Open Channel Design: Fundamentals and Applications [1 ed.] 1119664241, 9781119664246

Table of contents :
Cover
Title Page
Copyright Page
Contents
Preface
Acknowledgments
About the Companion Website
Chapter 1 Basic Principles and Flow Classifications
Fluid Mechanics Foundations
Hydrologic Foundations
Presentation Organization
Problems and Questions
References
Chapter 2 Channel Fundamentals*
Goals
Channel Elements and Nomenclature
General Flow Relationships
Uniform Flow Relationships
Theoretical Considerations
Natural, Compound, or Sustainable Channels
Lined Channels, Optimum Channels, and Velocity Constraints
Channel Installation
Summary
Problems and questions
References
Chapter 3 Vegetated Waterways and Bioswales*
Goals
Background
Channel Planning
Basic Design Procedures
Bioswales
Vegetated Filter Strips
Temporary Linings
Summary
Problems and Questions
References
Chapter 4 Tractive Force Methods for Earthen Channels
Goals
Riprap-lined or earthen waterways (Earthen II)
Tractive force for vegetated waterways
Details and Origins of the Parabolic Cross-section
Costing channel designs
Steady uniform flow conclusion
Problems and questions
References
Chapter 5 The Energy Equation and Gradually Varied Flows
Goals
Energy Preliminaries – Velocity Profiles and Boundary Effects
Longer Transitions – Gradually Varied Flow Analyses
Conclusions
Problems and Questions
References
Chapter 6 Momentum Equation for Analyzing Varied Steady Flows and Spatially Varied Increasing Flows
Goals
Rapidly Varying Steady Flows (dQ/dt = 0, dQ/dx = 0, dy/dx varies)
Spatially Varying Steady Flow (dQ/dt = 0, dQ/dx varies, dy/dx varies)
Conclusions
Problems and Questions
References
Chapter 7 Hydraulics of Water Management Structures*
Goals
Structure Types
Hydraulic Concepts
Stage–Discharge Relationships of Weir Inlets and Flumes
Discharge Relations of Orifices and Sluice Gates Inlet Devices
Flow Hydraulics of Closed Conduits
Stage–Discharge Curves for Culverts and Spillways
Closed Conduit Systems for Urban Stormwater Collection
Ecologic Suitability
Summary and Conclusions
Problems and Questions
References
Chapter 8 Gradually Varied Unsteady Flow
Goals
Hydrologic Routing Approaches
Kinematic Wave Method
Diffusion Wave Method
Dynamic Routing
Summary and Conclusions
Problems and Questions
References
Chapter 9 Rapidly Varying Unsteady Flow Applications – Waves
Goals
Surface Irrigation
Sluice Gate and Related Operations
The Dam-Break Problem2
Oscillatory Waves
Summary and Conclusions
Problems and Questions
References
Chapter 10 Channel Design Emphasizing Fine Sediments and Survey of Alluvial Channel Sediment Transport
Goals
Alluvial Channel vs. Earthen Channel and Other Preliminaries
Early Approaches to Sediment Transport
Incipient Motion
Riprap or Revetment Specification
Bedform Descriptions and Analysis
Sediment Fall Velocity
A Probabilistic Approach to Sediment Transport
Einstein (1950)–Laursen (1958)–Graf (1971) Stage–Discharge and Other Hydraulic Calculations
Van Rijn (1984) Stage–Discharge and Total Load
Total Load by Regression Approaches
Sediment Measurement
Sediment Routing Through Detention Ponds and Streams
Software Support for Estimating Sediment Transport
Empirical Channel Design Approaches Leading to Sustainable Channels
Forces Impacting Channel Cross Sections – Stream Restoration
Summary and Future Directions
Problems and Questions
References
Appendix A Software and Selected Solutions
Excel®
Mathematica®
HydroCAD
HY-8 culverts
HEC-RAS
Software Summary Tables
Selected Symbolic Solutions
References
Appendix B Solution Charts for Vegetated Waterways Using the Permissible Velocity Method
Reference
Appendix C Selected Cost Data for Channel Excavation and Lining Materials
Appendix D Design Strategy Summary for Uniform Flow Channels
Index
EULA

Citation preview

Open Channel Design

Open Channel Design Fundamentals and Applications

Ernest W. Tollner University of Georgia Athens GA, USA

This edition first published 2022 © 2022 John Wiley & Sons Ltd All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of Ernest W. Tollner to be identified as the author of this work has been asserted in accordance with law. Registered Offices John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, UK Editorial Office 9600 Garsington Road, Oxford, OX4 2DQ, UK For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print‐on‐demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty The contents of this work are intended to further general scientific research, understanding, and discussion only and are not intended and should not be relied upon as recommending or promoting scientific method, diagnosis, or treatment by physicians for any particular patient. In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of medicines, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each medicine, equipment, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging‐in‐Publication Data applied for Hardback ISBN: 9781119664246 Cover Design: Wiley Cover Image: © E W Tollner Set in 9.5/12.5pt STIXTwoText by Straive, Pondicherry, India 10  9  8  7  6  5  4  3  2  1

v

Contents Preface   ix Acknowledgments  xi About the Companion Website  xii 1 Basic Principles and Flow Classifications  1 Fluid Mechanics Foundations  2 Hydrologic Foundations  7 Presentation Organization  8 Problems and Questions  10 References  11 2 Channel Fundamentals  12 Goals  12 Channel Elements and Nomenclature  12 General Flow Relationships  17 Uniform Flow Relationships  17 Theoretical Considerations  23 Natural, Compound, or Sustainable Channels  25 Lined Channels, Optimum Channels, and Velocity Constraints  28 Channel Installation  43 Summary  43 Problems and Questions  47 References  51 3 Vegetated Waterways and Bioswales  53 Goals  53 Background  53 Channel Planning  54 Basic Design Procedures  56 Bioswales  60 Vegetated Filter Strips  62 Temporary Linings  62 Summary  66

vi

Contents

Problems and Questions  68 References  69 4 Tractive Force Methods for Earthen Channels  71 Goals  71 Riprap-Lined or Earthen Waterways (Earthen II)  71 Tractive Force for Vegetated Waterways  77 Details and Origins of The Parabolic Cross-section  82 Costing Channel Designs  92 Steady Uniform Flow Conclusion  94 Problems and Questions  95 References  97 5 The Energy Equation and Gradually Varied Flows  98 Goals  98 Energy Preliminaries – Velocity Profiles and Boundary Effects  98 Longer Transitions – Gradually Varied Flow Analyses  115 Conclusions  126 Problems and Questions  126 References  127 6 Momentum Equation for Analyzing Varied Steady Flows and Spatially Varied Increasing Flows  128 Goals  128 Rapidly Varying Steady Flows (dQ/dt = 0, dQ/dx = 0, dy/dx varies)  128 Spatially Varying Steady Flow (dQ/dt = 0, dQ/dx varies, dy/dx varies)  137 Conclusions  142 Problems and Questions  142 References  143 7 Hydraulics of Water Management Structures  144 Goals  144 Structure Types  145 Hydraulic Concepts  147 Stage–Discharge Relationships of Weir Inlets and Flumes  150 Discharge Relations of Orifices and Sluice Gates Inlet Devices  156 Flow Hydraulics of Closed Conduits  157 Stage–Discharge Curves for Culverts and Spillways  167 Closed Conduit Systems for Urban Stormwater Collection  169 Ecologic Suitability  171 Summary and Conclusions  177 Problems and Questions  179 References  182

Contents

8 Gradually Varied Unsteady Flow  185 Goals  185 Hydrologic Routing Approaches  187 Kinematic Wave Method  194 Diffusion Wave Method  199 Dynamic Routing  203 Summary and Conclusions  209 Problems and Questions  210 References  211 9 Rapidly Varying Unsteady Flow Applications – Waves  213 Goals  213 Surface Irrigation  213 Sluice Gate and Related Operations  217 The Dam-Break Problem  223 Oscillatory Waves  230 Summary and Conclusions  233 Problems and Questions  234 References  235 10 Channel Design Emphasizing Fine Sediments and Survey of Alluvial Channel Sediment Transport  236 Goals  236 Alluvial Channel vs. Earthen Channel and Other Preliminaries  237 Early Approaches to Sediment Transport  237 Incipient Motion  238 Riprap or Revetment Specification  243 Bedform Descriptions and Analysis  244 Sediment Fall Velocity  245 A Probabilistic Approach to Sediment Transport  249 Einstein (1950)–Laursen (1958)–Graf (1971) Stage–Discharge and Other Hydraulic Calculations  254 Van Rijn (1984) Stage–Discharge and Total Load  259 Total Load by Regression Approaches  264 Sediment Measurement  268 Sediment Routing Through Detention Ponds and Streams  268 Software Support for Estimating Sediment Transport  270 Implications of Sediment Transport on Infrastructure  271 Empirical Channel Design Approaches Leading to Sustainable Channels  274 Forces Impacting Channel Cross Sections – Stream Restoration  281 Summary and Future Directions  286 Problems and Questions  289 References  290

vii

viii

Contents

Appendix A  Software and Selected Solutions  294 Appendix B  Solution Charts for Vegetated Waterways Using the Permissible Velocity Method  305 Appendix C  Selected Cost Data for Channel Excavation and Lining Materials  310 Appendix D  Design Strategy Summary for Uniform Flow Channels  315 Index  317

ix

Preface With many excellent texts on Open Channel Hydraulics and Sediment Transport, why is another book needed? Available texts represent excellent tools for graduate instruction. We stand on the shoulders of giants in the field such as V.T. Chow, H.A. Einstein, Jr., to mention only a few. Undergraduates generally find the available books to be somewhat intimidating. Available texts tend not to have accessible supporting software. In a knowledge domain where most problems require iterative solutions, a need exists for software to fill a void, especially for undergraduates. The presentation of concepts in Open Channel hydraulics in available texts is more oriented to graduate students who have a solid knowledge of basic concepts. This text supports a split‐level class that is mostly undergraduate in composition. Planning for the book began just before the COVID‐19 pandemic. The move to online courses in response to the COVID‐19 pandemic caused many educators to rethink course delivery. In‐class and online education, in our experience, is most effective when content is delivered to undergraduates in modules sequentially build on the previously presented material. In our view, an online presentation stresses the need to be as sequential as possible because student interaction becomes more challenging than face‐to‐face delivery. Another guiding factor in the book organization was to present many design approaches for uniform flow as earlier as possible. Chapter 4 mostly completes the coverage of uniform flow. Early uniform flow presentation enables students to have a toolbox for solving many practical design problems early in the semester. The front‐loading of uniform flow allows students to begin working on design projects early in the term. We then present nonuniform flow and unsteady flow topics, enabling their addition to design projects as needed. Graduate students start to work on topics in Chapter  10, which flows from Chapter  4. Graduate students also do extra work on topics related to nonuniform and nonsteady flows as the course continues. A variety of Excel spreadsheets supports the concepts presented in each chapter. Public domain software examples (HEC‐RAS and HY‐8) support advanced analyses. Students may expect to use these analyses in subsequent classes, such as capstone design, when applicable. Mathematica notebooks support selected theoretical analyses. The Mathematica notebooks are useful for graduate student analysis of more advanced concepts. Chapter 10, covering alluvial transport processes, is written following a different approach compared to other chapters. Undergraduates could readily understand the incipient motion

x

Preface

concepts and the design of channels using Shields type analysis. They could readily grasp selected regression approaches for sediment transport computation. On the other hand, graduate students could spend considerable time looking at the more advanced sediment transport and river mechanics aspects. This text attempts to present a highly sequential course with affordable and challenging supporting software for undergraduate and graduate students. We leave advanced topics such as density currents, scour, and convection–diffusion of pollutant constituents to other texts. Emphasis on 3‐D computational modeling is left to other books as well. The author takes responsibility for the material presented. Please call my attention to any errors discovered. I look forward to learning about your experiences using the text and eagerly desire to hear suggestions for future improvements. Ernest W. Tollner

xi

Acknowledgments I am grateful for the can‐do attitude of my parental family, who immigrated from northeastern Germany late in the 1800s. They were dedicated to the proposition that the virtues in the US Declaration of Independence enabled improving one’s life compared to living on what bore resemblances to a feudal estate. The general farm and dairy background made possible by my parents, Ernest and Ruby Tollner, both now deceased, was of incalculable value to this undertaking. This view has sharpened as time passed. Likewise, the broad‐based agricultural and biological engineering and civil engineering experiences gained while studying under the guidance of B.J. Barfield, Tom Haan (Ag. Engineering, University of Kentucky), and Dr. David Gao (Civil Engineering, Univ of Kentucky), Drs. Charlie Busch and Dave Hill (Agric. Engineering, Auburn University) and Drs. Joseph Judkins and Fred Molz (Civil Engineering, Auburn) were inspiring and formative. Colleagues Brahm Verma and Dale Threadgill, who mentored me for 35 years at the University of Georgia, have been helpful in uncountable ways. Dr. Steve McCutcheon (Civil Engineer, USEPA, retired) continues to share many valuable insights. Many thanks to my CVLE 4210/6210 class for serving as guineas for the text’s trial run. They made valuable suggestions for improving the flow of the book. In particular, graduate students Will Mattison, Whitney Phelps, Shep Medlin, and Matthew Terrell come to mind. They also suggested improved figures and tables, for which I am grateful. I appreciate the patience and encouragement of my wife, Caren. To God be the Glory! I put this work forward as an offering to all. May all on His journey leave behind a more sustainable environment! May we dress and keep His magnificent creation in a sustainable way for all to better serve!

xii

About the Companion Website This book is accompanied by a companion website.

www.wiley.com/go/tollner/openchanneldesign This website includes: ●● ●● ●●

A variety of Excel spreadsheets to support the concepts presented in each chapter. Public domain software examples (HEC-RAS and HY-8) to support advanced analyses. Mathematica notebooks to support selected theoretical analyses.

1

1 Basic Principles and Flow Classifications Hydraulic Engineering has served humanity all through the ages by providing drinking water and protective measures against floods and storms. In the course of history, it has made the water resource available for human uses of many kinds. Biswas (1970) chronicles contributions since Hammurabi (c. 1700 bce) hydraulic engineering over the centuries. In a survey of the University of Georgia Libraries’ holdings under “land‐use change,” some 8000 articles discuss facets of the hydrologic cycle and associated runoff. Simons and Senturk (1992) provide a synopsis of contributions to sediment transport science in streams that date back to work in China to date back to 4000 bce. Students who seriously pursue open channel hydraulics and sediment transport should explore works such as those mentioned. Management of the world’s water is a complex task, and both its scope and importance continue to grow as we strive for sustainable stewardship of our abode. Over time humanity has not only diverted and used the waters of the world for its purposes but, by engaging nature into its service, has turned deserts into fertile land. Natural habitat is threatened in more and more parts of the world by an ever‐growing human population. Thus, long‐ term needs are food, water, shelter, and an aesthetically pleasing, healthy, nurturing environment. Open channel flow is, in brief, a flow where the fluid has a free surface, where the free surface of the flow is subject to atmospheric pressure. Problems covered include flow in a conduit when the conduit is not full, such as in a storm sewer. The primary fluid of interest is water, although any fluid could, in principle, be addressed. This chapter introduces concepts that are developed as we progress through the text. Why do we consider open channels when one can simply take an earthmover and create a conveyance? The hydraulic engineer meets engineering, economic, and social objectives in the client’s and society’s best interests. Sound engineering should result in sustainable design. Figure 1.1 shows a channel under construction with a channel design that conveys water while meeting aesthetic and sustainability goals. Here are some definitions: Ditch – an excavated water conveyance often installed without extensive advanced engineering design. Natural channel – watercourses that exist naturally, such as gullies, brooks, rivers, streams, or estuaries. These are often analyzed to determine flows associated with high‐water marks. Open Channel Design: Fundamentals and Applications, First Edition. Ernest W. Tollner. © 2022 John Wiley & Sons Ltd. Published 2022 by John Wiley & Sons Ltd. Companion website: www.wiley.com/go/tollner/openchanneldesign

2

Basic Principles and Flow Classifications

Figure 1.1  A trapezoidal waterway under construction (Source: Photo courtesy of Mr. Greg Jennings).

Ephemeral channel – a natural channel that does not continuously flow. Perennial channel – the natural channel that continually flows. Artificial channel  –  watercourse developed by humankind such as navigation, irrigation, drainage, or closed conduit (e.g., flow does not fill the entire channel such as a sewer or culvert) channels. Hydraulically designed conveyances with attributes strategically chosen to meet the client’s engineering, economic, and social objectives (Graf and Altinakar 1998). Designed channels not continuously flowing are intermittent. We address the hydraulic engineering needed to satisfy social, economic, and engineering needs in this text. We focus mainly on one‐dimensional solutions in this introductory text. Once solved using tabular solutions, the one‐dimensional problems (along the channel) are easily solved with spreadsheets and other equation processing software. A commonly used public domain software, HEC‐RAS, can, in some cases, facilitate two‐dimensional solutions.

­Fluid Mechanics Foundations Fluid statics is a crucial limiting case. It describes fluid pressures and forces when the fluid velocity is zero. The pressure at a point is the product of fluid unit weight, γ, and depth, symbolized as d, b, or y. The unit weight in SI and imperial units is 9810 N/m3 or 62.4 lb/ft3. Figure 1.2 shows a hydrostatic distribution on a channel wall and bottom. Figure 1.3 shows the hydrostatic forces on a sloping wall where one partitions the effects between vertical and horizontal vector sums. Acceleration causes deviations from the purely hydrostatic pressure distribution. Sluice gates and transitions due to structures and slope changes result in acceleration. Thus, the pressure is not hydrostatic near sluice gates and other facilities. One can easily show that the effective center of force is 1/3 up from the channel bottom. The 1/3 rule applies when the wall or gate object is rectangular. Symmetric gates of other shapes have force centers at the center of pressure, which deviates from the 1/3 point. In general, the center of pressure is computed using the following equation: I yp  G  y A y

(1.1)

Mechanics Foundation  3

­Fluid

Specific weight = g*ρ = γ

P = γ*h2 2

h

P = γ*h2 2

h/3

P = γh

Figure 1.2  Free body diagram of a static fluid.

Water surface

7 feet θ

4 feet b

c

90 – θ = φ

φ

te

Ga

wt

8 feet

θ a

d

e

4y 12γ

8y

θ

g

f

4 C

θ φ 90 – θ = φ

θ

8

Θ O 7

P = 4γ

P = 8γ

Figure 1.3  Fluid forces exerted on a submerged gate.

where IG is the moment of inertia (b * y3/12 for a rectangular gate where b is length and y = depth), y is the depth to the centroid of the object (half the distance to the bottom for a rectangle), A is the area of the port or gat object (length b × depth y for a rectangular gate), and yp is the distance below the surface to the centroid where the total hydrostatic force is concentrated.

4

Basic Principles and Flow Classifications

It is easily shown that yp is 2/3 the depth from the top or 1/3 of the bottom for the rectangular gate. Equation 1.1 gives the distance along the slope for non‐vertical gates. One can separate the forces into horizontal and vertical vector components and apply the 1/3 rule to the vertical component, giving a similar value to Equation 1.1. The value of Equation 1.1 comes to the fore when the gate is nonrectangular. One may then consult a statics text for the moment of inertia for the shape in question. Nonsymmetric gates (along the length b‐axis) have a product of inertia, which shifts the center of pressure a distance from the centroid along the width b of the gate. We are not concerned with nonsymmetric gates. One may apply fundamental statics analysis to compute forces to secure closed gates in a channel with ponded, static conditions. A spreadsheet is provided which analyzes the forces required to fasten a symmetric gate on sloping walls. A partially opened gate obviously does not represent a static condition. We evaluate forces associated with moving water in our chapter on rapidly varied flows. As in closed conduits flowing full, flows in open channels may be laminar or turbulent. The flow is laminar when viscous forces dominate inertial forces in determining flow behavior. Flows are turbulent when inertial flows dominate viscous forces. The Reynolds number, is expressed as follows: R

V L 



VL





(1.2)

where R is the Reynolds number (−), ν is the kinematic viscosity ((L/T2), typically 1.93E−06 m2/s or 1.93E−05 ft2/s), μ is the dynamic viscosity (FT/L2, typically 3.75E−05 lbf s/ft2 or 1.79E−03 N s/m2), ρ is the fluid density (M/L3), typically 1.94 slugs/ft3 or 1000 kg/m3, L is the characteristic length, typically depth (L), and V is the velocity (L/T). Units for viscosity are quite varied, depending on the usage of force or mass units. Dynamic viscosity may be expressed in mass units as M/(LT). One may perform an internet search to find these expressions in desired units. We include a spreadsheet showing viscosity and density as a function of temperature for SI and Imperial units. We generally ignore temperature effects in most open channel applications. As in closed conduit flow, flows with Re  10 000 are generally turbulent. The region 2000 > Re > 10 000 is a transition zone. Consider the Darcy–Weisbach formula given as follows: LV 2 hf = f do 2 g

(1.3)

where hf is the head loss (L), f is the friction factor (−), do is the diameter of the pipe (L), g is gravity (L/T2), and L is the length over which the head loss occurs (L). Defining the slope as hf/L, do as 4R (where hydraulic radius R is more fully defined later), one may write Equation 1.3 as follows: 8g R S f = V2

(1.4)

Simons and Senturk (1992) compare friction factor vs. Re pipes and channels with varying roughness heights. Their figure is reproduced in Figure  1.4. The Darcy–Weisbach

­Fluid

Mechanics Foundation  5

friction factor may be related directly to the Manning state equation friction term in the turbulent zone. The concept of channel roughness is more fully developed in the next chapter. Figure  1.4 shows notable similarities with the Darcy–Weisbach–Moody diagram for closed conduit pipe flow. Most flows generally occur in the regime where Re > 4000, which enables one to relate the friction factor to roughness height. Most flows of interest involve turbulent flows, except for shallow sheet flows and flow found in natural treatment ­systems. The above results generally apply to steady and unsteady flows.

Figure 1.4  Flow resistance in pipes (a) and channels (b) (Source: From Simons and Senturk (1992); used with permission of Water Resources Publishers).

6

Basic Principles and Flow Classifications

In addition to inertial flows and viscous flows, a third flow force, gravity forces, is vital in open channel flows. Another dimensionless number, the Froude number, is essential for further classifying turbulent flows. The Froude number appears as follows: V F= gL

(1.5)

Parameter L is a characteristic length, which is frequently taken as hydraulic depth. The hydraulic depth is the channel depth for a rectangular channel and is more fully defined later for nonrectangular channels. Graf (1971) provides Reynolds and Froude similitude relations, shown in Table 1.1. Similitude is a basis for studying scale models of various structures in a lab. Flowrate, velocity, time, and force may be measured and then scaled back to the lab’s prototype structure. An accurate model requires the satisfaction of multiple dimensionless numbers. However, without varying fluids properties and gravity, the satisfaction of true similarity is not practical, so we focus on key numbers in ranges that give reasonably good results. Most applications involve nondistorted models with predominant Reynolds or Froude similarity. For example, given a prototype structure with basic dimensions of 100 and 50 m, handling a turbulent 10 m3/s flow. If one chooses to scale it to 10 and 5 m, one can use Table 1.1 to see scale relations between the model and prototype. The required flow to operate the Froude similitude model is then Qm = Qp (Lm/Lp)5/2 or 10*(1/10)5/2 = 0.03 m3/s. Forces, velocities, and times can be measured in the model and scaled back to the prototype. For best results, Table 1.1  Modeling ratios for undistorted fixed bed models with Reynolds or Froude similarity. Model parameters Parameter of interest

M

Velocity, V VP Flowrate,

Froude number

LP  P  M

 LM   P  L 

M

L

QM QP

M

Force, F FP Time,

Reynolds number



M 2

 LM  P L

 P M  M P   

 LM  P L

  

3

2

tM tP M

 M  P 

2

 P  M  

 LM  P L

  

5/ 2

LM ρ M LP ρ P 1/ 2

 LM   M  P  P  P M L   

 LM   P  L 

–

 LP   M  L 

1/ 6

nP Roughness n, n



1/ 2

P

Nomenclature: μ is dynamic viscosity, ρ is density, L is length, t is time, Q is flowrate, V is velocity, and n is the Manning roughness, discussed in a subsequent chapter. Superscript P refers to the prototype, and M refers to the scaled model. Be sure to check that the Manning equation applies in the case of Froude similarity. Source: Based on Graf (1971) with modifications.

­Hydrologic Foundation  7

flows should be turbulent in the model, assuming the prototype is in the turbulent regime. If the prototype flow is laminar, then the model flow should also be laminar. One would use Reynolds’s similarity in laminar flow cases. If the flow in the prototype is laminar instead of turbulent, one can consider Reynolds’s similarity instead of Froude similarity. Henderson (1966) devotes an entire chapter to similitude studies. Currently, there is one remaining hydraulics laboratory in the United States1 that offers design services based on similitude. Flows with a Froude number greater than 1.1 are supercritical. A supercritical flow is typically a shooting flow. Placing one’s hand in a strongly shooting flow results in flow tending to go up one’s arm. Flows less than 0.9 are subcritical or tranquil. Moving one’s hand back and forth in a tranquil flow produces waves that move upstream. Flows with F ≈ 1 are denoted to be critical flows, which results in a standing wave. At the critical flow condition, V = Sqrt(g * D). The critical flow state is useful for the design of flow measurement devices. The critical flow condition gives a unique relationship between flow and depth. Flow measurement structures are further developed in a later chapter. It will be shown that Sqrt(g * D) is the velocity of a gravity wave. Gravity waves can lead to pulsations in channels, which can pose structural stability issues for lined channels and erosion hazards, particularly for unlined channels. To avoid the possibility of flow pulsations, we try to avoid designs leading to 0.9