Transfer of Substance in Vortex and Wave Flows in One-Component and Multi-component Environment 3031318552, 9783031318559

Table of contents :
Preface
Introduction
References
Contents
1 Vortex Flow in a Homogeneous Fluid
1.1 General Issues in the Theory of Vortex Flow
1.2 Experimental Modelling of Vortex Flow: Equipment, Methodology of Laboratory Experiments and Parameters of the Flow Under Study
1.3 General Model of Vortex in a Cylindrical Container
1.4 Surface Cavern Geometry and Critical Conditions for Restructuring the Flow in a Compound Vortex in Containers of Different Geometries
1.5 Cave Shapes Composite Vortex in Clear Water
1.6 Characteristic Processes at the Free Surface of a Compound Vortex Flow
1.7 Spiral Structure of Liquid Particle Trajectories Near the Vortex Surface
1.8 Flow Near the Disc
References
2 Solute Admixture Transport from a Compact Source in a Composite Vortex
2.1 Experimental Studies of Solute Transport in Vortex Flow
2.2 Structural Stability of Soluble Admixture Transfer Pattern from a Spot on the Surface of a Compound Vortex
2.3 Transfer of Miscible Admixture into the Thickness of the Composite Vortex
2.4 Visualisation and Qualitative Analysis of Flow Near the Disc Edge
References
3 Transfer of Immiscible Admixture in a Vortex Flow
3.1 Compound Vortex in a Liquid of Two Immiscible Components
3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow
3.3 Methodology for Comparing Data and Constructing Approximation Curves
3.4 Forms of Partial Disintegration of the Oil Body in a Compound Vortex, Formation of Forward and Backward Emulsions
References
4 Motion of Solid Markers in a Vortex Flow
4.1 Experimental Investigations of the Movement on the Surface of the Vortex Flow
4.2 Analytical Representation of Marker Movement on the Vortex Surface
4.3 Experimental Study of Miscible Admixture Transfer from a Solid Marker on a Vortex Flow Surface
References
5 Modelling Hydrocarbon Spillage on the Surface of Water
5.1 Evolution of Oil on the Water Surface
5.1.1 Spreading
5.1.2 Displacement (Advection)
5.1.3 Evaporation
5.1.4 Dissolution
5.1.5 Emulsification
5.1.6 Oxidation
5.1.7 Dispersing
5.1.8 Settling
5.2 Analytical and Numerical Modelling of the Hydrocarbon Slick Shape on the Water Surface
5.3 Experimental Investigation of Hydrocarbon Spreading on Water Surfaces
References
General Conclusions
Bibliography

Citation preview

Earth and Environmental Sciences Library

Tatiana Chaplina

Transfer of Substance in Vortex and Wave Flows in One-Component and Multi-component Environment

Earth and Environmental Sciences Library Series Editors Abdelazim M. Negm, Faculty of Engineering, Zagazig University, Zagazig, Egypt Tatiana Chaplina, Antalya, Türkiye

Earth and Environmental Sciences Library (EESL) is a multidisciplinary book series focusing on innovative approaches and solid reviews to strengthen the role of the Earth and Environmental Sciences communities, while also providing sound guidance for stakeholders, decision-makers, policymakers, international organizations, and NGOs. Topics of interest include oceanography, the marine environment, atmospheric sciences, hydrology and soil sciences, geophysics and geology, agriculture, environmental pollution, remote sensing, climate change, water resources, and natural resources management. In pursuit of these topics, the Earth Sciences and Environmental Sciences communities are invited to share their knowledge and expertise in the form of edited books, monographs, and conference proceedings.

Tatiana Chaplina

Transfer of Substance in Vortex and Wave Flows in One-Component and Multi-component Environment

Tatiana Chaplina Ishlinsky Institute for Problems in Mechanics Moscow, Russia

ISSN 2730-6674 ISSN 2730-6682 (electronic) Earth and Environmental Sciences Library ISBN 978-3-031-31855-9 ISBN 978-3-031-31856-6 (eBook) https://doi.org/10.1007/978-3-031-31856-6 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, 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 publisher, 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 publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The book is devoted to the experimental and theoretical study of the dynamics and structure of multiphase vortex flows and the nature of the transfer of three types of markers: solid (ice, plastic), immiscible with water (oil, oil, diesel fuel) and soluble (aniline dyes, uranyl). The author of the book has developed a technique for experimental studies of the dynamics of the formation and structure of a steady vortex flow, based on visualization and subsequent approximate mathematical description of the processes of matter transfer in vortex and wave flows in a wide range of parameters. A qualitative and quantitative description of the dynamics and structure of multiphase vortex flows and the nature of the transfer of three types of markers was carried out: solid (ice, plastic), water-immiscible (oil, oil, diesel fuel) and soluble (aniline dyes, uranyl). Experimental studies of the transfer of a soluble impurity in a vortex flow were carried out, which confirmed the linear nature of the dependence of the impurity penetration depth near the vertical axis of the flow on time. The structure of interfaces in a vortex flow was experimentally studied: oil body–water–air, including the regime of the beginning of emulsion formation. The problem of approximate analytical determination of the shape of an oil body in a compound vortex is solved based on the analysis of the equations of mechanics of fluids of different densities with physically justified boundary conditions. Dependences are obtained that describe the shape of the phase boundaries in a vortex flow of a liquid consisting of two components and characterize the shape of the zero approximation for the phase boundaries in a compound vortex, which are in satisfactory agreement with the experimental data. Experimental and theoretical studies of the transfer of solid-state markers of various shapes and sizes in a vortex flow in a single-component and multi-component liquid have been carried out. The experimental dependence of the marker rotation angle relative to the vortex axis on the marker rotation angle relative to its own axis is in good agreement with the theoretical dependence obtained on the basis of the proposed mathematical model. Experimental studies of the spreading of hydrocarbons from a compact spot over the water surface were carried out in a wide range of experimental parameters. Differential equations are presented that determine the shape of the oil body under the assumption that the angular velocities of rotation of the oil body and v

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the surrounding liquid are equal (the “oil body freezing in” hypothesis). Numerical solutions are obtained for the steady-state shape of a hydrocarbon spot on the surface of water at rest. Experimental studies of the sorption capacity of various materials in relation to oil products, their water absorption and use for the elimination of water surface pollution by hydrocarbons have been carried out. An original method of cleaning the water surface from oil pollution using natural sheep’s wool is proposed. The theoretical significance of the study is due to the fact that a theoretical model has been built that describes the universal geometry of vortex caverns, showing that the trajectories of liquid particles both near the vortex surface and near the disk surface are three-dimensional logarithmic spirals along which flow occurs from the periphery to the center of the vortex. A theoretical model is proposed to explain the movement of various markers in a vortex flow, and an equation is obtained that describes the movement of the center of mass of the marker, representing a logarithmic spiral on the surface of the vortex funnel, which coincides with the trajectories of liquid particles near the free surface. The significance of the results obtained by the applicant for practice is confirmed by the fact that models of the behavior of various impurities in circulation currents are presented, which can serve as a basis for modeling and predicting their distribution in natural conditions (in a stratified hydrosphere and atmosphere), for developing new models of multi-component flows, creating algorithms for numerical modeling of environmental problems associated with the spread and accumulation of pollutants in natural water bodies. The research results can be used to improve numerous oil– water separation plants and ways to preserve the quality of the environment. The effect of formation of flows in a liquid at rest during the sorption of petroleum products and oils on fibrous materials can be used in technologies for eliminating hydrocarbon spills and cleaning natural water bodies. The evaluation of the reliability of the results revealed the reproducibility of the results under various conditions, the correct formulation of the problems, the agreement between the results of the experiments and the data of independent experiments in the range of coincidence of parameters, as well as satisfactory agreement with the calculations based on the models based on the fundamental equations of fluid mechanics. Moscow, Russia

Tatiana Chaplina

Introduction

Vortex flows are widespread in natural conditions and are used in various technical applications. The problem of matter transfer in circulating flows is still very relevant and in demand for hydrophysics, ecology and industrial technologies and has many different applications. Despite a large number of studies, this area remains insufficiently studied, since such problems, as a rule, depend on a large number of random and difficult to control factors and are very difficult for experimental study, unambiguous interpretation and theoretical description. In this regard, it is very important to develop a methodology for experimental laboratory modeling, designed to ensure the stationarity of flows and reproducibility of results and the creation of an adequate theoretical model of the processes under study [1]. The study of vortex motion of fluids, initiated in the seminal work of Helmholtz [2] and continued by outstanding scientists of the century before last and the beginning of the last century Kelvin [3, 4], Prandtl [5], Poincaré [5], Zhukovsky [7] and others, remains relevant, until now, as evidenced by a large number of monographs and articles [8–12]. The general theory of vortex flow is described quite fully in monographs by Lamb [13] and Wille [14]. The originator of the vortex theory is considered to be H. Helmholtz who published in 1858 his work “On the integral of hydrodynamic equations corresponding to vortex motion” in which he first formulated the vortex conservation theorem [15]. According to this theorem, when forces arise that satisfy the energy conservation law, an existing vortex cannot be created or eliminated, much less change the tension of the latter. The integrals of hydrodynamic equations, which are the basis for appearance of the vortex conservation theorem, were derived in 1815 by the famous physicist Cauchy [16]. However, it is impossible not to mention the participation of the famous physicist Lagrange in creation of special cases in this vortex conservation theorem. In his “Analytic Mechanics” (1788), he proves that the motion of an ideal liquid, which has a potential of velocities at a certain moment in time, remains unchanged throughout the motion. Cauchy and Stokes later proved that any particle of an ideal fluid cannot obtain rotational motion assisted by its environment if it does not possess it at the starting point of time. vii

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Based on Helmholtz’s theorem, the Italian scientist Beltrami derived a rule for determining the velocities of particles in a compressible fluid that is in vortex motion and in a closed finite volume. This rule is called the Beltrami theorem and establishes electrodynamic analogies. The Hungarian scientist Theodor von Kármán was the first who in 1911 discovered the formation of a special sequence of vortices when flowing around a circular cylinder whose axis is perpendicular to the oncoming flow and described the conditions of its formation [17]. Carman’s vortex tracks continue to be studied until now because periodic emissions of such vortices can be so powerful that they can cause vibrations (resonance) in a variety of objects. The destruction of the Tacoma-Narrows Bridge (Washington State, USA) in 1940 by such vortex proves its danger. The current problem is the description of movement in a concentrated vortex of an object. Since a floating object has inherent physical and chemical characteristics that differ from those of a fluid medium, it is necessary to introduce corrections to account for the interaction between the medium and the object when describing its motion. In geophysics, such corrections can be useful, for example, in studies related to the problem of plastic debris accumulation in the centers of vortex formations in the open ocean, as well as for correcting the readings of various drifter probes that transmit information about sea and ocean flow. Consideration of the processes of substance transfer in such complex systems as natural water bodies is associated with many difficulties of methodological and principle nature: extreme complexity of conducting a full-scale experiment, complexity and variability of hydrophysical fields of the ocean and hydrometeorological conditions during studies and, in some cases, complexity and variability of the properties of the substance being transferred. In this regard, of particular interest is the study of marker transport in stationary eddies and wave flow, which can be formed in laboratory facilities with unchanged external conditions. In this case, one can avoid problems related to spatial and temporal variability of natural vortex sources and directly trace dependencies of characteristic flow parameters or displacement characteristics of solid or other objects placed in the flow. The author of this book experimentally investigated and created a theoretical description of the dynamics and structure of multiphase vortex flow and the transport of three types of markers: solid (ice, plastic), immiscible with water (oil, oil, diesel) and soluble (aniline dyes, uranyl). The following tasks have been carried out: • Experimental studies of vortex flow in containers of different geometry, as well as at different physical parameters of experiments on original installations upgraded to study the dynamics of formation, structure of the established current and the pattern of substance transfer in vortex flow in a wide range of defining parameters have been carried out. A methodology for collecting and processing experimental data has been worked out.

Introduction

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• A theoretical relationship is obtained which describes the universal geometry of vortex caverns occurring in cylindrical vessels during rotation of a coaxial disk and coincides both with experimental data and with the previously obtained model of vortex flow. • Analytical expressions showing that liquid particle trajectories near the vortex surface are three-dimensional spirals along which there is a flow from the vortex periphery to the vortex center were obtained for the first time. It is shown that the calculated and visualized trajectories of liquid particles are in good agreement and belong to the class of spatial logarithmic spirals. – The problem of flow near a disk serving as an inductor of a composite vortex current has been solved assuming that the rotating inductor is in contact with liquid only and that the physical fields are considered stationary and independent of the azimuthal angle ϕ. It is shown that with respect to the disk surface, the liquid elements move in logarithmic spirals. – Experimental studies of soluble admixture transport in vortex flow have been carried out and confirmed the linear character of time dependence of admixture penetration depth near the vertical axis of the current. It is established that the characteristic features of the vortex flow are set in the boundary layer area on the disk and then are transferred with preservation of the flow structure form into the whole area occupied by the liquid. This is confirmed by coincidence of types of spiral motion of liquid particles on the surface and near the disk. – Experimentally studied the fine structure of oil–water and liquid (water or immiscible hydrocarbons)–air interfaces in a compound vortex, including the mode of emulsion formation onset. Massively used fluids (sunflower and jet oils, oil, diesel fuel and their mixtures in different proportions) were chosen as the object of study. – For the first time, the problem of analytical definition of the shape of oil body in a composite vortex on the basis of analysis of equations of mechanics of divergent liquids with physically grounded boundary conditions is considered. The dependences, reflecting the form of interfaces in the vortex flow of the fluid, consisting of two components, are obtained. Analytical expressions describing the form of zero approximation for the phase boundaries in the composite vortex are in satisfactory agreement with the experimental data. – A theoretical model has been proposed to explain the motion of markers placed on the surface of a vortex flow in a one-component fluid. An equation describing the motion of the marker center of mass and representing a logarithmic spiral on the vortex surface, which coincides with the trajectories of liquid particles near the free surface, has been obtained. The experimental dependence of the rotation angle in the region of rotation of the solid body agrees well with the theoretical dependence derived from the proposed mathematical model. – Differential equations determining the shape of oil body in assumption of equality of angular speeds of rotation of oil body and surrounding liquid

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(hypothesis of “frozen oil body”) have been obtained. Numerical solutions have been obtained for the established shape of hydrocarbon slick on the resting water surface. Experimental studies of the process of spreading of hydrocarbons from a compact slick on water surface under different physical conditions—temperature and salinity—have been carried out. The validity and reliability of the results is ensured by the correct statement of tasks, confirmed by the reproducibility of the results within the accuracy of the experiment, the agreement of the results of experiments with data of independent experiments in the range of matching parameters, as well as a satisfactory agreement with the calculations based on models based on the fundamental equations of fluid mechanics. The book contains an introduction, five chapters, a conclusion and a list of references. The Introduction substantiates the relevance of the research topic, provides information on the degree of its development and formulates the purpose of the work and its scientific novelty. The theoretical and practical significance of the results is obtained, their novelty is given, the main provisions put forward for protection are given, and the summary of the book by chapters is presented. Chapter 1 presents the fundamentals of the theory of vortex flow in a free surface fluid and the requirements for experiments to study the processes of substance transfer in a steady-state vortex current. The description of experimental facilities used for modeling vortex flow is given, and equipment, methods of laboratory experiments and parameters of studied flow are described. The general model of the vortex in a cylindrical container is considered, and the shapes of the cavity of the compound vortex in pure water, the geometry of the surface cavity and the critical conditions for restructuring the flow in the compound vortex in containers of different geometries are determined. Processes at the free surface of the compound vortex are studied experimentally and analytically. Analytical expressions showing that the trajectories of liquid particles near the surface of the vortex are three-dimensional spirals along which the liquid flows from the periphery to the vortex center are presented. It is shown that the calculated and visualized trajectories of liquid particles are in good agreement and belong to the class of spatial logarithmic spirals. The problem of flow near a disk serving as an inductor of a compound vortex current is solved assuming that the rotating inductor is in contact with liquid only and assuming that the physical fields are considered stationary and independent of the azimuthal angle ϕ. It is shown that with respect to the disk surface, the liquid elements move in logarithmic spirals. Chapter 2 presents the results of experimental studies of soluble admixture transport from a compact spot on the free surface of liquid and inside the resting or involved in the composite vortex motion of liquid, as well as visualization and qualitative analysis of the flow near the disk edge. The rate of change of dye lowering into the liquid column depending on the inductor rotation speed was calculated. It is found that the characteristic features of vortex flow are determined in the region of

Introduction

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boundary layer on the disk and then transferred with preservation of the flow structure form to the whole area occupied by the liquid. This is confirmed by coincidence of types of spiral motion of liquid particles on the surface and near the disk. Chapter 3 presents the results of experimental studies of immiscible admixture transfer from a compact slick on the free surface of a liquid inside a resting or involved in a compound vortex motion of the liquid, including emulsion formation mode. Liquid unsaturated fatty acids (castor, sunflower and aviation oils) as well as diesel fuel, oil, fuel oil, etc. were used as immiscible mixture. Chapter 3 also considers the problem of analytical determination of the shape of an oil body in a composite vortex based on the analysis of the equations of mechanics of divergent liquids with physically justified boundary conditions. The dependences reflecting the form of interface in the vortex flow of a fluid consisting of two components were obtained. Analytical expressions describing the form of zero approximation for the phase boundaries in the composite vortex are in satisfactory agreement with the experimental data. Chapter 4 is devoted to the problem of visualization of vortex flows by introducing different kinds of markers into the moving fluid and the problem of measuring flow characteristics on the basis of observed marker displacements. Chapter 4 presents the results of experimental and theoretical investigations of solid markers of different shapes and sizes in vortex flow in multiphase liquids. A theoretical model to explain the motion of markers placed on the surface of the vortex flow in a one-component fluid is proposed. An equation describing the motion of the marker’s center of mass and representing a logarithmic spiral on the vortex surface, which coincides with the trajectories of liquid particles near the free surface, has been obtained. The experimental dependence of the rotation angle in the region of solid rotation agrees well with the theoretical dependence obtained from the proposed mathematical model. Also in Chap. 4, the results of experimental studies of spontaneous rotation of ice blocks of different sizes placed both on a solid surface (aluminum, polymethylmethacrylate, glass, foam plastic, ceramics) and on the surface of a basin with resting water of a certain depth at a given temperature are presented, and an explanation of the mechanism of spontaneous ice rotation is proposed. Chapter 5 classifies possible sources of oil and petroleum product spills, the likely risks of onshore and offshore oil production and the storage and transportation of oil and petroleum products due to accidents. The results of the investigation of the process of hydrocarbons spreading over water surface with different physical and chemical properties are given. Differential equations determining the shape of the oil body and their numerical solutions for the steady-state shape of the hydrocarbon slick on the resting water surface are obtained for the first time. The work was done within the framework of the state task 123021700046-4.

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References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17.

Gupta A, Lilly D, Sayred N (1987) Swirling flow. Moscow, Mir, p 590 Helmholtz H (1858) Uber Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen. J Fur Die Reine und Angewandte Mathematik 55:25–55 Kelvin L (1868) On vortex motion. Royal Soc Edinburgh 25:217–260 Kelvin L (1875) Vortex statics. Collected Works 4:115–128 Prandtl L (1918) Tragflügeltheorie. I Mitteilung. Nachrichten von der Gesellschaft der Wissenschaften Zu Göttingen. Mathematisch-Physikalische Klasse 151–177 Poincaré A (200) The theory of vortices. M, Izhevsk: RCD, p 160 Zhukovsky HE (1937) On attached vortices. Complete Works, vol 5, M.-L., State Publishing House of Technical Theoretical Literature Saffman F (2000) Vortex dynamics. Moscow, Scientific World, p 376 Alekseenko SV, Kuibin PA, Okulov VL (2003) Introduction to the theory of concentrated vortices. Novosibirsk, Institute of Thermal Physics SB RAS, p 504 Borisov AV, Mamaev IS, Sokolovsky IS (2003) Fundamental and applied problems of vortices. Moscow, Institute for Computer Research, pp 414–440 Gaifullin AM (2006) Investigation of vortex structures formed by fluid and gas flowing around bodies. Moscow, TsAGI Publishing House, p 139 Golovkin MA, Golovkin VA, Kalyavkin VM (2009) Vortex hydromechanics issues. Moscow, Fizmatlit, p 264 Lamb G (1947) Hydrodynamics - M.-L., State Publishing House of Technical Theoretical Literature, p 928 Ville G (2006) Theory of vortices. Moscow, ComBook, p 264 Helmholtz G (2002) Fundamentals of vortex theory. Moscow, IKI, p 82 Batchelor J (1973) Introduction to fluid dynamics. Moscow, Mir, p 760 von Kármán T, Rubach HL (1912) On the mechanisms of fluid resistance. Physik Z 13:49–59

Contents

1 Vortex Flow in a Homogeneous Fluid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 General Issues in the Theory of Vortex Flow . . . . . . . . . . . . . . . . . . . . 1.2 Experimental Modelling of Vortex Flow: Equipment, Methodology of Laboratory Experiments and Parameters of the Flow Under Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 General Model of Vortex in a Cylindrical Container . . . . . . . . . . . . . 1.4 Surface Cavern Geometry and Critical Conditions for Restructuring the Flow in a Compound Vortex in Containers of Different Geometries . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Cave Shapes Composite Vortex in Clear Water . . . . . . . . . . . . . . . . . . 1.6 Characteristic Processes at the Free Surface of a Compound Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Spiral Structure of Liquid Particle Trajectories Near the Vortex Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Flow Near the Disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Solute Admixture Transport from a Compact Source in a Composite Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Experimental Studies of Solute Transport in Vortex Flow . . . . . . . . . 2.2 Structural Stability of Soluble Admixture Transfer Pattern from a Spot on the Surface of a Compound Vortex . . . . . . . . . . . . . . 2.3 Transfer of Miscible Admixture into the Thickness of the Composite Vortex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Visualisation and Qualitative Analysis of Flow Near the Disc Edge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Transfer of Immiscible Admixture in a Vortex Flow . . . . . . . . . . . . . . . 3.1 Compound Vortex in a Liquid of Two Immiscible Components . . . . 3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1

3 11

12 17 25 28 35 41 43 43 44 48 54 61 63 64 69 xiii

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3.3 Methodology for Comparing Data and Constructing Approximation Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Forms of Partial Disintegration of the Oil Body in a Compound Vortex, Formation of Forward and Backward Emulsions . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88 92 99

4 Motion of Solid Markers in a Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Experimental Investigations of the Movement on the Surface of the Vortex Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Analytical Representation of Marker Movement on the Vortex Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Experimental Study of Miscible Admixture Transfer from a Solid Marker on a Vortex Flow Surface . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Modelling Hydrocarbon Spillage on the Surface of Water . . . . . . . . . . 5.1 Evolution of Oil on the Water Surface . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Spreading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Displacement (Advection) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Evaporation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Dissolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.5 Emulsification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.6 Oxidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.7 Dispersing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.8 Settling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Analytical and Numerical Modelling of the Hydrocarbon Slick Shape on the Water Surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Experimental Investigation of Hydrocarbon Spreading on Water Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123 125 126 127 128 128 128 129 129 130

102 107 119 121

130 137 139

General Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

Chapter 1

Vortex Flow in a Homogeneous Fluid

1.1 General Issues in the Theory of Vortex Flow In recent years much attention has been paid to the study of vortex formation, their fine structure, internal dynamics and decay. The results of experimental studies do not fit into a single model, which makes it difficult to compare them and isolate common properties. Of particular interest are vortex flow that allow a direct comparison with calculations based on fundamental equations, among which the main one is the current generated by a rotating disk in a free space [1], in a narrow fixed shell or in a cylindrical chamber of limited volume completely filled with liquid [2]. The study of flows in a transparent cylindrical container partially filled with liquid induced by a rotating disk significantly extends the range of application of various research methods. The free surface shape reflects the pressure distribution pattern, its perturbations are the characteristics of large-scale (inertial) and short spiral waves, especially difficult to study in filled containers, where they are also stably recorded, both at the disc surface and in the liquid column. In spite of a long history of theoretical and experimental studies of vortex motions, many questions remain unsolved. A significant achievement in the theoretical description of the vortex adjacent to the free surface of a semi-infinite ideal fluid belongs to Rankin [3]. The obtained solution for the pressure profile on the free surface in a fluid with a given swirl is one of the few that include a change in the shape of the free surface, while most solutions assume them to be negligibly small. For a more complete and accurate description of vortex flow, here is a definition of swirl. Swirl is a characteristic of fluid or gas flow in which the instantaneous speed of rotation of elementary volumes of the medium is not zero everywhere. A quantitative measure of vorticity is the vector Ω = rot v, where v is the velocity of the fluid. If in a boundless fluid there is a cylindrical vortex tube, the cross section of which is a circle of radius a, then the cross section of the vortex by the plane of motion is a circle. The region of solid rotation where the constant vorticity is maintained © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina, Transfer of Substance in Vortex and Wave Flows in One-Component and Multi-component Environment, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-31856-6_1

1

2

1 Vortex Flow in a Homogeneous Fluid

is usually called the vortex core, the outer region of the potential flow is called the vortex shell. In this kind of vortex, all vortex lines are straight and parallel to each other; all vortex tubes are cylinders whose formations are perpendicular to the plane of motion. Such vortices are called rectilinear vortices. In a plane perpendicular to the vortex lines, such flow can be considered as a circular vortex. Assuming that the vorticity Ω has a constant value in the region of radius a and zero outside this region, we write down the equations for the velocity of the fluid. (v∇)v = −∇ p ∇ · v = 0,

(1.1.1)

where p is the density-normalised pressure. The solution (1.1.1) for velocity is a continuous function of coordinates, but consists of two parts v = 21 Ωr , at r < a and, respectively, v = 21 Ω ar , at r > a or v=

  Ωa 1 Ωr θ(a − r ) + θ(r − a) , 2 r

where θ is the Heaviside function. Note that a circular vortex does not induce velocity at its centre. Thus, the centre of a circular vortex existing in a resting fluid remains stationary. The next step to construct a model vortex in the gravity field is the combined Rankine vortex. This is a stationary vortex adjacent to the surface of a semi-infinite ideal fluid (the flow is assumed to be independent of the azimuthal angle, radial and vertical velocity components are immediately assumed to be zero and a potentiality condition is additionally imposed on the flow in the vortex shell). In this case, the boundary conditions of equation system (1.1.1) consist of pressure continuity condition and normal component of velocity at the contact surface is equal to zero. The distribution of the tangential component of the flow velocity in the Rankine vortex and the pressure profile along the radial coordinate taken from [3] are shown in Fig. 1.1. In Fig. 1.1 the following symbols are introduced: h 2 —point of change of sign of pressure profile curvature, h—depth of free surface deflection, a—radius of vortex core. Reduced form of the equation for the shell pressure of such a vortex (r > a): p = p0 + gz −

Ω2 a 4 2r 2

While in the vortex core (r < a) the equations of motion ∂p ∂p = r Ω2 and =g ∂r ∂z

1.2 Experimental Modelling of Vortex Flow: Equipment, Methodology …

3

Fig. 1.1 Rankine vortex: a pressure profile and surface shape; b distribution of tangential velocity of liquid particles; from the centre of rotation along radial coordinate

For the pressure continuity condition to be fulfilled it is necessary that at r = a the pressures in the outer and inner parts of the vortex coincide. Based on this condition the pressure in the vortex core can be written as  p = p0 + gz − Ω a

2 2

r2 1− 2 2a



Consequently, the shape of the free surface where p = p0 , is determined by the pressure profile: z=

Ω2 a 2 g

   a2 r2 1 − 2 θ(a − r ) + 2 θ(r − a) 2a 2r

(1.1.2)

Accordingly, above the centre of the vortex the free surface subsides to a depth of h = Ω2 a 2 /g. Accordingly, the symbols on the pressure profile graph of the combined vortex become clear.

1.2 Experimental Modelling of Vortex Flow: Equipment, Methodology of Laboratory Experiments and Parameters of the Flow Under Study Several centres for geophysical flow modelling are currently active around the world. In Russia, centres for laboratory modelling of geophysical flows are located in Moscow (Institute for Problems in Mechanics of the Russian Academy of Sciences (IPMech RAS), Lomonosov Moscow State University, Shirshov Institute of Oceanology of the Russian Academy of Sciences, Obukhov Institute of Atmospheric Physics of the Russian Academy of Sciences), St. Petersburg, Novosibirsk (Lavrentiev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, S.S. Kutateladsky Institute of Thermophysics).

4

1 Vortex Flow in a Homogeneous Fluid

Experiments carried out in a cylindrical vessel of limited volume, where the flow was generated by rotation of both upper and lower lids [4–6], gave results similar in structure and main features to those obtained in works where the source of turbulence was only one end of the cylindrical chamber [2]. The realization under laboratory conditions of steady-state vortex flows is also described in [7] (a container with rotating inductors with a closed fluid volume), as well as in [8, 9], where a free surface flow is investigated in a container with different rotating inductors. Paper [10] is devoted to the investigation of the vortex flow occurring when a liquid flows out of a rotating cylinder. A series of experiments carried out using cylindrical containers in which fluid motion was initiated by rotation of both lids at once [11–13] showed results similar to those of experimental works where the source of turbulence was only one of the ends of the cylindrical chamber [14]. Experiments using miscible impurity as a marker were carried out by the group of Leveque and published in [15]. Studies of the behaviour of immiscible admixture in a cylinder with a rotating bottom and comparison with numerical simulations [16] showed the possibility of using a composite vortex flow as a separator for petroleum products. The lack of data on the properties of vortex flow with a free surface and the properties of the surface itself of flow of this kind makes the topic of the study topical. Experimental facilities and for modelling vortex flow In the course of the work, the experiments were carried out on three installations, allowing a stationary vortex to be created and its parameters to be monitored. 1. Vortex Flow with Torsion Experimental Setup The “VFT” (Vortex Flow with Torsion) unit was created at the Laboratory of Fluid Mechanics of IPMech RAS to study twisted flow, which was created by Professor Chashechkin. The experimental setup was created with the following requirements in mind: (1) Simplicity of the vortex generation mechanism; (2) Stationarity of the vortex generation conditions; (3) The limited volume of liquid involved in the vortex flow and its constancy during the experiment; (4) Observability of the flow pattern by optical methods in all projections; (5) Practical availability of a maximum range of flow areas for the application of probes or markers. The source of vortex motion is rotation of inductors—smooth discs, discs with ribs of different shapes, screws or other bodies of different geometry, which are placed at the bottom or surface of the container. Schematic diagram of the experimental setup is shown in Fig. 1.2a, photo in Fig. 1.2b. The vortex current was created by a rotating disc mounted at the bottom of a transparent cylindrical container. To reduce optical distortions, the container was placed in an open rectangular basin made of transparent polymethylmethacrylate with dimensions 0.6 × 0.45 × 0.7 m3 . All sides of the parallelepiped are made of 20 mm thick sheet organic glass. A shaft, directly

1.2 Experimental Modelling of Vortex Flow: Equipment, Methodology …

5

connected to an electric motor, runs through the geometric centre of the lower edge and its axis coincides with the axis of rotation of the electric motor. To keep the whole structure rigid, it is enclosed in a metal channel frame. The pool is secured inside the frame with screws. A cylindrical shell 2 with an inside diameter of 29.4 cm is inserted inside the basin. In the centre of the shell an axle is inserted through a gasket and a smooth disc 3 is inserted with a thickness of 2 mm and a diameter of 5, 10, 15, 20 and 29 cm, or another shape. The top edge of the disc is positioned 2 cm from the bottom of the pool. In some experiments, a false bottom 4 may be fitted at the level of the top edge. The electric motor can rotate the inductor uniformly at different angular speeds. On the axis there is a disc-mask 5 of the angular velocity meter. The disc is driven by an electric motor 6, the operation of which is controlled by the unit 7. Its rotational speed in the range from 200 to 2500 rpm is recorded by means of optical sensor 8 and signal conversion unit 9. The flow pattern is recorded by means of photo or video equipment 10. Experiment control and data recording is done by means of a computer 11. To visualise the flow, paint can be used which is injected into a selected part of the volume using a dosing pipette 12. Lighting for photography is provided by a white light source 13 with a scattering screen 14 or an ultraviolet light lamp 15, the basin is equipped with a hydraulic system 16. The pool is filled with water. The liquid level is set separately for each series of experiments. Illumination conditions (angle and height of the illuminator) are chosen so that during further image processing all details of free liquid surface are visible and distinguishable (free surface-air boundary should be seen very clearly throughout its length and at each value of activator rotation frequency). The shape of free liquid surface throughout the experiment is recorded by means of digital photo or video camera [17]. Three types of markers were used in the experiments: solid (ice, plastic), nonmiscible (oil, oil, diesel, jet fuel) and soluble (aniline dyes, uranyl). Experiments of each type were conducted in series. Throughout the series of experiments, the only parameter that remained unchanged was the shell radius R0 .

Fig. 1.2 General view of the torsional vortex simulator (VFT): a photo, b, c scheme

6

1 Vortex Flow in a Homogeneous Fluid

In each series, the depth of the fluid layer H and the radius of the activator disk R were recorded. The flow pattern was recorded for the fixed geometric parameters of the unit at different activator rotation frequencies. The main variable parameter, when changing from one series of experiments to another, was the radius of the activator disk R. The next variable parameter is the depth of liquid H. For each value of liquid depth, the frequency was changed Ω. In experiments where particles (aluminium powder with a 30 × 30 × 0.5 μm lamellar shape [18]) were used for visualisation, the investigated flow area was illuminated by means of a laser beam unfolded into a plane with a cylindrical lens (for the description of the laser beam unfolded into a plane we will hereinafter use the term laser knife). In the experiments, a diode-pumped solid-state laser operating in continuous mode with an emission wavelength of 532 nm and a power of 500 mW was used. The selected distance from the laser to the machine and the angle of the beam reversal by the lens allowed for a measuring area (the area illuminated by the laser knife) whose size exceeded the cross-sectional area of the working area of the machine. The plane of the laser cutter can be rotated vertically or horizontally. In the first case, it has been oriented in such a way that it passes through the geometrical centre of the cylinder inserted in the container at right angles to the wall of the outer container. In the horizontal positioning of the laser knife, its direction was set perpendicular to the wall of the outer container, and the distance from the bottom was a variable parameter. The visualised flow pattern was recorded using a video camera, the optical axis of which was perpendicular to the plane of the laser cutter. The range of variation of the problem parameters is limited by the available values of the geometric dimensions of the installation and the physical properties of the fluid. For the experiments carried out, the Reynolds number (Re) varies from 50 to 100,000 (similarity scale of flow) and the Froude number (Fr) varies from 0.1 to 15.0 (ratio of flow rate to gravity wave rate). The characteristic scale length δ is 0.2–0.6 mm (Stokes boundary layer, a layer in/which viscous effects affect flow velocity), the ratio of disk radius to shell radius R R /0 , may vary from 0.17 to 1.00, the ratio of shell radius to liquid column height R0 H may vary from 0.08 to 20.00. The medium is not stratified. 2. Magnetic Induced Vortex Machines (MIVM) Part of the experiments were carried out on current simulation setups in which the vortex flow is created by means of a magnetic armature coupled to a motor by a magnetic field. Magnetic stirrers are used as vortex current inductors: Intllab MS-500 and ES-6120 with heating. Magnetic stirrers consist of an enclosure made of metal or plastic, with an electric motor inside, which in turn drives the magnetic elements by means of an electric current. The desired substance is placed in a container, made of glass or special plastic, which is placed on a special platform of the magnetic stirrer. The anchor is placed in the container, with the help of which the liquid is stirred. The anchor for the magnetic stirrer is a magnetic rod with an inert coating made of polyethylene or Teflon. The anchor (stirrer) is driven by a rotating magnetic field from the drive in

1.2 Experimental Modelling of Vortex Flow: Equipment, Methodology …

7

Fig. 1.3 Magnetic Induced Vortex Machines (MIVM)

the housing of the device. The ES-6120 magnetic stirrer with heating is additionally equipped with a heating pad located above the electromagnet. Containers of different shapes and sizes (rectangular, square, triangular as well as cylindrical) were used in the experiments (Fig. 1.3). The characteristics of the containers are shown in Table 1.1. It is obvious that the volume of the container increases as the number of angles in the cross-section decreases with the same relative elongation. Side length a for all container configurations, r is the radius of the circle describing the polygonal container and the ratio of polygonal to cylindrical areas are given in Table 1.1. S N and S cyl are cross-sectional areas of polygonal and cylindrical containers respectively [19]. The experiments are carried out after the flow has been established and all transients have ended. First the flow pattern and the shape of the cavern on the surface of the liquid is recorded. Part of the experiments were carried out in closed containers of circular and square cross-section. A schematic and detailed description of the setup is given in [19]. Parameters of the vortex flow In a simple geometry experiment, a rather complex flow including both vortex and wave components both in the liquid column and at the free surface arose. A schematic of the flow occurring at the free surface is shown in Fig. 1.4. The evenly rotating disc, due to the sticking condition, spins the liquid around its vertical axis and simultaneously flings it along its surface towards the container wall. The accelerated liquid rises along the walls of the container, shifts towards the centre

8

1 Vortex Flow in a Homogeneous Fluid

Table 1.1 Geometric parameters of polygonal containers N H, mm

a, mm

1

5

6

Paral.

Cylinder

300

300

600

400

300

2

600

600

600

–

–

600

–

500

–

–

–

300

4

–

–

–

–

–

300

5

–

–

–

–

–

300

1

300

300

300

300

300

–

2

300

300

300

–

–

–

3

–

350

–

–

–

–

–

–

–

–

400

–

–

–

–

–

–

300

1 2

–

–

–

–

–

300

3

–

–

–

–

–

200

4

–

–

–

–

–

150

5

–

–

–

–

–

100

0.14

0.32

0.55

0.83

0.42

1

0.31

0.72

1.23

1.86

0.95

1

SN /Scyl

a/H

4

300

3

b, mm R, mm

3

0.55

1.27

2.19

3.31

1.70

1

1.24

2.86

4.93

7.44

3.82

1

–

0.43

–

–

–

–

–

0.97

–

–

–

–

–

1.73

–

–

–

–

–

3.90

–

–

–

–

1

1

1

1

0.5

0.75

2

0.5

0.5

0.5

3

0.7

along the free surface and sinks in the vicinity of the axis of rotation, forming a flow towards the centre of the disc which compensates for the constant transport of the substance along its surface. Immediately above the surface of the disc, the particles rotate and simultaneously shift from the centre to the edge. The observed flow pattern can be schematically reduced to a combination of two vortices, one is vertical cylindrical (around vertical axis, angular velocity ωc ) and the other is toroidal, with circular axis spanning the central vertical axis with local angular velocity ωt (Fig. 1.4a). Their combined action produces a composite flow in which fluid particles move along spiral and helical trajectories. The compositional flow formed by the superposition of two vortices is characterised by the combined frequency ω = ωc + ωt .

1.2 Experimental Modelling of Vortex Flow: Equipment, Methodology …

9

Fig. 1.4 Postulated scheme of the flow occurring in a cylindrical container under the action of a disc: a general view; b central section of the free surface of a compound vortex with wave perturbations

The vortex flow forms a surface cavern of a complex shape characterising the pressure distribution in the fluid (Fig. 1.4b). The shape of the central section of the cavern is characterised by a function ζ (r, ϕ), on which periodic perturbations— waves of various types (inertial, gravitational, capillary), amplitudes and lengths— are superimposed. To classify them, characteristic space–time scales are used to distinguish large-scale (inertial waves) and shorter spiral waves of smaller amplitude. The cavern depth depends on the radius of the cylindrical container R0 , the speed of the activator disc Ω, its dimensions R and roughness, the depth of the liquid layer H. Marker movements were recorded for two types of vortex flow. The first one is the flow of homogeneous liquid (Fig. 1.5a), the second one is the flow of base liquid with a small portion of immiscible light additive forming an “oil body” in the central part of the free surface (Fig. 1.5b).

Fig. 1.5 Flow pattern in a composite vortex: a H = 40 cm, Rd = 7.5 cm, Ωd = 14.2 s−1 , b side view, c top view

10

1 Vortex Flow in a Homogeneous Fluid

The base current is created by the disc rotating at a given angular velocity. At high inductor rotational speeds, a cavern is formed on the free surface, which is distorted by various types of waves as the rotational speed increases further (Fig. 1.5c). For the initial study of the marker movement patterns on the vortex flow surface, the inductor rotation frequencies were selected so that the surface cavern was the least pronounced. The intrinsic vertical vibrations of the liquid layer of depth H in the field of gravity g are characterized √ by the frequency of the main mode in the shallow water approximation Ω H = g/H . The equality of the gravitational g and centrifugal Ω2 R accelerations at the edge of the disk specifies the critical value of the inertial frequency √ √ Ω I = g/R. The other critical frequency is given by the condition Ω E = g H /R. The inertial frequencies are related by the relation Ω2I = Ω H Ω E . When the rotational speed of the disc passes through these critical values, the flow pattern may change. The system of fundamental equations for the flow of a homogeneous incompressible viscous fluid, including the Navier–Stokes and Dalamber continuity equations as well as the boundary conditions of adhesion on solid walls and pressure constancy at the free surface, is presented in [20]. In a fluid flow with instantaneous angular velocity ω, which is excited by a disk rotating with angular velocity Ω, the inertial scales given by the ratios of free fall acceleration to angular velocity of the disk—ΔΩ = g/Ω2 and fluid particles— Δω = g/ω2 and the microscales describing singularly perturbed components (Stokes √ √ boundary layers) on the disk—δΩ = ν/Ω and other contact surfaces—δω = ν/ω, including the free surface (Table 1.2). The dynamic experimental conditions  2are characterized by basic dimensionless Re = R Ω /v (also Ekman number Ek = 1/Re) parameters—Reynolds number   and Froude number Fr = R 2 Ω2 /g H . The two-layer medium flow is additionally characterized by Atwood numbers At = (ρ1 − ρ1 )/2(ρ1 + ρ1 ) and Bond numbers Bo = g H 2 (ρ1 − ρ1 )/σ , where ρ 1 and ρ 2 are densities of constituent fluids. Additional dimensionless parameters of the problem are determined by the ratios of Table 1.2 Main flow parameters Parameter Reynolds number Re = Froude number Fr =

(R 2 Ω)/ν

(R 2 Ω2 )/g H

Weber’s number W e =

ρL 3 Ω2 /σ

Minimum value

Maximum value

50

1 × 105

1 × 10−2

15

0.02

300

Bond’s number Bo = g H 2 (ρ1 − ρ2 )/σ

0.15

2.0

Atwood’s number A = (ρ1 − ρ2 )/(ρ1 + ρ2 )

15 × 103

40 × 103

ξ H = R0 /H

0.08

3.0

ξ R = R0 /R

0.3

1.0

ξb = 2R0 / h 0

0.01

3.0

LΩ =

25

4 × 103

2 × 10−4

6 × 10−4

g/Ω2

√ δΩ = ν/Ω

1.3 General Model of Vortex in a Cylindrical Container

11

characteristic linear dimensions: ξ H = R0 /H —relative depth of the container, ξ R = R0 /R—relative radius of the inductor. With values ξ H >> 1 the container is considered shallow and with ξ H > H the pool is defined as shallow, and at Ro 500 rpm) of disc rotation, the following feature was observed: the air volume in the ‘spout’ of the air part of the funnel, bounded horizontally by the solid rotation area, is constant and independent of the container geometry and dimensions. One of the obvious properties of the vortex is the conservation of mass of the rotating water. In the case of the zero approximation form, neglecting the surface tension effects     2 − c2 ω2 2 Q(r − c) (1.5.8) r Q(c − r ) + a 2 ζ(r ) = ζ0 + 2g r2 where c is the radius of solid rotation, the law of conservation of mass takes the form R0 ζ(r )r dr = 0

(1.5.9)

0

Here R0 is the radius of the container. Since ζ(r ) represents the deflection of the liquid surface from its unperturbed horizontal state, (1.5.9) means that the volume of water Vw in the region of positive deflections above the surface is equal to the volume of air Va in the funnel trough below the unperturbed surface, as shown in Fig. 1.16.

Fig. 1.16 Schematic (a) and photo (b) of vortex funnel Ω = 660 rpm

1.5 Cave Shapes Composite Vortex in Clear Water

23

Integration of (1.5.9) leads to the result which determines the sinkhole depth at the centre of the funnel ζ0 = −

 c 2 ω2  2 R0 − c2 (0.75 + ln R0 /c) < 0 2 g R0

(1.5.10)

The inequality in (1.5.10) follows from the fact that R02 − c2 (0.75 + ln R0 /c) > 0 at R0 ≥ c, as it occurs in the experiments. The angular frequency ω in relations (1.5.8–1.5.10) is the solid-state rotation frequency at the centre of the funnel. Substituting (1.5.10) into (1.5.8) results in an expression for the shape of the funnel with only one unknown parameter, the solid-state rotation frequency. The height of water rise on the container wall, i.e. at r = R0 , is determined by the expression ζ(R0 ) =

ω2 c 4 (1 − 4 ln c/R0 ) > 0. 4g R02

(1.5.11)

For the purpose of further research, it is convenient to present expression in the form   c2 c 2 ω2 (1.5.12) 1 − 2 (0.75 + ln R0 /c) ζ0 = − g R0 Experimental studies show that at significant values of disc speed (Ω ≥ ) the solid-state rotation radius decreases and becomes considerably 500 smaller than the container radius R0 , i.e. the condition is fulfilled c ≪1 R0

(1.5.13)

Simultaneously with this fact, experiments show that the depth of failure at the centre of the funnel (at Ω ≥ 500 rpm) is directly proportional to the speed of the disc, i.e. |ζ0 | ∼ Ω

(1.5.14)

If we assume the hypothesis that ω ∼ Ω, then on the basis of (1.5.12–1.5.14) it follows |ζ0 | ∼ Ω, |ζ0 | ∼ ω; c ∼ Ω−1/2 , c ∼ ω−1/2

(1.5.15)

and from (1.5.14) we obtain the following dependence on the height of the water rise at the container wall ζ(R0 ) ∼ A + B ln Ω, ζ(R0 ) ∼ A + B ln ω

(1.5.16)

24

1 Vortex Flow in a Homogeneous Fluid

Fig. 1.17 Air “spout” of the solid rotation vortex

where A and B are some values determined by the value of gravity acceleration and the geometric characteristics of the container, but independent of the speed of the disc. So, according to (1.5.15) c = c0 /Ω1/2

(1.5.17)

It is then possible to determine the value of c0 (this value has the dimension [c0 ] = cm/s1/2 and is constant for containers with a given diameter, height and for a given activator disk) from the experimental results and predict in advance the radius of solid-state rotation at other frequencies at which the experiment has not yet been set. Thus, at Ω ≥ 500 rpm the expression for the sinkhole depth at the centre of the funnel with a good approximation is (Fig. 1.17) ζ0 ≈ −c02 ω/g.

(1.5.18)

Based on (1.5.8) and the results obtained above, the air volume in the “spout” of the air part of the funnel, bounded horizontally by the solid rotation area, is calculated ω2 V = 2π 2g

c (c − r )(c + r)r dr =

π c04 π c 4 ω2 → . 4 g 4 g

(1.5.19)

0

The arrow sign means that the last value is obtained if Ω ≥ 500 rpm, when the formula (1.5.17) is valid. It can be seen that, at significant speeds, the volume of the air ‘spout’ is independent of frequency. Thus, it follows from (1.5.19) that c0 = (4gV /π)1/4

(1.5.20)

1.6 Characteristic Processes at the Free Surface of a Compound Vortex Flow

25

and on the basis of (1.5.17) this means that the radius of the solid rotation when the above conditions are fulfilled is 1/4  c = 4gV /πΩ2d

(1.5.21)

This is quite an engineering formula, applicable to practical calculations. For example, by measuring the radius of a solid rotation, we immediately determine the volume of the ‘spout’.

1.6 Characteristic Processes at the Free Surface of a Compound Vortex Flow Under different experimental conditions at the Vortex Torsion Flow (VTF) (Fig. 1.2), the free surface of the liquid has a different shape and consists of two parts: near the central axis the liquid level change ∼ r 2 ; further from the centre the character of the liquid level change becomes ∼ 1/r 2 (r —distance from central axis of cylindrical container to observation point). Also, within a certain range of installation parameters, systematic fluid perturbations are observed on the surface of the cavern. In the following we will use the term “waves” to refer to these structures, since the appearance of the observed surface disturbances has much in common with waves, in the classical sense of this term. For example, each individual disturbance has a profile similar in form to a surface wave. On the other hand, the features of the mutual arrangement of the observed disturbances in a certain range of plant parameters allow us to identify them as dissipative structures (similar to Benard cells) [34]. In a series of experiments, waves on the free surface of the composite vortex were visualised and it was found that the position of the waves relative to the laboratory coordinate system and to each other, as well as their number, did not change over time in a certain parameter range. In Fig. 1.18a–d it is clearly seen that the waves occupied an unchanged position in the coordinate system associated with the setup and relative to each other. Six disturbances were seen on the surface and their number was maintained. A spatial oscillation of the angular position of the wave crests was observed, which did not lead to their displacement. Based on the data obtained, it can be stated that the free surface shape admits a group of orthogonal eigen-transformations SO(6). In order to measure the spatial period and average frequency of oscillations of the waves generated on the free surface of the composite vortex created in the VTC installation, the photometry method is used [35]. The measurements are based on a video recording of the experiment in digital form, separate frames from which were subjected to processing. For each frame a section of the image was selected where waves on the flow surface were clearly distinguishable. From the photometric point of view, such an area of the free surface is an image area with strongly

26

1 Vortex Flow in a Homogeneous Fluid

Fig. 1.18 Perturbations on the free surface of a compound vortex current keep its geometrical position and quantity (R = 15.0 cm, H = 40 cm): a Ω = 190 rpm, b Ω = 235 rpm, c Ω = 255 rpm, d Ω = 275 rpm, e Ω = 290 rpm, f Ω = 310 rpm

differing brightness values, which correspond to crests and troughs of surface waves (Fig. 1.19). The selected area of the image should have a width of at least three points to eliminate random errors in determining the illuminance of the free surface area, but no more than five, in order to avoid excessive averaging of the wave characteristics along its crest. Typically, the image was three pixels wide, hence the corresponding matrix contained three columns of values. The luminance value was calculated as the arithmetic mean of the row elements. The values were normalized to the value of the maximum element of the resulting column of brightness values. The column element numbers corresponding to the pixel numbers of the selected image area were recalculated into length units. The conversion factor was defined as the ratio of the measured on the image (in pixels) diameter of the working area of the installation to its real value (in meters). The length values were normalized to the radius of the working area of the installation. Spatial period of waves was determined on the basis of the obtained graph of dependence of averaged and normalized brightness values on reduced coordinates (Table 1.4). In order to verify the claim of symmetry of the free surface elements the shape of all 6 waves was compared. For this purpose, the radius r of the point (Fig. 1.20a) belonging to each wave was measured with equal steps of angle ϕ. Subsequently, a graph of the dependence of r on ϕ in the polar coordinate system was plotted for each wave. Comparison of the plots showed the coincidence of the waveforms with an accuracy of 23% (Fig. 1.20b) [35].

1.6 Characteristic Processes at the Free Surface of a Compound Vortex Flow

27

Fig. 1.19 Schematic for determining the spatial period of surface disturbances, shown at R = 15.0 cm, H = 40 cm, Ω = 190 rpm Table 1.4 Measured spatial periods of surface disturbances at different experiment parameters (R = 15 cm)

H = 30 cm, Ω = 200 rpm

H = 40 cm, Ω = 310 rpm

λ, cm 1.02

1.57

1.10

1.38

1.11

1.14

Fig. 1.20 Schematic (a) plotting (b) the dependence of r on ϕ in the polar coordinate system for each wave

28

1 Vortex Flow in a Homogeneous Fluid

Fig. 1.21 Schematic of calculation (a) and graph (b) of dependence of relative illuminance (values were normalized to maximum in series) of the same section of each frame in series on time (normalized to frequency). Parameters of the experimental setup: R = 15 cm, H = 40 cm, Ω = 190 rpm

On the basis of the averaged brightness values obtained, a graph was plotted (Fig. 1.21), from which the average oscillation frequency of the waves was determined, which for the presented example (R = 14.0 cm, H = 30 cm) is ν = 1.52 s−1 . The waves have a sharp crest and a gentle wide (wider than the width of the crest) trough. In all conducted experiments it was found that the orders of measured instantaneous values of wave lengths do not change with time, which may serve as indirect evidence of a direct connection between the parameters of the vortex current and the disturbances occurring on its free surface.

1.7 Spiral Structure of Liquid Particle Trajectories Near the Vortex Surface The zero approximation form calculated in [33] agrees well with the experimental observations. At the same time, visualisation of liquid particle trajectories indicates that in the vortex flow near the free surface there is a velocity component normal to the azimuthal component and tangential to the vortex surface. In order to take into account this experimental result and in order to adequately describe the fluid motion, the study of the flow near the free surface is carried out in the accompanying coordinate system (L, ϕ, n), where s is the arc length along the zero approximation surface, counted from the point of greatest depressed vortex surface (whose coordinates in the cylindrical system are given by values (0, 0, θ(0))) to the point (r, 0, θ(r )). Since the shape of the zero approximation θ(r ) is known [36], the expression determining the arc length takes place

1.7 Spiral Structure of Liquid Particle Trajectories Near the Vortex Surface

29

Fig. 1.22 Surface related coordinate system

r L(r ) =

(1 + θr2 )1/2 dr

(1.7.1)

0

The coordinate ϕ coincides with the azimuthal coordinate of the original cylindrical coordinate system, and the coordinate n is counted along the zero approximation surface normal. Schematically, the associated coordinate system (more precisely, its unit orthodes) is shown in Fig. 1.22a. The unambiguous relation between the coordinates r and L allows one to consider the form of the zero approximation as a known function of the variable s, i.e. θ = θ(L). The radius-vector of a point lying on the surface of the zero approximation is defined by the expression rθ = R(L) er + θ(L) ez

(1.7.2)

where R(L) is a known function of the radius of a point on the surface. Since the zero approximation shape is a one-parameter surface of rotation, the expression eL =

∂rθ = R , er + θ ez ∂L

(1.7.3)

on the basis of which the condition es · es = 1 follows R, ≡

√ 1 − θ2

(1.7.4)

and a positive sign of the root is chosen. Hereafter, the derivatives of the functions θ(L) and R(L) are indicated by dashes. If a point with coordinates (r, ϕ, z) is given near the free surface, in (L, ϕ, n) its coordinates are determined according to the relations

30

1 Vortex Flow in a Homogeneous Fluid

r = R(L) − nθ, ϕ = ϕ, z = θ + n R ,

(1.7.5)

and the relations between the orthoses of the systems in question are given by the expressions er = R , e L − θ, en , eϕ = eϕ , ez = θ e L + R , en .

(1.7.6)

Relationship (1.7.5) gives rise to rules for differentiating an arbitrary function f R, f , − θ f n, , D(L, n) s θ f , + R , f n, , f z, = D(L, n) s

fr, =

D(L, n) = −

nθ +1 (1 − θ2 )1/2

(1.7.7)

The resulting relations (1.7.4–1.7.7) allow us to write the equations of motion in the coordinate system (L, n, ϕ) as follows:   ∂v L 1 ∂ D 1 v L ∂v L ∂v L + + vn + vL + p, ∂t D ∂L ∂n D ∂n D   vϕ ∂v L , − R vϕ − gθ =− R − nθ ∂ϕ L   vϕ ∂vϕ ∂vϕ ∂vϕ 1 v L ∂vϕ , + + vn + + R v L − θvn + p, = 0 ∂t D ∂L ∂n R − nθ ∂ϕ R − nθ ϕ     vϕ ∂vn vL ∂ D 1 ∂vn ∂vn ∂vn − vL − + vn + + θvϕ ∂t D ∂n D ∂L ∂n R − nθ ∂ϕ = −( pn, + g R , ) ∂vϕ ∂ ∂(R − nθ)vs +D + (1.7.8) [D(R − nθ) vn ] = 0 ∂L ∂ϕ ∂n The pressure at the point with coordinates (L, n) (there is no angular relationship in this problem) is determined by the expression   p(L, n) = g θ(L∗ ) − θ(L) − n R , (L) + p0 + p(L, ˜ n)

(1.7.9)

in which all pressure components are normalised to water density. The first term in (1.7.9) is the hydrostatic pressure of the water column above the point P(L, n), as shown in Fig. 1.22b. The second and third terms are the atmospheric pressure and the pressure due to water flow, respectively. Since, according to Fig. 1.22, the radial distance from the axis of the cylindrical coordinate system at P(L, n) and (L∗ , 0) is the same, the relationship R(L∗ ) = R(L) − n θ(L)

(1.7.10)

1.7 Spiral Structure of Liquid Particle Trajectories Near the Vortex Surface

31

In the cylindrical coordinate system, the equations of motion and the boundary conditions for the zero-order form are pz, = −g pr, =

v2ϕ r

p|z=θ = p0

(1.7.11)

which in the coordinate system (s, n, ϕ) take the form  θ(L) , , R (L) p + pn + g = 0 R , (L)D(L, n) L v2ϕ v2ϕ R , (L) , p L − θ(L) pn, = ≡ , p|n=0, L=L ∗ = p0 D(L, n) R(L) − n θ(L) R(L∗ ) R(L) − n θ(L) = R(L∗ ) (1.7.12) 

,

The coordinate L ∗ of the point on the surface corresponding to the point P(L , n) is a function of the coordinates L and n R , (L)D(L, n) ∂ L∗ = , ∂L R , (L∗ )

∂ L∗ θ(L) =− , ∂n R (L∗ )

(1.7.13)

Using (1.7.12) and (1.7.13) together, the system (1.7.13) can be transformed into p˜ , −

v2ϕ R(L∗ )

=−

θ(L∗ )g , R , (L∗ )

 p| ˜ n=0, L=L ∗ = σ

θ,, (L∗ ) θ(L∗ ) + , R(L∗ ) R (L∗ )

 (1.7.14)

∂ p˜ ∂ p˜ 1 = R , (L , using the zeroing condition for the viscous terms Since p˜ , = ∂ R(L ∗) ∗) ∂ L∗ in the Navier–Stokes equation leads to the result



 p=

R(L∗ )

vϕ R(L∗ )

2

R(L∗ )R , (L∗ )ds∗ −

gθ(L∗ ) R(L∗ )

whose substitution into the boundary condition (1.7.12) forms the equation 

v2ϕ (R(L))R , (L) R(L)

 d L − gθ(L) = σ

θ, (L) θ,, (L) + , R(L) R (L)

 (1.7.15)

For areas of central (solid) and peripheral flow, which are characterised by an azimuthal velocity of the form   1. vϕ = ω r = ω R(L) − n θ, (L) = ωR(L∗ ), 0 ≤ r ≤ a 2. vϕ =

ωa 2 ωa 2 ωa 2 = = , r ≥a r R(L) − n θ, (L) R(L∗ )

Equation (1.7.15) transforms into the equations

(1.7.16)

32

1 Vortex Flow in a Homogeneous Fluid

  , θ,, (L) 1 θ (L) (ωR(L))2 − gθ(L) + gθ0 = σ + , 2 R(L) R (L)   , 2  2 ωa θ (L) θ,, (L) 2. − − g(θ(L) − θ0 ) = σ + , 0.5R 2 (L) R(L) R (L)

1.

(1.7.17)

whose solutions are necessary to obtain an explicit form of dependence θ(L). The integration of (1.7.17) determines the shape of the zero approximation surface, neglecting the surface tension effects    2 2 ω2 a 2 2 3 θ(L∗ ) = θ0 + W(R(L∗ )) − a R (L∗ )W(a − R(L∗ )) + 2a − 2g R(L∗ ) (1.7.18) which is equivalent to the results of [36] presented in the cylindrical coordinate system in the same approximation. The obtained result allows one to investigate the structure of the velocity field near the free surface of the vortex in the coordinate system (L, n, ϕ). The condition of incompressibility of liquid is satisfied at velocity components defined by the relations vL =

ψ,L ψ,n , v = − n R − nθ, D(R − nθ, )

(1.7.19)

to which the expressions correspond in the cylindrical coordinate system 

, ,  R − nθ, L ψ,n − R − nθ, n ψ,L



D R − Dnθ, , ,  ψ,n θ + ψ,n n R , L − θ + n R , n ψ,L

vr = vz =

(1.7.20)

D R − Dnθ,

Satisfying the kinematic boundary conditions leads to their formal mathematical formulation | vn |n=0 = 0 → ψ,L |n=0 = 0,

vz /vr |n=0 =

θ, R,

(1.7.21)

indicating that near the zero approximation surface the velocity field must acquire a tangential (to this surface) character. The assumption is made that the nature of the flow near the surface is such that in the solid rotation region vr = A(L)(R − nθ), vn ∼ n and at the periphery of the vortex (also near the surface)

(1.7.22)

1.7 Spiral Structure of Liquid Particle Trajectories Near the Vortex Surface

vr =

B(L) , vn ∼ n R − nθ,

33

(1.7.23)

Substitute (1.7.22, 1.7.23) into (1.7.20) and find the solution of the resulting equations in the form vr =

B(L) , vn ∼ n R − nθ,

(1.7.24)

leads to the following results. For the solid rotation area:     , R ,, (R − nθ, ) , , , ,2 v L = −~ ω R (R − nθ ) − 2n R θ , vn = ~ ωn 2R + D ,     1  ~ ω R 2n D − θ, (R − nθ, ) vr = − ~ ω R − nθ, R ,2 1 − nθ,,, /R ,3 , vz = D D (1.7.25) but for the periphery: ~ ωn R ,, ~ ω R, , vn = − Y DY R ,2 + n R ,, /θ, R , θ, , vz = −~ vs = −~ ω ω DY DY , Y = R − nθ

vs = −

(1.7.26)

In the presented expressions ~ ω is some constant having dimension of frequency, the sign “minus” is chosen due to the fact that in experiments [37] in the nearsurface region motion of liquid to rotation axis and downwards is observed. Relations (1.7.25, 1.7.26) satisfy kinematic boundary conditions and also allow one to calculate trajectories of liquid particles near the vortex surface. Since v L = L , vϕ = R ϕ, ˙ vz = z˙

(1.7.27)

where the dot at the top denotes the time differential, the integration of (1.7.27) with account of (1.7.25, 1.7.26) leads to the following result. In the field of solid-state rotation:     z1 R = R1 e−εt , z = 2g − 1 − e−2εt (ωR1 )2 + 2g t = ϕ − ϕ1 (1.7.28) but in the peripheral area:

34

1 Vortex Flow in a Homogeneous Fluid

R = R2 e

−εt

2  1 ωa 2 εt , z = z2 − e 2 g R2

(1.7.29)

Here ε = ~ ω/ω (according to experimental data ε ≪ 1), and Ri , z i ϕi are integration constants (1.7.27) in the corresponding vortex flow regions. The stitching of trajectories at R = a (at the boundary of the solid and peripheral regions) relates the constants Ri , z i ϕi . Calculations using formulae (1.7.28, 1.7.29) show that liquid particles near the surface move in spirals from the periphery to the centre of the flow (Fig. 1.23) [37]. Experiments carried out with water-miscible admixture (ink, uranyl) showed that liquid particles move along the free surface along spiral trajectories (Fig. 1.24), while the transport of liquid particles in the liquid column follows spiral descending trajectories whose radius varies little along the vertical.

Fig. 1.23 Trajectories of liquid particles near the vortex surface

Fig. 1.24 Transfer of soluble marker on surface and in the liquid column (H = 30 cm, R = 5.0 cm). a, b t = 7, 23 s (Ω = 180 rpm), c Ω = 200 rpm

1.8 Flow Near the Disc

35

As can be seen from Fig. 1.24, the distance between the arms of the spirals grows with increasing radial distance. This means that the observed spirals are of the logarithmic type (the Archimedean spiral has constant arm spacing) in full agreement with the theoretical results (1.7.29). Thus, the experimental and theoretical study of the described vortex flow has shown that the trajectories of moving liquid particles near the water surface are spatial spirals along which these particles move from the periphery to the centre of the vortex. The good agreement between the shapes of the free surface described by the analytical relations and those observed experimentally at different vortex flow parameters indicates the applicability of the simplified theoretical description of this type of flow developed in this paper.

1.8 Flow Near the Disc The problem of flow near a disk serving as an inductor for a compound vortex is a problem which requires a separate consideration. This problem is considered under the assumption that the rotating inductor is in contact with the fluid only. Let v = uer + veϕ + wez be the velocity field in a cylindrical coordinate system. The physical fields are assumed to be stationary and independent of the azimuthal angle ϕ. Since the development of the flow in the composite vortex is due to the viscosity of the water, the sticking conditions on the disk are fulfilled. In solving the problem one assumes that near the disk surface the vertical component of the velocity w is independent of the radial coordinate r . Let w = −2 A(z), then from the continuity equation ur, + ru + w,z = 0 necessarily follow u = r A,z . The azimuthal velocity component appears in the form v = Ω r B(z). Then from the sticking conditions follows | A(z)|z=0 = A,z (z)|z=0 = 0,

B(z)|z=0 = 1

(1.8.1)

Since stationary flow in water obeys the equations of motion   1 v2 1 u ,, = − pr, + ν urr + ur, + u,,zz − 2 r ρ r r   uv 1 v ,, = ν vrr uvr, + wv,z + + vr, + v,,zz − 2 r r r   1 1 ,, uwr, + ww,z = − pz, + ν wrr + wr, + w,,zz − g ρ r

uur, + wu,z −

(1.8.2)

then, using the assumptions made, based on (1.8.2), the following equations are valid   ,, νBzz + 2 ABz, − B A,z = 0

36

1 Vortex Flow in a Homogeneous Fluid

  1 , ,, ,2 2 2 pr = r νA,,, zzz + 2 A A zz − A z + ω B ρ   1 − pz, = 2 νA,,zz + 2 A A,z + g ρ

−

(1.8.3)

Integration of the second and third equations of system (1.8.3) gives rise to the relations p=−

 ρr 2  ,,, 2 2 νA zzz + 2 A A,,zz − A,2 + f (z) z +ω B 2 p = −2ρ(ν A,z + A2 ) − ρgz + h(r )

(1.8.4) (1.8.5)

where f and h are arbitrary functions of their arguments. The joint condition (1.8.4) and (1.8.5) leads to the requirement of   f (z) = −2ρ νA,z + A2 − ρgz + c1 ,, ,2 2 2 νA,,, zzz + 2 A A zz − A z + ω B = c2

h(r ) = −c2

ρr 2 + c1 2

(1.8.6)

where c1,2 are some constants. With no disc rotation (Ω = 0) there is no flow, hence A = 0, p = −ρgz + c1 , c2 = 0 and the system (1.8.3) reduces to two equations   ,, ,, ,2 2 2 νBzz + 2 ABz, − B A,z = 0, νA,,, zzz + 2 A A zz − A z + Ω B = 0

(1.8.7)

at p = −2ρ(ν A,z + A2 ) − ρgz + c1 . When solving system (1.8.7) near the disk surface, the functions A and B are given by A(z) = a2 z 2 + a3 z 3 + a4 z 4 + a5 z 5 + · · · B(z) = 1 + b1 z + b2 z 2 + b3 z 3 + b4 z 4 + b5 z 5 + · · ·

(1.8.8)

which satisfy the sticking conditions (1.8.1). Substitution (1.8.8) into (1.8.7) defines the coefficients ai , bi , so that b1 Ω2 z 4 b2 Ω2 z 5 b3 Ω2 z 6 3 Ω2 z 3 νb3 z 2 − − − 1 − + ··· 2 6ν 12ν 60ν 120ν   b1 Ω2 z 5 Ω2 z 4 B(z) = 1 + b1 z + b3 z 3 + b1 b3 − 2 − + ··· 3ν 4 15ν2 A(z) =

(1.8.9)

The values b1 and b3 play the role of free solution parameters. Along with A(z) and B(z) it is convenient to determine the value of C(z) = A,z

1.8 Flow Near the Disc

C(z) = 3νb3 z −

37

b1 Ω2 z 3 b2 Ω2 z 4 b3 Ω2 z 5 Ω2 z 2 − − 1 − + ··· 2ν 3ν 12ν 20ν

(1.8.10)

The values b1 and b3 cannot be set to zero, as this would violate the physical meaning of the resulting solution. Let b3 = 0. Then the main term of expansion A(z) 2 3 equals − Ω6νz . Since w = −2 A(z), with b3 = 0 it follows that the vertical liquid flow is directed away from the disk, while the experiment shows a flow toward the disk. For the vertical flow to be directed towards the disk from the first Eq. (1.8.9) it follows that b3 > 0. This result agrees with expression (1.8.10) for the value C(z): since u = rC(z), then at b3 > 0 the liquid is thrown by the disk away from the rotation axis, in full agreement with the experiment. If we put b1 = 0, then it follows from the expression for B(z) that the azimuthal velocity component increases with vertical distance from the disk plane. In fact, it can only decrease, since its presence is determined by the viscous friction force, so that the maximum azimuthal velocity is reached only on the disk. Thus, from the second relation (1.8.9) it follows with necessity b1 < 0. To further analyse the results, a dimensionless vertical coordinate x is introduced, defined by the relation z x = , δν = δν

/

ν Ω

(1.8.11)

where δν is the thickness of the boundary layer formed on the rotating disk. Dimensionless parameters are also introduced at β1 , β3 , such that β1 =

b3 b1 < 0, β3 = 3 > 0 δν δν

(1.8.12)

As a result, the expressions for A(z), B(z), C(z) take the form   √ β1 x 4 β2 x 5 β3 x 6 3 x3 − − 1 − + · · · = Ων a(x) Ων β3 x 2 − 2 6 12 60 120  4  β1 x 5 1 x − + · · · = b(x) B(x) = 1 + β1 x + β3 x 3 + β1 β3 − 3 4 15   β1 x 3 β21 x 4 β3 x 5 x2 C(x) = Ω 3β3 x − − − − + · · · = Ω c(x) (1.8.13) 2 3 12 20 A(x) =

√

Using (1.8.13), the representation for the velocity field components takes the form √ u(r, x) = Ωr c(x), v(r, x) = Ωr b(x), w(r, x) = −2 Ων a(x)

(1.8.14)

The results obtained (1.8.14) show that rotating disk accelerates liquid particles in radial direction. Their radial velocity increases linearly with increasing distance

38

1 Vortex Flow in a Homogeneous Fluid

from rotation axis. This dependence provides equality in magnitudes of vertical flow of liquid flowing on disk and radial flow of liquid flowing away from rotation axis. Moving to the Lagrangian coordinates associated with the extracted fluid particle, the equations of motion of the fluid element follow from (1.8.14) r = Ωr (t)c(x(t)), ϕ˙ = Ωb(x(t)), x˙ = −2Ωa(x(t))

(1.8.15)

The use of the principal terms in the expansion (1.8.13) gives (1.8.15) the form r˙ = 3β3 Ωr (t)x(t), ϕ˙ = Ω(1 + β1 x(t)), x˙ = −3Ωβ3 x 2 (t)

(1.8.16)

Integration of the third equation of the system (1.8.16) leads to the result x(t) =

x0 1 + 3Ωβ3 x0 t

(1.8.17)

where x0 is the initial vertical coordinate of the liquid element. Substituting (1.8.17) into the first Eq. (1.8.16) defines the radial coordinate of the liquid element by the expression r (t) = r0 (1 + 3Ωβ3 x0 t)

(1.8.18)

where r0 is the initial radial coordinate of the liquid element. Substituting (1.8.17) into the second equation of the system (1.8.16) leads to the result ϕ − Ωt =

β1 ln(1 + 3Ωβ3 x0 t) + ϕ0 3β3

(1.8.19)

where ϕ0 is the initial azimuthal coordinate of the liquid element. Using (1.8.18), it is possible to give (1.8.19) the form 3β3 r (ϕ − Ωt − ϕ0 ) = ln β1 r0 hence 

3β3 r = r0 exp (ϕ − Ωt − ϕ0 ) β1

 (1.8.20)

The value ϕ − Ωt − ϕ0 is azimuthal coordinate of liquid element in coordinate system rotating with disk. Thus, relative to the disk surface the liquid elements, according to (1.8.20), move along logarithmic spirals. Since β3 /β1 < 0, the greater the radial coordinate of the liquid element, the more it lags behind the disk rotation.

1.8 Flow Near the Disc

39

Assessment of the physical parameters of the disc drop zone In estimating the thickness of the radial fluid outflow zone from the disc, the assumption is used that the kinetic energy of the vertical fluid flow is entirely converted into the kinetic energy of the radial outflow. This means that, as an approximation, the following equation is fulfilled d π R2

d w2 (x) d x = 2π

0

⎛ ⎝

0

R

⎞ r u2 (r, x) dr ⎠ d x

(1.8.21)

0

where d is the thickness of the drop zone. Substituting expressions (1.8.14) into (1.8.21) results in 8ν ω R2

d

d a (x) d x =

c2 (x) d x

2

0

(1.8.22)

0

Based on the assumption that at the upper limit of this zone, i.e. at x = d, the vertical inflow is maximum and the radial outflow turns to zero, the result follows but since a , (x) = c(x), the above conditions are converted to | a , (x)|x=d = 0,

| a ,, (x)|x=d < 0,

c(x)|x=d = 0

(1.8.23)

Using the first three terms of the expansion in (1.8.23) c(x) = 3β3 x −

β1 x 3 x2 − 2 3

(1.8.24)

leads to the fact that β3 =

d (2d β1 + 3), 4d β1 + 3 > 0 18

(1.8.25)

Since β3 > 0, it follows from (1.8.25) β1 > − 4d3 . Substituting (1.8.24, 1.8.25) into (1.8.22) results in ωR 2 2 2 214 d 2 β21 + 447 d β1 + 234 d = 2 2 3 ν 64 d β1 + 126 d β1 + 63 from which it follows ωR 2 2d ≈ ν 2

/ ⇒

d≈R

R ω =√ 2ν 2 δν

(1.8.26)

40

1 Vortex Flow in a Homogeneous Fluid

Since z = δν x, the transition to dimensional coordinates results in an estimate of the thickness of the drop zone of the current R zd ≈ √ 2

(1.8.27)

If we restrict the expansion (1.8.23) to the first two terms (1.8.13), the result is zd ≈

R 2

(1.8.28)

Thus, the value z d ≈ R acts as a characteristic size of the fluid throw area. The obtained result indicates that when introducing spatial scales used to reduce the equations of the problem to a dimensionless form, the radius of the exciting disk motion enters the sets of not only radial, but also vertical scales. Conclusions to this chapter Experimental investigations of vortex flows in homogeneous liquid in containers of different geometry, and also at various physical parameters of experiments on original installations for studying dynamics of formation and structure of vortex flows in a wide range of defining parameters are carried out. The methodology of collection and processing of experimental data is worked out. It is found that the shape of free surface depends on all physical parameters of the problem (depth of liquid layer, radius and shape of activator, rotation frequency). The cavern depth increases monotonically with increasing rotation frequency. A total of three types of surface caverns were observed in the experiments— smooth, with inertial (large-scale) waves and complex (with inertial and spiral waves simultaneously). Experimental results suggested that the shape of the free surface admits a group of SO(6) orthogonal eigenstrains. A theoretical relationship is obtained which describes the universal geometry of vortex caverns occurring in cylindrical vessels during rotation of a coaxial disk and coincides both with experimental data and with the previously obtained model of vortex flow. For the first time, analytical expressions have been obtained showing that the trajectories of liquid particles near the vortex surface are three-dimensional spirals along which the flow from the vortex periphery to the center occurs. It is shown that the calculated and visualized trajectories of liquid particles are in good agreement with each other and belong to the class of spatial logarithmic spirals. The problem of flow near a disk serving as an inductor of a compound vortex current has been solved assuming that the rotating inductor is in contact with liquid only and assuming that the physical fields are considered stationary and independent of the azimuthal angle ϕ. It is shown that with respect to the disk surface the liquid elements move in logarithmic spirals.

References

41

Good coincidence of free surface shapes described by analytical relations with experimentally observed at different parameters of vortex flow indicates the applicability of the simplified theoretical description of this type of flow developed in the presented work. The established form of vortex motion allows one to study the processes of substance transfer from a source with fixed properties.

References 1. Schlichting G (1969) Boundary layer theory. Nauka, GRFML, Moscow, 742 c 2. Escudier MP (1984) Observations of the flow produced in a cylindrical container by a rotating endwall. Exp Fluids 2:189–196 3. Milne-Thomson LM (1964) Theoretical hydrodynamics. Mir, Moscow, 660 c 4. Okulov VL, Sorensen JN, Voigt LK (2002) Alternation of right- and left-handed vortex structures with increasing swirl intensity. Tech Phys Lett 28(2):37–44 5. Naumov IV, Okulov VL, Sorensen JN (2007) Two scenarios of instability development in an intensely swirling flow. Tech Phys Lett 33(18):32–39 6. Naumov IV, Okulov VL, Sorensen JN (2010) Diagnosing the spatial structure of vortex multiplets in swirling flow. Teplofis Aeromech 17(4):585–593 7. Gushchin VA, Konshin VN (1992) Computational aspects of the splitting method for incompressible flow with a free surface. J Comput Fluids 21(3):345–353 8. Kremenetskiy VV, Stroganov OYu, Zatsepin AG et al (2004) Frontal currants in the rotating fluid over sloping bottom: influence of canyons. In: Selected papers of international conference “fluxes and structures in fluids—2003”. IPMech RAS, Moscow, pp 111–114 9. Karev VI, Pokazeev KV, Chaplina TO (2018) Actual problems of modelling processes in geo-environments. Process Geoenviron 1(14):818–822 10. Andersen A, Bohr T, Stenum B et al (2006) The bathtub vortex in a rotating container. J Fluid Mech 556:121–146 11. Brons M, Vougt LK, Sorensen JK (1999) Streamline topology of steady axisymmetric vortex breakdown in a cylinder with co- and counter-rotating end-covers. J Fluid Mech 401:275–292 12. Okulov VL, Sorensen JN, Voigt LK (2002) Alternation of right- and left-handed vortex structures with increasing swirling intensity. Tech Phys Lett 28(2):145–167 13. Okulov VL, Meledin VG, Naumov IV (2003) Experimental study of swirling flow in a cubic container. J Tech Phys 73(10):29–36 14. Hoerner SF, Borst HV (1985) Fluid-dynamic lift. Practical information on aerodynamic hydrodynamic lift. Hoerner Fluid Dynamics, Bakersfield, 376 p 15. Beguier C, Bousgarbies J-L, Leweke T (2001) Tourbillion, Instabilite et Decollement. CEPAD, 126 p 16. Popescu N, Robescu D (2011) Separation of petroleum residues using the vortex separation technique. UPB Sci Bull Ser D 73(1) 17. Chaplina TO, Stepanova EV (2015) Vortex flow with torsion. Laboratory simulation. In: Processes in geospheres, vol 1, pp 96–105 18. Kalinichenko VA, Chashechkin YD (2014) Structurization of suspended bottom sediments in periodic flow above vortex riffles. Izv RAN MZHG 2:95–106 19. Naumov IV (2013) Formation and optical-laser diagnostics of helical vortex structures in liquid. Doctoral thesis: 01.02.05, Mechanics of liquid, gas and plasma. Naumov Igor Vladimirovich, Novosibirsk, 313 c 20. Landau LD, Lifshitz EM (2001) Hydrodynamics, vol VI. Fizmatlit, Moscow, 731 c 21. Chashechkin YD, Baidulov VG, Bardakov RN, Vasiliev AY, Gorodtsov VA, Kistovich AV, Stepanova EV, Chaplina TO (2010) Simulation of stratified and rotating fluids flow. J Fluid Mech. Nauka, Moscow, pp 277–348

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22. Chaplina TO, Chashechkin YuD (2016) Coherent structure in oil body embedded in compound vortex. In: Advances in computation, modeling and control of transient and turbulent flows, pp 219–224 23. Stepanova EV (2009) Experimental study of fine structure of vortex flow in liquid with free surface. PhD thesis: 01.02.05, Moscow, 119 p 24. Alekseenko SV (1999) Helical vortices in swirl flow. In: Alekseenko SV, Kuibin PA, Okulov VL, Shtork SI (eds). J Fluid Mech 382:195 25. Basu P, Kefa C, Jestin L (2000) Boilers and burners. Springer, New York, 553 pp 26. Basu P (2000) Design and theory. Springer, New York, 130 pp 27. Zykova NG, Serant FA, Nozdrenko GV, Shinnikov PA (2003) Schematic-parametric optimization of TPP boilers with annular furnace. Teplofis Aeromech 3:477 28. Escudier MP, O’Leary J, Poole RJ (2007) Flow produced in a conical container by a rotating endwall. Int J Heat Fluid Flow 28:1418 29. Chiang TR, Sheu WH, Tsai SF (1999) Disk-driven vortical flow structure in cubical container. Comput Fluids 28:41 30. Okulov VL, Meledin VG, Naumov IV (2003) Experimental investigation of a swirling flow in a cubic container. Tech Phys 48:1249 31. Liberzon A, Feldman Y, Gelfgat AY (2011) Experimental observation of the steady-oscillatory transition in a cubic lid-driven cavity. Phys Fluids 23:084106 32. Anikin YuA, Naumov IV, Meledin VG, Okulov VL, Sadbakov OYu (2004) Experimental investigation of pulsation of swirl flow in cubic container (in Russian). Thermophys Aeromech 11:571 33. Kistovich AV, Chashechkin YD (2010) Deformation of free liquid surface in a cylindrical container by an attached compound vortex. Doklady RAN 432(1):50–54 34. https://dic.academic.ru/dic.nsf/ruwiki/99716 35. Shevtsov NI, Stepanova EV (2015) Application of image photometry method in some problems of hydrodynamics. Bull Mosc State Univ Ser 3 Phys Astron 3:44–48 36. Kistovich AV, Chaplina TO, Stepanova EV (2017) Vortex flow with free surface: comparison of analytical solutions with experimentally observed liquid particles trajectories. Int J Fluid Mech Res 44(3):215–227 37. Kistovich AV, Chaplina TO, Stepanova EV (2019) Spiral structure of liquid particle trajectories near vortex surface. Comput Technol 24(2):67–77

Chapter 2

Solute Admixture Transport from a Compact Source in a Composite Vortex

2.1 Experimental Studies of Solute Transport in Vortex Flow Only a few works have been devoted to problems of experimental study of transfer of marking admixture from a drop falling on the surface of rotating liquid. In the first of them [1] formation of a “wall of paint” from a droplet falling on the surface of a rotating liquid in a cylindrical or rectangular container, the uniformity of rotation of which was specially perturbed, was first described. The next work, in which the transfer of an admixture from a droplet falling on the surface of a rotating liquid and the analogy between the effects of rotation and stratification was visualized again, appeared more than thirty years later [2]. Although the results of the first publication [1] were included in the well-known monographs [3, 4] the process of substance transfer in vortex flow has not been studied in details up to now. The annular and spiral structure of vortex flow is visualised in the laboratory using soluble dyes [5, 6], smoke [7] and fine particles in electrolytic precipitation [8] (Fig. 2.1). In [12], experiments have been described in which it has been possible to generate and visualise clearly defined vortex filaments. These are generally not stable (Fig. 2.2). The effects of vortex pairing, local unsteady detachment of the flow from the wall and a new phenomenon—regular structure of turbulence on the scale of a large vortex—have been discovered by the author. The possibility of controlling the structure of turbulent flow by imposing periodic perturbations of negligibly small amplitude has been demonstrated. A number of works are devoted to experimental study of propagation of various impurities from a spot placed on free surface of composite vortex in cylindrical container [13, 14]. Specially designed experiments have shown that important elements of the substance transfer pattern are not predicted by existing models.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina, Transfer of Substance in Vortex and Wave Flows in One-Component and Multi-component Environment, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-31856-6_2

43

44

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

Fig. 2.1 Visualisation of vortex structures: a [9]; b [10]; c [11]

Fig. 2.2 Visualisation of coherent structures (large scale eddies) in a turbulent jet flowing on a barrier [12]

For example, the transformation of a dye spot in the core of a compound vortex [15], as well as the behaviour of a small amount of immiscible admixture on the rotating free surface of a liquid [16].

2.2 Structural Stability of Soluble Admixture Transfer Pattern from a Spot on the Surface of a Compound Vortex It is of practical interest to study the stability of the structure of spiral arms into which a compact spot placed on the surface of a composite vortex is transformed [17], which can be tested by successive application of the marker to the same or different flow regions.

2.2 Structural Stability of Soluble Admixture Transfer Pattern from a Spot …

45

Fig. 2.3 Change of area occupied by spiral arms on the surface of the composite vortex (H = 30 cm, Ω = 210 rpm, R = 7.5 cm): a–d t = 1, 12, 32, 29 s

a

c

b

d

The observed features of the initial evolution of the process of dye transfer from a spot arising when an ink drop falls on the surface of a compound vortex are maintained at longer times. The drop point is located at a distance of 4.7 cm from the centre of rotation of the free surface. The shape of the spot observed after 3 s (angular size of the structure relative to the centre of rotation of the free surface—79°, radial position—3.7 ÷ 3.8 cm) indicates partial immersion of the marker in the centre and stretching into spiral arms (Fig. 2.3a). The formation of the first coil of the spiral after the drop hits the free surface occurs rather quickly (Fig. 2.3b), after only t = 3 s. Here, the angular position of the edge of the spiral arm is offset from the initial spot position by 210°. By the time t = 32 s, 6 complete turns of the spiral are observed, the eccentricity of the enveloping oval √ (ε = 1 − (Tr /TR )2 , where TR and Tr are the maximum and minimum dimensions of the oval occupied by the spiral structure) is ε = 0.52 (Fig. 2.3c), the spiral arm thickness varies non-monotonically in the interval from 0.8 to 2.8 mm. At the periphery of the current the direction of the arm evolution changes and areas of return flow appear (section of curve 2, Fig. 2.3d), indicating a complex structure of the composite vortex. The eccentricity of the envelope of the central part of the spiral structure in this case is ε = 0.53, of the outer part—0.67, the thickness of the arms is from 0.4 to 2.5 mm. The rate of development of the spiral structure increases with the rotational speed of the disk. The spiral completely covers the centre of rotation at t = 1 s (τ = 3). By t = 32 s (τ = 70) more than 10 thin (0.9–2.2 mm) colored arms are formed (Fig. 2.3a), separated by bands of clear water.

46

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

As in the previous experiment, return flow loops appear at the periphery of the current, which are located in the directions of “12 h” and “6 h” (Fig. 2.3a). The distance of loop centres from the axis of rotation is 4.5 and 4.0 cm respectively. Gradually the basic centripetal current pulls the spiral strips into the central part of the composite vortex (Fig. 2.4b). By the time t = 51 s (τ = 170), the central region of 0.6 cm radius and a 1.5 cm wide clear water ring separated from it with an uneven inner edge of average thickness of 6.2 mm appear painted. On Fig. 2.5 shows photos illustrating the transfer of a soluble impurity in a vortex flow. 0.1 ml of purple ink was placed on the free surface of the vortex at a distance of 2.98 cm from the center. After 2 s, a drop of blue ink was introduced onto the surface of the vortex at a distance of 1.7 cm from the center of the vortex (Fig. 2.5a). Fig. 2.4 Evolution of the dye spot into spiral and concentric circles on the surface of the composite vortex: a, b t = 23, 51 s (H = 40 cm, Ω = 400 rpm, R = 7.5 cm)

a

b

Fig. 2.5 Change of area occupied by spiral arms on the surface of the composite vortex (H = 40 cm, Ω = 180 rpm, R = 5 cm): a–d t = 2, 4, 7, 23 s

a

c

b

d

2.2 Structural Stability of Soluble Admixture Transfer Pattern from a Spot …

47

The point of introduction of the second drop of admixture is at 0.9 cm, direction “at 8 o’clock”. From the second spot of admixture on the free surface of the liquid begins to extend its own spiral arm. The angular size of the outer spiral arm after t = 2 s (τ = 6) is Δϕ = 198°, the radial position ra is between 3.1 and 4.25 cm, the thickness δ is 1.5–0.5 mm; the outer Δϕ = 129°; ra is 1.23–0.81 cm, the thickness of δ is 5–3 mm (Fig. 2.5b). The rate of formation of the inner arm is higher than that of the outer arm: after t = 7 s (τ = 23) the inner spiral completes one and a half turns of it (angular size is Δϕ = 607°), the outer one slightly more than one (Δϕ = 367°) (Fig. 2.5c). The purple spiral arms vary in thickness from 1.5 to 0.8 mm and the blue ones from 2.5 to 1 mm. The eccentricity of the enveloping region of the inner helix is ε = 0.61, of the outer one ε = 0.37. As in the previous experiment, areas of return flow are observed at the periphery. The return loops of the outer marker are located in the “at 5 o’clock” and “at 12 o’clock” directions (Fig. 2.5d). The inner spiral has by this time more than 5 loops separated by bands of clear water. Return loops have also begun to appear in its centre (direction “at 10 o’clock”). The eccentricities of the enveloping regions occupied by the purple and blue ink are ε = 0.32 and ε = 0.25, respectively. At longer times, only the region of the centre of the small radius surface cavern, coloured by the internal structure marker, remains coloured. Figure 2.6 shows the change in thickness of the sleeve of a spiral structure extending from a drop of violet ink initially located 3.5 cm from the centre of rotation of the free surface. The sleeve elongates and thinning quite rapidly, at t = 5 s (Fig. 2.6a, broken line 1) the average width of the sleeve was 0.29 cm, a slight thickening in the central part—a trace of the original stain. After 1 s (Fig. 2.6a, broken line 2) the average width was 0.24 cm, the maximum − 0.34 cm and the minimum at the edge was 0.05 cm. After 7 s (Fig. 2.6a, broken line 3) the sleeve continues to thin, its width decreases from 0.2 cm (central part of the spiral) to 0.09 cm (outer edge). After another 9 s, the average width of the sleeve decreases to 0.12 cm (Fig. 2.6a, broken line 4). With a delay of 1 s, a double drop of blue ink was placed at a distance of 1.2 cm from the centre of rotation of the free surface (Fig. 2.6b) The width of the painted area after 2 s was 0.63 cm. Two arms began to extend from it. The width of the cyclonic arm advancing in the direction opposite to the main rotation of the free surface was 0.23 cm at t = 3 s, the maximum in the stained area was 0.9 cm and the minimum was 0.1 cm (Fig. 2.6b, broken line 1). After a drop of blue ink has been deposited on the free surface, the width of the spiral varies from 0.52 to 0.18 cm (Fig. 2.6b, broken line 2). At t = 5 s after the introduction of a new drop, the average width of the spiral structure is 0.18 cm and its central part remains thickened in comparison with neighboring parts by 0.2 cm (Fig. 2.6b, peaks on the broken line 3). With time, the arm becomes more homogeneous after 6 s, its width varies between 0.32 and 0.04 cm with an average of 0.15 cm (Fig. 2.6b, break 4). Thus, the length of the spiral arms increases and the width decreases monotonically with time.

48

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

Fig. 2.6 Spiral arm width (Ω = 180 rpm, H = 40 cm, R = 5 cm): a, b purple and blue ink, symbols 1, 2, 3, 4—t = 5, 6, 7, 9 s, respectively

2.3 Transfer of Miscible Admixture into the Thickness of the Composite Vortex The independent transfer of the dye into the thickness of the composite vortex is illustrated in Fig. 2.7. In the pictures shown, the experiment carried out with water dyeing using a water soluble fluorescent dye (uranyl), which emits green light in ultraviolet light. The penetration depth of the coloured central column increases with time (Fig. 2.7a–d). As soon as the main part of the dye descending in the central near-axial area of the flow to the activator disk reaches the bottom, intensive staining of the entire volume under investigation occurs (Fig. 2.7e, f). Intensely stained with uranium, the central part of the flow has a green colour, despite the general staining of the volume, due to the ultraviolet illumination. The penetration depth of the stained column for Fig. 2.7a–e is 13, 19, 22, 29, 35 cm from the bottom point of the surface cavern respectively. An algorithm has been developed to estimate the image brightness over the entire thickness of the volume under study, which is used to calculate the spatial distribution of the dye in the flow. In Fig. 2.8a the blue broken line 4 reflects the image brightness distribution along the horizontal level at 31 cm above the rotating activator disk; the broken line has a well-defined area in the middle of the pool where brightness is maximal and consequently there is a dye-filled area (picture is shown in Fig. 2.7b). The transverse dimension of the central stained column at this level is 3.8 cm. The intensely stained part has clear contours. The broken line 15 runs everywhere near the zero mark, this indicates that at the level of 3 cm above the rotating activator the total background brightness is uniform and, therefore, no more or less intensely coloured areas are observed compared to the total part of the flux. The broken line 7 in Fig. 2.8a (brightness distribution at the level of 22.5 cm above the activator) has a pronounced central peak, the magnitude of which is significantly smaller than that of all the lines above. If we take this level as the current dye

2.3 Transfer of Miscible Admixture into the Thickness of the Composite Vortex

a

d

b

e

49

c

f

Fig. 2.7 Transport of the admixture into the liquid column (H = 50 cm, Ω = 820 rpm, R = 7.5 cm): a–e t = 28, 38, 48, 58, 68, 78 s

penetration depth, it is 19.9 cm from the bottom point of the surface cavern, which is also confirmed by visual observation (Fig. 2.7b). The broken line 1 in Fig. 2.8b, reflecting the brightness of the image points located at a horizontal level of 42 cm above the rotating disk, shows a fairly high level of brightness fluctuation around the zero mark. The horizontal line just above the bottom point of the surface cavern corresponds to this area in the photograph, where no coloured volumes are observed. The remaining broken lines in Fig. 2.8b have a single pronounced maximum in the centre, corresponding to a bright central coloured column in the centre of the volume under investigation. Its average width here is about 4.5–5 cm. Experiments have shown that the width of the column is not constant along the vertical coordinate, while analysis of the left and right front motion has shown that the change of positions of the right and left boundaries of the painted region occurs simultaneously, i.e., the painted region performs bending oscillations. The average thickness of the coloured column is about 18 mm (i.e. the radius of the central coloured column is 0.12 of the activator size R = 7.5 cm) for the experiment at 225 rpm, at 435 rpm the average thickness of the coloured column is about 25 mm (i.e.

50

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

Fig. 2.8 Change of illuminance in the working area of the container (Ω = 820 rpm, H = 50 cm, R = 7.5 cm; 1–15—illuminance at 42 ÷ 3 cm in steps of 2.75 cm): time after dye introduction a 22 s, b 128 s

the radius of the central coloured column is 0.09 of the activator size R = 14.0 cm). Summarising the data from all experiments, it should be noted that the central column is wider in the higher frequency experiment than at lower frequencies; it is also worth noting that the coloured central part of the current is slightly wider near half the full depth of the current. A plot of the average deviation of the coloured column from the vertical over time is shown in Fig. 2.9. It is easy to see that the deviations from the vertical of the right and left boundaries of the stained region occur almost in phase, which confirms the assumption of a bending rather than varicose nature of the oscillations of the central part of the vortex flow, within which the marking admixture propagates (Fig. 2.10). The average thickness of the coloured column is about 17.7 mm (i.e. the radius of the central coloured column is about 0.06 of the container radius and 0.12 of the activator radius) for the experiment at Ω = 225 rpm. The average thickness of the coloured column is about 24.9 mm (i.e. the radius of the central coloured column is about 0.08 of the container radius and 0.09 of the activator radius) for the experiment at Ω = 435 rpm. The vortex column in the higher frequency experiment is wider than the column in the lower frequency experiment and has a thickening in the central part.

2.3 Transfer of Miscible Admixture into the Thickness of the Composite Vortex

51

Fig. 2.9 Dependence of the average deviation of the stained area boundary on time (H = 30 cm, Ω = 225 rpm): a R = 7.5 cm, b R = 14.0 cm (1—left side of the stained column, 2—right side, 3—total column width)

Fig. 2.10 Dependence of the change in thickness of the painted column on the immersion depth of the vortex (H = 20 cm): 1—Ω = 230 rpm, R = 20 cm; 2—Ω = 280 rpm, R = 32 cm; 3—Ω = 340 rpm, R = 32 cm

Analysis of the periodic component of variation of the coloured region position with time by the method of zero-crossing statistics analysis and counting of corresponding series lengths shows that at relatively low activator rotation frequency (see Figs. 2.11 and 2.12), longer waves (25 mm) appear, while for the case with higher rotation frequency, shorter waves (15 mm) appear. At high frequencies there are practically no oscillations with a wavelength greater than one third of the height of the undisturbed liquid layer, at lower frequencies there are waves with a characteristic length of half the height of the liquid column. Thus, the proposed approach to the experimental investigation of the behaviour of vortex structures has shown its high potential for its quantification of the various parameters of vortex structures. This technique allows to calculate all parameters calculated in this chapter independently of each other. Velocities, wavelengths,

52

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

Fig. 2.11 Dependence of trajectory deviation on time (H = 30 cm): a Ω = 225 rpm, R = 15 cm— blue line; b Ω = 435 rpm, R = 28 cm—red line

Fig. 2.12 Evolution of the vertical distribution of the dye of the two colours (H = 40 cm, Ω = 195 rpm, R = 5.0 cm): a, b t = 1, 12 s

trajectory deflections, thickness changes, each of these parameters can be calculated separately. The evolution of dye propagation into the thickness of the composite vortex is illustrated in Fig. 2.12 using the example of the development of structures formed by two dye droplets of different colours simultaneously placed on the free surface of the composite vortex. Both when droplets are placed smaller (Fig. 2.12a, b) and farther apart (Fig. 2.13a, b), a separate system of spiral arms on the free surface and a separate helical stained structure in the liquid column are drawn from each dye droplet. Each helical structure in the fluid column is stretched in the direction of gravity and the current created by the rotating disc. The dyes from the different helical structures do not mix; each one grows independently. The dyed structure closer to the axis of the flow elongates more slowly.

2.3 Transfer of Miscible Admixture into the Thickness of the Composite Vortex

53

Fig. 2.13 Evolution of the vertical distribution of the dye of the two colours (H = 40 cm, Ω = 195 rpm, R = 5.0 cm): a, b t = 1, 10 s

Similar behaviour is exhibited by helical structures formed from droplets of soluble dye of different composition. In all cases, helical structures form on the surface of the liquid, the degree of cohesion of which is determined by the duration of the process (Fig. 2.14). At the periphery of the flow, thin filaments are formed, separated by bands of clear water. The different dyes practically do not mix: in Fig. 2.14a, the inner core remains coloured with red ink and the outer core with blue ink; in Fig. 2.14b, on the contrary, the outer contour is coloured red. The coloured helical lines advance into the fluid column and gradually merge to form cylindrical shells (red inside blue in Fig. 2.14a, b). Fig. 2.14 Vertical stained dye structures in two colours (H = 40 cm, Ω = 400 rpm, R = 5.0 cm): a, b t = 1, 7 s

54

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

2.4 Visualisation and Qualitative Analysis of Flow Near the Disc Edge To visualise the flow at the bottom of the unit near the edge of the rotating disk, an admixture (soluble dye) was introduced, the direction of dye introduction being vertically upwards. Initially, the admixture was observed to spread in a limited area shaped like a sector with curved boundaries near the bottom, followed by a rise of the dye along the side wall of the cylindrical container. Using two cameras recording the flow pattern in mutually perpendicular directions, the size and shape of the dyed area boundary were recorded (Fig. 2.15), resulting in data on its size and shape as a function of flow parameters. For each experiment, a plot of the length of the radius vector drawn from the geometric centre of the rotating disk to the points on the painted sector boundary was plotted as a function of the angular position. The angles are counted from the position of the point where the dye is injected into the current. The approximation has been done by functions of the form r = r0 + Aeϕ/t , the coefficients in the approximation are given in Table 2.1. The errors of calculation of the coefficients do not exceed 10%. The problem statement is different for the second type of experiments, where the admixture was injected near the bottom through the side wall (the original direction of dye propagation was horizontal). When conducting experiments with the activator disk completely covering the bottom of the cylindrical container, the formation of an annular coloured layer along the vertical wall was observed. The thickness of the layer increased as the depth of the liquid layer under study increased. For example, for a layer with depth H = 20 cm, at angular speed of disc rotation Ω = 150 rpm (Fig. 2.15b, d), the thickness of annular layer averaged 1.75 cm, whereas at layer depth H = 30 cm, with the same rotation speed, the thickness of annular layer already equals 2.6 cm. Also, as the layer depth increases, a significant decrease in the linear velocity of propagation of the admixture (in this case, its horizontal component is considered) is observed. For layer depths H = 20, 30 cm tangential velocities are about 23 and 7 cm/s respectively. For comparison, the tangential velocity of the disk edge is about 100 cm/s (at disk radius R = 14 cm). The vertical velocity component also decreases with increasing depth of the fluid layer under investigation, but not as much as the horizontal component. For depths H = 20, 30 cm the velocity drops from 1.0 to 0.8 cm/s. The qualitative flow pattern observed in this series of experiments generally confirms the assumptions underlying the flow pattern in the cylindrical container and indicates the presence of a composite vortex flow in the installation. The technique of visualising the flow or its individual components with soluble dye had a number of drawbacks. Two main flow elements were stably visualised using soluble dyes: a central stained column with clear vertical boundaries and a surrounding cylindrical stained region—however, the concentration of dye in the thin cylindrical region did not allow its geometric characteristics to be stably determined at all flow parameters. Also, when the soluble dye was introduced into the flow

2.4 Visualisation and Qualitative Analysis of Flow Near the Disc Edge

55

Fig. 2.15 Sectors stained with miscible admixture when introduced near the edge of the rotating disc: a, d R = 5 cm, H = 20 cm, Ω = 350 rpm; b, e R = 5 cm, H = 30 cm, Ω = 300 rpm; c, f R = 5 cm, H = 40 cm, Ω = 250 rpm

studied, its amount and insufficient optical density did not allow conclusions to be drawn about the structure of the flow in the immediate vicinity of the flow inducer surface. In order to obtain data about the flow pattern in the whole investigated liquid volume, aluminium powder markers were injected; to highlight a part of the working volume of the installation, data about the flow in which was to be obtained, illumination was performed by a laser beam unfolded in a plane. Using this technique

56

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

Table 2.1 Approximation coefficients for angular position dependences of radius vector lengths drawn from the geometric centre of the rotating disk to points on the boundary of the coloured sector R = 5 cm H = 10 cm

H = 20 cm

A1

t1

A2

t2

Ω = 270 rpm

0.92

0.39

0.04

0.34

Ω = 590 rpm

0.28

0.26

0.11

0.42

Ω = 900 rpm

1.6

0.71

0.02

0.31

Ω = 350 rpm

1.04

0.78

0.01

0.36

H = 30 cm

Ω = 300 rpm

1.33

0.48

0.44

0.58

H = 40 cm

Ω = 250 rpm

1.34

0.46

0.47

0.55

it was possible to obtain flat “slices” of the flow pattern at different parameters of the experiment. As a result of recording the flow pattern on digital still and video cameras, images were obtained, the analysis of which provided information about the nature of admixture particle movement in different areas of the flow studied. All experiments performed can be divided into two groups. The first group includes experiments in which the particles were in the liquid volume before the vortex flow was created. The main purpose of these experiments was to visualise the structure of the vortex flow as a whole. The second group includes experiments where an admixture was introduced into individual regions of the steady flow, aimed at visualising the individual components of the flow. In order to investigate the movement of the markers in planes parallel to the plane of the bottom of the rig, experiments were conducted where the plane of the laser blade was deployed horizontally. Experiments were conducted where the secant plane was positioned at 2.0 and 15.0 cm from the bottom of the rig. Analysis of the experimental data with the laser knife at 2.0 cm showed a pronounced radial transfer of markers along spiral trajectories in the area above the edge of the rotating disk (Fig. 2.17). In the case where the disc occupied the entire area of the lower end of the unit, the most prominent radial movement was observed in the annular region near the walls of the container (Fig. 2.16a—area 2). Using the disk R = 7.5 cm, the radial motion was visualized above the disk in an annular region (Fig. 2.16b—region 2) with a thickness of 5.1 cm, whose outer contour diameter was equal to 8.3 cm. Based on these data, it is assumed that the central region (Fig. 2.16a, b—region 1), in which no radial particle transfer is observed, corresponds to the vortex core. For the experimental conditions H = 30.0 cm, R = 7.5 cm, Ω = 235 rpm, this assumption was confirmed after comparing the sizes of the vortex core obtained in the experiment with a vertical laser blade orientation and in the experiment with a horizontal laser plane arrangement. The sizes are 3.2 cm and 3.6 cm, respectively. Under other experimental conditions (H = 30 cm, R = 14 cm, Ω = 120 rpm) the comparison was not carried out due to impossibility to define the vortex core boundary exactly.

2.4 Visualisation and Qualitative Analysis of Flow Near the Disc Edge

57

Fig. 2.16 Visualisation of characteristic flow areas H = 30.0 cm at R = 14.0 cm, Ω = 120 rpm (a, c), R = 7.5 cm, Ω = 235 rpm (b, d)

Thus, in an experimental study of the transport of suspended solids (markers) introduced into a steady-state vortex through a free surface some of its characteristic features were observed. The markers were concentrated in a narrow cylindrical area around the axis of rotation, the horizontal size of this area being independent of the speed of the inductor. The radius of this area was not found to depend on the distance between the insertion point of the markers and the axis of rotation. Dependences of the depth of penetration of markers into the liquid column at various parameters of the experiment has a distinctly linear character. Experimental data were approximated by function z(t) = V t + z 0 . The dependencies are shown in Fig. 2.17. The calculated marker penetration rate characteristics are shown in Table 2.2. The experiments carried out to determine the main characteristics of suspended solids transport into the liquid column in the composite vortex flow confirmed the linear dependence of the admixture penetration depth near the vertical axis of the flow on time. Based on the results of these experiments, the shape and size of the area occupied by the particles were traced over time. An analytical solution for the flow near the disk generating a compound vortex is presented in Chap. 1 of this book. This problem is considered under the assumption that the rotating inductor is in contact with liquid only. From the solution of such a problem [18], it follows that, with respect to the disk surface, the liquid elements move in logarithmic spirals.

58

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

Fig. 2.17 Dependence of penetration depth on time. Calculated vertical velocity of the particles: a H = 20 cm, R = 7.5 cm, Ω = 150 rpm, V = 1.98 mm/s, b H = 20 cm, R = 7.5 cm, Ω = 300 rpm, V = 0.83 mm/s, c H = 30 cm, R = 7.5 cm, Ω = 140 rpm, V = 2.21 mm/s, d H = 30 cm, R = 7.5 cm, Ω = 340 rpm, V = 1.22 mm/s Table 2.2 Admixture penetration rates into the thickness of the composite vortex H, cm

R, cm

25

5

Ω, rpm 300

Approximation coefficients V, mm/s

V, mm/s

2.24

0.07

25

5

650

3.02

0.05

25

5

1000

5.32

0.09

25

5

1480

7.12

0.06

20

7.5

150

1.98

0.07

20

7.5

300

0.83

0.01

30

7.5

140

2.21

0.05

30

7.5

20

5

340

1.22

0.02

1600

5.58

0.04

2.4 Visualisation and Qualitative Analysis of Flow Near the Disc Edge

59

In order to investigate the movement of liquid particles near the bottom of the unit, along the surface of the flow as well as in the flow column, a series of relevant experiments were carried out using different types of markers. Soluble marker impurities (aniline ink, uranyl solution) were used to visualise the flow at the free surface and in the thickness. The structure of the flow near the disk was investigated by means of solid flat particles mixed with water and reflecting light (aluminium powder with an average particle size of 30 μm), with illumination by means of a laser knife. Analysis of the experimental data with the laser knife at a distance of 2.0 cm showed the presence of a pronounced radial transfer of markers along spiral trajectories in the area above the edge of the rotating disk (Fig. 2.18). In the case where the disk occupied the entire area of the lower end of the cylindrical container, the most noticeable radial movement was observed in the annular area near the walls of the container. Experiments have shown that liquid particles move along the free surface along spiral trajectories (Fig. 2.19), while the transfer of liquid particles into the liquid

Fig. 2.18 Flow visualisation (H = 20 cm, R = 7.5 cm, Ω = 240 rpm) with a laser knife positioned 1.0 cm above the rotating disk: a, b 114 and 120 s from the start of recording, c, d spiral structures detected for captured frames

60

2 Solute Admixture Transport from a Compact Source in a Composite Vortex

Fig. 2.19 Surface spiral structures (H = 20 cm, R = 7.5 cm, Ω = 240 rpm): a, b 9, 15 s after the dye drop falling on the free surface respectively

column follows helical downward trajectories, whose radius varies little along the vertical. All measured spiral structures interpolate most accurately with logarithmic functions. The coincidence of the types of spiral motion of liquid particles at the surface and near the disk indicates that the characteristic features of the vortex flow are set in the boundary layer on the disk and then transferred to the whole liquid volume. Conclusions to this chapter This chapter presents the results of experimental studies of the transfer of a soluble impurity from a compact spot on the free surface of a liquid and into a liquid at rest or involved in a compound vortex motion, as well as a visualization and qualitative analysis of the flow near the disk edge. The rate of change of the dye lowering into the liquid thickness is calculated depending on the frequency of rotation of the inductor. Here it is necessary to especially emphasize that the presented quantitative experimental data are the result of a new measurement method proposed by the author.

References

61

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

15. 16. 17. 18.

Taylor GI (1921) Experiments with rotating fluids. R Soc Proc A 100(2):114–121 Long RR (1954) Note on Taylor’s “ink walls” in a rotating fluid. J Atmos Sci 11(3):247–249 Batchelor J (1973) Introduction to fluid dynamics. Mir, Moscow, 760 c von Kármán T, Rubach HL (1912) On the mechanisms of fluid resistance. Phys Z 13:49–59 Chaplina TO, Stepanova EV, Chashechkin YD (2014) Structural stability of transfer pattern of immiscible fluids in vortex flow. Bull Mosc State Univ Ser 3 Phys Astron 6:122–127 Stepanova EV, Chashechkin YuD (2008) Anisotropic admixture transport in a compound vortex. Dokl RAN 423(4):474–478 Van Dyke M (1986) Album of fluid and gas flows. Mir, Moscow, 184 c Honji H (1988) Vortex motions in a stratified wake flows. Fluid Dyn Res 3(1–4):425–430 Kelso RM, Lim TT, Perry AE (1996) An experimental study of a round jet in a cross-flow. J Fluid Mech 306 http://www.efluids.com/efluids/gallery/gallery_pages/vortex_dislocate.jsp Stegner A (2007) Nonlinear dynamics of rotating shallow water: methods and advances. In: Advances in nonlinear science and complexity, vol 2, pp 323–379 Alekseenko S (2003) Waves, vortices and coherent structures in fluid streams. Sci Siberia (SB RAS Wkly) 49:24–35 Chaplina TO, Stepanova EV, Chashechkin YD (2012) Marker transfer patterns in a compound vortex. Nat Tech Sci 2(58):45–51 Chaplina TO (2013) Transport of oil in compound vortex. In: Procedia IUTAM. Special issue: IUTAM 2012 symposium on waves in fluids: effects of nonlinearity, rotation, stratification and dissipation, vol 8, pp 58–64 Chaplina TO, Stepanova EV, Chashechkin YuD (2012) Features of admixture transfer in a stationary vortex flow. Bull Mosc State Univ Ser 3 Phys Astron 4:69–75 Chaplina TO, Stepanova EV, Chashechkin YuD (2010) Deformation of a compact oil slick in a compound vortex cavern. Dokl RAN 432(2):185–189 Stepanova EV, Chaplina TO, Chashechkin YuD, Petrenko AI (2012) Experimental studies of admixture transfer in a compound vortex. Phys Probl Ecol 18:370–379 Kistovich AV, Chaplina TO, Stepanova EV (2019) Analytical and experimental study of the substance transport in the vortex flow. Theor Comput Fluid Dyn 1:1–16

Chapter 3

Transfer of Immiscible Admixture in a Vortex Flow

Intensive human activities to exploit the natural wealth of the World Ocean include exploration and extraction of minerals, transportation of goods, use of energy and biological resources, as well as many other aspects. One of the consequences of including the Ocean in the sphere of economic interests is the emergence and spread of pollution of various nature. The transport of pollutants on the surface and in the water column has been the subject of many studies and extensive data have been obtained, but there are still no general patterns and comprehensive descriptions of transport processes, which makes it difficult to solve scientific, environmental and economic problems. Laboratory simulations are used to establish the basic patterns and parameters affecting vortex flow to ensure that all flow characteristics are reproducible and to monitor the conditions under which they change. This chapter presents the results of experimental studies of immiscible admixture transfer from a compact slick on the free surface of a liquid inside a resting or involved in a compound vortex, including emulsion formation mode. Liquid unsaturated fatty acids (castor, sunflower, machine and aviation oils) as well as diesel fuel, oil, fuel oil and others have been used as immiscible mixture. The problem of analytical determination of the shape of an oil body in a composite vortex on the basis of the analysis of the equations of mechanics of dense liquids with physically grounded boundary conditions is considered. The dependences, reflecting the form of interface in the vortex flow of fluid, consisting of two components, are obtained. Analytical expressions describing the form of zero approximation for the phase boundaries in the composite vortex are in satisfactory agreement with the experimental data.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina, Transfer of Substance in Vortex and Wave Flows in One-Component and Multi-component Environment, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-31856-6_3

63

64

3 Transfer of Immiscible Admixture in a Vortex Flow

3.1 Compound Vortex in a Liquid of Two Immiscible Components In nature, vortex flow in the liquid column generate physical fields that curve the free surface of water. In experimental techniques, laboratory simulations of hydroelectric vortices, cyclones, vortices in separators, etc. are carried out in a closed cylindrical geometry. In this case, the distribution of physical characteristics in the vortex can be conveniently traced from the shape of the free surface. The surface shape is easily determined when the fluid is brought into solid rotation. However, this phenomenon is not always observed, the simplest example being the Rankine vortex. An even more complex flow is realized in the case of rotation of a fluid medium consisting of two immiscible components. The vortex formed in this case appears as if composed of several regions, the characteristics of the flow in each of which differ significantly from each other. In intensive flow modes, heterogeneous perturbations are observed on the free surface of such vortex, the description of which is a very complicated problem. At the same time the observations show that the free surface of such a complex vortex can be represented conditionally as a sum of a cylindrically symmetric rotation surface and a normal deflection from this surface superimposed on it. In experiments with immiscible admixture oil, diesel fuel, sunflower oil, machine oil and castor oil were used as additive. The central part of the volume of the immiscible additive is hereafter referred to as the “oil body”. The initial state, the final state, and the coordinate system are shown in Fig. 3.1. In the initial state the volumes of water Vw = πR 2 H and oil Vo are given and considered constant in all states of the system [1]. The densities of water ρw and oil

Fig. 3.1 Initial (a) and final (b) states of the two-component oil–water fluid system, c profile photo

3.1 Compound Vortex in a Liquid of Two Immiscible Components

65

ρo are constant throughout the process. In the stationary initial state S1 , S2 and So are the “water–air”, “water–oil” and “oil–air” interface respectively. The following assumptions are made when describing a compound vortex. The volumes and densities of water Vw , ρw and oil Vo , ρo are assumed to be constant in all states of the system. The symbols ςwa , ςwo and ςoa denote the water–air, water–oil and oil–air interface respectively. The problem is considered in a cylindrical coordinate system with the vertical axis z directed against the vector of gravitational acceleration g, and the origin coincides with the centre of the rotating disk. The system of hydrodynamic equations describing the motion is 1 vt, + (v∇)v = − ∇d + νΔv + g, ∇ · v = 0 ρ

(3.1.1)

The values v, ρ, ν refer to speed vw , density ρw and kinematic viscosity of water νw in the area occupied by water, or speed vo , density ρo and kinematic viscosity of oil νo in the area occupied by oil. The sticking conditions are fulfilled on hard surfaces vw |r =R = vw |z=0, r ∈[r D ,R] = 0,

vw |z=0, r ∈[0,r D ] = r ωeϕ

(3.1.2)

Here r D , R are the radii of the rotating disc and the shell respectively, ω is the angular frequency of the disc, eϕ is the azimuthal orth of the cylindrical coordinate system. On partition surfaces ςi j dynamic  n i (d1 − d2 ) − n i σ

1 1 + R1 R2



  ,(l) ,(2) = n k σik − σik

(3.1.3)

and kinematic | dςi j || =0 dt |Σi j =0

(3.1.4)

boundary conditions. Here n is the unit normal, σ is the surface tension coefficient, R1 , R2 are the principal radii of curvature. All these quantities refer to the respective interface. The symbols d1 , d2 denote the pressures on different sides of the selected surface, and ,(1,2) σik are the viscous stress tensors of the interfacing media. The system of relations (3.1.1–3.1.4) is extremely complicated for analysis and its exact solution has not been reached yet. At the same time it is possible to construct a “zero” approximation for the forms of interfaces of contacting media in which all physical fields near these surfaces do not depend either on time or on the azimuthal coordinate ϕ. Within this approximation, vw = vw eϕ , vo = vo eϕ ; surface tension effects are assumed to be small and are not taken into account. The interfaces are

66

3 Transfer of Immiscible Admixture in a Vortex Flow

given by the relations ςwa : z − ζ(r ) = 0, r ∈ [Ro , R]; ςwo : z − θ(r ) = 0, r ∈ [0, Ro ]; ςoa : z − η(r ) = 0, r ∈ [0, Ro ]

(3.1.5)

Here Ro is the radial coordinate of the triple water–oil–air contact point as shown in Fig. 3.1. As a result, system (3.1.1) takes the form dr, = ρ

v v2 v, ,, , dz, = −ρg, vrr + r − 2 + v,,zz = 0 r r r

(3.1.6)

where the incompressibility condition ∇ · v = 0 is identically satisfied. Representation of pressure in different areas of space in forms dw(1) = da + ρw g(τ − z) + qw(1) (r, z), r ∈ [Ro , R],

(3.1.7)

do = da + ρo g(η − z) + qo (r, z), r ∈ [0, Ro ],

(3.1.8)

dw(2) = da + ρo g(η − θ) + ρw g(θ − z) + qw(2) (r, z), r ∈ [0, Ro ],

(3.1.9)

where pa is atmospheric pressure, and substituting (3.1.7–3.1.9) into (3.1.6) and boundary conditions (3.1.3, 3.1.4) allows us to form two possible solutions for the velocity field. When the solid rotation radius Rc is smaller than Ro the following equations apply (the solid rotation zone refers to the central flow area where the linear velocity of the liquid particles depends linearly on the distance from the axis of rotation) Bo ϑ(r − Rc ), r ∈ [0, Ro ] r B2 ϑ(r − Rc ), r ∈ [0, Ro ] = A2 r ϑ(Rc − r ) + r B1 , r ∈ [Ro , R] = r

vo = Ao r ϑ(Rc − r ) + v(2) w v(1) w

(3.1.10)

where ϑ(x) is a Heaviside function. If the radius Rc of the solid rotation is greater than Ro there is vo = Ao r, r ∈ [0, Ro ] v(2) w = A2 r, r ∈ [0, Ro ] v(1) w

B1 ϑ(r − Rc ), r ∈ [Ro , R] = A1r ϑ(Rc − r ) + r

(3.1.11)

3.1 Compound Vortex in a Liquid of Two Immiscible Components

67

From the condition of equality of tangential stresses at the water–oil interface follows the relationship common to relations (3.1.10, 3.1.11) v(2) w = α vo ρo νo α= ρw νw

(3.1.12)

The shapes of the interfaces are determined on the basis of the equations (2)2 v2o = gr ηr, , v(1)2 w = gr τr , vw = gr



 ρo  , ηr − θr, + θr, ρw

 (3.1.13)

To determine the constants Ao , Bo , A1 , etc., that make up the expressions (3.1.10, 3.1.11), a set of natural conditions must be used, consisting of the contact point r = Ro η(Ro ) = θ(Ro ) = τ(Ro ) the laws of conservation of water are met ⎛ R ⎞  Ro 2π ⎝ ζ(r )r dr + θ(r )r dr ⎠ = Vw

(3.1.14)

(3.1.15)

0

Ro

and oils Ro (η(r ) − θ(r ))r dr = Vo

2π

(3.1.16)

0

The condition of continuity of the velocity field in water must also be fulfilled at r = Ro | | (2) | | v(1) w r =Ro = vw r =Ro

(3.1.17)

and continuity of velocity fields (3.1.10, 3.1.11) at r = Rc vo |r =Rc −0 = vo |r =Rc +0 ,

| | | | v(1,2) = v(1,2) w w r =Rc −0 r =Rc +0

(3.1.18)

In addition, for the steady motion of the compound vortex, the condition of equality of speed Wν of the viscous loss of kinetic energy of the vortex to the power input to the system Wext of the external source must be fulfilled

68

3 Transfer of Immiscible Admixture in a Vortex Flow

  Wext = ρo νo Vo

∂ vo vo − ∂r r

 

2 d Vo + ρw νw

Vw

∂ vw vw − ∂r r

2 d Vw

(3.1.19)

Integrating Eq. (3.1.13) and substituting the results into (3.1.14, 3.1.17, 3.1.18) defines a form of the “zero” approximation of the composite vortex surface by the following relations. In the case of Rc < Ro : η(r ) = H − 

r2 Rc2

θ (r ) = H + 

r2 Rc2

τ (r ) = H −

 2  α − 1 A2o Rc4 A2o Rc2 + 2g Ro2 2g   Rc2 − 2 ϑ(Rc − r ) − 2 ϑ(r − Rc ) , r ∈ [0, Ro ] r   2 α − 1 A2o Rc4 ρo α2 ρw − ρo A2o Rc2 + 2 2g Ro ρw − ρo ρw − ρo 2g   2 Rc − 2 ϑ(Rc − r ) − 2 ϑ(r − Rc ) , r ∈ [0, Ro ] r α2 A2o Rc4 , r ∈ [Ro , R] 2g r 2

(3.1.20)

where H is the thickness of the water layer before the rotating disc is switched on. In the case of Rc > Ro :   A2 r 2 A2o  2 2  2α Rc + 1 − α2 Ro2 + o , r ∈ [0, Ro ] 2g 2g     2 2 α − 1 ρo 2 A θ(r ) = H − o 2α2 Rc2 + R 2g ρw − ρo o

η(r ) = H −

α2 ρw − ρo A2o r 2 , r ∈ [0, Ro ] ρw − ρo 2g   2  r Rc2 α2 A2o Rc2 , − 2 ϑ(R − r − ϑ(r − R τ (r ) = H + ) ) c c 2g Rc2 r2 +

r ∈ [Ro , R]

(3.1.21)

Expressions (3.1.20, 3.1.21) change into each other at Rc = Ro , i.e. when the boundary of the region occupied by the oil coincides with the solid rotation boundary of the compound vortex. The values of Ao , Ro and Rc , entering relations (3.1.20, 3.1.21), are determined after substituting (3.1.20, 3.1.21) into (3.1.15, 3.1.16, 3.1.19). The resulting system of transcendental equations does not allow exact analytical solution but can be solved numerically. Comparison of the analytical model with experimental results is given in Sect. 3.3 of this book.

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow

69

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow The introduction of an immiscible viscous admixture into the flow significantly affects the shape of the free surface and the intensity of motions on it, despite the negligible relative volume of the additive itself (in some experiments, the volume of liquid exceeds the volume of the introduced admixture Vk by more than 200 times). Experiments were carried out with two-layer and three-layer liquids—underlying water and immiscible lighter markers. The depth of the water layer in all experiments ranged from Hw = 10 to 60 cm, the thickness of the marker layer Hm determined by its volume ranged from 2 mm to 20 cm. The physical characteristics of the working media are given in Table 3.1. The adjustable parameters of the problem were the thickness of the marker Hm , the radii of the container R0 and disc Rd , and its rotation frequency Ωd , the acceleration of gravity g. Each of the fluids is characterized by density ρ, kinematic viscosity ν, a a , marker–air σm , surface tension coefficients at the contact surfaces—water–air σw m m m water–marker σw and their specific values γw = σw /ρw (indices “w”, “a”, “d”, “m” refer to water, air, disc and markers respectively). The inertial length scales characterizing the global flow in the container are defined by the ratio of free fall acceleration to angular velocity of the disk rotation—L Ω = g/ Ω2d and liquid particles—L ω = g/ω2w serve as dynamic characteristics of the system. The fine structure of the emerging current (its singularly perturbed Stokestype boundary √ is characterized by microscales of the following √ layer components) form δΩ = ν/Ωd and δω = ν/ω. Important parameters of multilayer immiscible fluids are the relative density (analogous to Atwood number), the difference of surface tension coefficients, and the dynamic viscosities (see Table 3.2), whose influence on the transport pattern of matter has not been systematically investigated before. The structures of emerging flow are characterised by a large number of spatial and temporal scales. Table 3.1 Physical characteristics of the liquid media used Marker parameters, at T = 20 °C

Water Sunflower oil Diesel fuel Petroleum Aviation oil

Density (ρ), kg/m3

998.9 925.0

840.0

866.0

885.0

Kinematic viscosity (ν), × 10−6 , 1.05 m2 /s

60.60

1.50

8.14

20.50

Coefficient of surface tension at 73.0 a ), × the marker–air interface (σm 103 , N/m

33.06

22.0

30.0

32.0

Coefficient of surface tension at m ), × the marker–air interface (σw 103 , N/m

3.10

7.10

2.80

2.45

70

3 Transfer of Immiscible Admixture in a Vortex Flow

Table 3.2 Values of relative coefficients No.

Marker

The main relationship Dense, Rρm w

a Capillary, Rσw

Viscous, Rνm w

1

Sunflower oil

0.038

0.38

− 0.96

2

Diesel fuel

0.086

0.54

− 0.18

3

Oil

0.071

0.42

− 0.77

4

Aviation oil

0.060

0.39

− 0.91

Table 3.3 Characteristic scope of the task

ξw = Rd /Hw

0.05

0.75

ξ0 = Rd /R

0.2

1.0

LΩ =

16 × 10−3

1

2 × 10−4

5 × 10−4

δΩ =

m √ ν/Ωd , m g/Ω2d ,

The ratios of the characteristic scales form a set of dimensionless combinations including the conventional numbers: The ratios of the above quantities form the traditional set of dimensionless parameters including Reynolds numbers Re = (Rd2 Ωd )/νw , Froude numbers Fr = (Rd2 Ω2d )/g Hw , and for a two-layer medium m . (water—immiscible fluid), also Bond number Bo = g Hw2 (ρw − ρm )/σw The flow studied in these experiments are characterised by Reynolds numbers in the range 500 ÷ 50,000, Froude 50÷1800, Atwood − 0.009÷0.2, Bond − 1.0÷4.5. The characteristic scales of the problem as well as the relative coefficients are given in Table 3.3. After setting up the apparatus, the basin was filled with degassed water of a given depth. A selected volume of immiscible marker was applied to the surface of the resting liquid. After all disturbances were attenuated, the inductor was switched on and the marker distribution pattern on the surface and in the liquid column was recorded. Experimental conditions were changed in steps, the exposure time under the selected experimental conditions was determined by the duration of flow pattern stabilization and was not less than 20 min. The experiment was repeated under new conditions until the oil body or the emulsion tail reached the rotating disk, which was accompanied by the formation of a thin water–oil emulsion. A new experiment was carried out after the pool had been completely refilled. A picture of the distribution of immiscible admixture in the thickness of the composite vortex and the notation of its main geometrical parameters is shown in Fig. 3.2. A cavern is formed on the surface of the rotating working fluid—water— whose depth reaches a maximum value in the centre. The main part of the oil collects in the vicinity of the central vertical axis in a compact volume shaped like a body of rotation, which adjoins the bottom of the cavern. Here H is the maximum thickness of the rotating fluid layer (counted near the wall), h t is the height difference between the free surface and the bottom edge of the rotating oil volume, h k is the height of the rotation body, h = h t − h k is the free surface deflection arrow, Rk is the contact

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow

71

Fig. 3.2 Composite vortex with castor oil portion (Vk = 30 ml, H = 40 cm, R = 7.5 cm, Ω = 640 rpm): a, b photo (side view) and flow diagram

line radius of the oil volume with the cavern edge. Visually and in the side photos, all contact surfaces—water–oil, oil–air and water–air—are clearly distinguishable. At moderate angular speeds of disc rotation (Ω < 750 rpm) the flow pattern shown in Fig. 3.2 generally stabilised within 20 min. Under these experimental conditions, the bulk of the castor oil from a lens-shaped spot on the surface of the resting liquid with a characteristic transverse dimension of 3.5 cm and a thickness of about 4 mm, collects into a compact volume with a height of h k = 4.94 cm in the centre of the surface funnel, which rotates with the surrounding liquid. The tangential to the lateral surface of the oil body and the surface of the water changes as it passes through the contact line of the water and oil on the surface of the cavern. The volume of the rotating body containing the castor oil is Vk ≈ 29.5 ml, h t = 10.91 cm, Rk = 2.76 cm, h = 5.97 cm. The remaining oil (Vk ≈ 0.5 ml) remains in a thin layer on the surface of the cavern and forms spiral arms on the free surface, which will be presented below. Experiments have shown that the addition of even a small volume of oil (30 g per working volume of 54 L) significantly changes the vortex pattern and the shape of the free surface in general. In the first experiments with immiscible admixture, castor oil was used, placed on the surface of a resting liquid, only the volume of the admixture and flow parameters were changed, in order to reveal the basic patterns of insoluble substance transfer in the vortex flow. In further experiments other immiscible substances (oil, aviation oil, machine oil, sunflower oil and diesel fuel) were used. At moderate rotational speeds of the disc (R = 7.5 cm, Ω = 377 rpm) a cavern with a depth in the centre of 3.2 cm is formed on the surface of the clear liquid (H = 40 cm) and its walls remain smooth (Fig. 3.3a). A small amount of castor oil (Vk = 30 ml) almost completely fills such a cavern, the height of the oil body in the centre is 2.6 cm. The surface of the liquid–air boundary (outer part: water–air, inner part: oil–air) remains almost flat.

72

3 Transfer of Immiscible Admixture in a Vortex Flow

Vk = 30 ml

Vk = 60 ml

=370 rpm

Vk = 0 ml

Ω

Ω

b

c

f

g

=500 rpm

a

= 750 rpm

d

Ω

h

k

l

Fig. 3.3 Characteristic forms of the liquid surface in a container with a rotating disk (H = 40 cm, R = 7.5 cm): a, d, h pure water, Ω = 377, 500, 750 rpm; b, f, k Vk = 30 ml, 60 ml, Ω = 77, 500, 750 rpm; c, g, l Vk = 60 ml, Ω = 377, 500, 750 rpm

Adding a larger volume of castor oil (Vk = 60 ml, Fig. 3.3c) changes the size and shape of the body of rotation, however, the lateral surface and the lower edge of the oil volume remain smooth. The pattern of oil distribution over the free surface will be discussed below. When the rotational speed is increased up to 500 rpm, small inhomogeneities on the cavern walls can be observed even in pure liquid (Fig. 3.3d). These appear as distortions of the free surface at high magnifications and the boundaries of the bright spot in the centre of the cavern and its specular reflection (dark shadow over the cavern). The maximum perturbations with shallow troughs and pointed ridges are observed at the half depth of the cavern (2.37 cm). By increasing the amount of oil (60 ml), the free surface deflection is almost restored again (h = 4.5 cm, which is only 0.25 cm less than for the funnel in Fig. 3.3d). The lower edge of the oil volume is at a depth of H = 10 cm (Fig. 3.3e). Most of the added oil is concentrated inside a central area bounded by a rotation surface 4.4 cm deep, i.e. smaller than the cavern depth in the clear liquid (Fig. 3.3e). At high activator rotation frequencies (Ω = 750 rpm), two types of disturbances— large-scale (inertial) and fine-structure (spiral)—appear on the surface of the cavern in a homogeneous fluid (Fig. 3.3g). Its maximum depth is 12.3 cm. The fluid moving vigorously along the free surface drags gas bubbles into the centre of the cavern,

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow

73

clearly visible in Fig. 3.3g in the vicinity of the rotation axis Spiral waves are present on the entire surface of the cavern, while inertial waves are present in its lower part. The higher the speed of the disc, the more noticeable is the effect of the oil. When 30 ml of castor oil is added, the free surface deflection arrow is only 5.8 cm (Fig. 3.3), and individual drops of oil can be seen on the liquid surface. The distance of the lower edge of the oil volume from the free surface is 12.3 cm. When 60 ml of castor oil is added, the deflection arrow is reduced to 3.6 cm. Some of the oil is drawn into the fluid column and collected in a body of rotation in the vicinity of the vertical flow axis of the form h = Ar B (Fig. 3.3): A = 1.09 ± 0.49, B = 2.55 ± 0.07, [h] = cm, [r ] = cM. The total depth of the deformed surface in Fig. 3.3 is h t = 12.3 cm (the difference of liquid levels at the pool wall and the lower edge of the oil-filled paraboloid). The upper boundary of the oil volume in contact with air in the centre of the basin also has the shape of a body of rotation with parameters: A = 0.26 ± 0.06 and B = 1.91 ± 0.12 (Fig. 3.3i). The depth of the free surface deflection arrow (level difference at the edge and at the centre where the oil is in contact with the air) is only h = 5.8 cm. On the circular water–oil contact line with radius Rk = 1.56 cm there is a kink in the shape of the free surface (the angle between the tangents is α = 27◦ ). The outside of the funnel is approximated by a curve of the form h = B ln(r − A), where r is the radial coordinate counted from the rotation axis, A = 1.21 ± 0.43, B = 1.53 ± 0.07, [h] = cm, [r ] = cM. The distribution of a less viscous immiscible admixture (tinted sunflower oil) on the surface and in the thickness of the composite vortex was investigated to obtain data on the position of the admixture both in the thickness and on the free surface of the vortex in question. The photographs of the profile of the composite vortex with the addition of immiscible admixture shown in Fig. 3.4 illustrate the changing positions of the air–liquid and oil–water boundaries depending on the amount of admixture on the surface Vk and the water layer depth H . The images from the experiments clearly show the water–oil and oil–air contact surfaces. When the activator rotates at 320 rpm and 30 ml of sunflower oil is added, the deflection of the air–liquid surface is greatest, the peripheral part of the free surface is covered with small oil droplets extending in the direction of rotation of the liquid surface (Fig. 3.4a). As the added oil volume increases, the deflection of the free surface decreases and the area of individual oil droplets moving along the periphery increases (Fig. 3.4b). Further increasing the oil volume results in decreasing the deflection of the air–liquid surface, in addition the volume of oil not constricted in the central oil body increases (Fig. 3.4c). The greater depth of the liquid also significantly affects the distribution of the light admixture in the composite vortex. A small portion of oil (Fig. 3.4d) collects almost entirely in the vicinity of the axis of rotation, the oil body is 2.3 cm high and the air–liquid interface remains almost flat. The periphery of the surface is occupied by oil droplets. A large amount of oil Vk = 150 ml creates an oil body shaped like a hat, the edges of which are formed by spiral arms extending in the direction opposite to the activator rotation (Fig. 3.4e). The air–liquid interface surface consists almost entirely of oil and is very close to flat. An intermediate amount of oil Vk = 90 ml leads to

74

3 Transfer of Immiscible Admixture in a Vortex Flow

Vk = 30 ml

Vk = 90 ml

Vk = 150 ml

a

b

c

d

f

j

g

l

o

h

m

p

k

n

r

Fig. 3.4 Characteristic forms of the liquid surface in a container with a rotating disk (R = 7.5 cm): a–c H = 20 cm, Ω = 320 rpm, Vk = 30, 90, 150 ml; d–j H = 40 cm, Ω = 320 rpm, Vk = 30, 90, 150 ml Vk = 30 ml, 60 ml; g–k H = 20 cm, Ω = 480 rpm, Vk = 30, 90, 150 ml; l–n H = 40 cm, Ω = 480 rpm, Vk = 30, 90, 150 ml; o–r H = 40 cm, Ω = 770 rpm, Vk = 30, 90, 150 ml

a deflection of the free surface and draws the oil deep into the composite vortex (Fig. 3.4e). When the rotational speed is increased to 480–570 rpm, small inhomogeneities are observed on the walls of the cavern and the oil body. Most of the added oil is concentrated near the vertical axis of the cylindrical container. Adding a small amount of oil (30 ml) leads to a reduction in the deflection of the interface, compared with pure liquid (Fig. 3.4g). As the amount of oil increases (90 ml), the depth of deflection of the air–liquid interface decreases (h = 6.1 cm, Fig. 3.4b, h = 3.2 cm, Fig. 3.4i). The lower edge

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow

75

of the oil volume is at a depth of H = 11.8 cm (and 10.8 cm respectively). Spiral waves are present on the surface of the clear water, the effect of which is not seen on the surface of the oil body. Adding more oil (150 ml), which also collects in a conical area in the centre of the composite vortex (Fig. 3.4k), only leads to an increase in the height of the central oil body. At high activator rotation frequencies (Ω = 770 rpm), two types of disturbances—large-scale (inertial) and fine-scale (spiral)—appear on the cavern surface. Its maximum depth is 17.8 cm (Fig. 3.4o). The higher the speed of the disc, the more noticeable is the effect of the oil. When 90 ml of sunflower oil is added, the free surface deflection arrow is 7.1 cm (Fig. 3.4n), individual oil drops are visible on the liquid surface. The ratio of the width of the oil body to its height is 0.38. When 150 ml of sunflower oil is added, some of the oil is drawn into the liquid column and collected in a rotating body in the vicinity of the vertical axis of the current. The position and shape of the body is determined by the balance between the buoyancy forces pushing the light oil out and the drag forces due to the main circulating flow in the composite vortex. The total depth of the deformed surface in Fig. 3.4p (the difference between the fluid levels at the pool wall and the bottom edge of the oil-filled paraboloid) is 25.7 cm. The graphs illustrating the shape of the free surface and the shape of the water/castor oil interface are shown in Fig. 3.5. For ease of reference, the graphs are labelled: ◯ is the liquid/air interface, ∇ is the oil/water interface. The deflection depths of the air–liquid surface and the penetration of the light admixture buildup are given in Table 3.4. Dependence of the free surface deflection depth and the water–oil interface on the activator rotation frequency is approximated by a function of the same form as for the one-component liquid surface deflection, h = A r B . The coefficients of double logarithmic depth-frequency dependence obtained from the approximation are given in Table 3.5. When comparing the coefficients, a non-linear dependence on the volume of immiscible admixture is observed, which is further confirmed by flow pattern images (Figs. 3.3 and 3.4). A comparison of the shapes of the central cross-section of the oil body and the positions of the liquid interfaces and the different phases for the less viscous sunflower oil is shown in Fig. 3.7. At high rotational speeds of the activator disc (Ω = 750 rpm), a cavern is formed in the clear deep liquid (H = 40 cm, R = 7.5 cm) (symbols 1, Fig. 3.6a) with a depth of h t = 11.8 cm. When 30 ml of sunflower oil is added, most of the admixture is concentrated inside a central area bounded from above by a rotating surface of h = 10 cm depth (symbols 2, Fig. 3.6), i.e. smaller than the cavern depth in pure liquid, the height of the oil body is h k = 7.8 cm (symbols 3, Fig. 3.6a). Although the water surface is covered by a film of oil, the contact line of the oil body with the water surface is identified quite clearly. The addition of a large quantity of oil (Vk = 2000 ml) leads to the disappearance of the oil–water–air contact line at similar speeds (symbols 4, Fig. 3.6a). The height

76

3 Transfer of Immiscible Admixture in a Vortex Flow

Fig. 3.5 Surface geometry of the composite vortex with light admixture (H = 40 cm, R = 7.5 cm): a Ω = 240 rpm, b Ω = 500 rpm, c Ω = 750 rpm

Table 3.4 Depths of deflection of the air–liquid surface and penetration of admixture immiscible with water admixture

Ω, rpm 377

Vk , ml 0

h, cm 0 (2.51)

h t , cm 2.51

500

0 (5.27)

5.27

750

0 (12.30)

12.30

250

30

2.53

2.53

500

3.92

3.92

730

3.56

8.06

3.83

3.83

240

60

500

4.31

10.19

750

4.12

12.19

of the oil body of rotation is considerably greater than the cavern depth in pure liquid h t = 17.3 cm (symbols 4, Fig. 3.6a). The central cross sections of the surface cavern in the bilayer liquid at different speeds of the activator disk are shown in Fig. 3.6b. At low and moderate activator rotational speeds (Fig. 3.6b, symbols 1 and 2) the system is a two-layer liquid: the

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow

77

Table 3.5 Coefficients of the double logarithmic dependence of the surface deflection depth on the speed of the activator disc Vk , ml

Air–liquid interface A

Water–oil interface B

A

B

0

− 5.12 ± 0.25

1.72 ± 0.05

–

–

30

− 1.70 ± 0.24

1.40 ± 0.09

− 1.27 ± 0.24

1.48 ± 0.10

60

− 4.02 ± 1.10

2.06 ± 0.39

− 0.72 ± 0.54

1.20 ± 0.21

Fig. 3.6 Shapes of oil–water and liquid–air interfaces in a compound vortex (H = 40 cm, R = 7.5 cm): a 1—pure water Ω = 750 rpm, 2—Vk = 30 ml—oil body boundary Ω = 770 rpm, 3—Vk = 30 ml—liquid–air boundary Ω = 770 rpm, 4—Vk = 2000 ml—oil body boundary Ω = 720 rpm; b Vk = 2000 ml, 1–4 Ω = 220, 320, 520, 720 rpm Fig. 3.7 Dependence of the sharpness coefficient ξb of the oil body on the flow parameters

78

3 Transfer of Immiscible Admixture in a Vortex Flow

upper oil layer over the water layer. The system constituent boundary and the liquid– air contact surface are separated in space. In Fig. 3.6b, the oil layer thickness is indicated by a grey filler for 220 rpm, and a red dashed line for 320 rpm. At increase of frequency of rotation of the activator up to 520 rpm the height of an oil body increases to h k = 14.9 cm, the section of a contact line oil–water–air near a wall of a container is formed, which changes its position with rotation of system (Fig. 3.6b, symbols 3). Further increase of rotation speed (Fig. 3.6b, symbols 4) up to 720 rpm leads to drawing of oil into the central flow area and transition of the system from the state of two-layer liquid to water—admixture system on the surface. The preservation of the shape of the oil body as the angular velocity of the disc changes allows the relative change in its dimensions to be characterised by a single relationship ξb = 2Rk / h k as a function of Reynolds number. In the parameter range Re from 1000 to 5500 this dependence is represented by two linear empirical patterns ζb = aRe + b: at small Reynolds numbers (less than 2300) the curve is rapidly decreasing a = −1.9 × 10−3 ± 4.0 × 10−4 , b = 5.2 ± 0.7, at large numbers Re it is flat a = −1.7 × 10−4 ± 2.9 × 10−5 , b = 1.2 ± 0.1 (Fig. 3.7). Increasing the speed of rotation of the disc leads to an increase in the vertical component of the flow velocity at the centre of the container and consequently an increase in the frictional force pulling the oil body. In experiments where oil and diesel were used as an admixture, an oil body was also formed, the shape and size of which depended on all flow parameters. A comparison of the size of the oil body depending on the amount and physical properties of the immiscible additive added to the flow shows that increasing the viscosity of the marking additive leads to a decrease in the vertical size of the oil body. Also the vertical size of the area occupied by the immiscible liquid in the flow is affected by the surface tension coefficient. As it increases, the depth of admixture drawn into the flow increases (Fig. 3.8). For the vertical dimensions of the surface cavern and the oil body in the centre, the dependences on the activator rotation speed are plotted and interpolated by linear functions of the form h t = AΩ + B, the factors included in them are given in Table 3.6. There is a clear tendency for the vertical dimensions of the oil body to

a

b

c

d

Fig. 3.8 Distribution of immiscible impurities on the surface and in the thickness of the composite vortex Hw = 40 cm, Rd = 7.5 cm, Vm = 150 ml, Ωd = 770 rpm: a sunflower oil, b mixture of sunflower oil and diesel in equal proportions, c petroleum, d diesel fuel

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow Table 3.6 Linear interpolation factors for oil body sizes

79

A

B

Sunflower oil

2.33 ± 0.13

− 11.66 ± 1.30

Sunflower oil + diesel

2.09 ± 0.27

− 8.44 ± 0.67

Petroleum

1.88 ± 0.16

− 8.01 ± 0.55

Diesel

1.66 ± 0.21

− 7.74 ± 1.48

decrease when the density and surface tension factor of the admixture added to the flow decreases, which is confirmed by the obtained approximations. Preservation of the shape of the central part of the oil body when the angular velocity of the disk rotation changes makes it possible to characterize the relative change in its dimensions by a single ratio ξm = 2Rm / h m depending on the Reynolds number [2]. Increasing the rotational speed of the disc leads to an increase in the vertical component of the flow velocity at the centre of the container and, consequently, a shear rate stress at the boundary between the water and the oil body. In the parameter range Re from 500 to 12,000 the dependence is represented by two linear functions ξm = aRe + b: at small Reynolds numbers (less than 2500) the angular coefficient is large a = (− 10.3 ± 2.4) × 10−4 at b = 2.2 ± 0.4, there is a rapid change in the oil body height to diameter ratio near the free surface, at large Re the line becomes more gentle a = (4.0 ± 2.0) × 10−5 , b = 0.7 ± 0.1, the change in the oil body diameter to height ratio is small (Fig. 3.9). In experiments at close frequencies but with a different substance (aviation oil) a similar picture is observed (Fig. 3.10a, b). However, in experiments with high activator frequencies the oil body is completely immersed in the vortex, the free surface is even freed from individual admixture droplets (Fig. 3.10d). The stability of the water–oil–body interface and the shape of the liquid contact area at the bottom of the cavern depend on the type of marker. At the same angular velocity of disc rotation the water–petroleum boundary retains continuity (Fig. 3.10a), and droplets start separating from the oil body in the water–oil system Fig. 3.9 Dependence of sharpness coefficient ξb of oil body on flow parameters: 1—sunflower oil; 2—petroleum

80

3 Transfer of Immiscible Admixture in a Vortex Flow

a

b

c

d

Fig. 3.10 Different types of immiscible marker in vortex flow under different experimental conditions Hw = 40 cm, Rd = 7.5 cm, Vm = 150 ml: a petroleum Ωd = 770 rpm; b aviation oil Ωd = 820 rpm; c petroleum Ωd = 1020 rpm; d aviation oil Ωd = 1050 rpm

(Fig. 3.10b). The rotating oil body (petroleum) is preserved when the disc rotation frequency increases (Fig. 3.10c), while in the system water—aviation oil, an invert emulsion—cells with water outlined by an angular oil shell—begins to form on the flow axis. The general picture of flow in the area of contact of oil body with air cavern at various angular speeds of disc rotation is shown in Fig. 3.11. When oil layer covers all free surface, cavern walls are smooth, transition of oil from inner to outer surface of cavern occurs smoothly, without any features (Fig. 3.11a). As the disc rotation frequency increases, spiral waves are observed on the cavern surface and a step is formed in the transition area of the marker from the outside of the cavern to the inside (Fig. 3.11b), which is absent in experiments with other types of markers [3]. Spherical drops detached from the bottom edge of oil body join into groups and partially merge forming pear-shaped bodies. At further increase in rotation frequency the step in the area of oil transition line from external to internal side of liquid surface becomes more pronounced, and instead of straight emulsion drops on the bottom edge of the body invert cells are formed (enlarged image of flow pattern is shown on the inset on the right in Fig. 3.11c). The size of the emulsion cells ranges from 0.8 to 3.5 cm. The mechanisms for the formation of such structures will be presented below. Comparison of the size of the oil body depending on the amount and physical properties of the immiscible additive added to the flow for different initial values of the water layer depth shows that an increase in the viscosity of the marking additive leads to a decrease in the vertical size of the oil body h t . Dependencies of the height of the oil body on the rotational speed of the activator disc are shown in Fig. 3.12.

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow

a

b

81

c

Fig. 3.11 Shape change of the oil body (aviation oil) when changing the activator speed Hw = 40 cm, Rd = 7.5 cm, Vm = 150 ml: a–c Ωd = 670, 820, 1050 rpm respectively

Fig. 3.12 Dependencies of oil body height on rotational speed of the activator disc (R = 7.5 cm), 1—castor oil (H = 40 cm), 2 and 3—sunflower oil (H = 20, 40 cm respectively): a, b Vk = 30, 90 ml

The parameters of the linear approximations carried out by the formula h t = A · Ω + B are given in Table 3.7. Oil distribution over the surface of the composite vortex The free surface and the change in the distribution of immiscible admixture along it does not in all cases serve as a clear indicator of admixture distribution. Both in fresh water and in water with salinity of 0.35‰, the free surface shows the transformation of a spot of immiscible admixture into a system of spirals. The centre of such a system is located in the middle of the free surface of the liquid. Characteristic images of the free surface are shown in Fig. 3.13.

82 Table 3.7 Values of coefficients of linear approximation

3 Transfer of Immiscible Admixture in a Vortex Flow Oil volume, Vk (ml)

Depth of water A layer, H (cm)

B

30 ml, sunflower oil

20

0.042 ± 0.010

− 7.650 ± 3.635

30 ml, sunflower oil

40

0.034 ± 0.003

− 7.445 ± 2.080

30 ml, castor oil

40

0.014 ± 0.001

− 2.371 ± 1.029

90 ml, sunflower oil

20

0.031 ± 0.001

− 4.216 ± 0.421

90 ml, sunflower oil

40

0.029 ± 0.003

− 5.539 ± 1.778

90 ml, castor oil

40

0.009 ± 0.001

0.846 ± 1.225

Fig. 3.13 Free surface view with a portion of an immiscible admixture (H = 40 cm, R = 7.5 cm, Ω = 240 rpm, Vk = 50 ml): a petroleum, b aviation oil; c, d light intensity of image sections along the diameter in photo a and b respectively

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow

а

b

c

d

83

Fig. 3.14 Evolution of castor oil slick on the surface of a compound vortex cavern (H = 40 cm, Ω = 250 rpm, R = 7.5 cm, Vk = 30 ml): a, d t = 1, 99, 143, 147 s

Sequential photograms of the evolution of the castor oil slick shape of the cavern surface formed by the composite vortex are shown in Fig. 3.14. The differential rotation deforms the spot into an irregular shape, from which a thick “spiral arm” about 3 mm thick, separated from the spot by a contact line at the “15–17 o’clock” area (Fig. 3.14a), is pulled out. After a short time, under the action of gravitation and surface tension, the elongated “arm” is almost completely drawn into the compact central spot, on the contour of which corner points appear (Fig. 3.14b). Thin spiral arms emerge from the corner points. The process of forming angular points and forming thin spiral arms is almost continuous. The liquid at the leading edge of the enveloping thin irregularly shaped arm is constricted into an elongated drop (Fig. 3.14c), position at “8 o’clock”, drop length about 16 cm. At the same time, new arms grow out of the other corner points of the central spot; in Fig. 3.14d, one can see the formation of a four-entry spiral. It is important to emphasise that thin strips of clear water remain between the central spot and the arms. Droplets separate from the leading edge of the arms, the number of which increases over time. The remnants of the arms form a “spiral pattern” on the surface of the cavern of the composite vortex.

84

3 Transfer of Immiscible Admixture in a Vortex Flow

a

b

c

d

e

f

Fig. 3.15 Free surface of the compound vortex (R = 7.5 cm, H = 40 cm): a–c Ω = 310, 320, 260 rpm, Vk = 30, 90, 150 ml, d Ω = 470 rpm, Vk = 30 ml; e Vk = 1000 ml, Ω = 550 rpm; f Vk = 2000 ml, Ω = 720 rpm

On the surface of a rotating liquid with a small amount of sunflower oil added (30 ml), the central core—the upper surface of the oil body and the adjacent continuous oil film—is surrounded by a system of droplets extending in a tangential direction (Fig. 3.15a). The contour of the film extends in direction of “10–4” hours, its maximum size is 5.15 cm, minimum—4.88 cm. As the angular velocity of the disc increases, most of the oil on the free surface becomes concentrated in the central part; the outer edge of the oil film is pearshaped, oriented with the pointing downwards, the maximum size is 3.82 cm and the minimum is 3.73 cm (Fig. 3.15b). The core is surrounded by segments of six spiral arms separated from the central spot by bands of clear water. As in the case of mixed admixture, the directions of development of the spiral arms and the basic rotation of the free surface are opposite. As the amount of oil increases, the outer contour of the central spot loses its regular shape and individual protrusions, sharps and irregularities form, giving rise to extended spiral arms, some of which detach from the central film and exist as individual spiral filament structures (Fig. 3.15c). The arms can split and tear, irregularly filling the periphery of the current.

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow

а

b

85

c

Fig. 3.16 Surface spot shapes (R = 5.0 cm, H = 40 cm, Vk = 150 ml): a–c Ω = 750, 1370, 1470 rpm

As the angular velocity of the disc increases, most of the oil on the free surface becomes concentrated in the central part; the outer edge of the oil film has a pearshaped shape oriented with the pointing downwards, the maximum size is 3.82 cm, the minimum 3.73 cm (Fig. 3.15d). A series of experiments with a disk radius of R = 5.0 cm were performed to study the structural stability of the immiscible admixture distribution on the surface and in the thickness of the composite vortex. In this case, the shape of spiral arms, their number and size also change depending on the rotational speed of the activator. At medium disc speeds (Ω = 750 rpm) the edge of the oil slick on the air–liquid interface has an irregular shape. From the edge of the main slick, broad spiral arms (transverse size from 1.0 to 2.5 cm), extending in the direction opposite to the main flow, and partially adjoining back to the central slick (Fig. 3.16a) are detached. Increasing the activator speed to 1370 rpm leads to a decrease in the width of the oil arms (cross size from 0.5 to 1.5 cm) and an increase in the number of spiral windings around the centre. Oil droplets, of various shapes, are intensively separated from the extremities of the spiral arms (Fig. 3.16b). More intensive flow, in this case at rotational speed Ω = 1470 rpm (Fig. 3.16c), results in splitting of peripheral parts of spiral arms into separate drops having mostly elongated shape and thinning of short spiral arms adjacent to the central spot. The oil droplets range in size from 0.3 to 0.9 cm and the diameter of the central spot is 4.4 cm. Between the spiral arms, the individual droplets and the central slick are areas of clear water surface. The bulk of the experiments were carried out with the addition of an immiscible admixture on the resting water surface, before the vortex formation began. To study the structural stability of the insoluble admixture transfer process, additional series of experiments were conducted with a change in the conditions of marker introduction into the flow, namely, marker admixture of similar volume was introduced onto the surface of the steady vortex flow in a thin jet. The experiments performed to investigate the distribution of an immiscible admixture of volume Vk = 30 ml injected into a resting and rotating liquid at an activator speed Ω = 770 rpm showed no significant differences in the admixture distribution pattern.

86

3 Transfer of Immiscible Admixture in a Vortex Flow

In all previous experiments tap degassed water was used as the working fluid, it is of interest to repeat the same experiments in salt water in order to find out the differences in the substance transfer process in the working fluid with different salinity. In experiments with fresh water, when oil is added to the surface, an oil body is formed, which at high activator speeds is completely immersed in the vortex, the free surface not even occupied by admixture droplets (Fig. 3.17). In the experiment with salty water (salinity is 0.33‰) at the same frequencies the interface between pure water and the oil body is observed more clearly. Complete clearing of the free surface occurs at frequencies much lower than in pure water (Fig. 3.18). Observing the oil body and all its boundaries in experiments with petroleum is difficult: it has too high an optical density, light does not penetrate well even through a thin layer.

a

b

c

d

Fig. 3.17 Shapes of axial liquid surface cross-section at different frequencies in pure water (R = 7.5 cm, H = 40 cm, Vk = 100 ml): a Ω = 430 rpm, b Ω = 666 rpm, c Ω = 850 rpm, d Ω = 1050 rpm

a

b

c

d

Fig. 3.18 Shapes of axial liquid surface cross-section at different frequencies in salt water (R = 7.5 cm, H = 40 cm, Vk = 150 ml): a Ω = 430 rpm, b Ω = 680 rpm, c Ω = 800 rpm, d Ω = 1060 rpm

3.2 Experimental Studies of Immiscible Admixture Transport in a Vortex Flow Table 3.8 Coefficients for linear approximation of the results of the experiment

A

87 B

Oil

2.63 ± 0.33

− 11.56 ± 3.25

Petroleum

4.47 ± 0.41

− 15.11 ± 3.25

Petroleum in the salt water

3.66 ± 0.64

− 1.82 ± 1.24

For the dimensions of the surface cavern and the oil body in the centre, the dependencies on the rotational speed of the activator are plotted and interpolated by linear functions of the form h t = AΩ + B, the coefficients entering them are given in Table 3.8. There is a clear tendency for the vertical dimensions of the oil body to decrease as the density and surface tension coefficient of the admixture added to the flow decreases, which is confirmed by the approximations obtained. The air–liquid surface deflection and penetration depths of the light admixture accumulation are given in Table 3.9. At high speeds of the activator disc (Ω = 1370 rpm) a cavern is formed in the clear deep liquid (H = 40 cm, R = 7.5 cm) with a depth of h = 29 cm. When 150 ml of sunflower oil is added, most of the admixture is concentrated inside a central area bounded from above by a rotation surface with a depth h k = 3.3 cm, i.e. less than the depth of the cavern in pure liquid, the height of the oil body is h = 23.34 cm. Although the surface of the water is covered by a film of oil, the contact line of the oil body with the surface of the water is identified quite clearly. The course of the linear approximations (coefficients in Tables 3.8 and 3.9) and the points limiting the central sections of the oil body are shown in Fig. 3.19. Table 3.9 Depths of deflection of the air–liquid surface and penetration of the light immiscible admixture accumulation at different speeds of the activator disc

Ω, rpm

Vk , ml

550

150

h, cm

h t , cm

3.66

3.66

720

6.6

6.6

930

10.47

10.47

1170

18.51

18.51

1370

23.34

26.64

4.02

4.02

430

100

666

15.63

15.63

850

20.49

20.49

1050

26.58

32.64

2.97

2.97

430

15.12

15.12

680

16.83

25.74

220

100

800

19.34

31.29

1060

15.62

32.55

88

3 Transfer of Immiscible Admixture in a Vortex Flow

Fig. 3.19 The dependence of the cavity depth on activator speed (H = 40 cm, R = 7.5 cm): line 1—oil Vk = 150 ml; 2—petroleum Vk = 150 ml; 3—petroleum in salt water Vk = 150 ml

When comparing the approximation coefficients, a complex dependence on the volume of immiscible admixture is observed, which is vividly illustrated by flow pattern images (Figs. 3.19 and 3.20).

3.3 Methodology for Comparing Data and Constructing Approximation Curves The consistency of the analytical dependences for the shapes of the interface surfaces is determined by the ability to satisfactorily approximate the experimentally recorded surfaces using analytical expressions. The approximating coefficients are chosen by the least-squares method by minimising the functional  2 |  Σ  | | d 2 Φ= ari + b − yi |ri ≤Rc + c + 2 − yi || ri ri >Rc r

(3.3.1)

i

written for the surface z = η(r ), where a=

A20 , 2g

b=H− c=−

 2  α − 1 A20 Rc4

A20 Rc4

2g R02

−

, 2g  2  α − 1 A20 Rc4 d=H− . 2g R02

A20 Rc2 , g

(3.3.2)

3.3 Methodology for Comparing Data and Constructing Approximation …

89

Fig. 3.20 Composite vortex cavern central sections: a (R = 7.5 cm, H = 40 cm, Vk = 150 ml) 1—oil–water boundary Ω = 500 rpm, 2—oil–water boundary Ω = 720 rpm, 3—oil–water boundary Ω = 930 rpm, 4—oil–water boundary Ω = 1170 rpm, 5—oil–water boundary Ω = 1370 rpm; b (R = 75 cm, H = 40 cm, Vk = 100 ml) 1—oil–water boundary Ω = 430 rpm, 2—oil–water boundary Ω = 666 rpm, 3—oil–water boundary Ω = 850 rpm, 4—oil–water boundary Ω = 1050 rpm; c vortex (R = 7.5 cm, H = 40 cm, Vk = 100 ml): 1—oil–water boundary Ω = 220 rpm, 2—oil– water boundary Ω = 430 rpm, 3—oil–water boundary Ω = 680 rpm, 4—oil–water boundary Ω = 800 rpm, 5—oil–water boundary Ω = 1060 rpm

Since in each of the sum terms one of the components is zero, the functional is converted to the form Φ=

2  Σ 2 Σ d c + 2 − yi ari2 + b − yi + ri r ≤R r >R i

c

i

(3.3.3)

c

The parameters c and d can be expressed through a, b and Rc , which brings the function to the form Φ=

   2 Σ 2 Σ R2 a Rc2 2 − 2c + b − yi ari2 + b − yi + ri r ≤R r >R i

c

i

c

(3.3.4)

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3 Transfer of Immiscible Admixture in a Vortex Flow

where Rc is the radius of the solid rotation, ri is the radial coordinate of the measured point, yi is its vertical coordinate [1]. For the surfaces z = θ (r ) and z = ς (r ) the functionality is the same, only the expressions for a and b are different through the physical parameters. There are three unknown parameters in the resulting functional: a, b and Rc , and the parameter Rc not only directly affects the absolute value of the second sum in (3.3.4), but also determines the number of summable quantities in each sum. For this reason, the minimum of the proposed functional is searched as follows. First, we write down the conditions that the derivatives Φa, and Φ,b are zero: ⎞ ⎛ ⎞ 2 2 2 Σ Σ Σ R R 2 − 2c ⎠ 2 − 2c ⎠ + b⎝ ri2 + Rc2 a⎝ ri4 + Rc4 r ri i ri ≤Rc ri >Rc ri ≤Rc ri >Rc   Σ Σ R2 yi ri2 + Rc2 yi 2 − 2c = ri ri ≤Rc ri >Rc ⎛ ⎞  N Σ Σ Σ R2 2 − 2c ⎠ + N b = a⎝ ri2 + Rc2 yi (3.3.5) ri i=1 r ≤R r >R ⎛

Σ

i

c

i

c

where N is the number of measuring points. The system (3.3.5) is then analytically solved with respect to a and b, which turn out to be expressed through the experimental data and the unknown parameter Rc . These quantities are then substituted (analytically) into the functional (3.3.4), which turns out to be a function of only one parameter Rc . Finally, by numerically varying Rc the minimum of the functional is obtained. The parameters a and b are calculated after determining the value of Rc . The calculation is carried out separately for the shape of the oil–air interface and the complete water–air–oil interface. The calculated values a, b and Rc are indicated by the symbols ao , bo and Rco for oil and aw , bw and Rcw for water. An important feature of the calculation of the coefficients of the functional (3.3.3) should be noted here. Since the calculations are performed for each of the surfaces separately, the correspondence of the results to the real physical flow was achieved by imposing additional requirements. Firstly, the calculated values of the solid rotation radius Rc for different regions should coincide within a relative error of no more than 10%. Second, the values of coefficients a and b should ensure consistency with analytical relations (3.3.1). This means, for example, that in the expression for the surface η(r ), the ratio of the second term to the coefficient at r 2 is determined by the value 2α2 Rc2 + (1 − α2 )Ro2 (the value H is known in advance, and Ro is determined experimentally with sufficient accuracy). This condition also applies to the surfaces θ(r ), ζ(r ) of (3.1.21). In order to obtain satisfactory results, it was necessary to measure the coordinates of a sufficient number of points. A camera frame and computer image analysis tools

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91

were used. Knowing the physical dimensions of the working volume and their corresponding pixel dimensions, the vertical and horizontal scaling factors for coordinate conversion were calculated. An application was created for the NET, which allows a point image to be loaded and the coordinates recorded with the mouse as a list. This implementation speeds up the process of measuring the coordinates of key points many times over in comparison to the way in which point coordinates are manually determined using the status bar of a graphical editor (MS Paint, Adobe Photoshop). An application has been created which implements the search for minima of the functional in accordance with the described methodology. The system (3.3.5) was solved with respect to a and b for all values of the parameter Rc , varying in steps of 0.01 cm. Such a step of variation with a large margin overlaps the errors of determination of coordinates of points arising both at their visual determination from the image and fluctuations of the interface during the experiment, i.e. in the “raw” initial data. For each solution of this system, the value of functional F was calculated. In spite of the fact that the number of summed coordinates in system (3.3.5) depends on the parameter Rc , the value of functional F changed smoothly, which corresponds to the physical sense of the phenomenon in question. The value of Rc , corresponding to the minimum value of F from the list of solutions, is the radius of solid-state rotation. The parameter Rc for each experiment was determined twice—using a set of points lying on the water–oil, water–air boundary and a set of points lying on the oil–air boundary. In an experiment with a 150 ml portion of oil on the surface of a compound vortex with a liquid depth of 40 cm and an inductor with a radius of 7.5 cm, the change in the shape of the interface can be clearly traced over a wide range of disc-inductor rotation frequencies (Fig. 3.21). At different parameters of the experiment the frames corresponding to the steadystate flow regime were selected and then the points were sampled using the created application. An overlay of the selected points on the flow pattern images is shown in Fig. 3.22. The approximation curves constructed for the oil–air boundary using parameters ao , bo , Rco , and the full water–air and water–oil boundary using parameters aw , bw , Rwo , superimposed on the images are shown in Fig. 3.22. The main reason for this is its rotation around the vertical axis, which was taken into account when obtaining the analytical dependences of the interface shapes, performed in the absence of boundary layers and vertical flows in the flow volume, due to a much smaller contribution of these flow to the total energy of the vortex. The coincidence of the calculated values Rcw and Rco , along with good agreement between the approximating curves and the experimentally recorded interface shapes, proves the applicability of the proposed representation of the interface shape [4].

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3 Transfer of Immiscible Admixture in a Vortex Flow

Fig. 3.21 Composite vortex with a portion of tinted sunflower oil on the free surface (H = 40 cm, R = 7.5 cm, Vk = 150 ml): a Ω = 600 rpm, b Ω = 1000 rpm, c Ω = 1400 rpm

Fig. 3.22 Composite vortex with a portion of tinted sunflower oil on the free surface and superimposed dots (see Fig. 3.21 for experimental parameters): a Ω = 600 rpm, b Ω = 1000 rpm, c Ω = 1400 rpm

3.4 Forms of Partial Disintegration of the Oil Body in a Compound Vortex, Formation of Forward and Backward Emulsions Fluids in natural conditions and in industrial installations are usually multicomponent, containing suspended particles and dissolved gases which can form bubbles. In the field of mass forces, complex fluids stratify. The resulting stratification, even if weak, significantly affects the dynamics and structure of flows [5].

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Flow patterns become more complex when sharp interface surfaces exist in fluids. Interest in the study of the dynamics of multiphase and multicomponent media has been stimulated by environmental problems and by the increasing production and transportation of inert and chemically and biologically active substances. Of scientific and practical interest is the study of contact surface decay processes in vortex flow. The ability of fluid mixtures to fragment, form emulsions and foams is used in some industrial processes and is an undesirable factor in others [6]. The purpose of this paragraph is to investigate the stability of the water–oil interface in a composite vortex and to determine the characteristic shapes and conditions for the restructuring of the flow regimes. Experiments were carried out with a two-layer liquid—underlying water and an immiscible lighter marker. The depth of the water layer was Hw = 10 ÷ 60 cm, the thickness of the marker layer Hm determined by its volume was from 2 mm to 15 cm. Before the experiment, the container was filled with water (layer height Hw ), over which a layer of oil with the height Ho , covering the water surface at all flow regimes, was poured. In the experiments the oil volume was Vo = 1 or 2 L, the layer height was Ho = 2.5 and 4.9 cm respectively. As the disc rotates, the liquids are involved in a vortex flow which changes the shape of the contact and free surfaces. A picture of the steady vortex flow with a large amount of oil (Vo > 1000 ml, spin duration 20 min) and its diagram is shown in Fig. 3.23. At moderate angular velocities of rotation, the oil–water surface has a smooth shape with a small deflection in the centre (Fig. 3.23a). The oil–water surface has a more complex shape, with a central oil body adjacent to it—an oil-filled cavern that exists in a homogeneous liquid at close experimental parameters [3]. Observations have shown that each of the layers rotates with its own angular velocity relative to the vertical axis. The parameters of the flow geometry are shown in the diagram shown in Fig. 3.23b.

Fig. 3.23 Composite vortex with a portion of sunflower oil (Vo = 2000 ml, Hw = 40 cm, Rd = 7.5 cm, Ωd = 220 rpm): a, b photo (side view through the liquid–air interface) and flow diagram

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3 Transfer of Immiscible Admixture in a Vortex Flow

a

b

c

Fig. 3.24 Characteristic shapes of the oil–water contact surface in a compound vortex (Hw = 40 cm, Rd = 7.5 cm): a pure water, Ωd = 500 rpm; b, c water and sunflower oil: Vo = 30, 2000 ml, Ωd = 530, 330 rpm

The shape of the cavern, which changes markedly when oil is added, depends on the flow parameters and the amount of admixture. In clean water at Ωd = 500 rpm a cavern is formed whose cross section is close to the classical Rankine vortex parabola (h = A1 r B1 , A1 = 0.26 ± 0.06, B1 = 1.91 ± 0.12, the origin is placed at the lower pole of the cavern) (Fig. 3.24a). A family of spiral waves is observed at the liquid surface. Adding a small amount of oil (Vo = 30 ml) increases the cavern depth to 15.2 cm and changes its shape (Fig. 3.24b). The oil collects in a narrow body in the vicinity of the rotation axis h = A2 r B2 ( A2 = 1.09 ± 0.49, B2 = 2.55 ± 0.07) in all cases of oil body contour approximation the origin is placed on the lower pole, its lateral surface is distorted by large-scale waves (Fig. 3.24b). Part of the oil remains on the free surface where droplet systems gather in spiral arms. During spinning in two-layer liquid (Vo = 2000 ml of sunflower oil, Ho = 4.9 cm) a part of light admixture is drawn into water, where it forms an oil body of rotation (h = A3 r B3 , A3 = 3.44 ± 0.54, B3 = 7.48 ± 0.67) (Fig. 3.24c). The oil–air contact surface remains smooth at low angular velocities of disc rotation, with increase of rotation velocity a pronounced surface cavern, inertial and capillary waves appear on it. The initially smooth shape of the water–oil contact surface is disturbed as the angular velocity of the disc increases, and gradually fine spiral waves with sharp crests appear on it, from which the oil tears off into the water column in the form of individual droplets. Some of the oil droplets separate while others remain connected to the oil body by thin bridges (such elements can be seen in Fig. 3.24c). Gradually, the lower surface of the oil body loses its integrity and becomes covered by individual oil droplets, the highest density of which is observed at the lower pole of the body. The contact surface between immiscible liquids can fragment to form two types of emulsions—direct and invert [7]. The basis of direct emulsions is a more polar dispersion medium (usually water) in which there is a dispersed phase (oil). An example of a direct emulsion is oil droplets in water. In reverse (inverted) emulsions, the more polar medium is the dispersed medium distributed in the less polar medium (oil, petroleum products). In these experiments with two-layer liquids both types

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95

of emulsions were observed: at low angular velocities of disc rotation—direct, at high—inverted. Forming a straight emulsion At the initial loss of stability on the extended oil–water interface, which forms when a large volume of immiscible mixture is added (V0 = 2000 ml, Ho = 2.8 cm), direct emulsion formation was observed. The sunflower oil–water contact surface lost stability at moderate disc rotation speeds. Separate thickenings on crests of oil body surface at Ωd = 300 rpm began to tear off, both completely, forming isolated spherical drops, and partially, remaining connected with main oil body by thin bridges (“drops on legs” in Fig. 3.25a). Here the height of the oil body in the centre is 9.7 cm, and at the container wall it is 0.9 cm. A tubercle 0.8 cm long and 0.2 cm high is observed on the surface of the body at a distance of 0.3 cm from the lower pole (Fig. 3.25b). The enlarged image of the lower part of the oil body shown in Fig. 3.25b shows a set of irregularly shaped droplets reflecting the spatially non-uniform flow pattern in the vortex core. A layer of straight emulsion of variable thickness (0.6 cm at the lower pole) composed of individual oil droplets of 0.3 cm diameter covers the lateral surface of the body to a height of about 5 cm. The entire configuration, comprising the oil body and the adjoining straight emulsion droplets, rotates as a unit. At further increase of rotation frequency the flow pattern becomes more complicated and in addition to droplets forming direct emulsion, foam is formed in the lower part of the body, which consists of isolated volumes of water in the oil shell—invert emulsion (Fig. 3.26a). The individual oil cells, ranging in size from 0.5 to 1.3 cm, collect into a cloud 12.1 cm high. Fig. 3.25 Accumulation of emulsion droplets at the lower pole of the oil body (Rd = 7.5 cm, Hw = 40 cm, Vo = 2000 ml, Ωd = 300 rpm). a, b Full image and its enlarged part

а

b

a

b

Fig. 3.26 Straight (oil droplets) and inverted emulsion in a compound vortex (Rd = 7.5 cm, Hw = 40 cm, Vo = 2000 ml, Ωd = 520 rpm): a, b full image and its enlarged part

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3 Transfer of Immiscible Admixture in a Vortex Flow

Individual oil droplets, irregularly distributed along the lateral surface of the body, are retained on the crests of the waves. Individual droplets detach from the surface of the body and circulate around it, moving away and coming closer. The movement of isolated droplets is unsystematic. The foam cloud can be crushed into wisps made up of angular cells ranging in size from 0.5 to 4.0 cm, which collect on the underside of the oil body (an enlarged image of a section of such flow is shown in Fig. 3.26b). The foam also collects in separate wisps near the top of the body, which itself irregularly precesses around the axis of the container. Part of the foam forms a smooth continuation of the body, with a formed oil droplet 3.4 cm at the lower pole. The flow pattern stabilises as the rotational speed of the activator disk increases, when most of the oil is collected in the body. Under the conditions of this experiment (Fig. 3.26a), the lateral surface of the body (22.8 cm high and 12.7 to 8.4 cm transverse at 7.5 cm and 16.3 cm depths) is deformed and covered with large irregular waves and individual protrusions and depressions. Characteristic ridge heights are 0.8 cm and wave lengths range from 2.1 to 7.3 cm (Fig. 3.26a). The contour of the smoothed body section is approximated by the curve h = A7 r B7 (A7 = 1.41 ± 0.26, B7 = 1.38 ± 0.11). The lower part of the oil body is adjoined by a foam cloud of inverted emulsion, also in the shape of a body of rotation, 10.1 cm high and 2.1 cm wide in the area of contact with the oil body. The oil pattern on the surface of the liquid shows thin spiral arms oriented towards the main flow, as in earlier experiments [8]. Another system of spiral arms adjacent to the lateral surface of the oil body in the liquid column can be seen through the translucent oil. The fully submerged oil arms, the width of which is 0.4 ÷ 1.6 cm, are oriented in the direction of the main rotation of the liquid (Fig. 3.27b). When the disc rotation is switched off, the water velocity decreases rapidly and the pressure distribution in the liquid evens out. Shear stresses on the water–oil Fig. 3.27 Oil body and invert emulsion in a compound vortex (Hw = 40 cm, Rd = 7.5 cm, Vo = 2000 ml, Ωd = 700 rpm). a Oil distribution in the liquid column; b view of the oil body from above

b a

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Fig. 3.28 Disintegration of the oil body and the adjoining inverted foam cloud after disc stoppage t = 18, 47, 82, 112 s

a

b

c

d

contact surface also decrease, and the angular velocity of the oil body rotation falls. In process of attenuation of flow the surface of oil body smoothes, the height of ledges—decreases, the thickness of oil layer grows, and, by the moment t = 18 s, makes 1.8 cm (Fig. 3.28a). The remnants of the foam cloud, 11.9 cm high and 4.8 cm wide, are adjacent to the oil body in the area of contact with the oil body. The lower pole of the foam formation makes circular movements with deviations from the centre line of 1.8 cm (Fig. 3.28a). As the fluid velocity in the container decreases, the single inverted foam formation breaks into two clouds. The disintegration of the foam clouds (at t = 47 s, the height of the main one is 10.8 cm and the dimensions of the detached volume are di = 2.2 cm, h i = 4.5 cm) is accompanied by thickening of the oil cell borders, some of which are shrinking into droplets. The droplets, under the action of buoyancy forces, float up faster than the water-filled cells of the inverted foam and gradually pass from the detached cloud to the main one adjacent to the oil body (Fig. 3.28b). After complete stratification of liquid in water there remains foam formation in the form of body of rotation, the size of cells in which monotonically increases from 0.3 to 1.8 cm with distance from base to lower edge (Fig. 3.28c). The height of the foam cloud is 11.9 cm with an oil layer thickness of 2.8 cm. The 6.8 cm high inverted foam cloud, densely filled with cells ranging in size from 0.2 cm near the oil layer to 1.1 cm at the base, retains its body of rotation shape for a long time (Fig. 3.28d).

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3 Transfer of Immiscible Admixture in a Vortex Flow

Experiments have shown that a composite vortex in a transparent container makes it possible to study changes in the water–marker contact surface, tracing changes in its structure depending on the properties of the selected marker. The geometry of the contact surface depends on the vortex flow parameters given by the angular velocity of the inductor and the composition of the medium. It is found that a pronounced step is observed in the region of triple contact of oil with the carrier medium (water) and air when aviation oil is used as the marker. The conditions for loss of interface continuity depend on the type of admixture. At medium angular velocities direct emulsion is formed, with increasing velocity both types are observed, and at higher velocities invert emulsion is formed. Conclusions to this chapter A methodology has been developed for investigating the propagation of various impurities in vortex flow, allowing experiments to be carried out in pure water, in salt water, and in water containing hydrocarbons and alcohol. The fine structure of oil–water and liquid (water or immiscible hydrocarbons)–air interface surfaces in a compound vortex, including the mode of emulsion formation onset, was studied experimentally. Massively used fluids (sunflower, engine and aircraft oils, oil, diesel fuel and their mixtures in different proportions) were chosen as an object of study. Experiments have shown that in the vortex under study, in both fresh and salt water, immiscible fluids (such as oil, diesel fuel) partly remain on the free surface and partly are drawn into the liquid column, where they form a cohesive oil body in the vicinity of the rotation axis. The geometry of the contact surface depends on the parameters of the vortex flow, given by the angular velocity of the inductor, and the composition of the medium. At the interface between the oil body and the underlying fluid, as at the surface of a cavern in a pure fluid, two types of disturbances—large scale inertial waves and small scale spiral waves—are observed. The oil slick on the surface of the rotating liquid is adjoined by spiral arms oriented towards the main flow. The oil in the thick of the composite vortex collects into a compact body bounded by the rotating surface. The size and shape of the oil body and the adjoining arms depend on the flow parameters. A comparison of the size of the oil body depending on the amount and physical properties of the immiscible additive added to the flow for different initial values of the water layer depth shows that increasing the viscosity of the marking additive leads to a reduction in the vertical size of the oil body. In all the experiments performed, the admixture cannot be considered passive, the displacements of the individual strips, the position of the oil droplets and the orientation of the spiral arms do not reflect the flow pattern in the composite vortex. It is shown that oil–body decomposition can occur in several forms. During the decomposition of the oil body the loss of continuity of the interface depends on the type of admixture. The type of emulsion formed depends on the rotational speed: at medium angular velocities of rotation a straight emulsion is formed, with increasing speed both types are observed, and at high velocities an inverted emulsion is formed.

References

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For the first time the problem of analytical determination of the shape of oil body in a composite vortex on the basis of analysis of equations of mechanics of divergent liquids with physically grounded boundary conditions is considered. The dependences, reflecting the form of interfaces in the vortex flow of the fluid, consisting of two components, are obtained. The analytical expressions, which characterize the form of zero approximation for the phase boundaries in the composite vortex, are in a satisfactory agreement with the experimental data. Peculiarities of obtained analytical expressions for the shape of interface boundaries in vortex flow impose significant restrictions on properties of used immiscible additives because low optical density is necessary for exact determination of all parameters of interface boundaries and on applied imaging tools which should provide necessary image resolution for measurements of interface surface shape. All observed flow patterns are consistently reproducible when the experiments are repeated and conditions are maintained within the accuracy of the experiments.

References 1. Stepanova EV, Chaplina TO, Chashechkin YD (2013) Experimental study of oil transfer in a compound vortex. Appl Mech Tech Phys 54(3):79–86 2. Stepanova EV, Chaplina TO, Chashechkin YD (2014) Forms of partial disintegration of an oil body in a compound vortex. Izv RAN Fluid Gas Mech 5:52–64 3. Baidulov VG, Matyushin PV, Chashechkin YD (2007) Evolution of flow induced by diffusion on a sphere immersed in a continuously stratified fluid. Izv RAN Fluid Gas Mech 2:130–143 4. Chaplina TO (2019) Investigation of immiscible admixture transport in vortex flow in multiphase liquids. Process Geosp 3:282–291 5. Kutepov AM, Latkin AS (1999) Vortex processes for modifying disperse systems. Nauka, Moscow, 268 c 6. Karelin VY, Minaev AV (1986) Pumps and pumping stations. Stroyizdat, Moscow, 320 c 7. Stepanova EV, Chaplina TO, Chashechkin YD (2011) Oil transfer in a compound vortex. Izv RAN Fluid Gas Mech 2:52–64 8. W3C. Standard Rex-SVG11-20110816 scalable vector graphics (SVG) 1.1. URL: http://www. w3.org/TR/SVG

Chapter 4

Motion of Solid Markers in a Vortex Flow

The visualisation of fluid flow and the determination of hydrodynamic characteristics, particularly velocity, by introducing various marking objects into the flow have been and continue to be important tasks for researchers. However, different objects placed in a moving fluid behave differently due to their inherent physical and chemical properties. The movement of a solid object, especially in a vortex current, is complicated by the presence of a velocity shift at its boundaries, whereby the solid object, in addition to its basic motion around the centre of the vortex, begins to rotate around its own axis. Since a buoyant object has inherent physical and chemical characteristics that differ from those of a fluid medium, it is necessary to introduce corrections to account for the interaction between the medium and the object when describing its motion. In geophysics, such corrections can be useful, for example, in studies related to the problem of plastic debris accumulation in the centres of eddies in the open ocean, as well as for correcting the readings of various drifter probes that transmit information about sea and ocean flow. This chapter is devoted to the problem of visualization of vortex flows by introducing different kinds of markers into the moving liquid and the problem of measuring flow characteristics based on the observed movements of the markers. The method of automatic processing of the results of marker’s behaviour on the free surface of vortex flow is developed. This method is based on the raster image transformation into the vector representation that allows considerably accelerating the processing of the results of the experiments. The results of experimental and theoretical investigations of solid markers of different shapes and sizes in vortex flow in multiphase liquids are presented. In this chapter, a theoretical model to explain the motion of markers placed on the surface of the vortex flow in a one-component fluid is proposed.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina, Transfer of Substance in Vortex and Wave Flows in One-Component and Multi-component Environment, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-31856-6_4

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4 Motion of Solid Markers in a Vortex Flow

4.1 Experimental Investigations of the Movement on the Surface of the Vortex Flow In order to clarify the flow pattern and to estimate indirectly the rotation speed of the free surface of the liquid in the vortex flow, a series of experiments were conducted where solid markers were used as indicators (Fig. 4.1). The motion of a marker that moves along the surface of a fluid involved in a composite vortex is reduced to a combination of its circulation around the vertical axis of the flow and simultaneously rotation around its own centre of mass [1]. For convenience, several coordinate systems are introduced to describe the movements of the marker on the free surface of the flow. The origin of one of them XOY (rectangular Cartesian) coincides with the geometric centre of the free surface, the axes are directed along the sides of the video frame of the flow pattern, the plane of the system coordinate axes coincides with the level of the undisturbed free surface before the inductor rotation is activated (Fig. 4.2). Another coordinate system—polar (r, ψ)—is located in the same plane and is most convenient for recording the marker movements around the vertical flow axis. In this system, the coordinates of the marker centre are counted as a function of time. The centre of the marker is connected to the attached (movable) Cartesian coordinate system xOy, used to set the “twist” angle γ, the additional angle θ describes the deviation between the instantaneous position of the x-axis of the xOy coordinate system and the radius-vector of the marker r. It is worth noting that for all types of markers there are special marks on them to determine their angular position. The radial and angular coordinates of the marker in the polar coordinate system as well as the values of γ and ψ were independently determined from images taken from a video recording of the experiment with a known time step. Automation of experimental data processing Modern developments in video and computer processing of large data sets have greatly facilitated the processing of experimental data. Conducting an analysis of a series of experiments to record the movement of a tinted marker on the surface of

Fig. 4.1 Markers used in experiments

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103

Fig. 4.2 Moving the marker across a free surface: a Square marker (1 × 1 cm), Ω = 300 rpm; 1— 5 s, 2—20 s, 3—27 s, 4—35 s, 5—44 s—superimposed consecutive frames, b entered coordinate systems

a current requires the processing of a large amount of data. Motion of the marker along the free surface of the water was recorded using a video camera, so the input data for processing are video files in mp4 format with a resolution of 1920 × 1080, and shooting frequency is 25 frames per second. The amount of memory required to record 1 min of video is 250 MB. During processing such a video file, converted using frame-by-frame slicing, is converted into a set of still images, an array of 1500 images. Manual processing of such a large number of images is very time-consuming. The volume of raw data and the time spent on its processing significantly affect the efficiency of the experimental work carried out [2]. The task of reducing time by automating the machining process is relevant and its solution will improve the quality of the experimental work carried out. One possible way of solving the automation problem is the use of vector graphics [3, 4]. A single frame extracted from an original video recording is a bitmap image. At the same time, if the image is converted into vector representation, it will be a text file containing a set of commands by which this image is displayed on the screen. Firstly, such transformation allows to reduce considerably the volume of initial information, and secondly, the task of processing text files is many times easier than the task of processing of files with two-dimensional bitmap images. Estimation of marker movement parameters along the liquid surface on the basis of raster images requires the application of pattern recognition theory. Modern vector graphics is the result of many years of application of pattern recognition theory to such problems in the creation of geographic information systems, mapping and printing. As a result of the search for information on software tools for raster-to-vector conversion, the Potrace software product [5] was found capable of performing the task. The markers used in the experiments were: square, circular, ellipse-shaped, ringshaped, pentagonal and cross-shaped—the vertical size of all markers is 0.5 mm. Most experiments were performed with the following parameters: liquid depth 40 cm, inductor speed—220 rpm or less, which corresponds to a free surface with a slight bend in the centre for any size of the activator disk [6].

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For three-dimensional markers—a cylinder with a radius of 0.5 cm and a height of 0.3 cm, and a parallelepiped (round), one of whose faces is a square with side 1.0 cm, and the third dimension is 0.3 cm (square)—dependences of the “twist” angle relative to its own centre γ on the rotation angle around the flow axis ψ are more complex. A dependence on both marker shape and medium parameters (oneor two-component fluid) is observed. In clear water, a cylindrical marker moves with the fluid flow around a central vertical axis, slowly shifting towards the centre and simultaneously rotating around its own axis. The tangential component of the velocity of the centre of the marker in clear water at Ω = 9.2 s−1 at r = 8.1 cm from the centre is vϕ = 2.26 cm/s and changes slowly as it moves toward the centre of the current. During the observation time between the frames shown in Fig. 4.3a, b (40 s), the marker makes 24 revolutions and moves to the centre by 4.3 cm. At the same time marker makes 12 revolutions around its own axis. Under the same conditions (H = 40 cm, Ω = 9.2 s−1 , Rd = 7.5 cm) in the twolayer liquid water–oil, the cylindrical marker in time t = 40 s manages to make 16 revolutions around the centre (ψ = 16π), 8 revolutions around its own axis (γ = 34π) and hits the rotation centre at the bottom of the surface cavern, having traveled Δr = 81 cm (in a homogeneous liquid at the same angular velocity of rotation of the disk the marker is displaced by a smaller distance Δr = 4.3 cm) [7]. Each experiment was repeated at least three times, and the graphs show the average values for the whole set of experiments. Fig. 4.3 Circular marker transfer in a composite vortex (H = 40 cm, Ω = 9.2 s−1 , Rd = 7.5 cm): a, b Pure water (t = 10, 50 c), Rew Ω = 51562.5, Fr = 0.119; c, d bilayer liquid (water and petroleum Vo = 50 ml, unperturbed layer thickness −Δ = 15 mm) (t = 10, 50 c), Ataw = 0.998, Atao = 0.997, Atow = 0.099, Boaw = 0.055, Boao = 0.109, Boow = 0.257 (for clarity the risks are continued with white lines in the figures above)

a

b

c

d

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Fig. 4.4 Angle of rotation of the marker around its own axis versus time (H = 40 cm, R = 7.5 cm, Re = 18 750, 51 750, 93 750, Ω = 3.3, 9.2, 16.7 s–1 ): a Circular cylinder with radius 0.5 cm and height 0.3 cm, b parallelepiped 1.0 × 1.0 × 0.3 cm

a

b

When a marker is placed in a vortex current, viscous tangential stresses from the fluid side begin to act on its lower surface, carrying it as a whole along some trajectory lying on the surface and rotating it around its own centre of mass. As one moves towards the centre of the vortex, the marker enters the transition region, crosses the boundary between the solid-state and peripheral vortex rotation types and gradually reverses its own rotation relative to the centre of mass, which is maintained after the marker has fully entered the solid-state rotation region. The nature of the marker ‘spinning’ depends on its shape and the angular velocity of the activator disk Ω. . The circular marker after a short engagement interval (Δt = 5 s) spins with an almost constant angular velocity, the value of which monotonically increases with increasing disk rotation speed: ω1 = dϕ1 /dt = 66.6 s–1 , ω2 = dϕ2 /dt = 99.3 s–1 , ω3 = dϕ3 /dt = 134.8 s–1 . The square marker also spins at an almost constant angular velocity at low angular speed of disc rotation ω4 = dϕ4 /dt = 30.1 s–1 (at Ω = 3.3 s–1 ) and at high ω5 = dϕ5 /dt = 128.2 s–1 (Ω = 16.7 s–1 ). At intermediate values of angular velocity of the activator disk rotation Ω = 9.2 s–1 the square marker is believed to be uneven (curve 2, Fig. 4.4b), which reflects the complex nature of interaction of the entrained square marker with the main current. Probably, irregular character of the twirling is caused by interaction of the marker with inertial and spiral waves arising randomly on the surface of the rotating liquid, most pronounced at the given flow regime. The dependence of the square marker “twirl” angle on time is approximated by the functions ϕs = 40.8 t 1.4 (for frequency Ω = 3.3 s–1 ), ϕs = 0.77 t 2.68 (for frequency Ω = 9.2 s–1 ), ϕs = 1.4 t 2.4 (for frequency Ω = 16.7 s–1 ). Based on the results of automatic processing of the experimental data, the dependences of changes in the radius of circulation of markers of different shapes around the centre of the rotating free surface were obtained and the dependences of the radial position of the marker on time were calculated (see Figs. 4.5 and 4.6). The peculiarity of all presented graphs are periodic changes of coordinates (runout) of marker centres when approaching the centre of the rotating free surface. Small fluctuations of the marker centre coordinates are associated with elliptical

106

4 Motion of Solid Markers in a Vortex Flow

Fig. 4.5 Dependence of change of radius of circulation of markers of different shapes around the centre of rotating free surface (H = 40 cm, R = 7.5 cm, Ω = 200 rpm): 1—pentagonal, 2—oval and 3—circular markers with the largest size of 1 cm

Fig. 4.6 Dependence of change in radius of circulation of markers of different shapes around centre of rotating free surface at different liquid depths (R = 7.5 cm, Ω = 200 rpm): oval marker length L = 1.5 cm, liquid depth: 1—20 cm, 2—40 cm

distortions of the free surface shape, as well as with small fluctuations of the position of the free surface rotation centre relative to the stationary walls of the container. After being placed on a rotating free surface, the marker gradually gains speed and moves towards the centre—the radial coordinate decreases and the beating frequency increases. The treatment of these experiments clearly demonstrates the effect of the shape of the marker on the velocity of the marker and on the nature of the dependence of the change in the radius of the marker around the centre. The shape of the marker is related to the rate of speed gain at the start of the experiment—markers with smooth boundaries (round and oval) are slower to gain speed and slower to move towards the centre of the current than square and cross-shaped markers. Predictably, the slower the speed gain and approach to the centre of the current, the greater the mass of the marker, even when the shape is the same. At the same time, the rate of velocity

4.2 Analytical Representation of Marker Movement on the Vortex Surface

107

Fig. 4.7 Dependence of the change in the radius of circulation of markers of various shapes around the center of a rotating free surface at different depths of the liquid (R = 7.5 cm, Ω = 200 rpm): a Oval marker length L = 1.5 cm, b cruciform marker L = 1.5 cm, liquid depth H: 1—20 cm, 2—40 cm

acquisition and approach to the centre of the current is practically independent of the depth of the initial layer of liquid involved in the vortex flow (Fig. 4.7). Measurements of the radius of circulation of a large-scale solid marker from the centre of the free surface show that the best fit of the approximating function R(t) with the experimental data is obtained using the logarithmic relationship of the form Rm = A ln(t) + B, where Rm is the radial coordinate of the marker relative to the centre, t is the time elapsed since the marker was placed on the free surface of the current.

4.2 Analytical Representation of Marker Movement on the Vortex Surface In estimating the force impact of the water flow on the marker, the experimental fact that its depth of immersion in water is significantly less than the characteristic scale of the near-surface flow is used. This allows an approximation of no flow disturbance by placing the marker on its surface. On this basis it is assumed that the marker is flattened on the surface of the vortex but retains its flat shape in the calculations. The marker on the surface of the water is subjected to gravity, Archimedes’ force and the hydrodynamic force, which is the result of the addition of several forces generated by different physical phenomena. The hydrodynamic force is determined by the pressure distribution over the lower and upper surfaces of the marker, the capillary effects occurring at the water contact line on the edge of the marker, the flow entrainment and braking effects due to the immersion of the marker in water, and the viscous interaction effects with water over the entire contacting surface.

108

4 Motion of Solid Markers in a Vortex Flow

Obtaining exact analytical expressions for all the forces mentioned in the case of an arbitrarily shaped marker seems to be an unsolvable problem and therefore a number of approximations are used to obtain constructive results. The first approximation is that the motion of the marker is always described by the displacement of its centre of mass and rotation about an instantaneous axis always passing through it normal to the surface of the liquid. Since markers whose shape exhibits central symmetry (continuous or discrete) were used in the experiments, the tangential component of the total capillary force acting on the marker is zero. The force exerted by the water flow on the submerged part of the marker rib is determined by the expression ∫ c f (L)ν (ν · (v − r˙ c ))d L

Ff = L

where rc is the radius-vector of the centre of mass of the marker; v is the water velocity field, which due to the small thickness of the marker compared to the characteristic scales of the flow is considered to vary only along the contour L, bounding the edge of the marker; ν is the internal normal to the contour; c f (L) is the local shear coefficient of the form resistance (in case of an arbitrary shape of the bounding contour L this coefficient is a value difficult to determine); the points above the symbols here and below denote differs. Due to the central symmetry of the markers, the velocity head occurring on part of the contour and accelerating the marker is almost completely offset by the braking effect occurring on the remainder of the contour. The resulting second approximation is to neglect the force F f generated by the impact of the flow on the marker edge, allowing the marker to be considered as infinitely thin and its centre of mass located on the liquid surface. If the marker were placed on a horizontal surface, the resultant gravity force, the Archimedean force normal to the surface of the marker, the capillary force and the force due to the pressure difference between the upper and lower surfaces would be zero. In case of an inclined liquid surface this resultant force is directed along the tangent to the liquid surface, since the marker itself is not detached from this surface, and is given by the expression FR = λmget where m is the mass of the marker; g is the acceleration of gravity; λ is some dimensionless coefficient proportional to the tangent of the angle of the liquid surface to the horizon; et is the unit tangent vector to the liquid surface. When the marker is placed on the surface, viscous tangential stresses from the liquid side begin to act on its lower surface, carrying it as a whole along some trajectory lying on the surface and rotating it around its own centre of mass. Within the approximations made, the surface density of this viscous force is determined by

4.2 Analytical Representation of Marker Movement on the Vortex Surface

109

Fig. 4.8 Schematic representation of the marker on the surface of the vortex (a) and the velocity field adjacent to its underside (b)

the value fv = κ(vt − r˙ c − ω × r) where vt is the tangent component of the water velocity field at the point with coordinate rc + r; r is the local radius-vector of the point on the surface of the marker with origin in the centre of mass, as shown in Fig. 4.8; κ is some factor characterising the coupling of the marker surface (depending on its material, roughness, etc.) with liquid; term vt − r˙ c − ω × r is the speed of the point on the marker bottom relative to liquid flow. The total viscous force and the total momentum of the viscous forces acting on the marker about its axis of rotation passing through the centre of mass are defined by the expressions ∫ Fv =

∫ fv d S, Mv =

S

r × fv d S S

where S is the bottom surface of the marker. The momentum of the force F R applied to the centre of mass is zero. Thus, in the adopted approximate model, the equations and initial conditions describing the motion of the marker on the liquid surface are ∫ m r¨ c = Fv +FR = κ

(vw − r˙ c − ω × r) d S + λmget S

110

4 Motion of Solid Markers in a Vortex Flow

∫ r × (vw − r˙ c − ω × r) d S

J ω˙ = κ S

r˙ c |t=0 = 0,

rc |t=0 = rc0 ,

ω|t=0 = 0

(4.2.1)

Here J is the moment of inertia of the marker in relation to the specified axis of rotation. The motion of the marker is investigated in the coordinate system associated with the water surface [8] and shown in Fig. 4.8. The position of the centre of mass of the marker is described by the coordinates s, ϕ and n = 0 (since the marker is on the water surface), and its radius vector is given by the expression rc = (R R , + ζζ, )es + (ζR , − Rζ, )en

(4.2.2)

where es , en and eϕ are unit orthos, obeying the rules of differentiation R ,, en ∂es ζ ,, en =− , = , ∂s ζ R, ∂eϕ ∂eϕ = 0, = −R , es ∂s ∂ϕ ∂en ζ ,, es R ,, es =− , = , ∂s R ζ,

∂es ∂es = R , eϕ , =0 ∂ϕ ∂n ∂eϕ =0 + ζ , en , ∂n ∂en ∂en = −ζ , eϕ , =0 ∂ϕ ∂n

on the basis of which it follows from (4.2.2) r˙ c = s˙ es + R ϕ˙ eϕ , r¨ c = (¨s − R R , ϕ˙ 2 ) es + (Rζ, ϕ˙ 2 + ζ,, s˙ 2 /R , ) en + (R ϕ+2R ¨ s˙ ϕ) eϕ

(4.2.3)

as well as ∂ rc /∂ n = 0—this means that the marker’s centre of mass remains on the surface of the liquid throughout the entire movement. The angular velocity of the marker rotation in the adopted approximations is represented as ω ≈ ω(s, ϕ,t) en

(4.2.4)

where s and ϕ are the coordinates of the centre of mass, so that differentiating in time the expression (4.2.4) leads to the result ω˙ ≈ ω ˙ en

(4.2.5)

∫ ∫ Since for markers with central symmetry the relations S r × r˙ c d S = 0 and S ω × r d S = 0 are valid, the equations describing the motion, taking into account (4.2.3), take the form

4.2 Analytical Representation of Marker Movement on the Vortex Surface

111

     m s¨ − R R , ϕ˙ 2 es + R ϕ¨ + 2R , s˙ ϕ˙ eϕ ∫

  vt d S − S s˙ es + R ϕe ˙ ϕ − λmgζ, es =κ S ∫ 

∫

 . 2 r × vt − r · en d S − ω r d S Jω ˙ =κ c

S

(4.2.6)

S

In [5] it is shown that, neglecting the surface tension, the velocity of liquid particles on the water surface is described by the expression vt |n=0



a2 = (−Ωs R es + Ωϕ eϕ ) Rϑ(a − R) + ϑ(R − a) R ,

 (4.2.7)

where Ωs , Ωϕ are characteristic angular velocities along directions es and eϕ respectively, a is the radius of solid rotation in the vortex; ϑ(x) is the unit Heaviside function. The shape of the surface on which the marker is moving is given by the ratio ζ(s) = ζ0 +

Ω2ϕ  2g

 R 2 (s)θ(a − R(s)) + a 2 (2 − a 2 /R 2 (s))θ(R(s) − a)

(4.2.8)

and describes an emerging vortex. In the experiments [9], the angular velocity of the disk at the bottom of the container was 3.3 c−1 . Since the momentum transfer from the bottom to the surface of the container is due to viscous effects, the upper constraint is valid Ωϕ ≤ 3.3 c−1 . Since R ,2 + ζ, 2 = 1 [8], based on (4.2.8) the equations for the function R: √ R , 1 + k 2 R 2 /a 2 = 1,

R ≤ a;

√ R , 1 + k 2 a 6 /R 6 = 1,

R ≥ a,

where k = Ω2ϕ a/g. The evaluation from above in the limiting case at a = 7.5 cm (container radius) gives the value kmax ≤ 0.085, so with a good approximation we can put R , ≈ 1, R ,, ≈ 1, ζ, ≈ 0. Further investigation of the problem depends substantially on the particular kind of experimental marker and is related to the calculation of the integrals included in (4.2.6). It should be explained that the system (4.2.6) describing the motion of markers in a vortex current includes the equation for the change of kinetic momentum of the marker. In steady-state modes, especially when the marker is in the region of solidstate rotation, its frequency ω of rotation around its own axis is practically constant in time. Then from the equation for the kinetic momentum ⎡ ⎤ ∫ ∫ Jω ˙ = κ⎣ [r × vt ] · en d S − ω r 2 d S ⎦ S

S

112

4 Motion of Solid Markers in a Vortex Flow

Fig. 4.9 Marker impact diagram

at ω ˙ ≈0 ∫

∫ ˙ [r × vt ] · en d S − ω S

r2 dS ≈ 0

(4.2.9)

S

consequently, the result is ∫ ω≈

S

[r × vt ] · en d S ∫ 2 S r dS

(4.2.10)

Consider the situation where a simple centrally symmetric about the centre of mass figure is chosen as the marker—two small flat disks, each with area dΣ at a distance l from each other, connected by an infinitely thin absolutely rigid rod, as shown in Fig. 4.9. In this case ∫ l 2 dΣ l2 (4.2.11) r 2 d S = 2dΣ = 4 2 S

Let the marker at some point in time occupy the position shown in Fig. 4.9, with its centre of gravity at the coordinate s. Then, using an approximate expression for the vortex velocity field at the surface vt ≈ RΩϕ eϕ , an estimate is obtained ∫ [r × vt ] · en d S S

    l l l l − Ωϕ sin α dΣ s − sin α = Ωϕ sin α dΣ s + sin α 2 2 2 2

4.2 Analytical Representation of Marker Movement on the Vortex Surface

= Ωϕ sin2 α

l 2 dΣ 2

113

(4.2.12)

Substituting (4.2.11, 4.2.12) into (4.2.10) gives the result ω ≈ Ωϕ sin2 α

(4.2.13)

Since the angle α varies over time (and, strictly speaking, there is a relationship ˙ the average eigenrotation frequency over a period is estimated by the value ω = α), ∫2π ω ≈ Ωϕ

sin2 α dα =

Ωϕ 2

(4.2.14)

0

Only at certain moments of time when α = π2 + πn, n = 1, 2, . . ., the instantaneous natural rotation frequency is compared with the rotation frequency of the vortex surface in the solid domain. Since any marker having central symmetry with respect to the centre of mass can be broken down into pairs of such small dumbbell-shaped markers, we should expect the result that, whatever the shape, the eigenrotation of the marker in the solid vortex rotation region will obey the relation (4.2.14). Of course, the marker itself must satisfy the condition of smallness of its maximum size compared to the smallest (in absolute value) radius of curvature of the vortex surface in order to satisfy the condition of fairness of the system (4.2.14). Since the angles ψ and γ in the solid rotation domain obey the dependencies ψ = ψ0 + Ωϕ t, γ = γ0 + ωt, then at longer times there is ω dγ 1 = = . dψ Ωϕ 2 Disc-shaped marker with radius ρ In the approximate model used, the radius-vector r, as measured from the surface of the marker, is represented as r ≈ r (es sinβ+eϕ cosβ)

(4.2.15)

Then, where S = πρ 2 is the surface area of the disc, J = m ρ2 /2 is its moment of inertia. Based on (4.2.7), the expressions

114

4 Motion of Solid Markers in a Vortex Flow



∫ vt d S ≈ Rρ

2

S

f ± (x) =

   a2 f + (x) + 2 f − (x) − Ωs es + Ωϕ eϕ R

√ 2ρ sa − s π − (1 − x 2 )3/2 ± ar csin(x) + x 1 − x 2 , x = 2 3R ρ (4.2.16)   ∫ a2 [r × vt ] · en d S ≈ Ωϕ ρ3 R h + (x) + 2 h − (x) R S

 3/2 ρ arcsin(x) − x 1 − x 2 ± h ± (x) = 4R   3/2 sa − s 2 πρ , x= − 1 − x2 (4.2.17) ± 8R 3 ρ

where sa is such a value of the coordinate s, that R(sa ) = a is the radius of the solid vortex rotation. At x = −1 the marker is entirely on the periphery of the vortex, in this case f + = 0, h + = 0 so from (4.2.16, 4.2.17) it follows ∫ vt d S ≈

 a2  S − Ωs es + Ωϕ eϕ R

S

∫ [r × vt ] · en d S ≈ −Ωϕ

a 2 ρ2 S 4R

(4.2.18)

S

At x = 1 the marker is entirely in the solid vortex rotation region, in which case f − = 0, h − = 0 and from (4.2.16, 4.2.17) it follows ∫

  vt d S ≈ RS − Ωs es + Ωϕ eϕ

S

∫ [r × vt ] · en d S ≈ Ωϕ

Rρ2 S 4

(4.2.19)

S

In the transition zone −1 < x < 1, when part of the marker is located in the solid rotation region and part is located on the periphery, the required values are determined by the formulas (4.2.16, 4.2.17). A qualitative representation of the properties of the equations of motion (4.2.6) is given by writing these equations in the form / μΩϕ s¨ − s ϕ˙ 2 + μ˙s ≈ μas , ϕ¨ + 2˙s ϕ˙ s + μϕ˙ ≈ μaϕ , ω ˙ + μω ≈ M (4.2.20) 2

4.2 Analytical Representation of Marker Movement on the Vortex Surface

115

Fig. 4.10 Dependence plots of values aϕ (dashed), as (dots) and M (solid) in the transition region −1 < x < 1 for the disc-shaped marker

and shown in Fig. 4.10, and the dependences of the values as , aϕ and / M on x, / calculated according to the relations (4.2.16, 4.2.17) at ρ a = 0.1,. Ωs Ωϕ = 0.2 In (4.2.20) the notation μ = κS/m is introduced. In case of simultaneous reduction of all linear dimensions of marker μ → ∞, and in case of reduction of marker-water interaction (κ → 0) there is μ → 0. The vertical scale in Fig. 4.10 is not presented, because they show values of different dimensions, but the relations between as , aϕ are in exact accordance with (4.2.20). View of Eq. (4.2.20), limit relations (4.2.18, 4.2.19) and graphs as , aϕ of Fig. 4.10 show that after placing on vortex surface in vortex periphery area (x = −1) marker in form of disk is “caught” by liquid flow and starts moving towards vortex centre (as takes negative values) with simultaneous turning of its mass centre in the same direction as vortex rotates. At the same time, the marker itself starts rotating relative to the axis passing through its centre of mass in the opposite direction (M takes negative values), i.e. ω · Ωϕ < 0. As one moves towards the centre of the vortex, the marker enters the transition region (−1 < x < 1), crosses the boundary between the solid-state and peripheral vortex rotation types and gradually reverses its own rotation relative to the centre of mass, so that the ratio ω · Ωϕ > 0 begins to hold after the marker fully enters the solid-state rotation region (x = 1). When placing the marker in the peripheral region of the vortex, Eq. (4.2.20) when using (4.2.18) take the form a 2 Ωs =0 s / a 2 Ωϕ ϕ¨ + 2˙s ϕ˙ s + μϕ˙ − μ 2 = 0 s a 2 Ωϕ ω ˙ + μω + μ 2 = 0 2s

s¨ − s ϕ˙ 2 + μ˙s + μ

(4.2.21)

116

4 Motion of Solid Markers in a Vortex Flow

From the second Eq. (4.2.17) it follows   1 2  a Ωϕ 1 − e− μt + s02 ϕ˙ 0 e−μt 2 s s0 = s|t=0 , ϕ˙ 0 = ϕ| ˙ t=0 ϕ˙ =

(4.2.22)

Substituting (4.2.22) into the first equation of system (4.2.17) and integrating it by perturbation theory methods [9] leads to the result a2 σ(t) (1 + λ2 )ln + o(ε) σ(t) s0 / / ε = Ωs /μ ≪ 1, λ = Ωϕ Ωs , σ(t) = s02 − 2a 2 Ωs t

s = σ(t) + ε

(4.2.23)

Finally, substituting these results into the third Eq. (4.2.21) determines the frequency of rotation of the marker relative to its centre of mass by the expression    1 βμ  −μt ϕ˙ ϕ˙ 0 / / ϕe ω= − ϕ0 + − 2 2 1 + 4εϕ˙ 0 Ωϕ 1 + 4εϕ˙ Ωϕ β=

s02 ϕ˙ 0 , ϕ˙ 0 = ϕ|t=0 a 2 Ωϕ

(4.2.24)

The angle of rotation of the track marker in relation to its own axis is determined by the expression ∫t γ=

⎤ ⎡ ∫t 1 β⎣ ˙ −μt dt ⎦ − (ϕ − ϕ0 ) ω dt = ϕ0 − ϕe−μt − ϕe 2 2

0

(4.2.25)

0

The value ϕ − ϕ0 in expression (4.2.25) is the angle of rotation of the centre of mass relative to the rotation axis of the vortex. Since, according to (4.2.22, 4.2.23) estimates are valid, β ≃ 1, the main term in expression (4.2.25) looks like γ≈−

 1+β β  (ϕ − ϕ0 ) + ϕ 1 − e−μt 2 2

(4.2.26)

which means that when the marker is in the periphery, it rotates around its own axis in the opposite direction to that of the centre of mass of the marker relative to the vortex axis. At the initial moments of time, the rotation angle γ about its own axis is practically opposite in value to the rotation angle about the funnel axis, and then s 2 −(a+ρ)2 the growth of its absolute value gradually decreases until the time t∗ ≈ 0 2a 2 Ωs when the marker edge begins to enter the transition zone and its motion is no longer described by the system (4.2.21).

4.2 Analytical Representation of Marker Movement on the Vortex Surface

117

The process of marker movement in the transition zone is described by such cumbersome relations that it is not possible to give them an explicit form. In the kernel region, the motion of the marker is described by a system of equations s¨ − s ϕ˙ 2 + μ˙s + μΩs s = 0 / ϕ¨ + 2˙s ϕ˙ s + μϕ˙ − μΩϕ = 0 / ω + μω − μΩϕ 2 = 0

(4.2.27)

Integration of (4.2.27) leads to the result ϕ = ϕ∗ + Ωϕ (t − t ∗ )(1 + 2ε + 2ε2 (3 − λ2 )) s = s ∗ exp( − Ωs (t − t ∗ )(1 + ε(1 − λ2 ))) Ωϕ ∗ ∗ (1 − e− μ(t−t ) ) + ω∗ e− μ(t−t ) ω= 2   Ωϕ 1 Ωϕ ∗ ∗ ∗ ∗ (t − t ) − − ω (1 − e−μ(t−t ) ) γ=γ + 2 μ 2

(4.2.28)

where t ∗ is the time when the marker leaves the transition region and fully enters the solid rotation region; ϕ∗ , s ∗ , ω∗ and γ∗ are the values of the centre of mass coordinates, rotation frequency and angle of rotation around its own axis at the time t ∗ , respectively. As can be seen from (4.2.28) the eigenrotation of the marker coincides in direction with its rotation around the vortex axis, with γ − γ∗ ≈ 21 (ϕ − ϕ∗ ). An important feature of expressions (4.2.22, 4.2.23, 4.2.28) is that, using the principal terms in expressions for the coordinates s and ϕ of the centre of mass of the marker, these coordinates, both in the region of solid rotation and at the periphery of the vortex are related by a relation of the general form   Ωs ˜ s ≈ s˜ exp − (ϕ − ϕ) Ωϕ

(4.2.29)

where s˜ , ϕ˜ are some constants. Equation (4.2.29) describes a logarithmic spiral on the surface of the vortex, along which the centre of mass of the marker moves, and this spiral coincides with the trajectories of liquid particles near the free surface. Thus, the marker is entrained by the near-surface flow and performs an additional rotation due to a nonzero momentum of viscous forces acting on its lower surface from the liquid side. Both in the peripheral region and in the solid rotation region, the angles of rotation of the marker relative to its own axis and the centre of mass of the marker relative to the rotation axis of the funnel are related by the relation of the form γ − γ∗ ≈ c1 (ϕ − ϕ∗ ) + c0

(4.2.30)

118

4 Motion of Solid Markers in a Vortex Flow

Fig. 4.11 Dependence of the rotation angle of the marker around its own axis γ on the rotation angle around the centre ψ (Ω = 3.3 s–1 ): a Rectangular marker 1, 2—0.5 × 1.0 × 0.3 cm, H = 40 and 20 cm; 3, 4—1.0 × 2.0 × 0.3 cm, H = 20 and 40 cm; b cross-shaped marker 1.5 × 1.5 × 0.3 cm: 1, 2—H = 20 and 40 cm

where c0 , c1 are some constants, generally different for peripheral and solid rotation regions. The experiments showed that besides rotation of the marker around the centre of the free surface of the vortex flow, it also rotates around its own axis. The obtained data allow plotting the dependence of the marker rotation angles relative to its own centre (rotation angle γ) and the centre of the flow ψ (Fig. 4.11, Table 4.2.1) [10]. The following interesting feature should be noted: after being placed on the surface of the vortex at its periphery, the disc-shaped marker is “picked up” by the liquid flow and begins to move towards the centre of the vortex with simultaneous rotation of its centre of mass in the same direction as the vortex rotates. At the same time, the marker itself begins to rotate about the axis passing through its centre of mass in the opposite direction. As it moves towards the centre of the vortex, the marker enters the transition region, crosses the boundary between the solid-state and peripheral types of vortex rotation and gradually changes its own rotation relative to the centre of mass to the opposite one that remains after the marker has fully entered the region of solid-state rotation, which corresponds to the theoretical model. Thus, experiments have shown that the motion of the marker placed on the surface of the vortex cavity is complex and includes tangential displacement (rotation relative to the centre of the vortex cavity flow), radial displacement and rotation relative to its own axis. The nature of the movement depends on the experimental conditions (liquid depth and activator disc speed), type of liquid (homogeneous or bilayer with an oil film) and the shape of the marker. The experimental dependence of the rotation angle on the rotation angle in the solid-state rotation domain agrees well with the theoretical dependence (4.2.28) derived from the proposed mathematical model.

4.3 Experimental Study of Miscible Admixture Transfer from a Solid …

119

4.3 Experimental Study of Miscible Admixture Transfer from a Solid Marker on a Vortex Flow Surface In order to clarify the flow pattern and to measure the rotation velocity of the free surface of the liquid in a compound vortex, a series of experiments were conducted, where solid and soluble markers were used simultaneously as indicators. A solid marker was placed on the free surface of the steady-state vortex with the selected parameters, on which a layer of soluble dye of contrasting colour was additionally applied. Figure 4.12, a shows the photo from the experiment with the dyeing admixture (uranyl). Parameters of this experiment: depth H = 40 cm, disk radius R = 7.5 cm, disk rotation frequency Ω = 200 rpm. Figure 4.12b shows schematically the propagation of the leading and lagging arms from the marker, where Rm —radial coordinate of the marker relative to the centre of the free surface, radial positions R f i —end of the leading and Rbi —end of the lagging arm relative to the centre of the rotating free surface, ϕm —angle of the marker relative to the centre of the rotating free surface, ϕ f i and ϕbi —angular positions of the ends of the leading and lagging arms. After placing a solid floating marker with applied dye on the free surface of the composite vortex, the dye comes off (washes off) it in portions, each portion of the dye forming a coloured area—the arm. The data obtained from the movement of the solid-state marker over the free surface (not including the painted spiral structures) were processed automatically. The obtained data made it possible to find the dependences of changes in the angular positions of the leading and lagging paint arms on time and to approximate them by functions of the form ϕ f i = at + b and ϕbi = at + b where t is time elapsed since marker placement on the free surface of the current, ϕ f i is angular position of the leading and ϕbi is angular position of the lagging paint arms. The uranium

Fig. 4.12 Spreading of the tinting agent relative to the solid marker: a Photo, b schematic

120

4 Motion of Solid Markers in a Vortex Flow

Fig. 4.13 Dependence of angular positions of the leading painted arms on time, the arm moves away from the marker at time: 1—t = 0.12 s, 2—t = 0.84 s, 3—t = 1.4 s, 4—t = 2 s, 5—t = 2.6 s, 6—t = 3.16 s

Fig. 4.14 Dependence of angular positions of lagging paint hoses on time, the hose moves away from the marker at a point in time: 1—t = 1.56 s, 2—t = 2 s, 3—t = 2.6 s, 4—t = 3.16 s

arms come off the solid marker in portions. Figure 4.13 shows the dependency of the angular positions of the leading arms and Fig. 4.14 shows the lagging arms. The behaviour of the first dye portions on the free surface differs from the subsequent portions. The interval between sequences of dye portions is 0.6 s for leading arms and 0.5 s for lagging arms. Experiments performed on the free surface of a composite vortex of a solidstate marker with soluble dye applied clearly demonstrate the presence of dye arms leading and lagging relative to the solid-state marker. The dye admixture spreads across the free surface of the composite vortex in portions, departing from the marker both forward and backward at approximately the same time, and the time intervals between the formation of new portions are approximately equal. Conclusions to this chapter Experiments have shown that the motion of the marker placed on the surface of the vortex cavity is complex and includes tangential displacement (rotation relative to the centre of the vortex cavity flow), radial displacement and rotation about its own axis. The nature of the movement depends on the experimental conditions (liquid

References

121

depth and activator disc speed), the type of liquid (homogeneous or bilayer with an oil film) and the shape of the marker. A theoretical model has been proposed to explain the motion of markers placed on the surface of a vortex flow in a one-component fluid. It is shown experimentally that when a marker is placed on the surface of the vortex at its periphery, it starts moving towards the centre of the vortex with simultaneous reversal of its centre of mass in the same direction as the vortex rotates, while the marker itself starts rotating in the opposite direction about the axis passing through its centre of mass. As one moves towards the centre of the vortex, the marker enters the transition region, crosses the boundary between the solid-state and peripheral types of vortex rotation and gradually changes its own rotation relative to the centre of mass to the opposite, which is maintained after the complete transition of the marker into the solid-state rotation region, which corresponds to the theoretical model. An equation describing the motion of the centre of mass of the marker and representing a logarithmic spiral on the surface of the vortex, which coincides with the trajectories of liquid particles near the free surface, is obtained. Thus, the marker is entrained by the near-surface flow and performs an additional rotation due to the non-zero momentum of the viscous forces acting on its lower surface from the liquid side. Both in the peripheral region and in the solid-state rotation region, the angles of rotation of the marker relative to its own axis and the centre of mass of the marker relative to the rotation axis of the funnel are related by the relation of the form γ = ψ/2, which is valid for all marker forms having central symmetry. The experimental dependence of the rotation angle on the rotation angle in the solid rotation domain agrees well with the theoretical dependence derived from the proposed mathematical model. Experiments performed on the free surface of a composite vortex of a solidstate marker with soluble dye applied clearly demonstrate the presence of dye arms leading and lagging relative to the solid-state marker. The dye admixture spreads across the free surface of the composite vortex in portions, departing from the marker both forward and backward at approximately the same time, and the time intervals between the formation of new portions are approximately equal.

References 1. Budnikov AA, Chaplina TO, Pokazeev KV (2016) The movement of bodies of various sizes and shapes on the surface of the vortex. Int J Fluid Mech Res 43(4):368–374 2. https://dic.academic.ru/dic.nsf/ruwiki/99716 3. Eisenberg DJ (2014) SVG Essentials, 2nd edn. O’Relly Media, 366p 4. Tu J, Gonzalez R (1978) Principles of pattern recognition. Mir, Moscow, 411p 5. Kistovich AV, Chaplina TO, Stepanova EV (2019) Spiral structure of liquid particle trajectories near vortex surface. Comput Technol 24(2):67–77 6. Chaplina TO (2019) Transfer of substance in vortex and wave flow in single- and multicomponent media. Kim L.A. Publishing House, Moscow, 201p

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7. Chaplina TO (2019) Experimental study of substance transfer in vortex and wave flows in multicomponent media. In: Physical and mathematical modeling of Earth and environment processes, pp 159–174 8. Chaplina TO, Stepanova EV, Pakhnenko VP (2019) Investigation of features of tagging movement on the surface and in the vortex flow column. Process Geospheres 2:282–291 9. Naifeh A (1984) Introduction to perturbation methods. Mir, Moscow, 536p 10. Chaplina TO (2019) Transfer of matter in vortex and wave flow in single- and multicomponent media. Kim L.A. Publishing House, Moscow, 201p

Chapter 5

Modelling Hydrocarbon Spillage on the Surface of Water

Oil, gas and coal remain the main natural sources that satisfy humanity’s energy needs. Oil accounts for 10% of the world’s reserves of fossil fuels, and coal for 70%. Currently, about 10–15% of the proven reserves of coal deposits are being exploited, and about 65–70% of the reserves of oil deposits. Accidental oil releases are possible at all stages of oil transportation from wellheads to refineries. Accidents with oil leaks occur during oil extraction, collection and storage, from reservoirs, during oil loading operations, release of oil products to consumers, transportation by pipelines, etc. The magnitude of leakage reaches large values and, according to various sources, ranges from 5 to 17% of production [1]. At that not only valuable raw materials are lost but also considerable harm is caused to the environment. The reasons of ingress of hydrocarbons into the environment, characterization of sources and estimation of the value of pollution at oil and gas production facilities are covered in sufficient detail in the literature [2–11]. Getting into natural ecosystems, oil hydrocarbons cause disturbance of biological equilibrium for a long time. A list of major disasters related to the oil industry is given in Table 5.1. Ocean pollution by oil products and, in particular, the dynamics of oil slicks on the surface of water areas became actively studied in the sixties of the twentieth century. According to many researchers, the oil slick is one of the most common forms of oil as an ocean pollutant. One of the most important issues in assessing the environmental consequences of oil spills at sea is determining the duration of the oil slick on its surface. The criterion for identifying an oil slick and, therefore, estimating the time of its existence is the minimum slick thickness, upon reaching which the slick ceases to exist as a single whole, and further increase of the pollution zone occurs due to diffusion of individual slicks. For different grades of oil the estimation of this thickness is in the range from 4 to 100 μm [12]. By this time the slick may contain from 15 to 35% of the original volume of oil in the form of conglomerates of heavy tarry fractions. When planning and implementing hydrocarbon spill response operations, there is a need to predict the spread of oil at sea. Such forecasts allow, in particular, to warn © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 T. Chaplina, Transfer of Substance in Vortex and Wave Flows in One-Component and Multi-component Environment, Earth and Environmental Sciences Library, https://doi.org/10.1007/978-3-031-31856-6_5

123

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5 Modelling Hydrocarbon Spillage on the Surface of Water

Table 5.1 Major hydrocarbon spills in recent history The scene of the disaster

The year

Name of ship/spill source

Quantity of oil, thousand tonnes

Gulf of Mexico

2010

Deepwater Horizon, Platform

460

Arabian Sea

2003

Tasman Spirit

Bay of Biscay

2002

Prestige

90

Gulf of Aden

2002

Limburg

300

Mediterranean Sea, Italy

1991

Amoko Haven

144

Atlantic Ocean, shores of Angola

1991

ABT Summer

260

Indian Ocean, South Africa

1983

Castillo de Belver

252

Persian Gulf

1983

Nowruz

250

Gulf of Mexico

1980

Ixtoc I, platform

467

Navarino Bay, Greece

1980

Irenez Serenade

100

Caribbean Sea

1979

Atlantic Empress

290

The Atlantic coast of France 1978

Amoco Cadiz

223

Gulf of Oman

1972

C Star

115

Sealy Islands, UK

1967

Torey Canyon

119

60

about the possibility of oil pollution of the coastal area, about the oil slick crossing areas of intensive economic activity, ships’ courses, etc. The spreading of oil at sea during oil spills is a complex process that requires consideration of a great variety of factors. The lifetime of an oil slick is non-linearly dependent on wind speed and the volume of the initial spill. Results of numerical simulation showed that with wind velocity varying from 2.5 to 25 m/s slick existence time decreases 16 times and with 10 to 25 m/s only 3 times. In general, slick residence time is approximately proportional to the square root of the oil spill volume [13, 14]. In recent years, various states have made great efforts to improve oil and petroleum product spill prevention and response systems, but the problem still remains acute. In order to reduce the possible negative consequences, special attention should be paid to studying the methods of oil product spill containment and elimination and developing an additional set of measures to collect and dispose of hydrocarbons that have escaped into the external environment. This chapter classifies possible sources of oil and petroleum product spills, probable risks of oil spills during onshore and offshore production, as well as during storage and transportation of oil and petroleum products due to accidents. The results of studies of the spreading process of hydrocarbons with different physical and chemical properties on the water surface are given as well as comparison of expressions obtained analytically describing the established shape of the hydrocarbon slick surface on the water surface with the shapes observed in the experiments for different values of the experimental parameters. A review of existing techniques and sorbents

5.1 Evolution of Oil on the Water Surface

125

for the elimination of hydrocarbons from the water surface is given, their characteristics and the principle of operation of devices which are currently used for oil spill elimination are studied. An original method of elimination of hydrocarbon spills with the help of a natural sorbent—natural sheep’s wool is proposed. The fluorescent diagnostics of water purified from oil pollution by a sorbent based on sheep’s wool showed that the wool effectively sorbs pollution and can be successfully used for elimination of surface spills of various hydrocarbons in water bodies in two modes: from a surface vessel and from the shore of the reservoir.

5.1 Evolution of Oil on the Water Surface Oil is a natural oily combustible liquid, a mixture of hydrocarbons. The physical and chemical properties of oil can vary widely depending on its composition [1, 15, 16]: • • • •

Density 0.65–1.05 g/cm3 ; Viscosity 2–300 mm2 /s; Molecular weight 220–400 g/mol; It contains 82–87% carbon, 11–14% hydrogen and 0.4–1.0% various impurities such as oxygen, nitrogen, sulphur etc.; • The flash point—the minimum temperature at which an air mixture with petroleum product vapours forms and is capable of igniting briefly from an external source— ranges from 35 to 120 °C; • The crystallisation temperature ranges from − 60 to + 30 °C, and depends in particular on the paraffin content of the oil. The colour of the oil can vary from light yellow to almost black. On reaching the water surface, oil immediately begins to undergo changes due to the effects of various physical and chemical factors. This interaction of oil with the environment can be divided into several stages: spreading, evaporation, dissolution, emulsification, dispersion, oxidation and dispersion. All of these processes take different amounts of time and involve different fractions of oil—some occur with lighter components and others with heavier components, and the composition of the oil product changes as a result. Thus, on the first day after a spill the main processes are evaporation, dispersion, emulsification and spreading. In a week light fractions volatilize and the remaining hydrocarbons undergo biodegradation. The oil also tends to sorb the solid particles in the water, forming aggregates and precipitating to the bottom as a solid sediment. These processes can last from several weeks to several years. Let us consider the mechanisms of these processes in more detail.

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5 Modelling Hydrocarbon Spillage on the Surface of Water

5.1.1 Spreading In the case of oil on open water, spreading occurs under the action of wind, surface flow, surface tension forces and gravity, forming areas covered with thin oil films thousands of molecular layers thick [17]. In the absence of wind and flow, 1 m of crude oil3 can spread over an area with a radius of 50 m in 1.5 h [18]. Wind and surface flow are factors influencing the dynamics of oil spreading. For example, the higher the wind speed, the faster the slick spreads. It extends along the wind direction and thinning rapidly, after which it may break up into smaller slicks. The speed of this process depends on the relationship between the surface tension coefficients: σ = σwa − σwo − σoa where σwa is the surface tension at the water–air interface, σwo is the surface tension at the water–oil interface, σoa is the surface tension at the oil-air interface. The very notion of the surface tension coefficient is directly related to the isobaric potential, or Gibbs energy (G), by the formula [19]:  σ =

∂G ∂S

 p,T,n i

where S —contact area of the phases, P—pressure, T —temperature, n i —concentrations of substances in the phases. Consider an oil-on-water system at constant temperature and pressure. If the oil droplet area increases by d S, the water–oil and oil-air interface areas will increase by the same amount, and the water–air area will decrease. It follows that: Σ dG = σi d S dG = σwo d S + σoa d S − σwa d S Then for a spontaneous process of increasing the water–oil contact area (spreading) such that d S > 0, dG < 0, the Garkins spreading criterion can be introduced: K = σwo + σoa − σwa Spreading is possible at K > 0. There are several models of oil spreading on the water surface, based on hydrodynamic equations. In the model proposed by Fay in 1969, the hydrocarbon spreading process can be divided into the following phases [20]:

5.1 Evolution of Oil on the Water Surface

127

Table 5.2 Characteristics of hydrocarbon spill phases across the water Name of phase

Duration, s

Inertia

103 –104 (hours)

Spot length  1/3 gV Δ 1 k g−i t 2/3 d

Gravity-bound

104 –106 (days)

1 k g−v

Surface tension

106 (months)

1 kv−r

 

gV Δ 1/2 d 2 ρw vw σ 1/2 ρw vw

Spot radius 2 (gV Δ)1/4 t 1/2 k g−i

1/4

 t 3/8

2 k g−v

1/2

 t 3/4

1 kv−r

gρo ΔV 2 1/2 ρw vw σ 1/2 ρw vw

1/6 t 1/4

1/2 t 3/4

• Inertial—occurs due to the balance of the horizontal pressure gradient and inertial forces, the growth rate of the spot radius is determined by the relation: R = (Δgvt 2 )1/4 • gravity-viscous occurs due to the balance of horizontal pressure and viscous forces, R = (ΔgV 2 t 3/2 ν −1/2 )1/6 • surface tension phase—the main acting forces are viscosity and surface tension, R = (σ 2 t 3 ρ −2 ν)1/4 • the phase where the spot stops growing, where R is the radius of the slick, V is the volume of oil spilled, t is time, Δ is the relative density of water, defined by the formula Δ = ρωρ−ω ρo , ν—kinematic viscosity of water, the expression for σ has been defined above, g—acceleration of gravity, ρω and ρo —densities of water and oil respectively. Phase action times are calculated from Hoult diagrams depending on spill volume [21]. However, the slick radius calculated by Fay’s formulas turns out to be significantly smaller than that obtained experimentally. Table 5.2 shows the Fay characteristics of the phases. Here ki j is hydrocarbon parameter determined experimentally. Phase durations are given for 10 L of hydrocarbons. Fay’s work [20] also shows a graph of the dependence of the radius on time (Fig. 5.1).

5.1.2 Displacement (Advection) The drift of a slick is determined by external factors such as current, wind speed and direction. According to [1], the slick drift velocity is determined by the formula U = u w + 0.56 u c where u w is the wind drift velocity, u c is the drift velocity caused by flow. The wind drift velocity is about 3% of the wind speed. Due to the Coriolis force, the drift direction of the slick can deviate from the wind speed direction.

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5 Modelling Hydrocarbon Spillage on the Surface of Water

Fig. 5.1 Dependence of oil slick radius on water surface

5.1.3 Evaporation Starts almost immediately after the oil reaches the water surface. During evaporation, some of the spilled oil passes into the atmosphere. During this process, the oil slick loses up to 40% of its original volume [18]. During the first day after a spill, 50 percent of the compounds containing 13–14 carbon atoms evaporate, and by the end of the 3rd week, up to 50 percent of the compounds with 17 carbon atoms evaporate. This process can continue up to several years, but the lightest fractions (propane, butane) evaporate during the first hours, as a result of which viscosity of oil products remaining on water surface increases. The rate of evaporation depends on factors such as air temperature, wind speed, concentration and saturated vapour pressure of each component.

5.1.4 Dissolution The light oil fractions dissolve most readily in water. They are also actively involved in evaporation, which is much more active than dissolution. Therefore, it is considered that at normal temperatures oil is almost insoluble in water (a slick may lose only about 1% of its original volume). However, at temperatures above 200 °C the solubility increases dramatically [15].

5.1.5 Emulsification An emulsion is a mixture of liquids that are insoluble in each other and in a finely dispersed state. By emulsifying, a slick can increase in volume by up to 5 times. In this

5.1 Evolution of Oil on the Water Surface

129

case water-in-oil and oil-in-water emulsions can form. The predominant emulsion is “oil in water”. The formation of such an emulsion can lead to oil settling in the water column. However, when external influences (waves, wind, etc.) cease, the slick can recover [21]. The water-in-oil type, has a brownish tint, due to which it is called ‘chocolate mousse’. Emulsions can be divided into the following categories according to their persistence: persistent (persistence over a week), medium persistent (persistence a few days), and unstable (persistence a few hours) [3]. The stability of oil emulsions depends on the presence of emulsifiers such as asphaltenes and resins in the oil.

5.1.6 Oxidation The chemical alteration of oil and other substances by interaction with oxygen is called oxidation. Oxidation of hydrocarbons occurs not only through contact of the oil slick with the atmosphere, but also through exposure to microorganisms in the water [22]. For example, such microorganisms are Acinetobacter and Oceanospirillales. The biodegradation rate depends on the levels of dissolved nitrogen, phosphorus and oxygen in the water as well as on the ambient temperature and can last for years. However, different micro-organisms respond to different hydrocarbons, so full biodegradation requires the presence of many species of contaminant-degrading bacteria and fungi in one location, which is rarely performed in practice. The efficiency of biodegradation in a spill response can be improved by using dispersants, thereby increasing the contact area between the microorganisms and the oil. The consequence of oxidation is resinification: oil density and viscosity increase, and the amount of acid and asphalt components increases. Oxidation can also occur under the influence of sunlight (photo-oxidation) [23]. As a consequence, the proportion of water-soluble components increases.

5.1.7 Dispersing In windy conditions, an oil slick can fragment and become mixed with the water, causing small oil droplets to sink into the water column and spread out in the water column. The threshold droplet size at which they do not float to the surface but remain in the water column is ~ 70–150 μm [3]. Droplets of a larger size will rise back to the surface of the water body and may reestablish the film. Dispersed oil has a large surface area, so that such processes as oxidation, dissolution and absorption by microorganisms are accelerated. For this reason, dispersants—substances that accelerate the dispersal of a slick—are often used in hydrocarbon spill response.

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5 Modelling Hydrocarbon Spillage on the Surface of Water

5.1.8 Settling The final process after a spill is settling (or sedimentation). As mentioned above, the lightest oil fractions evaporate, dissolve and biodegrade. The heaviest fractions remain—resins and asphaltenes. They sorb the suspended solids in water and sink to the bottom of the reservoir as sediment.

5.2 Analytical and Numerical Modelling of the Hydrocarbon Slick Shape on the Water Surface As discussed above, surface tension forces act on a slick of oil in water. Figure 5.2 shows the interface of three surfaces: σwa , σwo , σoa are the surface tension forces acting normal to the contact line along the water–air, water–oil and oil–air boundary surfaces respectively. σoa + σwa + σwo = 0

(5.2.1)

The equilibrium condition (stationary position along the radial coordinate) of the contact line of the three substances is as follows: σwa = σoa cos αwa + σwo cos αwo σoa sin αwa − σwo sin αwo = 0 The shape of the oil slick on the water surface is of academic interest to us. We expect to see roughly the following picture (Fig. 5.3). Here ζ is the water–air boundary, η is the oil-air boundary, θ is the water–oil boundary, Ro is the slick radius. Here is the derivation of a system of equations describing the shape of the resting volume of hydrocarbon on the water surface. Expressed through the angles shown in Fig. 5.2, relation (5.2.1) breaks down into two equations: Fig. 5.2 Surface forces at the interface between the three media

5.2 Analytical and Numerical Modelling of the Hydrocarbon Slick Shape …

131

Fig. 5.3 Estimated surface shape of the hydrocarbon slick on the water surface

pw = pa + ρw g(ζ − z) + q1 npu R0 ≤ r ≤ R pw = pa + ρo g(η − θ ) + ρw g(θ − z) + q2 npu0 ≤ r ≤ Ro po = pa + ρo g(η − z) + qo npu0 ≤ r ≤ Ro It is required to set the boundary conditions. To do this, we define the interface between the substances as follows (hereinafter, the index oa means the oil-air interface, wa is the oil–water interface, and wo is the water–air interface): ⎧ ⎨ Swa : z = ζ (r ) S : z = η(r ) ⎩ oa Swo : z = θ (r )

(5.2.2)

In this model, water and oil are assumed to be incompressible and atmospheric pressure is assumed to be constant. Let’s write down the equations for the pressures of water ( pw ) and oil ( po ) in different regions of the system, where R is the radius of the cylindrical vessel surrounding the system. pw = pa + ρw g(ζ − z), at R0 ≤ r ≤ R

(5.2.3)

pw = pa + ρo g(η − θ) + ρw g(θ − z), at 0 ≤ r ≤ R0

(5.2.4)

po = pa + ρo g(η − z), at 0 ≤ r ≤ R0

(5.2.5)

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5 Modelling Hydrocarbon Spillage on the Surface of Water

To describe the boundary conditions, the interfaces between substances are given by relations of the form: Swa : z = ζ(r ); Soa : z = η(r ); Swo : z = θ(r )

(5.2.6)

Then the dynamic equations will take the following form: pw − σwa Kwa |z=ζ(r ) = pa , po − σoa Koa |z=η(r ) = pa , pw − po − σwo Kwo |z=θ(r ) = 0

(5.2.7)

Here K i are the curvatures of the interfaces. The curvatures of the interfaces are calculated using the formula 

frr,, + 1 + fr,2 fr, /r K( f ) = −

3/2 1 + fr,2

(5.2.8)

here the shape of the relevant surface is substituted as a function f : ζ(r ), η(r ) or θ(r ). At the boundary of the three media, i.e. at r = Ro , the condition of continuity of the physical fields is fulfilled. The model does not take evaporation into account, so we write down the volume conservation conditions for water and oil respectively: R

R0 ζ(r )r dr +

θ(r )r dr =

H R2 2

(5.2.9)

0

R0

R0 (η(r ) − θ(r ))r dr = Vo

2π

(5.2.10)

0

where Vo is a given volume of oil. At the interface with the shell, the water surface must form a fixed wetting angle αw , the value of which depends on the characteristics of the shell material, i.e. the boundary condition applies | ζr, |r =R = ctgαw

(5.2.11)

The relationship follows from the boundary conditions: σwa Kwa = σoa Koa + σwo Kwo

(5.2.12)

5.2 Analytical and Numerical Modelling of the Hydrocarbon Slick Shape …

133

The following can be written down: ρw gζ + σwa Kwa |z=ζ(r ) = a,

ρo gη + σoa Koa |z=η(r ) = b,

(ρw − ρo )gθ + σwo Kwo |z=θ(r ) = a − b

(5.2.13)

The relations (5.2.13) describe a lenticule of oil lying on the surface of the water, as shown in Fig. 5.3. Explicitly, the first boundary condition (5.2.13) is given by the expression 

σwa ζrr,, + 1 + ζr,2 ζr, /r a ζ− =

3/2 ρw g ρw g 1 + ζr,2

(5.2.14)

/ The introduction of a dimensionless radial coordinate r = ρσwwag r and a dimen/ sionless surface deflection ζ = ρσwwag ζ − √ρwagσwa transforms Eq. (5.2.14) to the form   ζτ ,, + 1 + ζr, 2 ζr, /r =0 (5.2.15) ζ−   , 3/2 1 + ζr From the second boundary condition / (5.2.14) follows exactly the same equation for a dimensionless quantity η = ρσooag η − √ρobgσoa when a dimensionless radial / coordinate r = ρσooag r is introduced. Similarly, from the third / boundary condition (5.2.15), the equation for a dimen-

(ρw −ρo )g sionless quantity θ = when a dimensionless radial θ − √(ρ a−b σwo w −ρo )gσwo / o )g coordinate r = (ρwσ−ρ r is introduced follows. wo Thus, all the introduced dimensionless surfaces satisfy a single equation of the form 

Fξ,,ξ + 1 + Fξ,2 Fξ, /ξ F− =0 (5.2.16)

3/2 1 + Fξ,2

where F refers to the corresponding dimensionless value and ξ to the corresponding dimensionless radial coordinate. Despite the fact that different regions of space are scaled differently, one important property is fulfilled as a result of these transformations: for the corresponding surfaces of the problem the relations are valid

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5 Modelling Hydrocarbon Spillage on the Surface of Water

⎧ ,⎫ ⎨ ζr ⎬ Fξ, = ηr, ⎩ ,⎭ θr

(5.2.17)

This property is necessary when obtaining a solution to Eq. (5.2.15). In the following, the symbols Fζ , Fη and Fθ refer to dimensionless surfaces corresponding to dimensional surfaces ζ, η and θ respectively, and the symbols Fζ, , Fη, and Fθ, refer to the derivatives of these dimensionless surfaces on the corresponding dimensionless radial coordinate ξ. From relations (5.2.1) at the point of contact of the three substances, it follows that / | 

2 σ2 − σ2 + σ2 − σ2 2 4σwa Fθ, − Fζ, || wo wa wo oa = tg αwo = | 2 + σ2 − σ2 σwa 1 + Fθ, Fζ, | wo oa r =R0 / | 2

2 σ2 − σ2 + σ2 − σ2 4σwa Fζ, − Fη, || wo wa wo oa tg αwa = = (5.2.18) | 2 + σ2 − σ2 σwa 1 − Fζ, Fη, | wo oa r =R0

Equalities ζ(R0 ) = η(R0 ) = θ(R0 ) at the point of contact are written in the form of the ratios /

a ρw g

+

| | / σwa || σoa || R0 b Fξ + Fη = ρw g |ξ =√ ρw g R0 ρo g ρo g |ξ =√ ρo g σwa

a−b + = (ρw − ρo )g

/

σoa

| / | σwo (ρw − ρo )g Fθ || R0 σwo (ρw − ρo )g ξ= (5.2.19)

From the laws of conservation follow the equations 

σwa ρw g

3/2 R

/

ρw g σwa



σwo (ρw − ρo )g

3/2 R0

√ (ρw −ρo )g

√ ρw g (ξ )ξ dξ + ∫ R0 σwa 

a R 2 − R02 (a − b)R02 H R2 − − = 2 2ρw g 2(ρw − ρo )g 

σoa ρo g

3/2 R0 / 0

ρo g Fη (ξ )ξ dξ − σoa

Vo b R02 (a − b)R02 = − + 2π 2ρo g 2(ρw − ρo )g



σwo (ρw − ρo )g

σwo

Fθ (ξ )ξ dξ 0

3/2 R0

√ (ρw −ρo )g σwo

Fθ (ξ )ξ dξ 0

(5.2.20)

5.2 Analytical and Numerical Modelling of the Hydrocarbon Slick Shape …

135

Now, in order to obtain the final expressions, we have to solve Eq. (5.2.16). This equation is non-linear, so its solutions are generally difficult to obtain, except for the trivial solution F = 0, which specifies a flat surface. Using a priori information about the flat surface of water at the interface | nearly | with air allows us to use the estimate | Fζ, |