pressure will be increased further, until it reaches a new pressure balance, where the amount of generated gas equals that discharged from the narrowed throat. It can be seen from this process that the increasing rate of chamber pressure increases, and also the thrust regulation time depends on the gas backflow rate and the volume of the combustion chamber cavity that needs to be filled. If the cavity is too large, the pressure adjustment time becomes unacceptable. Elsewhere [7, 8] studies of pintlecontrolled engines reveal a certain problem: due to the unreasonable experimental engine cavity design, pressure adjustment times reached more than 1 s, whereas actual thrust adjustment times need to be controlled to within milliseconds.

2.2 The Analytical Methods for Steady Aerodynamic Characteristics

31

2.2.2 Computational Fluidic Dynamics Computational fluidic dynamics is shown to be useful for simulation of the jet effect of fluidic-throat nozzles and the associated gas–liquid (for a liquid secondary working fluid flow), gas–particle flow phenomena. For a gas–gas fluidic throat, available information indicates that, compared to experimental values, the best accuracy of CFD simulation can reach about 1%, and as such it can be used to assist in the design and study of the internal flow regime in the fluidic throat. For hole-jet fluidic nozzle throat models, considering the symmetry of the layout, only the smallest symmetric unit of area of the flow field can be calculated (Fig. 2.9). A discrete computational domain, using a structured grid, can achieve higher accuracy and easier convergence. A better method is to discretize the computational domain of both jet and nozzle using a structured grid and undertake data transfer at the grid lap of the connecting face of the jet exit and the nozzle wall surface following interpolation (see Fig. 2.9 for a partially enlarged view) [9–13]. Fluidic-throat CFD simulation calculations showed that, to obtain higher accuracy, the number of 3-d grids in the computational domain is generally around one million, and adaptive mesh refinement should be used in regions of large Mach number gradient in the flow field. Figure 2.10a shows the choke performance curve of a hole-jet fluidic nozzle predicted using the S-A, k-ε, k-ω turbulence model before grid encryption (coarse mesh), and the comparison thereof to experimental results. Although the predicted values are slightly higher, the results reflect trends consistent with experimental data with a maximum error of c. 6.8%. Differences among the “ratio of flow coefficient−flow ratio” curves predicted by the three kinds of turbulence models are negligible. Figure 2.10b shows the prediction with encrypted (fine mesh) k-ε, k-ω turbulence models: it can be seen that even on the encrypted grid, the results of the predicted value of different turbulence models are still basically the same. Using the same number of grids, a

Fig. 2.9 Analysis model and grid

32

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

(a) Coarse grid results

(b) Encrypted grid results

Fig. 2.10 Flow coefficient−fixed flow ratio curves

single-equation SA turbulence model requires lower mesh quality near the wall and converges rapidly therewith. In practice, the SA turbulence model can be used to parameterize the study of fluidic-throat nozzles and their design. This study also shows that an SA turbulence model can reflect experimentally observed behavior and performance differences between nozzles with different structures. What was worth mentioning was that, in the numerical models, the roughness of the secondary flow jet inner wall is usually given as the ideal case, and the outlet edge of the jet is smooth. While for the actual processing engine, throat thermal protection materials have considerable thickness, which makes the secondary injection hole narrower and the wall roughness of the hole greater, thus causing the loss of total pressure and momentum in the secondary injection working fluid. In addition, if the outlet edge of secondary flow jet is of poor processing quality, it can also cause further loss of total pressure and momentum of the working fluid (secondary injection flow). Thus the choke performance of the secondary flow predicted by computational fluid dynamics methods is usually higher than that seen experimentally, as it can be seen that the ratio of flow coefficient curves turns downward in Fig. 2.10a. Usually, the results should be multiplied by an efficiency factor of less than 1 to characterize the actual situation.

2.3 Flow Characteristics To facilitate the discussion, main nozzle parameters are defined and shown in Fig. 2.11. The position of the nozzle is expressed as Di /Dt which is dimensionless, in which Di is the distance from center of the spray hole to the inner nozzle throat: a negative value indicates that the spray lies upstream of the throat. Dt is the diameter of the

2.3 Flow Characteristics

(a) Definition of distance

33

(b) Definition of incident angle

Fig. 2.11 Symbol definition

geometrical throat. The angle between the secondary and primary flows is defined as incident angle—α, as shown in Fig. 2.11.

2.3.1 Loop-Slot Fluidic Throat Some complicated flow phenomena appear in the spout of the secondary flow upon its being sprayed into the primary flow (Fig. 2.12a). After secondary flow ejection, it is sprayed by the primary flow into the nozzle exit. The separation of secondary flow appears slightly upstream of the secondary flow spout. Also, downstream of the secondary flow spout, there is a recirculation zone, which contains one kind of separation eddy in the boundary layer. At the same time, an induced shock is seen at the tail of the separation eddy. A shear layer of secondary flow forms as it is gradually mixed with the primary flow and exchanges energy and momentum on the wall of the nozzle. In addition, the place where the sonic line is educed has changed from the geometrical throat of the nozzle to one side of the secondary flow spout, which lies slightly downstream in the primary flow. The throat of the primary flow is also changed.

(b) Section position (a) Flow characteristics Fig. 2.12 Flow phenomena in a pneumatic throat

34

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

The formation of the induced shock is mainly due to shape changes experienced from the fluidic throat to the geometrical throat. After primary flow passes through the fluidic throat, it begins to re-expand on the pneumatic boundary thereof. However, after secondary flow through the recirculation zone, it begins to flow on the wall of the nozzle and is restricted by the wall. At the same time, the pneumatic boundary of the secondary and primary flows moves parallel to the wall of the nozzle whereafter, the direction of the secondary flow changes. All of these reasons cause the induced shock in the recirculation zone. Figure 2.13 shows the comparison of total pressure ratio distribution on different sections with, and without, secondary flow injection. The dashed lines represent the situation in which there was no secondary flow injection, and the total pressure ratio between the secondary flow and the main flow was greater than 1. The position of section D1–D4 is shown in Fig. 2.12b, where the vertical coordinate is a dimensionless radial distance (the reference length Y e is the distance between the local wall point and the axis) and the black dots represent the boundary between the secondary flow and the main flow. From the relative position of the black dots, it can be seen that the depth of the secondary flow penetrating into the main flow varies with axial position. During downstream flow mixing with of the secondary flow after injection from the nozzle, the depth of penetration into the main flow increases at first, then remains relatively stable, and finally decreases. Compared to the situation without secondary flow injection (dashed lines), the total pressure loss near the wall with secondary flow injection was greater; the total pressure loss existed in the shear layer of the secondary flow and the main flow and changed to the total pressure value of the main flow near the boundary.

Fig. 2.13 Total pressure ratio distribution on different sections

2.3 Flow Characteristics

35

Fig. 2.14 Static pressure distribution on different sections

Figure 2.14 shows the static pressure distribution in the latter three axial positions (D4–D6) in Fig. 2.13. With secondary flow injection, the pressure near the wall in the three positions was lower than that without secondary flow injection. The pressure variation in the center of the main flow was a little different in that the pressure was greater than that without secondary flow injection. This indicated that, when the secondary flow was injected, the nozzle throat moved downstream. Meanwhile, the Mach number in that position decreased and the pressure increased. Behind the recirculation zone, the pressure in the center of the main flow was lower than that without secondary flow injection (notice that the positions of D4 and D5 were near the center of a vortex core in the recirculation zone). In the recirculation zone, the pressure near the vortex core became lower and the pressure distribution appeared to be λ-shaped. The shape of the static pressure distribution behind the recirculation zone was similar to that without secondary flow, while the value of the static pressure decreased. It suggested that the secondary flow injection not only changed the size, shape, and position of the main flow throat, but also the interaction with the main flow to reform a new aerodynamic expanded profile. In addition, when the nozzle was in a state of full flow, the expansion ratio of the aerodynamic expanded profile was bigger than that of the original nozzle geometry, and the static pressure in the main flow at the nozzle exit was lower. Actually, the expansion ratio of the aerodynamic expanded profile increased with increasing total pressure ratio as shown in Fig. 2.15.

2.3.2 Fluidic Throat with the Rounded-Hole Injectors The typical 3-d flow field characteristics of a fluidic throat with a circular multijet are shown in Figs. 2.16 and 2.17. The Mach number of the secondary flow at the nozzle exit increased to the speed of sound with increasing total pressure ratio. Figure 2.16 shows that the circulation area of the mainstream decreased with increasing depth of secondary flow therein. Compared to the loop-slot injector, the fluidicthroat characteristics can be divided into two parts: similarities and differences. For their similarities, as the secondary flow sprayed in, the mainstream stagnated, and the total pressure increased to the total pressure in the combustion chamber at a location on the upstream side of the jet. The total pressure was reduced rapidly at the

36

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

Fig. 2.15 Static pressure distribution at the nozzle exit

(a) Ps/Pc = 1.25:1

(b) Ps/Pc = 2:1

Fig. 2.16 Three-dimensional flow characteristics of a pneumatic throat

(a) Ps/Pc = 1.25:1

(b) Ps/Pc = 2:1

Fig. 2.17 Mach number contours of the 3-d nozzle and its longitudinal section

2.3 Flow Characteristics

37

back of the secondary flow jet, and then slowly fell to that without secondary flow injection. However, the static pressure at the nozzle exit was smaller than that without secondary flow injection. For their differences, the backflow zone of the fluidic throat with a circular jet was not closed, which led to the area of pressure adjustment being smaller. Pressure adjustment near the fluidic throat tended to oscillate when the number of secondary flow circular jets was increased. In addition, because of the clearance of the jet, there was interference between the secondary flow from different jets, and a strong compression wave train was produced by jet interference thereat (Fig. 2.17).

2.3.3 Relationships Between Dimensionless Jet Position and Characteristics of the Fluidic Throat The pressure distribution along the nozzle wall and the mainstream parameters at the nozzle exit are affected by the shape and position of the fluidic throat (which were, in turn, decided by the position of the jet). In general, the characteristics of the fluidic throat will be obvious when the dimensionless jet position of the secondary flow Di /Dt is between −0.6 and 0.3, which corresponded to the high subsonic and transonic areas of the original nozzle. At Di /Dt > 0.3, the role of secondary flow disturbance was to induce shock waves, rather than to adjust the throat area (see Fig. 2.18). When changing the secondary flow nozzle position from high subsonic to the transonic zone, the choke performance of the fluid nozzle throat becomes sensitive to position, and the curve plotted for the ratio of flow coefficients shows a significant turning point: choke streaming performance will increase rapidly near the geometric

Fig. 2.18 The relationship between the ratio of flow coefficient and jet position

38

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

throat and reach its maximum in the vicinity of the geometric throat of the nozzle. The aforementioned inflection point position, and the largest choke performance position, will also be influenced by the chosen injection parameters, for example, when using a small spray angle, the best nozzle position for maximum performance of the choke will move to the right of the nozzle geometric throat. A reverse jet is actually equivalent to a vertical jet in cases where the nozzle position is advanced, but the choke performance of reverse jetting drops quickly in the high subsonic zone. Before the left-hand inflection point, choke performance is practically zero and the effective throat area barely changes. When using a large secondary flow ratio, the left, and right, sides of the inflection point generally move to the upstream and downstream sides of the geometric throat because, when the secondary stream flow ratio increases, the secondary flow disturbance of the mainstream will increase, as will the range of fluidic-throat features and the range of variation of the throat effective area. Similarly, increasing the total pressure ratio of the secondary flow will result in similar changes: there is a small difference in the high subsonic region before the left-hand inflection point, but the choking performance is more stable, and the effective throat area changes increased. Figure 2.19 shows Mach number contours and streamlines in a typical SRM nozzle throat for different secondary injection positions: changes in nozzle sonic lines for different secondary injection locations can be seen. In particular, Fig. 2.19c shows Mach number contours at two typical positions when the Di /Dt > 0.3: fluidic-throat characteristics had disappeared, and the position of geometric nozzle sonic lines cannot be changed by two ejection positions alone. When the ejection point moves downstream, the strength of secondary flow caused by the induced shock wave is stronger. Whether it formed a fluidic throat also affected the static pressure distribution at the nozzle outlet (Fig. 2.20). When forming a fluidic throat, the distribution parameters of the nozzle exit are similar to those in the situation with no secondary stream. When the secondary flow is ejected away from the throat in the supersonic region, there is zone in which the pressure increased in the nozzle outlet due to the shock wave: this differed from the distribution in the presence of a fluidic throat.

2.4 Relationship Between Secondary Flow Ratio and Total Pressure Ratio The curve which shows the relationship of the flux and total pressure ratio of the secondary flow nozzle is one of the basic characteristic curves for a nozzle, and it provides basic data for the design and selection of the nozzle parameters. Figure 2.21 shows the relationship of the flux and total pressure ratio of a secondary flow nozzle with different secondary flow nozzle area ratios, numbers of nozzles, and incident angles. When the nozzle position is fixed, the range of effective throat areas is mainly related to the ratio of secondary stream and mainstream flows. The example given in Fig. 2.21 includes a ring slit nozzle (nozzle area ratio, 30%), nozzle 8 (nozzle area ratio, 20%), and the fluidic-throat nozzle of nozzle 16. There, nozzle 8 includes three

2.4 Relationship Between Secondary Flow Ratio and Total Pressure Ratio

Fig. 2.19 Comparison of Mach number contours and streamlines at different jet positions Fig. 2.20 Parameter distribution: nozzle outlet section

39

40

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

Fig. 2.21 Flux changes with pressure ratio in the secondary flow nozzle

layout configurations (for specific layouts see Sect. 2.7.1) and two spray angles (90° and 45°). Nozzle 16 includes two nozzle configurations: contraction and straight types. The flux ratio of the secondary flow nozzle placed at the nozzle throat flow is mainly affected by nozzle area ratio and total pressure ratio, and is less affected by the number of nozzles and their layout. Different nozzle parameters generally correspond to a total pressure ratio threshold. When the total pressure ratio is below the threshold, the flow ratio of the secondary flow nozzle is quasi-constant as the total pressure ratio increases. When the total pressure exceeds the threshold, the flux gradually increases linearly with the total pressure ratio. After the total pressure ratio exceeds this threshold, to achieve the same flow ratio, the nozzle area ratio has to be larger, and the total pressure ratio will be smaller. At the same area ratio, to achieve the same flow ratio, the total pressure ratio of a shrunk nozzle is smaller than of a straight-hole nozzle, and the total pressure ratio of the reverse jet is smaller than that under vertical injection. To reduce the mass of the secondary flow system, the range of variation of the maximum effective throat area and the total pressure of the secondary flow system must be as small as possible. Usually, it is necessary to choose a nozzle whose area is relatively large, but there are many restrictions, such as mechanical requirements of ablation and structural integrity. The area of opening has an upper limit; if the area is too large, ablation and force conditions will worsen. Thus the opening area at the throat usually has an appropriate upper limit, and based on general requirements, the nozzle area ratio is deemed appropriate if it is in the vicinity of this upper limit. After determining the nozzle area ratio and configuration, according to throat requirements relating to the range of the maximum effective area, the desired maximum flow ratio can be obtained from the ratio of flow coefficient versus flow ratio curve for the

2.4 Relationship Between Secondary Flow Ratio and Total Pressure Ratio

41

corresponding nozzle. According to the maximum flow rate ratio, the design total pressure ratio can be read off Fig. 2.21, which shows curves of flux ratio versus total pressure ratio.

2.5 Secondary Flow Parameters and Effective Throat Area 2.5.1 Total Temperature of Secondary Flow and Equivalence Figure 2.22 shows the control performance comparison of fluidic nozzle throat with the same nozzle in two working conditions. In the first case, the working fluid of main flow is gas and the secondary fluid is high-pressure cold air. In the another case, the working fluid of the main flow and the secondary flow is cold air. In the example, the specific heat ratio of the gas is 1.2, the total temperature is 2900 K, and the total temperature of the cold air is 300 K. Figure 2.22a shows the flow coefficient ratio curve when the flow rate ratio is used as the independent variable. Figure 2.22b shows the curve when the modified √ flow ratio is used as the independent variable. The modified The modified flow rate is w = m˙ Tc (see Eq. (2.6) in Sect. 2.2.1). √ m˙ s √ Ts flow rate ratio of secondary flow and main flow is ws /wo = m˙ o T . o From the comparison of Fig. 2.22a, b, it can be seen that when the flow rate ratio is used as the independent variable, the flow coefficient ratio curve under the hot gas condition is quite different from the result when the cold flow is performed. However, when changing to the modified flow rate ratio as an independent variable, although the main working fluid properties are different and the total temperature is different, the results under the hot test conditions are in good agreement with the

Fig. 2.22 The flow coefficient curves. a Flow coefficient versus mass flow rate ratio. b Flow coefficient versus modified mass flow rate ratio

42

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

flow coefficient ratio curves of the cold-flow tests. This shows that the application of modified flow rate can extend the experimental and calculation results of cold flow to different main flow, secondary flow, and temperature conditions, so that the results under cold-flow conditions will be general. The significance of general results is that the working parameters to be investigated can be reduced, and the scope of research and the amount of hot test can be reduced. In the following, the modified flow rate ratio is uniformly used as the independent variable. When the total temperature of the main flow and the secondary flow is the same, the corrected flow ratio is equal to the flow rate ratio. In addition, from the modified flow rate ratio–flow coefficient ratio curve, it is found that under the same effective throat area, if the total temperature of the secondary flow is higher than the total temperature of the main flow, although the modified flow rate ratio required for different working fluids is the same, the mass flow rate ratio required is smaller. Therefore, to reduce the quality of carrying secondary fluids, high-temperature gas should be used as the working fluid of secondary flow. However, when high-temperature gas is used as the secondary fluid, the quality of the thermal protection structure of the secondary flow supply system will also increase and the valve requirements for the secondary flow will also be improved, so that comprehensive consideration should be made in the design.

2.5.2 The Mass Flow Rate Ratio Between Secondary Flow and Main Flow The law of effective throat area variation is a general rule of choke performance of fluidic nozzle throat. The change in the flow coefficient ratio C d /C do is actually equivalent to the change in the effective throat area before and after the secondary flow is introduced into the nozzle. The smaller the C d , the smaller the effective throat area (see Sect. 2.2.1). Figure 2.23 shows a typical curve of the flow coefficient ratio as a function of the modified flow ratio after the secondary flow injected into the fluidic-throat nozzle under different injector configurations. Summarizing curves of the flow coefficient ratio and flow rate ratio of fluidic nozzle throat, it can be seen that the curve shapes for different fluidic nozzle throat are similar. The effective throat area decreases with the increase of the secondary flow rate ratio, but the decreasing trend gradually slows down at the large flow rate ratio. It is because that, on the one hand, for the same increasing depth for injection penetration of secondary flow, the attained variation in throat area decreases as it is closer to the center of nozzle throat. On the other hand, the increasing penetration depth per unit of flow ratio decreases as flow rate ratio increases, so the decrease rate of the flow coefficient ratio decreases.

2.5 Secondary Flow Parameters and Effective Throat Area

43

Fig. 2.23 Flow coefficient versus the modified mass flow rate ratio

2.5.3 Comparison of Annular-Tuyere, and Rounded-Hole, Injectors Although the annular-tuyere injector has a different shape compared to the roundedhole type in actual use, it was shown to be consistent in performance terms. Therefore, parameterized numerical analysis, and injection parameter optimization, could be conducted on the basis of an annular-tuyere type, and the results remain justified for the design of rounded-hole injectors. The experimental data show that the injectors can perform as an ideal choke with annular-tuyere configurations as long as the right quantity of rounded-hole injectors is properly arranged (compare the data for the annular-tuyere-HF injector and 2008 injector in Fig. 2.23).

2.5.4 The Number of Secondary Flow Injectors At the same flow rate ratio, the more the rounded-hole injectors in the fluidic-throat nozzle, the larger the throat area variation (i.e., the better choked the system as seen in data from nozzles 2008 and 2016, Fig. 2.23). This advantage increases with increasing corrected flow rate ratio. As the corrected flow rate ratio is less than 0.2, the choked performances of the two injectors are similar. On the other hand, at the same injector area ratio, more secondary flow is required for the nozzle with fewer injectors to achieve the same effective throat area variation as one with more injectors. This could lead to an increase in the total pressure of the secondary flow. As a consequence, the mass secondary flow rate and that of the system also increase. However, given the same injector area ratio, a greater number of injectors confer no improvement. In

44

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

general, the throat insert and the heat-insulating layer in a solid rocket engine have considerable thickness. Punching in such a structure could generate a hole with a slenderness ratio larger than that of the fuel injector in a liquid-fuelled rocket engine. Due to friction, the loss of total pressure would increase as the secondary flow escaped through such a slender hole. As a result, the loss of the momentum of the injected secondary flow would degrade the choked performance of the fluid nozzle in a solid rocket engine. Therefore, for the same area ratio, the number of injectors should be designed in combination with the local erosion and stress conditions, as well as the choke performance of the nozzle.

2.5.5 Area Ratio of the Secondary Flow At the same flow rate ratio, the fluid nozzle injectors with a small area ratio can obtain better choke properties than those with large-area-ratio injectors (see nozzle data labeled 2008 and 1008 in Fig. 2.23). This improvement in performance is achieved by raising the total pressure of the secondary flow. Moreover, the advantage diminishes as the flow rate ratio increases. Thereby, at the same secondary flow rate, the high momentum of the flow injected by small injectors results in a good choke performance. As the flow rate ratio grows beyond 0.4, the performance of injectors with either small or large area ratios is similar, which implies that the effect of the total pressure becomes less significant. In reality, the secondary flow system in small injectors is required to provide a large total pressure to obtain the same flow rate ratio. In that case, the masses of the fluid tank and other structures increase significantly for only a limited improvement in the choke performance of the nozzle throat. On the other hand, to get the same area variation in the throat, the total pressure needed for larger injectors is much less than that in a small injector although the former deplete more of the secondary flow. Therefore, in practice, the area ratio of the injector is fixed according to the mass index of the secondary system, the erosion and stress state limits in the throat, and the thrust regulation index.

2.5.6 Injection Angle of the Secondary Flow There is an optimal angle for contra-stream injection to improve the choked performance of the throat nozzle: deviating from this angle results in performance degradation. Figure 2.24 shows the variation in throat choked performance for different secondary flow injection angles (90°, 60°, 45°, and 30°) from both annular-tuyere and rounded-hole injectors. Figure 2.24 shows that the choked performance of the fluidic-throat nozzle is improved as the injection angle decreased from 90° to 45°. However, as the angle further decreased, the performance degraded, as shown for an angle of 30°.

2.5 Secondary Flow Parameters and Effective Throat Area

45

Fig. 2.24 Flow coefficient versus the modified mass flow rate ratio at different injection angles

2.5.7 Effect of the Expansion Ratio on the Secondary Flow Injector The regulation ability of the fluidic throat performs as good as the expansion ratio of the secondary flow injector varied between 1 and 1.5. When the expansion ratio exceeded 1.5, a decreased choking action was observed. This was caused by a flow separation that arose near the outlet of the injectors at large expansion ratios, which led to a decrease in flow momentum. Figure 2.25 shows the profiles of the choking

Fig. 2.25 Flow coefficient versus the modified mass flow rate ratio (different K a )

46

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

action for different expansion ratios K a of the injector. The injected angle, in all such cases here, was 90°. The expansion ratio of the injector also affected the location of the liquid throat, i.e., the location of the main nozzle velocity line varied according to the injector expansion ratio. In general, the velocity line of the main flow from nozzles with large expansion ratio injectors migrated downstream compared to that with injectors with small expansion ratios. However, this effect was smaller than the effects of injected angle, injector location, etc. When the expansion ratio varied between 1 and 1.26, the deviation of the ratio of flow coefficient (effective throat area ratio) was about 0.04.

2.6 Effect of the Main Nozzle Parameters 2.6.1 Convergent Section of the Main Nozzle In the profile of the convergent section of the main nozzle, the convergent angle had the strongest effect on the choked performance of the fluidic-throat nozzle. In Fig. 2.26, “Nozzle2016” represents a profile of a fluidic-throat nozzle used in one experiment: this was used as a reference profile for comparative purposes. Due to the characteristics of the thermal protection structure of the solid rocket engine, the reference profile has a stepped face in the convergent section. The convergent halfangles before, and after, the step change were both 50°. The profile for “Nozzle 2016-1” has a convergent half-angle of 75° before the step change, while that after the step change was still 50°. The profile for “Nozzle 2016-2” had 75° half-angles

Fig. 2.26 Flow coefficient versus the modified mass flow rate ratio (different convergent sections)

2.6 Effect of the Main Nozzle Parameters

47

both before and after the step change. It was shown that “Nozzle 2016-2” had the smallest cavity volume, while the reference profile had the largest. The arc central angle of the nozzle convergent section, i.e., the convergent halfangle of the convergent section connected to the nozzle arc, also affected the choked property of the fluidic-throat nozzle. Too large an arc central angle could degrade performance. Therefore, in Fig. 2.26, the reference, and the “Nozzle 2016-1”, profiles have the best choked properties, while the performance of “Nozzle2016-2” was poorer. In addition, the response time to thrust regulation in the fluidic-throat nozzle and the cavity volume of the engine were correlated. A quick response requires a relatively small volume. Therefore, the “Nozzle2016-1” profile offers a better choice, combining both the requirements of response time and choked performance.

2.6.2 Back Pressure Ratio For a solid rocket engine nozzle with a relatively large expansion ratio, the ratio of the jet flow pressure and the background (nozzle pressure ratio, NPR) had little influence on the choked properties. Therefore, there was no need to consider this effect within the flight trajectory of a general solid-fuel engine. Only for small expansion nozzles, such as used in aircraft engines, the effect of the back pressure just becomes apparent. Figure 2.23 shows the data for a typical flow throat nozzle in over-expansion (NPR = 5), and full flow (design NPR of approximately 28), states, which indicated that the choked property in an over-expanded state with inverse flow appearing near the nozzle outlet differed from its designed state, but was consistent with the latter. This implied that inverse flow would not degrade the choked performance.

2.7 Comparison of Secondary Flow Injection Schemes in the Fluidic Throat 2.7.1 Layout of Secondary Flow Injectors In general, it was a good choice to adapt a radial layout scheme for the secondary flow injectors. Furthermore, the radially symmetric scheme outperformed its planesymmetric counterpart. A similar conclusion was obtained from the study of injector layout schemes for shock-induced vector control of secondary flow in a solid-fuel rocket engine. Figure 2.27 shows the performance profiles of three injector circumferential layout schemes in fluidic-throat nozzle with eight injectors. In these schemes, the injectors share the same position, have the same area ratio of 20%, and have the same geometry of their convergent holes. Scheme 1 is radially symmetrical. The nor-

48

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

Fig. 2.27 Flow coefficient versus the modified mass flow rate ratio

mal axes of the injectors in Scheme 2 are plane-symmetric. The injectors in Scheme 3 have radial injection but are disposed in a plane-symmetric layout. It was shown that Scheme 1 had the best choked performance, while Scheme 2 was the worst.

2.7.2 Negative and Active Schemes Under conditions with the same injector area ratio, the negative scheme outperformed the active scheme with regard to choke performance. While the injector area ratio increased, the choke performance gap increased for both negative and active schemes. Since secondary flow results from the main flow in a negative scheme, and the injector area is included in the main flow area, the bigger the injector area of the secondary flow, the bigger the main flow area. Therefore, the choke performance of an active scheme exceeded that of a negative scheme. Without carrying additional secondary flow, and given the simplicity of its secondary flow system, the negative scheme performed best. On the other hand, the total pressure ratio of the secondary flow cannot be changed therein, and the maximum secondary flow was dependent on the maximum injector area ratio alone: there were, therefore, some disadvantages. Hence, there was also a limit to the range of variation in effective throat area. The choke properties of both a negative and an active schemes under conditions of different injector area ratios are shown in Fig. 2.28. As the total pressure ratio was always 1 for a negative scheme, the flow ratio variation therefore depended on injector area ratio alone. The same Mach number nephograms of the corresponding negative schemes are shown on the right in Fig. 2.2.

2.7 Comparison of Secondary Flow Injection Schemes in the Fluidic Throat

49

Fig. 2.28 Flow coefficient ratio−modified flow rate ratio curves

2.7.3 Multi-injector Combination Scheme The choke performance curve of a multi-injector follows the trend seen in a single injector; however, the choke performance of the former is lower than that of the latter. Figure 2.29 shows the characteristic curve of a multi-injector and a single injector. Figure 2.29a shows the choke performance curve of the nozzle and Fig. 2.29b shows the injector flow ratio versus total pressure ratio curve for the two schemes.

Fig. 2.29 Curves of flow coefficient ratio and injector flow rate ratio

50

2 Steady Characteristics of a Gas–Gas Aerodynamic Throat

Fig. 2.30 Streamlines for two injector schemes

To reach the same flow ratio in the secondary flow, a multi-injector scheme needs a higher total pressure ratio in its secondary flow. Hence, a multi-injector scheme confers no performance advantage. The reason why the nozzle’s choke performance in multi-injector systems decreases lies in the interaction of much efflux of secondary flow and the backflow present among the flux in general. The increase of flow complexity in the backflow zone will increase the loss of momentum of the secondary flow, as shown in Fig. 2.30. However, the injector scheme combined with an active scheme and a negative scheme also has some engineering value. On one hand, a negative injector can decrease the carrying flow in the secondary flow. On the other hand, an active injector can use a high total pressure ratio to increase the accommodated range of effective throat area to adjust the total pressure of the negative scheme. In this sense, the combination of an active injector and a negative injector was still worth considering in engineering applications. This chapter has described the influence of different secondary flow, and nozzle, parameters on the flux parameter ratio of a fluidic-throat and fixed flow ratio curve under conditions of gas–gas fluidic-throat steady-state operations. Wherein, the most sensitive parameters of the fluid effective area of the throat were the flow ratio of the secondary flow, the absolute temperature ratio of the secondary flow, the number of injectors, the area ratio of the injectors, the spray angle, and the spray position. These were followed by the convergence angle of the primary nozzle constriction, the nozzle expansion ratio, and the layout of the nozzles. Unlike the nozzle of aeroengine with its small expansion ratio, there was little influence from the back pressure ratio on the effective area of the fluidic throat of a solid-fuel rocket engine.

References

51

References 1. Miller, D.N., Catt, J.A.: Conceptual development of fixed-geometry nozzles using fluidic injection for throat-area control. AIAA-95-2603 2. McArdle, J.G.: Internal characteristics and performance of an aerodynamically controlled, variable discharge convergent nozzle. NACA-TN4312, July 1958 3. Damiel, Y., Miller, N., Patrick, J.Y., Jeffrey, W.H.: Fluidic throat skewing for thrust vectoring in fixed-geometry nozzles. AIAA-99-0365 4. Baruzzini, D., Domel, N., Miller, D.N., et al.: Pulsed injection flow control for throttling in supersonic nozzles—a computational fluid dynamics design study. AIAA-2007-4215 5. Xie, K.: Numerical and Experimental Study on Fluid Throat of Solid Rocket Engine. Beihang University, Beijing 6. Kan, X., Yu, L., Wang, Y.B.: Steady numerical study of aerodynamic throat formed by round injectors. J. Aerosp. Power 26(4), 924–930 (2011) 7. Wang, Y.L., Guo-Qiang, H.E., Jiang, L.I., et al.: Experiment on non-coaxial variable thrust pintle solid motor. J. Solid Rocket Technol. (2008) 8. Juan, L.I., Jiang, L.I., Wang, Y.L., et al.: Study on performance of pintle controlled thrust solid rocket motor. J. Solid Rocket Technol. 30(6), 505–509 (2007) 9. Wang, F.: Computational Fluid Dynamics Analysis. Tsinghua University Press, Beijing (2004) 10. Tao, W.: Computational Fluid Dynamics and Heat Transfer. China Architecture & Building Press, Beijing (1991) 11. Zheng, Ya., Chen, J., et al.: Solid Rocket Motor Heat Transfer. Beihang University Press, Beijing (2006) 12. Ma, T.: Computational Fluid Dynamics. Beijing Aviation Academy Press, Beijing (1986) 13. Yan, C.: Computational Fluid Dynamics Method and Application. Beihang University Press, Beijing (2006)

Chapter 3

The Characteristic Function and Nozzle Efficiency

Abstract The characterization function and thrust efficiency are discussed in this chapter. A linear characterization function is applied. It is convenient to learn the parameters which affect the throttling capability obviously, by comparing the fitting slope of characteristic function curves from combining the results of hot firing tests, cold-flow tests, and simulations. Besides, the linear form of the characteristic function curve is convenient for engineering application and reference, which results in less experiments and simulations than before. The relationship between secondary flow parameters and thrust efficiency is also discussed.

3.1 Performance Characterization and Calculation Method for the Fluidic Throat in Solid Rocket Motors 3.1.1 Characteristic Function Before introducing the characteristic function of the fluidic throat, three characteristic flows should be defined as follows: Mass flux:  m ˙ =

ρV · dA

(3.1)

A

Momentum flux:  f =

 ρ|V|V · dA +

A

(P − Pref ) dA

(3.2)

A

© Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_3

53

54

3 The Characteristic Function and Nozzle Efficiency

Energy flux: ⎛ mH ˙ =⎝



⎞ ρV · dA⎠cp Tc

(3.3)

A

where m ˙ is the mass flow rate of working medium, H is the enthalpy, Pref is the reference pressure, V is the velocity vector, and ρ is the density of working medium.  Pref = Pc,m

2 γ +1

 γ γ+1 (3.4)

where Pc,m is the total pressure of primary flow and γ is the specific heat ratio. It is difficult to identify any variable of the three fluxes without knowing the other two. To compare the effective throat area under different pressures, temperatures, and Mach numbers, a composite flux function is defined for the three characteristic fluxes, which can make the dimensionless coefficient function of effective throat area a linear function. The composite flux function is defined as

f m ˙ s (mH ˙ )s s x= fm m ˙ m (mH ˙ )m

(3.5)

where the subscripts s and m represent the quantities of secondary flow and primary flow, respectively. The coefficient function y of the effective throat area is defined as y=

 2 At −1 At

(3.6)

At is the effective throat area of primary where At is the geometric throat area and flow. As a result, the xy curve composes characteristic function of the fluidic throat.

3.1.2 Thrust Efficiency of FNTs It is known that the pressure difference on the inside and outside surface of the motor composes the thrust [1–5]. Figure 3.1 shows the pressure at working state inside the motor, while the moving direction is taken as the positive direction. The thrust is calculated by Eq. (3.7). Unlike the original solid rocket motor, the thrust is provided by two parts: the secondary flow and the gas produced by the burning of the propellant. According to Fig. 3.1, the secondary flow contributes to the thrust by acting on the

3.1 Performance Characterization and Calculation Method …

55

Fig. 3.1 Schematic diagram for pressure distributed inside and outside the motor

divergent section, while the primary flow by acting on the top of the combustor and the convergent section. Besides, the secondary flow of a symmetrical fluidic throat has no contribution to the total thrust because of the symmetrical structure of the supply system. It is necessary to establish a model involving the part between section-1 and section-2 to study the fluidic throat. The thrust can be divided into two parts in Eq. (3.8), where F in is the integration of internal pressure acting on the wall, F b is that of the environment pressure. F in is divided into two parts in Eq. (3.11), where F 0–1,in is the thrust provided by the part between section-0 and section-1 calculated by Eq. (3.9), and F 1–2,in is the integration of internal pressure on the wall of nozzle model calculated by Eq. (3.10). The integration of environment pressure is calculated by Eq. (3.12).   Pin dA − Pb dA (3.7) Ft = wall

F0−1,in

wall

Ft = Fin + Fb   = vx dm + p dA 1−1

(3.9)

1−1



F1−2,in =

(3.8)

pin dA

(3.10)

1−2

Fin = F0−1,in + F1−2,in

(3.11)

Fb = Pa Ae

(3.12)

The one-dimensional ideal thrust of a certain flow refrigerant to environment pressure is defined as

56

3 The Characteristic Function and Nozzle Efficiency

Fi = m ˙

  1 2γ RT ∗ 1− γ −1 NPR

NPR = Pc /P b

(3.13) (3.14)

where m ˙ is the mass flow of the working medium; γ is the specific heat ratio; R is the ideal gas constant; and T ∗ is the total gas temperature. It is notable that for the primary flow, NPRc = Pc /Pb , and for secondary flow, NPRs = Ps /Pb . The thrust efficiency of a solid rocket motor (SRM) with fluidic throat is defined in two ways as η1 = Ft /Fi,o

(3.15)

  η2 = Ft / Fi,o + Fi,s

(3.16)

where F t is the actual total thrust calculated by Eqs. (3.7)–(3.12), which is provided by the primary flow and the secondary flow; F i is the ideal thrust of a certain flow gas and NPR; the subscripts o and s represent the primary flow and the secondary flow, respectively. The thrust efficiency η1 indicates the contribution of secondary flow to the thrust of primary flow; the thrust efficiency η2 indicates the total thrust loss of actual fluidicthroat nozzle directly.

3.2 Curves of Characteristic Function for FNTs Generally, only the condition of composite flux function x < 1 is considered, which means that the characteristic flux of the secondary flow is smaller than that of the primary flow. Figure 3.2 shows that the effective throat area y increases as the composite flux function x with the same amount of injector and different area ratios, which indicates that a linear function y = ax can be fitted. In physics, with the same composite flux function, a big slope a is related to a wide range of effective throat area and a strong throttling capability of secondary flow. In other words, a certain range of effective throat area requires a small x with a big slope a, which means consuming a small characteristic flux of secondary flow. It is convenient to learn the parameters which affect the throttling capability obviously, by comparing the fitting slope of characteristic function curves from combining the results of hot firing tests, cold-flow tests, and simulations. Besides, the linear form of the characteristic function curve is convenient for engineering application and reference, which results in less experiments and simulations than before. According to Fig. 3.2, the conditions of injector area ratio 10 and 20% have basically same characteristic function slopes with same nozzle amount 8, which means that the injector area ratio has few influences on characteristic function slope

3.2 Curves of Characteristic Function for FNTs

57

Fig. 3.2 Characteristic function curves with same injector amount while different area ratios

with the same amount and structure of injectors. The same characteristic flux of secondary flow (the same x) is required to make a certain range of fluidic-throat area no matter how large the area ratio of injector is. But a small area ratio requires high pressure ratio of the secondary flow to reach the intended characteristic flux. A high pressure ratio means a large momentum flux of secondary flow and results in the requirement of increasing pressure withstanding ability and system mass addition, which is consistent with the results discussed in previous chapter using the curve showing the relationship between flow coefficient ratio and modified flow rate ratio. The characteristic curves of a convergent injector and a straight-hole injector (nozzles of types B and C in Fig. 2.6) with area ratio 20% are shown in Fig. 3.3. According to the figure, the convergent injector has a greater slope which indicates a better performance of fluidic throat than straight-hole injector. But the straight-hole injector is worthy to be studied in practice because of the advantages of simple structure, easy processing, and good mechanical property. The equi-Mach line nephograms of the cross section of the convergent injector and the straight-hole injector at the same position of nozzle are shown in the right side of Fig. 3.3. The nephogram indicates that the secondary flow reaches sonic out of two kinds of injectors. However, after passing through the convergent injector, the permeation depth of secondary flow into the primary flow is larger than that through the straight-hole injector. One reason is that the total pressure loss of the narrow straight-hole injector is larger, which is why the throttling capability of the convergent injector is stronger than that of the straight-hole nozzle. According to the previous chapter, the use of reverse injection has a contribution to the throttling capability of the fluidic throat, and this conclusion can also be seen in Fig. 3.4. The characteristic function curves under conditions of the injection angles 90° and 60°, the area ratio 20%, and 8 injectors, are shown in the figure, in which the characteristic function slope is significantly increased with the used reverse injection.

58

3 The Characteristic Function and Nozzle Efficiency

Fig. 3.3 Characteristic function and flow field of two kinds of injector structure Fig. 3.4 Function curve with different injection angles

In fact, to characterize the throttling capability of the fluidic throat, the characteristic function curve can be fitted in a function y = ax, no matter what situation of the injector structure, injector count, injector area ratio, and injector profile parameters is. By using the characteristic function to compare the throttling capability of fluidic throat with different injection parameters and structure parameters, the conclusion is always consistent with that by using the relationship curve of flow coefficient ratio and modified flow rate ratio.

3.2 Curves of Characteristic Function for FNTs

59

Table 3.1 Characteristic function slopes a of typical fluidic throats Nozzle

Slope a

Injection angle

Nozzle structure

Nozzle area ratio (%)

Nozzle count n

Explanation

HF

4.96

90°

–

20

–

Annular nozzle

2008-1

6.19

90°

B

20

8

Radially symmetrically distributed nozzle

2008-2

5.86

90°

B

20

8

See Fig. 2.26

2008-3

6.05

90°

B

20

8

See Fig. 2.26

1008

6.72

90°

B

10

8

Radially symmetrically distributed nozzle

2016

6.01

90°

B

20

16

Radially symmetrically distributed nozzle

2016-T

5.67

90°

C

20

16

Radially symmetrically distributed nozzle

2016-1

5.53

90°

B

20

16

See Fig. 2.27

2016-2

4.52

90°

B

20

16

See Fig. 2.27

Both of the characteristic function curve and the curve of flow rate ratio or modified flow rate ratio can be used to characterize the throttling capability of fluidic throat. The former is simple in form, but the latter is more intuitive in design and engineering. One reason is that the momentum flux at injector outlet in composite flux is generally difficult to be measured directly in experiment. And the momentum flux at injector outlet has to be estimated while designing the fluidic throat by using characteristic function curve. However, in theoretical analysis and preliminary design, the characteristic function is still a good method. Typical characteristic function slopes related to previous chapter are shown in Table 3.1, which are convenient for lookup and interpolation in preliminary engineering design.

3.3 The Thrust Efficiency of the Fluidic-Throat Nozzle 3.3.1 Modified Mass Flow Rate Ratio Versus Thrust Efficiency As discussed above, on one hand, the pressure in the combustion chamber changes during the thrust adjustment by using the fluidic-throat nozzle. On the other hand, the expansion ratio of the aerodynamic expansion boundary is also different from that of the original geometric nozzle. Therefore, even if the fluidic-throat engine works at the design height of the geometric nozzle, it is difficult for the fluidic-throat nozzle to work just at the design pressure ratio. Therefore, it is more realistic to discuss the

60

3 The Characteristic Function and Nozzle Efficiency

Fig. 3.5 η1 -modified flow rate ratio curves with different back pressures

Fig. 3.6 η2 -modified flow rate ratio curves with different back pressures

thrust efficiency within a certain pressure ratio range. Figures 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, and 3.15 show the two thrust efficiency curves of the fluidic-throat nozzle 1008 with a circular orifice injector under different injection parameters and working conditions. The design pressure ratio of this geometric nozzle is 33.6. The thrust efficiencies η1 and η2 are defined by Eqs. (3.15) and (3.16). Since η1 is only referenced to the isentropic expansion thrust of primary flow, η1 could be greater than 1; but η2 is less than 1 because it is referenced to the total ideal thrust of secondary flow and primary flow. The contribution of secondary flow to the primary thrust increases as the flow rate ratio increases, and η1 generally increases

3.3 The Thrust Efficiency of the Fluidic-Throat Nozzle

61

Fig. 3.7 Curves of η1 -NPR

Fig. 3.8 Curves of η2 -NPR

linearly as the modified flow ratio increases, as shown in Fig. 3.5. The curves of η1 modified flow rate ratio have little differences around the design pressure ratio. When the working pressure ratio is large (NPR = 200 in Fig. 3.5), η1 will decrease due to the under expansion of the nozzle, but the curve slope is the same near the design pressure ratio. The curves of η2 -modified flow rate ratio with different pressures are shown in Fig. 3.6. When the modified flow rate ratio is less than 10%, η2 increases slightly or substantially unchanged with the increase of the modified flow rate ratio; when the modified flow rate ratio is greater than 10%, η2 decreases with the increase of the modified flow rate ratio, and the total thrust loss increases. The thrust efficiency η2 decreases obviously around the design pressure ratio (in the case of NPR = 30–60

62

3 The Characteristic Function and Nozzle Efficiency

(a) Curves of thrust efficiency–pressure ratio

(b) Curves of thrust efficiency–flow rate ratio

Fig. 3.9 Experimental curves of nozzles with a small expansion ratio

Fig. 3.10 η1 -modified flow rate ratio curves with different area ratios

in Fig. 3.6), but it becomes slowly at the high pressure ratio (NPR = 100 and 200). At the condition of NPR = 30, as an example, the thrust efficiency of the secondary flow decreases by 1.4% at the modified flow rate ratio of 30% compared with the case of without a fluidic throat, but it is much less at a high pressure ratio.

3.3 The Thrust Efficiency of the Fluidic-Throat Nozzle

63

Fig. 3.11 η2 -modified flow rate ratio curves with different area ratios

Fig. 3.12 η1 -NPR curves with different injection angles

3.3.2 Back Pressure Ratio NPR Versus Thrust Efficiency The thrust efficiency reaches a maximum at the design pressure ratio and decreases with the pressure ratio departing from the design value. The curves of thrust efficiency η1 and η2 with pressure ratio NPR of typical fluidic-throat nozzle of solid rocket motors are shown in Figs. 3.7 and 3.8. According to Fig. 3.8, with the increase of modified flow rate ratio, the pressure ratio will move away from the design value of the geometric nozzle toward the higher side after η2 reaches a maximum, which is

64

3 The Characteristic Function and Nozzle Efficiency

Fig. 3.13 η2 -NPR curves with different injection angles

Fig. 3.14 η2 -modified flow ratio curves with different injector numbers

caused by the greater expansion ratio of the aerodynamic expansion profile of the fluidic-throat nozzle compared with that of the geometric nozzle (see Fig. 2.15). Similar rules exist in the nozzles of aeroengines with small geometric expansion ratios (1.1–1.2). The curves of η2 -NPR for nozzles with a small expansion ratio applying air as the medium are shown in Fig. 3.9. The experimental nozzle has a design pressure ratio of about 3.3, a secondary fluidic nozzle with slot injectors, and an injection angle of 60°. Although the profile and structure of the experimental nozzle are different from those of the secondary fluidic nozzle in the form of a roundhole injector of solid rocket motors, the variation of thrust efficiency with NPR is consistent with that in Fig. 3.7. The thrust efficiency reaches a maximum at the design

3.3 The Thrust Efficiency of the Fluidic-Throat Nozzle

65

Fig. 3.15 η2 -modified flow ratio curves with different injector numbers

pressure ratio and decreases as the pressure ratio departs from the design value; with the increase of the modified flow ratio, the pressure ratio moves away from the design value of the geometric nozzle toward the higher side after η2 reaches a maximum. The experimental and CFD results of the thrust efficiency with flow rate ratio for the small expansion-ratio nozzle under a certain pressure ratio are shown in Fig. 3.9b. The thrust efficiency–flow rate ratio curve in the figure is also consistent with the law in Fig. 3.6: when the flow rate ratio is within 10%, η2 increases slightly or decreases slowly with the increase of the modified flow rate ratio; when the flow rate ratio is greater than 10%, thrust efficiency decreases with the increase of flow rate ratio.

3.3.3 Injector Area Ratio Versus Thrust Efficiency With the same flow rate ratio, the thrust efficiency with a large area ratio of an injector is greater than that with a small area ratio, and the critical secondary flow rate ratio is also larger when the efficiency curve begins to decrease significantly. Figures 3.10 and 3.11 show the thrust efficiency of the example nozzle 2008 and 1008 which have the same injector numbers while different area ratios of injectors. Figure 3.10 indicates that the injector with a large area ratio (nozzle 2008) has the approximately same η1 -modified flow rate ratio as that with a small area ratio (nozzle 1008) at near the design pressure ratio, implying that area ratio has little influence on η1. Figure 3.11 shows the curves of thrust efficiency η2 -modified flow ratio at about design pressure ratio. It can be seen that with the same secondary flow ratio, the fluidic-throat nozzle with a large area ratio has higher thrust efficiency than that with a small area ratio. Therefore, as the condition of ablation and structural force allows,

66

3 The Characteristic Function and Nozzle Efficiency

using the fluidic-throat nozzle with a large injector area ratio can not only adopt a small total pressure ratio of secondary flow but also obtain high nozzle efficiency. Besides, the efficiency of the nozzle with a large injector area ratio and a small injector area ratio begins to drop significantly when the secondary flow ratio was about 20% and 10%, respectively. The discrete points in Fig. 3.11 show that the nozzle efficiency varies with the flow rate ratio changed by altering the injector area ratio under conditions of fixed total pressure ratio of secondary flow (Ps /Pc = 1 and 1.5), and the thrust efficiency changes much slower than that by changing total pressure ratio of secondary flow. Compared to the nozzle without a secondary flow, a higher secondary total pressure leads to a greater decrease of thrust efficiency. It indicates that the total pressure ratio of secondary flow is also the main factor affecting the efficiency of the fluidic-throat nozzle.

3.3.4 Injector Angles Versus Thrust Efficiency With reverse injection, the thrust efficiency of fluidic-throat nozzle is reduced, and a smaller reverse injection angle causes a lower nozzle efficiency. The thrust efficiencies of the example nozzle 2008 with injection angles 90° and 60° of secondary flow under different conditions are shown in Figs. 3.12 and 3.13. The injection angle has little influence on η1 at near the design pressure ratio. The thrust efficiency η2 at conditions of a small injection angle is smaller than that of a large injection angle. According to Fig. 3.13, at the modified flow rate ratio 30%, the injection with the angle of 60° is about 0.4% less efficient than that with the angle of 90° at near the design pressure ratio.

3.3.5 Injector Numbers Versus Thrust Efficiency The nozzles with different injector numbers have similar thrust efficiency η2 at conditions that the modified flow rate ratio is less than 20%. More injectors cause lower thrust efficiency η2 at a large modified flow rate ratio, as shown in Fig. 3.14. The curves of η2 -modified flow rate ratio are compared in the figure under the conditions of the same injector area ratio while different injector numbers at near the design pressure ratio. For a typical fluidic-throat nozzle scheme, the thrust efficiency changes little as the effective throat area varies within 20%; while if the effective throat area decreases more than 20%, the thrust efficiency begins to drop significantly. In general, when the effective throat area decreases 40%, the thrust efficiency η2 of the fluidic-throat nozzle is reduced by 1–2% compared to that without the secondary flow injection. The curves of η2 -effective throat area (1 − C d /C do ) with different injector area ratios, injector numbers, and injection angles are shown in Fig. 3.15.

References

67

References 1. Li, Y., Zhongqin, Z., Zhao, Y.: Theory of Solid-Propellant Rocket, pp. 111–128. National Defense Industry Press, Beijing (1985) 2. Jirui, Z., Qingtang, Y., Tianyou, W., et al.: Design Basis for the Motor of Solid-Propellant Rocket (first volume). No. 210 Research Institute of China Ordnance Industry, Beijing (1982) 3. Wang, Y.: The Design of the Solid Rocket Motor. National Defense Industry Press, Beijing (1984) 4. Chongzhi, R., Zhang, Y., et al.: Design and Research on Motors of Solid-Propellant Rockets (first volume). National Defense Industry Press, Beijing (1990) 5. Ruxun, C., Li, Z., et al.: Design and Research on Motors of Solid-Propellant Rockets (second volume). National Defense Industry Press, Beijing (1990)

Chapter 4

The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

Abstract The fluidic-throat nozzles will encounter a new problem when they are applied to solid rocket motors. In order to improve the specific impulse of solid rocket motors, metal powders are often added to the propellant; the particle phase in the gas flow will cause the disturbance on the fluidic throat and change nozzle performance. Therefore, it is necessary to reveal the influence of the gas–particle two-phase flow on the performance of a fluidic-throat nozzle in the design. The disturbance of the particle phase on the primary flow and the secondary flow is divided into two aspects. On the one hand, due to the inertia of the particle phase, there is a speed difference between the gas phase and the particle phase. Generally, the particle phase lags behind the gas phase, so the particle phase will drag the gas phase. On the other hand, there is also a temperature difference between the particle phase and the gas phase, and the temperature of the particle phase is higher than that of the gas phase, so there is heat exchange between the two phases.

4.1 Disturbance of the Particle Phase on the Fluidic Throat The fluidic-throat nozzles will encounter a new problem when they are applied to solid rocket motors. In order to improve the specific impulse of solid rocket motors, metal powders are often added to the propellant; the particle phase in the combustion gas flow will cause the disturbance on the fluidic throat and change nozzle performance. Therefore, it is necessary to reveal the influence of the gas–particle two-phase flow on the performance of a fluidic-throat nozzle in the design. Figure 4.1 shows the disturbance of the particles with respect to the primary flow of the nozzle and the fluidic throat. The disturbance of the particle phase on the primary flow and the secondary flow is divided into two aspects. On the one hand, due to the inertia of the particle phase, there is a speed difference between the gas phase and the particle phase. Generally, the particle phase lags behind the gas phase, so the particle phase will drag the gas phase. On the other hand, there is also a temperature difference between the particle phase and the gas phase, and the temperature of the particle phase is higher than that of the gas phase, so there is heat exchange between the two phases. © Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_4

69

70

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

Fig. 4.1 Schematic diagram of particle turbulence on an aerodynamic throat

The above disturbance of the particle phase will influence the momentum ratio between the secondary flow and the primary flow at the throat, and thus affects the choking characteristics of the fluidic throat. The drag of the particle phase on the primary flow reduces the momentum of primary flow at the fluidic throat. Although the secondary flow is close to the wall of the nozzle, the “necking” phenomenon usually begins to appear to the particle trajectory at the throat, and the drag of the particle phase on the secondary flow is mainly concentrated on the edge of the fluidic throat; However, the change of parameters of the primary flow at the fluidic throat actually changes the injection environment of the secondary flow injector (such as the back pressure of the injector), which affects the injection parameters of the secondary flow injector exit. From this point of view, parameters of the particle phase also change the jet momentum of the secondary flow indirectly.

4.2 Two-Phase Flow Theory and the FLNT-V1.1 Analytical Codes It is difficult to separate the loss caused by two-phase flow from the other losses of the nozzle, such as viscous loss, diffusion loss, etc., from the results of the hot test, so the gas–particle two-phase theory and numerical model are mainly used to analyze the effect of the particles on the performance of the nozzle of solid rocket motors, and then the gas–particle two-phase flow loss of the nozzle is determined by comparing with the hot test results.

4.2.1 Particle–Gas Two-Phase Flow Models The calculation methods of gas–particle two-phase flow can be divided into two types: particle orbit model (Lagrange) [1–6] and pseudo-fluid (Euler) model [7–11]. The main difference between the two types of calculation methods lies in the treatment of the condensed phase. The former uses the Lagrangian method to track the

4.2 Two-Phase Flow Theory and the FLNT-V1.1 Analytical Codes

71

movement of the particles. The details of the particle motion are easier to grasp, which helps to understand the two-phase flow field phenomenon; while as the calculation number of particle orbits increases, the computational complexity increases greatly. The latter considers the particle phase as a pseudo-fluid from a macroscopic point of view, and establishes control equations using a similar method to the gas phase to solve the flow field of particle phase. For the pseudo-fluid method, the control equations of the gas phase and the particle phase are similar, so that the turbulent diffusion of the condensed phase can be better simulated without introducing large computational complexity. The two-phase flow calculation of nozzles adopts the particle orbit model, including the US Solid Rocket Engine Industry Standard Procedure SPP [12], which has been proved to be able to describe the main characteristics of the flow field at the nozzle of solid rocket motors and can better predict its performance. The Beijing Institute of Technology developed the FLNT-V1.1 analysis code for the fluidic throat of solid rocket motors. The code uses the particle orbital model to calculate the two-phase flow field of the fluidic-throat nozzle and predict the effect of the particle phase on the performance of the fluidic-throat nozzle. In addition, the code includes a one-dimensional swirling ideal model that can analyze the effect of particle phase on the swirling fluidic throat, as in the case of a vortex valve engine [13]. We take the fluidic-throat nozzles: the axial-gap injector and the round-hole injector, as an example to explain the effect of the particle phase on the performance of fluidic throats of the solid rocket motor. The particle phase in the example is Al2 O3 particles. The particles have an average diameter of 1–10 µm; the particle size varies from 1 to 100 µm and the particle mass fraction ranges from 5 to 30%. The above parameters are typical ranges for particle parameters in solid rocket motors.

4.2.2 Control Functions of Two-Phase Flow The particle phase in the solid rocket engine is generally considered to be the “dilute two-phase flow” because of its small content, and the volume occupied by the particles and its contribution to the pressure can be neglected. Here we only consider resistance and convective heat transfer as the interaction between the two phases, and there is no mass exchange. There is also no mass exchange between the two-phase flow and the wall, and the wall is adiabatic. In addition, the phase transition of the particles is not considered at the nozzle, and the particles are considered to be solid. Under the above assumption, the two-dimensional N-S equations of the source terms of coupled particle phase are ∂E ∂F ∂ Ev ∂ Fv ∂U + + = + + Sp ∂t ∂x ∂y ∂x ∂y

(4.1)

72

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

where S p is the gas–particle source term and expressed as

Sp =

⎧ ⎫ 0⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎨S ⎪ u

⎪ Sv ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ ⎪ Se   Su = n˙ k m p u p in − m p u p out /V k 

 Sv = n˙ k m p u p in − m p u p out /V k      u 2 +v2 u 2 +v2 − mp hp + p2 p Se = n˙ k m p h p + p 2 p in

k

out

(4.2)

 /V

(4.3)

where n˙ k is the particle number flux; m p is the mass of a single particle; u p and v p are the velocity components of the particle in the x and y directions, respectively; k is the particle group; the subscripts in and out represent the mesh inlet and the mesh exit, respectively; V is the grid volume. The three-dimensional N-S equations of the source terms of coupled particle phase can be derived from the two-dimensional equation. The orbit of the discrete phase particles is solved by integrating the differential equation of particle force in the Laplace coordinate system. The force balance equation of the particle, that is, the kinetic description equation of the particle phase is particle inertia = various forces acting on the particle. For example, the form of the component form in the x-direction in the Cartesian coordinate system is   gx ρ p − ρ du p = FD u − u p + + Fx (4.4) dt ρp  where FD u − u p is the drag force per unit mass of the particle. FD =

18μ C D Re ρ p d 2p 24

(4.5)

where u is the fluid phase velocity; up is the particle velocity; μ is the dynamic viscosity of the fluid; ρ is the fluid density; ρ p is the particle density; d p is the particle diameter; Re is the relative Reynolds number (particle Reynolds number),   ρd p u p − u  (4.6) Re ≡ μ The calculation method of the coefficient C D in the Eq. (4.5) can be found in the literature [1–6]. F x is expressed as other force in the kinetic equation of the particle phase. These possible forces are the additional mass force, thermophoresis, Brown force, Saffman lift, etc., which can be added to the equation depending on the needs of the specific simulation problem. Generally speaking, only the drag force of the

4.2 Two-Phase Flow Theory and the FLNT-V1.1 Analytical Codes

73

particles on the gas phase is considered in the two-phase flow simulation of the solid rocket motor. The calculation area and grid of the two-phase flow field of the fluidic-throat nozzle are the same as in Chap. 2. The convection term in the gas-phase equation adopts the second-order upwind scheme to discrete, and the turbulence model still uses the S-A turbulence model. The total temperature and total pressure of the gas are given at the inlet of the secondary flow injector and the nozzle. The pressure outlet is applied as the right boundary condition of the nozzle, and the wall surface is the adiabatic and non-slip condition. In addition, the mass flow, temperature and particle diameter distribution of the particle phase are also needed to be given at the nozzle inlet. During the calculation, the particles are assumed to completely rebound when reaching the solid wall and are assumed to leave the calculation area when reaching the right boundary.

4.2.3 Nozzle Loss and Standard Calculation Cases The main factors affecting the efficiency of the nozzle are the properties of the gas, the components and the structural shape of the nozzle. Losses of nozzles of solid rocket motors include chemical kinetic loss ηk , boundary layer loss ηB , two-phase flow loss and diffusion loss ηTD , submergence loss ηs , and nozzle ablation loss ηE . Therefore, the actual thrust coefficient of the nozzle is C F = ηC E × C F 

where C F is the theoretical thrust coefficient and ηC E is the nozzle efficiency coefficient. ηC E can be expressed as ηC E = 1 − (η K + η B + ηT D + η S + η E ) The two-phase flow and diffusion loss ηTD of the nozzle is the most important loss among all of the above losses. The US Solid Rocket Engine Industry Standard Procedure SPP mainly predicts two-phase flow loss and nozzle diffusion loss through numerical simulation of two-phase flow field of nozzles, and other losses are predicted by using empirical formulas. The purpose of the standard calculation cases is to test the relative accuracy of the numerical model and the code used to predict the twophase loss and diffusion loss of the nozzle of the solid rocket motor. The loss of two-phase flow is mainly related to particle size and mass percentage. The diameter of the Al2 O3 particles in the standard calculation cases uses the empirical formula in the SPP empirical method.   −8 (4.7) D p = 9.3722Dt0.2932 1 − e−0.0816×10 ξ τ

74

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

where Pc is the combustion chamber pressure; Dt is the diameter of nozzle throat; ξ is the mole fraction of Al2 O3 particles (mol/100 g); τ is the average residence time of the Al2 O3 particles in the combustion chamber. Next, we use the FLNT-V1.1 code developed to predict the performance of the US AM engine and the French SEP engine, and compare with the experimental results and other prediction results. I. The predictive result of the AM engine The nozzle profile of the AM solid rocket motor is shown in the left side of Fig. 4.2, and its coordinates are dimensionless using the ratio of it to the throat radius Rt . The nozzle profile consists of a three-heart arc convergent section (the radius of the inlet section arc is 14.478 mm and the curvature radius of the throat is 53.086 mm) and a conical divergence section (divergence half-angle is 19.7°). The initial divergence ratio of the nozzle is 103.2, the average pressure in the combustion chamber is 5.04 MPa, and the hydroxy-terminated polybutadiene propellant (HTPB) with an aluminum powder content of 16.4% is used. The burning rate and the working time of the propellant are 6.25 mm/s and 33.12 s, respectively. The throat diameter before and after the engine test is 51.708 mm and 53.374 mm, respectively. The nozzle is a submerged nozzle and its submergence is 44.45 mm. The vacuum-specific impulse of the engine test is 2874.5 N s/kg. The two-phase flow field of the AM engine with a corresponding particle size of 2.48–2.95 µm was calculated according to the formula (4.3–4.7). Figure 4.2 shows the equi-Mach line diagram and particle orbit diagram for d p = 2.95 µm, and these diagrams were compared with the flow field diagram under pure gas-phase conditions. Because of the disturbance of the particle phase, the position of the sonic line at the throat moves downstream compared to that under pure gas-phase conditions. In addition, due to the inertia of the particle phase, the particle trajectory has a “necking” phenomenon at the throat of the nozzle, and there is a limit particle trajectory.

Fig. 4.2 Flow field diagram and particle orbit diagram for d p = 2.95 µm

4.2 Two-Phase Flow Theory and the FLNT-V1.1 Analytical Codes

75

Table 4.1 The predictive result of the AM engine impulse Process

DE BC

US SPP

IT SN IA BPD

GB RO

FR GB Aerospa- SBS tiale

The FLNT Fourth Institute of Aerospace [5]

Theoretical vacuum-specific impulse

3177.35 3183.5

3185.2

3185.2

3185.2

3185.2

3185.4

3187.39

Diffusion loss

95.12

237.6

93.33

239.3

223.1–233.74

Two-phase flow loss

100.02

Boundary layer loss

68.65

20.27

20.13 29.32

29.32

50

24.2

7.35

16.08

15.6

7.84 0

9.81

7.96

7.14

Use empirical formula in SPP for prediction

Chemical dynamic loss

0

24.49

20.45 0

Ablation loss of nozzle throat

10.79

8.74

Nozzle submergence loss

14.71

19.99

20.83 0

24.22

20.69

22.55

Characteristic velocity loss

0

6.84

15.93 30.99

0

0

0

Total loss

289.29

314.63

294.06 344.7

308.32

301.33

308.79

Predicted specific impulse

2888.1

(Two-phase loss + diffusion loss)/total loss (%)

67.4

234.3

208.88 284.39

113.27

2868.9

2891.1

74.5

71.0

2840.5

2876.88 2883.87 2876.6

82.5

77.1

68.6

77.5

74.1

The two-phase flow and divergence loss predicted by using the FLNT-V1.1 code are listed in Table 4.1 (other loss values are consistent with the predicted results of the SPP). For the convenience of comparative analysis, the predictive results of the software such as German BC, American SPP, Italian SNIABPD, French SEP, British RO, French Aerospatiale, German Landsbaum, and Italian SBS are also listed. It can be seen from the table that the two-phase flow loss and diffusion loss of the nozzle of the AM engine solid rocket motor predicted by all prediction software accounts for 67.45–82.6% of the total loss, indicating that the two-phase flow and diffusion loss contributes the most to the total loss. The result of the prediction means that the ability to predict two-phase flow loss and diffusion loss is the key to accurately predict nozzle efficiency. Among them, the software of SPP and the Fourth Institute adopt the flow field simulation method to predict the loss of two-phase flow and diffusion loss, which is more accurate than the prediction using the empirical method. Therein,

76

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

the predicted value of two-phase flow loss and diffusion loss is 234.3, 239.3, and 223.1–233.74 by SPP, the Fourth Institute of Aerospace of China, and this model (corresponding particle size ranging from 2.48 µm to 2.95 µm [5]), respectively. II. The predictive result of the SEP engine The SEP solid rocket motor is supplied by the France SEP, and the nozzle profile is shown in Fig. 4.3. The convergent section of the nozzle is conical, the half-angle of the convergent nozzle is 45°, the curvature radius of the upstream of the nozzle is 30.75 mm, and the profile of the downstream of the nozzle is a special expansion section. The throat radius before and after the engine test was 20.5 mm and 21.075 mm, respectively, and the external radius of the nozzle is 158.85 mm. The combustion chamber pressure is 7.34 MPa and the propellant contains 20% of aluminum powder. The vacuum specific impulse of the engine based on the experiment is 2909.63 N s/kg. The two-phase flow field of the SEP engine with a corresponding particle size of 2–2.5 µm was calculated according to the formula (4.7). Figure 4.3 shows an equi-Mach line diagram and a particle trajectory diagram. The flow field obtained is similar to that of the AM engine. The theoretical vacuum-specific impulse and losses predicted by this model and other software are listed in Table 4.2. It can be seen from the table that the two-phase flow loss and diffusion loss predicted by all the software account for 64.91–86.92% of the total loss, which is still the majority of the total loss. Among them, the values of two-phase flow loss and diffusion loss predicted by SPP and the Fourth Institute are 249 and 273.11, respectively, while it is in the range of 258.24–274.63 by this model (corresponding particle size range from 2 to 2.5 µm [5, 6]). The calculation of the above two standard calculation cases indicates that the accuracy of the two-phase flow loss and the diffusion loss predicted by using FLNTV1.1 code developed by the Beijing Institute of Technology is acceptable.

Fig. 4.3 Flow field diagram and particle orbit diagram for d p = 2.5 µm

4.3 The Influence of Particle Sizes

77

Table 4.2 Predictive result of the SPE engine Project

DE BC

Theoretical vacuum-specific impulse

IT SN IA BPD

GB RO

FR IT Aerospa- SBS tiale

The Fourth Institute

FLNT

3194.03 3186.5

3185.2

3185.2

3185.2

3190.1

3193.11

3193.11

Diffusion loss

94.14

269.00

232.42

241.24

83.94

273.11

258.24–274.63

Two-phase flow loss

99.05

Boundary layer loss

62.76

37.3

25.99

39.23

35.3

93.46

26.45

Chemical dynamic loss

4.9

8.33

14.32

0

6.96

14.32

11.01

Use empirical formula in SPP for prediction

1.87

3.92

0

4.41

3.92

3.62

Ablation loss of nozzle throat

US SPP

249.00

122.68

Nozzle submergence loss

0

0

20.83

0

24.22

20.69

22.55

Characteristic velocity loss

0

1.27

0

40.21

0

0

0

Total loss

260.85

297.77

313.23

344.7

287.91

318.32

314.21

Predicted specific impulse

2933.18 2888.7

2876.87 2882.16 2898.66

2871.78 2878.9

(Two-phase loss + diffusion loss)/total loss (%)

74.1

85.9

64.9

83.2

67.4

83.8

86.9

87.0

4.3 The Influence of Particle Sizes The particles disturb the fluidic throat, resulting in the change to the curve of flow coefficient ratio with flow rate ratio under the two-phase flow condition deviating from the result under pure gas-phase condition. In the range of 0–40% modified flow rate ratio and 5–30% content of particle phase, the flow coefficient ratio of the fluidicthroat nozzle has a deviation (C d /C do ), which is generally within ±4%, compared with the pure gas phase. The value of (C d /C do ) is related to the size, content and modified flow rate ratio of the particle phase. When the particle size is relatively small, the performance curve of the fluidic throat is not apparently different from that of the pure gas phase. When the particle size increases and the particle content is large, there is a significant deviation between the two-phase flow and the pure gas flow. For a fluidic-throat nozzle in the form of a round hole, this deviation also tends to increase as the modified flow ratio becomes larger. Therefore, when designing a fluidic-throat nozzle under two-phase flow conditions, it is necessary to correct the experimental and theoretical calculation data of the pure gas phase that used, or set a margin, depending on the specific situation.

78

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

4.3.1 Cases of Axial-Gap Nozzles Figure 4.4 shows the curve of flow coefficient ratio with the modified flow rate ratio of a fluidic-throat nozzle with an axial gap. The definition of the modified flow rate ratio under two-phase flow conditions is the same as that under pure gas-phase conditions, see formula (4.2–4.6); except for the condition that the flow rate of the primary flow m˙ o contains that of the particle phase, that is m˙ o = m˙ g + m˙ p , m˙ g is the flow rate of the gas phase in the primary flow. In the example, the refrigerant of the secondary flow is high-pressure cold gas, the primary flow is gas, the nozzle area ratio is 15%, the mass content of the particle phase is fixed at 20%, and the particle size here refers to the uniform size. Under the two-phase flow condition, the trend of the curve of flow coefficient ratio with the modified flow rate ratio of the fluidic throat with an axial gap is basically consistent with that under pure gas-phase condition. When the average diameter of the particle phase d p = 1 µm, the curve substantially coincides with the curve under pure gas-phase condition; however, as the particle size increases, the curve deviates from the curve under pure gas-phase condition. Therefore, when the average particle size is small, the experimental and calculated results obtained under the pure gasphase condition can still be used to estimate the result under the two-phase condition. However, when the particle size is large, it is necessary to modify the value of pure gas phase or set a certain margin in the design (see the thick dotted line in Fig. 4.4). Figure 4.5 further shows the changes of the flow coefficient ratio C d /C do of the fluidic-throat nozzle with particle size under different modified flow rate ratios of the secondary flow. It can be seen that the flow coefficient ratio presents approximately the same deviation trend, that is, the flow coefficient ratio decreases at first as the particle size increases, and then increases. Moreover, in the common average particle

Fig. 4.4 Curve of flow coefficient ratio with the modified flow rate ratio

4.3 The Influence of Particle Sizes

(a) ws/wo=13%

79

(b) ws/wo=20%

(c) ws/wo=30%

Fig. 4.5 Changes of flow coefficient ratio with particle size under different secondary flow ratios

size range of the solid rocket motors, the predictive value of the flow coefficient ratio is smaller than that under the pure gas-phase condition, that is, the choking performance increases. The analytical curve here provides correction and reference for extending the data under the pure gas-phase condition to two-phase flow condition. The effect of average particle size on the two-phase flow field and sonic line position of the fluidic-throat nozzle is shown in Fig. 4.6. Figure 4.6a compares the equi-Mach line nephograms of the fluidic-throat nozzle (f = 20%) under the pure gas-phase condition and the double-phase flow condition that the average particle diameter is d p = 5 µm. It can be seen that the presence of the particle phase causes the sonic line at the fluidic throat to move downstream compared with the pure gas phase. Figure 4.6b shows the equi-Mach line nephogram under conditions that the particle sizes are d p = 1 and 10 µm. It can be seen that the sonic line under condition of d p = 1 µm appears at the position further downstream than that under d p = 10 µm. The particle size also has a significant effect on the particle trajectory in the fluidicthroat nozzle of the solid rocket motor. When the particle size is small, there is a

(a) pure gas phase and dp=5 μm Fig. 4.6 Equi-Mach line nephogram

(b) dp=1 μm and dp=10 μm

80

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

(a) dp=1 μm

(b) dp=10 μm

Fig. 4.7 Particle trajectories with and without secondary flow

significant difference between the limit trajectories of the particles under conditions with and without secondary flow, see Fig. 4.7a. When d p = 1 µm, the limit particle trajectory is significantly deflected toward the axis at the fluidic throat due to the influence of the secondary flow. Compared with the situation without secondary flow, the effect of secondary flow and particles in the nozzle causes the particles near the physical wall of the nozzle to flow along the “aerodynamic wall” of the shear layer formed by the secondary flow. The limit particle trajectory away from the physical wall indicates that the collision probability of the particle to the nozzle is reduced. With the increase of particle size, as shown in Fig. 4.7b, the difference between the particle trajectories at the throat with and without secondary flow is reduced, and the positions of the limit particle trajectory at the divergent section are also substantially the same. This is because after the particle size enlarges, the inertia of the particles increases, the tendency of the particle trajectory to converge toward the axis enhances, and the necking phenomenon at the throat becomes more obvious. As a result, the area and extent of the particle interacting with the secondary flow at the throat are reduced. In addition, it can be seen from the static pressure distribution along the nozzle wall in Fig. 4.8 that the disturbance of the particle phase to the aerodynamic throat is concentrated on the upstream side of the nozzle and the recirculation zone of the downstream side. Besides, the difference in static pressure in this area is more obvious as the content of the particle phase increases.

4.3.2 Cases of Round-Hole Nozzles Similar to the case of the axial-gap nozzle, the choke curve of the fluidic throat of a round-hole nozzle under conditions of small particle size is not different much from that under pure gas-phase condition; when the particle size increases, the difference between them increases. For the round-hole fluidic nozzle, the difference in choking

4.3 The Influence of Particle Sizes

(a) f=20%

81

(b) f=30%

Fig. 4.8 Static pressure distribution along the wall of the nozzle under different contents of particle phase Fig. 4.9 Performance of the nozzle under two-phase flow condition

performance always increases, but the trend slows down when the particle size enlarges (see d p = 20 µm in the figure). Under the general condition of the modified flow rate ratio (less than 20%), the choking performance of the round-hole nozzle basically enhances as the particle size increases (Fig. 4.9). When the modified flow rate ratio is small, the trend of how the flow ratio coefficient of round-hole fluidic-throat nozzle varying with particle size is similar to that of the axial-gap nozzle. But the trends of these two are slightly different when there is a large flow ratio coefficient. Here, as the particle size increases, the C d /C do value first decreases, then increases and finally decreases. Figure 4.10 shows the curve of the

82

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

(a) ws/wo=8%

(b) ws/wo=17.5%

(c) ws/wo=23.4%

Fig. 4.10 Changes of the flow coefficient ratio with particle size

C d /C do value with particle size at the fluidic throat with a round-hole nozzle under typically corrected flow coefficients. It is worth noting that, unlike the axial-gap nozzle, due to presence of the gap between the round-hole injectors, the secondary flow injected will mix with the primary gas in the gap during the downstream flow, and the primary gas flow will have a tendency to be entrained to the wall by the secondary flow, see Fig. 4.11. Therefore, when the secondary flow ratio is large, the limit trajectory of the particles at the round-hole fluidic-throat nozzle is entrained toward the wall of the nozzle, so it is not as far away from the nozzle wall as expressed in the axial-gap nozzle. This may also be the reason why the variation of the round-hole fluidic-throat nozzle and the axial-gap nozzle are similar in trend but not completely consistent under two-phase flow condition. It indicates that the disturbance of the particle phase to the fluidic throat also depends on the morphology of fluidic throat and the distribution of particles in the flow field.

4.4 The Influences of Mass Fraction of Particle Phase 4.4.1 Cases of Axial-Gap Nozzles When the particle size increases and the particle content is large, the choking performance curve of the axial-gap fluidic-throat nozzle is significantly deviated compared with that under the pure gas-phase condition; when the particle content is small, the deviation is small. However, within the range of parameters of particle phase commonly found in solid rocket motors, usually the maximum deviation (C d /C do ) is still within ±4%, see Fig. 4.12.

4.4 The Influences of Mass Fraction of Particle Phase

83

Fig. 4.11 Flow field diagram under conditions of different particle sizes

(a) dp=1 μm

(b) dp=5 μm

(c) dp=10 μm

Fig. 4.12 Curves of flow coefficient ratio with modified flow rate ratio under different particle sizes

84

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

4.4.2 Cases of Round-Hole Nozzles Similar to the results of the axial-gap nozzle, the curve of the round-hole fluidicthroat nozzle on conditions of small particle size has little difference with that under pure gas-phase conditions; when the particle size and content are large, there is a significant deviation (Fig. 4.13). In addition, the regularity of the round-hole fluidic-throat nozzle is better than the axial-gap nozzle, that is, when the particle size is large, the choking performance of the nozzle enhances as the particle content increases, and the choking performance is better than that on the pure gas-phase condition when f > 10%. Figure 4.14 shows the typical nephograms of Mach number distribution near the wall, mass content of the secondary flow and Mach number distribution under conditions of different particle contents at d p = 5 µm. In the secondary flow (highpressure nitrogen) in the figure, it can be seen from the nephogram of nitrogen content distribution that after the secondary flow is injected, the core flow of the secondary flow does not flow along the wall after passing through the recirculation zone, but leaves a certain distance from the wall of the nozzle and flows downstream, which is different from the result of the axial-gap nozzle; and there is a lateral mixing of the secondary flow and the primary flow in the nozzle gap. It can be seen from the nephogram of Mach number distribution near the wall of the nozzle that the interference shock system is formed between the secondary flow jets. Under the same nozzle area ratio, the more the nozzles there are, the more details the interference shock system. Moreover, the transverse flow of the nozzle increases, and the momentum loss of the secondary flow will also grow. This explains why the choking performance of the round-hole fluidic-throat nozzle reduces and is lower than that of the axial-gap nozzle when there are too many nozzles (see Chap. 2).

(a) dp=1 μm

(b) dp=5 μm

(c) dp=20 μm

Fig. 4.13 Curves of flow coefficient ratio with modified flow rate ratio under different particle sizes

4.5 Influences of Profiles of Particle-Size Distribution

85

Fig. 4.14 Flow field diagrams under different contents of particle phase

4.5 Influences of Profiles of Particle-Size Distribution 4.5.1 Control Performance The influence of main particle parameters on the choking performance of fluidic throat discussed above is based on the uniform-size model. In actual solid rocket motors, the size of the particles is continuously distributed in a certain range, not a single value. The study of traditional nozzle shows that the flow field characteristics calculated by the multi-size model are closer to the actual situation [14]. The influence of particle-size distribution on the performance of fluidic-throat nozzle is further introduced below. The particle-size distribution at the fluidic-throat nozzle of solid rocket motors can be roughly divided into two types—unimodal distribution and bimodal distribution [5, 6, 14]. For example, the normal distribution is a typical unimodal distribution. Figure 4.15 shows the continuous  distribution density of three typical unimodal , where dm is the particle mass whose distribution models Y (d p ). Y d p = m1p d dm (d p ) diameter is from d p − d(d p ) to d p + d(d p ) and mp is total particle mass. In the calculation, the particle size is discrete into 10 groups. The mass fraction of each

86

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

Fig. 4.15 Three types of size distribution

particle-size group is C i which is obtained by the integration of Y (d p ). Each group corresponds to a mass-weighted average particle diameter d¯ p,i . C i also satisfies the following relation formula: 

Ci = 1(i = 1 . . . 10)

The universal mean of the three size-distribution models is 10 µm. In Fig. 4.16, the chocking performance curves of the fluidic-throat nozzle in these three sizedistribution models are showed and compared with the results of the pure gas-phase situation. When the average particle size is the same, the different particle-size distributions unobviously influence the flow coefficient ratio of the fluidic-throat nozzle. The choking performance curves of the three models of particle-size distribution coincide basically. The result shows that the accuracy of the calculation by the uniform-size model is the same to that by multi-size model when predicting the interference of the particle to the choking performance of the fluidic throat.

4.5.2 Two-Phase Flow Fields and Particle Tracks Although the particle-size distribution has little influence on the choking performance curve of the fluidic-throat nozzle, it will affect the flow characteristics, flow details and the thrust efficiency of the nozzle. Figure 4.17 gives the comparison of Mach number distribution between uniform-size model and the multi-size model with the three ones which are shown in Fig. 4.15. It can be seen that the flow field parameters calculated by the multi-size model are different from that by uniform-size model.

4.5 Influences of Profiles of Particle-Size Distribution

87

Fig. 4.16 Discrete distribution (size classification model)

Fig. 4.17 Mach number distributions under different size distributions

The flow field parameters, especially those near the tube wall and at the axis, also vary in different size distributions. Figure 4.18 shows the particle trajectories near the throat under size-distribution model 1 and model 2. When the particles pass through the fluidic throat, the large ones converge to the axis while the small ones are relatively close to the nozzle wall. The probability of interaction between small-size particles and secondary flow is larger than that of large-sized particles. The collision between part of the particles and the injector exit can be observed from the partially enlarged drawing of the secondary flow injector. The calculation of the two size distribution models shows that the size of the particles which collide with the injector exit ranges from 50 to 70 µm, while there is no collision observed for particles beyond the size range. This detail cannot be captured when the uniform-size model is employed. The results show that when

88

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

Fig. 4.18 Particle trajectories under different size distributions

the flow rate or the total pressure ratio of the secondary flow is not large, some particles are more likely to collide with the injector. This phenomenon will block the injector or affect the injector flow rate and the jet momentum if it is serious, which will further influence the choking performance and working process of the fluidic nozzle. Therefore, when designing the fluidic-throat nozzle under two-phase flow condition, the position of the injector should be carefully determined to reduce the probability of the collision between the particles and the injector exit to prevent the blocking.

4.5.3 Choosing Injector Positions in Two-Phase Flow Conditions From Fig. 4.19, the injection of secondary flow from the throat will change the boundary position of the particle streamline in the nozzle diffusion section. This influence is the same as the result of small-sized particles when the uniform-size model is employed which is discussed above. In Fig. 4.19, multi-size particle model 3 is used. Compared with the different positions for the injection of the secondary flow in Fig. 4.19, if the injection position is selected at the upstream of the throat, the boundary of the particle streamline will be very close to the injection exit, which means that the particles will be more likely to collide with the exit (in the top of Fig. 4.19b). While if the position is chosen at the downstream of the throat (in the bottom of Fig. 4.19b), the boundary position will be far away from the injection exit, leading to a lower collision probability. This is worth noticing when designing the injection position of secondary flow.

4.6 Efficiency of Fluidic-Throat Nozzle in Two-Phase Flow Conditions

(a) Comparison of particle trajectories under conditions without secondary flow and with secondary flow at nozzle position A

89

(b) Comparison of particle trajectories at nozzle positions B and C of secondary flow

Fig. 4.19 Particle trajectories under different nozzle positions

4.6 Efficiency of Fluidic-Throat Nozzle in Two-Phase Flow Conditions Since the presence of the particle phase changes the flow rate of the nozzle, when comparing the efficiency of the fluidic-throat nozzle under the two-phase flow condition, we define the efficiency by the ratio of the actual specific impulse to the theoretical specific impulse, see Eqs. (4.8)–(4.10). To distinguish from the definition of the previous nozzle efficiency, it is recorded as η3 . The theoretical ratio is the specific impulse value corresponding to the same ambient pressure under pure gasphase conditions. The calculation method of the thrust of the fluidic-throat nozzle is still obtained according to the method described in Chap. 3. η3 = I p /Ig

(4.8)

 Ig = Ft,g / m˙ g + m˙ s

(4.9)

 I p = Ft, p / m˙ p + m˙ g + m˙ s

(4.10)

where F t,g and F t,p are the combined thrusts in the gas-phase condition and the twophase flow condition, respectively; m˙ p , m˙ g , and m˙ s represent the flow rates of the particle phase, the primary gas phase, and the secondary flow, respectively. As the modified flow rate ratio grows, the efficiency of the two-phase fluidicthroat nozzle shows an overall increase trend, and the nozzle efficiency gradually

90

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

Fig. 4.20 Change of η3 with modified flow rate ratio

increases and approaches that under the pure gas phase, as shown in Fig. 4.20. This is because the specific gravity of the secondary flow in the flow field increases and the direct influence of the particles on the secondary flow is reduced (only small-sized particles directly interact with the secondary flow at the edge of the aerodynamic throat). In addition, as the proportion of the mainstream decreases, the particle phase also decreases correspondingly, so that the macroscopic two-phase flow loss caused by the particles relative to the gas phase is reduced. As the modified flow rate ratio grows, the efficiency of the two-phase fluidicthroat nozzle shows an overall increase trend, and the nozzle efficiency gradually increases and approaches that under the pure gas phase, as shown in Fig. 4.20. This is because the specific gravity of the secondary flow in the flow field increases and the direct influence of the particles on the secondary flow is reduced (only small-sized particles directly interact with the secondary flow at the edge of the aerodynamic throat). In addition, as the proportion of the mainstream decreases, the particle phase also decreases correspondingly, so that the macroscopic two-phase flow loss caused by the particles relative to the gas phase is reduced. As the particle size increases, the efficiency of the fluidic-throat nozzle decreases, and the downward trend for the annular injector and the circular injector is similar, as shown in Fig. 4.21. In addition, the efficiency of both throat nozzles decreases significantly with increasing content of the particulate phase, see Fig. 4.22. However, the downward trend of the two is slightly different. The efficiency curve of the fluidic nozzle of the annular injector is outward convex, while the circular injector is inward concave. The particle distribution also has an effect on the specific impulse of the nozzle. Figure 4.23 compares the efficiency of the fluidic-throat nozzle at particle-size distributions 1 and 3. Among them, the ratio of the distribution under the distribution

4.6 Efficiency of Fluidic-Throat Nozzle in Two-Phase Flow Conditions

91

Fig. 4.21 Change of η3 with particle diameter

Fig. 4.22 Change of η3 with content of particle-phase mass

3 is lower than that under the distribution 1. The results here are consistent with the laws obtained in studies on traditional solid-state nozzles. Based on the above analysis, it is known that the flow rate ratio of the secondary flow to primary gas flow and the content of the particle phase have the greatest influence on the efficiency of the fluidic-throat nozzle, followed by the average size of the particles. The distribution of the final particle size has an effect on the performance of the nozzle, but its influence is much smaller than the first few factors. As the size or content of the particle phase increases, the efficiency of the fluidic-throat nozzle decreases, which is the same as the conventional nozzle. In addition, it is sufficient

92

4 The Fluidic Throat in Gas–Particle Two-Phase Flow Conditions

Fig. 4.23 Curve of η3 -flow ratio

to use a uniform-size model to predict the control performance of the fluidic-throat nozzle under two-phase flow conditions. However, to more accurately predict the specific impulse of the nozzle, the result of the size classification model is closer to the actual situation.

References 1. Sun, M., Fang, D., Zhang, C.: The experimental and theoretical studies of two-dimensional two-phase nozzle flows. Acta Aeronaut. Astronaut. Sin. 9, A572–A576 (1988) 2. Kan, X., Yu, L., Jun-Xue, R., et al.: An ideal method for the two-phase ring plug nozzle design. J. Solid Rocket Technol. (2007) 3. Kan, X., Liu, Yu., Junxue, R., Yunfei, L.: Design methods of two-phase axial plug nozzle. Acta Aeronaut. Astronaut. Sin. 28(6), 1339–1344 (2007) 4. Yadi, L., Linquan, C., Zequn, J.: Numerical study of two-phase nozzle flow with classified particles. J. Solid Rocket Technol. 26(3), 32–34 (2003) 5. Dingyou, F., Zhixun, X., Weihua, Z.: Performance prediction of solid-propellant rocket motors. J. Solid Rocket Technol. 23(1), 1–5 (2000) 6. Linquan, C., Xiao, H., Yanfang, L., Xingang, D.: Calculation of nozzle efficiency for solid rocket motor. J. Solid Rocket Technol. 25(4), 9–11 (2002) 7. Tang, X., Wang, F., Yulin, W.: An improved large eddy simulation of two-phase flows in a pump impeller. Acta. Mech. Sin. 23, 635–643 (2007) 8. Guo, Y.C., Chen, X.G., Xu, C.M.: Applying theory of dense gas to model gas-particle flow. Chem. React. Eng. Technol. 115(4), 416–423 (1999) 9. Tang, X.L., Xu, Y., Wu, Y.L.: Kinetic model for silt-laden solid liquid two-phase flow. Acta. Mech. Sin. 34(6), 956–962 (2002) 10. Tang, X.L., Qian, Z.D., Wu, Y.L.: Improved subgird scale model for dense turbulent solid–liquid two-phase flows. Acta. Mech. Sin. 20(4), 354–365 (2004) 11. Chen, S., Liu, Z.H., Shi, B.C., et al.: A novel incompressible finite difference lattice Boltzmann equation for particle-laden flow. Acta. Mech. Sin. 21(6), 574–581 (2005)

References

93

12. Coats, D.E., Nickerson, G.R., Dang, A.L., et al.: Solid performance program (SPP). AIAA-871701 13. Guo, C., Wei, Z., Xie, K., Wang, N.: One-dimensional theoretical analysis of dilute particle-gas swirling flow in the Laval nozzle. AIAA J. 56(3), 1277–1283 (2018) 14. Hongtao, X.: Sampling Method for Particles in Combustion Chamber of Solid-Propellant Rockets. The Fourth Academy of China Aerospace Science and Technology Corporation, Xi’an (2007)

Chapter 5

Secondary Flow TVC for Fluidic-Throat Nozzles

Abstract This chapter mainly discusses the injector arrangement concepts with the combined secondary flow vector control, including work modes and TVC manners. The following issues are also discussed and introduced: performance characterization of TVC, six-component forces tests of secondary flow TVC, injector positions and TVC characters, influences of fluidic-throat jet on TVC, and arrangement concepts of round-hole injectors.

The previous chapters discussed the basic regulation of the fluidic throat of solid rocket motors. In practical applications, the fluidic-throat nozzle system using secondary flow injection can also be naturally integrated with the TVC system [1–21], so that the secondary flow injection can achieve both the thrust magnitude control and the TVC of solid rocket motors. This integrated solution can effectively improve the performance of the system, making the fluidic-throat nozzle be of higher engineering application value. This chapter focuses on the TVC method and injection parameters suitable for fluidic-throat nozzles of solid rocket motors. The working characteristics, performance, and interaction between jet flows of fluidic-throat nozzles under the simultaneous control over throat area and primary flow deflection based on secondary flow injection are introduced. This chapter provides a reference for the overall design and selection of the fluidic-throat nozzle systems of solid rocket motors.

5.1 Work Modes and TVC Manners There are two injection mechanisms of secondary flow suitable for deflecting the primary flow in the fluidic-throat nozzle of solid rocket motors to produce lateral forces. One is to introduce an asymmetrical secondary flow near the throat to deflect the sonic surface in the subsonic region, while achieving flow control and TVC (sonicline tilt vector control), as shown in Fig. 5.1a; The other is to inject a secondary flow into the divergent section of the nozzle to generate an induced shock wave, thereby causing deflection of the primary flow to realize TVC, as shown in Fig. 5.1b. © Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_5

95

96

5 Secondary Flow TVC for Fluidic-Throat Nozzles

(a) Throat tilt vector control scheme

(b) SVC Scheme

Fig. 5.1 Schematic diagrams of two TVC schemes

Fig. 5.2 Physical image of the injection of secondary gas flow

Among them, the shock-wave-induced TVC based on secondary flow injection is more mature, and has been used on multiple rocket engine models. Figure 5.2 shows the physical process and flow field characteristics of this TVC approach. After the secondary flow is injected into the side wall of the divergent section of the nozzle, the secondary flow expands rapidly and turns into a Coanda flow, forming an obstacle to the supersonic main airflow near the side of the injection port, similar to the flow of supersonic flow around a blunt object, creating a bow shock wave in upstream side of the injection port. According to the aerodynamic theory, the flow direction of the main flow after the bow shock wave is deflected, and the flow parameters are also changed. This causes the entire exhaust flow from the nozzle no longer passing through the center line of the nozzle when exiting the nozzle outlet, but leaves the nozzle with a deflexion angle. As a result, the thrust deflects and its lateral component are the desired lateral control force. By injecting secondary flows in different directions, lateral forces in different directions can be obtained. By adjusting some parameters of the secondary flow, such as the secondary flow rate, the secondary flow angle, etc., the intensity and

5.1 Work Modes and TVC Manners

97

angle of the bow shock wave can be changed, and the deflection degree of the main flow is also changed, thereby achieving the purpose of controlling the lateral force. Therefore, as long as a set of devices for injecting secondary flow are arranged in each of the four quadrants of the cross-section of the nozzle, as with the full-axis swinging movable nozzle, both the full-axis TVC and the aircraft pitch and yaw control can be guaranteed. Since the fluidic-throat nozzle itself has injection holes for secondary flow at the nozzle throat, it is also conceivable to use the sonic-line tilt TVC method to conduct the full-axis TVC of the fluidic-throat nozzle.

5.2 Performance Characterization of TVC Regardless of the thrust vector mechanism of the secondary flow injection, for a solid rocket motor using the fluidic-throat nozzle, the lateral force generated by the injection of the secondary flow is composed of two parts: one is the reaction force of the fluid injection, and the other is the induced force generated by the pressure difference between the disturbed flow zone and the undisturbed flow zone where the fluid and the gas are mixed and interacted with each other. Therefore, the formula for calculating the lateral force can be expressed as Eq. (5.1): Fn = Fn,s + Fn,p

(5.1)

where F n is the total lateral force generated by the secondary injection of the fluid; F n,s is the reaction force generated by the injected fluid; F n,p is the induced lateral force between the disturbed flow zone and the undisturbed flow zone. From the calculation formula of the lateral force, the lateral force is composed of the reaction force F n,s generated by injection of the secondary flow and the induced lateral force F n,p , and the two components can be expressed as Eqs. (5.2) and (5.3): Fn,s = m ˙ s · Vs + (pse − psa )As  Fn,p = p dsn

(5.2) (5.3)

where m ˙ s is the flow rate of the secondary flow; V s is the velocity of the secondary flow at the injection outlet; pse is the pressure of the secondary flow at the injection outlet; psa is the ambient pressure of the injection system of the secondary flow; As is the area of the injection outlet of the secondary flow; p is the pressure on the nozzle wall with the secondary flow disturbance; dsn is the projection of the integral micro-element of the nozzle wall in the direction of the lateral force. It is easy to know from the expression of the reaction force F n,s generated by the injection of the secondary flow that the injection process of the secondary flow is similar to the working process of a rocket engine. The main influence parameters are

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

the secondary flow rate m ˙ s , the speed V s of the outlet of injection port, the pressure pse of the outlet of the injection port, and the outlet area As of the secondary flow injection As . In the case where the other parameters of the secondary flow are constant, the velocity V s and the pressure pse at the outlet of injection port of the secondary flow are related to the outlet area of the injection port. As can be seen from the discussion in Chap. 2, the flow rate of the secondary flow injector is related to the total pressure ratio of the secondary flow to primary flow and the injector position. There are two main parameters affecting the induced lateral force F n,p : one is the pressure increment in the disturbance region after the secondary flow disturbance, and the other is the area of the pressure disturbance region. Under the condition that the working pressure ratio of the fluidic-throat nozzle and the position of the secondary flow injection port are constants, the factors affecting these two parameters are mainly related to the working parameters of the secondary flow, such as the flow rate, the injection angle, the total temperature and total pressure of the secondary flow, and the number of injection ports of secondary flow. In characterizing the magnitude and performance of the lateral force of the thrust vector, it is expressed by the thrust declination θ , which is defined as the arctangent of the ratio of the lateral force F n to the axial force F n and measured in degree (°).   Fn (5.4) θ = tg −1 Fa In the small range of secondary/primary flow ratio, thrust vector efficiency is often used to characterize the lateral force performance, as defined below: Kva =

θ ˙ o ) × 100% ˙ s /m (m

(5.5)

˙ o is the flow of primary flow; θ is the where m ˙ s is the flow of the secondary flow; m thrust declination at the corresponding flow ratio. The thrust vector efficiency K va means the thrust declination value per unit percentage of flow rate ratio. The thrust efficiency of the fluidic-throat nozzle combined with the secondary flow TVC system can still be represented by two efficiency characterization methods η1 and η2 , as defined in Eqs. (3.15) and (3.16) in Chap. 3. The resultant is calculated by the following formula: Ft =

  Fn2 + Fa2

Among them, the axial force can still be calculated by Eqs. (3.7)–(3.12), and the normal force is calculated by Eqs. (5.1)–(5.3).

5.3 Six-Component Forces Tests of Secondary Flow TVC

99

5.3 Six-Component Forces Tests of Secondary Flow TVC The thrust vector of a solid rocket motor is actually a space vector. The ground test mainly measures the magnitude and direction of the thrust vector. In early days, a thrust test bench of three-point-support type was used together with a light oscilloscope to form a test system. This method has low measurement accuracy and the post-processing of the data should be handled manually with the development of the multidimensional force sensor. The theory of the six-component force test bench has gradually matured and begun to be applied to the aerospace field. Scholars first used the six-component force test bench to measure the thrust vector of an engine with vector thrust nozzles in the 1960s. In the 1970s, China began the theoretical research and practical application of the six-component force test bench. At present, the six-component test bench (thrust stand) is divided into vertical and horizontal ones. The vertical thrust stand is taken as an example in the section to introduce the measurement principle, measurement error, and calibration of the thrust vector of the fluidic-throat nozzle, as shown in Fig. 5.3. The structure of the fluidic-throat nozzle engine tested in the figure can refer to Chap. 2.

5.3.1 Six-Component Force Test Methods and Thrust Stand For the thrust vector of the fluidic-throat nozzle engine to be measured, the direction and action point are usually unknown. On the condition, the space force is composed of six unknown components: three vertical force components and three

Fig. 5.3 Six-component force thrust stand and its measurement principle

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

vertical moment components. The task of the six-component force test bench is to measure the six components through a certain number of sensors to solve the thrust vector of the engine. In principle, the number of sensors on the six-component force test bench is not less than that of unknown force components to be measured. In addition, the layout of the sensors should be independent of each other, and the interference between the measured components should be as small as possible, such as using a flexible rod structure. The six-component force test bench is mainly composed of a moving frame, a static frame, a force measuring component, a calibration component, and a computer data acquisition system. The diagram of the main compositions is shown in Fig. 5.3. The moving frame is an important structure for positioning and fixing the experimental engine. Relatively high requirements are put forward for the moving frame in the design. The main points are as follows: (1) Under the condition of ensuring high strength and rigidity, it needs to reduce its mass as much as possible, make the mass distribution as uniform as possible, and make the inertia moment as small as possible; (2) With the axial and radial cooperation of the rocket engine, it is necessary to ensure a high precision, so that the error of the coaxiality between the axes of the rocket engine and the moving frame is as small as possible; (3) The position to be coupled to the flexure assembly, including the angle and distance, has to be of sufficient precision. The static frame is composed of two parts, a reinforced concrete base and a metal column. It is mainly used for installing sensors, flexible components, and in situ calibration components. The requirements for the static frame are to ensure sufficient strength and rigidity. The force-measuring component mainly comprises a set of components for measuring main thrust force and five sets of components for measuring lateral force. The force measuring assembly includes a sensor and a flexible component, wherein the round bar flexible member is a commonly used flexible component. The calibration components mainly include pulleys, brackets, and weights. The calibration components are primarily used to calibrate the overall static precision of the thrust stand before testing.

5.3.2 Mechanical Models of the Six-Component Force Test The structure of the six-component test bench in Fig. 5.3 can be simplified to the mechanical model shown in Fig. 5.4. The spatial rectangular coordinate system and the planar rectangular coordinate system are established with O as the origin, and the positive direction of each axis is as shown in the figure. Where C is the mass center of the engine, the sensor reads positive under pulling and negative under press. The positive direction of the moment is in the right-hand spiral rule. The positive directions of each force of F 1 –F 6 are shown in figure. 2R is the horizontal distance between F 1 and F 2 , and L is the distance between the planes of the two coordinate systems.

5.3 Six-Component Forces Tests of Secondary Flow TVC

101

Fig. 5.4 Simplified mechanical model of the six-component force test bench

I. Component force and component moment Let point O be the center of the simplified mechanical model, and the equilibrium equations of the component force and component moment can be obtained from the equilibrium condition of the rigid body: ⎧ ⎨ FX = −(F1 + F2 + F4 ) F = −(F3 + F5 ) ⎩ Y FZ = −F6 ⎧ ⎨ MX = F3 × L M = −(F1 + F2 ) × L ⎩ Y MZ = (F1 − F2 ) × R

(5.6)

(5.7)

where F 1 –F 6 are the values of the forces in corresponding directions measured by the sensor; F X , F Y, and F Z are the projections of the main thrust in the positive direction of X, Y, and Z axes, respectively; M X , M Y , M Z are the projections of the moment of the main thrust to the O point in the positive direction of the X, Y, and Z axes, respectively. II. Calculation of thrust eccentricity and argument As shown in Fig. 5.5, ρ is the thrust eccentricity,  is the angle between CD and X axis, which is called the argument, and γ is the thrust eccentric angle. Thrust eccentricity, thrust eccentric angle, and argument are parameters that describe the degree of eccentricity of the main thrust vector of the engine. It can be seen from the figure that ρ = CD =

  CM 2 + CN 2 = ρX2 + ρY2

(5.8)

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

Fig. 5.5 Thrust eccentricity and amplitude

 = arctan

ρY ρX

(5.9)

According to the translation theorem of force, a principal vector F, and a principal moment M which are simplified by the main thrust of the engine at point O are obtained by Eqs. (5.6) and (5.7). The moment M can be decomposed into a moment M parallel to F and a moment M⊥ perpendicular to F. If the coordinates of point C are (0, 0, Z C ) and the coordinates of point D are (ρX , ρY , Z C ), then we have → →− →− − i j k −→ −→ − → M⊥ = OD × F = ρX ρY ZC FX FY FZ − → − → − → = (ρY FZ − FY ZC ) i + (FX ZC − ρX FZ ) j + (ρX FY − FX ρY ) k (5.10) 





M = M − M⊥  − → − → = (MX + ZC FY − ρY FZ ) i + (MY + ρX FZ − ZC FX ) j +(MZ + ρY FX − ρX FY ) k

(5.11)

Since M F, we can get MX MY MZ M = = = FX FY FZ F

(5.12)

5.3 Six-Component Forces Tests of Secondary Flow TVC

103

Solutions have to be ⎧ ⎨ MX = M = ⎩ Y MZ =

FX M F FY M  F FZ M F

(5.13)

From Eqs. (5.11) and (5.13): ⎧ ⎨ MX + ZC FY − ρY FZ = M + ρX FZ − ρY FZ = ⎩ Y MZ + ρY FX − ρX FY =

FX M F FY M  F FZ M  F

⎧ FX MY FY ⎪ ⎨ ρX = FZ ZC − FZ + FFZ M FY MX FX ρY = F ZC + F − FF M ⎪ ⎩ M = 1Z (M F Z+ M FZ + M F )  X X Y Y Z Z F

(5.14)

(5.15)

Under normal circumstances, the magnitude of MP is between 10−4 and 10−6 , which can be neglected, so Eq. (5.15) can be simplified as  ρX = FFXZ ZC − MFZY (5.16) ρY = FFYZ ZC + MFZX By substituting the Eqs. (5.6) and (5.7) into the equation above, we can get  ρX = F16 [(F1 + F2 )(ZC + L) + ZC F4 ] (5.17) ρY = F16 [F3 (ZC + L) + ZC F5 ] In the equations above, both Z c and L are constant, so the values of ρX and ρY can be solved. Thus, the values of ρ and  can be solved according to Eqs. (5.8) and (5.9). The thrust eccentric angle γ can be calculated according to the following equations: ⎧ √ 2 2 ⎪ ⎨ tan γ = FX +FY  FZ√ (5.18) FX2 +FY2 ⎪ ⎩ γ = arctan FZ The principal vector and the principal moment are  F = FX2 + FY2 + FZ2  M = MX2 + MY2 + MZ2

(5.19) (5.20)

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

5.3.3 Measurement Error and Calibration For the thrust vector experiment based on secondary flow injection, since the measured force is a non-alternating force, the system error of the thrust stand can be obtained by using the static calibration method. I. Static calibration method The main purpose of the static calibration of the six-component test system is to obtain the linear relationship between the output of the six-way sensor and the main thrust of the engine in six directions, which can be expressed in following matrixes: X = KF + B ⎡ ⎤ k11 x1 ⎢ k21 ⎢ x2 ⎥ ⎢ ⎢ ⎥ X = ⎢ . ⎥, K = ⎢ . ⎣ .. ⎣ .. ⎦ x6 k61 ⎡

(5.21)

⎡ ⎤ ⎡ ⎤ ⎤ F1 b1 k16 ⎢ F2 ⎥ ⎢ b2 ⎥ k26 ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ .. ⎥, F = ⎢ .. ⎥, B = ⎢ .. ⎥ ⎣ ⎣ . ⎦ ⎦ ⎦ . . · · · k66 F6 b6

k12 · · · k22 · · · .. . k62

where the force in the ith direction indicates the output of the sensor in the direction, and k ij indicates the magnitude of the influence of the jth direction force on the output of the sensor in the ith direction. The matrix K is the calibration matrix, and the vector B is the equivalent system error of the six-component force test system. The static linear calibration method can be divided into two types according to the calibration method. One is the more common method called independent calibration. It separately calibrates each way, that is, respectively, applies the calibration force in different directions to the moving frame, reads the output voltage value of the six-way sensor, and then calculates the calibration parameter matrix. The other is called simultaneous calibration. The method adopts six-way simultaneous calibration method, that is, simultaneously loads the calibration force in each direction onto the moving frame, reads the output reading of the six-way sensor, and then solves the parameters of the calibration matrix. A simple in situ static calibration device for the six-component force test system is shown in Fig. 5.6. The device includes a main thrust calibration device, a lateral force calibration pulley, a bracket, a rope, and a weight. For the scheme in the figure, road 1 and 2 can be simultaneously calibrated, and roads 3–6 can be calibrated separately. During the calibration process, the weight is connected to the moving frame through the pulley rope. Taking a single channel as an example, if the system is calibrated twice and the calibration order is 3, the calibration process is shown in Fig. 5.7. II. Data processing of static calibration system The six channels of the six-component test system can be viewed as a linear system, whose static performance involves nonlinearity, repeatability, hysteresis and other

5.3 Six-Component Forces Tests of Secondary Flow TVC

105

Fig. 5.6 Independent calibration method

Fig. 5.7 Loading and unloading of the calibration process

indicators. The static index characteristics of the system can be derived from the measured static characteristic curve. If the independent calibration method is applied to the six channels of the sixcomponent force, assuming that the standard force F is applied to the j direction, then the relationship between the output readings of the sensor x i in each direction and F is ⎧ x1 = k1j Fj + b1 ⎪ ⎪ ⎪ ⎨ x2 = k2j Fj + b2 .. ⎪ ⎪ . ⎪ ⎩ x6 = k6j Fj + b6

(5.22)

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

The fitting curve of the six-channel calibration data can be obtained by the linear fitting method, and it is a static characteristic curve. The fitting method usually uses the least squares method. According to the principle of least squares, the processing of Eq. (5.22) can be obtained: ⎧ n (xik −¯xi )(Fjk −Fj ) ⎨ k = j=1 n ij 2 (5.23) j=1 (Fjk −F j ) ⎩ B = x¯ − k F ij

ij

j

where Bij represents the value obtained by the least squares method when the source of the calibration force is added to the j direction. The Bi for a single channel can average Bij to get 1 Bij Bi = 6 j=1 6

(5.24)

Thus, the linear regression equation is xˆ i = Kij Fj + Bi

(5.25)

X = KF + B

(5.26)

The matrix form is

where K is the sensitivity matrix. Equation (5.22) is the relationship between the sensor output voltage and the force in each direction. In the experiment, we need to find the value of each force through the output voltage of each sensor, so we rewrite Eq. (5.26) as F = K −1 (X − B)

(5.27)

In the equation, K −1 is the decoupling matrix. III. Uncertainty analysis of test system The main causes for error (error sources) in the measurement of the six-component test system are (1) Measurement error of the sensor; (2) Errors caused by different axial characteristics of the sixth sensor axis and the moving frame axis; (3) The error caused by the actual position of the first to fifth sensors deviating from the axis of the theoretical rectangular coordinate system; (4) Errors caused by the dissimilarity between the axis of the rocket engine and the axis of the moving frame and measuring coordinates; (5) Errors caused by the reaction and the mutual interference between the disturbing component assembly and the moving frame;

5.3 Six-Component Forces Tests of Secondary Flow TVC

Channel 2

107

Channel 3

Fig. 5.8 Coupling interference between channels

(6) Calibration error (accuracy error of calibration sensor, error of calibration device, etc.); (7) Electrical measurement system error. Figure 5.8 shows the calibration data plots for typical interference between channels. When channel 2 is loading, the zero point of channel 3 drifts, and the amount of drift is related to the force value in the loading of channel 2. The properties, magnitude, and impact of the several major errors on the thrust and eccentricity angles are analyzed as follows. A. Installation error The measurement error caused by installation includes the measurement error caused by the parallel movement of the axis of rocket engine and the measured z-axis (plumb line) by dL, and the measurement error caused by the angle d θ between the axis of rocket engine and the measured z-axis. These errors are independent of each other. (1) Measurement error caused by dL First, the error of the rocket thrust axis under the projection of measuring coordinate system on the YOZ plane is analyzed, as shown in Fig. 5.9. In the figure, F 3 , F 5 , and F 6 are the reaction forces of the third, fifth, and sixth flexible rods, respectively; F is the thrust of the rocket; L is the distance of the rocket thrust line from the O point on the OXY plane; θ is the angle between the rocket thrust line and the axis. In addition, l is the distance between the measuring surface of the upper flexible rod and that of the lower flexible rod. According to the above analysis, the calculation formula is F5 · l = F · L cos θ So, after derivation calculus to L, we can get dF5 = Fcosθ dL. l Similarly, dF3 = Fcosθ dL and dF = 0. 3 l In the XOY plane, there are dF4 = dF5 and dF1 = dF2 = 0.5dF3 .

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

Fig. 5.9 Measurement error caused by installation position

Fig. 5.10 Error caused by installation angle

(2) Measurement error caused by d θ From the relationship in Fig. 5.10, there is F6 = −F cos θ . Therefore, dF6 = F sin θ d θ ; If F5 × l = F × 0, there is dF5 = 0; If F3 × l = F × l × sin θ , there is dF3 = F cos θ d θ ; Similarly, in the XOZ plane, there is dF4 = dF5 and dF1 = dF2 = 0.5dF3 . B. Calibration error In situ calibration of the sensor will result in a static calibration error. This error is a composite value that can be obtained from multiple calibration data and the fitted static characteristic curve.

5.3 Six-Component Forces Tests of Secondary Flow TVC

109

Table 5.1 Error sources and values of the typical six-component thrust stand Channel 1 (%)

Channel 2 (%)

Channel 3 (%)

Channel 4 (%)

Channel 5 (%)

Channel 6 (%)

dL

0.07

dθ

0.29

0.07

0.14

0.14

0.14

0.00

0.29

0.58

0.00

0.00

Calibration

0.00

4.62

4.83

3.61

1.6

2.83

0.95

Load source

0.50

0.5

0.5

0.5

0.50

0.50

Comprehensive uncertainty

4.60

4.80

3.66

1.70

2.87

1.07

C. Uncertainty of loading force source The in situ calibration system error depends to a large extent on the measurement error of the calibration system and the calibration force loading error. The measurement error of the calibration system is mainly caused by the signal detection of the data acquisition system and the drift of the amplification circuit, and the errors caused by various interferences. The calibration force loading error can be divided into the deviation of the calibration force value and that of the calibration force position. The positional deviation of the calibration force can be reduced or even eliminated by instruments such as a level meter and a dial indicator. D. Comprehensive uncertainty The comprehensive uncertainty of the six-component force test system consists of the uncertainties above, and the individual uncertainties are independent of each other. The comprehensive uncertainty of the six-component force test system can be calculated using the following equation.   n  dFi2 (5.28) dF1−6 =  i=1

Table 5.1 shows the main sources and values of errors of a six-force thrust stand. The calibration error is the main source. This is because the error is already a composite value. It should be noted that for fluidic-throat nozzles, the secondary flow line connected to it also affects the six-component force measurement. Flexible connections, such as longer bellows, can be used to minimize the impact of the pipeline; in addition, the engine is required to be connected to the secondary flow line for calibration, and the influence of the pipeline is included as a systematic error in the calibration error.

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

5.4 Injector Positions and TVC Characters When there is a secondary flow jet in the throat and the divergent section at the same time, the action characteristics and mechanism of the secondary flow and the primary flow have a great relationship with the relative position of the injector. Under the premise that the ratio of the secondary flow to the primary flow is constant, the flow field characteristics of the asymmetric injection at different injector positions are shown in Fig. 5.11. In the figure, the secondary flow injector at the throat Di1 is fixed and located at the origin of the reference coordinate system, and Di2 is the position of the secondary flow injector on the divergent section of the nozzle, and the distance between the two is Di . When the injector of the throat and the divergent section are close to each other and generally the Di /Dt is between 0.12 and 0.5, the mechanism of the secondary flow causing deflection of the primary flow is expressed as the throat inclination mode. The asymmetric secondary flow jet on the wall of the throat and the divergent section causes the throat sonic surface to tilt, and the primary flow begins to deflect in the contraction section of the nozzle. This is the main feature of the throat-tilting TVC. At this time, the lateral force is mainly caused by the inclination of the sonic surface.

Fig. 5.11 Nephograms of equivalent Mach number at different injector positions

5.4 Injector Positions and TVC Characters

111

When Di /Dt continues to increase, the characteristic of the inclination of the throat sonic surface is weakened, and a distinct bow shock wave begins to appear before the secondary flow injector on the divergent section. At this time, the action mechanism of the secondary flow and the primary flow begins to appear as the SVC. The lateral force is generated by the inclination of the sonic surface and the induction of shock wave, and the degree of inclination of the sonic surface is reduced. If the Di /Dt continues to increase, the inclination characteristic of the throat disappears, and the secondary flow of the throat only plays a role in reducing the throat area. On the condition, the lateral force is mainly generated by the secondary flow SVC on the divergent section. Figure 5.12 shows the variation curve of the thrust declination angle with the flow ratio for different fluidic-throat nozzles at different pressure ratios and injector positions, including two-dimensional nozzle and three-dimensional axisymmetric nozzle models. The three-dimensional axisymmetric nozzle 3D-1 corresponds to “nozzle-B2” in Table 5.2 in Sect. 5.7, and “nozzle 3D-2” corresponds to “nozzleB1” in the section. For the asymmetric secondary jet flow in the throat and the divergent section, no matter which generation mechanism of lateral force is dominant, the two-dimensional nozzle and the three-dimensional nozzle show a relatively consistent trend. The thrust declination increases linearly with the increase of the secondary flow ratio. Then, when the secondary flow ratio continues to increase to a certain value, the slope of the thrust declination curve gradually slows down. When the injector at the divergent section is closer to the throat, the corresponding flow of the nozzle is larger than that at the turning point (about 30%), and the corresponding maximum thrust declination θmax is smaller. When the injector at the divergent section is near the middle of the divergent section, the corresponding flow is smaller than that at the turning point (about 12%), and the corresponding maximum thrust declination θmax is larger.

Fig. 5.12 Changes of the vector angle with flow ratio

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

Table 5.2 Combination injection scheme of injectors in six types of axisymmetric nozzles Number

Direction of secondary flow

Ratio of nozzles for secondary flow to throat area of primary flow

Distribution diagram

Diameter and number of nozzle throats for secondary flow

A1

60° reverse jet

Ratio of nozzles for secondary flow in throat to those in throat area is 15%

The diameter and number of nozzle throats for the secondary flow are 1.52 × 8

A2

90° vertical jet

Ratio of nozzles for secondary flow in expansion section to throat area is 10%

The diameter and number of nozzles for the secondary flow in the expansion section are 2.15 × 2

A3

Ratio of nozzles for secondary flow in the throat to throat area is 10% Ratio of nozzles for secondary flow in expansion section to throat area is 10%

The diameter and number of the nozzle throats for secondary flow are 1.75 × 3 The diameter and number of the nozzles for secondary flow in the expansion section are 2.15 × 2

A4

Ratio of nozzles for secondary flow in throat to throat area is 20%, ratio of nozzles for secondary flow in expansion section to throat area is 10%

The diameter and number of the nozzle throats for secondary flow are 1.52 × 8 The diameter and number of the nozzles for secondary flow in the expansion section are 2.15 × 2

A5

Ratio of nozzles for secondary flow in the throat to throat area is 10%, ratio of nozzles for secondary flow in expansion section to throat area is 10%

The diameter and number of the nozzle throats for secondary flow are 1.75 × 3 The diameter and number of nozzles for the secondary flow in the expansion section are 2.15 × 2

A6

Ratio of nozzles for secondary flow in the throat to throat area is 10%, ratio of nozzles for secondary flow in expansion section to throat area is 10%

The diameter and number of nozzle throats for the secondary flow are 1.75 × 3 The diameter and number of nozzles for the secondary flow in the expansion section are 3 ×1

5.4 Injector Positions and TVC Characters

113

The law of asymmetric jetting of the fluidic-throat nozzle is similar to the shock induced by a single jet. That is, the position of each injector corresponds to a maximum deflection angle θmax . The closer the injector is to the throat, the smaller the maximum thrust angle. Whether it is a single jet or an asymmetric double jet, the throat-tilting mechanism mainly appears as the induction mechanism of shock wave, and the variation of lateral force with flow is similar to that of traditional SVC, that is, the thrust declination increases within a certain flow range as the injection flow rate increases. When the injection flow rate reaches a certain value, the thrust deflection angle reaches a maximum value; and then as the flow rate continues to increase, the thrust deflection angle decreases. For SVC, this is because when the secondary flow rate reaches a certain value, the disturbance gradually expands to the entire nozzle wall (the arcuate shock hits the opposite nozzle wall), and some of the lateral forces cancel each other out, causing the decrease of the whole lateral force, as shown in Fig. 5.13. For the throat-tilting thrust vector mode, the secondary flow increases to a certain value and continues to increase. The zone affected by the secondary flow on the upper and lower sides of the divergent section expands toward the nozzle axis and tends to be symmetrical, so that the inclination of the primary flow in the divergent section is reduced and the flow field and pressure distribution on the wall tend to be symmetrical. In practical applications, the condition where the flow ratio is greater than the value at the turning point should be avoided because the further increase of the secondary flow will result in a negative gain or zero gain of the thrust deflection. In addition, Fig. 5.12 also shows that the vector control method of the two mechanisms both can use the thrust vector efficiency K va , a linearity parameter, to characterize the benefit of changing the thrust deflection angle before the turning point. Figure 5.14a shows the thrust vector efficiency of the fluidic-throat nozzle as a function of injector position at different pressure ratios. It can be seen that the efficiency of nozzle thrust vector increases as the injector on the divergent section gradually moves away from the nozzle throat. For the nozzle with a large area expansion ratio, the traditional shock wave induction method is more suitable. Because of the acceptable nozzle efficiencies, the injectors in the divergent section can be placed a little further downstream to achieve greater thrust deflection and thrust vector efficiency. At the same time, it is noted that under the condition of design pressure ratio and high-pressure ratio, less thrust loss is found in the throat inclination mode than while using the shock wave induction method, as shown in Fig. 5.14b. This is because the

Fig. 5.13 The relationship between the lateral force of the traditional axially symmetric nozzle with secondary flow injection based TVC and the secondary flow

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5 Secondary Flow TVC for Fluidic-Throat Nozzles

(a) Thrust vector efficiency as a function of injector position

(c) Di/Dt=0.25

(b) Thrust efficiency η2 varies with nozzle position

(d) Di/Dt=0.25

Fig. 5.14 Thrust vector efficiency and nozzle efficiency

primary flow in the throat inclination mode starts to deflect in the subsonic zone, and there is no obviously strong shock wave in the flow field. In the case of a low-pressure ratio, the thrust efficiency changes in the direction opposite to that at a high-pressure ratio due to the complicated flow separation and backflow in the divergent section of the nozzle. Therefore, a suitable TVC mode of secondary flow injection can be selected depending on the overall requirements and specific nozzle parameters. In addition, when the injector position is fixed, the thrust efficiency decreases with the increase of the pressure ratio in any TVC mode under the design pressure ratio and high-pressure ratio, see Fig. 5.14c, d. This is similar to the case where there is only a secondary flow in the throat in Chap. 2. The fluidic throat formed by the two mechanisms also shows different choking performances, see Fig. 5.15. In the throat-tilting mechanism of TVC, the sonic surface is tilted, so both choke and vector control are coupled in this method. The sonic surface in the shock-induced TVC mechanism of secondary flow is basically not affected by

5.4 Injector Positions and TVC Characters

115

Fig. 5.15 Curves of flow coefficient ratio

the SVC on the downstream supersonic divergent section. Therefore, the influence of jet from the throat on the SVC performance is mainly concerned in this mechanism. Since the nozzle expansion ratio of the general solid rocket motor is relatively large (relative to the nozzles of battle planes and unmanned aerial vehicles as well as those used in ramjet engines) and the shock-induced TVC technology based on secondary flow injection is more mature, the following discussion focuses on the characteristics of the shock-induced TVC based on secondary flow injection for the fluidic-throat nozzle. Especially when the fluidic-throat nozzle has to not only adjust the thrust but also provide lateral force, a secondary flow will be injected in both the throat and the divergent section. If the injection method used is unreasonable, it can be known from Eqs. (5.1)–(5.3) that on the one hand, the lateral force F n,s generated by the secondary flow injection is reduced. On the other hand, the jet of the throat will interfere with the jet on the downstream divergent section, which may reduce the pressure difference caused by the shock induction and reduce the lateral force.

5.5 Influences of Fluidic-Throat Jet on SVC 5.5.1 Asymmetric Injection in 2D FNTs When the system performs throat area adjustment and TVC at the same time, there will be a jet at both the throat and the expansion section [22]. Under this condition, there are three types of throat injection modes that work in conjunction with the secondary flow of the SVC. (1) The secondary flow is only injected in half-side of the throat for adjustment of the throat area, while the jets in the throat and expansion section are on different sides, i.e., asymmetric injection; (2) the secondary flow is only injected in half-side of the throat for throat area adjustment, but the jets in the throat

116

5 Secondary Flow TVC for Fluidic-Throat Nozzles

Fig. 5.16 Effect of jets in the throat

and the expansion section are on the same side; (3) the secondary flow is sprayed into the entire throat and injected in one-side of the expansion section. This section first takes the first injection mode as an example to illustrate the interference and influence of the jet method at the throat on the SVC. In the next section, a more detailed comparison of the above three injection modes and nozzle combination schemes will be made with an axisymmetric nozzle as an example to help select the appropriate injection mode so that both throat adjustment and thrust vector performance can be achieved. The secondary flow in the throat will affect the performance of the SVC, see Fig. 5.16. The secondary flow at the throat is ws1 , the flow at the expansion is ws2 , the total secondary flow is ws , and “TVC-1” represents the SVC when there exists a jet at the throat, “TVC-2” represents the traditional SVC when there is no secondary flow in the throat. The left side of the figure shows the curve of the thrust vector angle with the total secondary flow ratio. It can be clearly seen that the curve of the thrust vector angle and flow ratio is different when there is no secondary flow in the throat. In general, the use of an asymmetric injection scheme in the gas–gas fluidic throat will offset the lateral force generated by the injection of the secondary flow in the expansion section to a certain extent, resulting in a decrease in the maximum deviation angle of thrust vector that can be achieved, and the corresponding turning flow ratio is increased. Figure 5.17 shows the ratio of the lateral force F ns generated by the secondary flow injection of “TVC-1” and “TVC-2” at different pressure ratios to the total lateral force F n . When the same total secondary flow rate ratio is used, the direct proportion of lateral force generated by the conventional injection system of secondary flow for SVC is greater than that of the secondary flow in the throat, and it also reflects the influence of asymmetric secondary flow in the throat on downstream SVC. Figure 5.18 compares the thrust efficiency versus flow rate ratio for SVC under conditions of different pressure ratios with or without secondary flow in the throat.

5.5 Influences of Fluidic-Throat Jet on SVC

117

Fig. 5.17 Contribution of lateral force generated by secondary flow injection

(a) η2-flow ratio

(b) Kva-NPR

Fig. 5.18 Curves of basic performance

In the case of the same ratio of total secondary flow rate to primary flow rate, there are generally two cases: (1) near the design pressure ratio, the thrust efficiency of the throat is equivalent on conditions with or without secondary flow; (2) at the high pressure ratio when ws /wo > 10%, the thrust efficiency of “TVC-2” will be significantly lower than “TVC-1”. This is because at the same total secondary flow rate ratio, a part of the secondary flow of “TVC-1” is distributed in the throat for disturbance, while the entire secondary flow of “TVC-2” is injected into the expansion section for vector control, and the shock loss is large. Figure 5.18b shows the effect of jet flow at the throat on the SVC thrust vector efficiency at different pressure ratios.

118

5 Secondary Flow TVC for Fluidic-Throat Nozzles

5.5.2 Actual Lateral Forces Modification When comparing the SVC performance on conditions with or without jets at the throat, the small value of the maximum vector angle does not mean that the maximum lateral force obtained is also small. When the secondary jet is injected into the throat, the pressure value of the combustion chamber of solid rocket motors will be larger than that before the injection of the secondary flow. Equations (5.29)–(5.37) show how to compare the actual lateral force under two working conditions of thrust vectors. The equilibrium pressure of the combustion chamber of solid rocket motors can be calculated by the following formula:  1  Pc = aρp C ∗ Ab /At 1−n

(5.29)

where n is the pressure index of the propellant used, and the chamber pressure ratio before and after the jet injected in the throat is  1  1 At / At 1−n = (Cd /Cdo ) 1−n Pc /Pc = 

(5.30)

At , and C do represent the parameters before the pressure of the chamber where Pc ,  rises. C d /C do is the flow rate coefficient ratio frequently mentioned above, which can be obtained from a series of curves of flow coefficient ratio in the second to fourth chapters and corresponding operating conditions. If the pressure of the combustion chamber is adjusted after the secondary flow is injected, as on the above-mentioned “TVC-1” operating condition, the corresponding working pressure ratio NPR of the engine will change, and the combined thrust corresponding to a certain pressure ratio can be obtained by the following formula: Ft,NPR = η2,NPR (Fci,NPR + Fsi,SPR×NPR )

(5.31)

The thrust efficiency η2,NPR corresponding to a certain NPR and secondary flow ratio in the formula, which can be obtained from a series of thrust efficiency curves as shown in Fig. 5.13a; FEi,NPR and Fsi,SPR×NPR are the ideal thrusts of the primary and secondary flows, respectively. For the calculation formula, see Eqs. (3.13) and (3.14) in Chap. 3. It is worth noting that the back pressure ratio NPR of the secondary flow system is equal to the product of the back pressure ratio NPR of the primary flow and the total pressure ratio SPR of the secondary flow. The working pressure ratio before and after the injection of the flow is calculated by Eqs. (5.32) and (5.33), respectively. NPR = Pc /Pa (No Injection)

(5.32)

NPR = Pc /Pa

(5.33)

5.5 Influences of Fluidic-Throat Jet on SVC

119

The combined thrust before and after the secondary flow injection is represented by Eqs. (5.34) and (5.35), respectively. FT ,NPR = η2,NPR Fci,NPR

(5.34)

FT ,NPR = η2,NPR (Fci,NPR + Fsi,SPR×NPR )

(5.35)

Then, the lateral force F n corresponding to the current NPR can be obtained from the combined thrust calculated by the Eq. (5.35) and the thrust vector angle θ under the corresponding operating conditions. Here, θ can be obtained from the series of curves in Fig. 5.7. The ratio of the lateral force after the secondary flow injection to the combined thrust before the secondary flow injection is calculated by the Eq. (5.37).

Fn,NPR /FT ,NPR =

Fn,NPR = FT ,NPR × SinθNPR,ws/wo

(5.36)

η2,NPR (Fci,NPR + Fsi,SPR×NPR ) × SinθNPR,ws/wo η2,NPR Fci,NPR

(5.37)

The change of the primary flow before and after the secondary flow injection is calculated by the Eq. (5.38):     At At Pc Cd  Pc Cdo Pc Cd wo /wp =   (5.38) /   =  Pc Cdo RTf RTf where wp is the primary flow rate before the secondary flow injection; wo is the primary flow rate after the injection of the secondary flow. For the SVC when there exists a secondary flow in the throat, there are different chamber pressures and the primary flow before and after the secondary flow injection; and for the conventional SVC, the flow in the chamber before and after the secondary flow injection is not changed.   ws wo (5.39) ws /wp = wo wp To make the above two working conditions with the same primary flow and resultant values as a reference and the actual secondary flow and lateral force in the two conditions be compared, the flow rate ratio is based on the primary flow rate wp before injection of the secondary flow, see Eq. (5.39). This is different from the previously used flow rate ratio ws /wo . Figure 5.19a shows the variation of the typical dimensionless lateral force with the dimensionless secondary flow in the expansion section, and F to is the combined thrust before the secondary flow injection. The magnitude of the lateral force in the

120

5 Secondary Flow TVC for Fluidic-Throat Nozzles

Fig. 5.19 Actual lateral force

“TVC-1” mode is related to the pressure index n of the propellant and augments as n increases. For “TVC-2”, that is, SVC with no secondary flow in the throat, the lateral force curve is not associated with the pressure index; and after the flow turning point, it is impossible to obtain a larger lateral force by further increasing the secondary flow. The “TVC-1” mode can increase the thrust of primary flow by adjusting the throat area, so that the secondary flow can be increased at the turning point corresponding to the maximum vector angle, and the increased lateral force can still be obtained. Especially when the total ratio of the two flows is ws /wp > 12%, the lateral force in the “TVC-1” mode under the high-pressure index can continue to increase and begin to exceed that in the “TVC-2” mode, as shown in Fig. 5.19b. As mentioned in the thrust regulation principle in Chap. 1, in order to obtain a large thrust change when the chamber pressure varies slightly, a propellant with a higher pressure index is usually used. Therefore, the curves of n = 0.6 and 0.8 in the figure are closer to the actual situation of the fluidic-throat nozzle of solid rocket motors. In practice, the above two TVC methods can be used in combination (note the dotted line in Fig. 5.19b). For example, the traditional SVC method is used to obtain the lateral force under a small secondary flow rate while the SVC method of simultaneously injecting the secondary flow in the throat at the same time after the threshold of the secondary flow rate of the conventional SVC is used to obtain a larger lateral direction, so that better lateral force performance can be achieved throughout the flight envelope. Or when one needs to obtain the maximum thrust vector angle to intercept attack or pursue a target (such as a missile to make the maneuver of shoulder turn), the secondary flow at the throat can be closed at first to use SVC alone. Afterward, the posture is adjusted quickly, and then the fluidic throat is opened to force the target.

5.6 Influence Factors of TVC

121

5.6 Influence Factors of TVC 5.6.1 Influence of Back Pressure NPR In the previous study of the effect of nozzle position on vector efficiency, the effect of back pressure has been briefly analyzed. Figure 5.20 shows the thrust vector efficiency and nozzle thrust efficiency as a function of back pressure ratio at a fixed nozzle position. The primary flow is fully expanded under conditions of the design pressure ratio and the high-pressure ratio and the acting ability of the secondary flow is relatively weakened at the meantime. Because of this, the thrust vector efficiency is higher at the low-pressure ratio, and then sharply decreases to a certain value near that at the design pressure ratio. It gradually stabilizes at the high-pressure ratio. The nozzle thrust efficiency of the vector control method of asymmetric injection at the throat and the expansion section is similar to that of the third section where only the secondary flow exists in the throat. The nozzle thrust efficiency reaches the maximum near the design pressure ratio of the geometric nozzle; the thrust efficiency decreases under the condition that the pressure ratio deviates from the design pressure ratio. As the flow rate ratio increases, the pressure ratio at which the thrust efficiency reaches the maximum value gradually deviates from the design pressure ratio of the geometric nozzle and moves toward the side of high-pressure ratio. At the same pressure ratio, the trend of the nozzle thrust efficiency is basically reduced as nozzle appears at the position further downstream in the expansion section.

5.6.2 Influence of Injection Angle Taking the asymmetric jet mode as an example, Fig. 5.21a shows three combinations of injection angles: (1) 90° mode. The injection angle of the injector at the throat and in the expansion section is 90°; (2) 45° mode-1. The injector injection angle at

(a) Kva-NPR Fig. 5.20 Effects of back pressure

(b) η2-NPR

122

5 Secondary Flow TVC for Fluidic-Throat Nozzles

(b) Kva-NPR (a) Three combination modes Fig. 5.21 The effect of the injection angle under condition of Di /Dt = 0.875

the throat is 90° while 45° in the expansion section; (3) 45° mode-2. The injector injection angles at the throat and the expansion section are both 45°. The positions of the injector in all of the above modes are the same. From the perspective of thrust vector efficiency, the mode with a reverse injection angle is slightly better than the vertical injection mode, especially in the case of nozzle design pressure ratio and high-pressure ratio, see Fig. 5.21b. However, the curve of the thrust vector angle with the flow rate ratio given in Fig. 5.22a shows that the maximum thrust vector angle in the mode of a reverse jet angle is not much larger than that in the vertical mode. The maximum vector angles of the three schemes are equivalent, but the reverse injection mode has a smaller critical mass flow rate than that in the vertical injection mode. Under the asymmetric injection condition, when the throat is reversely injected, the maximum vector angle of the fluidic-throat nozzle will increase slightly, and the turning flow ratio will also increase. The use of reverse injection in the expansion section causes an increase in thrust loss, as shown in Fig. 5.22b. The nozzles with the two reverse injections are equally efficient, and the nozzle in vertical injection mode is more efficient than that in the reverse injection. In summary, the reverse injection mode can obtain a slightly better thrust vector efficiency, but does not get a much larger maximum vector angle, and the nozzle efficiency is reduced more apparently when the reverse injection is used.

5.6 Influence Factors of TVC

(a) θ-ws/wo

123

(b) η2-NPR and Ps/Po=1

Fig. 5.22 K va -flow curve

5.6.3 Influence of Injector Area Ratio The thrust vector performance and the nozzle efficiency of the mode of asymmetric secondary flow injection with the nozzle area ratios of 10 and 20% are compared. It can be seen that the thrust vector efficiencies of the nozzle are equivalent with the two nozzle areas, but the maximum vector angle of nozzle with a smaller area ratio is larger than that of the nozzle with a larger area ratio; and the thrust efficiency of the former nozzle is lower than that of the nozzle with a larger area ratio, especially the thrust loss is more obvious in the case of having a large secondary flow ratio, see Fig. 5.23b.

(a) Curve of vector angle and flow ratio Fig. 5.23 Effect of injector area ratio

(b) η2-flow ratio curve

124

5 Secondary Flow TVC for Fluidic-Throat Nozzles

5.7 Arrangement Concepts of Round-Hole Injectors The above is based on a binary nozzle (which may be used in a solid scramjet engine). The effect of one of the throat jet modes on the performance of SVC of solid rockets is discussed. The following is an example of a solid-state axisymmetric nozzle, which is used to further discuss the effects of different modes of jet flow and the combination mode of the round-hole injectors on the characteristics and performance of the SVC. Figure 5.24 shows a typical injection system of secondary flow for SVC used on existing solid rocket motors. The tanks of the secondary flow system may be spherical, annular and split cylindrical, and the nozzles for

Fig. 5.24 Typical injector layouts

5.7 Arrangement Concepts of Round-Hole Injectors

125

the secondary flow are generally distributed in a radially symmetric manner. The influence of the number of injectors and the non-radial injection on the performance of SVC has been qualitatively investigated in previous studies. Careful research on the influence of number and distribution of injectors on the performance disturbance of the fluidic throat has also been done in Chap. 3. Therefore, the influence of the above factors on the working condition, where there exists simultaneous injection of the secondary flow in the throat and the expansion section is not described here. Instead, the section emphasizes the interference caused by the mode of jet flow in the throat to the downstream SVC. Axisymmetric fluidic throat combined with SVC have two basic options for injector layout, see Fig. 5.25. The black dots in the figure represent the injection positions of secondary flow. Unlike previous pure SVC systems, there are layout problems relating phase differences between the two sets of injectors on the throat and the expansion section. In the figure, there is no phase difference between the front and rear nozzles in the “Phase 1” scheme, and the phase difference between these two groups of nozzles in the “Phase 2” scheme is 22.5°. From Figs. 5.26 and 5.27, the thrust vector performance and thrust efficiency of the two nozzle phases under asymmetric jet conditions can be compared. In general, the thrust efficiency and thrust vector efficiency of a nozzle with phase difference are slightly better than those without phase difference. In addition, at the high-pressure ratio, the thrust vector angle and the thrust efficiency of the two schemes are lower than those at the design pressure ratio, as shown in Fig. 5.26a. The curve of nozzle efficiency in Fig. 5.26b is consistent with the analysis of the binary nozzle (see Fig. 5.18a). The thrust efficiency of the two-phase schemes is shown in Fig. 5.27. Table 5.2 provides details of the typically combined injection schemes of nozzles available for the six types of fluidic-throat nozzles of solid rocket motors. The black points on the inner circle in the distribution diagram represent injection positions of

Phase1 Fig. 5.25 Two typical injection phases

Phase2

126

5 Secondary Flow TVC for Fluidic-Throat Nozzles

(a) Vector angle-flow ratio

(b) η2-flow ratio

Fig. 5.26 Effect of injector phases

(a) η1-flow ratio

(b) Kva-NPR

Fig. 5.27 Effect of injector phases

the secondary flow at the throat, and the black points on the outer circle represent the injection points on the expansion section. Regardless of the injection combination of secondary flow, the gain of the axial thrust of the engine generally varies linearly with the total secondary flow ratio. The gain of axial thrust obtained when there is only a secondary flow in the throat is the largest, as shown in the slope of the experimental curve of the A1 injection scheme in Fig. 5.28. Figure 5.29 is a plot of the effective area ratios of the dimensionless fluidic throats (compared to the area of nozzle throat without secondary flow) and the secondary flow rate ratios for the six injection schedules in Table 4.2. For pure SVC, the dimensionless effective area of the fluidic throat in the experiment is always around 1,

5.7 Arrangement Concepts of Round-Hole Injectors

127

Fig. 5.28 Axial thrust of A1–A6 nozzles

Fig. 5.29 Curves of the effective area of the dimensionless fluidic throats and the flow rate ratio of secondary flow to primary flow

that is, the secondary flow does not play a role in regulating the size of the throat, as shown in the A2 scheme. Unlike the A2 scheme, other injection schemes have secondary flow injections at the throat position, and these nozzles act to control the throat area. Among them, the change in the effective area of the fluidic throat of the A1 scheme is the largest under the same working conditions. The variation law of the effective throat area in the combined injection mode is similar to that in the second and third chapters when only the secondary flow is injected into the throat. From the comparison of the data in Fig. 5.30, it can be seen that the thrust deflection angle generated by the A2 scheme is the largest. When the ratio of the secondary flow to the primary flow is 10%, the thrust deflection angle of the A2 scheme reaches

128

5 Secondary Flow TVC for Fluidic-Throat Nozzles

Fig. 5.30 Relationship between thrust deflection angle and flow rate ratio

15°, and it reaches 10° for the A3 and A5 nozzles. The thrust deflection angles of the A4 and A6 schemes are small, below 6.5°. Combining the above experimental results of axial thrust gain, the variation of fluidic-throat area, and thrust deflection, it can be known that the A3 and A5 combined injection schemes are better than A6 scheme, when there are secondary jets in the throat and the expansion sections. The combination of A3 and A5 injection schemes can take into account the throat control and vector control performance, and the interference between the jets can be minimized. Table 5.3 shows the values of thrust vector efficiency for some typical combinations of injectors at low flow rate ratios. The values of thrust vector efficiency K va in the table are obtained with reference to the total secondary flow rate, and the values in parentheses are based on the secondary flow in the expansion section. Comparing the schemes B1, B2, B3, B4, B5 and schemes A5, A2, A4, and B6 in Table 5.3, it can be seen that increasing the injection points near the horizontal boundary line in the expansion section only reduces the thrust vector efficiency of the nozzle. Thus only injection units at two positions are used in the expansion section, which is the same as the qualitative conclusions obtained in pure SVC studies. It is worth noting that the injection unit can be split into a multi-nozzle structure that is closely juxtaposed at a corresponding position for better vector performance; but for ease of discussion, each injection position has only a single injector. The nozzle efficiency of the four schemes with better thrust vector efficiency in Table 5.3 is given below, and compared with the A2 scheme (traditional SVC method). In the actual design, the maximum vector angle required can be determined according to the overall needs according to the working condition when the throat has no jet flow. Because of this, the position of the injector in the expansion section is determined by the maximum vector angle (each injector position corresponds to a maximum vector angle). Then the influence of the throat jet on the performance of the SVC is carefully examined in all of the operating envelopes of the aircraft. Finally, injector arrangement and throat injection mode are determined.

5.7 Arrangement Concepts of Round-Hole Injectors

129

Table 5.3 NPR = 40 and Ps/Po = 1:1 K va

η2

C d /C d0

ws /wo

B1

0.261

0.970

0.940

0.0815

B2

0.281

0.975

0.942

0.0702

B3

0.425 (0.677)

0.976

0.9631

0.0548

B4

0.601

0.983

0.997

0.0340

B5

0.224 (0.556)

0.973

0.909

0.0948

B6

0.246 (0.699)

0.978

0.940

0.0534

A2

0.870

0.983

0.999

0.0170

A4

0.207 (0.8148)

0.976

0.909

0.0757

A5

0.471 (1.02)

0.981

0.9625

0.0378

Number

Combination mode

130

5 Secondary Flow TVC for Fluidic-Throat Nozzles

For the injection mode in the A5 scheme, the injection of the secondary flow in the throat is advantageous for downstream SVC performance. Under the condition of the same secondary flow rate ratio of injector, the maximum thrust vector angle and thrust vector efficiency of the scheme are better than those in the pure SVC control scheme. This is in contrast to the results of a binary nozzle where the jets in different sides at the throat present a negative effect on the SVC. In addition to the difference in the flow fields of the two nozzles, an important factor is that there is a phase difference in the nozzle of this combined injection scheme of the axisymmetric injectors, and the jet interference is small. Figure 5.31 shows the thrust efficiency of the A5 and A2 schemes as a function of the secondary flow rate ratio in the whole expansion section. Under the condition of the secondary flow and flow rate ratio of the same expansion section, the thrust efficiency when there is a jet flow in the throat is lower than that in the throat without the jet flow. This is not inconsistent with the previous result of the ratio of the total secondary flow rate as an independent variable. Under the same ratio of the secondary flow rate in the expansion section, there is an additional interaction between the jet flow in the throat and primary flow in A5 scheme, so the thrust loss is larger than that of pure SVC. When all the injection units at the throat are turned on to inject the secondary flow (A4 scheme), the thrust vector efficiency of the SVC is lower than that of the A2 scheme. In addition, due to the presence of more secondary jets in the throat of the A4 scheme, the thrust loss will be greater, see Fig. 5.32. Generally, when the secondary flow is injected in the same side of the throat and the expansion section, the thrust vector efficiency of the SVC is lower than that of the pure SVC. In this mode, the effect of jets in the throat on downstream SVC is negative. The thrust loss of scheme B6 for injection on the same side is equivalent to that of scheme A5 for injection on the different side, see Fig. 5.33.

Fig. 5.31 η2 -flow rate ratio of scheme A5

5.7 Arrangement Concepts of Round-Hole Injectors

131

Fig. 5.32 η2 -flow rate ratio of the A4 scheme

Fig. 5.33 η2 -flow rate ratio of scheme B6

In summary, the jets in the throat have a complex effect on the SVC performance in the expansion section and it is related to the nozzle position in the expansion section, the combination of nozzles, and the jet pattern in the throat. In general, for the axisymmetric nozzle, the scheme of nozzle position with jets in different sides of the throat and the expansion section and the phase difference can simultaneously obtain better turbulence performance of the fluidic throat and thrust vector performance. Figure 5.34 shows the range of throat area and the thrust vector angle that can be achieved by the above-mentioned axisymmetric nozzles through different injection schemes. In particular, when there is a jet only in the throat (only the throat adjustment

132

5 Secondary Flow TVC for Fluidic-Throat Nozzles

Fig. 5.34 Vector angle-throat area reduction for 4 schemes

is required), the thrust vector angle under this scheme is 0°, when there is no jet in the throat and there is a jet only in the expansion section (only vector control is required), the throat area change is zero. When the fluidic-throat nozzle performs both throat adjustment and vector control, the variation ranges of throat area and vector angles that can be achieved corresponding to the middle portion of Fig. 5.34. It can be seen from the example in the figure that the combined jet can achieve a thrust vector angle of about 14° while achieving a 50% change in the throat area.

References 1. Zhi, C.: Design Rule of Dual-Throat Fluidic Thrust Vectoring Nozzles and Exploration of Rear Fuselage Integration. Nanjing University of Aeronautics and Astronautics, Nanjing (2007) 2. Yongjiu, L.: Fluidic thrust vectoring control technology. Aircr. Des. 28(2) (2008) 3. Yongsheng, Z., Yankui, W., Xiaowei, Y., Xueying, D.: Design of secondary-divergent vectoring nozzle based on secondary fluidic injection. J. Beijing Univ. Aeronaut. Astronaut. 33(3) (2007) 4. Wang, Q., Fu, Y., Eriqitai: Computation of three dimensional nozzle flow field with fluidic injection. J. Propul. Technol. 23(6), 441–444 (2002) 5. Zhang, Q., Lv, Z., Wang, G., Liu, Z., Jin, J.: Numerical simulation of an axisymmetric fluidic vectoring nozzle. J. Propul. Technol. 25(2), 139–143 (2004) 6. Deng, Y., Zhong, Z., Song, W.: Computational investigation of secondary flow thrust vector control technology used in a convergent-divergent nozzle. J. Solid Rocket Technol. 28(1), 29–32 (2004) 7. Qiao, W., Cai, Y.: A study on the two-dimensional thrust vectoring nozzle with secondary flow injection. J. Aerosp. Power 16(3), 273–278 (2001) 8. Lu, B., Xu, X., Zhou, M.: Numerical simulation on rectangular jet vector nozzle. Aeroengine 34(1), 16–18 (2008) 9. Dechuan, S.: Study on Supersonic Flow Field with Secondary Injection and Its Control Parameters. Northwestern Polytechnical University, Xi’an (2002)

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10. Deer, K.A.: Summary of fluidic thrust vectoring research conducted at NASA langley research center. AIAA 2003-3800 (2003) 11. Mason, M.S., Crowther, W.J.: Fluidic thrust vectoring for low observable air vehicles. AIAA 2004-2210 (2004) 12. Deere, K.A.: Computational investigation of the aerodynamic effects on fluidic thrust vectoring. AIAA 2000-3598 (2000) 13. Wing, D.J.: Static investigation of two fluidic thrust-vectoring concepts on a two-dimensional convergent-divergent nozzle. In: NASA TM-4574 (1994) 14. Chiarelli, C., Johnsen, R.K., Shieh, C.F., et al.: Fluidic scale model multi-plane thrust vector control test results. AIAA 93-2433 (1993) 15. Miller, D.N., Yagle, P.J., Hamstra, J.W.: Fluidic throat skewing for thrust vectoring in fixed geometry nozzles. AIAA-99-0365 16. Zhang, X., Wang, R., Yang, F.: Influence of double gas flow on fluid control vectoring nozzle. J. Solid Rocket Technol. 30(4), 295–298 (2007) 17. Zhou, M., Wang, R., Zhang, X., Xu, X.: Effect of jet flow distribution on fluidic throat skewing nozzle. J. Propul. Technol. 29(1), 58–61 (2008) 18. Luo, J., Wang, Q., Eriqitai, : Computational analysis of two fluidic thrust-vectoring concepts on nozzle flow field. J. Beijing Univ. Aeronaut. Astronaut. 30(7), 597–601 (2004) 19. Richard, J.Z.: Thrust vector control by liquid injection for solid propellant rockets. AIAA 75-1225 (1975) 20. Berdoyes, M.: Hot gas thrust vector control motor. In: 28th JPC, AIAA 92-3551 21. Green, C.J.: Liquid injection thrust vector control. AIAA J. 1(3), 573–578 (1963) 22. Zhang, J., Xie, K.: Secondary flow thrust vector control study for fluidic throat nozzle. J. Beijing Univ. Aeronaut. Astronaut. 38(3), 309–313 (2012)

Chapter 6

Gas–Liquid Fluidic Throat

Abstract In the engine, if the secondary flow is a liquid that can react with the fuel gas, the gas–liquid fluidic throat in the nozzle can be roughly divided into four regions: an extrusion zone of secondary flow, a gasification zone, a chemical reaction zone, and a mainstream area. The liquid phase state injected through the secondary flow injector, such as a liquid belt or droplet, directly affects the extrusion of the gas–liquid fluidic throat and the size of the gasification zone. If the liquid secondary flow is well atomized at the nozzle to form fine droplets, in addition to its own volume extrusion, the interaction of the droplets with the mainstream flow will increase, increasing the flow resistance of the mainstream flow. In addition, the surface area of the liquid secondary flow increases after the formation of the droplets, the evaporation enhances, and the gasification zone enlarges. The mixing rate of the gas phase in the reaction zone and the reaction rate are also dependent on the atomization and evaporation of the secondary flow at the nozzle. The effect of different injectors on the atomization characteristics under conditions with different ratios of secondary flow rate to mainstream flow rate in different secondary flow injection modes is introduced in this chapter.

6.1 Flow Characters of Gas–Liquid Throat In the engine, if the secondary flow is a liquid that can react with the fuel gas, the gas–liquid fluidic throat in the nozzle can be roughly divided into four regions: an extrusion zone of secondary flow, a gasification zone, a chemical reaction zone, and a mainstream area, as shown in Fig. 6.1. The extrusion zone of secondary flow: in this zone, the chemical composition of the secondary flow is not affected by the main flow and only exhibits flow characteristics. Since the secondary flow in this region is basically an incompressible liquid, the size of the fluidic throat formed by the layer is mainly determined by the volume of the secondary flow. After the mainstream flow (gas) is squeezed by the secondary flow, the flow area begins to become smaller and a gas–liquid two-phase flow is formed in the nozzle.

© Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_6

135

136

6 Gas–Liquid Fluidic Throat

Fig. 6.1 Schematic diagram of gas–liquid fluidic throat

The gasification zone: this area is formed by the evaporation and gasification of the liquid secondary flow which absorbs the heat of the mainstream flow. At the same time, due to the difference in the components of the secondary flow injected, the secondary flow may also undergo decomposition reaction in this area. The chemical reaction zone: it is a region where the component of the gasified secondary flow and high-temperature and high-pressure gas of the main flow mix and react. In this region, the heat released by the chemical reaction affects the transonic flow of the main flow and also affects the performance and location of the fluidic throat. The performance of the gas–liquid fluidic throat depends on the contribution of each of the above three regions to the flow resistance produced by the main flow. In the conventional hot test, only the overall performance of the gas–liquid fluidic throat can be obtained, and it is difficult to separate the contributions of the above three regions. A feasible method is to use the experimental method of cold flow to investigate the extrusion effect of the liquid on the mainstream flow, and then evaluate the effects of gasification and chemical reaction exotherm on fluidic throats by numerical simulation or theoretical calculation [1, 2]. Finally, the results of the hot test are compared and verified. This book only describes the extrusion effect of secondary fluid on the effective throat area. The influence on the gasification zone and the chemical reaction zone depends on the type of the selected secondary fluid, and its behavior is more complicated. If a more volatile oxidizing liquid is used, the influence of the gasification zone and the chemical reaction zone will be obvious. However, if an inert liquid is used, only the extrusion and gasification of the secondary flow will work. The liquid phase state injected through the secondary flow injector, such as a liquid belt or droplet, directly affects the extrusion of the gas–liquid fluidic throat and the size of the gasification zone. If the liquid secondary flow is well atomized at the nozzle to form fine droplets, in addition to its own volume extrusion, the interaction of the droplets with the mainstream flow will increase, increasing the flow resistance of the mainstream flow. In addition, the surface area of the liquid secondary flow increases after the formation of the droplets, the evaporation enhances, and the gasification zone enlarges. The mixing rate of the gas phase in the reaction zone and the reaction rate are also dependent on the atomization and evaporation of the secondary flow at the nozzle.

6.1 Flow Characters of Gas–Liquid Throat

137

In the cold flow experiment, a solution for the engine with a two-dimensional nozzle and an observation window can be used to visually observe the flow details of the mainstream flow, the liquid secondary flow, and the abovementioned atomization, as shown in Fig. 6.2. Figure 6.3 shows the flow details of a typical gas–liquid fluidic throat observed in a two-dimensional nozzle. The liquid working fluid used in the observation is water. Figure 6.3a, b shows pictures taken by a high-speed camera for experimental use (an ordinary light source is used as the laser source in the figures), and Fig. 6.3c displays the numerical simulation (using the volume of fluid (VOF) model for liquid level tracking).

Fig. 6.2 Schematic and physical map of an engine with a two-dimensional nozzle

Fig. 6.3 Diagram of flow in the nozzle

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It can be seen that the simulation and experimental phenomena are consistent. Under the interaction of the mainstream flow (gas) and the secondary flow (water), the flow direction of the water is deflected, forming a new aerodynamic boundary similar to that in the gas–gas fluidic throat, and the effective throat area of the primary flow becomes smaller. The water content in Fig. 6.3c also shows that the secondary flow will rapidly mix with the mainstream gas flow downstream of the fluidic throat, and the gas–liquid interface will gradually blur.

6.2 Liquid Injection Forms and Atomization The effect of different injectors on the atomization characteristics under conditions with different ratios of secondary flow rate to mainstream flow rate in different secondary flow injection modes is introduced in this chapter. The cold flow test was carried out using high-pressure air as the medium.

6.2.1 Test Methods The cold flow test system for fluidic nozzles can be divided into four parts, namely, an equipment system, a gas supply system, a measurement and control system, and an engine system. The equipment used in the atomization test was the Malvern particle size analyzer M300 and the six-component test bench. The particle size analyzer can measure the particle size in the range of 0.2–2000 µm, and the obtained particle size is the equivalent diameter of the droplet in the measurement area. The operation is simple and convenient, and is most commonly used in steady-state spray measurement with high precision. When measuring particle size in real time, the test equipment is installed in the structure shown in Fig. 6.4.

Fig. 6.4 Installation diagram of the particle size analyzer in the test

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The gas supply system mainly supplies working fluids for mainstream and secondary flow. The gas supply system consists of a gas source, a gas distribution cabinet, and supply lines for the mainstream and the secondary flows. The working medium used in the mainstream flow test is high-pressure air, and high-pressure water is used as the working fluid of the liquid secondary flow. The whole pneumatic system is shown in Figs. 6.5, and 6.6 shows the layout of the air pipeline of the equipment in the test scene. The mainstream flow is defined by the sonic nozzle on the pipeline. The secondary flow is measured by the flowmeter with an accuracy of 0.1%. The sonic nozzle is calibrated before the test, so that the mainstream flow in the test can be calibrated by P4, P6, and the corresponding sonic nozzle. The curves are determined together. The high-pressure water tank is used when the medium of secondary flow is in liquid state. The gas distribution cabinet is used to adjust the pressure of the mainstream flow and the secondary flow to change the flow rates of the two flows. The high-pressure cylinder has a volume of 4 m3 and a design pressure of 10 MPa. The safety pressure of the pipeline and various valves in the gas supply system is 15 MPa, which ensures the safety of the test, as shown in Fig. 6.5. The design thrust of the test engine is 100 N, the mainstream pressure is 1 MPa; the nozzle throat is rectangular with a size of 16 mm × 9 mm; the expansion ratio is 2; the convergence half angle is 45°; the expansion half angle is 15°; and the design safety pressure is 10 MPa. The structure is shown in Fig. 6.7. Figure 6.7a illustrates the physical map of the engine, and Fig. 6.7b shows the half section of the nozzle. The engine can be designed to move or block the position of the injector, so that different injection schemes can be easily realized to achieve different working conditions.

Fig. 6.5 Air supply pipeline system

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Fig. 6.6 Equipment layout in the test scene

(a) Engine

(b) Half section of the nozzle

1-nozzle gasket, 2-nozzle pressure plate, 3-secondary jet device, 4-blocking, 5-nozzle, 6-quartz glass, 7-visual window Fig. 6.7 Diagram of the engine system

6.2 Liquid Injection Forms and Atomization

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A key factor in the success of the cold flow test is the seal. If the gas leaks, it seriously affects the collection of test data, resulting in erroneous test results, the airtightness of the test engine was tested before the test. The engine is inspected to be airtight and pressurized to 10 MPa without gas leakage. The designed sealing structure fully meets the requirements of the cold flow test. In order to study the atomization characteristics of different combinations of fluidic-throat nozzles, a variety of secondary injection methods were designed. Different injection schemes were implemented by replacing injectors or moving plugs. As shown in Fig. 6.8, there are three types of test nozzles. The left-hand side of the figure is the isometric view of the nozzle, and the right-hand side is the full section of the nozzle. The nozzle is a rectangular parallelepiped with bosses on both sides, and two identical deep holes are symmetrically opened on the bottom surface as a concentrating cavity for the jet; the length-to-diameter ratio of the holes is 2; the bottom surface of the concentrating cavity is provided with three through holes, among which the middle hole is arranged symmetrically for the axis. The mutual impact angle of the interoperating nozzle is set to be 30° by referring to the angle between the two inclined holes, the horizontal distance between the intersections of the three through holes and the end face of the throat is 1 mm. The injection angle of the reverse nozzle, that is, the angle between the direction of the secondary jet and the axial direction of the nozzle is 60°. The horizontal jet nozzle sprays a secondary flow in the horizontal direction, and the length of the intermediate orifice is 3 mm and the diameter is 0.4 mm. Figure 6.8d is the physical map of the processing of the DC nozzles during the test. When a secondary liquid flow with a stable pressure is finally injected into the nozzle throat or the expansion flow passage through the jet orifice, the liquid secondary flow interacts with the high-pressure mainstream gas to generate a two-phase flow jet flow field and the mist is produced by the aerodynamic force. Under the action, the mist is further broken into finer droplets. Three nozzles were used in the test, and each nozzle has four injection schemes, see Table 6.1. The jet diagram in Table 6.1 has a peripheral rectangle representing the expansion section (the direction used to control the thrust) and an internal rectangle representing the throat injection (used to control the thrust or the mainstream throat area). The three short segments in the figure represent three secondary jet orifices, with arrows pointing to the direction of the mainstream. By changing the position of the blockage (see Fig. 6.9) in the sump, the O-ring seals the outside of the blockage and acts as a seal to achieve a symmetrical injection scheme for the expansion section or a symmetric jet scheme for the throat or an injection scheme for one-way jet of the throat expansion. On this basis, the flow rate ratio of the secondary flow to the mainstream and the injection medium of the secondary flow are changed, thereby obtaining the atomization characteristics under different working conditions.

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(a) Inter-jet injector

(b) Reverse-jet injector

(c) Horizontal-jet injector

(d) Physical map of DC nozzles with the mutual attack jet

Fig. 6.8 Diagrams of the nozzle structures

6.2 Liquid Injection Forms and Atomization Table 6.1 Injection schemes

Program

Secondary flow mode

1

Jets hit each other at the same time expanding jet

2

Jet expansion under mutual impact

3

Mutually symmetric jet-jet throat

4

Jets hit each other while expanding in single shots

5

Reverse jet-diffuser throat while spraying

6

Injective and reverse jet expansion section

7

Symmetric and reverse jet-injector throat

8

Reverse jet. Jets are expanding in single shots

9

Horizontal jet throat expanding while spraying

10

Expansion section of horizontal jets in single shots

11

Symmetric horizontal jet-injector throat

12

Horizontal jet. Jets are expanding in single shots

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Jet diagram

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Fig. 6.9 Blocking diagram

6.2.2 Atomization in the Nozzle Plume In real-time measurement, the timing of the test is set as follows: the solenoid valve for the primary flow opens first so as to pump the primary flow in the high-pressure gas, and after that, the solenoid valve for the secondary flow opens. After pumping the secondary flow in the high-pressure liquid to a steady state, the Malvern particle size analyzer is turned on to trigger the acquisition module at the frequency of 2,000 Hz. Then the real-time data acquisition is performed. At the end of the collection, the solenoid valves are closed and the data acquisition is stopped. The total duration lasts for 18 s. The area where the test droplet is measured is the nozzle outlet plume, see Fig. 6.10.

Fig. 6.10 Test and measurement device

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The secondary liquid flow with stable pressure and the high-pressure primary gas flow collide with each other in the fluidic throat, resulting in an atomized flow field. At this point, it can be observed that the shape of the throat passage has changed, and the throat area has changed significantly. Under the action of aerodynamic force and surface interaction force, the droplets in the atomization field are further broken into finer droplets, and the atomization field further forms a more uniform atomization area. Figure 6.11 shows the experimental scene of the jet plume for the symmetrical jet from the throat of the impinging injector of the secondary flow, the symmetrical jet in the divergent section and the simultaneous symmetrical jet in the throat and the divergent section at a pressure ratio of 4.12.

6.2.2.1

The Cumulative Distribution of Droplets in the Test Plume

Under cross-flow conditions, the secondary liquid flow with a stable pressure collides with the high-pressure primary gas flow in the fluidic throat, forming a stable spray field at the nozzle outlet. Figure 6.12 shows the real-time particle size measurement and acquisition of the atomized steady-state plume at the nozzle outlet on condition of simultaneous injection of the impinging injector with an angle of 30° in the engine divergent section and throat. Figure 6.12 shows the cumulative distribution of particle sizes collected during steady-state jets. There are three distribution curves of particle sizes in the figure, i.e., the lower, upper, and middle lines represent the distributions of d10 (lower limit diameter), d90 (upper limit diameter), and d50 (medium diameter). It can be observed that the amplitude of the oscillation of the whole particle is small, and d50 and d90 tend to rise slowly with time and then tend to be stable, and d10 is relatively stable. The medium diameter d50 fluctuates around 60 µm, which means that after the pressurestable secondary liquid flow strongly collides with the high-pressure mainstream gas, a well-atomized flow field can be formed at the nozzle outlet in a short time, indicating favorable atomization fineness and the overall high quality of atomization. In order to further illustrate the atomization quality of the nozzle plume at the steady-state jet, it is now described by the Sauter mean diameter (SMD) and the liquid mist distribution index N. In the cumulative data distributions collected in Fig. 6.12, the size distributions of liquid mist at the three sampling points of the steady-state jets at the first, fourth, and last one second were randomly selected to analyze the atomization quality. The particle size distribution map is shown in Fig. 6.13. The curves in Fig. 6.13 show the cumulative volume distributions of the droplets, the size of which corresponds to the ordinate on the left-hand side. The bars in the figure represent the distribution of volumetric frequency of the droplets, the size of which corresponds to the ordinate of the right measurement; the abscissa is the diameter of the droplet. It can be observed from the figure that in the random sampling results at the first, fourth, and eighth second, the droplet diameters are not much different, and the diameter is about 52 µm; the liquid mist distribution index is equivalent, and the value is around 2. The droplet diameter and atomization

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(a) Symmetrical jet from the throat

(b) Symmetrical jet in the divergent section

(c) Simultaneous symmetrical jets in the throat and divergent section Fig. 6.11 Spray field plume

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Fig. 6.12 Time–diameter distribution of droplets in the plume

distribution index of the integrated spray field are high, and the atomization quality of the plume at the outlet of the nozzle is high.

6.2.2.2

Droplet Diameter Analysis at the Nozzle Outlet

The relationship between the SMD and the pressure ratio measured by the Malvern particle size analyzer is shown in Fig. 6.14. It can be seen from Fig. 6.14 that the droplet diameter varies from 35 to 90 µm, and the atomization fineness under conditions of the three test injectors shows the same trend, and the droplet diameter becomes smaller as the pressure ratio increases. At the same time, in the various injection schemes, those showing the smallest droplet diameter are schemes 1, 5, and 9 in the same pressure ratio, all of which are jetted simultaneously in the throat and the divergent section. The injection schemes in which the diameter of the droplets is the largest are schemes 4, 8, and 12, all of which are throat-expanded one-way jets. The atomization fineness in the one-way jet in the divergent section is better than that of the throat symmetric injection, both of which are between the throat-expansion injection scheme and the single expansion injection scheme. This is because when the pressure ratio is increased, the mutual squeeze action between the secondary liquid flow and the primary flow is more intense, and the gas–liquid two-phase flow produces a strong mixing effect and accelerates the speed of the droplet breakage, so that the diameter of the droplet becomes smaller. The throat-expansion injection scheme introduces a symmetrical jet in the divergent section on the basis of the throat symmetric injection scheme, so as to further increase the impact of the secondary flow injected from the divergent section and the flow field that has been atomized in the throat, to enhance the degree of breakage of the droplets, and to reduce the diameter of the droplets. However, the mixing effect of the gas–liquid two-phase flow under the throat-expansion one-way injection scheme is not strong and the atomization quality is poor, which is consistent with the experimental results.

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Fig. 6.13 Distributions of droplet diameter

(a) SMD=53.83 μm and N=2.83

(b) SMD=52.19 μm and N=2.23

(c) SMD=51.82 μm and N=2.24

Figure 6.15 shows the atomization characteristics of the three test injectors under the throat-expansion injection scheme. It can be found from Fig. 6.15a that when the pressure ratio is constant, the injector with the smallest droplet diameter is the impinging orifice injector, and the nozzle with the largest droplet diameter is the horizontal orifice injector. It can be seen from Fig. 6.15b that as the pressure ratio increases, the liquid mist distribution index N of the spray field becomes larger, indicating that the uniformity of atomization is improved. When the pressure ratio

6.2 Liquid Injection Forms and Atomization

(a) Impinging orifice injector

(b) Reverse-type orifice injector

(c) Horizontal orifice injector

Fig. 6.14 Relationships of SMD, pressure ratio, and injection scheme

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Fig. 6.15 Comparison of atomization under the throat-expansion injection scheme

(a) Droplet diameter SMD

(b) Distribution index N

is constant, the injector with the largest distribution index is the impinging orifice injector. This is because the greater the pressure ratio is, the more intense the turbulent interaction between gas and liquid and the faster the droplet breaks. Especially when the secondary flow is injected from the impinging orifice injector, the liquid is atomized by the three-pronged impinging jet, then mixed with the primary gas flow, and further broken into smaller droplets under the action of aerodynamic force. However, the mutual squeezing effect of the secondary liquid flow and the primary flow under conditions of the horizontal orifice injector is relatively weak, with a low breaking degree of the droplets and the poor quality of the atomization. The diameter of the droplets and the uniformity of the atomization are combined to compare the three test nozzles, which reveals that the nozzle with the best atomization effect is the impinging orifice injector.

6.2 Liquid Injection Forms and Atomization

6.2.2.3

151

Comparison of the Atomization Effects of Impinging Injectors

Three different types of injectors were used in the test and the atomization effects of the three injectors were compared under different working conditions. By analyzing, it is found that the atomization effect of the impinging injector is best. A flow field of small diameter droplets can be obtained by using the impinging injector, and the atomization distribution is also quite uniform. There are three types of impinging injectors studied in the test, with the impinging angles of 30°, 45°, and 60°, respectively. Figure 6.16 shows the atomization characteristics of the impinging injectors with different angles under the scheme of the throat-expanding section injection. The relationship between droplet diameter and secondary flow impinging injection angle under different compression and flow ratios is also studied. It can be seen from Fig. 6.16 that when the compression ratio

Fig. 6.16 Atomization characteristics of the impinging injectors with different angles under the throat-expanding section injection

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and flow rate ratio are constant, the droplet diameter ranges from 30 to 70 µm; the injector with the smallest droplet diameter is the impinging injector with the impinging angle of 60°; and the injector with the largest droplet diameter is the one with the impinging angle of 30°. As the compression ratio and flow rate ratio increase, the diameter of the droplets under conditions of the three different impinging angles is gradually reduced, indicating that the impact strength between the secondary flow and the high-pressure primary flow is more intense, and the droplets are broken faster. It is easy to know that among the three injectors with different impinging angles, the atomization effect of the one with 60° is the best.

6.2.2.4

Atomization and Weber Number

Weber number (We) is widely used as the criterion of gas–liquid atomization and crushing. For a given flow field, the larger the Weber number is, the finer the atomization will be. According to the condition of the test injector, we analyzed the Weber number at the throat where the secondary flow impacts with the primary flow under the horizontal jet injector. The flow rate of the pipeline is measured by the flowmeter and the structural dimensions of the three vias are already known. By using the pipeline flow formula, the exit velocity at the orifice and the gas velocity at the throat are obtained, and both of them are sonic. Figure 6.17 shows the relationship between the Weber number and SMD for the four injection schemes. It can be observed from the figure that the droplet diameter becomes smaller as the Weber number increases, and the test results are consistent with the theoretical analysis.

Fig. 6.17 Relationship between SMD and We

6.2 Liquid Injection Forms and Atomization

6.2.2.5

153

Atomization and Thrust Characteristic

Figure 6.18 shows the relationship of the flow rate ratio, SMD, and thrust characteristics of atomization field at the nozzle exit of the three test injectors under the scheme of the throat-expanding section injection. It can be observed from Fig. 6.18 that for the schemes of throat-expanding section injection corresponding to the three injectors, the effective throat area ratio ranges from 0.89 to 0.99 and the thrust ratio ranges from 1.04 to 1.16. As the flow rate ratio increases, the diameter of the droplet decreases, the effective throat area ratio decreases, and the thrust ratio increases. This is because when the pressure increases, the flow rate of the secondary jet increases, and the squeeze on the primary flow becomes stronger. The more intense the mixing effect between gas and liquid, the faster the droplet broken, the smaller the droplet diameter, the smaller the loss of gas–particle two-phase, and the larger the flow resistance generated by the secondary flow, which will result in the higher pressure of the gas collecting chamber, enhance the choking ability and improve the thrust ratio. In practical applications, for a given flow rate, the smaller the diameter of the droplet is when the secondary flow interacts with the primary flow, the larger the atomization surface, the more complete the secondary combustion, and the more the heat released by combustion of secondary flow. Consequently, the choking ability and the thrust ratio may also further improve. When the flow rate is given, the choking ability and the thrust ratio of the impinging injector are higher, and both effects of them are better than those using horizontal injector and reverse-type injector. This is because when the impinging injector is used, the mixing between gas and liquid is more intense, the flow resistance generated is larger, the pressure in the gas collecting chamber increases faster, the choking ability is stronger, and the thrust ratio is higher. When the diameter of the droplet is constant, the thrust characteristics of the three types of injectors have no fixed regularity due to the different flow rates and structures. In particular, when the diameter of the droplet is about 70 µm, the three types of injectors have nearly the same thrust ratio. This is because they have different flow rates of the secondary jet and dissimilar structures. But it is possible that they have similar choking ability. In general, for the spray field at the exit of the nozzle, as the pressure ratio increases, the diameter of the droplet becomes smaller. Under the scheme of throat-expanding section injection, the secondary flow injector with the best atomization quality is the impinging injector. It was found that the atomized droplet diameter of the injector with an impinging angle of 60° was the smallest in the given examples. The larger the flow rate ratio is, the more intense the mixing between the secondary flow and the primary flow, the faster the droplet is broken, the smaller the diameter of the droplet, and the higher the choking ability and thrust ratio.

154

Fig. 6.18 Relationship between SMD and thrust characteristics

6 Gas–Liquid Fluidic Throat

6.2 Liquid Injection Forms and Atomization

155

6.2.3 Atomization in the Throat When a liquid secondary jet having a stable pressure is finally injected into the nozzle throat or the expanded flow passage through the jet orifice, the liquid secondary jet interacts with the high-pressure mainstream gas to generate a two-phase flow jet flow field. Under the action of aerodynamic force, the droplets are further broken into finer droplets. During the steady-state spray, the diameter of the throat droplets was measured by the Malvern particle size analyzer. Figure 6.19 shows the droplet diameter distribution of the throat flow field in the throat and expansion section of the impinging injector. According to Fig. 6.19, the overall oscillation amplitude of the particles is large, with values of d10 , d50 , and d90 changing a lot and showing a large fluctuation range over time. In terms of numerical values, d10 fluctuates at 100 µm; d90 fluctuates up and down at 300 µm; and the pore size of the secondary flow injection hole is 400 µm, indicating that the gas and the liquid are interlaced when the secondary liquid flow interacts with the high-pressure mainstream flow near the throat. In the atomization area, there may be a large number of droplets that are re-agglomerated while being broken in a short time, and medium diameter floats up and down at 200 µm. Therefore, the atomization at the throat of the engine nozzle is not ideal, and the diameter of the droplet is large. The main reason is that the breaking of the droplet on the condition mainly depends on the interaction force between the gas and the liquid, and the aerodynamic force that further breaks the droplet has not played a large role at the throat, so the atomization effect near the throat is not very good. At the same time, the diameter of the droplets fluctuates greatly and quantitative comparison and research under different working conditions cannot be performed based on the collected test data. In order to further explain the atomization effect of the interaction between the secondary flow and the mainstream flow in the throat,

Fig. 6.19 Distributions of droplet diameter

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we can randomly select the distributions of droplet diameter at the first, second, third, and fourth s (the total steady-state jet for 4 s) during the steady-state jet as the sampling point under the throat-expansion injection scheme for comparative analysis. Figure 6.20 shows the distributions of the droplet diameter on condition of pressure ratio of r = 3.2, interpenetrating DC expansion jet at the nozzle throat. It can be seen from Fig. 6.20 that the diameter of the droplets at the four sampling points changes greatly during the steady-state jet process. Most of the droplet diameters are distributed between 200 and 300 µm, and other segments only have a small amount of droplets. About 125 µm, N is about 1.0, which indicates that the atomization flow field of the nozzle throat is very uneven. This is because the secondary flow is injected from the vicinity of the throat. When the secondary flow collides with the mainstream flow, the mixing effect of gas and liquid is not strong. The interaction between the two plays a leading role, and the aerodynamic force does not play a significant role. As a result, the degree of breakage of the droplets is limited, so that the atomization area is very uneven and the diameter of the droplets is large. When the pressure ratio is increased, the interaction between the secondary liquid flow and the high-pressure transverse mainstream flow is more intense, and a stronger gas–liquid mixing occurs. Figure 6.21 shows the stochastic jets at first, second, third, and fourth s (the total steady-state jet for 4 s) of the countercurrent nozzles at steady-state jets at a pressure ratio of r = 5.1. The droplet diameter distribution was used as a sampling point to compare the atomization quality. It can be seen from Fig. 6.21 that the diameter of the droplets at the four sampling points changes greatly during the steady-state jet process, and most of the droplet diameters fluctuate around 100–300 µm, and other segments occupy a small proportion of droplets. The diameter of the droplets is generally large. Compared with the sample analysis results when the pressure ratio is 3.2, the sample diameter distribution is looser under the pressure ratio of 5.1, and the droplets with a diameter below 100 µm occupy a certain proportion. It indicates that when the pressure is increased, the secondary flow and the high-pressure mainstream flow collide with each other at the moment, and the degree of particle breakage is larger, so that the distribution range of particle size is larger and the atomization is uneven. The sampling point analysis of the atomization under steady-state jet of the mutual impact nozzle is summarized with different pressure ratios. It is found that the throat has moderate atomization effects and uneven atomization. Compared with the atomization effect of the simultaneous injection of the throat and the expansion section in the previous section, the droplet diameter near the nozzle outlet is small and the flow field is uniform in the mutual impact nozzle. In the same scheme and pressure ratio, the diameter of the droplet measured at the throat is large, and the flow field is less uniform. Therefore, in the flow field from the throat to the outlet of the nozzle, the aerodynamic force produced by the high-pressure transverse mainstream flow plays an increasingly important role, so that the droplets are broken in the flow field of the throat under the action of aerodynamic forces, becoming more and more smaller.

6.2 Liquid Injection Forms and Atomization

(a) SMD=128.83μm,N=1.23

(b) MD=124.91 μm and N=0.98

(c) SMD=127.09 μm and N=1.14

(d) SMD=129.13 μm and N=1.08

Fig. 6.20 Distributions of droplet diameter

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(a) Distribution of sample particle size at t=1 s

(b) Distribution of sample particle size at t=2 s

(c) Distribution of sample particle size at t=3 s

(d) Distribution of sample particle size at t=4 s

Fig. 6.21 Droplet diameter distribution

6.3 Comparison Between Gas and Liquid Secondary Flows

159

6.3 Comparison Between Gas and Liquid Secondary Flows The supersonic mainstream gas flow can also generate oblique shock waves at the gas–liquid interaction boundary, so that the mainstream flow is deflected to generate the thrust vector. As shown in Fig. 6.22, the mainstream flow has a significant deflection in the direction of the outlet of the nozzle after the secondary fluid is injected into the divergent section of the nozzle, thus producing a significant deflection. Therefore, similar to the gas–gas fluidic throat, the gas–liquid fluidic throat can also adjust the magnitude and direction of the thrust at the same time. Here, six combined injection schemes are taken as an example to illustrate the effects and differences in the disturbance performance and thrust vector control performance of the gas–liquid fluidic throat and the gas–gas fluidic throat. The six combined injection schemes are shown in Table 5.1 in Chap. 5 (the injection schemes correspond to the nozzles A1–A6 in the table in sequence).

6.3.1 Injection Concept 1— The No. 1 injection scheme is a reverse symmetrical injection of a secondary flow at the throat of 60° (the lateral force and the yaw angle of the thrust are zero). At the same flow rate ratio, the contribution of the secondary gas flow to the axial thrust is greater than the secondary liquid flow. And the gradient of the thrust growth of the secondary gas flow is also larger than that of the secondary liquid flow, as shown in Fig. 6.23. The increase ratio of thrust in the figure is based on the thrust value without

Fig. 6.22 Vector control of gas–liquid fluidic throat

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Fig. 6.23 Increase and decline of thrust with flow rate ratio

Fig. 6.24 Increase of thrust with pressure ratio of secondary flow

secondary flow. This is because when the secondary flow is a gas, it still flows as a transonic flow in the nozzle and reaches the supersonic velocity (large impulse) at the outlet of the nozzle. Therefore, the contribution of the secondary flow to the main thrust is high. When the secondary flow is a liquid, it still flows much slower in the nozzle (small impulse) relative to the main flow (gas), and thus contributes relatively slightly to the main thrust because the liquid is incompressible. It is worth noting that under the same injector area ratio and the total pressure ratio of the secondary flow, the contribution to the main thrust of the nozzle is generally large when the secondary flow is a liquid, as shown in Fig. 6.24. This is because under this condition, since the density of the liquid is much larger than that of the gas, the flow rate of the secondary liquid flow is large, and the generated momentum is large.

6.3 Comparison Between Gas and Liquid Secondary Flows

161

6.3.2 Injection Concept 2— In No. 2 injection scheme, the secondary flow is injected vertically into the nozzle on one side of the divergent section (pure SVC) (Figs. 6.25 and 6.26). Similar to the case where the secondary flow is injected into the throat, at the same flow ratio, the contribution of the secondary gas flow to the axial thrust and the lateral force is greater than the secondary liquid flow. And the gradient of the component force growth of the former is also larger than that of the latter. Under the same injector area ratio and total pressure ratio of the secondary flow, the secondary liquid flow is superior in terms of the contribution of the lateral force and the axial force. The change in yaw angle of thrust has a similar pattern.

6.3.3 Injection Concept 3— In No. 3 injection scheme, the secondary flow is injected into the nozzle vertically at the same time on the same side of the throat and the divergent section, as shown in Figs. 6.27 and 6.28 (while adjusting the thrust magnitude and direction). By comparing Figs. 6.25 and 6.27, it can be seen that after the secondary flow is injected into the throat, the available thrust vector angle is reduced, and it is reduced more apparently when the secondary flow is gas than when the secondary flow is a liquid. In the combined injection scheme, the performance gap between the gas–gas fluidic throat and the gas–liquid one is reduced at the same secondary flow ratio. At the same pressure ratio, the difference in the lateral force performance between the two increases. This is because the jet interference between the jets of the same side when the secondary flow uses gas is larger, while the influence of the jet interference when using water is smaller.

Fig. 6.25 Magnitude increase and deflection of thrust with flow rate ratio

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Fig. 6.26 Magnitude increase and deflection of thrust with pressure ratio of the secondary flow

Fig. 6.27 Magnitude increase and deflection of thrust with flow rate ratio

Fig. 6.28 Magnitude increase and deflection of thrust with pressure ratio of the secondary flow

6.3 Comparison Between Gas and Liquid Secondary Flows

163

6.3.4 Injection Concept 4— In No. 4 injection scheme, the secondary flow is uniformly symmetrically and vertically injected into the throat, and the secondary flow in the divergent section is vertically injected into the nozzle. It can be seen from the curve trend that under the same flow ratio, it is still better for the secondary flow to be a gas than a liquid. In the case of the same pressure ratio, the opposite is true. At the same flow ratio of the secondary gas, the thrust gain and maximum vector angle of this scheme are smaller than that of the third scheme (Figs. 6.29 and 6.30).

6.3.5 Injection Concept 5— In No. 5 injection scheme, the secondary flow is simultaneously injected perpendicularly into the nozzle on one side of the throat and on the other side of the divergent section (different side injection). The ipsilateral injection scheme differs greatly from

Fig. 6.29 Magnitude increase and deflection of thrust with flow rate ratio

Fig. 6.30 Magnitude increase and deflection of thrust with pressure ratio of secondary flow

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Fig. 6.31 Magnitude increase and deflection of thrust with flow rate ratio

Fig. 6.32 Magnitude increase and deflection of thrust with pressure ratio of secondary flow

the schemes of different side injection in that the performance of the secondary flow is larger when the secondary flow is a gas. Due to the injections in different sides, the interference between the secondary flows in the throat and the divergent section is reduced, and the different injection performance of the secondary flow as a gas is improved. When the secondary flow is a liquid, the performance of injection schemes on the same side and different sides is not much different. This is a difference between a gas–liquid fluidic throat and a gas–gas fluidic throat (Figs. 6.31 and 6.32).

6.3.6 Injection Concept 6— In No. 6 injection scheme, the secondary flow is injected into the nozzle on the same side of the throat and the divergent section. The difference between this solution and No. 3 solution is that there is no phase difference in the circumferential direction between the injection points of the divergent section and the throat. In this injection scheme, the interference between the secondary flows in the throat and the divergent section is greater, and the performance is worse than that of the third

6.3 Comparison Between Gas and Liquid Secondary Flows

165

Fig. 6.33 Magnitude increase and deflection of thrust with flow rate ratio

Fig. 6.34 Magnitude increase and deflection of thrust with pressure ratio of secondary flow

scheme. Especially when the secondary flow uses water, the performance is more degraded. Therefore, the practical application should avoid the use of such a spray scheme regardless of whether the secondary flow is a high-pressure gas or a liquid (Figs. 6.33 and 6.34).

6.3.7 General Comparison Since the density of the liquid is much larger than that of the gas, the secondary liquid flow can obtain a larger multiple flow rate at the same injector area ratio and the total pressure ratio, which is the advantage of using the liquid as the secondary flow. However, the disadvantage is that the liquid is substantially incompressible, and it is not possible to continue to expand as a gas in the nozzle to obtain a larger impulse; thus, at the same flow rate ratio, the performance gain obtained by the secondary flow using the liquid is inferior to that using the gas. Of course, the above conclusion is only obtained by considering the extrusion action of the secondary liquid flow on the mainstream flow. Further research is needed on the gains from liquid gasification and chemical reaction exotherm.

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When the fluidic-throat nozzle needs to adjust the engine thrust (throat area) and the thrust direction at the same time, it is necessary to select a combined injection scheme with a small total thrust loss and a superior control performance over thrust vector according to the general types of the secondary flow. I. Total thrust change Under the same flow rate ratio, for the case where the secondary flow is a gas, the total thrust gain is optimal in the first injection scheme (the thrust loss is the smallest), is the worst in the second scheme, and the third, fourth, and fifth schemes are not much different. However, for the case where the secondary flow is a liquid, under the same flow rate ratio, No. 1 nozzle is the best, followed by No. 2, No. 3, No. 5, and No. 4 ones, and the worst is No. 6 nozzle. This is because the interference between the two jets of the two working fluids is different, and the thrust loss caused by them is also different. Especially when a liquid is used as the secondary flow, the thrust loss will be large if there is no phase difference between the secondary flow injectors at the throat and those at the divergent section (Figs. 6.35 and 6.36).

Fig. 6.35 Diagrams of total thrust variation ratio using a gas as the secondary flow

Fig. 6.36 Diagrams of total thrust variation ratio using a liquid as the secondary flow

6.3 Comparison Between Gas and Liquid Secondary Flows

167

At the same pressure ratio, for the case where the secondary flow is a gas, the total thrust gain is optimal for No. 4 nozzle, while those of No. 1, No. 3, No. 5, and No. 6 ones are basically the same, and No. 2 nozzle is the worst. For the case where the secondary flow is a liquid, under the same pressure ratio, No. 2 is the best while No. 1 is the worst. However, in terms of the growth slope, No. 2 is smaller and No. 1 is the largest. It can be seen that the performance of the combined injection scheme of the gas–liquid fluidic throat is different from that of the gas–liquid fluidic throat. Therefore, while designing the combined injection scheme of the gas–liquid fluidic throat, it is not possible to simply refer to the gas–gas fluidic throat. II. Thrust angle variation The curves in Figs. 6.37 and 6.38 show that whether the secondary flow is a gas or a liquid does not have much effect on the thrust vector control performance of the six nozzles. When the secondary flow is a gas or a liquid, the thrust angle of No. 2 scheme is always the largest, followed by No. 3 and No. 5 ones, and then No. 4 and No. 6 schemes. In general, when the secondary flow is a liquid, it is particularly suitable to adopt No. 2 scheme for thrust vector control, and No. 3 or No. 5 scheme can simultaneously control the thrust magnitude and direction; when the secondary flow is a gas, it is

Fig. 6.37 Thrust deflection angle diagram using a gas as the secondary flow

Fig. 6.38 Thrust deflection angle diagram using a liquid as the secondary flow

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particularly suitable for adopting No. 4 or No. 5 scheme for control over thrust vector, and a larger vector angle can be obtained by the second scheme, but the thrust loss is larger than when the secondary flow is a liquid. III. Variation of the effective throat area Compared to the left-hand side of Figs. 5.29 and 6.40, when the secondary flow is a liquid, if the fluidic throat is formed by its extrusion (that is, regardless of the influence of gasification and chemical reaction), to achieve the same variation range of the fluidic-throat area, the required liquid flow should be much larger than the secondary gas flow. By comparing the curve with that applying pressure ratio of the secondary flow as the independent variable, the difference between conditions of using a gas or a liquid as the secondary flow is not too great. For the gas–liquid fluidic throat, an injector of a smaller area ratio and a higher total pressure ratio of the secondary flow can be used, and a volatile liquid oxidant is used to increase the gasification zone and the zone heated by chemical reaction to reduce the mass of the secondary liquid flow that needs to be carried (Figs. 6.39 and 6.40).

Fig. 6.39 Effective throat area ratio using a gas as the secondary flow

Fig. 6.40 Effective throat area ratio using a liquid as the secondary flow

References

169

References 1. Lixing, Z.: Dynamics of Multiphase Turbulent Reacting Fluid Flows. National Defense Industry Press, Beijing (2002) 2. Jun, H., Cunde, F.: Testing Technology of Solid-Propellant Rocket Engines. Astonautic Publishing House, Beijing (1989)

Chapter 7

Thrust Modulation Process of Fluidic Throat for Solid Rocket Motors

Abstract This chapter discusses the time-dependent thrust modulation process of the fluidic throat for engineering applications. It is necessary to learn the thrust modulation of the fluidic throat and the unsteady process of the mainstream flux. It is directly related to the thrust modulation characteristic of the fluidic-throat nozzle and its proper operating mode. Typical internal ballistic results under different operating conditions are introduced in the following sections, including calculation method of internal ballistics, modulation process at fixed mass flow rates, unsteady process for the axial-gap fluidic throat, and thrust modulation for the hole-type fluidic throat in hot test conditions.

7.1 Calculation Method of Internal Ballistics The thrust modulation process for solid-propellant rocket motors (SRM) is realized by the calculation of internal ballistics. Here, we will introduce the computational fluid dynamics (CFD) method. The structured grids used for the fluidic throat refer to Chap. 2. A few differences exist between the boundary conditions of the domain in Chap. 2 and this chapter. After the burning surface of the solid propellant is ignited, the combustion gas of the propellant is injected into the chamber at a certain velocity and temperature, forming the additive mass flow [1].

7.1.1 Description of Mass Sources When simulating the propellant combustion, we can ignore the specific process and the reaction time, but just consider the chemical reaction result, namely, the high-temperature product. The aspect of fluid control volumes clinged to the burning surface is accounted as the energy source, mass source, and momentum source [2, 3]. The terms of the mass source, the momentum source, and the energy source can be respectively expressed as:

© Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_7

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The term of mass source: m˙ = ρ p r A

(7.1)

mv ˙ = ρ p r Av

(7.2)

The term of momentum source:

ρ

where v is the radial injection velocity of combustion gas, v = −r ρgp . The term of energy source: mh ˙ = ρ p r Ac p Tg

(7.3)

where ρp A p n cp Tg r

The density of the propellant; The area of burning surface; The pressure in the chamber; The burning rate pressure exponent of combustion gas; The constant-pressure specific heat of combustion gas; The gas temperature; Burning rate.

The value of the source terms can be attained by calculating the burning rate and the influence of pressure gradient on the burning rate. The burning rate formula can be given as r = a Pcn

(7.4)

where Pc is in the unit of Pa; the burning rate r is in m/s.

7.1.2 Discrete Method for N-S Equations The N-S equation describing the combustion gas of the fluidic throat for SRM and the secondary jet flow ignores the body forces. For the inert secondary flow, we ignore the chemical reaction and assume that the combustion gas is totally ideal gas. In the calculation, diffusion and transfer between the components of the combustion gas and the secondary jet flow should be considered. The second-order upwind scheme is used for the convection term of the average N-S Reynolds equation, the centraldifference discrete method for the viscidity term, and S-A model for the turbulence model. The unsteady term utilizes the second-order accurate scheme, while the time step could be specified as 10−6 for the timescale of the thrust modulation in the fluidic throat.

7.1 Calculation Method of Internal Ballistics

173

Theoretically, the domain inlet corresponds to the burning surface of the propellant of motors that changes its position in the unsteady calculation, which suggests that the dynamic mesh is supposed to be exploited to reflect the real process. But the typical timescale of modulation is required to be less than the order of 0.1 s. Within this time scale, the mass inlet of the propellant surface has a small amount of movement, so it is assumed that the propellant surface is fixed while simulating the modulation process of the fluidic throat. When simulating ignition boosting process of SRM, the aforementioned calculation models, boundary conditions, and mass transfer method have been proved to fit the experimental curve quite well [2].

7.2 Modulation Process at Fixed Mass Flow Rates The practical regulation mechanism in cold-flow test introduced in Chap. 2 is the same as the one in the hot test, which indicates that the overstock of the mainstream working media and the filling of the motor chamber causes the imbalance of dynamic flux in the nozzle while decreasing the throat area; until the dynamic flux in the nozzle rebalances, the chamber pressure and the thrust are stabilized at new balance values. For real engines, the difference lies in that the flux at the inlet would still increase as the chamber pressure rises (the burning rate of the propellant accelerates as the chamber pressure rises, which leads to the multiplication effect). However, the flux at the inlet, affected by the upstream injector controlling, of the engine in the cold-flow test is a fixed value, which means that the chamber pressure in the cold test has a lower increasing value than that in the hot test (Fig. 7.1).

Fig. 7.1 CFD and experimental curves of pressure modulation in cold-flow test

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Therefore, we use the cold test as an example for the comparison, through which we can also check if the model established can describe the modulation mechanism and specific details of this kind of engines in some degree. Compared to the experimental values, the error of final balancing pressure in the CFD prediction is about 10%, which is close to the accuracy of the steady model, and the error analysis is shown in Sect. 2.1.2. The experimental pressure-boosting transition time is longer than the numerical model, which is related to the higher balancing pressure in the CFD model, while the CFD prediction has the same trend of pressure boosting and the boosting rate as the experimental curve.

7.3 Unsteady Process for the Axial-Gap Fluidic Throat It is necessary to learn the thrust modulation of the fluidic throat and the unsteady process of the mainstream flux. It is directly related to the thrust modulation characteristic of the fluidic-throat nozzle and its proper operating module. Typical internal ballistic results under different operating conditions are introduced in the following sections.

7.3.1 Typical Unsteady Work Process Figure 7.2 presents the chamber pressure-boosting curve during the settingup progress of the fluidic throat. The burning rate formula in the figure is r = 7.796 × 10−7 P0.6 c , the total pressure of the secondary flow at the inlet is 15 MPa. At the initial moment, there is no secondary flow injected, and the balanced pressure in the chamber is 3 MPa. The initial flow field for the calculation of the internal ballistic is a steady fluid field under a pressure of 3 MPa. In Fig. 7.2, t = 0 s corresponds to the moment at which the injector is opened. After opening the injector of the secondary flow (the boundary condition at the injector inlet changes from the wall condition into the pressure inlet condition), the secondary flow injected perpendicularly extrudes the mainstream; as the fluidic throat gradually forms, the effective area of the nozzle decreases, and the chamber pressure starts to rise and becomes stable at another balanced pressure. During the starting period, the chamber pressure responds quickly and increases fast; in the final period, the chamber pressure smoothly transits to about 10 MPa, in which there is no overshoot oscillation phenomenon. Figure 7.3 gives streamlines and contours of the gas component corresponding to the critical moment under the pressure in Fig. 7.2. The corresponding end surface in Fig. 7.3 is the burning surface at the moment of the 3rd second under the chamber pressure of 3 MPa, and the cylinder section on the right side is the cavity formed under the respective operating condition. Figure 7.3a is the initial moment, at which there is no secondary flow injected. Therefore, there is only combustion gas component

7.3 Unsteady Process for the Axial-Gap Fluidic Throat

175

Fig. 7.2 Typical pressure–time curve

in the fluid field contour and the gas produced by the left burning surface totally flows out through the nozzle throat. Figure 7.3b shows that the secondary flow can be quickly injected into the mainstream flow after the secondary flow nozzle is opened (see partially enlarged schematic of the nozzle). Since the total pressure of the secondary flow is much higher than that of the mainstream in the initial stage, the mainstream flow is strongly squeezed after the secondary flow is injected, and the secondary flow completely blocks the mainstream throat at 0.2 ms. At this time, the gas generated on the left side cannot flow out through the throat, and the reverse flow begins to form at the throat (see the streamline diagram in Fig. 7.3c). Figure 7.3d shows that because the pressure of the secondary flow is much higher than that in the combustion chamber, a part of the secondary flow flows into the cavity of the combustion chamber, and the remaining secondary flow flows out of the nozzle; The high-pressure secondary flow mixes with the generated gas to fill the cavity near to the end face of the propellant to form a vortex. Due to the filling of the gas and part of the high-pressure secondary flow, the pressure in the combustion chamber rises rapidly and the mass of gas generated by the left burning surface increases. At about 6 ms, it can be seen from Fig. 7.3e that as the pressure in the combustion chamber increases, the blocking action of the secondary flow on the mainstream flow is relatively weakened, and the high-pressure secondary flow at the throat begins to be shouldered out; a small portion of the gas begins to flow through the nozzle, and the high-pressure secondary flow flowing to the cavity also begins to decrease. It can be seen from Fig. 7.3f, g that the fluidic throat formed by the high-pressure secondary flow is further enlarged as the pressure in the combustion chamber increases, and the high-pressure secondary flow does not flow back to the combustion chamber and completely flows out through the nozzle; since there is more generated gas than the gas flowing out of the throat, the filling vortex still exists in the flow field, with difference lies in that the diameter of the vortex gradually reduces, and its position

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Fig. 7.3 Streamlines and contours of mass fraction of the gas

moves from the wall of the combustion chamber toward the axis of the combustion chamber. In Fig. 7.3h, the filling vortex begins to disappear at 27 ms, corresponding to the pressure curve in Fig. 7.3, in which it is seen that the pressure in the combustion chamber changes from a fast near-linear increase to a slow smooth transition. At about 68 ms, the pressure in the combustion chamber is approximate to equilibrium, and the fluidic throat formed by the interaction between the secondary flow and the mainstream flow is also stabilized.

7.3.2 Influence of Chamber Volume The modulation time for pressure in the combustion chamber of the fluidic throat is related to the cavity volume of the chamber. If the cavity volume of the chamber is small and the time for filling the cavity with the gas and the secondary flow is shortened, the pressure-boosting time is shortened. Figure 7.4 shows the pressure–rise curve for a chamber with a smaller cavity volume, with the other conditions the same as those in Fig. 7.2. Compared to the volume of the cylindrical cavity in Fig. 7.2, the volume of the chamber in this figure is reduced by 2/3. Figure 7.4 shows the streamlines and contours of the gas component at two moments. In the figure, the corresponding position of the end surface of the column

7.3 Unsteady Process for the Axial-Gap Fluidic Throat

177

Fig. 7.4 Pressure–time curve

is that of the burning surface of the engine when it is operated for 1 s under the chamber pressure of 3 Mpa, and the cylindrical section on the right side is a cavity corresponding to the moment. The process of increasing the pressure in the combustion chamber after injecting the secondary flow is similar to that in Fig. 7.2, while the time to reach the new equilibrium is reduced by about 50%. This is similar to the rule obtained by adding a cork in the cold-flow test to simulate the effects of different positions and cavity volumes of the propellant.

7.3.3 Influence of Injection Angle Steady studies have shown that using reverse jetting under conditions of the same parameters of secondary flow results in a larger variation range of effective throat area. It can also be seen from the pressure curve in Fig. 7.5 that after reverse injection (reverse injection angle α = 30°), the pressure in the combustion chamber of the example is increased from 3 MPa to over 13 MPa, which is higher than that of the vertical injection in Fig. 7.2. It does have a larger change range of chamber pressure. In addition, with reverse injection, the time for the pressure to rise to a new equilibrium is slightly shorter than the time shown in Fig. 7.2. One possible reason is that after the reverse injection, the high-pressure secondary flow in the backstream cavity is increased, which allows the cavity to be rapidly filled at the initial stage, suggesting that the pressure rises rapidly. The streamlines on the right side of Fig. 7.5 shows that the pneumatic throat formed by the reverse injection has been basically stable at about 59 ms.

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Fig. 7.5 Pressure–time curve (α = 30◦ )

The aforementioned fluidic-throat nozzle utilizing the axially lapped nozzle discloses the pressure regulation process of the fluidic throat and the main factors affecting the modulation time. The following is an example of a fluidic-throat nozzle of an SRM round-injector nozzle that is more approximate to the actual application. The variation law of the thrust increment obtained by the fluidic-throat nozzle is emphasized when the total pressure of the secondary flow increases.

7.4 Thrust Modulation for the Active Hole-Type Fluidic Throat 7.4.1 Characteristics of Mass Flow Rate Modification As mentioned in Chap. 1, the fluidic-throat nozzle can be used as a technical realization method of attitude control systems and the main power systems with the variable thrust, and can also be used as a flow regulation technology for a gas generator of solid propellants. The structure of an engine in the ground hot test is used as a calculation model. The main nozzle and gas parameters are shown in Table 7.1. Table 7.1 Parameters of the nozzle and the combustion gas Diameter of the throat

Expansion Expansion Convergence ratio semisemi-angle angle

Area ratio of secondary flow

Jet angle

Injector configuration

Specific heat ratio of gas

Total temperature of gas

9

3.5

15%

90°

Convergence

1.31

~3000 K

50°

16°

7.4 Thrust Modulation for the Active Hole-Type Fluidic Throat

179

Fig. 7.6 Variation curve of dimensionless mainstream flux with dimensionless total pressure of the secondary flow

The burning rate formula for the DB propellant used in the model is r = 1.6 × 10−6 P0.58 c . The equilibrium pressure of the combustion chamber is 2 MPa, when no secondary flow is injected. Figure 7.6 shows the main flux variation in the engine corresponding to a series of secondary flow pressures. The dimensionless reference parameter of the total pressure of the secondary flow in the figure is the equilibrium pressure value of the combustion chamber when there is no secondary flow injected, and the dimensionless reference flux of the mainstream flow is the mainstream flux when there is no secondary flow. It can be seen that as the total pressure of the secondary flow system continues to increase, the mainstream flow rate increases. If the dimensionless total pressure of the secondary flow is less than 1, the dimensionless flow rate of the mainstream flow increases slowly. For gas generators, the range of flow regulation is more interesting than the variation in thrust. If the secondary flow used is not an inert gas, certain gas that can participate in the combustion in the combustion chamber is connected to the gas generator (such as the case of a solid ram engine), the flux change at the outlet of the gas generator of the fluidic throat should include the flux of the secondary flow. In this way, the gas generator can have a larger modulation range of flow than the calculated value in the figure.

7.4.2 Thrust Modification Characteristic Figure 7.7 shows the increment curve of the flux ratio of the dimensionless chamber pressure generated by the SRM fluidic-throat engine with the flux (that is, the

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Fig. 7.7 Increment of the dimensionless thrust with steady flux ratio

secondary flux/mainstream flux after stabilization). It can be observed that the dimensionless chamber pressure is approximately linear increases with the flow rate ratio. In practical applications, a large increment of chamber pressure means that the maximum pressure that the engine shell is subjected to is large, and the mass of the shell is correspondingly increased. Therefore, it is desirable to obtain the largest possible variation range of thrust at the minimum increment of the chamber pressure. Figure 7.8 shows the corresponding increments of the dimensionless thrust, in which the total pressure ratio at steady state is a function of the dimensionless total pressure of the secondary flow. When the dimensionless total pressure of the secondary flow is Ps /P0,i ≥ 1, the total pressure ratio increases linearly to the dimensionless total pressure of the secondary flow. The thrust increment increases nonlinearly with the increase of the total pressure of the secondary flow. Normally, under the acceptable total pressure and mass flux of the secondary flow, the maximum dimensionless thrust increment of the gas–gas throat nozzle can reach more than three times, that is, the maximum thrust can reach more than four times of the initial thrust. The variation curve of the thrust increment with the total pressure ratio of the secondary flow after stabilization is parabolic, as shown in Fig. 7.9. When the total pressure ratio of secondary flow is Ps /P0,f ≤ 0.5, the thrust increment is basically 0; when the total pressure ratio of secondary flow is Ps /P0,f ≥ 1, the thrust increment starts to increase significantly. Generally, for the gas–gas fluidic-throat nozzle, when the total secondary flow pressure is selected, the fluidic-throat nozzle system will have obvious thrust modulation effect only if the total pressure ratio of the steady-state secondary flow of the system is greater than 1.

7.4 Thrust Modulation for the Active Hole-Type Fluidic Throat

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Fig. 7.8 Increment curve of dimensionless thrust with secondary total pressure

Fig. 7.9 Variation of thrust with pressure ratio

7.4.3 Filling Process Figure 7.10 shows the content distribution of the secondary flow in an engine model with a hole-type fluidic-throat nozzle at different moments before setting up a new equilibrium pressure. In the mass distribution map of the secondary flow, the process of the reaction between the secondary flow and the mainstream flow, the backstream, the filling, and the finally stable pneumatic throat can be clearly observed. The process of the pneumatic throat of the hole-type injector shown in the above figure is similar to that of the axially lapped injector in Fig. 7.3. The difference is that, due to the gap between the hole-type injector, at the initial injection stage of the secondary flow,

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Fig. 7.10 Distribution of mass content in the unsteady process

the fluidic throat cannot be completely closed like the axially lapped injector. As a result, part of combustion gas in the combustion chamber can still pass through the gap between the injectors and mixes with the secondary flow, and a small amount of gas leaks out of the nozzle.

References 1. Juntao, C., Zequn, J., Linquan, C.: Axisymmetric numerical analysis for ignition transient interior flow field of SRMs. J. Solid Rocket Technol. 27(3), 173–176 (2004) 2. Linquan, C., Xiao, H., Yanfang, L., Xingang, D.: Calculation of nozzle efficiency for solid rocket motor. J. Solid Rocket Technol. 25(4), 9–11 (2002) 3. Yadi, L., Linquan, C., Zequn, J.: Numerical study of two-phase nozzle flow with classified particles. J. Solid Rocket Technol. 26(3), 32–34 (2003)

Chapter 8

Erosion Characteristics of Fluidic Throat in Solid-Rocket Motors

Abstract The nozzle-throat insert of an engine is the most important part of the nozzle. In high-performance engines, the nozzle-throat insert cannot only resist highspeed and high-temperature (up to 3,000 K) gas with metal particles, but also produce extremely high-temperature gradient and thermal stress. In order to ensure pressure of the combustion chamber to guarantee the performance of engines, it is necessary to strictly control the ablation of the nozzle-throat insert. This chapter discusses the throat erosion of the fluidic-throat nozzle, and proposes an active thermal protection technology and the throat lining surface compensation technology based on the fluidic throat.

8.1 Material of Nozzle Lining The nozzle-throat insert of an engine is the most important part of the nozzle. In high-performance engines, the nozzle-throat insert cannot only resist high-speed and high-temperature (up to 3,000 K) gas with metal particles, but also produce extremely high-temperature gradient and thermal stress [1, 2]. In order to ensure pressure of the combustion chamber to guarantee the performance of engines, it is necessary to strictly control the ablation of the nozzle-throat insert. For solid rocket motors with high combustion temperature, high flow rate, and long working hours, the ablation of the nozzle-throat insert is particularly prominent. At present, ablation-resistant materials suitable for making nozzle-throat insert of solid rock engines mainly include graphite, carbon-carbon (C/C) composite, refractory metals, reinforced plastic, ceramic matrix composites, etc. C/C composite material is a multiphase pure carbon composite, which uses carbon fiber or fabric as prefabricated reinforcements [3–5], and is the composites of pyrolysis carbon or pitch carbon (resin carbon). In addition to being resistant to graphite ablation, having low thermal expansion coefficient and low density, C/C composites also overcome the shortcomings of low strength and thermal shock resistance of graphite materials. At present, C/C composites have evolved from being produced using a single ablative material to thermal structural materials and is been applied in a variety of solid rocket motors. The use of this material can greatly simplify the © Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_8

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nozzle-throat structure of solid rocket motors and improve the reliability and the total-impulse-to-mass ratio of the nozzle. The ablation characteristics of the fluidic nozzle-throat insert will be discussed below by taking this throat insert made of C/C composite material with good comprehensive performance as an example. Graphite and C/C composites have the same ablation mechanism (note that the tungsten–copper material is different). After the carbon-based throat insert is ablated by high-temperature gas, its inner surface is retracted (throat diameter is enlarged). The process can be divided into two parts according to the ablation mechanism [1–5]: one part is thermochemical ablation, during which material loss is caused by the reaction of the oxidized component in the gas and the wall surface; another part is particle erosion and mechanical ablation. These two parts are caused by the erosion of the wall surface by gas carrying metal oxide particles (e.g., Al2 O3 ) respectively. Previous studies have shown that the ablation of a nozzle-throat insert is mostly caused by thermochemical ablation [1–5]. Calculation of the thermochemical ablation rate of carbon-based nozzle-throat inserts will be described below.

8.2 Calculation Method for Predicting Erosion Rate of the Nozzle Throat in Solid Rocket Motors A nozzle-throat insert of solid rocket motors usually adopts a composite structure, typically involving two or three layers of materials. The thermal protection structure of a typical fluidic-throat spray insert is shown in Fig. 8.1a, where the throat insert is made of C/C composite material. Figure 8.1b shows the computational domain and the setting of grids and boundary conditions. The calculation area for ablation is divided into a throat-lined solid-phase region and a fluid region. The boundary conditions of the fluid area are described in Chap. 2. In the heat conduction calculation, the solid-phase zone only contains the part of the C/C throat insert, without insulation layer connected (such as high silicon oxynitride), and the outer boundary of the C/C throat insert is set as an adiabatic wall or a constant-temperature boundary. The precision of calculating the ablation rate of the throat insert has proven to be sufficient while adopting this boundary condition [1, 2]. The thermochemical ablation process of the nozzle-throat insert can be described by the N-S equation with chemical reaction source terms and heat conduction equations of solid-phase region. The N-S equation with the chemical reaction source term, the gas component transport equations, and the heat conduction equation of the solid-phase region can refer to the literature [3], and will not be described here. The convection term in the Reynolds average N-S equation can be set to a secondorder upwind scheme, and the source term adopts the central difference discrete method. The turbulence model generally uses a two-equation model, such as the k-ε model.

8.2 Calculation Method for Predicting Erosion Rate …

(a) Thermal protection for a fluidic nozzle-throat insert

185

(b) Computational domain and boundary conditions

Fig. 8.1 Calculation model for a fluidic nozzle-throat insert

8.2.1 Boundary Conditions at Gas–Solid Boundary In heat transfer calculation, the quality of the mesh near the wall of the nozzle-throat insert is important. Accuracy is guaranteed only when the mesh in the boundary layer is sufficiently dense. The y + value of the mesh can be used to verify the mesh quality [3]. In the ablation calculation of the nozzle-throat insert, generally y + < 2, which needs to be reassured after each calculation is converged. If y + > 2, further calculation will be required. The adaptive mesh is suggested to be used for local grid refinement (refer to Chap. 2), followed by the repetition of the above steps until it is satisfied. In particular, except for the nonslip boundary conditions, the gas/solid interface in the computational domain has to meet the following boundary conditions, due to mass and energy exchange at the interface: Mass conservation: ρ¯g u˜ r = ρc r˙c

(8.1)

Component conservation:   dYk ˜ + ρ¯g Yk u˜ r = ω¯˙ i −ρ¯g Dkm dr r

(8.2)

Energy conservation:  λc

∂T ∂r



 + r˙c ρc h c = λg

∂ T˜ ∂r

 + g

N  i=1

ω¯˙ i h˜ g,i

(8.3)

186

8 Erosion Characteristics of Fluidic Throat in Solid-Rocket Motors

where r˙c is ablation rate; ρc is material density in the solid-phase zone; ρ¯g is density of gas component in ablation; λ is thermal conductivity; h is specific enthalpy. Radiation heat transfer is not considered in the energy conservation above. In addition, suppose that the nozzle-throat insert’s solid-phase zone is at a given temperature: Tc = Tc,0 at the initial moment. If the outer boundary of the nozzle-throat insert adopts a wall boundary condition, then ∂∂rTc = 0.

8.2.2 Gas Composition Table 8.1 shows the gas composition and parameters at the inlet of a typical nozzle throat of solid rocket motors. It should be noted that generally, OH will still exist in the gas component. However, when pressure is high, it is so small that it is ignored in the ablation calculation of the nozzle-throat insert.

8.2.3 Chemical Reaction System The chemical reaction mechanism of ablation of the C/C nozzle-throat insert can be characterized by a two-element reaction system on the ablation boundary. Table 8.2 lists the values of kinetic coefficient of the chemical reaction in the heterogeneous reaction system. These parameters have been proven to be reliable by Chelliah and Lee [4].

Table 8.1 Gas composition and parameters at the pressure inlet boundary Component mass content/inlet parameter

Value

YH2 O

0.29

YCO2

0.22

YCO

0.11

YH2

3.4×10−4

YN2

0.10

YHCl

1.0 − (YH2 O + YCO2 + YCO + YH2 + YN2 )

Total temperature T o

3500 K

Total pressure Po

10 MPa

Initial temperature T c,0 of the nozzle-throat insert

300 K

8.2 Calculation Method for Predicting Erosion Rate … Table 8.2 Heterogeneous reaction system at the interface

187

Surface reaction

Ai

b

Ei (kal/mol)

C(s) + H2 O → CO + H2

4.8×105

0

68.8

C(s) + CO2 → 2CO

9.0×103

0

68.1

(1) Heterogeneous reaction at the gas–solid interface The equation of the chemical reaction rate in Table 8.2 is r˙i = ki pi0.5

(8.4)

where ki = Ai Tsb exp(−E i /R0 Ts ); R0 is the general gas constant. (2) Reaction in the gas phase The reaction in the gas phase only considers the following reversible reaction: Kf

CO + H2 O ⇔ CO2 + H2 Kb

The mass rate at which the components are generated is calculated by the following formulas: w˙ co2 = k f [CO]0.5 [H2 O] w˙ co = kb [H2 ]0.5 [CO2 ]

8.2.4 Calculation Methods of Physical Parameters of Gas The gas of SRM is a mixture, which contains various components, as shown in Table 8.1. In the calculation of the ablation flow field, it is necessary to know the physical properties of the gas, while these parameters are related to the composition of the gas. The attribute parameters of the mixed gas can be calculated by the following method, after obtaining the gas component and mass content. (1) Viscosity coefficient The viscosity coefficient of the mixed gas is μ = μl + μt , where μl and μt are the viscosity coefficients of laminar flow and turbulence of the mixed gas, respectively. μt is given by the turbulence model. The laminar viscosity coefficient μli of i is obtained using the Enskog–Chapman formula:  μli = 2.6693 × 10−6 Mi T /(σi2 i )

188

8 Erosion Characteristics of Fluidic Throat in Solid-Rocket Motors

where i = 1.147T ∗−0.145 + (T ∗ + 0.5)−2.0 T ∗ = T /(εi /k0 ) where εi k0 T∗ i σi Mi μi

characteristic energy between gas molecules; the Boltzmann constant; transition temperature; converted collision integrals; the diameter of the collision cross section of the gas molecule, in units of Å (Angstrom); the molar mass of the component gas i, in g/mol; the viscosity coefficient of the component gas i, in kg/m s. The viscosity coefficient of the mixed gas is μ=

ns  i=1

1+

1 xi

μi ns j=1 j=i

x j ϕi j

(8.5)

where

 M 1/4 2 1 + μμij Mij ϕi j =   8 1 + Mi /M j where xi is the mole fraction of the component i in the mixed gas. (2) Thermal conductivity coefficient According to the Enskog–Chapman formula, the heat transfer coefficient of a single gas molecule is λi =

R0 μi (0.45 + 1.32c pi /R0 ) Mi

where c pi is the constant-pressure specific heat of the component i in the mixed gas. For a mixed gas composed of a plurality of gas components, the heat transfer coefficient is calculated by the following formula: λ=

ns  i=1

λi  1 + 1.065 ns j=1 j=i

xj xi

ϕi j

(8.6)

8.2 Calculation Method for Predicting Erosion Rate …

189

(3) Diffusion coefficient The laminar diffusion coefficient Di j between the components i and j is given by the following formula: T

3/2



Di j = 1.858 × 10−7

1 Mi

+

1 Mj



pσi2j  D

where p in atm. The remaining physical quantities in the above formula are determined by the following formulas:  −0.2  D = T ∗−0.145 + T ∗ + 0.5  T ∗ = T / Tεi Tεj Tεi = εi /k0 σi j =

 1 σi + σ j 2

For multicomponent diffusion systems, the laminar diffusion coefficient of the mixed gas is calculated using the following formula: 1 − xi Dim =  xj

(8.7)

j=i Di j

8.3 Erosion of Traditional SRM Nozzles—The Borie Nozzle Model First, the ablation calculation results of the nozzle-throat insert of the conventional solid rocket motors are given, taking the Borie nozzle model as the object. The throat insert of the Borie nozzle is made of C/C composite [4], with the density of 1,900 kg/m3 , the thickness of 90 mm, and the diameter of 25.4 mm. The gas compositions and parameters of the nozzle inlet are shown in Table 8.3.

Table 8.3 Gas compositions at the inlet and parameters of Borie nozzle YH2 O

YCO2

YCO

YH2

YHCl

YAl2 O3

YN2

Pc /MPa

Tt /K

0.075

0.035

0.20

0.02

0.17

0.5

0.1

4.9

3390

190

8 Erosion Characteristics of Fluidic Throat in Solid-Rocket Motors

Fig. 8.2 Calculated ablation rate of the Borie engine and comparison with the experimental result

After calculating ablation of the nozzle by using the method in Sect. 7.2, computed and experimental results are shown in Fig. 8.2. It can be seen that the calculation of the predicted ablation amount with time is basically consistent with the experimental results, especially for enough time, these two results become much more similar. During the initial working hours, the predicted value differs greatly from the experimental measurement. It might because the temperature of the motor case wall rises fast with time initially in reality, while the computation method adopts the adiabatic or isothermal wall conditions. Although there is a certain difference between them, it is still accurate to use this model to predict the ablation of the carbon-based nozzle-throat insert of solid rocket motors, and this is enough for engineering design stage.

8.4 Erosion of FNT The following three cases are compared to understand the ablation change of the fluidic nozzle-throat insert made of C/C composite: (1) the ablation of the throat has a non-hole structure (or the injector is closed, so there is no secondary flow injected); (2) the secondary flow injected is a low-temperature inert gas; (3) the secondary flow injected is a high-temperature gas. In the following examples, the low-temperature inert secondary working flow is exemplified by nitrogen gas. If the secondary flow is high-temperature gas, it is assumed that its gas composition is the same as the mainstream (similar to the passive injection scheme, see Chap. 2), but the temperature is slightly lower. Since the pressure is a major parameter affecting the ablation rate and to make it more

8.4 Erosion of FNT

191

Fig. 8.3 Ablation of the ordinary throat

comparable, the total pressure, total temperature and gas component content at the inlet in the above three cases are given the same values, as shown in Table 8.3. Figure 8.3 shows the distribution of the steady-state ablation rate without secondary flow injection (ordinary nozzle-throat insert). The left side of the figure is the cloud map of the corresponding flow field. The upper half of the figure shows the static distribution of the temperature, and the lower half is the nephogram of corresponding Mach numbers. It can be seen from the figure that the isothermal lines of the solid-phase region are hierarchically distributed in a macroscopic manner. From the distribution of ablation rate on the right side, the maximum ablation rate occurs near the geometric throat, roughly corresponding to the position of the sonic line, and the ablation rate rapidly decreases in the wall area away from the throat, which is consistent with previous research results. Figure 8.6 shows the content distribution of CO2 and H2 O at a section in downstream of the geometry nozzle throat in three cases. By comparing Figs. 8.3 and 8.4, it can be seen that when the cold inert gas N2 is injected into the throat, the maximum ablation rate of the nozzle-throat insert is reduced (by about 22%, from 0.46 to 0.36 mm/s in the figure). The distribution of ablation rate along the wall of the nozzle-throat insert also changes, showing a sudden decrease in the ablation rate at the secondary flow injection and downstream. This is mainly because the fluidic throat formed by N2 changes the parameters of gas composition near the wall of the throat insert, and the CO2 and H2 O contents participating in the heterogeneous reaction in the near-wall region decrease, as shown in Fig. 8.6a, b, also changing the temperature of the wall and the pressure distribution near the wall (see the nephogram of flow field on the left side of Figs. 8.3 and 8.4). And after the secondary flow is injected, the position of the sonic line has moved to

192

8 Erosion Characteristics of Fluidic Throat in Solid-Rocket Motors

Fig. 8.4 Case of secondary flow of N2

Fig. 8.5 Case of the high-temperature secondary flow gas

the new fluidic throat, and the heat flux value at the original geometric throat, the maximum heat flux density, is lowered. A comparison of Figs. 8.5 and 8.6 reveal that when the secondary flow injected is a high-temperature gas, the maximum ablation rate of the nozzle-throat insert will still decrease slightly, but the ablation rate of the downstream fluidic throat will increase, and the distribution of ablation rate will be a double-peak structure. As it is analyzed previously, after the secondary flow is injected, the original sonic line moves to the fluidic throat, and the heat flux value of the front wall of the fluidic throat changes, which causes the maximum ablation rate at the original geometric throat to decrease. However, after the high-temperature gas is injected into the throat,

8.4 Erosion of FNT

(a) no secondary flow

193

(b) secondary flow is N2

(c) secondary flow is gas

Fig. 8.6 Mass fraction of CO2 and H2 O for different cases

the temperature at the injection point and the downstream near-wall region increases, and the parameters and pressure distribution of gas components change, resulting in an increase in the local ablation rate.

8.5 Active Thermal Protection Method Based on FNT At present, the thermal protection technology of solid rocket motors is basically passive. Figure 8.7 shows the ablation of a typical nozzle-throat insert made of C/C composite after a long period of operation (longer than 80 s), from which the nozzle is ablated severely and the throat profile has been significantly retreated. Figure 8.8 shows the experimental variation of chamber pressure and throat diameter for several typical engines after long-time operation. It can be seen that after operating for more than 20 s, the throat diameters of the three engines begin to become significantly larger, and the combustion chamber pressure is rapidly reduced. It shows that passive thermal protection is difficult to meet the overall index requirements under long working, and the ablation will be more significant in engines of high chamber pressure and high gas flux. Although the nozzle-throat insert with tungsten-coated copper material suffers a lower degree of ablation, the density of the material is large. Due to the limitation of the overall mass index, this material is hard to be used for producing nozzle-throat insert of solid rocket motors with a large thrust. From the ablation analysis of the fluid nozzle-throat insert above, if a lowtemperature secondary flow (gas or liquid) is injected near the throat, the ablation rate at the injection and downstream will be greatly reduced, which provides an idea of an active thermal protection scheme based on fluidic throats. That is, injecting a small amount of secondary flow in upstream of the nozzle-throat insert of solid rocket motors to form an isolation belt to reduce the ablation rate near the nozzlethroat insert, thereby ensuring the size of the throat to be steady for a long time. This solution is particularly suitable as an active technique for protecting nozzle-throat insert of large engines from ablation.

194

8 Erosion Characteristics of Fluidic Throat in Solid-Rocket Motors

(a) Nozzle profile before and after ablation

(b) Experimental photo

Fig. 8.7 Ablative carbonization after long-time operational engine test

(a) Engine 1

(b) Engine 2

(c) Engine 3

Fig. 8.8 Variations of pressure and throat diameter of typical engines with ablation time of nozzlethroat insert

In addition, Northwestern Polytechnical University of China has proposed an active thermal protection scheme for nozzle-throat insert of solid rocket motors. The solution uses circularly distributed spray of water [6] at the inlet of the converging section of the nozzle to alleviate the problem of throat ablation of large engines during long-time operation. Their experimental work proves that this active thermal protection technology can effectively reduce the ablation rate at the throat. The fluidic-throat solution idea here is similar; and the fluidic-throat technology can also realize the active compensation for the throat size, which can accurately maintain the throat area to be approximate to the design value, keep the thrust and chamber pressure stable, and then precisely control the thrust value, see Fig. 8.9. If it is only used to compensate retracted profile of throat ablation, without the need for thrust adjustment in a wide range, the mass flow rate of secondary flow required for the fluidic-throat nozzle will be significantly reduced.

References

195

Fig. 8.9 Schematic diagram of active thermal protection based on FNT

References 1. Kexiu, W.: Foundation of Composite Materials for Producing Solid-Propellant Rocket Engines. China Astronautic Publishing House, Beijing (1994) 2. Hongqing, H., Xu, Z.: Control mechanism of ablation in nozzle of solid-propellant rockets. J. Propul. Technol. 1993(4) 3. Thakre, P., Yang, V.: Graphite nozzle material erosion in solid-propellant rocket motors. AIAA, 2007-778 4. Borie, V., Brulard, J., Lengelle, G.: Aerothermochemical analysis of carbon-carbon nozzle regression in solid-propellant rocket motors. J. Propul. Power 5(6), 665–673 (1989) 5. Thakre, P., Yang, V.: Graphite nozzle material erosion in solid-propellant rocket motors. In: 45th AIAA Aerospace Sciences Meeting and Exhibit, AIAA 2007-778 6. Jinli, L., Guoqiang, H., Jiang, L., Peijin, L., Fei, Q.: Exploratory research on active cooling of SRM by means of liquid injection. J. Solid Rocket Technol. 31(3), 239–242 (2008)

Chapter 9

Nozzle Damping of the Fluidic Nozzle

Abstract For a thrust-adjustable rocket motor, to avoid unstable combustion, we not only need to consider nozzle damping before thrust adjustment, but also need to consider nozzle damping variation thereafter, and it is important to learn about the influence of nozzle damping before, and after, thrust adjustment. This chapter discusses wave attenuation method for predicting nozzle damping of the fluidic nozzle, validation of numerical prediction method for nozzle damping, and gives the nozzle damping variation with different secondary flow parameters.

At present, the traditional fixed-thrust rocket motor, in order to increase nozzle damping, suppresses unstable combustion, and the nozzle geometry will be taken fully considered at the beginning of motor design. For a thrust-adjustable rocket motor, to avoid unstable combustion, we not only need to consider nozzle damping before thrust adjustment, but also need to consider nozzle damping variation thereafter, and it is important to learn about the influence of nozzle damping before, and after, thrust adjustment.

9.1 Wave Attenuation Method for Predicting Nozzle Damping of the Fluidic Nozzle The given nozzle geometry without secondary injection is the same as that in the research by Buffum [1]. The computational zones generally consist of a combustion chamber, a convergent–divergent nozzle, and secondary injection area. The inlets of the primary and secondary flows are set to be the mass flow rate inlet. The experimental data under a pressure of 137.9 kPa and a temperature of 300 K are adopted as an example condition of the primary flow. At the exit, all conditions are extrapolated from their counterparts within the nozzle. At the walls, no-slip conditions are applied along with an adiabatic wall and zero normal pressure gradient. As shown in Fig. 9.1, the chamber length L is 304.8 mm, with a port radius Dc and nozzle-throat radius Dt (listed in Table 9.1).

© Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_9

197

198

9 Nozzle Damping of the Fluidic Nozzle

Fig. 9.1 Computational model and boundary conditions Table 9.1 Experimental cases in the study of Buffum [1]

Case no.

Dc (mm)

Dt (mm)

J = (Dt /Dc )2

1

25.4

12.7

0.250

2

38.1

12.7

0.111

3

50.8

12.7

0.0625

4

63.5

12.7

0.0400

5

76.2

12.7

0.0278

6

88.9

12.7

0.0204

In Table 9.1, J is the ratio of the nozzle-throat area to the grain port area. For the sake of comparison, all cases with secondary flow injection have the same boundary conditions of primary flow. Without the consideration of combustion kinetics, the thermodynamic properties are assumed constant. The gas is assumed ideal. In a solid rocket engine, when the engine is impulsed by excitation, the pressure p in the combustion chamber can be divided into the mean pressure p¯ and the pressure fluctuation p . In this case, the pressure fluctuation p satisfies the one-dimensional wave equation. Then, the combustion chamber pressure p can be expressed as follows: p = p¯ + p  = p¯ + p0 eiωt+αt

(9.1)

where p0 is initial pressure amplitude, ω is the angular frequency, and α is growth constant of sound energy. The value of coefficient α determines whether the oscillation will grow or decay with time. If coefficient α > 0, the amplitude of the small oscillations will increase with time and the motor is unstable. If coefficient α < 0, the amplitude of the small oscillations will decrease with time and the motor is stable. The coefficient α contains the contributions of the various relevant processes, and it can be expressed as follows: α = αpc + αvc + αdc + αn + αp + αmf + αg + αw + αst

(9.2)

where, α pc , α vc , and α dc are the driving terms due to pressure coupling, velocity coupling, and distributed combustion, respectively; α n , α p , α mf , and α st are nozzle

9.1 Wave Attenuation Method for Predicting Nozzle …

199

damping, particle damping, mean flow interactions, and structural damping, respectively; and α g and α w are dissipation damping coefficients of gas phase and wall, respectively. If there is no other source of acoustic energy present in the chamber and the acoustic energy of consumption due to viscous effects is ignored, the acoustic energy is not affected by other factors than the nozzle itself. On this condition, the nozzle is the only factor that can affect the growth or decay of a pressure oscillation inside the combustion chamber. The temporal behavior of the combustion chamber oscillation can be expressed in the following form: p  = p0 eiωt+αn t

(9.3)

According to Eq. (9.3), the pressure oscillation decays exponentially, and therefore the nozzle damping decay coefficient can be obtained by plotting the peak-to-peak amplitude–time curve in a logarithmic-time coordinate system. By fitting the line to the plot and calculating its slope, the decay constant will be determined as given below: αn =

ln p2 − ln p1 t2 − t1

(9.4)

In the numerical calculation (see Chap. 2), for an example, the steady calculation is first completed to obtain a convergent solution. Then, the unsteady calculation process begins. When a good convergent result is obtained once again, a periodic oscillation of mass flow rate with frequency equaling the first acoustic mode is superimposed on a steady flow at the inlet. After that, the mass flow rate returns to its original value and meanwhile, the chamber pressure decays. A typical decay process of the fluctuating pressure is demonstrated in Fig. 9.2a (Case 1 in Table 9.1), and the attenuation constant after decay of pressure oscillation can be obtained by plotting the peak-to-peak amplitude–time curve in a logarithmic-time coordinate system, as shown in Fig. 9.2b. The slope of the curve in Fig. 9.2b is the attenuation constant α n , based on Eq. (9.4).

9.2 Validation of Numerical Prediction Method for Nozzle Damping To confirm the rationality and effectiveness of the above numerical methods, a theoretically evaluated formula is adopted [2] as supported by many different cold-flow experiments and widespread recognition, which is given below: αn = −λ1

a J L

(9.5)

200

9 Nozzle Damping of the Fluidic Nozzle

(a) Fluctuating pressure, Case 1

(b) Ln p´ - t curve after decay of pressure oscillation, Case 1

Fig. 9.2 A typical example of data processing

where a is the actual speed of sound in the combustion chamber of a motor, L is the length of the closed tube, J is the ratio of the nozzle-throat area to the grain port area, and λ1 is a correction factor, which is weighted with different coefficients while using different experimental methods, and is 0.69 in accordance with data obtained in cold-flow test [3]. Six cases, in Table 9.1, having the same diameter as that in the research by Buffum [1] are applied to access the reliability of the numerical method. Based on the steadystate decay method, numerical simulations are carried out. Experimental, calculated, and empirical results of attenuation constants under different J of Cases 1–6 are shown in Fig. 9.3. From Fig. 9.3, it can be found that the attenuation constants of experimental values are higher than the numerical and theoretical counterparts and there are differences between them. This is because the experimental results not only incorporate the Fig. 9.3 Comparison of the attenuation constants from three different methods, Cases 1–6

9.2 Validation of Numerical Prediction Method for Nozzle Damping

201

effects of damping factor of the nozzle but also external damping factors, such as viscosity loss of wall and absorption of acoustic energy by porous media at the inlet, which makes the coefficient of Eq. (9.3) much larger than that applicable to the numerical simulation and actual engineering experience. Although there are discrepancies between the experimental results and numerical results, the trends of the numerical and experimental values are in agreement with empirical data. Both the experimental and numerical results have shown good linear characteristics: with increasing J, the absolute value of nozzle attenuation constant increases linearly. In summary, the numerical method is able to provide an efficient, convenient way to evaluate the characteristics of nozzle damping of solid rocket motors.

9.3 Effect of the Secondary Flow Injection on Nozzle Damping To simplify the analysis, a vertical injection theoretical model for the Cd of FNT is introduced first. After the secondary flow is injected vertically into the throat of the nozzle, the typical steady-state flow field is shown in Fig. 9.4. Before deriving the vertical injection model of the FNT, the following assumptions are made. The gas in the combustion chamber is an ideal gas, which satisfies the state equation of the ideal gas. The primary gas reaches a stagnation state on the upstream edge of the inlet. The secondary flow reaches its maximum penetration depth before it enters the expansion section of the nozzle. At the maximum penetration depth, the pressure of the primary flow is the same as that of the secondary flow, and the velocity reaches the speed of sound. In the nozzle, the section from the upstream edge to the aerodynamic throat (position 1–2 in Fig. 9.4, where the secondary flow reaches its maximum penetration depth) is selected as the control body. From position 1–2, the momentum equation of the secondary flow in the x-direction is given below: ( p1 − p2 )(At − A˜ t ) = m˙ s (vs2 − vs1 )

Fig. 9.4 Schematic diagram of the vertically injected secondary flow

(9.6)

202

9 Nozzle Damping of the Fluidic Nozzle

At position 1, the axial velocity of the primary flow is zero, so pressure p1 = po0 , and the axial velocity of the secondary flow is also zero, that is, vs1 = 0. At position 2, the primary flow regains its previous sonic speed, and the primary flow pressure is the same as that of the secondary flow: p2 =

∗ po0

 =

2 γo + 1

o  γ γ−1 o

po0

(9.7)

Similarly, velocity of the secondary flow at position 2 is given below: vs2 =

as∗

 = γs Rs Ts2 =

 2γs R0 Ts0 γs + 1 μs

(9.8)

The total pressure of the primary gas flow can be given by the following equation: po0

m˙ o = ˜ A t o



R0 To w˙ o  = R0 μo A˜ t o

(9.9)

By substituting this into Eq. (9.6), one can obtain the following: 

At −1 A˜ t

   o  γ γ−1 √  o R0 2γs R0 2 w˙ o 1 − = w˙ s o γo + 1 γs + 1

(9.10)

Then, Cd = 1 + o



1 −1 o

 γ γ−1 o 2γs 2 1 − fw γs +1 γo +1

(9.11)

Let  λ 0 = o

−1  o  γ γ−1  o 2γs 2 1− γs + 1 γo + 1

(9.12)

where λ0 is a constant associated with primary and secondary flows, so this can be simplified as Cd =

1 1 + λ0 f w

(9.13)

Figure 9.5 shows a comparison of the ratio of effective throat area calculated using the theoretical method with the CFD simulation results and previous experimental data [4]. It can be seen from Fig. 9.5 that the theoretical results are in good agreement

9.3 Effect of the Secondary Flow Injection on Nozzle Damping

203

Fig. 9.5 Comparison of theoretical results with numerical and experimental results

with the numerical and experimental results, which verifies that the FNT theoretical model is reliable when it is used to analyze the choking performance of the FNT vertical injection conditions. During operation of a solid rocket motor, after injection of the secondary flow, the aerodynamic throat is formed, as expressed by Eq. (2.1): simultaneous solution with Eq. (2.4) gives the actual ratio of throat area of the primary flow to grain port area, which is given below: A˜ t = J Cd J˜ = Ac

(9.14)

At the same time, according to the theoretical Eq. (9.13) for the ratio of effective throat area and the ratio of modified mass flow at vertical injection, the relationship between the actual ratio of the throat area of primary flow to grain port area and the ratio of the modified mass flow rate can be obtained as follows: J˜ =

J 1 + λ0 f w

(9.15)

According to the aforementioned data processing method, the pressure decay curve after the mass flow rate returns to its original value and the attenuation constant under different ratios of modified mass flow rate is shown in Fig. 9.6. Figure 9.6a shows a comparison of the theoretical and numerical results of the actual ratio of throat area of the primary flow to grain port area with the ratio of modified mass flow rate. Figure 9.6b shows the variation of the nozzle damping as the actual ratio of throat area of the primary flow to grain port area changes. From Fig. 9.6a, after injecting the secondary flow into the nozzle to adjust the thrust, the effective throat area of the nozzle gradually decreases as the ratio of the modified mass flow rate increases, which results in a decrease in the ratio J. In

204

9 Nozzle Damping of the Fluidic Nozzle

(a) The actual ratio of throat area of the primary flow to grain port area

(b) Nozzle attenuation constant

Fig. 9.6 The effects of different ratios of modified mass flow rate

addition, it can be observed from Fig. 9.6a that the numerical results are in good agreement with the theoretical results. As the ratio J decreases, the fluidic nozzle damping exhibits slightly decreasing damping and it exhibits linear variation afterward (Fig. 9.6b). Meanwhile, the following formula can be obtained for the linear fitting of the nozzle damping (Fig. 9.6b):



 a (9.16) αn = λ2 J˜ − J0 + α0 = λ2 J˜ − J0 − λ1 J0 L where λ3 is the constant associated with primary and secondary flows, J 0 and α 0 are the ratio of nozzle-throat area to grain port area and nozzle dumping, respectively, when there is no secondary flow injected. Combining Eqs. (9.15) and (9.16), the relationship between the nozzle damping and the ratio of the modified mass flow rate can be obtained as follows: αn =



a λ2 J0 − λ1 + λ2 J0 1 + λ0 f w L

(9.17)

From Fig. 9.6, it can be observed that when the ratio of the modified mass flow rate of secondary flow is less than 0.5, the nozzle damping changes within 8%. It means that using the FNT method to adjust the engine thrust will slightly reduce nozzle damping. Compared to traditional mechanical methods used to change nozzle-throat area and thrust (e.g., using a moving pintle to adjust the throat area), the reduction in geometric throat area and the ratio of nozzle-throat area to grain port area will cause a significant decrease in nozzle damping (Figs. 9.3 and 9.6b) without injection of the secondary flow.

9.3 Effect of the Secondary Flow Injection on Nozzle Damping

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Therefore, when designing a solid rocket motor of adjustable thrust by regulating throat area in real time, taking into account the suppression of unsteady combustion of the engine, FNT method has a smaller reduction of nozzle damping, which is more advantageous than the mechanical method.

9.4 Effect of Temperature of the Secondary Flow on Nozzle Damping In actual operations, fluid consumption of the secondary flow will pose a significant problem. To reduce the consumption of secondary fluid flow, it is necessary to increase the temperature of the secondary flow (see Chaps. 2 and 3). This section studies the effect of the temperature of the secondary flow on nozzle damping. Four different temperatures within the range from 300 to 2,100 K, in increment of 600 K, are selected as an example, and the attenuation constants under different gas temperatures in Case 5 are illustrated in Fig. 9.7a and b, respectively. According to Eqs. (2.6) and (2.7), the ratio of the modified mass flow rate is used to unify the conditions of different jet temperatures. That is, the ratio J, throat area of primary flow to grain port area, at different jet temperatures can be also calculated according to Eq. (9.15). Figure 9.7 shows that at different jet temperatures, the fluidic nozzle damping continues to decrease in a quasi-linear manner as the ratio J decreases, as shown in Eq. (9.17). Therefore, at the same ratio of the modified mass flow rate, different temperatures of the injected secondary flow do not have an obvious effect on nozzle damping,

Fig. 9.7 The effects of different temperatures of the secondary flow

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9 Nozzle Damping of the Fluidic Nozzle

that is to say, Eqs. (2.6), (9.15), and (9.17) can be extended to working conditions at different jet temperatures.

References 1. Buffum, J.F.G., Dehority, G.L., Price, E.W., et al.: Acoustic attenuation experiments on subscale, cold-flow rocket motors. AIAA J. 5(2), 272–280 (1967) 2. Janardan, B.A., Zinn, B.T.: Rocket nozzle damping characteristics measured using different experimental techniques. AIAA J. 15(3), 442–444 (1977) 3. Zinn, B.: Nozzle damping in solid rocket instabilities. AIAA J. 11(11), 1492–1497 (1973) 4. Guo, C.C., Wei, Z.J., Xie, K., et al.: Thrust control by fluidic injection in solid rocket motors. J. Propul. Power 33(4), 815–829 (2017)

Chapter 10

System Application Modes and Key Technologies

Abstract From the general perspective of the propulsion system, this chapter introduces the feasible scheme and working mode of the FNT for the attitude and orbit control system and main power propulsion system. Since FNT technology is an emerging technology which is still in the development stage, characteristics of some typical solutions and ways to improve the application value of the project and the improvement measures are mainly introduced. Moreover, the basic performance requirements to meet the actual application requirements are also discussed. Typical thermal experimental results are given at the end of this chapter.

10.1 System Types The fluidic-throat nozzle system is generally classified according to the task requirements of the application object and the working performance of the secondary flow. Because of different mission requirements, the system generally has different forms and indicator requirements. In addition, apart from its volume and weight, the secondary flow system and the characteristics of the secondary flow also determine suitable application objects, and therefore need to be discussed separately.

10.1.1 ACS System The advanced representative of the thrust-adjustable attitude and orbit control system is the attitude and orbit control system of solid rocket motors on the kinetic kill vehicle (KKV) of the US “Standard” series of missiles. The first attitude and orbit control system for “Standard” missiles—SDACS—consists of six attitude control engines and four orbit control engines. It adopts HTPB/AP propellant weighing 4.5 kg and an engine case fabricated with graphite/epoxy, and working time is 20 s. The propellant grains are arranged concentrically around the central tube made of C/C composite, and the central tube transfers the gas to the valve device. The hot gas valve is the core technology of SDACS, working at the opening frequency of 2,000 Hz and © Springer Nature Singapore Pte Ltd. and National Defense Industry Press 2019 K. Xie et al., Fluidic Nozzle Throats in Solid Rocket Motors, https://doi.org/10.1007/978-981-13-6439-6_10

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response time less than 1 ms. The throat area of each nozzle in the first-generation system cannot be changed, and the thrust generated by each nozzle is substantially unchanged. Subsequently, a new generation of attitude and orbit control system, i.e., TDACS with a solid gas generator, with which the thrust can be adjusted between high thrust and endurance thrust, is adopted. It includes ten balanced pintle thrusters (four for orbit control and six for attitude control), and can change the throat area of each nozzle by adjusting the position of the pintle, thereby changing the working pressure and thrust of the gas generator. Figures 10.1 and 10.2 show a 6-degree-of-freedom attitude and orbit control system concept using fluidic-throat nozzle technology. The system uses FNT technology in the sonic nozzle throat, at the bottom of the central gas conduit and in front of throttle and the attitude and orbit control nozzles, to regulate the gas flow rate and chamber pressure, and further realize the adjustment between high thrust and endurance thrust. The attitude stabilization and trimming of the pintle-controlled solid rocket motor system have to rely on the attitude control system, as mass center of the engine changes with the consumption of the propellant. Even if the thrust of the orbital control nozzle will pass the initial centroid of the system, the thrust of the orbit control engine will generate torque as the centroid moves during operation. At this time, the torque needs to be balanced by the attitude control motor to prevent the system from rolling. One of the benefits of the fluidic-throat system is that the number and layout of the tanks can be reasonably designed at the beginning of the design to ensure minimum centroid changes during operation, or by dynamic mass trimming during operation. Taking the system in Fig. 10.2 as an example, when the mass center of the engine deviates to the right, the secondary flow in the right tank can be drained to move the mass center to the left, back to original position.

Fig. 10.1 Typical FNT—ACS

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Fig. 10.2 System profile

Fig. 10.3 Adjustment modes of the thrust

The working mode of the TDACS system is shown in Fig. 10.3: (1) Zero thrust. When the four nozzles are fully open, the throat area of the nozzles is the largest and the pressure in the combustion chamber is the smallest. (2) Full thrust. When one nozzle is open and the throat area is the smallest, the remaining three nozzles are closed. (3) Off-axis thrust. Both the nozzle 1 and the nozzle 2 are open. The magnitude and direction of the thrust depend on the opening of pintles of the two nozzles (i.e., the throat area of the nozzles). The working mode of the FNT attitude and orbit control system is similar to the TDACS system, but the implementation and system structure are not completely consistent. In mode (1), the secondary flow is not injected in the fluidic-throat sonic nozzle, when the chamber pressure and flow rate of the gas generator are minimized; for mode (2), a secondary flow is injected into the fluidic-throat sonic nozzle, the sonic nozzle has the smallest throat area, and the gas generator chamber pressure and flow rate are maximum at this time. To achieve mode (3), the fluidic-throat attitude and orbit control nozzles still require a high-temperature gas valve or a secondary flow valve to open and close the main nozzle. For systems with simpler structural requirements, one-time nozzle cover can be used instead of high-temperature gas valve. In mode (3), the required thrust direction

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can be controlled by opening the plugs and changing the time intervals and the average thrust size [1]. In addition, passive solutions can be used to further simplify the fluidic-throat system described above. The secondary flow also generates thrust together with the gas produced by the gas generator and the system mass of the secondary flow gradually decreases during operation. Despite this, compared with the pintle-based attitude and orbit control engine, the fluidic-throat attitude and orbit control system still requires additional supply systems and tanks for the secondary flow, which will cause difficulties in miniaturization and simplified construction of the system. Thus, the fluidic-throat attitude and orbit control system in Fig. 10.2 is particularly suitable for improving systems that have used high-pressure gas-adjustment systems, such as certain satellite systems that have used nitrogen attitude controlled engines, or manned return cabin systems (the original oxygen in the system is available for secondary flow), landing cabin system, etc. Thus, the secondary flow used in the fluidic-throat sonic nozzle can be shared with the original attitude control system, including sharing existing supply systems and the storage tank. This can improve the performance of the original attitude control system, without adding too many extra components. Figure 10.4 is a schematic diagram of the fluidic-throat attitude control system when the return or landing cabin is landing. When the thrusts of the attitude control engines in left and right are asymmetrical, a corresponding attitude adjustment torque will be generated to ensure the attitude accuracy at the time of landing. For this fluidicthroat attitude control system, there are no many additional system components of secondary flow required, and it is not necessary to develop the pintle and the gas valve that requires high anti-ablative and dynamic sealing properties. In addition, since the fluidic throat can achieve thrust vector control on a single engine, lesser units are needed to achieve three-axis control of aircrafts. For example, the differential thrust of the two symmetrical elements in Fig. 10.3 can produce a rolling moment about its own axis. The idea of using the fluidic-throat technology to control the flow rate of the gas generator can also be applied in solid ramjet engines to change the fuel flow rate and further adjust the thrust. Moreover, other fuel or oxidant can also be used as the secondary flow, entering the afterburning chamber of the ramjet engine with the gas generated by the solid gas generator together for secondary combustion, thereby further improving the thrust adjustment range of the ramjet engine.

Fig. 10.4 Return cabin system using fluidic-throat ACS

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10.1.2 Main Propulsion System In addition to the attitude control system, the fluidic-throat nozzle can also be used in the main power system of the missile. Figure 10.5 shows a fluidic-throat power system intended to be used on ground-launched rocket. The nozzle of the engine is a long tailpipe, and the secondary flow tank is arranged at the tail of the engine, so that the tank can be closest to the secondary injection point on the fluidic-throat nozzle, and the conveying pipe of the secondary flow can be the shortest. With the eight injector units set on the throat and the expansion section of the fluidic-throat nozzle, the system can realize the control of thrust magnitude and the SVC of the secondary flow.

Fig. 10.5 SRM fluidic-throat engine system as the main power

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There are two purposes of using the long tailpipe; one is to easily adjust the centroid of the whole missile and the other is to leave enough space for arranging the secondary flow tank within the small tactical missile. By applying vector control technology in the fluidic throat of solid rocket motor, the pneumatic control system of the missile can be simplified (only retaining the roll control rudder), or the air rudder control channels and equipments can be completely eliminated, which will reduce the number of moving parts and the mass of systems, and further improve the whole reliability. In addition, design and research process of the aerodynamic configuration will be greatly simplified after the airfoil is set as a fixed wing or omitted. As the main power, the fluidic-throat system can draw on some design experience and technology of many existing system schemes, such as the existing shock-induced secondary flow system on solid rocket engines and the solid–liquid engine system. And also the structure of the secondary flow tank of these systems, the layout of the tank, the connection structure of the secondary injector and the main nozzle, the thermal protection structure, and the valve components can all be used for reference. For smaller tactical missiles, the fluidic-throat system can be simpler, such as adopting passive systems. Although the passive system can only achieve a limited range of thrust adjustment, it can eliminate the supply system and the tank of the secondary flow, so it can serve as a propulsion system for small tactical missiles, such as air-to-air missiles. The internal ballistics of a missile can generally be divided into two parts: the booster section and the active cruising section. In order to enable the missile to quickly obtain the initial speed, the thrust and overload of the booster section are relatively large, and the endurance section generally only needs to maintain certain speed and ensure sufficient motive power, so the thrust requirements of the two sections are quite different. Sometimes it is necessary to design a separate booster engine that will be separated from the missile body after work. However, in order to make the structure more compact and increase the number of units that the weapon system can carry, it is desirable to design an advanced solution that integrates the booster engine and the endurance engine. At present, there are two main methods for realizing this scheme: one is to adopt the single-chamber double-thrust engine, only redesign the burning surface of propellant grains and use the same nozzle to satisfy the thrust requirements of both the booster section and the endurance section, respectively. However, sometimes, the required thrust ratio of the boost/cruise cannot be met with only designing profile of propellant grain, under the constraint of the outer diameter of the projectiles and the requirement of grain’s structural integrity. Therefore, another method is proposed using double nozzles or even two kinds of propulsion. For example, if the booster section uses a small-throat nozzle and after the booster is burned, the booster nozzle will be blown out, leaving the large-throat nozzle and the endurance cartridge to continue working. By using fluidic-throat technology, different boost/cruise thrust ratios can be achieved with maximum range and flexibility in the case of a fixed single nozzle. The fluidic throat of solid rocket motors can be designed as a single-chamber double-thrust combustor structure, which can achieve a certain thrust through the ini-

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tial profile design of the propellant grain or using different propellants, even without throat adjustment. The internal ballistics without throat adjustment are called the reference internal ballistics. The fluidic-throat engine can introduce throat adjustment during the booster phase based on actual operational needs, so that a higher boost thrust and a required boost/endurance thrust ratio can be obtained in the boost section compared to the reference trajectory. Compared with the previous two solutions, this one has a smaller constraint on the overall size, an achievable high boost/endurance thrust ratio, and requires only one fixed nozzle. In addition, fluidic throat of solid rocket motors can also introduce the magnitude and direction adjustment of the thrust in the endurance section according to the specific flight path of the incoming target, making the ballistics “flexible”, thereby adapting to a wider mission profile. The flexibility of changing the ballistics can be achieved by simply adjusting the binding control parameters without changing or replacing the hardware configuration, which can increase the battlefield adaptability and response speed of the weapon. The fluidic-throat engine provides an effective technical approach for the control system to achieve flexible energy management.

10.1.3 System Types of Different Work Media The fluidic-throat system can be divided into two types according to the property of secondary fluid carried, gas–liquid fluidic-throat system and gas–gas fluidic-throat system, see Fig. 10.6.

Fig. 10.6 Scheme of gas–liquid fluidic-throat and gas–gas fluidic-throat engine systems

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I. Gas–liquid fluidic-throat nozzle system The obvious benefit of using a liquid medium is that the tank is small in size and can carry more secondary flow medium under the same volume requirement, allowing a larger secondary flow to increase adjustment range of throat area. At present, the secondary flows used in solid rocket motors are generally oxidizing, so that the liquid refrigerant will react with the gas after it is ejected into the nozzle. The heat generated by the secondary combustion will also have a “hot congestion” effect, and the interception effect will also increase. In addition, unlike gas–liquid fluidic-throat systems, the startup and shutdown of the gas–liquid fluidic throat are simpler. For a gas–gas fluidic throat with using a solid gas generator, once the solid gas generator is activated, it will continue to work and generally cannot stop midway. Therefore, the fluidic-throat system using the liquid medium as the secondary flow is generally given priority when it requires long working time, small effective missile volume, and wide throat adjustment range. II. Gas–gas fluidic-throat nozzle system In the case where it only requires the end correction of the ballistics, the instantaneous large thrust or lateral force, and the short thrust adjustment time, the gas–gas fluidicthroat system may be preferentially considered. In addition, as described above, for the system originally containing high-pressure nitrogen and oxygen, the fluidic-throat attitude and orbit control system—using the gas medium as the secondary flow is also suitable. The gaseous secondary flow can also use the gas generated by the solid gas generator, and the gas generator scheme can reduce the volume of the fluidic-throat system; however, the heat protection structure of the conveying pipeline of the secondary flow has higher requirements, and switching control also does not have the flexibility of an air adjustment system. A feasible mode of operation is shown in Fig. 10.7. Once the solid gas generator is started, the secondary flow will be injected into the nozzle when a secondary flow is required to form a fluidic throat or vector control, and the secondary flow will be symmetrically emitted when no thrust is required for extensive adjustment. The conversion of the secondary flow passage will be completed by the high-temperature reversing valve.

10.2 Typical Engine Structure A typical fluidic-throat nozzle engine structure and the secondary flow gas generator used above are shown in Fig. 10.8a and b, respectively. The fluidic-throat nozzle engine mainly consists of a combustion chamber shell, a propellant grain, a heatinsulating layer, a fluidic-throat nozzle, a convergent section of the combustionresistant layer, a plenum insulation layer of the secondary flow, a plenum housing of the secondary flow, a pressure plate, and a one-way valve. For the solid rocket motor for a tactical missile with a small diameter, the plenum housing of the secondary

10.2 Typical Engine Structure

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Fig. 10.7 “Work” and “Off” modes of a gas–gas fluidic throat

flow and the combustion chamber may be screwed, and the convergent section of the combustion-resistant layer is fixed by the plenum housing of secondary flow. The gland is connected to the plenum housing of secondary flow and able to fix the nozzle-throat insert. The plenum housing opening is welded to the adapter and connected to the secondary flow source. Since the secondary flow generally does not work immediately after the engine is ignited, the mainstream will flow back into the concentrating chamber of the secondary flow. To prevent backflow, a check valve is added between the fitting and the air supply. There are two kinds of seals on the engine, the O-ring seal is used for the joint of the parts and high-temperature putty is applied in the contact surfaces between the throat insert, the converging section, and the combustion layer. The grain configuration is also selectable according to different overall thrust requirements. Whether it is a gas–liquid fluidic-throat or a gas–gas fluidic-throat nozzle system, it is necessary to use a solid gas generator to generate high-pressure gas, to pressurize the secondary liquid flow or directly serve as a secondary gas flow. Therefore, the solid gas generator of secondary flow is one of the most important key technologies in fluidic-throat nozzle of solid rocket motors. Section 10.4 will further describe the structure of the gas generator of secondary flow of solid rocket motors and the key technologies involved.

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(a)

(b)

1-combustion chamber housing; 2-thermal insulation layer; 3-propellant grain; 4-convergent section of combustion resistant layer; 5-sealing ring; 6-plenum insulation layer of the secondary flow; 7-check valve; 8-plenum housing of the secondary flow; 9-pressure ring; 10-secondary flow plenum; 11-fluidic nozzle-throat insert

Fig. 10.8 Structure of a typical fluidic-throat engine

10.3 Injector Structure Figure 10.9 shows a typical nozzle configuration commonly used in secondary flow injection systems on solid rocket motors. Figure 10.9a shows a secondary flow injector using a liquid working medium, and Fig. 10.9b shows a secondary flow injector using a gaseous working fluid. A valve can also be installed on the injector to control the injection and blocking of the secondary flow. In addition, it is noted that the injectors used on solid rocket motors, especially those used near the throat, are slightly different from the injectors in the head of the liquid rocket engine. Due to the thickness requirements of the thermal protection structure of the solid rocket motor, the injector depth at the throat is large.

10.4 Solid Generator of Secondary Flow

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(a) Secondary flow system and injector using a liquid medium

(b) Secondary flow system and injector using a gaseous medium Fig. 10.9 Injectors used on a secondary flow system of a solid rocket motor

10.4 Solid Generator of Secondary Flow In addition to the characteristics of a conventional solid rocket engine, the solid gas generator used in the fluidic-throat nozzle has additional four key technologies: gas temperature control, ignition reliability, gas cleanliness, and pressure stability control. I. Ignition reliability Ignition reliability is also a practical problem that needs to be solved by solid gas generators used on fluidic-throat nozzles mainly in the following aspects: (1) Ignition difficulties Due to the limitation of the gas temperature, the propellant of low burning temperature has great difficulty in ignition due to the large content of inert additives, and it frequently cannot be ignited normally. The problem will become much more serious for propellant grains of low initial temperature. (2) The pressure in the combustion chamber rises slowly In addition, the low temperature of the propellant will cause the pressure in the combustion chamber to rise slowly and the phenomenon of “climbing hill” to occur. The lower the ambient temperature is, the more serious the climbing phenomenon. But for thrust adjustment, the time taken is an important indicator. If the initial thrust build-up time of the secondary flow is too long, the thrust adjustment time

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will increase. Therefore, it is necessary to take effective measures during the ignition phase to minimize the climbing pressure and ensure the build-up time of initial pressure. (3) Ignition shock One solution to problems of ignition difficulties and serious climbing is to increase the amount of ignition grains, but this usually results in excessive pulses of ignition pressure and excessive ignition shock. For secondary gas systems that not adopt pressure stabilized solution, the excessive ignition shock means sudden increase and drop of the total pressure of the secondary flow at the beginning of the work, which will affect the stability of fluidic throat and the accuracy of the thrust control. II. Gas temperature control In practical applications, if the temperature of the secondary gas flow is too high; on the one hand, the secondary gas pipe, throat insert, and other parts will be ablated severely, and overheating and even burn-through problems are very likely to occur; on the other hand, the high-temperature gas will cause deformation of the parts, and then the gas will not be smoothly emitted, which will further increase the pressure in the combustion chamber of the solid gas generator and cause an explosion accident. Therefore, temperature of the gas generated by the solid gas generator needs to be controlled. III. Gas cleanliness The installations of a solid rocket engine generally do not require strict requirement for gas cleanliness, but as a secondary solid gas source, there is a high demand for gas cleanliness. During the combustion of solid propellant, a large amount of solid particles are generated, together with debris and impurities generated by the coating layer and the heat-insulating layer, etc. Considering the small size of the nozzle channel for the secondary flow of the fluidic throat, the passage of the secondary flow is easily to be blocked if there are too many impurities or big solid particles, then causing fluidic throat to be out of control and generate unnecessary lateral force. In severe cases, the pressure of the solid gas generator rises and even cause explosion. Therefore, it is necessary to filter the gas before it is injected to the secondary nozzle of the fluidic throat. The filtration technology and device applied in the industry are relatively mature, but cannot be directly applied to the filtration of gas produced by solid propellant. The main difficulties are erosion of high-temperature gas, clogging of the filter device, and high requirements on quality, volume, and reliability. IV. Pressure stability control In order to ensure the control accuracy, it is generally required that the gas pressure of secondary flow maintains relatively stable. However, the working pressure of the solid gas generator is greatly affected by environmental conditions, propellant properties, burning surface changes, shell temperature, and other factors, so it is often difficult to maintain it at a certain stable value. For example, when the initial temperature of

10.4 Solid Generator of Secondary Flow

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the gas devices is high, the working pressure is generally high, and when the initial temperature is low, the working pressure will be significantly reduced, which will lead to a dramatic change in the pressure, flow rate, and so on of the secondary flow, and further affect the control accuracy of the fluidic throat. Therefore, it is necessary to implement a pressure stabilization technology for a gas generator with a large pressure fluctuation, and maintain the gas pressure relatively stable by designing a gas pressure regulator. The implementation of pressure regulation technology for high-temperature gas has the following difficulties: (1) The ablation resistance of the pressure regulator, (2) gas cleanliness affects the pressure regulation effect, (3) decrease in pressure regulation capability after long-time operation: high-temperature gas will influence performance of the operating components during long-time operation, which will reduce the pressure regulation capability, (4) the missile system has high requirements on the quality, volume, and reliability of the pressure regulator, and it is difficult to adopt complex pressure control technology and structure.

10.4.1 Reliable Ignition Technology The solid gas power plant generally is too small to arrange the ignition engine, and ignition adopting powder will cause high ignition pressure peak and poor ignition reliability. Here, an ignition structure scheme of combining the electric ignition device, ignition kit, and compensatory grain is given, which can effectively solve the difficulties, and the specific structure of the grain box is shown in Fig. 10.10. The compensatory grain is made of propellant block, and its formula is generally the same as the main grain formula, or other formulas are adopted. The function of the compensatory grain is to extend the ignition time of the ignition device, to lengthen the reaction time of ignition gas and main propellant grain, and to further facilitate the ignition of the main propellant grain, which can solve the difficulty of igniting the low-temperature propellant grain. On the other hand, the energy released by the combustion of the compensatory grain provides additional energy to the pressure ramp caused by the large heat loss in the initial main propellant grain. This ignition scheme can effectively suppress the climbing phenomenon of gas pressure, rapidly rise the pressure in combustion chamber, and shorten the response time of the system. In addition, the ignition scheme using the compensatory grain can avoid excessive initial shock due to relatively slow energy release. Apart from arranging the compensatory grain in the ignition kit, a plurality of compensatory grain can be arranged outside the ignition kit according to structure of the combustion chamber, so that the space of the combustion chamber can be fully utilized and the system becomes more flexible in design. If the time for reaching initial pressure of the gas generator is required to be shorter, the support component of the compensatory grain can be used in the design to separate the compensatory grain. In this way, the installation of the compensatory grain has great flexibility (as shown in Fig. 10.11), and the internal space can be fully utilized

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Fig. 10.10 The ignition kit of gas source

1—the cartridge holder; 2—the strip-shaped ignition kit; 3—the cartridge bottom plate; 4—the compensatory cartridge

Fig. 10.11 Position of the holder for compensatory propellant grain

Fig. 10.12 Gas generator ignition device

without changing the configuration parameters of the gas device, thereby shortening the time for reaching initial pressure. Figure 10.12 shows another practical ignition device structure combining an electric igniter, an ignition kit, and a compensatory cartridge. As can be seen from the figure, in order to make full use of the space of the combustion chamber, the ignition grain and the compensatory grain are arranged in a cavity composed of the filter device, the combustion chamber housing, and the perforated cover. The actual hot test shows that the ignition scheme given here can reliably and rapidly ignite propellant in the solid gas generator.

10.4 Solid Generator of Secondary Flow

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10.4.2 Gas Filtering Technology In order to prevent impurities from clogging the filter channel too much, the filter device is usually divided into two parts. One part is a rough filter device, which is mainly used to filter large diameter particles (for example, particles larger than 0.5 mm in diameter), cladding fragments, and other large diameter impurities; the other part is a thin filtration device for secondary filtration of the gas stream (for example, filtering fine particles larger than 30 µm in diameter), so that the gas cleanliness meets the requirement of the secondary flow of the fluidic throat. (1) Filtration technology of gas generating device for short-time working The design of the rough filtering device can adopt a common filter, when the gas working time is short (such as a few seconds), and the heat intensity requirement of the filtering device is not too high. Arranging the rough filtering device, which directly adopts the single-layer filter in the gas passage, can effectively block the large-sized particles and impurities. The filter mesh is woven from a single layer of filamentous refractory metals, and the filtration precision is determined by the size of the void of the woven fabric. A typical rough filter device for short-time operation is shown in Fig. 10.13. The structure mainly consists of a rough filter disk, a rough filter cartridge, a filter mesh, and a connecting fastener. According to the combustion characteristics of the propellant grain in the gas generator, the gas is filtered when flowing through the rough filter device in both axial and circumferential directions. For the fine filtration device, a multi-layer composite filter is adopted, and the filtration precision is determined by the finest filter holes. Both ablation resistance and thermal strength should be considered in selection of the filter material. A fine filtration device is shown in Fig. 10.14, which is mainly composed of a medium filter screen made of a multi-mesh molybdenum wire mesh, a composite filter layer, a fine filter screen made of a stainless steel wire mesh, and a connection support member. In practical applications, the rough filter and the fine filter are usually combined. (2) Filtration technology of gas generating device for long-time working When the gas needs to work for a long time (such as several tens of seconds), it is required that the filtering device cannot be broken under the long-time high-

Fig. 10.13 Structure of the rough filter device

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Fig. 10.14 Structure of the fine filter device

temperature gas flushing, or the impurities will pass freely, and also it needs to make sure that the excessively filtered impurities cannot block the gas passage. The rough filtering device for long-time working uses a certain thickness of filter cake as a rough filtering device. The filter cake is formed by multi-layer winding of the refractory metal wires, and the pores of the multi-layer composite determine the filtering precision in which the rough filter screen is formed by winding a molybdenum wire. The rough filter screen is placed together with the porous screen support plate in an annular cavity formed by the ignition device and the combustion chamber cylinder and is fastened by a gland. The long-time working fine filtering device can adopt a cyclone-type gas fine filter, which can withstand high temperature, high pressure, and long-time work, and the filtering precision is determined by the device geometry and working parameters. The cyclone gas fine filter is designed according to the principle of the cyclone separator, as shown in Fig. 10.15. The flow with particles enters the separator tangentially through the inlet, rotates in the separator, and flows downward. Due to the centrifugal force, the solid particles are smashed toward the cylinder wall and enter the collecting hopper downing along the conical portion. At the same time, the flow rotates upward and exits the separator through the outlet pipe. This kind of separation device has been widely used in the industry, and it can reach a good filtering effect if it is applied to the fine filtration of solid propellant gas and the filtration precision can be generally controlled within several micrometers. Since it has a special particle-collecting device which is separated from the flow passage, the flow passage blocking problem can be effectively solved. The cyclone gas filter has the following characteristics: (1) Simple structure and no moving parts, (2) good high temperature tolerance (up to 1000 °C), (3) wide range of processing gas flow, (4) strong pressure-bearing capacity (up to 50 Mpa), and (5) high dust removal efficiency (50% efficiency when the small diameter cyclone dust collector capture 1–2 µm particles). Since the device is disposable, the particle-collecting portion can be closed and does not need to be separately assembled and disassembled, so it can be welded, which reduces the use of components and increases the reliability of operation.

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Fig. 10.15 Structure of the cyclone separator

10.4.3 Gas Cooling Technology There are two ways to decrease gas temperature. Using grain with low burning temperature and designing a cooling device and by adding an inert component to the propellant formula and reducing the propellant energy, the burning temperature of the propellant grain can be effectively reduced. However, if blindly pursuing a low fuel temperature, the energy efficiency will be greatly reduced, the combustion residue will be increased, and the propellant grain will even not able to be burned normally. Therefore, it is necessary to control temperature using a cooling device. This book gives several reliable gas cooling schemes such as labyrinth, cyclone separation, and chemical cooling with plexiglass. (1) The labyrinth scheme Physical cooling with labyrinth is mainly used in situations where the working time is short, the cooling is required to be rapid, and has low requirements of the mass and volume. The cooling method mainly adopts the principle of heat absorption of materials. The structure with good heat absorption capability is arranged on both sides of the gas passage and heat is absorbed to decrease the temperature of the gas flowing through the passage. In order to effectively implement the cooling, metal is selected as the base material to increase the thermal conductivity, and a labyrinth structure is adopted as the gas passage to increase the heat exchange area. Since the temperature of the substrate gradually rises with the working time, the gas cooling capacity is gradually reduced, and the structure is not suitable for long-time gas cooling. Figure 10.16 shows a labyrinth cooling scheme in a typical solid gas generator. The cooling component in the figure comprises an inner collar, a middle collar, and a throttle ring, and each side of the collars has many small holes, so that a plurality of impinging jets are formed when the flow is discharged out through the

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Fig. 10.16 Cooling component

small holes, which will enhance heat transfer and cool the flow. The small holes of the adjacent two collars are staggered from each other, so that the jets discharged from the small holes directly impact the inner wall surface of the lower layer of the collar and redirect the flow, thereby increasing local heat exchange and the contact area. In order to increase the ablation resistance, the inner collar is made of refractory metals, and the middle collar and throttle ring are made of ordinary steel. (2) Cyclone separation cooling Cyclone separation cooling is mainly used in cases where it is not in high demand of a cooling range. The cooling method takes the advantage of a certain cooling ability when a cyclone-based fine filter is used to filter the gas. When the gas is rotated and separated through the inside of the fine filter, a vortex cooling effect is generated inside the filter, so that the temperature of the center of the gas stream in the filter differs from the ambient temperature, with lower temperature in the core and higher temperature on the outer edge, which enhances the heat dissipation of the filter through the wall to the outside, thereby significantly reducing the temperature of the outlet gas stream. The main advantage of cyclone separation cooling is that the fine gas filter can act as a cooling device, unnecessary to add additional components. (3) Chemical cooling with plexiglass Chemical cooling with plexiglass can be applied to gas cooling in the case of longtime working. The basic principle is that when the reduced gas flows through the inner hole of the plexiglass tube, the plexiglass is thermally degraded under the action of the oxygen-deficient gas flow, and a certain amount of heat is absorbed during the degradation. As a result, the temperature of gas flowing through the plexiglass is reduced. Polymethyl methacrylate resin is commonly known as plexiglass, or PMMA, and its molecular structure is

It is obtained by radical polymerization of methyl methacrylate (MMA, C5 H8 O2 ). In an oxygen-deficient environment where the outside temperature is higher than

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225

443–573 K, PMMA undergoes thermal degradation reaction while absorbing a certain amount of heat. The chemical reaction equation for thermal degradation is [−C5 H8 O2 −]n −→ nC5 H8 O2 − E a where E a is the amount of heat absorbed when per unit mole of plexiglass polymer [−C5 H8 O2 −]n is completely degraded to its monomer C5 H8 O2 . The reference value of E a is between 158.5 and 214.79 kJ/mol. The main components of the gas produced by the combustion of solid propellant are CO2 , H2 O, CO, H2 , and N2 , wherein the elemental oxygen is very poor, and reducibility of the gas is much lower than its oxidability. Therefore, PMMA experiences a thermal degradation reaction rather than a combustion exothermic reaction in the gas environment, which can cool it down. In addition, the chemical cooling technology with plexiglass can also be combined with the physical cooling technology with labyrinth to achieve the purpose of both rapid cooling and long-time operation, as shown in Fig. 10.17. The structure of physical cooling with labyrinth is the same as that in Fig. 10.16, and a plexiglass tube is arranged in the gas passage outside the structure of the physical cooling with labyrinth, to cool the gas by the thermal degradation endothermic reaction of the plexiglass.

10.4.4 Pressure Regulation Technology In order to solve the problem that the control precision is greatly influenced by the changes in pressure of the gas source, a solution using a self-operated gas pressure

Fig. 10.17 Gas cooling device

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1—outer casing and locking assembly; 2—outer; 3—large washer; 4—washer; 5—fixing bolt Fig. 10.18 Structure of the mechanical regulator valve

regulator is introduced here. It can effectively solve the technical difficulties in the ablation performance, the pressure regulation effect affected by cleanliness, and the pressure regulation capability after long-time working, and it also has simple structure and good reliability. The structure of the self-operated pressure regulator is mainly composed of a valve core, an elastic component, and a valve body. It does not require external energy as a power, just relying on its own elastic element as a source of accommodation force, working under the pressure of the fluid on the pipeline. The spool is made of a refractory material that acts as both a sensitive component and an actuator and moves with the elastomeric component under the pressure of the gas. The valve does not need to be specifically controlled after installation. It will automatically work when the pipeline gas meets a certain pressure condition. The gas generator produces a fixed flow of gas, which causes a certain pressure inside the pipe. When the pressure exceeds a limit value, the valve is opened and part of the gas is diverted to reduce the internal pressure. After a series of dynamic response processes, the equilibrium state is finally reached. At this time, the spool reaches a force balance and the valve opening increases as the balance pressure increases. Thereby, the internal pressure of the gas generator is stabilized near a value and the gas flow rate into the pneumatic component is stabilized within a certain range. The structure of a typical valve of the self-operated regulator is shown in Fig. 10.18, which consists of a housing, latching assemblies, and adjustment assemblies. The amount of pressure for stability and adjustment depends on the stiffness of the adjustment spring and the preload of the adjustment screw.

10.5 Thrust Modulation and Performance 10.5.1 Modulation Modes The key to achieve thrust regulation in a fluidic-throat system scheme is to control the flow rate of the secondary flow. At present, there are two modes for adjusting the thrust

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227

Fig. 10.19 Principle of pulse width control

(both for the secondary flow): the continuous adjustment of the parameters within a certain range and the pulse width adjustment and control of the transformation between two states. In order to simplify the structure of the secondary flow system on the solid rocket motor, pressure supplied to the secondary flow system is generally stable, and the continuous adjustment of the flow rate is achieved by adopting the needle plug injector (see Fig. 10.9a). When the needle bolt is moved to different positions, the throat of the secondary flow injector has different opening degrees, thereby achieving continuous adjustment of the flow rate. For the second mode, the nozzle throat has only two state transitions of “on” and “off”, and the corresponding thrust can only be switched between “high” and “low” ones. The principle of the pulse width control method for the switching valve is shown in Fig. 10.19. During each pulsation cycle, the average secondary flow rate through the injector can be varied by adjusting the dwell time of the valve in the open or closed position, thereby changing the average thrust of the engine. The modulated quantity M can be expressed as M = t 1 /(t 1 + t 2 ), where t 1 and t 2 represent the durations of opening and closing the shutter within a single time T, respectively. The modulated quantity M can vary from 0 to 100% by changing t 1 and t 2 . The pulse width control method is theoretically more suitable for small fluidicthroat propulsion systems (such as attitude and orbit control systems and small tactical missiles) because it has fewer components and the structure is easier to implement. However, it is worth noting that the switching frequency of the nozzle valve of the secondary flow should avoid the natural frequency of the engine and the missile to avoid resonance damage to the components. For large fluidic-throat engine systems, the continuously regulated mode is available. For pulse width modulation, the switching frequency of the valve needs to be carefully chosen. On the one hand, the switching frequency of the valve should be generally greatly different from the first- and second-order natural frequencies of the missile, because the damage caused by low-frequency resonance is the severest. The natural frequency of a missile is related to the overall size and structure of the missile. The first-order natural frequency of a general strategic missile is generally low, while the tactical missile has a higher natural frequency due to its small volume and large

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rigidity. On the other hand, if the required thrust can be adjusted in a short period of time, the frequency of the pulse width modulation method is also required to be large, such as the KKV warhead on the US “standard” series mentioned above. Considering the two aspects, the switching frequency of the valve is generally required to be larger, that is, the high-frequency mode. However, excessively high frequency will cause two problems. On the one hand, the injected secondary flow may form a pulsating pneumatic throat [2, 3] which cannot be simply estimated from the results of the steady fluidic throat. This is because the disturbance of the pulsating fluidic throat is beneficial compared with the steady one. On the other hand, the thrust response requires a certain reaction time t r (the transition time when the thrust value reaches a new equilibrium value from the previous one, see Chap. 7). If the frequency of the switching valve is too fast and the switching period of the valve is less than t r , the thrust will not be able to respond in time, and the square wave of the thrust output will be distorted. Figure 10.20 shows a comparison of the input–output square wave in the ideal state of pulse width modulation and its distortion (the preset modulated quantity M for both high and low frequencies is 50%). When the frequency is appropriate, the thrust can be the approximate square wave of high–low conversion applying signals of pulse square wave, and also achieve the preset modulated quantity of the input signal. When the frequency is too high, first, the narrowing range of the pulsating throat may be higher than the steady one; second, the thrust response also takes time, the comprehensive result will cause instable thrust output and fluctuation of the peak thrust, making the modulated quantity deviated greatly from the reset value. For example, the modulated quantities of both frequencies in Fig. 10.20 are 50%, but the modulated quantities of the thrust output are indeed different. Figure 10.21 is a flow diagram of a secondary flow injected at high-frequency pulse into a binary contraction–expansion nozzle, and it is simulated by the large eddy simulation method (subgrid viscosity is described by the Smagorinsky model [4–6]), where S tr = 0.2 and preset modulated quantity is 50%. S tr is related to the modulation frequency and its definition is shown in Eq. (10.1). In Fig. 10.21, the vortex structure and evolution of the supersonic jet during highfrequency pulsation can be observed. It is the interaction of the vortex that causes the secondary flow to penetrate into the mainstream deeper than that in the steady fluidic throat, so the narrowing range of the pulsating fluidic throat is also larger than that of the steady one (getting gain); on the other hand, after the valve is shut down, it takes time for the injected secondary flow to flow out of the nozzle. In small timescales (corresponding to high frequencies), this flow process cannot be ignored. This is because even if the valve is closed, the secondary flow will still act as a turbulent flow during the outflow of the nozzle. Str = f · L/Us ≥ 0.1

(10.1)

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229

Fig. 10.20 The input–output relationship of the fluidic throat and the frequency of pulse width control method

where U s is the injecting flow rate of the secondary flow, L is the reference length of cross section of local cross-flow, and f is the switching frequency (modulation frequency) of the valve. Equation (10.1) is also a judgment formula for forming an upper frequency limit of the pulsating fluidic throat. When the modulation frequency satisfies Eq. (10.1), the gain of the pulsating fluidic throat cannot be ignored. Therefore, modulation frequency is required to satisfy f < 0.1 to be away from the boundary of the pulsating fluidic throat. Meanwhile, considering the response time of the thrust, the final upper frequency limit is determined by Eq. (10.2). The constant-state results can still be used to predict and design the thrust square wave of the fluidic throat within the upper frequency limit. A distorted waveform of thrust square wave should be avoided because it can complicate the control system, causing the thrust adjustment to be less than the expected amount of control and resulting in flight failure.   0.1Us 1 (10.2) , f max = min L tr 1 > f n,1 (10.3) tr From the formulas above, the frequency boundary of the pulsating fluidic throat is generally pretty high, so f max is mainly determined by t r .

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steady fluidic throat

Pulsating fluidic throat

(a) Experimental pictures

(b) Logarithmic distribution of volatility

(c) Cloud map of temperature

Fig. 10.21 Comparison of the flow fields in the pulsating and the steady jet fluidic throats

In summary, it can be seen that in order to avoid low-frequency resonance and improve the response time of missiles, the modulation frequency is suggested to be higher, but the thrust response time of the fluidic throat also determines the upper limit of the modulation frequency. Therefore, if the pulse width modulation method is used to control the thrust in practice, the reciprocal of the thrust response time t r of the fluidic throat is required to be greater than the first-order natural frequency of the missile at least, see Eq. (10.3), where f n,1 represents the first-order natural frequency of the missile [7, 8]. Therefore, t r is an important engineering application

10.5 Thrust Modulation and Performance

231

index and the thrust response time of the fluidic throat should be increased as much as possible.

10.5.2 Typical Thermal Experimental Results The practical thermal experimental system is similar to the cold-flow experimental system. The thermal motor should be designed for thermal protection from ablation during the injection of secondary flow, which makes the FNT structure for thermal tests more complex than that for cold-flow test. The parameters of the applied CMDB propellant in the given example are listed in Table 10.1. The mass flow rate of gas during the thermal tests can be predicted by Eq. (10.4). The mass flow rate of secondary flow can be calculated using Eq. (2.3). Then, the ratio C d of effective throat area can be calculated from Eqs. (2.1), (2.4), and (10.4). m˙ o =ρp Abr˙ = ρp Ab apcn

(10.4)

Figure 10.22 shows the plume variation before and after injection of the secondary flow in a typical thermal test on ground. With secondary flow injection, the plume region of the mainstream becomes narrower. The data of the effective throat area ratio versus the modified mass flow rate ratio at four injection angles (90°, 60°, 45°, and 30°) are plotted in Fig. 10.23. The control performance of throat area of 16 injectors decreases slightly compared to the results

Table 10.1 Propellant parameters

Parameter Density ρ p

Value (g/cm3 )

Burning temperature T c (K)

1.71 ~3000

Al

~15%

Burning rate exponent n

0.31

Burning rate constant a (mm/(s MPan ))

8.36

Injector numbers

8

Fig. 10.22 Primary flow plumes before and after injection of the secondary flow

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Fig. 10.23 The effective throat area ratio versus the modified mass flow rate ratio

of 8 injectors, which implies that applying more small injectors may result in a larger loss in the choking performance of secondary flow. The CFD prediction results show similar trends to the cold-flow test results and the typical thermal test results. The agreement also indicates that the steady coldflow results can be extended to hot gas conditions by introducing the modified mass flow rate, as in Eq. (2.6). Therefore, to decrease the mass flow rate consumption of the secondary flow while attaining the same C d , based on Eq. (2.6), the methods of decreasing the consumption of the secondary flow include using hot gas as the secondary flow, which can be generated by a solid gas generator or directly from the main gas flow; or using oxidizing/catalytic gas, liquid, or a mixture thereof as the secondary flow to form secondary combustion at the injection place and release a large amount of heat to enhance the choking performance per unit mass of secondary flow. Define the adjustment ratio of thrust of the hot tests as follows: η=

Fs,o F

(10.5)

The thrust F is the thrust of an FNT solid rocket motor without injection of secondary flow: n   1 F = CF aρ p C ∗ Ab 1−n Atn−1

(10.6)

The burning area Ab , density ρ p , and burning rate constant a are all constants if constant burning surface of grain is applied. At a low or medium modified mass flow

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233

rate ratio and pressure ratio, it can be supposed that the thrust coefficient C F and characteristic velocity C* do not change to significant extent for the main flow gas before and after injection of secondary flow. Then, the thrust generated by the main flow after secondary flow injection can be predicted as follows: n   1 Fo = CF aρ p C ∗ Ab 1−n A˜ tn−1

(10.7)

If it is assumed that both primary and secondary flows can expand closely to the same pressure at nozzle exit under ideal conditions, for a large expansion ratio nozzle, taking into account the mixing of two flows, the gas velocity at the nozzle exit can be treated as approximately equal. Then, the thrust produced by the secondary flow can be calculated as follows: Fs = m˙ s u e =

m˙ s m˙ s m˙ o u e = Fo m˙ o m˙ o

(10.8)

Then, the ideal adjustment ratio by the secondary flow injection can be expressed as follows: Fs + Fo ηT = = F



A˜ t At

n  n−1 

√  w˙ s Ts 1+ √ w˙ o To

(10.9)

The value of C d can be found from cold-flow test results or steady CFD simulation as shown in Fig. 10.23.

Fig. 10.24 Adjustment ratios ηT of thrust with modified mass flow rate ratio f m with different n

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Figure 10.24 shows that, when the burning rate exponent is 0 < n < 0.55, the slope of the adjustment ratio ηT of thrust increases slowly as f m increases. The ηT tends to increase linearly with f m at 0 < n < 0.55. When the burning rate exponent n > 0.5, the variation rate of ηT increases obviously with the rise of f m . Equation (10.9) can predict ηT of thermal test conditions with an acceptable error compared to the values found from the typical thermal tests. If a propellant with a larger burning rate exponent is applied, a much larger ηT can be attained than the values of the given thermal tests of the FNT engine. For example, when n increases to 0.8, the thrust adjustment ratio can rise to about nine at a modified mass flow rate ratio of 30%.

10.5.3 Modification Process with Gas-Catalysis Particle Two-Phase Flow Injection Based on the simulation analysis in Chap. 4, copper powders with a particle size of 2 µm were added to the nitrogen jet in the thermal test of FNT engine. The effect of the jet added with copper powders on the thrust control of the engine was studied. Table 10.2 shows the steady-state data in the experiment and Fig. 10.25 shows the pressure and thrust dynamics of the thermal test. The experiment is divided into four stages: after the engine is ignited normally with no jet injected, pure nitrogen at normal temperature is injected, then the jet is closed, and nitrogen mixed with copper powder is injected. The content of copper powders in the mixed jet was relatively low, merely accounting for about 1.72% of the mixed jet. The results show that the adjustment ratio of the thrust produced by the injected jet in the two stages is also substantially the same, indicating that the jet mixing a small amount of copper powder has less influence on the steady-state performance of the thrust adjustment of the engine. In terms of dynamic performance, as shown in Fig. 10.25, after the copper powders are added in the jet, the response delay of thrust of the engine is significantly shortened, from 140 ms under condition pure gas jet to 100 ms. In practice, the par-

Table 10.2 Thermal experiment data Pc (MPa)

m˙ 0 (g/s)

m˙ s (g/s) gas

m˙ s (g/s) particle

F (N)

fw

Cd

ηT

ηE

No injection flow

2.03

224

–

–

428

–

–

–

–

Pure gas jet

2.74

247

178

–

569

0.228

0.775

1.329

0.98

Jet mixed with copper powder

2.82

249

171

3

570

0.216

0.758

1.332

0.96

10.5 Thrust Modulation and Performance

235

Fig. 10.25 Pressure and thrust curves in the hot gas test experiment

ticles of copper powder form a reflow upstream of the nozzle at the initial stage of injection, and are mixed with the mainstream gas, and a chemical reaction occurs. The exothermic reaction increases the thermal resistance of the throat of the nozzle, further increasing the variation rate of pressure rise in the combustion chamber. Eventually, the equilibrium pressure of the combustion chamber does not change much, while resulting in a significant reduction in response delay. Figure 10.26 shows photographs of the plume from the pure gas of secondary flow and the mixed gas of secondary flow. The comparative observation shows that the copper powder in the mixed gas continues to burn in the plume, which indicates that the copper powder has a secondary combustion reaction with the gas.

Fig. 10.26 Photographs of plume in the experiment

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10.5.4 Performance and Thrust Modification Arrange This section uses specific ballistic examples to illustrate the performance gains and advantages of applying thrust-adjustable technology in missiles with fixed launch pads. In the example of Fig. 10.27, the missile has an outer diameter of 178 mm, a total weight of about 46 kg, and a pusher mass of about 14 kg. After the application of the thrust-control technology, the trajectory of the two different mission profiles of “boost-endurance” and “endurance-boost” can be realized without changing the hardware. From the example of Fig. 10.27, the first trajectory is similar to the parabolic trajectory of ballistic missiles, mainly for attacking close-range targets, and the maximum Mach number is larger than 1. The second type of trajectory is similar to the trajectory of traditional cruise missiles. The range is increased to six times the long range and the maximum Mach number is about 0.7. These examples show that the thrust-control technology can make the trajectory of missiles more flexible. Its application mode is very different from that of the missile using the traditional solid rocket engine, and it can adapt to the rapid changes in the modern battlefield. It is noted that both the fluidic-throat nozzle technology and the pintle engine technology can be used to realize the thrust-controlled trajectory mentioned above.

(a) “Boost-endurance” mode

(b) “Endurance-boost” mode Fig. 10.27 Calculation results of ballistic missiles

10.5 Thrust Modulation and Performance

237

In theory, whether it is the injection of secondary flow or mechanical scheme based on pintles, the adjustment range of nozzle throat can be very large, and the thrust ratio (maximum thrust value to minimum thrust value) can be 10–100. However, in practice, the thrust ratio greater than 10 is mainly suitable for small attitude and orbit control systems to provide a large lateral force. As the main power, it is unrealistic and unnecessary to simply pursue high thrust ratio. Because the range of the main thrust varies widely, on the one hand, the control difficulty increases; on the other hand, the maximum overload that the structure needs to bear is increased, and the impact resistance and structural mass of the electronic components on the missile are greatly increased. Some researches using the pintle-controlled solid rocket motor as main thrust indicate that the controllable thrust scheme of the missiles can have great engineering practical value and efficiency improvement [9–12], when the maximum thrust ratio of the pintle engine reaches 4–5. The previous chapters have introduced the feasibility of achieving a maximum thrust ratio of 4–5 in the fluidic-throat system.

References 1. Shaobo, H.: Design of the Gas-Generator with Multi-nozzle for the Direct Lateral Force Control in Air-to-Air Missile. Beihang University (2006) 2. Baruzzini, D., Domel, N., Miller, D.N., et al.: Pulsed injection flow control for throttling in supersonic nozzles—a computational fluid dynamics design study. AIAA Paper 2007-4215 3. Domel, N.D., Baruzzini, D., Miller, D.N., et al.: Pulsed injection flow control for throttling in supersonic nozzles—a computational fluid dynamics based performance correlation. AIAA Paper 2007-4214 4. Bing, F., Zhengxi, G.: Investigation on influence of particular phase to stability of solid rocket motor by large eddy simulation. Shanghai Aerosp. 24(6), 43–47 (2008) 5. Calhoon, W.H., Menon, S.: Subgrid modeling for reacting large-eddy simulations. In: 34th AIAA Aerospace Sciences Meeting, AIAA 96-0516 6. Calhoon, W.H., Menon, S.: Linear-eddy subgrid model for reacting large-Eddy simulations: heat release effects. In: 35th AIAA Aerospace Sciences Meeting, AIAA 97-0368 7. Zongmei, Z.: Militia Intercontinental Ballistic Missile. Astronavigation Press, Beijing (1997) 8. China Aerospace Industry Corporation (ed.): World Missile and Aerospace Engine. Military Science Press, Beijing (1999) 9. Yilin, W., Guoqiang, H., Jiang, L., et al.: Experiment on non-coaxial variable thrust pintle solid motor. Solid Rocket Technol. 31(1) (2008) 10. Juan, L., Jiang, L., Yilin, W., et al.: Study on performance of pintle controlled thrust solid rocket motor. 30(6) (2007) 11. Juan, L., Weiping, T., Xiaosong, G., et al.: Numerical simulation on steady flow field of variable thrust motor nozzle with pintle. Solid Rocket Technol. 31(4) (2008) 12. Juan, L.: Characteristics of Pintle Controlled Thrust Solid Rocket Motor. Northwestern Polytechnical University (2007)