Kinematic-Wave Rainfall-Runoff Formulas

KINEMATIC-WAVE RAINFALL-RUNOFF FORMULAS

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KINEMATIC-WAVE RAINFALL-RUNOFF FORMULAS

TOMMY S.W. WONG

Nova Science Publishers, Inc. New York

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Wong, Tommy S. W. Kinematic-wave rainfall-runoff formulas / Tommy S.W. Wong. p. cm. Includes index. ISBN 978-1-61470-182-8 (eBook) 1. Runoff--Mathematical models 2. Rain and rainfall--Mathematical models. 3. Hydrologic models. I. Title. GB980.W66 2009 627'.042--dc22 2008051332

Published by Nova Science Publishers, Inc.

New York

DEDICATED TO my parents, Sze Fong Wong and En Yueh Woo my parents-in-law, Chip Shing Sum and Luk Ying Ko my darling wife, Christina and my wonderful sons, Alston, Lester and Hanson

CONTENTS Preface

ix

Acknowledgements

xi

List of Symbols

xiii

Chapter 1

Introduction

1

Chapter 2

General Formulas for Flow on Overland Plane

3

Chapter 3

Working Formulas for Flow on Overland Plane

29

Chapter 4

General Formulas for Flow in Open Channel

39

Chapter 5

Working Formulas for Flow in Circular Channel

63

Chapter 6

Working Formulas for Flow in Parabolic Channel

77

Chapter 7

Working Formulas for Flow in Rectangular (Deep) Channel

91

Chapter 8

Working Formulas for Flow in Rectangular (Square) Channel

101

Chapter 9

Working Formulas for Flow in Rectangular (Wide) Channel

113

Chapter 10

Working Formulas for Flow in Trapezoidal Channel with Equal Side Slopes

125

Working Formulas for Flow in Trapezoidal Channel with One Side Vertical

137

Chapter 12

Working Formulas for Flow in Triangular Channel

151

Chapter 13

Working Formulas for Flow in Vertical Curb Channel

163

Chapter 11

Appendices

175

References

233

Index

237

PREFACE This is not an ordinary book on rainfall and runoff. All the general and working formulas in this book are theoretically derived. The formulas are therefore globally and eternally applicable, as long as the situations under consideration are within the assumptions and limitations of the theory. This epitomizes the powerful nature of the physically-based approach in hydrology. This book covers formulas for flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of concentration; rising, equilibrium and falling phases of a hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage for flow on an overland plane, and flow in nine different channel shapes, which are (i) circular, (ii) parabolic, (iii) rectangular (deep), (iv) rectangular (square), (v) rectangular (wide), (vi) trapezoidal with equal side slopes, (vii) trapezoidal with one side vertical, (viii) triangular, and (ix) vertical curb.

ACKNOWLEDGEMENTS I would like to thank Bhagavan Sri Sathya Sai Baba for selecting me as his instrument in authoring this book for the benefit of mankind. I would also like to thank Emeritus Professor Charng Ning Chen for introducing the subject to me. Last but not least, I would like to thank the following who have contributed to this book in one way or another: Syed-Alwi-BinSheikh-Bin-Hussien Alkaff, Jason Christopher Chan, Lee Ching Chung, Joel Ore Iwanaga, Ahmed Nusrath Bilal Izzath, Xiao Chi Koh, Heng Yein Kong, Yunjie Li, Jessie Su Hui Lim, Teik Peng Lim, Xiaoxie Liu, Teng Tsair Ma, Chia Li Ow, and Maichun Zhou.

LIST OF SYMBOLS A flow area (m2) Ac flow area in channel (m2) Ac’ parameter relating Ac to H for parabolic channel Ae flow area in channel corresponding to equilibrium discharge (m2) Afull flow area in circular channel under full flow condition (m2) Ai flow area in channel at inflection point (m2) Ao area of overland plane (ha) Ap flow area in channel corresponding to partial equilibrium discharge (m2) AQmax flow area in circular channel under maximum flow condition (m2) Au flow area in channel corresponding to upstream discharge (m2) a parameter relating i to tr b parameter relating i to tr Cr runoff coefficient c parameter relating i to tr cav average kinematic wave celerity (m·s-1) ck kinematic wave celerity (m·s-1) D diameter of circular channel (m) Dec equilibrium detention storage for a channel of length Lc (m3) Deo equilibrium detention storage for an overland plane of length Lo (m3⋅m-1) Deu equilibrium detention storage for a plane or a channel of length Lu (m3⋅m-1 or m3) Deuc equilibrium detention storage for a channel of length (Lu + Lc) (m3) Deuoequilibrium detention storage for a plane of length (Lu + Lo) (m3⋅m-1) Fe Froude number at the end of the plane at equilibrium g acceleration due to gravity (m⋅s-2) H height of focal point above parabolic channel invert (m) i rainfall intensity (mm⋅h-1) id design rainfall intensity for overland plane (mm⋅h-1) K kinematic flow number La arc length of parabola (m) Lc length of channel (m) Ld length of overland plane or channel contributing to duration of partial equilibrium discharge (m)

xiv

Tommy S.W. Wong Lf length of overland plane or channel in which the flow equals to upstream inflow during falling phase (m) Lo length of overland plane (m) Lp length of overland plane or channel contributing to partial equilibrium discharge (m) Lu length of upstream plane or channel (m) n Manning’s roughness coefficient (s·m-1/3) nc Manning’s roughness coefficient for channel surface (s·m-1/3) no Manning’s roughness coefficient for overland surface (s·m-1/3) P wetted perimeter (m) P’ parameter relating P to H for parabolic channel Q discharge (m3·s-1) Qc discharge in channel (m3·s-1) Qd design discharge of overland plane (m3·s-1) Qe discharge at the end of channel at equilibrium (m3·s-1) Qfull discharge in circular channel under full flow condition (m3·s-1) Qmax discharge in circular channel under maximum flow condition (m3·s-1) Qo discharge on overland plane (m3·s-1) Qp discharge at the end of channel at partial equilibrium (m3·s-1) Qu upstream inflow to channel (m3·s-1) q discharge per unit width of overland plane (m2·s-1) qe unit discharge at the end of overland plane at equilibrium (m2·s-1) qL lateral inflow per unit length of channel (m2·s-1) qp unit discharge at the end of overland plane at partial equilibrium (m2·s-1) qu unit upstream inflow to overland plane (m2·s-1) R hydraulic radius (m) S bed slope (m·m-1) Sc slope of channel bed (m·m-1) Sf friction slope (m·m-1) So slope of overland plane (m·m-1) T top width (m) Tw wave period (min) t time (min) td duration of partial equilibrium discharge (min) to time of concentration of overland flow (min) tq duration of lateral inflow (min) tr duration of rainfall (min) tt time of travel in channel (min) tu time of travel in upstream plane or channel (min) v flow velocity (m·s-1) vav average flow velocity (m·s-1) vs steady-state, uniform, mean flow velocity in channel (m·s-1) W base width of rectangular or trapezoidal channel (m) w width of overland plane (m) x semi-width of parabolic channel at height y (m) x’ parameter relating x to H for parabolic channel xc distance along a channel in the direction of flow (m)

Kinematic-Wave Rainfall-Runoff Formulas

xv

xi distance xo or xc of the inflection point (m) xo distance along an overland plane in the direction of flow (m) y height above parabolic channel invert (m) yc flow depth in channel (m) ye flow depth at the end of overland plane at equilibrium (m) yi flow depth at inflection point (m) yo flow depth on overland plane (m) yp flow depth at the end of overland plane at partial equilibrium (m) yQmax flow depth in circular channel under maximum flow condition (m) ys steady-state, uniform, flow depth in channel (m) yu flow depth on overland plane corresponding to upstream inflow (m) Z parameter relating x to H for parabolic channel (m) Z’ parameter relating Z to H for parabolic channel z reciprocal of channel side slope of trapezoidal, triangular, or vertical curb channel (m·m-1) αc parameter relating Qc to Ac for open channel αo parameter relating q to yo for overland plane βc parameter relating Qc to Ac for open channel βo parameter relating q to yo for overland plane γ parameter relating AQmax to D for circular channel µ parameter relating yc to W θ water surface angle for circular channel (rad) τ dimensionless wave period ψ parameter relating Ac to W

In: Kinematic-Wave Rainfall-Runoff Formulas Tommy S.W. Wong, pp. 1-2

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 1

1. INTRODUCTION Ever since Lighthill and Whitham (1955) showed that the main body of a natural flood wave moves as the kinematic wave, there has been continual interest in the application of the kinematic wave theory to hydrologic engineering. The greatest strength in this application is the feasibility of obtaining physically-based analytical formulas. The values of this strength are two-fold: 1. It enables hydrologists and engineers to have a clear understanding of the contribution by each parameter in the physical process. 2. Without the need for any experimental data, it offers formulas that can be applied to practical situations, including ungauged catchments. Further, these formulas have great advantages: 1. As the formulas are theoretically derived, the assumptions and limitations involved in the formulas can be clearly stated. 2. As the formulas are general in nature, they are globally and eternally applicable, as long as the situation under consideration is within the assumptions and limitations of the theory. 3. As the formulas are analytical, they can be used without the need for computer programming. Since the formulas are not hidden in some computer program, the steps leading to each answer can easily be traced. To enable hydrologists and engineers to have ready access to the kinematic wave formulas, the objectives of this book are: 1. To show the derivation of the kinematic wave formulas for the rainfall-runoff process, and to highlight the assumptions and limitations in the derivations. 2. To present the kinematic wave formulas in a form that can be readily used by practitioners.

2

Tommy S.W. Wong

1.1. HOW TO USE THIS BOOK The Chapters in this book are more or less self-contained; hence, they can be read fairly independently. The topics covered may be grouped under four phases of the rainfall-runoff process for an overland plane subject to uniform rainfall excess and with a constant upstream inflow, and for a channel subject to uniform lateral inflow and with a constant upstream inflow. For a catchment comprising a network of overland planes and channels, the outflow from the overland planes can become the lateral inflow to the channels. The four phases of the rainfall-runoff process are: 1. General phase covering (i) flow depth, (ii) flow velocity, (iii) average flow velocity, (iv) wave celerity, and (v) average wave celerity. 2. Rising phase covering (i) time of concentration or time of travel, (ii) rising phase of hydrograph, (iii) forward characteristic, and (iv) rising phase of water surface or flow area profile. 3. Equilibrium phase covering (i) design discharge of an overland plane, (ii) duration of partial equilibrium discharge, (iii) equilibrium phase of hydrograph, (iv) equilibrium phase of water surface or flow area profile, and (v) equilibrium detention storage. 4. Falling phase covering (i) falling phase of hydrograph, and (ii) falling phase of water surface or flow area profile. This book may be read in the following ways: 1. Readers who are interested in the assumptions and background of the formulas may refer to Chapter 2 for flow on an overland plane, and Chapter 4 for flow in an open channel. 2. Readers who are interested in the working formulas may refer to Chapter 3 for flow on an overland plane, and Chapters 5-13 for flow in nine different channel shapes, which are (i) circular, (ii) parabolic, (iii) rectangular (deep), (iv) rectangular (square), (v) rectangular (wide), (vi) trapezoidal with equal side slopes, (vii) trapezoidal with one side vertical, (viii) triangular, and (ix) vertical curb. 3. Readers who are interested in the assumptions and background of the formulas and the working formulas may refer to all the Chapters. For ease of reference, the applicability of the kinematic wave theory is summarized in Appendix A, the general formulas in Appendices B-C, the kinematic wave parameters in Appendix D, and the working formulas in Appendices E-S. The units for the working formulas are contained in the List of Symbols. Finally, the values for the runoff coefficient may be selected from the American Society of Civil Engineers (1992), the values for the Manning’s roughness coefficient for overland surface may be selected from Engman (1986), and the values for the Manning’s roughness coefficient for channel surface may be selected from Chow (1959) or Arcement and Schneider (1989).

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 3-28

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 2

2. GENERAL FORMULAS FOR FLOW ON OVERLAND PLANE In this Chapter, based on the kinematic wave theory; the general formulas for flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of concentration; design discharge; rising, equilibrium and falling phases of a hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage for flow on an overland plane are derived.

2.1. FLOW CONDITIONS Consider an overland plane of length Lo, subject to a uniform rainfall intensity i, and with a constant upstream inflow qu, the unit discharge, q, along the equilibrium water surface profile for a unit width of the plane is:

q = qu + Cr ixo

(2.1)

where Cr = runoff coefficient, xo = distance along the plane in the direction of flow. Substituting xo = Lo into Eq. (2.1) gives the discharge qe at the end of the plane at equilibrium, i.e.

qe = qu + Cr iLo

(2.2)

Further, the upstream inflow, qu, can be considered to be contributed by an imaginary overland plane of length Lu, which is situated immediately upstream of the overland plane of length Lo. This imaginary upstream plane is also subject to a uniform rainfall intensity i, but with zero upstream inflow. At the outlet point of this upstream plane and at equilibrium, the discharge is qu. Substituting qe = qu, qu = 0 and Lo =Lu into Eq. (2.2) gives the length of the upstream plane, Lu, in terms of the upstream inflow, qu, as follows:

Tommy S.W. Wong

4

Lu =

qu Cr i

(2.3)

Figure 2.1 shows the upstream plane, the overland plane, and the equilibrium water surface profile.

Figure 2.1. Upstream and Overland Planes with Equilibrium Water Surface Profile.

2.2. DYNAMIC WAVE EQUATIONS The mechanics of unsteady flow on an overland plane with a rainfall contribution can be expressed mathematically by the Saint Venant equation. Derived from the principles of continuity and momentum, the equations are (Chow et al 1988):

∂y o ∂q + = Cr i ∂t ∂x o

(2.4)

1 ∂v v ∂v ∂yo + + − (So − S f ) = 0 g ∂t g ∂xo ∂xo

(2.5)

where yo = overland flow depth, t = time, g = acceleration due to gravity, v = flow velocity, So = overland slope, and Sf = friction slope. The assumptions inherent in Eqs. (2.4) and (2.5) are:

Kinematic-Wave Rainfall-Runoff Formulas

5

1. The flow is one dimensional (i.e. velocity varies in the longitudinal direction only). This implies that the velocity is constant and the water surface is horizontal across any section perpendicular to the longitudinal axis. 2. All flows are gradually varied with hydrostatic pressure prevailing at all points in the flow such that all vertical acceleration within the water column can be neglected. 3. The longitudinal axis of the overland plane can be approximated by a straight line (i.e. there is no secondary circulation). 4. The slope of the overland plane is small. 5. The overland plane is fixed (i.e. the effects of scour and deposition are negligible). 6. Resistance to flow can be described by empirical resistance formulas, such as Manning’s equation. 7. The fluid is incompressible and homogeneous in density. 8. The momentum carried to the fluid from the rainfall is negligible. The momentum equation (Eq. 2.5) consists of five terms; namely local acceleration, convective acceleration, pressure force, gravity force and friction force; each representing a physical process that governs the flow momentum described as follows: 1. The acceleration terms represent the effect of velocity change over time and space. 2. The pressure force term represents the effect of flow depth change. 3. The gravity force term So is proportional to the overland slope and accounts for the change in bed level. 4. The friction force term Sf is proportional to the friction slope and accounts for the friction loss for the flow on an overland plane.

2.3. KINEMATIC WAVE EQUATIONS If the backwater effect is negligible and there is no rapid change in flow, the acceleration and pressure terms in Eq. (2.5) may be neglected (Stephenson 1981, Wong 1992), and the momentum equation reduces to:

So = S f

(2.6)

Equations (2.4) and (2.6) are called the “kinematic wave equations”. Equation (2.6) shows that the overland slope is parallel to the friction slope, which means that the kinematic wave is under the uniform flow condition. Thus, Eq. (2.6) can be replaced by the general uniform flow equation, which is:

q = α o y oβ o where αo and βo = kinematic wave parameters relating q to yo.

(2.7)

Tommy S.W. Wong

6

2.4. FLOW DEPTH Rearranging Eq. (2.7) gives the equation for the flow depth for a plane with and without upstream inflow:

⎛ q y o = ⎜⎜ ⎝αo

⎞ ⎟⎟ ⎠

1 βo

(2.8)

2.5. FLOW VELOCITY From continuity, the flow velocity, v, is related to the unit discharge, q, as follows:

v=

q yo

(2.9)

Substituting Eq. (2.7) into Eq. (2.9) and the velocity, v, becomes (Wong 2003):

v = α o y oβ o −1

(2.10)

Substituting Eq. (2.8) into Eq. (2.10) gives the equation for the velocity, v, in terms of unit discharge, q (Wong 2003):

(

v = α o q β o −1

)

1 βo

(2.11)

Substituting Eq. (2.1) into Eq. (2.11) gives the equation for the flow velocity along the equilibrium profile for a plane with upstream inflow:

[

v = α o (qu + C r ixo )

]

β o −1 1 β o

(2.12)

For a plane with zero upstream inflow (qu= 0), Eq. (2.12) reduces to:

[

v = α o (C r ix o )

]

β o −1 1 β o

(2.13)

2.6. AVERAGE FLOW VELOCITY Further, the average flow velocity, vav , over the length of the plane, Lo, can be derived as follows (Wong 2003):

Kinematic-Wave Rainfall-Runoff Formulas

vav =

Lo

7

(2.14)

Lo

1 ∫0 v dxo

Substituting Eq. (2.12) into Eq. (2.14) and integrating (1/v) gives the equation for the average flow velocity for a plane with upstream inflow:

vav =

α o 1 β Cr iLo o

[

β o (qu + Cr iLo )1 β − qu 1 β o

o

]

(2.15)

For a plane with zero upstream inflow (qu = 0), Eq. (2.15) reduces to:

v av =

1

βo

[α (C iL ) ]

β o −1 1 β o

o

r

o

(2.16)

2.7. KINEMATIC WAVE CELERITY Differentiating Eq. (2.7) with respect to t gives:

∂q ⎛ ∂y ⎞ = α o β o y oβo −1 ⎜ o ⎟ ∂t ⎝ ∂t ⎠

(2.17)

Rearranging Eq. (2.17) gives:

∂y o 1 ⎛ ∂q ⎞ = ⎟ β o −1 ⎜ ∂t α o β o y o ⎝ ∂t ⎠

(2.18)

Substituting Eq. (2.18) into Eq. (2.4), the continuity equation becomes:

⎡ 1 ⎛ ∂q ⎞⎤ ⎛ ∂q ⎟⎥ + ⎜⎜ ⎢ β o −1 ⎜ ∂ t α β y ⎝ ⎠⎦ ⎝ ∂xo ⎣ o o o

⎞ ⎟⎟ = C r i ⎠

(2.19)

Kinematic wave results in changes in q, which is dependent on both xo and t, and the increment in flow rate dq can be written as:

dq =

∂q ∂q dt + dxo ∂t ∂xo

Dividing Eq. (2.20) by dxo:

(2.20)

Tommy S.W. Wong

8

dq ⎛ ∂q =⎜ dxo ⎜⎝ ∂xo

⎞ ⎡ ∂q ⎛ dt ⎟⎟ + ⎢ ⎜⎜ ⎠ ⎣ ∂t ⎝ dxo

⎞⎤ ⎟⎟⎥ ⎠⎦

(2.21)

If

dq dxo

(2.22)

1 dt = dxo α o β o y o β o −1

(2.23)

Cr i = and

then Eq. (2.19) and Eq. (2.21) are identical. Differentiating Eq. (2.7) with respect to yo:

dq β −1 = α o β o yo o dy o

(2.24)

Comparing Eq. (2.23) and Eq. (2.24) gives:

dq dxo = dy o dt

(2.25)

Since kinematic wave celerity, ck, is:

ck =

dxo dt

(2.26)

Substituting Eq. (2.25) into Eq. (2.26) gives:

ck =

dx o dq β −1 = = α o β o yo o dt dy o

(2.27)

Substituting Eq. (2.8) into Eq. (2.27) gives:

(

c k = β o α o q β o −1

)

1 βo

(2.28)

Substituting Eq. (2.1) into Eq. (2.28) gives the equation for the wave celerity along the equilibrium profile for a plane with upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

[

c k = β o α o (qu + C r ixo )

]

β o −1 1 β o

9 (2.29)

For a plane with zero upstream inflow (qu = 0), Eq. (2.29) reduces to:

[

c k = β o α o (C r ix o )

]

β o −1 1 β o

(2.30)

2.8. AVERAGE WAVE CELERITY The average wave celerity, cav, over the length of the plane, Lo, can be derived as follows (Wong 1996):

cav =

Lo Lo 1 ∫0 ck dxo

(2.31)

Substituting Eq. (2.29) into Eq. (2.31) and integrating (1/ck) gives the equation for the average wave celerity for a plane with upstream inflow:

cav =

α o1 β Cr iLo o

(qu + Cr iLo )1 β

o

− qu

1 βo

(2.32)

For a plane with zero upstream inflow (qu = 0), Eq. (2.32) reduces to:

[

c av = α o (C r iLo )

]

β o −1 1 β o

(2.33)

2.9. TIME OF CONCENTRATION The time of concentration for flow on an overland plane, to, can be obtained by dividing length of the plane, Lo, by the average wave celerity, cav, as follows:

to =

Lo c av

(2.34)

Substituting Eq. (2.32) into Eq. (2.34) gives the equation for the time of concentration for a plane with upstream inflow (Wong 1995):

Tommy S.W. Wong

10

to =

1

α o1 β

o

⎡ (qu + C r iLo )1 β o − qu 1 β o ⎢ Cr i ⎢⎣

⎤ ⎥ ⎥⎦

(2.35)

For a plane with zero upstream inflow (qu = 0), Eq. (2.35) reduces to (Henderson and Wooding 1964, Wong 1995):

⎡ ⎤ Lo to = ⎢ β o −1 ⎥ ⎣⎢ α o (C r i ) ⎦⎥

1 βo

(2.36)

Further, for the upstream plane of length Lu, substituting Lo = Lu into Eq. (2.36) gives the time of concentration, tu, of the upstream plane:

⎡ ⎤ Lu tu = ⎢ β o −1 ⎥ ⎣⎢ α o (C r i ) ⎦⎥

1 βo

(2.37)

2.10. DESIGN DISCHARGE For estimating the design discharge of a desired recurrence interval, the rainfall intensityduration curve of the same recurrence interval is used. For a given rainfall recurrence interval, the design concept is to choose a storm from the rainfall intensity-duration curve such that it produces the maximum peak discharge. This maximum discharge is the design discharge (Wong 2005a).

2.10.1. Rainfall Intensity-Duration Relationship Analyses of the total rainfall curves show that for a given recurrence interval, the rainfall intensity varies inversely with the rainfall duration, and it can be mathematically described by (American Society of Civil Engineers 1992):

i = a (c + t r )

b

(2.38)

where tr = rainfall duration, and a, b and c = constants. To facilitate the derivation of an explicit expression for the design discharge, Eq. (2.38) is reduced to (Wong 1995):

i = at r

−b

(2.39)

Although the use of Eq. (2.39) with a single set of a and b values cannot fit the entire rainfall intensity-duration curve, Chen and Evans (1977), and Wong (1992) showed that by dividing

Kinematic-Wave Rainfall-Runoff Formulas

11

the rainfall curve into segments, it is possible to fit the entire rainfall curve with different values of a and b for each segment.

2.10.2. Design Discharge For the purpose of estimating the design discharge, Wong (2005a) showed that the critical rainfall duration is the time of concentration. Eq. (2.36) and Eq. (2.39) are therefore solved simultaneously by equating to = tr, resulting in an explicit expression for the design rainfall intensity, id, for a plane with zero upstream inflow: bβ o

⎡ 1 b ββo −1 ⎤ b + β o −bβ o ⎢ a Cr o ⎥ id = ⎢ (L α )1 βo ⎥⎥ ⎢ o o ⎦ ⎣

(2.40)

Figure 2.2 shows a graphical solution for obtaining id. The design discharge, Qd, is related to the design rainfall intensity, id, and the area of the plane, Ao, as follows:

Qd = C r i d Ao

(2.41)

i

i d (Eq. 2.40)

tr = to Figure 2.2. Design Rainfall Intensity for a Plane without Upstream Inflow.

t r or to

Tommy S.W. Wong

12

For a rectangular plane, the area Ao, is related to the dimensions of the plane as:

Ao = Lo w

(2.42)

where w = width of the plane. Substituting Eq. (2.40) into Eq. (2.41) gives the equation for the peak discharge per unit area of the plane:

(

)

bβ o

⎡ aC 1 b ⎤ b + β o −bβ o r Qd Ao = ⎢ 1 βo ⎥ ( L α ⎣⎢ o o ) ⎦⎥

(2.43)

2.11. HYDROGRAPH - RISING PHASE Expanding the partial derivative, (∂q ∂xo ) , into total derivative results in:

∂q dq ⎛ ∂y o ⎜ = ∂x o dy o ⎜⎝ ∂x o

⎞ ⎟⎟ ⎠

(2.44)

Substituting Eq. (2.27) into Eq. (2.44) gives:

∂y ∂q = ck o ∂x o ∂x o

(2.45)

Substituting Eq. (2.45) into Eq. (2.4) gives:

∂y o ∂y + ck o = C r i ∂x o ∂t

(2.46)

Differentiating yo with respect to xo and t:

dy o =

∂y o ∂y dt + o dxo ∂t ∂xo

(2.47)

Dividing Eq. (2.47) by dt and substituting Eq. (2.27) into it:

dy o ∂y o ∂y = + ck o dt ∂t ∂x o Comparing Eqs. (2.46) and (2.48) gives:

(2.48)

Kinematic-Wave Rainfall-Runoff Formulas

dy o = Cr i dt

13

(2.49)

Integrating Eq. (2.49) from (qu

α o )1 β to yo for yo and 0 to t (where t ≤ to) for t gives: o

1 βo

⎛q ⎞ yo = ⎜⎜ u ⎟⎟ ⎝ αo ⎠

+ Cr it

(2.50)

Substituting Eq. (2.50) into Eq. (2.7) gives the equation for the rising phase (rising limb) of the hydrograph for a plane with upstream inflow for t ≤ to:

⎡⎛ q ⎞1 β o ⎤ q = α o ⎢⎜⎜ u ⎟⎟ + Cr it ⎥ ⎢⎣⎝ α o ⎠ ⎥⎦

βo

(2.51)

For a plane with zero upstream inflow (qu = 0), Eq. (2.51) reduces to:

q = α o (C r it )

βo

(2.52)

Figures 2.3 and 2.4 show the rising phase (rising limb) of an equilibrium and a partial equilibrium runoff hydrographs for a plane without and for a plane with upstream inflow, respectively. If the hydrographs in figure 2.4 are shifted by a distance tu to the right, they become the same as those in figure 2.3.

Figure 2.3. Equilibrium and Partial Equilibrium Runoff Hydrographs for a Plane without Upstream Inflow.

Tommy S.W. Wong

14

Figure 2.4. Equilibrium and Partial Equilibrium Runoff Hydrographs for a Plane with Upstream Inflow.

2.12. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eq. (2.1) into Eq. (2.51) gives the equation for the forward characteristic for a plane with upstream inflow:

t=

1

α o1 β

o

⎡ (qu + Cr ixo )1 β o − qu1 β o ⎤ ⎢ ⎥ Cr i ⎣ ⎦

(2.53)

The forward characteristic traces the time it takes for the wave to travel downstream. With the kinematic wave equations (Eqs. 2.4 and 2.6), there is no backward characteristic, and this is why the kinematic wave approximation cannot simulate the backwater effect (Section 2.3). For a plane with zero upstream inflow (qu = 0), Eq. (2.53) reduces to:

⎡ ⎤ xo t=⎢ β o −1 ⎥ ⎣α o (Cr i ) ⎦

1 βo

(2.54)

As shown in figure 2.5, the forward characteristic commences at the upstream end of the overland plane (xo = 0), the time it takes for the wave to travel the length of the plane, Lo, equals to the time of concentration, to (Eqs. 2.35 and 2.36).

Kinematic-Wave Rainfall-Runoff Formulas

15

t to (Eq. 2.36)

t o (Eq. 2.35)

0

Lo

xo

Figure 2.5. Forward Characteristics for Planes without and with Upstream Inflow.

2.13. WATER SURFACE PROFILE – RISING PHASE Figure 2.6 shows the successive water surface profiles during the rising phase for a plane subject to a uniform rainfall intensity only, without upstream inflow. At t = 0, the profile is the line O-A, corresponding to q = 0 and yo = 0. At time interval 0 < t < to, the flow depth increases and the profile becomes the curve O-B-C. This is a partial equilibrium profile corresponding to the partial equilibrium discharge, qp. The length, Lp, contributes to the discharge, qp, which corresponds to the flow depth, yp. Finally, at t ≥ to, the flow depth increases even further and the profile reaches equilibrium. The equilibrium profile is the curve O-B-D. The length, Lo, contributes to the equilibrium discharge, qe, which corresponds to the flow depth, ye. Figure 2.7 shows successive water surface profiles during the rising phase for a plane subject to a uniform rainfall intensity and with a constant upstream inflow. The upstream inflow, qu, which corresponds to the flow depth, yu, is considered to be contributed by an upstream plane of length, Lu. Hence at t = 0, the water surface profile is the curve O-Ou-A. At time interval 0 < t < to, the flow depth increases and the profile becomes the curve O-Ou-B-C. This is a partial equilibrium profile corresponding to the partial equilibrium discharge, qp. The length (Lu + Lp) contributes to the discharge, qp, which corresponds to the flow depth, yp. Finally, at t ≥ to, the flow depth increases even further and the profile reaches equilibrium. The equilibrium profile is the curve O-Ou-B-D. The length (Lu + Lo) contributes to the equilibrium discharge, qe, which corresponds to the flow depth, ye.

16

Tommy S.W. Wong

Figure 2.6. Successive Water Surface Profiles during Rising Phase for a Plane without Upstream Inflow.

Figure 2.7. Successive Water Surface Profiles during Rising Phase for a Plane with Upstream Inflow.

From figure 2.7, it is apparent that the water surface profile (curve O-Ou-B) within the length, (Lu + Lp) is identical to the equilibrium water surface profile (curve O-Ou-B-D). Substituting Eq. (2.1) into Eq. (2.7), gives the equation for the profile between –Lu ≤ xo ≤ Lp:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ q + Cr ixo ⎞ ⎟⎟ yo = ⎜⎜ u αo ⎠ ⎝

17

1 βo

(2.55)

Substituting yo= yp and xo = Lp into Eq. (2.55) gives the equation for the profile between Lp ≤ xo ≤ Lo:

⎛ qu + Cr iL p ⎞ ⎟⎟ y p = ⎜⎜ α o ⎠ ⎝

1 βo

(2.56)

Substituting q = qp and xo = Lp into Eq. (2.1) and rearranging gives the distance Lp:

Lp =

q p − qu

(2.57)

Cr i

If the profiles in figure 2.7 are shifted by a distance Lu to the right, they become the same as those in figure 2.6 which are for a plane with zero upstream inflow (qu = 0). For such a case, Eqs. (2.55)-(2.57) reduce to:

⎛ C ix y o = ⎜⎜ r o ⎝ αo

⎞ ⎟⎟ ⎠

1 βo

(2.58)

which is valid for 0 ≤ xo ≤ Lp,

⎛ C r iL p y p = ⎜⎜ ⎝ αo

⎞ ⎟⎟ ⎠

1 βo

(2.59)

which is valid for Lp ≤ xo ≤ Lo, and

Lp =

qp Cr i

(2.60)

Equation (2.59) can also be derived by substituting yo = yp and xo = Lp into Eq. (2.58).

2.14. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE In figure 2.7, the curve O-Ou-B-C is a partial equilibrium water surface profile corresponding to the partial equilibrium discharge, qp. The duration of the partial equilibrium

Tommy S.W. Wong

18

discharge, td, is the time taken for the water particle to travel from B to C, and is therefore related to the length, Ld, and the kinematic wave celerity, ck , as follows:

td =

Ld ck

(2.61)

As shown in figure 2.7, Ld is related to Lo as follows:

Ld = Lo − L p

(2.62)

Substituting Eq. (2.62) into Eq. (2.61) gives:

td =

Lo − L p

(2.63)

ck

Substituting Eq. (2.57) into Eq. (2.63) gives:

⎛ q p ⎞ ⎛ qu ⎞ ⎟+⎜ ⎟ Lo − ⎜⎜ C r i ⎟⎠ ⎜⎝ C r i ⎟⎠ ⎝ td = ck

(2.64)

Substituting q = qp and yo = yp into Eqs. (2.7) and (2.27) give:

q p = α o y βp o

(2.65)

and

ck = α o β o y p

β o −1

(2.66)

Substituting Eqs. (2.65) and (2.66) into Eq. (2.64) gives:

⎛ α o y βp o ⎞ ⎛ qu ⎞ ⎟+⎜ ⎟ Lo − ⎜ ⎜ C r i ⎟ ⎜⎝ C r i ⎟⎠ ⎠ ⎝ td = α o β o y p βo −1 Substituting yo = y p and t = t r into Eq. (2.50) gives:

(2.67)

Kinematic-Wave Rainfall-Runoff Formulas

19

1 βo

⎛q ⎞ y p = ⎜⎜ u ⎟⎟ ⎝ αo ⎠

+ Cr it r

(2.68)

Substituting Eq. (2.68) into Eq. (2.67) and rearranging gives the equation for the duration of partial equilibrium discharge for a plane with upstream inflow:

⎡⎛ q ⎞1 βo ⎤ Cr iLo + qu − α o ⎢⎜ u ⎟ + Cr itr ⎥ ⎢⎣⎝ α o ⎠ ⎥⎦ td = βo −1 ⎡⎛ q ⎞1 βo ⎤ u α o β oCr i ⎢⎜ ⎟ + Cr itr ⎥ ⎢⎣⎝ α o ⎠ ⎥⎦

βo

(2.69)

For a plane with zero upstream inflow (qu = 0), Eq. (2.69) reduces to:

td =

Lo − α o (C r i )

β o −1

α o β o (C r it r )β

tr

βo

o −1

(2.70)

Substituting Eq. (2.36) into Eq. (2.70) gives td in terms of to:

td =

to

βo

− tr

β otr β

βo

o −1

(2.71)

The duration of partial equilibrium discharge, td, for a plane without and for a plane with upstream inflow are shown in figures 2.3 and 2.4, respectively.

2.15. HYDROGRAPH – EQUILIBRIUM PHASE As shown in figures 2.3 and 2.4, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of rainfall tr. If tr < to, the hydrograph reaches partial equilibrium with a constant discharge qp. If tr ≥ to, the hydrograph reaches equilibrium with a constant discharge qe.

2.15.1. Partial Equilibrium Discharge Substituting t = tr (where tr < to) into Eq. (2.51) gives the equation for the partial equilibrium discharge for a plane with upstream inflow:

Tommy S.W. Wong

20

⎡⎛ q q p = α o ⎢⎜⎜ u ⎢⎣⎝ α o

⎞ ⎟⎟ ⎠

1 βo

⎤ + C r it r ⎥ ⎥⎦

βo

(2.72)

which is valid for tr ≤ t ≤ (tr + td). For a plane with zero upstream inflow (qu = 0), Eq. (2.72) reduces to:

q p = α o (C r it r )

βo

(2.73)

2.15.2. Equilibrium Discharge Substituting t = to into Eq. (2.51) gives the equilibrium discharge qe:

⎡⎛ q q e = α o ⎢⎜⎜ u ⎢⎣⎝ α o

⎞ ⎟⎟ ⎠

1 βo

⎤ + C r it o ⎥ ⎥⎦

βo

(2.74)

which is valid for to ≤ t ≤ tr. Substituting Eq. (2.35) into Eq. (2.74) gives the equation for the equilibrium discharge for a plane with upstream inflow:

qe = qu + Cr iLo

(2.75)

For a plane with zero upstream inflow (qu = 0), Eq. (2.75) reduces to:

q e = C r iLo

(2.76)

2.16. WATER SURFACE PROFILE – EQUILIBRIUM PHASE As shown in figure 2.7, the curve O-Ou-B-D is the equilibrium water surface profile. Substituting Eq. (2.1) into Eq. (2.7) gives the equation for the profile between –Lu ≤ xo ≤ Lo :

⎛ q + Cr ixo ⎞ ⎟⎟ yo = ⎜⎜ u αo ⎠ ⎝

1 βo

(2.77)

Equation (2.77) is identical to Eq. (2.55) because the equilibrium profile and the partial equilibrium profile are identical for -Lu ≤ xo ≤ Lp (figure 2.7). For a plane with zero upstream inflow (qu = 0), Eq. (2.77) reduces to:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ C ix y o = ⎜⎜ r o ⎝ αo

⎞ ⎟⎟ ⎠

21

1 βo

(2.78)

Equation (2.78) is the equation for the curve O-B-D in figure 2.6, which is valid for 0 ≤ xo ≤ Lo.

2.17. EQUILIBRIUM DETENTION STORAGE The amount of water that is detained under the equilibrium condition is known as the equilibrium detention storage (Wong and Li 2000). As the equilibrium detention storage can be evaluated from a water surface profile or from a rising phase of a hydrograph, the general formula for the equilibrium detention storage of an overland plane with upstream inflow is derived using both approaches.

2.17.1. Water Surface Profile Approach Rearranging Eq. (2.3) gives

qu = C r iLu

(2.79)

Substituting Eq. (2.79) into Eq. (2.55) gives:

⎡ C i (L + x o ) ⎤ yo = ⎢ r u ⎥ αo ⎦ ⎣

1 βo

(2.80)

As shown in figure 2.8, integrating Eq. (2.80) from -Lu to Lo for xo gives the equilibrium detention storage, Deuo, for an overland plane of length (Lu + Lo), which is the shaded areas A and B:

Deuo

βo = 1+ βo

⎛ Cr i ⎞ ⎟⎟ ⎜⎜ ⎝ αo ⎠

1/ βo

(Lu + Lo )(1+ β ) / β o

o

(2.81)

Similarly, integrating Eq. (2.80) from -Lu to 0 for xo gives the equilibrium detention storage, Deu, for an overland plane of length Lu, which is the shaded area A in figure 2.8:

βo Deu = 1+ βo

⎛ Cr i ⎞ ⎟⎟ ⎜⎜ α ⎝ o ⎠

1/ βo

Lu

(1+ β o ) / β o

(2.82)

Tommy S.W. Wong

22

The difference between Eqs. (2.81) and (2.82) is the equilibrium detention storage, Deo, for an overland plane of length Lo, which is the shaded area B in figure 2.8:

βo Deo = 1 + βo

⎛ Cr i ⎞ ⎟⎟ ⎜⎜ ⎝ αo ⎠

1 / βo

[(L

u

+ Lo )

(1+ β o ) / β o

− Lu

(1+ β o ) / β o

]

(2.83)

Substituting Eq. (2.3) into Eq. (2.83) gives the equation for the equilibrium detention storage for a plane with upstream inflow (Wong and Li 2000):

Deo =

[

βo (qu + C r iLo )(1+ β 1/ β (1 + β o )α o C r i o

o

) / βo

− qu

(1+ β o ) / β o

]

(2.84)

For a plane with zero upstream inflow (i.e. qu = 0), Eq. (2.84) reduces to:

βo Deo = 1+ βo

⎛ Cr i ⎞ ⎟⎟ ⎜⎜ α o ⎠ ⎝

1 / βo

L(o1+ β o ) / β o

(2.85)

Figure 2.8. Determination of Equilibrium Detention Storage using Water Surface Profile Approach for a Plane with Upstream Inflow.

2.17.2. Hydrograph Approach Similar to the derivation using the water surface profile, the upstream inflow, qu, is considered to be produced by an upstream plane with a time of concentration, tu, subject to

Kinematic-Wave Rainfall-Runoff Formulas

23

rainfall intensity, i. Substituting q = qu and t = tu in Eq. (2.52) gives the upstream inflow, qu, in terms of tu as follows:

qu = α o (C r i t u )

βo

(2.86)

Rearranging Eq. (2.86) gives the time of concentration tu in terms of qu:

tu =

qu1 / β o C r iα o1 / β o

(2.87)

Equation (2.87) can also be derived by substituting Eq. (2.3) into Eq. (2.37). As shown in figure 2.9, integrating (qe - q) from -tu to to for t gives the equilibrium detention storage, Deuo, for an overland plane of length (Lu + Lo), which is the shaded areas A and B:

Deuo = ∫

to

−t u

(qe − q )dt

(2.88)

Substituting Eqs. [(2.2), (2.35), (2.51) and (2.87)] into Eq. (2.88) and integrating gives:

Deuo =

βo 1+ βo

⎛ 1 ⎜ ⎜ C iα 1 / β o ⎝ r o

⎞ ⎟(qu + C r iLo )(1+ β o ) / β o ⎟ ⎠

(2.89)

Similarly, by integrating (qu - q) from -tu to 0 gives the equilibrium detention storage, Deu, for an overland plane of length, Lu, which is the shaded area A in figure 2.9:

Deu = ∫

0

−tu

(qu − q )dt

(2.90)

Substituting Eqs. [(2.51), (2.86) and (2.87)] into Eq. (2.90) and integrating gives:

Deu =

βo 1+ βo

⎛ 1 ⎜ ⎜ C iα 1 / β o ⎝ r o

⎞ (1+ β o ) / β o ⎟qu ⎟ ⎠

(2.91)

The difference between Eqs. (2.89) and (2.91) is the equilibrium detention storage, Deo, for an overland plane of length, Lo, which is Eq. (2.84). It is the shaded area B in figure 2.9.

Tommy S.W. Wong

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Figure 2.9. Determination of Equilibrium Detention Storage using Hydrograph Approach for a Plane with Upstream Inflow.

2.18. WATER SURFACE PROFILE - FALLING PHASE During the falling phase, rainfall ceases (i.e. i = 0 for 0 ≤ xo ≤ Lo), Eq. (2.49) becomes (Henderson and Wooding 1964, Overton and Meadows 1976):

dy o =0 dt

(2.92)

Integrating Eq. (2.92) gives: yo = constant

(2.93)

Equation (2.93) signifies that water flows out at constant depth. The celerity at which the water flows out is governed by the kinematic wave celerity, ck (Eq. 2.27). Figure 2.10 shows the successive water surface profiles during the falling phase for a plane without upstream inflow. Curve O-D is the equilibrium profile at t = tr ≥ to, which is identical to the curve O-BD in figure 2.6. After a time increment at t = tr + ∆t, the profile falls and becomes curve O-C. During the time increment ∆t, the water particle a1 travels a distance ∆xo to a2 at constant flow depth. The distance, ∆xo, between points a1 and a2 can be derived from the kinematic wave celerity, ck. Rearranging Eq. (2.27) gives:

Δx o = α o β o y o

β o −1

Δt

(2.94)

Kinematic-Wave Rainfall-Runoff Formulas

25

The distance between points b1 and b2 is also given by Eq. (2.94). Since the flow depth for the b points are larger than those for the a points, the corresponding wave celerity, ck, is greater, and the corresponding distance ∆xo is therefore longer, as shown in figure 2.10. At t > tr + ∆t, the profile falls further and becomes curve O-B. Finally, at t >> tr + ∆t, when all the water flows out of the plane, the profile falls to the line O-A, which is identical to that in figure 2.6.

Figure 2.10. Successive Water Surface Profiles during Falling Phase for a Plane without Upstream Inflow.

Further, figure 2.11 shows the successive water surface profiles for a plane with a constant upstream inflow during the falling phase. The curve O-Ou-G-D is the equilibrium profile at time tr, which is identical to curve O-Ou-B-D in figure 2.7. If the rainfall stops over the entire length (Lu + Lo), after a time interval ∆t, the water surface profile falls and becomes curve O-E-C. However, since the upstream inflow is constant, the curve O-Ou is fixed. Hence, only the curve Ou-G-D falls. At time t = tr + ∆t, the water surface profile on the plane with a constant upstream inflow is the curve Ou-E-C, and the curve O-E does not exist. At time t > tr + ∆t, the water surface profile falls further and becomes the curve Ou-E-F-B. Finally, at time t >> tr + ∆t, the discharge reduces to the upstream discharge qu. The water surface profile is the line Ou-E-F-A, which is identical to the line Ou-A in figure 2.7.

Tommy S.W. Wong

26

Figure 2.11. Successive Water Surface Profiles during Falling Phase for a Plane with Upstream Inflow.

As shown in figure 2.11, at time tr, the distance xo of any point on the equilibrium profile (curve O-Ou-G-D) can be expressed in terms of flow depth yo by substituting Eq. (2.7) into Eq. (2.1):

xo =

α o y oβ − qu o

(2.95)

Cr i β

Integrating Eq. (2.27) from (αo yo o - qu)/Cri (Eq. 2.95) to xo for xo and from tr to t for t gives the equation for the curve O-E-C:

⎛ α y β o − qu xo = α o β o yoβ o −1 (t − t r ) + ⎜⎜ o o Cr i ⎝

⎞ ⎟⎟ ⎠

(2.96)

For a plane with a constant upstream inflow, Eq. (2.96) is only valid for Lf ≤ xo ≤ Lo, where Lf = length of plane in which the flow is equal to upstream inflow during the falling phase. For the profile between 0 ≤ xo ≤ Lf, it is the line Ou-E, i.e. 1 βo

⎛q ⎞ yo = yu = ⎜⎜ u ⎟⎟ ⎝ αo ⎠

Substituting yo = yu and xo = Lf into Eq. (2.96) gives the equation for Lf in terms of yu:

(2.97)

Kinematic-Wave Rainfall-Runoff Formulas

L f = α o β o yuβ o −1 (t − t r )

27 (2.98)

Substituting Eq. (2.97) into Eq. (2.98) gives the equation for Lf in terms of qu:

L f = α o1 β o β o qu( β o −1) β o (t − t r )

(2.99)

For a plane with zero upstream inflow (qu = 0), Eq. (2.96) reduces to (Wong 2008a):

⎛ α y βo xo = α o β o yoβ o −1 (t − tr ) + ⎜ o o ⎜ Cr i ⎝

⎞ ⎟ ⎟ ⎠

(2.100)

which is valid for 0 ≤ xo ≤ Lo (figure 2.10). Equations (2.96)-(2.100) are only valid for t ≥ tr.

2.18.1. Inflection Line As shown in figure 2.10, the equilibrium water surface profiles (curve O-D) is concave downwards, while the water surface profile at time t > tr + ∆t (curve O-B) is concave upwards. Similarly, in figure 2.11, the curve O-Ou-D is concave downwards, and the curve OF-B is concave upwards. The equation for the inflection line can be derived by first obtaining the second derivative of Eq. (2.95) with respect to yo:

⎡α β (β − 1) β o − 2 ⎤ d 2 xo yo ⎥ = α o β o (β o − 1)(β o − 2) yoβ o −3 (t − t r ) + ⎢ o o o 2 dyo Cr i ⎣ ⎦

(2.101)

Next, by equating Eq. (2.101) to zero, and equating yo = yi results in:

yi = (2 − β o )(t − t r )Cr i

(2.102)

where y i = flow depth of the inflection point. Substituting Eq. (2.102) into Eq. (2.96) and equating xo = xi and yo = yi gives the equation for the inflection line for a plane with upstream inflow:

⎛ 2 ⎞⎛ α o yiβo ⎟⎟⎜⎜ xi = ⎜⎜ − β 2 o ⎠⎝ Cr i ⎝

⎞ ⎛ qu ⎞ ⎟⎟ − ⎜⎜ ⎟⎟ C i ⎠ ⎝ r ⎠

(2.103)

where x i = distance xo of the inflection point. For a plane with zero upstream inflow (qu = 0), Eq. (2.103) reduces to (Wong 2008a):

Tommy S.W. Wong

28

⎛ 2 ⎞⎛ α o yiβo ⎟⎟⎜⎜ xi = ⎜⎜ ⎝ 2 − β o ⎠⎝ Cr i

⎞ ⎟⎟ ⎠

(2.104)

Equations (2.103) and (2.104) have been superimposed respectively onto figures 2.10 and 2.11 as dashed lines which are labeled as an inflection line.

2.19. HYDROGRAPH - FALLING PHASE As shown by Eq. (2.93), during the falling phase, water flows out at constant depth. Hence, the water particle at G flows out to C at constant depth (figure 2.11). The time required for the water particle to flow from G to C is in fact the same as the duration of partial equilibrium discharge, td, as shown in figure 2.4. Substituting td = t – tr and qp = q into Eq. (2.64) gives:

⎛ q − qu Lo − ⎜⎜ ⎝ Cr i t − tr = ck

⎞ ⎟⎟ ⎠

(2.105)

Equation (2.105) may also be derived by integrating Eq. (2.26) from tr to t (where t ≥ tr) for t and from [(q-qu)/Cri] to Lo for xo. Since the discharge on the overland plane cannot be less than upstream discharge, Eq. (2.105) is only valid for q ≥ qu. Substituting Eq. (2.28) into Eq. (2.105) gives the equation for the falling phase (falling limb) of the hydrograph, which is only valid for q ≥ qu:

⎛ q − qu ⎞ ⎟ Lo − ⎜ ⎜ Cr i ⎟ ⎝ ⎠ +t t= r 1 β o [1− (1 β o )] β oα o q

(2.106)

For a plane with zero upstream inflow (qu = 0), Eq. (2.106) reduces to:

⎛ q ⎞ ⎟⎟ Lo − ⎜⎜ C i ⎝ r ⎠ +t t= r 1 β o [1− (1 β o )] β oα o q

(2.107)

Figures 2.3 and 2.4 show the falling phase (falling limb) of an equilibrium and a partial equilibrium runoff hydrograph for a plane without and for a plane with upstream inflow, respectively.

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 29-38

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 3

3. WORKING FORMULAS FOR FLOW ON OVERLAND PLANE In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow on an overland plane are derived. By applying these parameters to the general formulas in Chapter 2, working formulas for the flow depth, flow velocity, average flow velocity, wave celerity, average wave celerity, time of concentration, design discharge, rising and falling phases of hydrograph, forward characteristic, rising, equilibrium and falling phases of water surface profiles, duration of partial equilibrium discharge, and equilibrium detention storage are also derived.

3.1. KINEMATIC WAVE PARAMETERS The Manning’s equation is defined as:

Q=

AR 2 3 S 1 2 n

(3.1)

where Q = discharge, A = cross-sectional flow area, R = hydraulic radius, S = bed slope and n = Manning’s roughness coefficient. The hydraulic radius, R, is related to the flow area, A, as follows:

R=

A P

(3.2)

where P = wetted perimeter. By considering the overland plane as a rectangular channel, the flow area, A, and the wetted perimeter, P, are related to the flow depth, yo, as follows:

A = wy o

(3.3)

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P = w + 2 yo

(3.4)

Substituting Eqs. (3.3) and (3.4) into Eq. (3.2) gives:

R=

wy o w + 2 yo

(3.5)

Since overland flow depth is usually small as compared to the width of the overland plane, Eq. (3.5) reduces to:

R=

yo = yo 2 yo 1+ w

(3.6)

Substituting Eqs. (3.3) and (3.6) and Q = Qo, S = So, n = no into Eq. (3.1) gives:

⎛ S o1 2 Qo = ⎜⎜ ⎝ no

⎞ ⎟⎟ wy o 5 3 ⎠

(3.7)

where Qo = discharge of the overland plane, and no = Manning’s roughness coefficient of the overland surface. Dividing Eq. (3.7) by w gives the discharge per unit width of the overland plane, q:

⎛ S1 2 ⎞ q = ⎜⎜ o ⎟⎟ yo5 3 ⎝ no ⎠

(3.8)

A comparison of Eq. (3.8) with Eq. (2.7) gives the kinematic wave parameters (Chen and Evans 1977):

αo =

S o1 2 no

βo =

5 3

(3.9)

(3.10)

3.2. FLOW DEPTH Rearranging Eq. (3.8) gives the equation for the flow depth for a plane with and without upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

⎛n q⎞ yo = ⎜⎜ o1 2 ⎟⎟ ⎝ So ⎠

31

35

(3.11)

3.3. FLOW VELOCITY Substituting Eqs. (3.9) and (3.10) into Eq. (2.12) gives the equation for the flow velocity along the equilibrium profile for a plane with upstream inflow:

⎛ S1 2 ⎞ v = 0.00238⎜⎜ o ⎟⎟ ⎝ no ⎠

35

(3.6 x10 q 6

u

+ Cr ixo )

25

(3.12)

For a plane with zero upstream inflow (qu = 0), Eq. (3.12) reduces to:

⎛ S1 2 ⎞ v = 0.00238⎜⎜ o ⎟⎟ ⎝ no ⎠

35

(Cr ixo )2 5

(3.13)

3.4. AVERAGE FLOW VELOCITY Substituting Eqs. (3.9) and (3.10) into Eq. (2.15) gives the equation for the average flow velocity for a plane with upstream inflow: 35 ⎤ ⎛ So1 2 ⎞ ⎡ Cr iLo ⎟⎟ ⎢ vav = 0.00143⎜⎜ 35 35⎥ 6 6 ⎝ no ⎠ ⎢⎣ (3.6 × 10 qu + Cr iLo ) − (3.6 × 10 qu ) ⎦⎥

(3.14)

For a plane with zero upstream inflow (qu = 0), Eq. (3.14) reduces to:

⎛ S o1 2 v av = 0.00143⎜⎜ ⎝ no

⎞ ⎟ ⎟ ⎠

35

(C r iLo )2 5

(3.15)

3.5. KINEMATIC WAVE CELERITY Substituting Eqs. (3.9) and (3.10) into Eq. (2.29) gives the equation for the kinematic wave celerity along the equilibrium profile for a plane with upstream inflow:

Tommy S.W. Wong

32

⎛ S1 2 ⎞ ck = 0.00397⎜⎜ o ⎟⎟ ⎝ no ⎠

35

(3.6 ×10 q 6

u

+ Cr ixo )

25

(3.16)

For a plane with zero upstream inflow (qu = 0), Eq. (3.16) reduces to:

⎛ S1 2 ⎞ ck = 0.00397⎜⎜ o ⎟⎟ ⎝ no ⎠

35

(Cr ixo )2 5

(3.17)

3.6. AVERAGE WAVE CELERITY Substituting Eqs. (3.9) and (3.10) into Eq. (2.32) gives the equation for the average wave celerity for a plane with upstream inflow:

⎛ S1 2 ⎞ c av = 0.00238⎜⎜ o ⎟⎟ ⎝ no ⎠

35

⎡ C r iLo ⎢ 35 ⎢ 3.6 × 10 6 q u + C r i Lo − 3.6 × 10 6 q u ⎣

(

)

(

⎤ ⎥ 35 ⎥ ⎦

)

(3.18)

For a plane with zero upstream inflow (qu = 0), Eq. (3.18) reduces to:

⎞ ⎟⎟ ⎠

⎛ S1 2 c av = 0.00238⎜⎜ o ⎝ no

35

(C r iLo )2 5

(3.19)

3.7. TIME OF CONCENTRATION Substituting Eqs. (3.9) and (3.10) into Eq. (2.35) gives the equation for the time of concentration for a plane with upstream inflow:

(

)

(

35 35 ⎛ no ⎞ ⎡ 3.6 ×10 6 qu + Cr iLo − 3.6 ×106 qu ⎜ ⎟ t o = 6.988⎜ 1 2 ⎟ ⎢ Cr i ⎝ S o ⎠ ⎢⎣

)

35

⎤ ⎥ ⎦⎥

(3.20)

For a plane with zero upstream inflow (qu = 0), Eq. (3.20) reduces to (Woolhiser and Liggett 1967):

6.988 ⎛ no Lo ⎞ ⎜ ⎟ to = (Cr i )2 5 ⎜⎝ So1 2 ⎟⎠

35

(3.21)

Kinematic-Wave Rainfall-Runoff Formulas

33

3.8. DESIGN DISCHARGE Substituting Eqs. (3.9) and (3.10) into Eq. (2.43) gives the equation for the design discharge per unit area of the plane for a plane with zero upstream inflow: 5b

Qd

⎤ 5− 2b ⎡ ⎥ ⎢ 1b aC r 1 ⎢ ⎥ Ao = ⎥ ⎢ 35 360 ⎢ 6.988⎛⎜ no ⎞⎟ L3 5 ⎥ o ⎥ ⎜ S1 2 ⎟ ⎢ ⎝ o ⎠ ⎦ ⎣

(

)

(3.22)

3.9. HYDROGRAPH - RISING PHASE Substituting Eqs. (3.9) and (3.10) into Eq. (2.51) gives the equation for the rising phase (rising limb) of a hydrograph for a plane with upstream inflow:

S o1 2 q= no

⎡⎛ n q ⎞ 3 5 Cr it ⎤ ⎢⎜⎜ o1 2u ⎟⎟ + ⎥ 60 × 103 ⎥ ⎢⎣⎝ S o ⎠ ⎦

53

(3.23)

For a plane with zero upstream inflow (qu = 0), Eq. (3.23) reduces to:

q=

S o1 2 ⎛ C r it ⎞ ⎜ ⎟ no ⎝ 60 × 10 3 ⎠

53

(3.24)

Equations (3.23) and (3.24) are valid for t ≤ to.

3.10. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (3.9) and (3.10) into Eq. (2.53) gives the equation for the forward characteristic for a plane with upstream inflow:

⎛ n ⎞ t = 6.988⎜⎜ 1o2 ⎟⎟ ⎝ So ⎠

35

(

)

(

⎡ 3.6 × 10 6 q + C ix 3 5 − 3.6 × 106 q u r o u ⎢ Cr i ⎢⎣

)

35

⎤ ⎥ ⎥⎦

For a plane with zero upstream inflow (qu = 0), Eq. (3.25) reduces to:

(3.25)

Tommy S.W. Wong

34

6.988 ⎛ no xo ⎞ ⎜ ⎟ t= (Cr i )2 5 ⎜⎝ So1 2 ⎟⎠

35

(3.26)

3.11. WATER SURFACE PROFILE - RISING PHASE Substituting Eqs. (3.9) and (3.10) into Eq. (2.55) gives the equation for the rising phase of the water surface profile for a plane with upstream inflow, which is valid for 0 ≤ xo ≤ Lp:

⎡⎛ n ⎞ ⎤ y o = 0.116 × 10 ⎢⎜⎜ 1o2 ⎟⎟ 3.6 × 10 6 qu + C r ix o ⎥ ⎥⎦ ⎣⎢⎝ S o ⎠

(

−3

)

35

(3.27)

Substituting Eqs. (3.7) and (3.8) into Eq. (2.56) gives the equation for the rising phase of the water surface profile for a plane with upstream inflow, which is valid for Lp ≤ xo ≤ Lo.

⎡⎛ n y p = 0.116 × 10 ⎢⎜⎜ 1o2 ⎣⎢⎝ S o −3

⎤ ⎞ ⎟⎟ 3.6 × 10 6 qu + C r iL p ⎥ ⎥⎦ ⎠

(

)

35

(3.28)

From Eq. (2.57), the distance Lp is:

⎛ q p − qu L p = 3.6 × 10 6 ⎜⎜ ⎝ Cr i

⎞ ⎟⎟ ⎠

(3.29)

For a plane with zero upstream inflow (qu = 0), Eqs. (3.27)-(3.29) reduce to:

⎛ n C ix ⎞ yo = 0.116 ×10 ⎜⎜ o 1r 2 o ⎟⎟ ⎝ So ⎠

35

−3

(3.30)

which is valid for 0 ≤ xo ≤ Lp,

⎛ noCr iL p ⎞ ⎟⎟ y p = 0.116 × 10 ⎜⎜ 12 ⎝ So ⎠ −3

35

(3.31)

which is valid for Lp ≤ xo ≤ Lo, and

⎛ qp ⎞ ⎟⎟ L p = 3.6 × 10 6 ⎜⎜ ⎝ Cr i ⎠

(3.32)

Kinematic-Wave Rainfall-Runoff Formulas

35

3.12. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (3.9) and (3.10) into Eq. (2.69) gives the equation for the duration of partial equilibrium discharge for a plane with upstream inflow: 53 35 ⎧ 12 ⎡ ⎤ ⎫ ⎛ ⎞ C iL S n q C it r r ⎪ r o + q − o ⎢⎜ o u ⎟ + ⎥ ⎪ u 12 ⎟ 3 ⎜ ⎪ 3.6 ×106 60 × 10 ⎥ ⎪⎪ no ⎢⎝ S o ⎠ ⎣ ⎦ 3⎪ td = 36 ×10 ⎨ ⎬ 23 35 12 ⎡ ⎤ ⎪ ⎪ ⎛ ⎞ S o Cr i no qu C it ⎢⎜⎜ 1 2 ⎟⎟ + r r 3 ⎥ ⎪ ⎪ 60 ×10 ⎥ no ⎢⎝ S o ⎠ ⎪⎭ ⎪⎩ ⎣ ⎦

(3.33)

For a plane with zero upstream inflow (qu = 0), Eq. (3.33) reduces to: 53 ⎡ Cr iLo S o1 2 ⎛ Cr it r ⎞ ⎤ − ⎜ ⎟ ⎥ ⎢ 6 no ⎝ 60 × 103 ⎠ ⎥ 3 ⎢ 3.6 × 10 t d = 36 ×10 23 ⎥ ⎢ S o1 2Cr i ⎛ Cr it r ⎞ ⎥ ⎢ ⎜ 3 ⎟ no ⎝ 60 × 10 ⎠ ⎥⎦ ⎢⎣

(3.34)

3.13. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 2.3 and 2.4, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of rainfall tr. If tr < to, the hydrograph reaches partial equilibrium with a constant discharge qp. If tr ≥ to, the hydrograph reaches equilibrium with a constant discharge qe.

3.13.1. Partial Equilibrium Discharge Substituting Eqs. (3.9) and (3.10) into Eq. (2.72) gives the equation for the partial equilibrium discharge for a plane with upstream inflow:

S o1 2 qp = no

⎡⎛ n q ⎞3 5 Cr it r ⎤ ⎢⎜⎜ o1 2u ⎟⎟ + ⎥ 60 ×103 ⎥ ⎢⎣⎝ S o ⎠ ⎦

53

(3.35)

For a plane with zero upstream inflow (qu = 0), Eq. (3.35) reduces to:

qp =

S o1 2 ⎛ C r it r ⎞ ⎜ ⎟ no ⎝ 60 × 10 3 ⎠

53

(3.36)

Tommy S.W. Wong

36

Equations (3.35) and (3.36) are valid for tr ≤ t ≤ (tr + td).

3.13.2. Equilibrium Discharge From Eq. (2.75), the equation for the equilibrium discharge for a plane with upstream inflow is:

qe = qu +

Cr iLo 3.6 × 106

(3.37)

For a plane with zero upstream inflow (qu = 0), Eq. (3.37) reduces to:

qe =

C r iLo 3.6 × 10 6

(3.38)

Equations (3.37) and (3.38) are valid for to ≤ t ≤ tr.

3.14. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (3.9) and (3.10) into Eq. (2.77) gives the equation for the equilibrium water surface profile for a plane with upstream inflow between 0 ≤ xo ≤ Lo:

⎡⎛ n ⎞ ⎤ yo = 0.116 × 10 ⎢⎜⎜ 1o2 ⎟⎟ 3.6 × 106 qu + Cr ixo ⎥ ⎣⎝ S o ⎠ ⎦ −3

(

)

35

(3.39)

For a plane with zero upstream inflow (qu = 0), Eq. (3.39) reduces to:

⎛ n C ix ⎞ yo = 0.116 ×10 ⎜⎜ o 1r 2 o ⎟⎟ ⎝ So ⎠

35

−3

(3.40)

which is also valid for 0 ≤ xo ≤ Lo.

3.15. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (3.9) and (3.10) into Eq. (2.84) gives the equation for the equilibrium detention storage for a plane with upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

72.8 × 10 −6 Deo = Cr i

⎛ no ⎞ ⎜⎜ 1 2 ⎟⎟ ⎝ So ⎠

35

[(3.6 × 10 q 6

u

+ C r iLo

)

8/5

(

− 3.6 × 10 6 qu

37

)

8/5

]

(3.41)

For a plane with zero upstream inflow (qu = 0), Eq. (3.41) reduces to:

⎛ n Deo = 72.8 × 10 ⎜⎜ 1o2 ⎝ So −6

⎞ ⎟ ⎟ ⎠

35

(C r i )3 5 Lo 8 / 5

(3.42)

3.16. WATER SURFACE PROFILE - FALLING PHASE Substituting Eqs. (3.9) and (3.10) into Eq. (2.96) gives the equation for the falling phase of the water surface profile for a plane with upstream inflow, which is valid for Lf ≤ xo ≤ Lo:

⎡ ⎛ ⎛ So1 2 yo 5 3 ⎞ ⎜⎜ ⎟ − qu ⎢ ⎜ ⎜ no ⎟⎠ ⎛ S o1 2 yo 2 3 ⎞ ⎢ 6 ⎝ ⎟(t − t r ) + 3.6 ×10 ⎜ xo = 100.0⎜⎜ ⎢ ⎟ n Cr i o ⎝ ⎠ ⎜ ⎢ ⎜ ⎢⎣ ⎝

⎞⎤ ⎟⎥ ⎟⎥ ⎟⎥ ⎟⎥ ⎟⎥ ⎠⎦

(3.43)

From Eq. (2.97), the equation for the profile between 0 ≤ xo ≤ Lf, is:

⎡⎛ n ⎞ ⎤ yo = 0.116 × 10 ⎢⎜⎜ 1o2 ⎟⎟ 3.6 × 106 qu ⎥ ⎣⎝ So ⎠ ⎦

(

−3

)

35

(3.44)

Substituting Eqs. (3.9) and (3.10) into Eq. (2.99) gives the equation for the distance Lf for a plane with upstream inflow, which is valid for t ≥ tr:

⎛ S1 2 ⎞ L f = 0.238⎜⎜ o ⎟⎟ ⎝ no ⎠

35

(3.6 ×10 q ) (t − t ) 25

6

u

r

(3.45)

For a plane with zero upstream inflow (qu = 0), Eq. (3.43) reduces to:

⎛ S o1 2 yo 2 3 ⎞ ⎡ ⎛ y ⎞⎤ ⎟ ⎢100(t − t r ) + 3.6 × 10 6 ⎜ o ⎟⎥ xo = ⎜⎜ ⎜C i⎟ ⎟ ⎝ r ⎠⎦ ⎝ no ⎠ ⎣ which is valid for 0 ≤ xo ≤ Lo, and t ≥ tr.

(3.46)

Tommy S.W. Wong

38

3.17. HYDROGRAPH - FALLING PHASE Substituting Eqs. (3.9) and (3.10) into Eq. (2.106) gives the equation for the falling phase (falling limb) of a hydrograph for a plane with upstream inflow.

⎛ n ⎞ ⎡ C iL − 3.6 × 106 (q − qu ) ⎤ t = 0.0100⎜⎜ 1o2 ⎟⎟ ⎢ r o ⎥ + tr Cr iq 2 5 ⎦ ⎝ So ⎠ ⎣ 35

(3.47)

For a plane with zero upstream inflow (qu = 0), Eq. (3.47) reduces to:

⎛ n ⎞ t = 0.0100⎜⎜ 1o2 ⎟⎟ ⎝ So ⎠

35

⎛ Cr iLo − 3.6 × 106 q ⎞ ⎜⎜ ⎟⎟ + t r Cr iq 2 5 ⎝ ⎠

(3.48)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 39-62

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 4

4. GENERAL FORMULAS FOR FLOW IN OPEN CHANNEL In this Chapter, based on the kinematic wave theory, the general formulas for flow area; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of flow area profiles; duration of partial equilibrium discharge; and equilibrium detention storage for flow in an open channel are derived.

4.1. FLOW CONDITIONS Consider an open channel of length Lc, subject to a uniformly distributed lateral inflow qL, and with a constant upstream inflow Qu, the discharge in channel, Qc, along the equilibrium water surface profile of the channel is:

Qc = Qu + q L x c

(4.1)

where xc = distance along the channel in the direction of flow. Substituting xc = Lc into Eq. (4.1) gives the discharge Qe at the end of the channel at equilibrium, i.e.

Qe = Qu + q L Lc

(4.2)

Further, the upstream inflow, Qu, can be considered to be contributed by an imaginary channel of length Lu, which is situated immediately upstream of the channel of length Lc. This imaginary upstream channel is also subject to a uniformly distributed lateral inflow qL, but with zero upstream inflow. At the outlet point of this upstream channel and at equilibrium, the discharge is Qu. Substituting Qe = Qu, Qu = 0 and Lc = Lu into Eq. (4.2) gives the length of the upstream channel, Lu, in terms of the upstream inflow, Qu, as follows:

Lu =

Qu qL

(4.3)

Tommy S.W. Wong

40

Figure 4.1 shows the upstream channel, the open channel, and the equilibrium water surface profile.

Figure 4.1. Upstream and Open Channels with Equilibrium Water Surface Profile.

4.2. DYNAMIC WAVE EQUATIONS The mechanics of unsteady open channel flow with a lateral inflow contribution can be expressed mathematically by the Saint Venant equation. Derived from the principles of continuity and momentum, the equations are (Chow et al 1988):

∂Ac ∂Qc + = qL ∂t ∂xc

(4.4)

1 ∂Qc 1 ∂ ⎛ Qc2 ⎞ ∂y c ⎜ ⎟+ + − (S c − S f ) = 0 gAc ∂t gAc ∂x c ⎜⎝ Ac ⎟⎠ ∂x c

(4.5)

where Ac = channel flow area, t = time, g = acceleration due to gravity, Sc = channel bed slope and Sf = friction slope. The assumptions inherent in Eqs. (4.4) and (4.5) are (DeVries and MacArthur 1979): 1. The flow is one dimensional (i.e. velocity varies in the longitudinal direction only). This implies that the velocity is constant and the water surface is horizontal across any section perpendicular to the longitudinal axis. 2. All flows are gradually varied with hydrostatic pressure prevailing at all points in the flow such that all vertical acceleration within the water column can be neglected.

Kinematic-Wave Rainfall-Runoff Formulas

41

3. The longitudinal axis of the channel can be approximated by a straight line (i.e. there is no secondary circulation). 4. The slope of the channel bed is small. 5. The bed of the channel is fixed (i.e. the effects of scour and deposition are negligible). 6. Resistance to flow can be described by empirical resistance formulas, such as Manning’s equation. 7. The fluid is incompressible and homogeneous in density. 8. The momentum carried to the fluid from the lateral inflow is negligible. The momentum equation (Eq. 4.5) consists of five terms, namely local acceleration, convective acceleration, pressure force, gravity force and friction force, each representing a physical process that governs the flow momentum described as follows: 1. The acceleration terms represent the effect of velocity change over time and space. 2. The pressure force term represents the effect of flow depth change. 3. The gravity force term Sc is proportional to the channel bed slope and accounts for the change in bed level. 4. The friction force term Sf is proportional to the friction slope and accounts for the friction loss for the flow in an open channel.

4.3. KINEMATIC WAVE EQUATIONS If the backwater effect is negligible and there is no rapid change in flow, the acceleration and pressure terms in Eq. (4.5) may be neglected (Stephenson 1981, Wong 1992), and the momentum equation reduces to: Sc = Sf

(4.6)

Equations (4.4) and (4.6) are called the “kinematic wave equations”. Equation (4.6) shows that the channel bed slope is parallel to the friction slope, which means that the kinematic wave is under the uniform flow condition. Hence, Eq. (4.6) can be replaced by the general uniform flow equation, which is:

Qc = α c Acβ c

(4.7)

where αc and βc = kinematic wave parameters relating Qc to Ac.

4.4. FLOW AREA Rearranging Eq. (4.7) gives the equation for the flow area, Ac , in terms of the discharge, Qc , as follows:

Tommy S.W. Wong

42 1/ β c

⎛Q ⎞ Ac = ⎜⎜ c ⎟⎟ ⎝ αc ⎠

(4.8)

4.5. FLOW VELOCITY From continuity, the flow velocity, v, is related to the channel discharge, Qc, as follows:

v=

Qc Ac

(4.9)

Substituting Eq. (4.7) into Eq. (4.9), the velocity, v, becomes:

v = α c Ac

β c −1

(4.10)

Substituting Eq. (4.8) into Eq. (4.10) gives the equation for the velocity, v, in terms of channel discharge, Qc:

(

v = α c Qcβ c −1

)

1 βc

(4.11)

Substituting Eq. (4.1) into Eq. (4.11) gives the equation for the flow velocity along the equilibrium profile for a channel with upstream inflow:

[

v = α c (Qu + q L x c )

]

β c −1 1 β c

(4.12)

For a channel with zero upstream inflow (Qu = 0), Eq. (4.12) reduces to:

[

v = α c (q L x c )

]

β c −1 1 β c

(4.13)

4.6. AVERAGE FLOW VELOCITY Further, the average flow velocity, vav, over the length of the open channel, Lc, can be derived as follows:

vav =

Lc Lc

1 ∫0 v dxc

(4.14)

Kinematic-Wave Rainfall-Runoff Formulas

43

Substituting Eq. (4.12) into Eq. (4.14) and integrating (1/v) gives the equation for the average flow velocity for a channel with upstream inflow:

v av =

α c 1 β q L Lc c

[

β c (Qu + q L Lc )1 β − Qu 1 β c

c

]

(4.15)

For a channel with zero upstream inflow (Qu = 0), Eq. (4.15) reduces to:

vav =

α c 1 β qL Lc β c (qL Lc )1 β

[

c

]

c

(4.16)

4.7. KINEMATIC WAVE CELERITY Differentiating Eq. (4.7) with respect to t gives:

∂Qc β −1 ⎛ ∂A ⎞ = α c β c Ac c ⎜ c ⎟ ∂t ⎝ ∂t ⎠

(4.17)

Rearranging Eq. (4.17) gives:

∂Ac 1 ⎛ ∂Qc ⎞ = ⎜ ⎟ ∂t α c β c Ac β c −1 ⎝ ∂t ⎠

(4.18)

Substituting Eq. (4.18) into Eq. (4.4), the continuity equation becomes:

⎡ 1 ⎛ ∂Qc ⎢ β −1 ⎜ ⎢⎣ α c β c Ac c ⎝ ∂t

⎞⎤ ⎛ ∂Qc ⎟⎥ + ⎜⎜ ⎠⎥⎦ ⎝ ∂xc

⎞ ⎟⎟ = q L ⎠

(4.19)

Kinematic waves result in changes in Qc which is dependent on both xc and t, and the increment in flow rate dQc can be written as:

dQc =

∂Qc ∂Q dt + c dx c ∂t ∂xc

(4.20)

Dividing Eq. (4.20) by dxc:

dQc ⎛ ∂Qc =⎜ dxc ⎜⎝ ∂xc

⎞ ⎡ ∂Qc ⎟⎟ + ⎢ ⎠ ⎣ ∂t

⎛ dt ⎜⎜ ⎝ dxc

⎞⎤ ⎟⎟⎥ ⎠⎦

(4.21)

Tommy S.W. Wong

44 If

qL =

dQc dx c

(4.22)

and

1 dt = dxc α c β c Ac β c −1

(4.23)

then Eq. (4.19) and Eq. (4.21) are identical. Differentiating Eq. (4.7) with respect to Ac:

dQc β −1 = α c β c Ac c dAc

(4.24)

Comparing Eq. (4.23) and Eq. (4.24) gives:

dQc dxc = dAc dt

(4.25)

Since kinematic wave celerity, ck, is:

ck =

dx c dt

(4.26)

Substituting Eq. (4.25) into Eq. (4.26) gives:

ck =

dx c dQc β −1 = = α c β c Ac c dt dAc

(4.27)

Substituting Eq. (4.8) into Eq. (4.27) gives:

(

c k = β c α c Qcβ c −1

)

1 / βc

(4.28)

Substituting Eq. (4.1) into Eq. (4.28) gives the equation for the wave celerity along the equilibrium profile for a channel with upstream inflow:

[

c k = β c α c (Qu + q L xc )

]

β c −1 1 / β c

For a channel with zero upstream inflow (Qu = 0), Eq. (4.29) reduces to:

(4.29)

Kinematic-Wave Rainfall-Runoff Formulas

[

c k = β c α c (q L x c )

]

β c −1 1 / β c

45 (4.30)

4.8. AVERAGE WAVE CELERITY The average wave celerity, cav, over the channel length, Lc, can be derived as follows:

cav =

Lc

(4.31)

Lc

1 ∫0 ck dxc

Substituting Eq. (4.29) into Eq. (4.31) and integrating (1/ck) gives the equation for the average wave celerity for a channel with upstream inflow:

cav =

α c1 / β q L Lc c

(Qu + qL Lc )1 / β

c

− Qu

1/ βc

(4.32)

For a channel with zero upstream inflow (Qu = 0), Eq. (4.32) reduces to:

[

c av = α c (q L Lc )

]

β c −1 1 / β c

(4.33)

4.9. TIME OF TRAVEL The time of travel in channel, tt, can be obtained by dividing channel length, Lc, by the average wave celerity, cav, as follows:

tt =

Lc cav

(4.34)

Substituting Eq. (4.32) into Eq. (4.34) gives the equation for the time of travel for a channel with upstream inflow, (Wong 2001):

tt =

1

α c1 / β

c

⎡ (Qu + q L Lc )1 / β c − Qu 1 / β c ⎤ ⎢ ⎥ qL ⎣⎢ ⎦⎥

(4.35)

For a channel with zero upstream inflow (Qu = 0), Eq. (4.35) reduces to, (Wooding 1965, Wong and Chen 1989):

Tommy S.W. Wong

46 1/ β c

⎛ Lc ⎞ ⎟ tt = ⎜⎜ β c −1 ⎟ ⎝ α c qL ⎠

(4.36)

Further, for the upstream channel of length Lu, substituting Lc = Lu into Eq. (4.36) gives the time of travel, tu, of the upstream channel: 1/ β c

⎛ Lu ⎞ ⎟ tu = ⎜⎜ β c −1 ⎟ ⎝ α c qL ⎠

(4.37)

4.10. HYDROGRAPH - RISING PHASE Expanding the partial derivative, (∂Qc ∂x c ) , into total derivative results in:

∂Qc dQc = ∂xc dAc

⎛ ∂Ac ⎜⎜ ⎝ ∂xc

⎞ ⎟⎟ ⎠

(4.38)

Substituting Eq. (4.27) into Eq. (4.38) gives:

∂A ∂Qc = ck c ∂xc ∂xc

(4.39)

Substituting Eq. (4.39) into Eq. (4.4) gives:

∂A ∂Ac + ck c = q L ∂xc ∂t

(4.40)

Differentiating Ac with respect to xc and t:

dAc =

∂A ∂Ac dt + c dxc ∂xc ∂t

(4.41)

Dividing Eq. (4.41) by dt and substituting Eq. (4.27) into it:

dAc ∂Ac ∂A = + ck c dt ∂t ∂xc Comparing Eqs. (4.40) and (4.42) gives:

(4.42)

Kinematic-Wave Rainfall-Runoff Formulas

dAc = qL dt

(4.43)

Integrating Eq. (4.43) from (Qu

⎛Q Ac = ⎜⎜ u ⎝ αc

47

α c )1 / β to Ac for Ac and 0 to t (where t ≤ tt) for t gives: c

1/ βc

⎞ ⎟⎟ ⎠

+ qLt

(4.44)

Substituting Eq. (4.44) into Eq. (4.7) gives the equation for the rising phase (rising limb) of the hydrograph for a channel with upstream inflow for t ≤ tt:

⎡⎛ Q Qc = α c ⎢⎜⎜ u ⎢⎣⎝ α c

1/ βc

⎞ ⎟⎟ ⎠

⎤ + q Lt ⎥ ⎥⎦

βc

(4.45)

For a channel with zero upstream inflow (Qu = 0), Eq. (4.45) reduces to:

Qc = α c ( q L t ) β c

(4.46)

Figures 4.2 and 4.3 show the rising phase (rising limb) of an equilibrium and a partial equilibrium runoff hydrographs for a channel without and for a channel with upstream inflow, respectively. If the hydrographs in figure 4.3 are shifted by a distance tu to the right, they become the same as those in figure 4.2.

Figure 4.2. Equilibrium and Partial Equilibrium Runoff Hydrographs for a Channel without Upstream Inflow.

Tommy S.W. Wong

48

Figure 4.3. Equilibrium and Partial Equilibrium Runoff Hydrographs for a Channel with Upstream Inflow.

4.11. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eq. (4.1) into Eq. (4.45) gives the equation for the forward characteristic for a channel with upstream inflow:

t=

1

α c1 / β

c

⎡ (Qu + q L xc )1 / β c − Qu 1 / β c ⎤ ⎢ ⎥ qL ⎣⎢ ⎦⎥

(4.47)

The forward characteristic traces the time it takes for the wave to travel downstream. With the kinematic wave equations (Eqs. 4.4 and 4.6), there is no backward characteristic, and this is why the kinematic wave approximation cannot simulate the backwater effect (Section 4.3). For a channel with zero upstream inflow (Qu = 0), Eq. (4.47) reduces to: 1/ β c

⎛ xc ⎞ ⎟ t = ⎜⎜ β c−1 ⎟ α q ⎝ c L ⎠

(4.48)

As shown in figure 4.4, the forward characteristic commences at the upstream end of the channel (xc = 0), the time it takes for the wave to travel the length of the channel, Lc, equals to the time of travel in channel, tt (Eqs. 4.35 and 4.36).

Kinematic-Wave Rainfall-Runoff Formulas

49

t t t (Eq. 4.36)

t t (Eq. 4.35)

0

Lc

xc

Figure 4.4. Forward Characteristics for Channels without and with Upstream Inflow.

4.12. FLOW AREA PROFILE - RISING PHASE Figure 4.5 shows the successive flow area profiles during the rising phase for a channel subject to a uniform lateral inflow only, without upstream inflow. At t = 0, the profile is the line O-A, corresponding to Qc = 0 and Ac = 0. At time interval 0 < t < tt, the flow depth increases and the profile becomes the curve O-B-C. This is a partial equilibrium profile corresponding to the partial equilibrium discharge, Qp. The length, Lp, contributes to the discharge, Qp, which corresponds to the flow area, Ap. Finally, at t ≥ tt, the flow area increases even further and the profile reaches equilibrium. The equilibrium profile is the curve O-B-D. The length, Lc, contributes to the equilibrium discharge, Qe, which corresponds to the flow area, Ae.

Figure 4.5. Successive Flow Area Profiles during Rising Phase for a Channel without Upstream Inflow.

Tommy S.W. Wong

50

Figure 4.6 shows successive flow area profiles during the rising phase for a channel subject to a uniform lateral inflow and with a constant upstream inflow. The upstream inflow, Qu, which corresponds to the flow area, Au, is considered to be contributed by an upstream channel of length, Lu. Hence at t = 0, the water area profile is the curve O-Ou-A. At time interval 0 < t < tt, the flow area increases and the profile becomes the curve O-Ou-B-C. This is a partial equilibrium profile corresponding to the partial equilibrium discharge, Qp. The length (Lu + Lp) contributes to the discharge, Qp, which corresponds to the flow area, Ap. Finally, at t ≥ tt, the flow area increases even further and the profile reaches equilibrium. The equilibrium profile is the curve O-Ou-B-D. The length (Lu + Lc) contributes to the equilibrium discharge, Qe, which corresponds to the flow area, Ae. From figure 4.6, it is apparent that the flow area profile (curve O-Ou-B) within the length, (Lu + Lp), is identical to the equilibrium flow area profile (curve O-Ou-B-D). Substituting Eq. (4.1) into Eq. (4.7) gives the equation for the profile between –Lu ≤ xc ≤ Lp: 1/ β c

⎛ Q + q L xc ⎞ ⎟⎟ Ac = ⎜⎜ u α c ⎝ ⎠

(4.49)

Substituting Ac = Ap and xc = Lp into Eq. (4.49) gives the equation for the profile between Lp ≤ xc ≤ Lc: 1/ β c

⎛ Qu + q L L p ⎞ ⎟⎟ Ap = ⎜⎜ αc ⎝ ⎠

(4.50)

Substituting Qc = Qp and xc = Lp into Eq. (4.1) gives the distance Lp:

Lp =

Q p − Qu

(4.51)

qL

If the profiles in figure 4.6 are shifted by a distance, Lu, to the right, they become the same as those in figure 4.5 which are for a channel with zero upstream inflow (Qu = 0). For such a case, Eqs. (4.49)-(4.51) reduce to:

⎛q x Ac = ⎜⎜ L c ⎝ αc

⎞ ⎟⎟ ⎠

1 / βc

(4.52)

which is valid for 0 ≤ xc ≤ Lp,

⎛ qL Lp A p = ⎜⎜ ⎝ αc

⎞ ⎟⎟ ⎠

1 / βc

(4.53)

Kinematic-Wave Rainfall-Runoff Formulas

51

which is valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp qL

(4.54)

Equation (4.53) can also be derived by substituting Ac = Ap and xc = Lp into Eq. (4.52).

Figure 4.6. Successive Flow Area Profiles during Rising Phase for a Channel with Upstream Inflow.

4.13. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE In figure 4.6, the curve O-Ou-B-C is the partial equilibrium flow area profile corresponding to the partial equilibrium discharge, Qp. The duration of the partial equilibrium discharge, td, is the time taken for the water particle to travel from B to C, and is therefore related to the length, Ld, and the kinematic wave celerity, ck, as follows:

td =

Ld ck

(4.55)

As shown in figure 4.6, Ld is related to Lc as follows:

Ld = Lc − L p Substituting Eq. (4.56) into Eq. (4.55) gives:

(4.56)

Tommy S.W. Wong

52

td =

Lc − L p

(4.57)

ck

Substituting Eq. (4.51) into Eq. (4.57) gives:

⎛ Q p ⎞ ⎛ Qu ⎟+⎜ Lc − ⎜⎜ q L ⎟⎠ ⎜⎝ q L ⎝ td = ck

⎞ ⎟⎟ ⎠

(4.58)

Substituting Qc = Qp and Ac = Ap into Eqs. (4.7) and (4.27) gives:

Q p = α c A pβ c

(4.59)

and

ck = α c β c Ap

β c −1

(4.60)

Substituting Eqs. (4.59) and (4.60) into Eq. (4.58) gives:

⎛ α c A pβ c ⎞ ⎛ Qu ⎟+⎜ Lc − ⎜ ⎜ q L ⎟ ⎜⎝ q L ⎠ ⎝ td = β c −1 αβ Ap

⎞ ⎟⎟ ⎠

(4.61)

Substituting Ac = Ap and t = tq into Eq. (4.44) gives: 1 βc

⎛Q ⎞ Ap = ⎜⎜ u ⎟⎟ ⎝ αc ⎠

+ qL tq

(4.62)

where tq = duration of lateral inflow. Substituting Eq. (4.62) into Eq. (4.61) and rearranging gives the duration of partial equilibrium discharge for a channel with upstream inflow:

⎡⎛ Q ⎞1/ β c ⎤ Lc qL + Qu − α c ⎢⎜⎜ u ⎟⎟ + qL t q ⎥ ⎢⎣⎝ α c ⎠ ⎥⎦ td = β − 1 c ⎡⎛ Q ⎞1/ β c ⎤ u α c β c qL ⎢⎜⎜ ⎟⎟ + q Lt q ⎥ ⎢⎣⎝ α c ⎠ ⎥⎦

βc

(4.63)

Kinematic-Wave Rainfall-Runoff Formulas

53

For a channel with zero upstream inflow (Qu = 0), Eq. (4.63) reduces to:

td =

Lc − α c qL

β c −1

tq

α c β c (q Lt q )β

βc

(4.64)

c −1

Substituting Eq. (4.36) into Eq. (4.64) gives td in terms of tt: β

td =

tt c − t q

β ct q β

βc

(4.65)

c −1

The duration of partial equilibrium discharge, td, for a channel without and for a channel with upstream inflow are shown in figures 4.2 and 4.3, respectively.

4.14. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium, or equilibrium depending on the duration of lateral inflow, tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge, Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge, Qe.

4.14.1. Partial Equilibrium Discharge Substituting t = tq (where tq < tt) into Eq. (4.45) gives the equation for the partial equilibrium discharge for a plane with upstream inflow:

⎡⎛ Q Q p = α c ⎢⎜⎜ u ⎢⎣⎝ α c

⎞ ⎟⎟ ⎠

1 / βc

⎤ + qLtq ⎥ ⎥⎦

βc

(4.66)

which is valid for tq ≤ t ≤ (tq + td). For a channel with zero upstream inflow (Qu = 0), Eq. (4.66) reduces to:

Q p = α c ( q L t q ) βc

4.14.2. Equilibrium Discharge Substituting t = tt into Eq. (4.45) gives the equilibrium discharge Qe:

(4.67)

Tommy S.W. Wong

54

⎡⎛ Q Qe = α c ⎢⎜⎜ u ⎢⎣⎝ α c

⎞ ⎟⎟ ⎠

1 / βc

⎤ + qL tt ⎥ ⎥⎦

βc

(4.68)

which is valid for tt ≤ t ≤ tq. Substituting Eq. (4.35) into Eq. (4.68) gives the equation for the equilibrium discharge for a plane with upstream inflow:

Qe = Qu + q L Lc

(4.69)

For a channel with zero upstream inflow (Qu = 0), Eq. (4.69) reduces to:

Q e = q L Lc

(4.70)

4.15. FLOW AREA PROFILE - EQUILIBRIUM PHASE As shown in figure 4.6, the curve O-Ou-B-D is the equilibrium water surface profile. Substituting Eq. (4.1) into Eq. (4.7) gives the equation for the profile between -Lu ≤ xc ≤ Lc : 1/ β c

⎛ Q + q L xc ⎞ ⎟⎟ Ac = ⎜⎜ u αc ⎝ ⎠

(4.71)

Equation (4.70) is identical to Eq. (4.49) because the equilibrium profile and the partial equilibrium profile are identical for -Lu ≤ xc ≤ Lp (figure 4.6). For a channel with zero upstream inflow (Qu = 0), Eq. (4.71) reduces to:

⎛q x Ac = ⎜⎜ L c ⎝ αc

⎞ ⎟⎟ ⎠

1 / βc

(4.72)

Equation (4.71) is the equation for the curve O-B-D in figure 4.5, which is valid for 0 ≤ xc ≤ Lc.

4.16. EQUILIBRIUM DETENTION STORAGE The amount of water that is detained under the equilibrium condition is known as the equilibrium detention storage (Wong and Li 2000). As the equilibrium detention storage can be evaluated from a flow area profile or from a rising phase of a hydrograph, the general formula for the equilibrium detention storage of a channel with upstream inflow is derived using both approaches.

Kinematic-Wave Rainfall-Runoff Formulas

55

4.16.1. Flow Area Profile Approach Rearranging Eq. (4.3) gives:

Qu = q L Lu

(4.73)

Substituting Eq. (4.73) into Eq. (4.49) gives:

⎡ q (L + x c ) ⎤ Ac = ⎢ L u ⎥ αc ⎣ ⎦

1/ βc

(4.74)

As shown in figure 4.7, integrating Eq. (4.74) from -Lu to Lc for xc gives the equilibrium detention storage for an open channel of length (Lu + Lc), which is the shaded areas A and B:

Deuc

β c ⎛ qL ⎞ ⎜ ⎟ = 1 + β c ⎜⎝ α c ⎟⎠

1 / βc

(Lu + Lc )(1+ β ) / β c

(4.75)

c

Similarly, integrating Eq. (4.73) from –Lu to 0 for xc gives the equilibrium detention storage for an open channel of length, Lu, which is the shaded area A in figure 4.7:

β c ⎛ qL ⎞ ⎜ ⎟ Deu = 1 + β c ⎜⎝ α c ⎟⎠

1 / βc

Lu

(1+ β c ) / β c

(4.76)

The difference between Eqs. (4.75) and (4.76) is the equilibrium detention storage for an open channel of length, Lc, which is the shaded area B in figure 4.7:

β c ⎛ qL ⎞ ⎜ ⎟ Dec = 1 + β c ⎜⎝ α c ⎟⎠

1 / βc

[(L

u

+ Lc )

(1+ β c ) / β c

− Lu

(1+ β c ) / β c

]

(4.77)

Substituting Eq. (4.3) into Eq. (4.77) gives the equation for the equilibrium detention storage for a channel with upstream inflow:

Dec =

⎡ (Qu + qL Lc )(1+ β ) / β − Qu (1+ β βc (1 + β c )α c1 / β ⎢⎣ qL c

c

c

c

)/ βc

⎤ ⎥ ⎦

(4.78)

For a channel with zero upstream inflow (i.e. Qu = 0), Eq. (4.78) reduces to:

βc Dec = 1+ βc

⎛ qL ⎜⎜ ⎝αc

⎞ ⎟⎟ ⎠

1/ βc

L(c1+ β c ) / β c

(4.79)

Tommy S.W. Wong

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Figure 4.7. Determination of Equilibrium Detention Storage using Flow Area Profile Approach for a Channel with Upstream Inflow.

4.16.2. Hydrograph Approach Similar to the derivation using the water surface profile, the upstream inflow, Qu, is considered to be produced by an upstream channel with time of travel, tt, subject to a uniform lateral inflow into the channel, qL. The upstream inflow, Qu, is then related to tt as follows:

Qu = α c (q L t t )

βc

(4.80)

Substituting Eq. (4.3) into Eq. (4.37):

tu =

Qu1 / β c q Lα c1 / β c

(4.81)

At equilibrium (xc = Lc), Eq. (4.1) becomes:

Qe = Qu + q L Lc

(4.82)

where Qe = equilibrium channel discharge. As shown in figure 4.8, integrating (Qe - Qc) from -tu to tt for t gives the equilibrium detention storage for an open channel of length (Lu + Lc), which is the shaded areas A and B:

Kinematic-Wave Rainfall-Runoff Formulas

Deuc = ∫

tt

− tu

(Qe − Qc )dt

57

(4.83)

Substituting Eqs. [(4.35), (4.45), (4.81) and (4.82)] into Eq. (4.83) and integrating gives:

Deuc =

βc 1 + βc

⎛ 1 ⎜⎜ 1 / βc ⎝ q Lα c

⎞ ⎟⎟(Qu + q L Lc )(1+ β c ) / β c ⎠

(4.84)

Similarly, by integrating (Qu - Qc) from -tu to 0 gives the equilibrium detention storage for an open channel of length Lu, which is the shaded area A in figure 4.8:

Deu = ∫

0

−tu

(Qu − Qc )dt

(4.85)

Substituting Eqs. [(4.45), (4.79) and (4.80)] into Eq. (4.85) and integrating gives:

Deu =

βc 1 + βc

⎛ 1 ⎜ ⎜ q α 1/ βc ⎝ L c

⎞ (1+ β c ) / β c ⎟Qu ⎟ ⎠

(4.86)

The difference between Eqs. (4.84) and (4.86) is the equilibrium detention storage, Dec for an open channel of length Lc, which is Eq. (4.78). It is the shaded area B in figure 4.8.

Figure 4.8. Determination of Equilibrium Detention Storage using Hydrograph Approach for a Channel with Upstream Inflow.

Tommy S.W. Wong

58

4.17. FLOW AREA PROFILE - FALLING PHASE During the falling phase, lateral inflow ceases (i.e. qL = 0 for 0 ≤ xc ≤ Lc), Eq. (4.43) becomes:

dAc =0 dt

(4.87)

Integrating Eq. (4.87) gives: Ac = constant

(4.88)

Equation (4.88) signifies that water flows out at constant flow area. The celerity at which the water flows out is governed by the kinematic wave celerity, ck (Eq. 4.27). Figure 4.9 shows the successive flow area profiles during the falling phase for a channel without upstream inflow. Curve O-D is the equilibrium profile at t = tq ≥ tt, which is identical to the curve O-BD in figure 2.6. After a time increment at t = tq + ∆t, the profile falls and becomes curve O-C. During the time increment ∆t, the water particle a1 travels a distance ∆xo to a2 at constant flow area. The distance, ∆xo, between points a1 and a2, can be derived from the kinematic wave celerity, ck. Rearranging Eq. (4.27) gives:

Δxc = α c β c Ac

β c −1

Δt

(4.89)

The distance between points b1 and b2 is also given by Eq. (4.89). Since the flow area for the b points are larger than those for the a points, the corresponding wave celerity, ck, is greater, and the corresponding distance ∆xo is therefore longer, as shown in figure 4.9. At t > tq + ∆t, the profile falls further and becomes curve O-B. Finally, at t >> tq + ∆t, when all the water flows out of the channel, the profile falls to the line O-A, which is identical to that in figure 4.5. Further, figure 4.10 shows the successive flow area profiles for a channel with a constant upstream inflow during the falling phase. The curve O-Ou-G-D is the equilibrium profile at time tr, which is identical to curve O-Ou-B-D in figure 4.6. If the lateral inflow stops over the entire length (Lu + Lo), after a time interval ∆t, the flow area profile falls and becomes curve O-E-C. However, since the upstream inflow is constant, the curve O-Ou is fixed. Hence, only the curve, Ou-G-D, falls. At time t = tq + ∆t, the flow area profile on the channel with a constant upstream inflow is the curve Ou-E-C, and the curve O-E does not exist. At time t > tq + ∆t, the flow area profile falls further and becomes the curve Ou-E-F-B. Finally, at time t >> tq + ∆t, the discharge reduces to the upstream discharge, Qu. The flow area profile is the line Ou-E-F-A, which is identical to the line Ou-A in figure 4.6.

Kinematic-Wave Rainfall-Runoff Formulas

59

Figure 4.9. Successive Flow Area Profiles during Falling Phase for a Channel without Upstream Inflow.

Figure 4.10. Successive Flow Area Profiles during Falling Phase for a Channel with Upstream Inflow.

As shown in figure 4.10, at time tr, the distance xc of any point on the equilibrium profile (curve O-Ou-G-D) can be expressed in terms of flow area Ac by substituting Eq. (4.7) into Eq. (4.1):

xc =

α c Ac β − Qu c

qL

(4.90)

Tommy S.W. Wong

60 Integrating Eq. (4.27) from

[(α A c

βc

) ]

− Qu qL (Eq. 4.90) to xc for xc and from tq to t (where t

≥ tq) for t gives the equation for the curve O-E-C:

xc = α c β c Ac

β c −1

(t − t ) + ⎛⎜⎜ α q

β

Ac c − Qu qL

c

⎝

⎞ ⎟ ⎟ ⎠

(4.91)

For a channel with a constant upstream inflow, Eq. (4.91) is only valid for Lf ≤ xc ≤ Lc, where Lf = length of channel in which the flow equals to upstream inflow during the falling phase. For the profile between 0 ≤ xc ≤ Lf, it is the line Ou-E, i.e.

⎛Q Ac = Au = ⎜⎜ u ⎝ αc

1 βc

⎞ ⎟⎟ ⎠

(4.92)

Substituting Ac = Au and xc = Lf into Eq. (4.91) gives the equation for Lf in terms of Au:

L f = α c β c Auβ o −1 (t − t q )

(4.93)

Substituting Eq. (4.92) into Eq. (4.93) gives the equation for Lf in terms of Qu:

L f = α c1 β c β c Qu( β c −1) β c (t − t q )

(4.94)

For a channel with zero upstream inflow (Qu = 0), Eq. (4.91) reduces to:

xc = α c β c Ac

β c −1

(t − t ) + α q

Ac qL

βc

c

(4.95)

which is valid for 0 ≤ xc ≤ Lc. Equations (4.91)-(4.95) are valid for t ≥ tq.

4.17.1. Inflection Line As shown in figure 2.10, the equilibrium flow area profile (curve O-D) is concave downwards, while the flow area profile at time t > tq + ∆t (curve O-B) is concave upwards. Similarly, in figure 4.10, the curve O-Ou-D is concave downwards, and the curve O-F-B is concave upwards. The equation for the inflection line can be derived by first obtaining the second derivative of Eq. (4.90), with respect to Ac:

⎡ α c β c (β c − 1) Acβ c − 2 ⎤ d 2 xc βc −3 ( ) ( )( ) α β β β 1 2 = − − − + A t t c c c c c q ⎢ ⎥ 2 qL dAc ⎣ ⎦

(4.96)

Kinematic-Wave Rainfall-Runoff Formulas

61

Next, by equating Eq. (4.96) to zero and equating Ac = Ai results in:

Ai = (2 − β c )(t − t q )qL

(4.97)

where Ai = flow area of the inflection point. Substituting Eq. (4.97) into Eq. (4.91) gives the equation for the inflection line for a channel with upstream inflow:

⎛ 2 ⎞⎛ αAiβ c ⎟⎟⎜⎜ xi = ⎜⎜ ⎝ 2 − β c ⎠⎝ qL

⎞ ⎛ Qu ⎟⎟ − ⎜⎜ ⎠ ⎝ qL

⎞ ⎟⎟ ⎠

(4.98)

where xi = distance xc of the inflection point. For a channel with zero upstream inflow (Qu = 0), Eq. (4.98) reduces to:

⎛ 2 ⎞⎛ αAiβ c ⎟⎟⎜⎜ xi = ⎜⎜ β 2 − c ⎠⎝ q L ⎝

⎞ ⎟⎟ ⎠

(4.99)

Equations (4.98) and (4.99) have been superimposed respectively onto figures 4.9 and 4.10 as dashed lines, which are labeled as inflection lines.

4.18. HYDROGRAPH - FALLING PHASE As shown by Eq. (4.88), during the falling phase, water flows out at constant flow area, hence, the water particle at G flows out to C at constant flow area (figure 4.10). The time required for the water particle to flow from G to C is in fact the same as the duration of partial equilibrium discharge, td, as shown in figure 4.3. Substituting td = t – tq and Qp = Qc into Eq. (4.58) gives:

⎛ Q − Qu Lc − ⎜⎜ c ⎝ qL t − tq = ck

⎞ ⎟⎟ ⎠

(4.100)

Equation (4.99) may also be derived by integrating Eq. (4.26) from tq to t (where t ≥ tq) for t and from [(Qc-Qu)/qL] to Lc for xc. Since the discharge in the channel cannot be less than upstream discharge, Eq. (4.100) is only valid for Qc ≥ Qu. Substituting Eq. (4.28) into Eq. (4.100) gives the equation for the falling phase (falling limb) of the hydrograph, which is only valid for Qc ≥ Qu:

⎛ Q − Qu ⎞ ⎟ Lc − ⎜⎜ c qL ⎟⎠ ⎝ t = 1 / βc + tq α c β c Qc( βc −1) / βc

(4.101)

Tommy S.W. Wong

62

For a channel with zero upstream inflow (Qu = 0), Eq. (4.101) reduces to

⎛Q ⎞ Lc − ⎜⎜ c ⎟⎟ q t = 1 / β c ⎝ (βLc −1⎠) / β c + t q α c β c Qc

(4.102)

Figures 4.2 and 4.3 show the falling phase (falling limb) of an equilibrium and a partial equilibrium runoff hydrograph for a channel without and for a channel with upstream inflow, respectively.

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 63-76

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 5

5. WORKING FORMULAS FOR FLOW IN CIRCULAR CHANNEL In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a circular channel are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

5.1. KINEMATIC WAVE PARAMETERS For flow in a circular channel, the flow area Ac, and the wetted perimeter, P, are related to the diameter of circular channel D, and the water surface angle θ, as follows:

D2 Ac = (θ − sin θ ) 8 P=

Dθ 2

(5.1)

(5.2)

Figure 5.1 shows the circular channel with diameter D, water surface angle θ, and flow depth yc. Substituting Eqs. (5.1) and (5.2) and Q = Qc, S = S c, A = Ac, n = nc into Eq. (3.1) gives:

⎛ S 1 2 D 8 3 ⎞ ⎡ (θ − sin θ )5 / 3 ⎤ ⎟⎟ ⎢ Qc = 0.0496⎜⎜ c ⎥ θ 2/ 3 ⎝ nc ⎠ ⎣ ⎦

(5.3)

Tommy S.W. Wong

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centre D θ yc

Figure 5.1. Cross-section of Circular Channel.

For full flow condition in a circular channel (i.e. θ = 2π), Eqs. (5.1) and (5.3) become:

A full

πD 2 = 4

(5.4)

Q full

⎛ S c1 2 D 8 3 ⎞ ⎟⎟ = 0.312⎜⎜ n c ⎠ ⎝

(5.5)

where Afull = flow area under full flow condition, and Qfull = discharge under full flow condition. Dividing Eq. (5.3) by Eq. (5.5) and Eq. (5.1) by Eq. (5.4) give: 5/ 3 Qc 1 ⎡ (θ − sin θ ) ⎤ = ⎥ ⎢ Q full 2π ⎣ θ 2/3 ⎦

(5.6)

(θ − sin θ ) Ac = A full 2π

(5.7)

Equations (5.6) and (5.7) are considered to be the true relationship between discharge and flow area for flow in a circular channel. This true relationship is shown in figure 5.2, and it is apparent that the discharge reaches a maximum under the partially full flow condition. Differentiating Qc with respect to θ in Eq. (5.3) gives: 23

⎛ S c1 2 D 8 3 ⎞⎛ θ − sin θ ⎞ ⎛ 2 sin θ dQc ⎞ ⎟⎟⎜ − 5 cos θ + 3 ⎟ = 0.0165⎜⎜ ⎟ ⎜ dθ ⎠ ⎝ θ ⎠ ⎝ nc ⎠⎝ θ

(5.8)

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65

Figure 5.2. Comparison between True and Kinematic Wave Relationships for Flow in Circular Channel.

Equating Eq. (5.8) to zero shows that the maximum discharge, Qmax, occurs at θ = 5.278 rad (or 302.4°). Substituting θ = 5.278 rad into Eq. (5.6) gives:

Qmax = 1.076 Q full

(5.9)

Substituting Eq. (5.5) into Eq. (5.9) gives:

Qmax = 0.335 D

8/ 3

⎛ S c1 2 ⎜⎜ ⎝ nc

⎞ ⎟⎟ ⎠

(5.10)

To evaluate the kinematic wave parameters αc and βc using the same method that was used by Harley et al (1970), the parameters αc and βc are related to Qmax and Amax as follows:

Qmax = α c AQβ cmax

(5.11)

where AQmax = flow area under maximum discharge condition. Further, relating AQmax to D through a parameter γ:

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γ =

AQ max

(5.12)

D2

Substituting D in Eq. (5.4) into Eq. (5.12) gives a relationship between γ and Afull:

γ=

πAQ max

(5.13)

4 A full

Substituting Eqs. (5.10) and (5.12) into Eq. (5.11) gives a relationship between αc and γ:

⎡ 0.335 D (8 / 3)− 2 β c ⎤⎛ S c1 2 ⎥⎜⎜ γ βc ⎣ ⎦⎝ nc

αc = ⎢

⎞ ⎟⎟ ⎠

(5.14)

Equation (5.14) shows that the value of αc is dependent on the value of γ, which is dependent on the flow area AQmax (Eq. 5.16). To identify the values of αc and βc, Eq. (4.7) is divided by Eq. (5.11):

⎛ A Qc =⎜ c Qmax ⎜⎝ AQ max

⎞ ⎟ ⎟ ⎠

βc

(5.15)

Substituting Eqs. (5.9) and (5.13) into (5.15) gives:

Qc ⎛π = 1.076⎜⎜ Q full ⎝ 4γ

⎞ ⎟⎟ ⎠

βc

⎛ Ac ⎜ ⎜A ⎝ full

⎞ ⎟ ⎟ ⎠

βc

(5.16)

Wong and Zhou (2003) fitted the kinematic wave relationship (Eq. 5.19) to the true relationship (Eqs. 5.7 and 5.8), and found that the best fit occurs at AQmax/Afull = 0.923 (which corresponds to yQmax/D = 0.87 where yQmax = flow depth under maximum discharge condition), γ = 0.725, and βc = 5/4, as shown in figure 5.2. Substituting βc = 5/4 and γ = 0.725 into Eq. (5.14), gives the kinematic wave parameters, which are valid for yc ≤ 0.87D:

⎛ S c1 2 D1 / 6 ⎞ ⎟⎟ α c = 0.501⎜⎜ ⎝ nc ⎠

βc =

5 4

(5.17)

(5.18)

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67

5.2. FLOW DEPTH For flow in a circular channel, the flow depth, yc, is related to D and θ, as follows:

yc =

D⎡ ⎛ θ ⎞⎤ 1 − cos⎜ ⎟⎥ ⎢ 2⎣ ⎝ 2 ⎠⎦

(5.19)

Equating θ in Eq. (5.19) to that in Eq. (5.1), and by curve fitting results in the following equation relating Ac to yc:

Ac =

πD 2 4

3 2 ⎡ ⎛ yc ⎞ ⎛ yc ⎞ ⎛ y ⎞⎤ − . + . 1 195 1 801 ⎜ ⎟ ⎜ ⎟ + 0.397⎜ c ⎟⎥ ⎢ ⎝D⎠ ⎝D⎠ ⎝ D ⎠⎥⎦ ⎢⎣

(5.20)

Substituting Eqs. (5.17) and (5.18) into Eq. (4.7) gives:

⎛ S c1 2 D 1 6 Qc = 0.501⎜⎜ ⎝ nc

⎞ 54 ⎟⎟ Ac ⎠

(5.21)

Substituting Eqs. (5.20) into Eq. (5.21) gives: 3 2 ⎛ S 1 2 D8 3 ⎞⎡ y y y ⎤ ⎟⎟ ⎢− 1.195⎛⎜ c ⎞⎟ + 1.801⎛⎜ c ⎞⎟ + 0.397⎛⎜ c ⎞⎟⎥ Qc = 0.370⎜⎜ c ⎝D⎠ ⎝D⎠ ⎝ D ⎠⎥⎦ ⎝ nc ⎠ ⎢⎣

54

(5.22)

Rearranging Eq. (5.22) gives the equation for the flow depth for a channel with and without upstream inflow: 3

2

⎛ nQ ⎞ ⎛y ⎞ ⎛y ⎞ ⎛y ⎞ − 1.195⎜ c ⎟ + 1.801⎜ c ⎟ + 0.397⎜ c ⎟ = 2.213⎜⎜ 1 2c c8 3 ⎟⎟ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝ Sc D ⎠

45

(5.23)

5.3. FLOW VELOCITY Substituting Eqs. (5.17) and (5.18) into Eq. (4.12), gives the equation for the flow velocity along the equilibrium profile for a channel with upstream inflow:

⎛ S c1 2 D1 6 ⎞ ⎟⎟ v = 0.575⎜⎜ ⎝ nc ⎠

45

(Qu + qL xc )1 5

(5.24)

Tommy S.W. Wong

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For a channel with zero upstream (Qu = 0), Eq. (5.24) reduces to:

⎛ S 1 2 D1 6 ⎞ ⎟⎟ v = 0.575⎜⎜ c n c ⎝ ⎠

45

(qL xc )1 5

(5.25)

5.4. AVERAGE FLOW VELOCITY Substituting Eqs. (5.17) and (5.18) into Eq. (4.15) gives the equation for the average flow velocity for a channel with upstream inflow: 45

⎛ S 1 2 D1 6 ⎞ ⎟ qL Lc 0.460⎜⎜ c nc ⎟⎠ ⎝ vav = (Qu + qL Lc )4 5 − Qu 4 5

(5.26)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.26) reduces to:

⎛ S 1 2 D1 6 ⎞ ⎟⎟ vav = 0.460⎜⎜ c n c ⎝ ⎠

45

(qL Lc )1 5

(5.27)

5.5. KINEMATIC WAVE CELERITY Substituting Eqs. (5.17) and (5.18) into Eq. (4.29) gives the equation for the wave celerity along the equilibrium profile for a channel with upstream inflow:

⎛ S c1 2 D1 6 ⎞ ⎟⎟ ck = 0.719⎜⎜ ⎝ nc ⎠

45

(Qu + q L xc )1 5

(5.28)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.28) reduces to:

⎛ S 1 2 D1 6 ⎞ ⎟⎟ ck = 0.719⎜⎜ c ⎝ nc ⎠

45

(q L xc )1 5

(5.29)

5.6. AVERAGE WAVE CELERITY Substituting Eqs. (5.17) and (5.18) into Eq.(4.32) gives the equation for the wave celerity for a channel with upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

69

45

⎛ S 1 2 D1 6 ⎞ ⎟ qL Lc 0.575⎜⎜ c nc ⎟⎠ ⎝ cav = (Qu + qL Lc )4 5 − Qu 4 5

(5.30)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.30) reduces to:

⎛ S c1 2 D1 6 ⎞ ⎟⎟ cav = 0.575⎜⎜ n c ⎠ ⎝

45

(qL Lc )1 5

(5.31)

5.7. TIME OF TRAVEL Substituting Eqs. (5.17) and (5.18) into Eq. (4.35) gives the equation for the time of travel for a channel with upstream inflow:

⎛ nc ⎞ ⎡ (Qu + qL Lc )4 5 − Qu4 5 ⎤ tt = 0.0290⎜⎜ 1 2 1 6 ⎟⎟ ⎢ ⎥ qL ⎝ Sc D ⎠ ⎣ ⎦ 45

(5.32)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.32) reduces to:

⎛ 0.0290 ⎞⎛ n L ⎞ tt = ⎜⎜ 1 5 ⎟⎟⎜⎜ 1 2c c1 6 ⎟⎟ ⎝ qL ⎠⎝ S c D ⎠

45

(5.33)

5.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (5.17) and (5.18) into Eq. (4.45) gives the equation for the rising phase (rising limb) of the hydrograph for a channel with upstream inflow:

⎛ S c1 2 D1 6 ⎞ ⎡ ⎛ nQ ⎞ ⎟⎟ ⎢1.738⎜⎜ 1 c2 u1 6 ⎟⎟ Qc = 0.501⎜⎜ ⎝ nc ⎠ ⎢⎣ ⎝ Sc D ⎠

45

⎤ + 60qL t ⎥ ⎥⎦

54

(5.34)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.34) reduces to:

⎛ S 1 2 D1 6 ⎞ ⎟⎟(q L t )5 4 Qc = 83.66⎜⎜ c n c ⎝ ⎠ Equations (5.34) and (5.35) are valid for t ≤ tt.

(5.35)

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5.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (5.17) and (5.18) into Eq. (4.47) gives the equation for the forward characteristic of the rising phase for a channel with upstream inflow:

⎛ n ⎞ ⎡ (Q + q L xc )4 5 − Qu4 5 ⎤ t = 0.0290⎜⎜ 1 2 c 1 6 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc D ⎠ ⎣ ⎦ 45

(5.36)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.36) reduces to:

⎛ 0.0290 ⎞⎛ n x ⎞ t = ⎜⎜ 1 5 ⎟⎟⎜⎜ 1 2c c1 6 ⎟⎟ ⎝ qL ⎠⎝ S c D ⎠

45

(5.37)

5.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (5.17) and (5.18) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎤ ⎡⎛ n ⎞ Ac = 1.738⎢⎜⎜ 1 2 c 1 6 ⎟⎟(Qu + q L xc )⎥ ⎦ ⎣⎝ S c D ⎠

45

(5.38)

Substituting Eqs. (5.20) into Eq. (5.38) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp: 3 2 ⎡⎛ n ⎤ ⎞ ⎛y ⎞ ⎛y ⎞ ⎛y ⎞ − 1.195⎜ c ⎟ + 1.801⎜ c ⎟ + 0.397⎜ c ⎟ = 2.213⎢⎜⎜ 1 2 c 8 3 ⎟⎟(Qu + qL xc )⎥ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎣⎝ S c D ⎠ ⎦

45

(5.39)

Substituting Eqs. (5.16) and (5.17) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎤ ⎡⎛ n ⎞ Ap = 1.738⎢⎜⎜ 1 2 c 1 6 ⎟⎟(Qu + q L L p )⎥ ⎦ ⎣⎝ S c D ⎠

45

(5.40)

Substituting Ac = Ap, and yc = yp into Eq. (5.20), and then substituting it into Eq. (5.40) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ yp − 1.195⎜⎜ ⎝D

3

⎞ ⎛y ⎟⎟ + 1.801⎜⎜ p ⎠ ⎝D

2

⎞ ⎛y ⎟⎟ + 0.397⎜⎜ p ⎠ ⎝D

71

⎤ ⎡⎛ n ⎞ ⎞ ⎟⎟ = 2.213⎢⎜⎜ 1 2 c 8 3 ⎟⎟(Qu + q L L p )⎥ ⎠ ⎦⎥ ⎣⎢⎝ S c D ⎠

45

(5.41)

From Eq. (4.51), the distance Lp is:

Lp =

Q p − Qu

(5.42)

qL

For a channel with zero upstream inflow (Qu = 0), Eqs. (5.38)-(5.42) reduce to:

⎛ nq x ⎞ Ac = 1.738⎜⎜ 1c 2 L 1c6 ⎟⎟ ⎝ Sc D ⎠

45

(5.43)

3

2

⎛ nq x ⎞ ⎛y ⎞ ⎛y ⎞ ⎛y ⎞ − 1.195⎜ c ⎟ + 1.801⎜ c ⎟ + 0.397⎜ c ⎟ = 2.213⎜⎜ 1c 2 L 8c3 ⎟⎟ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝ Sc D ⎠

45

(5.44)

which are valid for 0 ≤ xc ≤ Lp,

⎛ nc q L L p ⎞ Ap = 1.738⎜⎜ 1 2 1 6 ⎟⎟ ⎝ Sc D ⎠ ⎛ yp − 1.195⎜⎜ ⎝D

3

45

⎞ ⎛y ⎟⎟ + 1.801⎜⎜ p ⎠ ⎝D

(5.45)

2

⎞ ⎛y ⎟⎟ + 0.397⎜⎜ p ⎠ ⎝D

⎛nq L ⎞ ⎞ ⎟⎟ = 2.213⎜⎜ 1c 2 L 8 p3 ⎟⎟ ⎠ ⎝ Sc D ⎠

45

(5.46)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp qL

(5.47)

5.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (5.17) and (5.18) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow:

Tommy S.W. Wong

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td =

45 ⎤ ⎛ S c1 2 D1 6 ⎞ ⎡ ⎛ ncQu ⎞ ⎟⎟ ⎢1.738⎜⎜ 1 2 1 6 ⎟⎟ + 60q Lt q ⎥ Lc q L + Qu − 0.501⎜⎜ ⎥⎦ ⎝ nc ⎠ ⎢⎣ ⎝ Sc D ⎠ 45 ⎤ ⎛ S D qL ⎞⎡ ⎛ nc Qu ⎞ ⎟⎟ ⎢1.738⎜⎜ 1 2 1 6 ⎟⎟ + 60q Lt q ⎥ 37.58⎜⎜ nc ⎥⎦ ⎝ ⎠ ⎢⎣ ⎝ Sc D ⎠ 12 c

16

54

14

(5.48)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.41) reduces to:

⎛ S c1 2 D1 6 ⎞ 1 4 5 4 ⎟q L t q Lc − 83.66⎜⎜ nc ⎟⎠ ⎝ td = ⎛ S c1 2 D1 6 ⎞ ⎟⎟(q Lt q )1 4 104.59⎜⎜ ⎝ nc ⎠

(5.49)

5.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

5.12.1. Partial Equilibrium Discharge Substituting Eqs. (5.17) and (5.18) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎛ S c1 2 D1 6 Q p = 0.501⎜⎜ ⎝ nc

⎛ nQ ⎞⎡ ⎟⎟ ⎢1.738⎜⎜ 1 c2 u1 6 ⎝ Sc D ⎠ ⎢⎣

⎞ ⎟⎟ ⎠

45

⎤ + 60q L t q ⎥ ⎥⎦

54

(5.50)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.50) reduces to:

⎛ S c1 2 D 1 6 ⎞ ⎟⎟(q L t q )5 4 Q p = 83.66⎜⎜ ⎝ nc ⎠ Equations (5.50) and (5.51) are valid for tq ≤ t ≤ (tq + td).

(5.51)

Kinematic-Wave Rainfall-Runoff Formulas

73

5.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc

(5.52)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.52) reduces to:

Q e = q L Lc

(5.53)

Equations (5.52) and (5.53) are valid for tt ≤ t ≤ tq.

5.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (5.17) and (5.18) into Eq. (4.71) gives the equation for the equilibrium flow area profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lc:

⎤ ⎡⎛ n ⎞ Ac = 1.738⎢⎜⎜ 1 2 c 1 6 ⎟⎟(Qu + q L xc )⎥ ⎦ ⎣⎝ S c D ⎠

45

(5.54)

Substituting Eqs. (5.20) into Eq. (5.54) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lc: 3 2 ⎡⎛ n ⎤ ⎞ ⎛y ⎞ ⎛y ⎞ ⎛y ⎞ − 1.195⎜ c ⎟ + 1.801⎜ c ⎟ + 0.397⎜ c ⎟ = 2.213⎢⎜⎜ 1 2 c 8 3 ⎟⎟(Qu + qL xc )⎥ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎣⎝ S c D ⎠ ⎦

45

(5.55)

For a channel with zero upstream inflow (Qu = 0), Eqs. (5.54) and (5.55) reduce to:

⎛ nq x ⎞ Ac = 1.738⎜⎜ 1c 2 L 1c6 ⎟⎟ ⎝ Sc D ⎠

45

3

(5.56)

2

⎛ nq x ⎞ ⎛y ⎞ ⎛y ⎞ ⎛y ⎞ − 1.195⎜ c ⎟ + 1.801⎜ c ⎟ + 0.397⎜ c ⎟ = 2.213⎜⎜ 1c 2 L 8c3 ⎟⎟ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝ Sc D ⎠ which are valid for 0 ≤ xc ≤ Lc.

45

(5.57)

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5.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (5.17) and (5.18) into Eq. (4.78) gives the equation for the equilibrium detention storage for a channel with upstream inflow:

⎛ n ⎞ ⎡ (Q + qL Lc )9 5 − Qu9 5 ⎤ Dec = 0.966⎜⎜ 1 2 c 1 6 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc D ⎠ ⎣ ⎦ 45

(5.58)

For a channel with zero upstream inflow (i.e. Qu = 0), Eq. (5.58) reduces to:

⎛ nc qL L9c 4 ⎞ Dec = 0.966⎜⎜ 1 2 1 6 ⎟⎟ ⎝ Sc D ⎠

45

(5.59)

5.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (5.17) and (5.18) into Eq. (4.91) gives the equation for the falling phase of the flow area profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

⎧⎡ ⎫ ⎛ S c1 2 D1 6 Ac5 4 ⎞⎤ ⎜⎜ ⎟⎟⎥ − Qu ⎪ 0 501 . ⎪ ⎢ nc ⎛ S 1 2 D1 6 Ac1 4 ⎞ ⎪ ⎪ ⎝ ⎠⎦ ⎟⎟(t − t q ) + ⎨ ⎣ xc = 37.58⎜⎜ c ⎬ nc qL ⎝ ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭

(5.60)

Substituting Eq. (5.20) into Eq. (5.60) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc: 3 2 ⎛ Sc1 2 D 2 3 ⎞ ⎡ ⎛ yc ⎞ ⎛ yc ⎞ ⎛ y ⎞⎤ ⎟⎟ ⎢− 1.195⎜ ⎟ + 1.801⎜ ⎟ + 0.397⎜ c ⎟⎥ xc = 35.38⎜⎜ ⎝D⎠ ⎝D⎠ ⎝ D ⎠⎥⎦ ⎝ nc ⎠ ⎢⎣

14

(t − t ) q

54 3 2 ⎛⎧ ⎛ S c1 2 D 8 3 ⎞ ⎡ yc ⎞ yc ⎞ yc ⎞⎤ ⎫⎪ ⎜⎪ ⎛ ⎛ ⎛ ⎟⎟ ⎢− 1.195⎜ ⎟ + 1.801⎜ ⎟ + 0.397⎜ ⎟⎥ ⎬ − Qu ⎜ ⎨0.370⎜⎜ n ⎝D⎠ ⎝D⎠ ⎝ D ⎠⎥⎦ ⎪ c ⎠ ⎢⎣ ⎝ ⎜ ⎪⎩ ⎭ +⎜ qL ⎜ ⎜⎜ ⎝

From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎟ ⎠

(5.61)

Kinematic-Wave Rainfall-Runoff Formulas

⎛ nQ Ac = 1.738⎜⎜ 1 c2 u1 6 ⎝ Sc D

⎞ ⎟⎟ ⎠

75

45

(5.62)

Substituting Eq. (5.20) into Eq. (5.62) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf: 3

2

⎛ nQ ⎞ ⎛y ⎞ ⎛y ⎞ ⎛y ⎞ − 1.195⎜ c ⎟ + 1.801⎜ c ⎟ + 0.397⎜ c ⎟ = 2.213⎜⎜ 1 2c u8 3 ⎟⎟ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝ Sc D ⎠

45

(5.63)

Substituting Eqs. (5.17) and (5.18) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq: 45

⎛ S 1 2 D1 6 ⎞ ⎟⎟ Qu1 5 (t − t q ) L f = 43.15⎜⎜ c n c ⎠ ⎝

(5.64)

For a channel with zero upstream inflow (Qu = 0), Eqs. (5.60) and (5.61) reduce to:

⎡ ⎛ S c1 2 D1 6 Ac5 4 ⎞ ⎤ ⎟⎟ ⎥ ⎜⎜ 0 501 . ⎢ nc ⎛ S c1 2 D1 6 Ac1 4 ⎞ ⎠⎥ ⎝ ⎢ ⎟⎟(t − tq ) + xc = 37.58⎜⎜ ⎢ ⎥ nc qL ⎠ ⎝ ⎢ ⎥ ⎣⎢ ⎦⎥

(5.65)

14

3 2 ⎛ S c1 2 D 2 3 ⎞ ⎡ ⎛ yc ⎞ ⎛ yc ⎞ ⎛ y ⎞⎤ ⎟ ⎢− 1.195⎜ ⎟ + 1.801⎜ ⎟ + 0.397⎜ c ⎟⎥ (t − t q ) xc = 35.38⎜⎜ ⎟ ⎝ D ⎠⎥⎦ ⎝D⎠ ⎝D⎠ ⎝ nc ⎠ ⎣⎢ 54 3 2 ⎛⎧ ⎫⎞ ⎛ S c1 2 D 8 3 ⎞ ⎡ ⎜⎪ ⎛ yc ⎞ ⎛ yc ⎞ ⎛ y c ⎞⎤ ⎪ ⎟ ⎜ ⎟ . . . . − + + 0 370 1 195 1 801 0 397 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎬⎟ ⎜⎨ ⎜ n ⎟ ⎝D⎠ ⎝D⎠ ⎝ D ⎠⎥⎦ ⎪ ⎟ c ⎝ ⎠ ⎢⎣ ⎜ ⎪⎩ ⎭ +⎜ ⎟ qL ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎠ ⎝

(5.66)

which are valid for 0 ≤ xc ≤ Lc.

5.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (5.17) and (5.18) into Eq. (4.101) gives the equation for the falling phase (falling limb) of a hydrograph for a channel with upstream inflow:

Tommy S.W. Wong

76 45

⎛ 0.0232 ⎞⎛ n ⎞ ⎡ ⎛ Q − Qu t = ⎜⎜ 1 5 ⎟⎟⎜⎜ 1 2 c 1 6 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎝ qL ⎝ Qc ⎠⎝ S c D ⎠ ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

(5.67)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.67) reduces to: 45

⎛ 0.0232 ⎞⎛ n ⎞ ⎡ ⎛ Q ⎞⎤ t = ⎜⎜ 1 5 ⎟⎟⎜⎜ 1 2 c 1 6 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎟⎟⎥ + t q ⎝ qL ⎠⎦ ⎝ Qc ⎠⎝ S c D ⎠ ⎣

(5.68)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 77-89

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 6

6. WORKING FORMULAS FOR FLOW IN PARABOLIC CHANNEL In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a parabolic channel are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

6.1. KINEMATIC WAVE PARAMETERS For the parabolic channel as shown in figure 6.1, the channel section can be described mathematically by:

x2 y= 4H

(6.1)

where y = height above the channel invert, x = semi-width at height y, and H = height of focal point above channel invert. From mathematics, the flow area Ac can be related to x and H, as follows:

Ac =

x3 3H

(6.2)

Tommy S.W. Wong

78

focal point

H

yc 2H

Figure 6.1. Cross-section of Parabolic Channel.

The arc length, La, of the parabola can be derived by integrating Eq. (6.1), as follows:

[

La = ∫ dx + dy 2

2

12

]

12

⎡ ⎛ dy ⎞ 2 ⎤ = ∫ ⎢1 + ⎜ ⎟ ⎥ dx ⎣⎢ ⎝ dx ⎠ ⎦⎥

(6.3)

Upon integration, the arc length, La, of the parabola is:

[

]

⎡ 1 ⎤ 2 La = ⎢ xZ + (2 H ) ln ( x' + Z' ) ⎥ ⎣ 2(2 H ) ⎦

(6.4)

where

[

Z = (2 H ) + x 2 2

]

12

x' =

x 2H

Z' =

Z = 1 + x' 2 2H

(6.5)

(6.6)

and

(

)

12

(6.7)

In Eq. (6.4), the trigonometric equivalent of the logarithmic term is:

ln( x' + Z' ) = sinh −1 ( x' )

(6.8)

Equation (6.6) is defined as the dimensionless ratio of the flow semi-width to focal semiwidth. As derived from Eq. (6.4), the wetted perimeter P being twice the arc length La is:

Kinematic-Wave Rainfall-Runoff Formulas

[

79

]

⎛ 1 ⎞ 2 P=⎜ ⎟ xZ + (2 H ) ln ( x' + Z' ) ⎝ 2H ⎠

(6.9)

Dividing Eq. (6.9) by (2H) gives the equation in a dimensionless form, as follows:

P' =

P = x' Z' +ln ( x' + Z' ) 2H

(6.10)

To eliminate x in Eqs. (6.2)-(6.10), Eq. (6.2) is converted to a dimensionless form, as follows:

Ac ' =

Ac

(2 H )2

x3

=

3(2 H ) H 2

=

x' 3 1.5

(6.11)

Rearranging Eq. (6.11) gives: 13

⎛3 ⎞ x' = ⎜ Ac ' ⎟ ⎝2 ⎠

= 1.145( Ac ' )

13

(6.12)

Substituting Eq. (6.12) into Eq. (6.7) gives:

[

Z' = 1 + 1.311( Ac ' )

]

2/ 3 1 2

(6.13)

Substituting Eqs. (6.8), (6.12) and (6.13) into Eq. (6.10) gives:

[

P' = 1.145( Ac ' ) 1 + 1.311( Ac ' ) 13

]

2/3 1 2

[

+ sinh −1 1.145( Ac ' )

13

]

(6.14)

To expand Eq. (6.14) into a series, the following series expansions are used:

(1 + x' )

2 12

= 1+

1 2 1 4 1 6 x' − x ' + x' −...... 2 8 16

1 3 5 7 sinh −1 ( x') = x'− x' 3 + x' 5 − x' +...... 6 40 112

(6.15)

(6.16)

Equations (6.15) and (6.16) are only valid for x’ < 1. Substituting Eqs. (6.15) and (6.16) into Eq. (6.14), and cancelling the higher order terms, the equation becomes:

P' ≈ 2 x' Substituting Eq. (6.12) into Eq. (6.17) gives:

(6.17)

Tommy S.W. Wong

80

P' = 2.290( Ac ' )

13

(6.18)

Substituting Eqs. (6.10), and (6.11) into Eq. (6.18) gives a relationship between P and Ac, as follows:

P = 2.885(HAc )

13

(6.19)

Brady (1983) showed that Eq. (6.19) is valid for x’ < 0.6

(6.20)

As x is related to the top width, T, as:

x=

T 2

(6.21)

and T is related to the flow depth, yc, as:

T = 4(Hy c )

12

(6.22)

Substituting Eqs. (6.20)-(6.22) into Eq. (6.6) gives:

yc < 0.18B

(6.23)

Substituting Eq. (6.19) and Q = Qc, S = S c, A = Ac, n = nc into Eq. (3.1) gives:

⎛ S1 2 ⎞ Qc = 0.493⎜⎜ c 2 9 ⎟⎟ Ac13 9 ⎝ nc H ⎠

(6.24)

A comparison of Eqs. (6.24) with Eq. (4.7) gives the kinematic wave parameters (Brady 1983), which are valid for yc < 0.18H:

⎛ S c1 2 ⎞ ⎟ 29 ⎟ ⎝ nc H ⎠

α c = 0.493⎜⎜

βc =

13 9

(6.25)

(6.26)

Kinematic-Wave Rainfall-Runoff Formulas

81

6.2. FLOW DEPTH For flow in a parabolic channel, the flow area Ac, is related to the flow depth yc, and the parabola’s focal height H, as follows (Jan 1979):

Ac =

8H 1 2 y c3 2 3

(6.27)

Substituting Eq. (6.27) into Eq. (6.24) gives:

⎛ S 1 2 H 1 2 ⎞ 13 6 ⎟⎟ y c Qc = 2.033⎜⎜ c ⎝ nc ⎠

(6.28)

Rearranging Eq. (6.28) gives the equation for the flow depth for a channel with and without upstream inflow:

⎛ nQ y c = 0.721⎜⎜ 1 2c c1 2 ⎝ Sc H

⎞ ⎟⎟ ⎠

6 13

(6.29)

6.3. FLOW VELOCITY Substituting Eqs. (6.25) and (6.26) into Eq. (4.12) gives the equation for the flow velocity along the equilibrium profile for a channel with upstream inflow:

⎛ S c1 2 v = 0.613⎜⎜ 29 ⎝ nc H

⎞ ⎟⎟ ⎠

9 13

(Qu + q L xc )4 13

(6.30)

For a channel with zero upstream inflow (Qu = 0), Eq. (6.30) reduces to:

⎛ S1 2 v = 0.613⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟⎟ ⎠

9 13

(q L xc )4 13

(6.31)

6.4. AVERAGE FLOW VELOCITY Substituting Eqs. (6.25) and (6.26) into Eq. (4.15) gives the equation for the average flow velocity for a channel with upstream inflow:

Tommy S.W. Wong

82 9 13

⎛ S1 2 ⎞ 0.424⎜⎜ c 2 9 ⎟⎟ q L Lc ⎝ nc H ⎠ v av = (Qu + q L Lc )9 13 − Qu9 13

(6.32)

For a channel with zero upstream inflow (Qu = 0), Eq. (6.32) reduces to:

⎛ S1 2 v av = 0.424⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟⎟ ⎠

9 13

(q L Lc )4 13

(6.33)

6.5. KINEMATIC WAVE CELERITY Substituting Eqs. (6.25) and (6.26) into Eq. (4.29) gives the equation for the wave celerity along the equilibrium profile for a channel with upstream inflow:

⎛ S c1 2 ⎞ ⎟ c k = 0.885⎜⎜ 29 ⎟ n H ⎝ c ⎠

9 13

(Qu + q L xc )4 13

(6.34)

For a channel with zero upstream inflow (Qu = 0), Eq. (6.34) reduces to:

⎛ S1 2 c k = 0.885⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟⎟ ⎠

9 13

(q L xc )4 13

(6.35)

6.6. AVERAGE WAVE CELERITY Substituting Eqs. (6.25) and (6.26) into Eq. (4.32) gives the equation for the average wave celerity for a channel with upstream inflow: 9 13

⎛ S1 2 ⎞ 0.613⎜⎜ c 2 9 ⎟⎟ q L Lc ⎝ nc H ⎠ c av = (Qu + q L Lc )9 13 − Qu9 13

(6.36)

For a channel with zero upstream inflow (Qu = 0), Eq. (6.36) reduces to:

⎛ S c1 2 c av = 0.613⎜⎜ 29 ⎝ nc H

⎞ ⎟⎟ ⎠

9 13

(q L Lc )4 13

(6.37)

Kinematic-Wave Rainfall-Runoff Formulas

83

6.7. TIME OF TRAVEL Substituting Eqs. (6.25) and (6.26) into Eq. (4.35) gives the equation for the time of travel for a channel with upstream inflow:

⎛n H29 t t = 0.0272⎜⎜ c 1 2 ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

⎡ (Qu + q L Lc )9 13 − Qu9 13 ⎤ ⎢ ⎥ qL ⎢⎣ ⎥⎦

(6.38)

For a channel with zero upstream inflow (Qu = 0), Eq. (6.38) reduces to:

0.0272 ⎛ n H 2 9 L t t = 4 13 ⎜⎜ c 1 2 c qL ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

(6.39)

6.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (6.25) and (6.26) into Eq. (4.45) gives the equation for the rising phase (rising limb) of the hydrograph for a channel with upstream inflow:

⎛ S1 2 Qc = 0.493⎜⎜ c 2 9 ⎝ nc H

⎛ n H 2 9Q ⎞⎡ ⎟⎟ ⎢1.639⎜⎜ c 1 2 u ⎝ Sc ⎠ ⎢⎣

⎞ ⎟⎟ ⎠

9 13

⎤ + 60q L t ⎥ ⎥⎦

13 9

(6.40)

For a channel with zero upstream inflow (Qu = 0), Eq. (6.41) reduces to:

⎛ S1 2 Qc = 182.5⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟⎟(q L t )13 9 ⎠

(6.41)

Equations (6.40) and (6.41) are valid for t ≤ tt.

6.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (6.25) and (6.26) into Eq. (4.47) gives the equation for the forward characteristic of the rising phase for a channel with upstream inflow:

⎛ nc H 2 9 t = 0.0272⎜⎜ 1 2 ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

⎡ (q L xc + Qu )9 13 − Qu9 13 ⎤ ⎢ ⎥ qL ⎢⎣ ⎥⎦

For a channel with zero upstream inflow (Qu = 0), Eq. (6.42) reduces to:

(6.42)

Tommy S.W. Wong

84

⎛ 0.0272 ⎞⎛ n H 2 9 x t = ⎜⎜ 4 13 ⎟⎟⎜⎜ c 1 2 c ⎝ qL ⎠⎝ S c

⎞ ⎟⎟ ⎠

9 13

(6.43)

6.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (6.25) and (6.26) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎡⎛ n H 2 9 Ac = 1.632 ⎢⎜⎜ c 1 2 ⎣⎢⎝ S c

⎤ ⎞ ⎟⎟(Qu + q L xc )⎥ ⎠ ⎦⎥

9 13

(6.44)

Substituting Eq. (6.27) into Eq. (6.44) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎡⎛ n y c = 0.721⎢⎜⎜ 1 2 c 1 2 ⎣⎢⎝ S c H

⎤ ⎞ ⎟⎟(Qu + q L xc )⎥ ⎠ ⎦⎥

6 13

(6.45)

Substituting Eqs. (6.25) and (6.26) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎡⎛ n H 2 9 A p = 1.632 ⎢⎜⎜ c 1 2 ⎢⎣⎝ S c

⎤ ⎞ ⎟⎟(Qu + q L L p )⎥ ⎥⎦ ⎠

9 13

(6.46)

Substituting Ac = Ap, and yc = yp into Eq. (6.27) and then substituting it into Eq. (6.46) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎡⎛ n ⎤ ⎞ y c = 0.721⎢⎜⎜ 1 2 c 1 2 ⎟⎟(Qu + q L L p )⎥ ⎣⎢⎝ S c H ⎠ ⎦⎥

6 13

(6.47)

From Eq. (4.51), the distance Lp is:

Lp =

Q p − Qu qL

For a channel with zero upstream inflow (Qu = 0), Eqs. (6.44)-(6.48) reduce to:

(6.48)

Kinematic-Wave Rainfall-Runoff Formulas

⎛ n H 2 9q x Ac = 1.632⎜⎜ c 1 2 L c Sc ⎝ ⎛ nq x y c = 0.721⎜⎜ 1c 2 L 1c2 ⎝ Sc H

⎞ ⎟⎟ ⎠

⎞ ⎟⎟ ⎠

85

9 13

(6.49)

6 13

(6.50)

which are valid for 0 ≤ xo ≤ Lp,

⎛ nc H 2 9 q L L p A p = 1.632⎜ ⎜ S c1 2 ⎝ ⎛ nc q L L p y c = 0.721⎜⎜ 1 2 1 2 ⎝ Sc H

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

9 13

(6.51)

6 13

(6.52)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp

(6.53)

qL

6.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (6.25) and (6.26) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow:

td =

9 13 ⎤ ⎛ S c1 2 ⎞ ⎡ ⎛ nc H 2 9Qu ⎞ ⎟ ⎜ ⎟ ⎢ Lc q L + Qu − 0.493⎜⎜ 1 . 632 + 60 q t L q⎥ 29 ⎟ ⎜ S1 2 ⎟ ⎥⎦ c ⎝ nc H ⎠ ⎢⎣ ⎝ ⎠ 9 13 ⎤ ⎛ S q ⎞⎡ ⎛ nc H 2 9Qu ⎞ ⎟⎟ ⎢1.632⎜⎜ ⎟ 42.73⎜⎜ + 60 q t L q⎥ 12 ⎟ ⎥⎦ ⎝ n H ⎠ ⎢⎣ ⎝ Sc ⎠ 12 c L 29 c

13 9

49

For a channel with zero upstream inflow (Qu = 0), Eq. (6.54) reduces to:

(6.54)

Tommy S.W. Wong

86

⎛ S1 2 ⎞ Lc − 182.5⎜⎜ c 2 9 ⎟⎟q L4 9t q13 9 ⎝ nc H ⎠ td = ⎛ S1 2 ⎞ 49 263.6⎜⎜ c 2 9 ⎟⎟(q Lt q ) ⎝ nc H ⎠

(6.55)

6.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

6.12.1. Partial Equilibrium Discharge Substituting Eqs. (6.25) and (6.26) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎛ S c1 2 Q p = 0.493⎜⎜ 29 ⎝ nc H

⎛ n H 2 9Q ⎞⎡ ⎟⎟ ⎢1.639⎜⎜ c 1 2 u ⎝ Sc ⎠ ⎢⎣

⎞ ⎟⎟ ⎠

9 13

⎤ + 60q L t q ⎥ ⎥⎦

13 9

(6.56)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.56) reduces to:

⎛ S1 2 Q p = 182.5⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟(q L t q )13 9 ⎟ ⎠

(6.57)

Equations (5.56) and (5.57) are valid for tq ≤ t ≤ (tq+ td).

6.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc

(5.58)

For a channel with zero upstream inflow (Qu = 0), Eq. (5.58) reduces to:

Q e = q L Lc

(5.59)

Kinematic-Wave Rainfall-Runoff Formulas

87

Equations (5.58) and (5.59) are valid for tt ≤ t ≤ tq.

6.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (6.25) and (6.26) into Eq. (4.71) gives the equation for the equilibrium flow area profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lp:

⎡⎛ n H 2 9 Ac = 1.632 ⎢⎜⎜ c 1 2 ⎣⎢⎝ S c

⎤ ⎞ ⎟⎟(Qu + q L xc )⎥ ⎠ ⎦⎥

9 13

(6.60)

Substituting Eq. (6.27) into Eq. (6.60) gives the equation for the equilibrium water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lp:

⎡⎛ n y c = 0.721⎢⎜⎜ 1 2 c 1 2 ⎢⎣⎝ S c H

⎤ ⎞ ⎟⎟(Qu + q L xc )⎥ ⎥⎦ ⎠

6 13

(6.61)

For a channel with zero upstream inflow (Qu = 0), Eqs. (6.60) and (6.61) reduce to: 9 13

⎛ n H 2 9q x Ac = 1.632⎜⎜ c 1 2 L c Sc ⎝

⎞ ⎟ ⎟ ⎠

⎡⎛ n q x y c = 0.721⎢⎜⎜ 1c 2 L 1c2 ⎢⎣⎝ S c H

6 13

⎞⎤ ⎟⎟⎥ ⎠⎥⎦

(6.62)

(6.63)

which are valid for 0 ≤ xc ≤ Lc.

6.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (6.25) and (6.26) into Eq. (4.78) gives the equilibrium detention storage for a channel with upstream inflow:

⎛n H29 Dec = 0.964⎜⎜ c 1 2 ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

⎡ (Qu + q L Lc )22 13 − Qu22 13 ⎤ ⎥ ⎢ qL ⎣⎢ ⎦⎥

For a channel with zero upstream inflow (Qu = 0), Eq. (6.64) reduces to:

(6.64)

Tommy S.W. Wong

88

⎛ n H 2 9q Dec = 0.964⎜⎜ c 1 2 L ⎝ Sc

⎞ ⎟ ⎟ ⎠

9 13 13 L22 c

(6.65)

6.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (6.25) and (6.26) into Eq. (4.91) gives the equation for the falling phase of the flow area profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc.

⎛ S c1 2 Ac4 9 xc = 42.73⎜⎜ 29 ⎝ nc H

⎫ ⎧⎡ ⎛ S c1 2 Ac13 9 ⎞⎤ ⎟ Q − ⎪ ⎪ ⎢0.493⎜⎜ ⎥ u nc H 2 9 ⎟⎠⎦⎥ ⎞ ⎪ ⎪ ⎣⎢ ⎝ ⎟⎟(t − t q ) + ⎨ ⎬ qL ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

(6.66)

Substituting Eq. (6.27) into Eq. (6.66) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc: 6 ⎧⎡ ⎛ S 1 2 H 1 2 y 13 c 2.033⎜⎜ c ⎪ ⎢ nc ⎛ S1 2 y2 3 ⎞ ⎪⎢ ⎝ xc = 66.08⎜⎜ c c ⎟⎟(t − t q ) + ⎨ ⎣ qL ⎝ nc ⎠ ⎪ ⎪ ⎩

⎫ ⎞⎤ ⎟⎟⎥ − Qu ⎪ ⎪ ⎠⎦⎥ ⎬ ⎪ ⎪ ⎭

(6.67)

From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

⎛ n H 2 9Q Ac = 1.632⎜⎜ c 1 2 u ⎝ Sc

⎞ ⎟ ⎟ ⎠

9 13

(6.68)

Substituting Eq. (6.27) into Eq. (6.68) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf:

⎛ nQ y c = 0.721⎜⎜ 1 2c u1 2 ⎝ Sc H

⎞ ⎟ ⎟ ⎠

6 13

(6.69)

Substituting Eqs. (6.25) and (6.26) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ S1 2 L f = 53.12⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟ ⎟ ⎠

9 13

Qu4 13 (t − t q )

89

(6.70)

For a channel with zero upstream inflow (Qu = 0), Eqs. (6.66) and (6.67) reduce to:

⎛ S 1 2 A4 9 xc = 42.73⎜⎜ c 2c 9 ⎝ nc H

⎡ ⎛ S c1 2 Ac13 9 ⎜⎜ . 0 493 ⎢ nc H 2 9 ⎞ ⎝ ⎢ ⎟⎟(t − t q ) + ⎢ qL ⎠ ⎢ ⎢⎣

⎞⎤ ⎟⎟ ⎥ ⎠⎥ ⎥ ⎥ ⎥⎦

⎡ ⎛ S c1 2 H 1 2 y c13 6 ⎜⎜ . 2 033 ⎢ nc ⎛ S c1 2 y c2 3 ⎞ ⎝ ⎟⎟(t − t q ) + ⎢ xc = 66.08⎜⎜ ⎢ qL ⎝ nc ⎠ ⎢ ⎢⎣

(6.71)

⎞⎤ ⎟⎟ ⎥ ⎠⎥ ⎥ ⎥ ⎥⎦

(6.72)

which are valid for 0 ≤ xc ≤ Lc.

6.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (6.25) and (6.26) into Eq. (4.101) gives the equation for the falling phase of a hydrograph for a channel with upstream inflow:

0.0188 ⎛ nc H 2 9 ⎜ t= Qc4 13 ⎜⎝ S c1 2

⎞ ⎟⎟ ⎠

9 13

⎡ ⎛ Qc − Qu ⎢ Lc − ⎜⎜ ⎝ qL ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

(6.73)

For a channel with zero upstream inflow (Qu = 0), Eq. (6.73) reduces to:

0.0188 ⎛ nc H 2 9 ⎜ t= Qc4 13 ⎜⎝ S c1 2

⎞ ⎟ ⎟ ⎠

9 13

⎡ ⎛ Qc ⎞⎤ ⎢ Lc − ⎜⎜ ⎟⎟⎥ + t q ⎝ q L ⎠⎦ ⎣

(6.74)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 91-100

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 7

7. WORKING FORMULAS FOR FLOW IN RECTANGULAR (DEEP) CHANNEL In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a rectangular (deep) channel are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

7.1. KINEMATIC WAVE PARAMETERS For flow in a rectangular channel, the flow area Ac, and the wetted perimeter P, are related to the channel width W, and the flow depth yc, as follows:

Ac = Wy c

(7.1)

P = W + 2 yc

(7.2)

Substituting Eqs. (7.1) and (7.2) and A = Ac into Eq. (3.2) gives:

R=

Wy c W + 2 yc

(7.3)

Rearranging Eq. (7.3) gives:

R=

W W +2 yc

(7.4)

Tommy S.W. Wong

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For a rectangular deep channel, as shown in figure 7.1, yc >> W and Eq. (7.4) reduces to:

R≈

W 2

(7.5)

yc

W Figure 7.1. Cross-section of Rectangular (Deep) Channel.

Substituting Eq. (7.5) and Q = Q c, S =S c, A = Ac, n = nc into Eq. (3.1) gives:

⎛ S 1 2W 2 3 ⎞ ⎟⎟ Ac Qc = 0.630⎜⎜ c ⎝ nc ⎠

(7.6)

A comparison of Eq. (7.6) with Eq. (4.7) gives the kinematic wave parameters (Wong 2002, Wong and Zhou 2006):

⎛ S c1 2W 2 3 ⎞ ⎟⎟ ⎝ nc ⎠

α c = 0.630⎜⎜

(7.7)

βc = 1

(7.8)

Kinematic-Wave Rainfall-Runoff Formulas

93

7.2. FLOW DEPTH Substituting Eqs. (7.1) into Eq. (7.6) gives:

⎛ S 1 2W 5 3 yc ⎞ ⎟⎟ Qc = 0.630⎜⎜ c nc ⎝ ⎠

(7.9)

Rearranging Eq. (7.9) gives the equation for the flow depth for a channel with and without upstream inflow:

⎛ nQ ⎞ yc = 1.587⎜⎜ 1 2c c5 3 ⎟⎟ ⎝ Sc W ⎠

(7.10)

7.3. FLOW VELOCITY Substituting Eqs. (7.7) and (7.8) into Eq. (4.12) gives the equation for the flow velocity along the equilibrium profile for a channel with and without upstream inflow:

⎛ S 1 2W 2 3 ⎞ ⎟⎟ v = 0.630⎜⎜ c ⎠ ⎝ nc

(7.11)

7.4. AVERAGE FLOW VELOCITY Substituting Eqs. (7.7) and (7.8) into Eq. (4.15) gives the equation for the average flow velocity for a channel with and without upstream inflow:

⎛ S 1 2W 2 3 ⎞ ⎟⎟ vav = 0.630⎜⎜ c ⎝ nc ⎠

(7.12)

7.5. KINEMATIC WAVE CELERITY Substituting Eqs. (7.7) and (7.8) into Eq. (4.29) gives the working equation for the wave celerity along the equilibrium profile for a channel with and without upstream inflow:

⎛ S c1 2W 2 3 ⎞ ⎟⎟ ck = 0.630⎜⎜ n c ⎝ ⎠

(7.13)

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7.6. AVERAGE WAVE CELERITY Substituting Eqs. (7.7) and (7.8) into Eq. (4.32) gives the working equation for the average wave celerity for a channel with and without upstream inflow:

⎛ S c1 2W 2 3 ⎞ ⎟⎟ cav = 0.630⎜⎜ n c ⎝ ⎠

(7.14)

7.7. TIME OF TRAVEL Substituting Eqs. (7.7) and (7.8) into Eq. (4.35) gives the formula for the time of travel for a channel with and without upstream inflow:

⎛ nL ⎞ tt = 0.0265⎜⎜ 1 2c c2 3 ⎟⎟ ⎝ Sc W ⎠

(7.15)

7.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (7.7) and (7.8) into Eq. (4.45) gives the equation for the rising phase of the hydrograph for a channel with upstream inflow:

⎤ ⎛ S 1 2W 2 3 ⎞ ⎡ ⎛ nQ ⎞ ⎟⎟ ⎢1.587⎜⎜ 1 2c u2 3 ⎟⎟ + 60qL t ⎥ Qc = 0.630⎜⎜ c ⎝ nc ⎠ ⎣ ⎝ Sc W ⎠ ⎦

(7.16)

For a channel with zero upstream inflow (Qu = 0), Eq. (7.16) reduces to:

⎛ S 1 2W 2 3 ⎞ ⎟⎟ qL t Qc = 37.80⎜⎜ c n c ⎝ ⎠

(7.17)

Equations (7.16) and (7.17) are valid for t ≤ tt.

7.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (7.7) and (7.8) into Eq. (4.47) gives the equation for the forward characteristic of the rising phase for a channel with and without upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ nx ⎞ t = 0.0265⎜⎜ 1 2c c2 3 ⎟⎟ ⎝ Sc W ⎠

95

(7.18)

7.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (7.7) and (7.8) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎛ n ⎞ Ac = 1.587⎜⎜ 1 2 c 2 3 ⎟⎟(Qu + q L xc ) ⎝ Sc W ⎠

(7.19)

Substituting Eq. (7.1) into Eq. (7.19) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎛ n ⎞ yc = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟(Qu + q L xc ) ⎝ Sc W ⎠

(7.20)

Substituting Eqs. (7.7) and (7.8) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎛ n ⎞ Ap = 1.587⎜⎜ 1 2 c 2 3 ⎟⎟(Qu + q L L p ) ⎝ Sc W ⎠

(7.21)

Substituting Ac = Ap, and yc = yp into Eq. (7.1) and then substituting it into Eq. (7.21) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎛ n ⎞ y p = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟(Qu + qL L p ) ⎝ Sc W ⎠

(7.22)

From Eq. (4.51), the distance Lp is:

Lp =

Q p − Qu qL

For a channel with zero upstream inflow (Qu = 0), Eqs. (7.19)-(7.23) reduce to:

(7.23)

Tommy S.W. Wong

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⎛ n ⎞ Ac = 1.587⎜⎜ 1 2 c 2 3 ⎟⎟q L xc ⎝ Sc W ⎠

(7.24)

⎛ n ⎞ yc = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟q L xc ⎝ Sc W ⎠

(7.25)

which are valid for 0 ≤ xo ≤ Lp , and

⎛ n ⎞ Ap = 1.587⎜⎜ 1 2 c 2 3 ⎟⎟qL L p ⎝ Sc W ⎠

(7.26)

⎛ n ⎞ y p = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟qL L p ⎝ Sc W ⎠

(7.27)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp qL

(7.28)

7.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (7.7) and (7.8) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow:

⎤ ⎛ S 1 2W 2 3 ⎞ ⎡ ⎛ nQ ⎞ ⎟⎟ ⎢1.587⎜⎜ 1 2c u2 3 ⎟⎟ + 60q Lt q ⎥ Lc q L + Qu − 0.630⎜⎜ c ⎝ nc ⎠ ⎣ ⎝ Sc W ⎠ ⎦ td = 12 23 ⎛ S W qL ⎞ ⎟⎟ 37.80⎜⎜ c nc ⎝ ⎠

(7.29)

For a channel with zero upstream inflow (Qu = 0), Eq. (7.29) reduces to:

⎛ S 1 2W 2 3 ⎞ ⎟tq Lc − 37.80⎜⎜ c nc ⎟⎠ ⎝ td = ⎛ S 1 2W 2 3 ⎞ ⎟⎟ 37.80⎜⎜ c ⎝ nc ⎠

(7.30)

Kinematic-Wave Rainfall-Runoff Formulas

97

7.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

7.12.1. Partial Equilibrium Discharge Substituting Eqs. (7.7) and (7.8) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎤ ⎛ S c1 2W 2 3 ⎞ ⎡ ⎛ nQ ⎞ ⎟⎟ ⎢1.587⎜⎜ 1 2c u2 3 ⎟⎟ + 60q L t q ⎥ Q p = 0.630⎜⎜ ⎝ nc ⎠ ⎣⎢ ⎝ Sc W ⎠ ⎦⎥

(7.31)

For a channel with zero upstream inflow (Qu = 0), Eq. (7.31) reduces to:

⎛ S c1 2W 2 3 ⎞ ⎟⎟q L t q Q p = 37.80⎜⎜ ⎠ ⎝ nc

(7.32)

Equations (7.31) and (7.32) are valid for tq ≤ t ≤ (tq + td).

7.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc

(7.33)

For a channel with zero upstream inflow (Qu = 0), Eq. (7.33) reduces to:

Q e = q L Lc

(7.34)

Equations (7.33) and (7.34) are valid for tt ≤ t ≤ tq.

7.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (7.7) and (7.8) into Eq. (4.71) gives the equation for the equilibrium flow area profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lp:

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⎛ n ⎞ Ac = 1.587⎜⎜ 1 2 c 2 3 ⎟⎟(Qu + q L xc ) ⎝ Sc W ⎠

(7.35)

Substituting Eq. (7.1) into Eq. (7.35) gives the equation for the equilibrium water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lp:

⎛ n ⎞ yc = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟(Qu + q L xc ) ⎝ Sc W ⎠

(7.36)

For a channel with zero upstream inflow (Qu = 0), Eqs. (7.35) and (7.36) reduce to:

⎛ n ⎞ Ac = 1.587⎜⎜ 1 2 c 2 3 ⎟⎟q L xc ⎝ Sc W ⎠

(7.37)

⎛ n ⎞ yc = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟q L xc ⎝ Sc W ⎠

(7.38)

which are valid for 0 ≤ xc ≤ Lc.

7.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (7.7) and (7.8) into Eq. (4.78) gives the equilibrium detention storage for a channel with upstream inflow:

⎛ n ⎞ ⎡ (Q + qL Lc )2 − Qu2 ⎤ Dec = 0.794⎜⎜ 1 2 c 2 3 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc W ⎠ ⎣ ⎦

(7.39)

For a channel with zero upstream inflow (Qu = 0), Eq. (7.39) reduces to:

⎛ n ⎞ Dec = 0.794⎜⎜ 1 2 c 2 3 ⎟⎟qL L2c ⎝ Sc W ⎠

(7.40)

7.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (7.7) and (7.8) into Eq. (4.91) gives the equation for the falling phase of the flow area profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc.

Kinematic-Wave Rainfall-Runoff Formulas

⎫ ⎧⎡ ⎛ S c1 2W 2 3 Ac ⎞⎤ ⎜⎜ ⎟⎟⎥ − Qu ⎪ . 0 630 ⎪ ⎢ nc ⎛ S c1 2W 2 3 ⎞ ⎪ ⎪ ⎝ ⎠⎦ ⎟⎟(t − t q ) + ⎨ ⎣ xc = 37.80⎜⎜ ⎬ qL ⎝ nc ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

99

(7.41)

Substituting Eq. (7.1) into Eq. (7.41) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

⎫ ⎧⎡ ⎛ S c1 2W 5 3 yc ⎞⎤ ⎜⎜ ⎟⎟⎥ − Qu ⎪ 0 630 . ⎪ ⎢ nc ⎛ S 1 2W 2 3 ⎞ ⎪ ⎪ ⎝ ⎠⎦ ⎟⎟(t − t q ) + ⎨ ⎣ xc = 37.80⎜⎜ c ⎬ qL ⎝ nc ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

(7.42)

From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

⎛ nQ ⎞ Ac = 1.587⎜⎜ 1 2c u2 3 ⎟⎟ ⎝ Sc W ⎠

(7.43)

Substituting Eq. (7.1) into Eq. (7.43) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf:

⎛ nQ ⎞ y c = 1.587⎜⎜ 1 2c u5 3 ⎟⎟ ⎝ Sc W ⎠

(7.44)

Substituting Eqs. (7.7) and (7.8) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq:

⎛ S 1 2W 2 3 ⎞ ⎟⎟(t − t q ) L f = 37.80⎜⎜ c ⎝ nc ⎠

(7.45)

For a channel with zero upstream inflow (Qu = 0), Eq. (7.41) and (7.42) reduce to:

⎡ ⎛ S c1 2W 2 3 Ac ⎞ ⎤ ⎜⎜ ⎟⎟ ⎥ . 0 630 ⎢ n ⎛ S c1 2W 2 3 ⎞ c ⎝ ⎠⎥ ⎟⎟(t − t q ) + ⎢ xc = 37.80⎜⎜ ⎢ ⎥ qL ⎝ nc ⎠ ⎢ ⎥ ⎣⎢ ⎦⎥

(7.46)

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⎡ ⎛ Sc1 2W 5 3 yc ⎞ ⎤ ⎟⎟ ⎥ ⎢ 0.630⎜⎜ n ⎛ Sc1 2W 2 3 ⎞ c ⎝ ⎠⎥ ⎟⎟(t − tq ) + ⎢ xc = 37.80⎜⎜ ⎢ ⎥ n q c L ⎠ ⎝ ⎢ ⎥ ⎢⎣ ⎥⎦

(7.47)

which are valid for 0 ≤ xc ≤ Lc.

7.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (7.7) and (7.8) into Eq. (4.101) gives the equation for the falling phase (falling limb) of a hydrograph for a channel with upstream inflow:

⎛ n ⎞⎡ ⎛ Q − Qu t = 0.0265⎜⎜ 1 2 c 2 3 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎝ qL ⎝ Sc W ⎠⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

(7.48)

For a channel with zero upstream inflow (Qu = 0), Eq. (7.48) reduces to:

⎛ n ⎞⎡ ⎛ Q ⎞⎤ t = 0.0265⎜⎜ 1 2 c 2 3 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎟⎟⎥ + t q ⎝ q L ⎠⎦ ⎝ Sc W ⎠⎣

(7.49)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 101-111

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 8

8. WORKING FORMULAS FOR FLOW IN RECTANGULAR (SQUARE) CHANNEL In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a rectangular (square) channel are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

8.1. KINEMATIC WAVE PARAMETERS For a rectangular square channel, as shown in figure 8.1, yc = W and Eq. (7.3) reduces to:

R=

yc 3

(8.1)

yc

W Figure 8.1. Cross-section of Rectangular (Square) Channel.

Tommy S.W. Wong

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Substituting Eq. (7.1) into Eq. (8.1) gives:

R=

A 3W

(8.2)

Substituting Eq. (8.2) and Q = Q c , S =S c, A = Ac, n = nc into (3.1) gives:

⎛ S1 2 Qc = 0.481⎜⎜ c ⎝ nc

⎞ 43 ⎟⎟ Ac ⎠

(8.3)

A comparison of Eq. (8.3) with Eq. (4.7) gives the kinematic wave parameters (Wong 2002, Wong and Zhou 2006):

⎛ S c1 2 ⎞ ⎟ ⎟ n c ⎝ ⎠

α c = 0.481⎜⎜

(8.4)

4 3

(8.5)

βc =

8.2. FLOW DEPTH Substituting W = yc into Eq. (7.1) gives:

Ac = yc2

(8.6)

Substituting Eq. (8.6) into Eq. (8.3) and rearranging gives the equation for the flow depth for a channel with and without upstream inflow:

⎛nQ yc = 1.316⎜⎜ c 1 2c ⎝ Sc

⎞ ⎟⎟ ⎠

38

(8.7)

8.3. FLOW VELOCITY Substituting Eqs. (8.4) and (8.5) into Eq. (4.11) gives the equation for the flow velocity along the equilibrium profile for a channel with upstream inflow:

⎛ S1 2 ⎞ v = 0.578⎜⎜ c ⎟⎟ ⎝ nc ⎠

34

(Qu + qL xc )1 4

(8.8)

Kinematic-Wave Rainfall-Runoff Formulas

103

For a channel with zero upstream inflow (Qu = 0), Eq. (8.8) reduces to:

⎛ S1 2 ⎞ v = 0.578⎜⎜ c ⎟⎟ ⎝ nc ⎠

34

(qL xc )1 4

(8.9)

8.4. AVERAGE FLOW VELOCITY Substituting Eqs. (8.4) and (8.5) into Eq. (4.15) gives the equation for the average flow velocity for a channel with upstream inflow: 34

⎛ S1 2 ⎞ 0.433⎜⎜ c ⎟⎟ q L Lc ⎝ nc ⎠ vav = (Qu + qL Lc )3 4 − Qu 3 4

(8.10)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.10) reduces to:

⎛ S c1 2 v av = 0.433⎜⎜ ⎝ nc

⎞ ⎟⎟ ⎠

34

(q L Lc )1 4

(8.11)

8.5. KINEMATIC WAVE CELERITY Substituting Eqs. (8.4) and (8.5) into Eq. (4.29) gives the equation for the wave celerity along the equilibrium profile for a channel with upstream inflow:

⎛ S1 2 ⎞ ck = 0.770⎜⎜ c ⎟⎟ ⎝ nc ⎠

34

(Qu + qL xc )1 4

(8.12)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.12) reduces to:

⎛ S1 2 ⎞ ck = 0.770⎜⎜ c ⎟⎟ ⎝ nc ⎠

34

(qL xc )1 4

(8.13)

8.6. AVERAGE WAVE CELERITY Substituting Eqs. (8.4) and (8.5) into Eq. (4.32) gives the equation for the wave celerity for a channel with upstream inflow:

Tommy S.W. Wong

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⎛ S1 2 ⎞ 0.578⎜⎜ c ⎟⎟ q L Lc ⎝ nc ⎠ cav = (Qu + qL Lc )3 4 − Qu 3 4

(8.14)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.14) reduces to:

⎛ S1 2 c av = 0.578⎜⎜ c ⎝ nc

⎞ ⎟⎟ ⎠

34

(q L Lc )1 4

(8.15)

8.7. TIME OF TRAVEL Substituting Eqs. (8.4) and (8.5) into Eq. (4.35) gives the equation for the time of travel for a channel with upstream inflow:

⎛ n t t = 0.0289⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎡ (Qu + q L Lc )3 4 − Qu3 4 ⎤ ⎥ ⎢ qL ⎥⎦ ⎢⎣

(8.16)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.16) reduces to:

0.0289 ⎛ nc Lc ⎞ ⎜ ⎟ tt = q1L 4 ⎜⎝ S c1 2 ⎟⎠

34

(8.17)

8.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (8.4) and (8.5) into Eq. (4.44) gives the equation for the rising phase of the hydrograph for a channel with upstream inflow:

⎛ S 1 2 ⎞⎡ ⎛n Q Qc = 0.481⎜⎜ c ⎟⎟ ⎢1.731⎜⎜ c 1 2u ⎝ nc ⎠ ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎤ + 60qL t ⎥ ⎥⎦

43

(8.18)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.18) reduces to:

⎛ S c1 2 Qc = 113.0⎜⎜ ⎝ nc

⎞ ⎟⎟(q L t )4 3 ⎠

Equations (8.18) and (8.19) are valid for t ≤ tt.

(8.19)

Kinematic-Wave Rainfall-Runoff Formulas

105

8.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (8.4) and (8.5) into Eq. (4.47) gives the equation for the forward characteristic of the rising phase for a channel with upstream inflow:

⎛ n t = 0.0289⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎡ (Qu + q L xc )3 4 − Qu3 4 ⎤ ⎥ ⎢ qL ⎥⎦ ⎢⎣

(8.20)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.20) reduces to:

0.0289 ⎛ nc xc ⎞ ⎜ ⎟ t= q1L 4 ⎜⎝ S c1 2 ⎟⎠

34

(8.21)

8.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (8.4) and (8.5) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎤ ⎡⎛ n ⎞ Ac = 1.731⎢⎜⎜ 1c2 ⎟⎟(Qu + qL xc )⎥ ⎦ ⎣⎝ Sc ⎠

34

(8.22)

Substituting Eq. (8.6) into Eq. (8.22) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎤ ⎡⎛ n ⎞ yc = 1.316 ⎢⎜⎜ 1c2 ⎟⎟(Qu + q L xc )⎥ ⎦ ⎣⎝ S c ⎠

38

(8.23)

Substituting Eqs. (8.4) and (8.5) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc.

⎤ ⎡⎛ n ⎞ Ap = 1.731⎢⎜⎜ 1c2 ⎟⎟(Qu + qL L p )⎥ ⎦ ⎣⎝ S c ⎠

34

(8.24)

Substituting Ac = Ap, and yc = yp into Eq. (8.6) and then substituting it into Eq. (8.24) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

Tommy S.W. Wong

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⎤ ⎡⎛ n ⎞ y p = 1.316 ⎢⎜⎜ 1c2 ⎟⎟(Qu + q L L p )⎥ ⎦ ⎣⎝ S c ⎠

38

(8.25)

From Eq. (4.51), the distance Lp is:

Lp =

Q p − Qu

(8.26)

qL

For a channel with zero upstream inflow (Qu = 0), Eqs. (8.22)-(8.26) reduce to:

⎤ ⎡⎛ n ⎞ Ac = 1.731⎢⎜⎜ 1c2 ⎟⎟q L xc ⎥ ⎦ ⎣⎝ S c ⎠ ⎤ ⎡⎛ n ⎞ yc = 1.316⎢⎜⎜ 1c2 ⎟⎟qL xc ⎥ ⎦ ⎣⎝ Sc ⎠

34

(8.27)

38

(8.28)

which are valid for 0 ≤ xc ≤ Lp ,

⎤ ⎡⎛ n ⎞ Ap = 1.731⎢⎜⎜ 1c2 ⎟⎟qL L p ⎥ ⎦ ⎣⎝ S c ⎠ ⎤ ⎡⎛ n ⎞ y p = 1.316⎢⎜⎜ 1c2 ⎟⎟qL L p ⎥ ⎦ ⎣⎝ Sc ⎠

34

(8.29)

38

(8.30)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp qL

(8.31)

8.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (8.4) and (8.5) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

td =

⎛ S 1 2 ⎞⎡ ⎛n Q Lc q L + Qu − 0.481⎜⎜ c ⎟⎟ ⎢1.731⎜⎜ c 1 2u ⎝ nc ⎠ ⎢⎣ ⎝ Sc

⎤ + 60qL t q ⎥ ⎥⎦ 1 / 3 34 ⎤ ⎛ S c1 2 q L ⎞ ⎡ ⎛ ncQu ⎞ ⎟⎟ ⎢1.731⎜⎜ 1 2 ⎟⎟ + 60qL t q ⎥ 38.48⎜⎜ ⎥⎦ ⎝ nc ⎠ ⎢⎣ ⎝ Sc ⎠ ⎞ ⎟⎟ ⎠

34

107

43

(8.32)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.32) reduces to:

⎛ S c1 2 ⎞ 1 3 4 3 ⎟qL t q Lc − 0.481⎜⎜ nc ⎟⎠ ⎝ td = ⎛ S c1 2 ⎞ ⎟⎟(qL t q )1 3 38.48⎜⎜ ⎝ nc ⎠

(8.33)

8.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

8.12.1. Partial Equilibrium Discharge Substituting Eqs. (8.4) and (8.5) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎛ S c1 2 Q p = 0.481⎜⎜ ⎝ nc

⎞⎡ ⎛n Q ⎟⎟ ⎢1.731⎜⎜ c 1 2u ⎠ ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎤ + 60q L t q ⎥ ⎥⎦

43

(8.34)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.34) reduces to:

⎛ S c1 2 Q p = 113.0⎜⎜ ⎝ nc

⎞ ⎟⎟(q L t q )4 3 ⎠

Equations (8.34) and (8.35) are valid for tq ≤ t ≤ (tq + td).

(8.35)

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8.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc

(8.36)

For a channel with zero upstream inflow (Qu = 0), Eq. (8.36) reduces to:

Q e = q L Lc

(8.37)

Equations (8.36) and (8.37) are valid for tt ≤ t ≤ tq.

8.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (8.4) and (8.5) into Eq. (4.71) gives the equation for the equilibrium flow area profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lc.

⎤ ⎡⎛ n ⎞ Ac = 1.731⎢⎜⎜ 1c2 ⎟⎟(Qu + qL xc )⎥ ⎦ ⎣⎝ Sc ⎠

34

(8.38)

Substituting Eq. (8.6) into Eq. (8.38) gives the equation for the equilibrium water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lc:

⎤ ⎡⎛ n ⎞ yc = 1.316 ⎢⎜⎜ 1c2 ⎟⎟(Qu + q L xc )⎥ ⎦ ⎣⎝ S c ⎠

38

(8.39)

For a channel with zero upstream inflow (Qu = 0), Eqs. (8.38) and (8.39) reduce to:

⎤ ⎡⎛ n ⎞ Ac = 1.731⎢⎜⎜ 1c2 ⎟⎟q L xc ⎥ ⎦ ⎣⎝ S c ⎠ ⎤ ⎡⎛ n ⎞ yc = 1.316⎢⎜⎜ 1c2 ⎟⎟qL xc ⎥ ⎦ ⎣⎝ Sc ⎠ which are valid for 0 ≤ xc ≤ Lc.

34

(8.40)

38

(8.41)

Kinematic-Wave Rainfall-Runoff Formulas

109

8.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (8.4) and (8.5) into Eq. (4.78) gives the equilibrium detention storage for a channel with upstream inflow:

⎛ n Dec = 0.989⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

3/ 4

⎡ (Qu + q L Lc )7 / 4 − Qu7 / 4 ⎤ ⎥ ⎢ qL ⎥⎦ ⎢⎣

(8.42)

For a channel with zero upstream inflow (i.e. Qu = 0), Eq. (8.42) reduces to:

⎛ n Dec = 0.989⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

3/ 4

q L3 / 4 Lc

7/4

(8.43)

8.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (8.4) and (8.5) into Eq. (4.91) gives the equation for the falling phase of the flow area profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

⎫ ⎧⎡ ⎛ S c1 2 Ac4 3 ⎞⎤ ⎜⎜ ⎟⎟⎥ − Qu ⎪ 0 481 . ⎪ ⎢ ⎛ S 1 2 A1 3 ⎞ ⎪ ⎪ ⎝ nc ⎠⎦ xc = 38.48⎜⎜ c c ⎟⎟(t − t q ) + ⎨ ⎣ ⎬ qL ⎝ nc ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

(8.44)

Substituting Eq. (8.6) into Eq. (8.44) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

⎫ ⎧⎡ ⎛ S c1 2 yc 8 3 ⎞⎤ ⎟⎥ − Qu ⎪ ⎜ 0 481 . ⎪ ⎢ ⎟ ⎜ n ⎛ S1 2 y 2 3 ⎞ ⎪ ⎪⎢ c ⎠⎦⎥ ⎝ xc = 38.48⎜⎜ c c ⎟⎟(t − t q ) + ⎨ ⎣ ⎬ qL ⎝ nc ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

(8.45)

From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

⎛n Q Ac = 1.731⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟⎟ ⎠

34

(8.46)

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Substituting Eq. (8.6) into Eq. (8.46) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf:

⎛n Q y c = 1.316⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟⎟ ⎠

38

(8.47)

Substituting Eqs. (8.4) and (8.5) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq: 34

⎛ S1 2 ⎞ L f = 46.21⎜⎜ c ⎟⎟ Qu1 4 (t − t q ) ⎝ nc ⎠

(8.48)

For a channel with zero upstream inflow (Qu = 0), Eqs. (8.44) and (8.45) reduce to:

⎡ ⎛ S c1 2 Ac4 3 ⎞ ⎤ ⎜⎜ ⎟⎟ ⎥ 0 481 . ⎢ n ⎛ Sc1 2 Ac1 3 ⎞ c ⎝ ⎠⎥ ⎟⎟(t − t q ) + ⎢ xc = 38.48⎜⎜ ⎢ ⎥ qL ⎝ nc ⎠ ⎢ ⎥ ⎣⎢ ⎦⎥

(8.49)

⎡ ⎛ S c1 2 yc 8 3 ⎞ ⎤ ⎟⎥ ⎢ 0.481⎜⎜ nc ⎟⎠ ⎥ ⎛ S c1 2 yc 2 3 ⎞ ⎢ ⎝ ⎟(t − t q ) + xc = 38.48⎜⎜ ⎟ ⎢ ⎥ n qL c ⎝ ⎠ ⎢ ⎥ ⎢⎣ ⎥⎦

(8.50)

which are valid for 0 ≤ xc ≤ Lc.

8.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (8.4) and (8.5) into Eq. (4.101) gives the equation for the falling phase (falling limb) of a hydrograph for a channel with upstream inflow: 34

⎛ 0.0216 ⎞⎛ n ⎞ ⎡ ⎛ Q − Qu ⎞⎤ ⎟⎟⎥ + tq t = ⎜⎜ 1 4 ⎟⎟⎜⎜ 1c2 ⎟⎟ ⎢ Lc − ⎜⎜ c Q S q L ⎝ ⎠⎦ ⎝ c ⎠⎝ c ⎠ ⎣ For a channel with zero upstream inflow (Qu = 0), Eq. (8.51) reduces to:

(8.51)

Kinematic-Wave Rainfall-Runoff Formulas

111

34

⎛ 0.0216 ⎞⎛ n ⎞ ⎡ ⎛ Q ⎞⎤ t = ⎜⎜ 1 4 ⎟⎟⎜⎜ 1c2 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎟⎟⎥ + t q ⎝ qL ⎠⎦ ⎝ Qc ⎠⎝ S c ⎠ ⎣

(8.52)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 113-123

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 9

9. WORKING FORMULAS FOR FLOW IN RECTANGULAR (WIDE) CHANNEL In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a rectangular (wide) channel are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

9.1. KINEMATIC WAVE PARAMETERS Rearranging Eq. (7.3) gives:

R=

yc 2y 1+ c W

(9.1)

For a rectangular wide channel, as shown in figure 9.1, yc << W and Eq. (9.1) reduces to:

R ≈ yc

Figure 9.1. Cross-section of Rectangular (Wide) Channel.

(9.2)

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Substituting Eq. (7.1) into Eq. (9.2) gives:

R=

A W

(9.3)

Substituting Eq. (9.3) and Q = Q c , S =S c, A = Ac, n = nc into (3.1) gives:

⎛ S1 2 ⎞ Qc = ⎜⎜ c 2 3 ⎟⎟ Ac5 3 ⎝ ncW ⎠

(9.4)

A comparison of Eq. (9.4) with Eq. (4.7) gives the kinematic wave parameters (Wong 2002, Wong and Zhou 2006):

⎛ S1 2 ⎞

α c = ⎜⎜ c 2 3 ⎟⎟ ⎝ ncW ⎠

βc =

(9.5)

5 3

(9.6)

9.2. FLOW DEPTH Substituting Eqs. (7.1) into Eq. (9.4) gives:

⎛ S1 2 ⎞ Qc = ⎜ c ⎟ Wyc5 3 ⎝ nc ⎠

(9.7)

Rearranging Eq. (9.7) gives the equation for the flow depth for a channel with and without upstream inflow:

⎛ nQ yc = ⎜⎜ 1c 2 c ⎝ Sc W

⎞ ⎟⎟ ⎠

35

(9.8)

9.3. FLOW VELOCITY Substituting Eqs. (9.5) and (9.6) into Eq. (4.12) gives the equation for the flow velocity along the equilibrium profile for a channel with upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ S1 2 ⎞ v = ⎜⎜ c 2 3 ⎟⎟ ⎝ ncW ⎠

35

(Qu + qL xc )2 5

115

(9.9)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.9) reduces to:

⎛ S1 2 ⎞ v = ⎜⎜ c 2 3 ⎟⎟ ⎝ ncW ⎠

35

(qL xc )2 5

(9.10)

9.4. AVERAGE FLOW VELOCITY Substituting Eqs. (9.5) and (9.6) into Eq.(4.15) gives the equation for the average flow velocity for a channel with upstream inflow: 35

⎛ S1 2 ⎞ 0.600⎜⎜ c 2 3 ⎟⎟ q L Lc ⎝ nW ⎠ vav = (Qu + qL Lc )3 5 − Qu 3 5

(9.11)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.11) reduces to:

⎛ S1 2 ⎞ vav = 0.600⎜⎜ c 2 3 ⎟⎟ ⎝ nW ⎠

35

(qL Lc )2 5

(9.12)

9.5. KINEMATIC WAVE CELERITY Substituting Eqs. (9.5) and (9.6) into Eq. (4.29) gives the equation for the wave celerity along the equilibrium profile for a channel with upstream inflow:

⎛ S1 2 ⎞ ck = 1.667⎜⎜ c 2 3 ⎟⎟ ⎝ ncW ⎠

35

(Qu + q L xc )2 5

(9.13)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.13) reduces to:

⎛ Sc1 2 ⎞ ⎟ ck = 1.667⎜⎜ 23 ⎟ n W ⎝ c ⎠

35

(q L xc )2 5

(9.14)

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9.6. AVERAGE WAVE CELERITY Substituting Eqs. (9.5) and (9.6) into Eq. (4.32) gives the equation for the wave celerity for a channel with upstream inflow: 35

⎛ S c1 2 ⎞ ⎜⎜ ⎟ q L Lc ncW 2 3 ⎟⎠ ⎝ cav = (Qu + qL Lc )3 5 − Qu 3 5

(9.15)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.15) reduces to:

⎛ S1 2 ⎞ cav = ⎜⎜ c 2 3 ⎟⎟ ⎝ ncW ⎠

35

(qL Lc )2 5

(9.16)

9.7. TIME OF TRAVEL Substituting Eqs. (9.5) and (9.6) into Eq. (4.35) gives the equation for the time of travel for a channel with upstream inflow:

⎛ n W 2 3 ⎞ ⎡ (Q + qL Lc )3 5 − Qu3 5 ⎤ tt = 0.0167⎜⎜ c 1 2 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc ⎠ ⎣ ⎦ 35

(9.17)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.17) reduces to:

0.0167 ⎛ ncW 2 3 Lc ⎞ ⎜ ⎟ tt = qL2 5 ⎜⎝ S c1 2 ⎟⎠

35

(9.18)

9.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (9.5) and (9.6) into Eq. (4.45) gives the equation for the rising phase (rising limb) of the hydrograph for a channel with upstream inflow: 35 ⎤ ⎛ S c1 2 ⎞ ⎡⎛ nW 2 3Qu ⎞ ⎜ ⎟ ⎥ ⎢ ⎟ 60 Qc = ⎜⎜ + q t L 12 23 ⎟ ⎜ ⎟ ⎥ ⎝ ncW ⎠ ⎢⎣⎝ S c ⎠ ⎦

53

For a channel with zero upstream inflow (Qu = 0), Eq. (9.19) reduces to:

(9.19)

Kinematic-Wave Rainfall-Runoff Formulas

⎛ S1 2 ⎞ 53 Qc = 919.6⎜⎜ c 2 3 ⎟⎟(qL t ) ⎝ ncW ⎠

117

(9.20)

Equations (9.19) and (9.20) are valid for t ≤ tt.

9.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (9.5) and (9.6) into Eq.(4.47) gives the equation for the forward characteristic of the rising phase for a channel with upstream inflow:

⎛ n W 2 3 ⎞ ⎡ (Q + qL xc )3 5 − Qu3 5 ⎤ t = 0.0167⎜⎜ c 1 2 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc ⎠ ⎣ ⎦ 35

(9.21)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.21) reduces to:

0.0167 ⎛ ncW 2 3 xc ⎞ ⎜ ⎟⎟ t= 25 ⎜ 12 qL ⎝ Sc ⎠

35

(9.22)

9.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (9.5) and (9.6) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎤ ⎡⎛ ncW 2 3 ⎞ Ac = ⎢⎜⎜ 1 2 ⎟⎟(Qu + q L xc )⎥ ⎦ ⎣⎝ S c ⎠

35

(9.23)

Substituting Eq. (7.1) into Eq. (9.23) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎡⎛ n yc = ⎢⎜⎜ 1 2c ⎣⎝ S c W

⎤ ⎞ ⎟⎟(Qu + q L xc )⎥ ⎠ ⎦

35

(9.24)

Substituting Eqs. (9.5) and (9.6) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc.

Tommy S.W. Wong

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⎤ ⎡⎛ n W 2 3 ⎞ Ap = ⎢⎜⎜ c 1 2 ⎟⎟(Qu + qL L p )⎥ ⎦ ⎣⎝ S c ⎠

35

(9.25)

Substituting Ac = Ap, and yc = yp into Eq. (7.1) and then substituting it into Eq. (9.25) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎡⎛ n y p = ⎢⎜⎜ 1 2c ⎣⎝ S c W

⎤ ⎞ ⎟⎟(Qu + q L L p )⎥ ⎠ ⎦

35

(9.26)

From Eq. (4.51), the distance Lp is:

Lp =

Q p − Qu

(9.27)

qL

For a channel with zero upstream inflow (Qu = 0), Eqs. (9.23)-(9.27) reduce to:

⎤ ⎡⎛ n W 2 3 ⎞ Ac = ⎢⎜⎜ c 1 2 ⎟⎟q L xc ⎥ ⎦ ⎣⎝ S c ⎠ ⎡⎛ n yc = ⎢⎜⎜ 1 2c ⎣⎝ S c W

⎤ ⎞ ⎟⎟q L xc ⎥ ⎠ ⎦

35

(9.28)

35

(9.29)

which are valid for 0 ≤ xc ≤ Lp, and

⎤ ⎡⎛ n W 2 3 ⎞ Ap = ⎢⎜⎜ c 1 2 ⎟⎟q L L p ⎥ ⎦ ⎣⎝ S c ⎠ ⎡⎛ n y p = ⎢⎜⎜ 1 2c ⎣⎝ S c W

⎤ ⎞ ⎟⎟qL L p ⎥ ⎠ ⎦

35

(9.30)

35

(9.31)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp qL

(9.32)

Kinematic-Wave Rainfall-Runoff Formulas

119

9.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (9.5) and (9.6) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow:

td =

⎛ S 1 2 ⎞ ⎡⎛ n W 2 3Q Lc q L + Qu − ⎜⎜ c 2 3 ⎟⎟ ⎢⎜⎜ c 1 2 u ⎝ ncW ⎠ ⎢⎣⎝ S c

⎤ + 60qL t q ⎥ ⎥⎦ 2 3 35 ⎤ ⎛ S c1 2 q L ⎞ ⎡⎛ ncW 2 3Qu ⎞ ⎟ ⎜ ⎟ ⎢ + 100.0⎜⎜ 60 q t L q⎥ 23 ⎟ ⎜ 12 ⎟ ⎥⎦ ⎝ ncW ⎠ ⎢⎣⎝ S c ⎠ ⎞ ⎟⎟ ⎠

35

53

(9.33)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.29) reduces to:

⎛ S1 2 ⎞ Lc − 919.6⎜⎜ c 2 3 ⎟⎟qL2 3t q5 3 ⎝ ncW ⎠ td = ⎛ S1 2 ⎞ 23 1532.6⎜⎜ c 2 3 ⎟⎟(qL t q ) ⎝ ncW ⎠

(9.34)

9.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

9.12.1. Partial Equilibrium Discharge Substituting Eqs. (9.5) and (9.6) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎛ S 1 2 ⎞ ⎡⎛ nW 2 3 Qu Q p = ⎜⎜ c 2 3 ⎟⎟ ⎢⎜⎜ 12 ⎝ ncW ⎠ ⎢⎣⎝ S c

⎞ ⎟ ⎟ ⎠

35

⎤ + 60q L t q ⎥ ⎥ ⎦

53

(9.35)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.35) reduces to:

⎛ S1 2 ⎞ 53 Q p = 919.6⎜⎜ c 2 3 ⎟⎟(q L t q ) n W ⎠ ⎝ c

(9.36)

Tommy S.W. Wong

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Equations (9.35) and (9.36) are valid for tq ≤ t ≤ (tq + td).

9.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc

(9.37)

For a channel with zero upstream inflow (Qu = 0), Eq. (9.37) reduces to:

Q e = q L Lc

(9.38)

Equations (9.37) and (9.38) are valid for tt ≤ t ≤ tq.

9.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (9.5) and (9.6) into Eq. (4.71) gives the equation for the equilibrium flow area profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lc:

⎤ ⎡⎛ ncW 2 3 ⎞ Ac = ⎢⎜⎜ 1 2 ⎟⎟(Qu + q L xc )⎥ ⎦ ⎣⎝ S c ⎠

35

(9.39)

Substituting Eq. (7.1) into Eq. (9.39) gives the equation for the equilibrium water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lc:

⎡⎛ n yc = ⎢⎜⎜ 1 2c ⎣⎝ S c W

⎤ ⎞ ⎟⎟(Qu + q L xc )⎥ ⎠ ⎦

35

(9.40)

For a channel with zero upstream inflow (Qu = 0), Eqs. (9.39) and (9.40) reduce to:

⎤ ⎡⎛ ncW 2 3 ⎞ Ac = ⎢⎜⎜ 1 2 ⎟⎟q L xc ⎥ ⎦ ⎣⎝ S c ⎠ ⎡⎛ n yc = ⎢⎜⎜ 1 2c ⎣⎝ S c W

⎤ ⎞ ⎟⎟q L xc ⎥ ⎠ ⎦

35

(9.41)

35

(9.42)

Kinematic-Wave Rainfall-Runoff Formulas

121

which are valid for 0 ≤ xc ≤ Lc.

9.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (9.5) and (9.6) into Eq. (4.78) gives the equilibrium detention storage for a channel with upstream inflow:

⎛ n W 2 3 ⎞ ⎡ (Q + q L Lc )8 5 − Qu8 5 ⎤ Dec = 0.625⎜⎜ c 1 2 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc ⎠ ⎣ ⎦ 35

(9.43)

For a channel with zero upstream inflow (i.e. Qu = 0), Eq. (9.43) reduces to: 35

⎛nW23 ⎞ Dec = 0.625⎜⎜ c 1 2 ⎟⎟ qL3 5 L8c 5 ⎝ Sc ⎠

(9.44)

9.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (9.5) and (9.6) into Eq. (4.91) gives the equation for the falling phase of the flow area profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

⎡ ⎛ S c1 2 Ac5 3 ⎞ ⎤ ⎟ − Qu ⎥ ⎢ ⎜⎜ 23 ⎟ 12 23 nW ⎠ ⎛S A ⎞ ⎥ xc = 100.0⎜⎜ c 2c 3 ⎟⎟(t − t q ) + ⎢ ⎝ c ⎢ ⎥ q n W L ⎝ c ⎠ ⎢ ⎥ ⎣⎢ ⎦⎥

(9.45)

Substituting Eq. (7.1) into Eq. (9.45) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

⎡ ⎛ S c1 2Wyc5 3 ⎞ ⎤ ⎟⎟ − Qu ⎥ ⎢ ⎜⎜ 12 23 nc ⎛S y ⎞ ⎠ ⎥ xc = 100.0⎜⎜ c c ⎟⎟(t − t q ) + ⎢ ⎝ ⎢ ⎥ qL ⎝ nc ⎠ ⎢ ⎥ ⎣⎢ ⎦⎥ From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

(9.46)

Tommy S.W. Wong

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⎛ n W 2 3Q Ac = ⎜⎜ c 1 2 u ⎝ Sc

⎞ ⎟⎟ ⎠

35

(9.47)

Substituting Eq. (7.1) into Eq. (9.47) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf:

⎛nQ y c = ⎜⎜ 1c 2 u ⎝ Sc W

⎞ ⎟⎟ ⎠

35

(9.48)

Substituting Eqs. (9.5) and (9.6) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq: 35

⎛ S1 2 ⎞ 25 L f = 100.0⎜⎜ c 2 3 ⎟⎟ Qu (t − t q ) ⎝ ncW ⎠

(9.49)

For a channel with zero upstream inflow (Qu = 0), Eqs. (9.45) and (9.46) reduce to:

⎡ ⎛ S c1 2 Ac5 3 ⎞ ⎤ ⎟⎥ ⎢ ⎜⎜ ncW 2 3 ⎟⎠ ⎥ ⎛ S c1 2 Ac2 3 ⎞ ⎝ ⎢ ⎟(t − t q ) + xc = 100.0⎜⎜ 23 ⎟ ⎢ ⎥ n W qL ⎝ c ⎠ ⎢ ⎥ ⎣⎢ ⎦⎥

(9.50)

⎧ ⎡⎛ S c1 2Wyc5 3 ⎞⎤ ⎫ ⎟⎟⎥ ⎪ ⎪ ⎢⎜⎜ nc ⎛ S c1 2 yc2 3 ⎞ ⎪ ⎣⎝ ⎠⎦ ⎪ ⎟⎟(t − t q ) + ⎨ xc = 100.0⎜⎜ ⎬ qL ⎝ nc ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

(9.51)

which are valid for 0 ≤ xc ≤ Lc.

9.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (9.5) and (9.6) into Eq. (4.101) gives the equation for the falling phase (falling limb) of a hydrograph for a channel with upstream inflow: 35

⎛ 0.0100 ⎞⎛ ncW 2 3 ⎞ ⎡ ⎛ Q − Qu ⎟⎜ 1 2 ⎟⎟ ⎢ Lc − ⎜⎜ c t = ⎜⎜ 2 5 ⎟⎜ ⎝ qL ⎝ Qc ⎠⎝ S c ⎠ ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

(9.52)

Kinematic-Wave Rainfall-Runoff Formulas

123

For a channel with zero upstream inflow (Qu = 0), Eq. (9.52) reduces to: 35

⎛ 0.0100 ⎞⎛ ncW 2 3 ⎞ ⎡ ⎛ Q ⎞⎤ ⎟⎜ 1 2 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎟⎟⎥ + t q t = ⎜⎜ 2 5 ⎟⎜ ⎝ q L ⎠⎦ ⎝ Qc ⎠⎝ S c ⎠ ⎣

(9.53)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 125-136

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 10

10. WORKING FORMULAS FOR FLOW IN TRAPEZOIDAL CHANNEL WITH EQUAL SIDE SLOPES In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a trapezoidal channel with equal side slopes are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

10.1. KINEMATIC WAVE PARAMETERS For flow in a trapezoidal channel with equal side slopes, the flow area Ac, and the wetted perimeter P, are related to the base width W, reciprocal of channel side slope z, and the flow depth yc, as follows:

Ac = zyc2 + Wyc

(

P = 2 yc 1 + z 2

(10.1)

)

12

+W

(10.2)

Figure 10.1 shows the trapezoidal channel with base width W, reciprocal of channel side slope z, and flow depth yc. Next, defining two dimensionless variables, ψ and µ, as:

ψ = Ac W 2

(10.3)

μ = yc W

(10.4)

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1

1 z

yc

z W

Figure 10.1. Cross-section of Trapezoidal Channel with Equal Side Slopes.

Substituting Eqs. (10.3) and (10.4) into Eq. (10.1) gives the following relationship between ψ and µ:

zμ 2 + μ − ψ = 0

(10.5)

For z ≠ 0, the positive solution for Eq. (10.5) is:

μ=

(1 + 4 zψ )1 2 − 1

(10.6)

2z

Substituting Eq. (10.4) into Eq. (10.6) to eliminate µ results:

⎡ (1 + 4 zψ )1 2 − 1⎤ yc = ⎢ ⎥W z 2 ⎦ ⎣

(10.7)

Substituting Eq. (10.7) into Eq. (10.2) gives an expression for P:

(

⎧⎪ 1+ z2 P = ⎨1 + ⎪⎩

) [(1 + 4 zψ ) 12

12

z

]

− 1 ⎫⎪ ⎬W ⎪⎭

(10.8)

Substituting Eqs. (10.3) and (10.8) into Eq. (3.1) results in a dimensionless equation in terms of ψ, which can be considered as the "true" relationship between Qc and Ac for a trapezoidal channel of equal side slopes:

⎧⎪ nc Qc = ⎨ Sc1 2W 8 3 ⎪ z + 1 + z 2 ⎩

(

23

⎫⎪ 53 ⎬ ψ 12 (1 + 4 zψ ) − 1 ⎪⎭

) [ 12

z

]

(10.9)

As shown in figure 10.2, by mathematical fitting to the true relationships for 0.1 ≤ z ≤ 5.0, Wong and Zhou (2006) obtained the following kinematic wave parameters:

Kinematic-Wave Rainfall-Runoff Formulas

⎛

S c1 2 ⎞ ⎟ 0.0909 ⎟ ⎝ ncW ⎠

127

α c = 0.340⎜⎜

(10.10)

β c = 1.379

(10.11)

Figure 10.2. Comparison between True and Kinematic Wave Relationships for Flow in Trapezoidal Channel with Equal Side Slopes.

10.2. FLOW DEPTH Substituting Eqs. (10.10) and (10.11) into Eq. (4.7) gives:

⎛ S c1 2 ⎞ 1.379 ⎟A Qc = 0.340⎜⎜ 0.0909 ⎟ c ⎝ ncW ⎠

(10.12)

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Substituting Eqs. (10.1) into Eq. (10.12) gives:

⎛ S c1 2 ⎞ 2 1.379 ⎟(zyc + Wyc ) Qc = 0.340⎜⎜ 0.0909 ⎟ ⎝ ncW ⎠

(10.13)

Rearranging Eq. (10.13) gives:

⎛ n W 0.0909 Q zy + Wy c − 2.187⎜⎜ c 1 2 c Sc ⎝ 2 c

⎞ ⎟⎟ ⎠

0.725

=0

(10.14)

Solving Eq. (10.14) gives the equation for the flow depth for a channel with and without upstream inflow:

⎡ ⎛ n W 0.0909 Q − W + ⎢W 2 + 8.748 z ⎜⎜ c 1 2 c Sc ⎢⎣ ⎝ yc = 2z

⎞ ⎟⎟ ⎠

0.725 1 2

⎤ ⎥ ⎥⎦

(10.15)

10.3. FLOW VELOCITY Substituting Eqs. (10.10) and (10.11) into Eq. (4.12) gives the equation of flow velocity for a channel with upstream inflow:

⎤ ⎡ ⎛ S c1 2 ⎞ ⎟(Qu + qL xc )0.379 ⎥ v = ⎢0.340⎜⎜ 0.0909 ⎟ ⎝ ncW ⎠ ⎦ ⎣

0.725

(10.16)

For a channel with zero upstream inflow (Qu = 0), Eq. (10.16) reduces to:

⎡ ⎛ S c1 2 ⎞ 0.379 ⎤ ⎟ ( ) v = ⎢0.340⎜⎜ q x ⎥ 0.0909 ⎟ L c ⎝ ncW ⎠ ⎦ ⎣

0.725

(10.17)

10.4. AVERAGE FLOW VELOCITY Substituting Eqs. (10.10) and (10.11) into Eq. (4.15) gives the equation of average flow velocity along the equilibrium profile for a channel with upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

129

0.725

⎛ S c1 2 ⎞ ⎟ qL Lc 0.332⎜⎜ ncW 0.0909 ⎟⎠ ⎝ vav = (Qu + qL Lc )0.725 − Qu 0.725

(10.18)

For a channel with zero upstream inflow (Qu = 0), Eq. (10.18) reduces to:

⎛ S c1 2 ⎞ ⎟ vav = 0.332⎜⎜ 0.0909 ⎟ n W ⎝ c ⎠

0.725

(qL Lc )0.275

(10.19)

10.5. KINEMATIC WAVE CELERITY Substituting Eqs. (10.10) and (10.11) into Eq. (4.29) gives the equation of wave celerity along the equilibrium profile for a channel with upstream inflow:

⎤ ⎡⎛ S c1 2 ⎞ ⎟(Qu + q L xc )0.379 ⎥ ck = 0.630 ⎢⎜⎜ 0.0909 ⎟ ⎠ ⎦ ⎣⎝ ncW

0.725

(10.20)

For a channel with zero upstream inflow (Qu = 0), Eq. (10.20) reduces to:

⎤ ⎡⎛ S c1 2 ⎞ ⎟(q L xc )0.379 ⎥ ck = 0.630⎢⎜⎜ 0.0909 ⎟ ⎠ ⎦ ⎣⎝ ncW

0.725

(10.21)

10.6. AVERAGE WAVE CELERITY Substituting Eqs. (10.10) and (10.11) into Eq. (4.32) gives the equation of wave celerity for a channel with upstream inflow: 0.725

⎛ S c1 2 ⎞ ⎟ qL Lc 0.457⎜⎜ ncW 0.0909 ⎟⎠ ⎝ cav = (Qu + qL Lc )0.725 − Qu 0.725

(10.22)

For a channel with zero upstream inflow (Qu = 0), Eq. (10.22) reduces to:

⎛ Sc1 2 ⎞ ⎟ cav = 0.457⎜⎜ 0.0909 ⎟ ⎝ ncW ⎠

0.725

(qL Lc )0.275

(10.23)

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10.7. TIME OF TRAVEL Substituting Eqs. (10.10) and (10.11) into Eq. (4.35) gives the equation of time of travel for a channel with upstream inflow:

⎛ n W 0.0909 ⎞ tt = 0.0364⎜⎜ c 1 2 ⎟⎟ ⎝ Sc ⎠

0.725

⎡ (Qu + qL Lc )0.725 − Qu0.725 ⎤ ⎢ ⎥ qL ⎣ ⎦

(10.24)

For a channel with zero upstream inflow (Qu = 0), Eq. (10.24) reduces to:

⎛ 0.0364 ⎞⎛ ncW 0.0909 ⎞ ⎟⎟ tt = ⎜⎜ 0.275 ⎟⎟⎜⎜ 12 q S c ⎝ L ⎠⎝ ⎠

0.725

(10.25)

10.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (10.10) and (10.11) into Eq. (4.45) gives the equation for the rising phase of the hydrograph for a channel with upstream inflow:

⎛ S c1 2 ⎞ ⎡ ⎛ ncW 0.0909Qu ⎟ ⎜⎜ ⎢ Qc = 0.340⎜⎜ 2 . 186 0.0909 ⎟ S c1 2 ⎝ ncW ⎠ ⎢⎣ ⎝

⎞ ⎟⎟ ⎠

0.725

⎤ + 60q L t ⎥ ⎥⎦

1.379

(10.26)

For a channel with zero upstream inflow (Qu = 0), Eq. (10.26) reduces to:

⎛ S c1 2 ⎞ ⎟(q t )1.379 Qc = 96.28⎜⎜ 0.0909 ⎟ L ⎝ ncW ⎠

(10.27)

Equations (10.26) and (10.27) are valid for t ≤ tt.

10.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (10.10) and (10.11) into Eq. (4.47) gives the equation for the forward characteristic of the rising phase for a channel with upstream inflow:

⎛ ncW 0.0909 ⎞ ⎟⎟ t = 0.0364⎜⎜ 12 S c ⎝ ⎠

0.725

⎡ (Qu + qL xc )0.725 − Qu0.725 ⎤ ⎢ ⎥ qL ⎣ ⎦

For a channel with zero upstream inflow (Qu = 0), Eq. (10.28) reduces to:

(10.28)

Kinematic-Wave Rainfall-Runoff Formulas

0.0364 ⎛ n W 0.0909 x ⎞ t = 0.275 ⎜⎜ c 1 2 c ⎟⎟ qL Sc ⎠ ⎝

131

0.725

(10.29)

10.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (10.10) and (10.11) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎡⎛ n W 0.0909 Ac = 2.186⎢⎜⎜ c 1 2 ⎢⎣⎝ S c

⎤ ⎞ ⎟⎟(q L xc + Qu )⎥ ⎥⎦ ⎠

0.725

(10.30)

Substituting Eq. (10.1) into Eq. (10.30), and solving it gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp: 12

0.725 ⎧⎪ ⎫⎪ ⎡ ncW 0.0909 (q L x c + Qu )⎤ 2 − W + ⎨W + 8.748 z ⎢ ⎬ ⎥ S c1 2 ⎪⎩ ⎪⎭ ⎣ ⎦ yc = 2z

(10.31)

Substituting Eqs. (10.10) and (10.11) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎤ ⎡⎛ n W 0.0909 ⎞ Ap = 2.186 ⎢⎜⎜ c 1 2 ⎟⎟(qL L p + Qu )⎥ ⎠ ⎦ ⎣⎝ S c

0.725

(10.32)

Substituting Ac = Ap, and yc = yp into Eq. (10.1), and then substituting it into Eq. (10.32), and solving it gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc: 0.725 ⎧ ⎫ ⎡ ncW 0.0909 (q L L p + Qu )⎤ ⎪ 2 ⎪ − W + ⎨W + 8.748 z ⎢ ⎥ ⎬ 12 Sc ⎪⎩ ⎪⎭ ⎣⎢ ⎦⎥ yp = 2z

12

From Eq. (4.51), the distance Lp is:

(10.33)

Tommy S.W. Wong

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Lp =

Q p − Qu

(10.34)

qL

For a channel with zero upstream inflow (Qu = 0), Eqs. (10.30)-(10.34) reduce to:

⎛ n W 0.0909 q L xc ⎞ ⎟⎟ Ac = 2.186⎜⎜ c S c1 2 ⎝ ⎠

0.725

(10.35)

⎡ ⎛ n W 0.0909 q L x c − W + ⎢W 2 + 8.748 z ⎜⎜ c S c1 2 ⎢⎣ ⎝ yc = 2z

⎞ ⎟ ⎟ ⎠

0.725 1 2

⎤ ⎥ ⎥⎦

(10.36)

which are valid for 0 ≤ xc ≤ Lp ,

⎛ ncW 0.0909 q L L p A p = 2.186⎜ ⎜ S c1 2 ⎝

⎞ ⎟ ⎟ ⎠

0.725

⎡ ⎛ ncW 0.0909 q L L p − W + ⎢W 2 + 8.748 z ⎜ ⎜ S c1 2 ⎢ ⎝ ⎣ yp = 2z

(10.37)

⎞ ⎟ ⎟ ⎠

0.725 1 2

⎤ ⎥ ⎥ ⎦

(10.38)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp

(10.39)

qL

10.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (10.10) and (10.11) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow:

td =

0.725 ⎤ ⎛ Sc1 2 ⎞ ⎡ ⎛ ncW 0.0909Qu ⎞ ⎜ ⎟ ⎜ ⎟ ⎢2.186⎜ ⎥ + Lc qL + Qu − 0.340⎜ q t 60 L q 0.0909 ⎟ 12 ⎟ Sc ⎥⎦ ⎝ ncW ⎠ ⎢⎣ ⎝ ⎠

⎛ S q 28.13⎜⎜ ⎝ ncW

12 c L 0.0909

⎞⎡ ⎛ n W 0.0909Q ⎟⎟ ⎢2.186⎜⎜ c 1 2 u Sc ⎠ ⎢⎣ ⎝

⎞ ⎟⎟ ⎠

0.725

⎤ + 60qL t q ⎥ ⎥⎦

0.379

1.379

(10.40)

Kinematic-Wave Rainfall-Runoff Formulas

133

For a channel with zero upstream inflow (Qu = 0), Eq. (10.40) reduces to:

⎛ S c1 2 ⎞ 0.379 1.379 ⎟qL t q Lc − 96.41⎜⎜ ncW 0.0909 ⎟⎠ ⎝ td = ⎛ S c1 2 ⎞ ⎟(q t )0.379 133.0⎜⎜ 0.0909 ⎟ L q ⎝ ncW ⎠

(10.41)

10.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

10.12.1. Partial Equilibrium Discharge Substituting Eqs. (10.10) and (10.11) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎛ S c1 2 Q p = 0.340⎜⎜ 0.0909 ⎝ ncW

⎞⎡ ⎛ n W 0.0909 Q ⎟⎟ ⎢2.186⎜⎜ c 1 2 u Sc ⎠ ⎢⎣ ⎝

⎞ ⎟⎟ ⎠

0.725

⎤ + 60q L t q ⎥ ⎥⎦

1.379

(10.42)

For a channel with zero upstream inflow (Qu = 0), Eq. (10.42) reduces to:

⎛ S c1 2 Q p = 96.28⎜⎜ 0.0909 ⎝ n cW

⎞ ⎟⎟(q L t q )1.379 ⎠

(10.43)

Equations (10.42) and (10.43) are valid for tq ≤ t ≤ (tq + td).

10.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc For a channel with zero upstream inflow (Qu = 0), Eq. (10.44) reduces to:

(10.44)

Tommy S.W. Wong

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Q e = q L Lc

(10.45)

Equations (10.44) and (10.45) are valid for tt ≤ t ≤ tq.

10.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (10.10) and (10.11) into Eq. (4.71) gives the equation for the equilibrium flow area profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lp:

⎡⎛ n W 0.0909 Ac = 2.186⎢⎜⎜ c 1 2 ⎢⎣⎝ S c

⎤ ⎞ ⎟⎟(q L xc + Qu )⎥ ⎥⎦ ⎠

0.725

(10.46)

Substituting Eq. (10.1) into Eq. (10.40), and solving it gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lp: 12

0.725 ⎧⎪ ⎫⎪ ⎡ ncW 0.0909 (q L x c + Qu )⎤ 2 − W + ⎨W + 8.748 z ⎢ ⎬ ⎥ S c1 2 ⎪⎩ ⎪⎭ ⎣ ⎦ yc = 2z

(10.47)

For a channel with zero upstream inflow (Qu = 0), Eqs. (10.46) and (10.47) reduce to:

⎛ n W 0.0909 q L xc ⎞ ⎟⎟ Ac = 2.186⎜⎜ c 12 S c ⎝ ⎠

0.725

⎡ ⎛ n W 0.0909 q L xc − W + ⎢W 2 + 8.748 z⎜⎜ c S c1 2 ⎢⎣ ⎝ yc = 2z

(10.48)

⎞ ⎟⎟ ⎠

0.725 1 2

⎤ ⎥ ⎥⎦

(10.49)

which are valid for 0 ≤ xc ≤ Lc.

10.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (10.10) and (10.11) into Eq. (4.78) gives the equilibrium detention storage for a channel with upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ n W 0.0909 ⎞ Dec = 1.268⎜⎜ c 1 2 ⎟⎟ ⎝ Sc ⎠

0.725

135

⎡ (Qu + q L Lc )1.725 − Qu1.725 ⎤ ⎢ ⎥ qL ⎣ ⎦

(10.50)

For a channel with zero upstream inflow (i.e. Qu = 0), Eq. (10.50) reduces to:

⎛ ncW 0.0909 qL ⎞ ⎟⎟ Dec = 1.268⎜⎜ 12 S c ⎝ ⎠

0.725

L1c.725

(10.51)

10.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (10.10) and (10.11) into Eq. (4.91) gives the equation for the falling phase of the flow area profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc.

⎡ ⎤ ⎛ S c1 2 Ac1.379 ⎞ ⎜ ⎟ − 0 340 . Q ⎢ ⎥ u . 0 0909 ⎜nW ⎟ ⎛ S c1 2 Ac0.379 ⎞ c ⎝ ⎠ ⎢ ⎥ ⎟(t − t q ) + xc = 28.14⎜⎜ 0.0909 ⎟ ⎢ ⎥ q n W L ⎝ c ⎠ ⎢ ⎥ ⎢⎣ ⎥⎦

(10.52)

Substituting Eq. (10.1) into Eq. (10.46) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

⎡ S1 2 xc = 28.14⎢ c ⎣⎢

(

⎧ ⎡ S 1 2 zy 2 + Wy c c 0 340 . ⎪ ⎢ c 0.0909 0.379 2 n W ⎤ ⎪ c zyc + Wyc ⎣⎢ ⎥ (t − t q ) + ⎨ 0.0909 ncW qL ⎪ ⎦⎥ ⎪ ⎩

(

)

)

1.379

⎫ ⎤ ⎥ − Qu ⎪ ⎪ ⎦⎥ ⎬ ⎪ ⎪ ⎭

(10.53)

From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

⎛ n W 0.0909 Q Ac = 2.186⎜⎜ c 1 2 u Sc ⎝

⎞ ⎟⎟ ⎠

0.725

(10.54)

Substituting Eq. (10.1) into Eq. (10.50) and solving it gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf:

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⎡ ⎛ n W 0.0909 Q − W + ⎢W 2 + 8.748 z ⎜⎜ c 1 2 u Sc ⎢⎣ ⎝ yc = 2z

⎞ ⎟⎟ ⎠

0.725 1 2

⎤ ⎥ ⎥⎦

(10.55)

Substituting Eqs. (10.10) and (10.11) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq:

⎛ S c1 2 ⎞ ⎟ L f = 27.84⎜⎜ 0.0909 ⎟ ⎝ ncW ⎠

0.725

Qu0.275 (t − t q )

(10.56)

For a channel with zero upstream inflow (Qu = 0), Eqs. (10.52) and (10.53) reduce to:

⎡ ⎛ S c1 2 Ac1.379 ⎞ ⎤ ⎜⎜ ⎟ . 0 340 ⎢ 0.0909 ⎟ ⎥ n W ⎛ S c1 2 Ac0.379 ⎞ c ⎝ ⎠⎥ ⎟(t − tq ) + ⎢ xc = 28.14⎜⎜ 0.0909 ⎟ ⎢ ⎥ qL ⎝ ncW ⎠ ⎢ ⎥ ⎣⎢ ⎦⎥

⎡ S1 2 xc = 28.14 ⎢ c ⎢⎣

(10.57)

(

⎧ ⎡ S 1 2 zy 2 + Wy c c ⎪ . 0 340 ⎢ c 0.0909 0.379 2 n W ⎤ ⎪ c zyc + Wyc ⎢⎣ ⎥ (t − t q ) + ⎨ 0.0909 ncW qL ⎥⎦ ⎪ ⎪ ⎩

(

)

)

1.379

⎤⎫ ⎥⎪ ⎥⎦ ⎪ ⎬ ⎪ ⎪ ⎭

(10.58)

which are valid for 0 ≤ xc ≤ Lc.

10.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (10.10) and (10.11) into Eq. (4.101) gives the equation for the falling phase of a hydrograph for a channel with upstream inflow:

0.0264 ⎛ n W 0.0909 ⎞ t = 0.275 ⎜⎜ c 1 2 ⎟⎟ Qc ⎝ Sc ⎠

0.725

⎡ ⎛ Qc − Qu ⎞⎤ ⎟⎟⎥ + t q ⎢ Lc − ⎜⎜ q L ⎝ ⎠⎦ ⎣

(10.59)

For a channel with zero upstream inflow (Qu = 0), Eq. (10.59) reduces to:

0.0264 ⎛ ncW 0.0909 ⎞ ⎟⎟ t = 0.275 ⎜⎜ 12 Qc ⎝ Sc ⎠

0.725

⎡ ⎛ Qc ⎞⎤ ⎢ Lc − ⎜⎜ ⎟⎟⎥ + t q ⎝ qL ⎠⎦ ⎣

(10.60)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 137-149

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 11

11. WORKING FORMULAS FOR FLOW IN TRAPEZOIDAL CHANNEL WITH ONE SIDE VERTICAL In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a trapezoidal channel with one side vertical are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

11.1. KINEMATIC WAVE PARAMETERS For flow in a trapezoidal channel with one side vertical, the flow area Ac, and the wetted perimeter P, are related to the flow depth yc, and the reciprocal of channel side slope z, as follows:

Ac = 0.5 zyc2 + Wyc

[

(

P = 1+ 1+ z2

)

12

(11.1)

]y + W c

(11.2)

Figure 11.1 shows the trapezoidal channel with base width W, reciprocal of channel side slope z, and flow depth yc. Next, defining two dimensionless variables, ψ and µ, as:

ψ = Ac W 2

(11.3)

μ = yc W

(11.4)

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Substituting Eqs. (11.3) and (11.4) into Eq. (11.1) gives the following relationship between ψ and µ:

0.5 zμ 2 + μ −ψ = 0

(11.5)

For z ≠ 0, the positive solution for Eq. (11.5) is:

μ=

(1 + 2 zψ )1 2 − 1

(11.6)

z

Substituting Eq. (11.4) into Eq. (11.6) to eliminate µ results in:

⎡ (1 + 2 zψ )1 2 − 1⎤ yc = ⎢ ⎥W z ⎦ ⎣

(11.7)

Substituting Eq. (11) into Eq. (6) gives an expression for P:

[

(

⎧⎪ 1 + 1 + z 2 P = ⎨1 + ⎪⎩

) ][(1 + 2 zψ ) 12

12

z

]

− 1 ⎫⎪ ⎬W ⎪⎭

(11.8)

Substituting Eqs. (11.3) and (11.8) into Eq. (3.1) results in a dimensionless equation in terms of ψ, which can be considered as the "true" relationship Qc and Ac for a trapezoidal channel with one side vertical: 23

⎧⎪ ⎫⎪ ncQc z 53 = ⎨ ⎬ ψ 12 12 83 12 2 Sc W ⎪⎩ z + 1 + (1 + z ) (1 + 2 zψ ) − 1 ⎪⎭

[

][

]

(11.9)

As shown in figure 11.2, by mathematical fitting to the true relationships for 0.1 ≤ z ≤ 5.0, Wong and Zhou (2006) obtained the following kinematic wave parameters:

⎛

S c1 2 ⎞ ⎟ 0.0526 ⎟ n W c ⎝ ⎠

α c = 0.323⎜⎜

(11.10)

β c = 1.360

(11.11)

Kinematic-Wave Rainfall-Runoff Formulas

1

139

yc

z

W Figure 11.1. Cross-section of Trapezoidal Channel with One Side Vertical.

Figure 11.2. Comparison between True and Kinematic Wave Relationships for Flow in Trapezoidal Channel with One Side Vertical.

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11.2. FLOW DEPTH Substituting Eqs. (11.10) and (11.11) into Eq. (4.7) gives:

⎛ S c1 2 Qc = 0.323⎜⎜ 0.0526 ⎝ ncW

⎞ 1.360 ⎟⎟ Ac ⎠

(11.12)

Substituting Eqs. (11.1) into Eq. (11.12) gives:

⎛ S c1 2 Qc = 0.323⎜⎜ 0.0526 ⎝ n cW

⎞ ⎟⎟ 0.5 zy c2 + Wy c ⎠

(

)

1.360

(11.13)

Rearranging Eq. (11.13) gives:

⎛ ncW 0.0526 Qc 0.5 zy + Wy c − 2.296⎜⎜ S c1 2 ⎝ 2 c

⎞ ⎟⎟ ⎠

0.735

=0

(11.14)

Solving Eq. (11.14) gives the equation for the flow depth for a channel with and without upstream inflow:

⎡ ⎛ n W 0.0526 Q − W + ⎢W 2 + 4.592 z⎜⎜ c 1 2 c Sc ⎢⎣ ⎝ yc = z

⎞ ⎟⎟ ⎠

0.735 1 2

⎤ ⎥ ⎥⎦

(11.15)

11.3. FLOW VELOCITY Substituting Eqs. (11.10) and (11.11) into Eq. (4.12) gives the equation of flow velocity along the equilibrium profile for a channel with upstream inflow:

⎤ ⎡⎛ S c1 2 ⎞ ⎟(Qu + qL xc )0.360 ⎥ v = 0.436 ⎢⎜⎜ 0.0526 ⎟ ⎠ ⎦ ⎣⎝ ncW

0.735

(11.16)

For a channel with zero upstream inflow zero upstream inflow (Qu = 0), Eq. (11.16) reduces to:

⎤ ⎡⎛ S c1 2 ⎞ ⎟(q x )0.360 ⎥ v = 0.436⎢⎜⎜ 0.0526 ⎟ L c ⎠ ⎦ ⎣⎝ ncW

0.735

(11.17)

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141

11.4. AVERAGE FLOW VELOCITY Substituting Eqs. (11.10) and (11.11) into Eq. (4.15) gives the equation of average flow velocity for a channel with upstream inflow: 0.735

⎛ S c1 2 ⎞ ⎟ q L Lc 0.321⎜⎜ ncW 0.0526 ⎟⎠ ⎝ vav = (Qu + qL Lc )0.735 − Qu 0.735

(11.18)

For a channel with zero upstream inflow (Qu = 0), Eq. (11.18) reduces to:

⎛ S c1 2 ⎞ ⎟ vav = 0.321⎜⎜ 0.0526 ⎟ n W ⎝ c ⎠

0.735

(qL Lc )0.265

(11.19)

11.5. KINEMATIC WAVE CELERITY Substituting Eqs. (11.10) and (11.11) into Eq. (4.29) gives the equation of wave celerity along the equilibrium profile for a channel with upstream inflow:

⎤ ⎡⎛ S c1 2 ⎞ ⎟(Qu + q L xc )0.360 ⎥ ck = 0.593⎢⎜⎜ 0.0526 ⎟ ⎠ ⎦ ⎣⎝ ncW

0.735

(11.20)

For a channel with zero upstream inflow (Qu = 0), Eq. (11.20) reduces to:

⎡⎛ S c1 2 ⎞ 0.360 ⎤ ⎟ ck = 0.593⎢⎜⎜ ( q x ) ⎥ L c 0.0526 ⎟ ⎠ ⎦ ⎣⎝ ncW

0.735

(11.21)

11.6. AVERAGE WAVE CELERITY Substituting Eqs. (11.10) and (11.11) into Eq. (4.32) gives the equation of wave celerity for a channel with upstream inflow: 0.735

⎛ S c1 2 ⎞ ⎟ q L Lc 0.436⎜⎜ ncW 0.0526 ⎟⎠ ⎝ cav = (Qu + qL Lc )0.735 − Qu 0.735 For a channel with zero upstream inflow (Qu = 0), Eq. (11.22) reduces to:

(11.22)

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⎛ Sc1 2 ⎞ ⎟ cav = 0.436⎜⎜ 0.0526 ⎟ ⎠ ⎝ ncW

0.735

(qL Lc )0.265

(11.23)

11.7. TIME OF TRAVEL Substituting Eqs. (11.10) and (11.11)) into Eq. (4.35) gives the equation of time of travel for a channel with upstream inflow:

⎛ n W 0.0526 ⎞ tt = 0.0382⎜⎜ c 1 2 ⎟⎟ ⎝ Sc ⎠

0.735

⎡ (Qu + qL Lc )0.735 − Qu0.735 ⎤ ⎢ ⎥ qL ⎣ ⎦

(11.24)

For a channel with zero upstream inflow (Qu = 0), Eq. (11.24) reduces to:

⎛ 0.0382 ⎞⎛ ncW 0.0526 Lc ⎞ ⎟⎟ tt = ⎜⎜ 0.265 ⎟⎟⎜⎜ 12 q S c ⎝ L ⎠⎝ ⎠

0.735

(11.25)

11.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (11.10) and (11.11) into Eq. (4.45) gives the equation for the rising phase of the hydrograph for a channel with upstream inflow:

⎛ S c1 2 ⎞ ⎡ ⎛ ncW 0.0526Qu ⎟ ⎜⎜ ⎢ Qc = 0.323⎜⎜ 2 . 295 0.0526 ⎟ S c1 2 ⎝ ncW ⎠ ⎢⎣ ⎝

⎞ ⎟⎟ ⎠

0.735

⎤ + 60qL t ⎥ ⎥⎦

1.360

(11.26)

For a channel with zero upstream inflow (Qu = 0), Eq. (11.26) reduces to:

⎛ S c1 2 ⎞ ⎟(q t )1.360 Qc = 84.62⎜⎜ 0.0526 ⎟ L ⎝ ncW ⎠

(11.27)

Equations (11.26) and (11.27) are valid for t ≤ tt.

11.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (11.10) and (11.11) into Eq. (4.47) gives the equation for the forward characteristic of the rising phase for a channel with upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ n W 0.0526 ⎞ t = 0.0383⎜⎜ c 1 2 ⎟⎟ ⎝ Sc ⎠

0.735

⎡ (Qu + q L xc )0.735 − Qu0.735 ⎤ ⎢ ⎥ qL ⎣ ⎦

143

(11.28)

For a channel with zero upstream inflow (Qu = 0), Eq. (11.28) reduces to:

0.0383 ⎛ n W 0.0526 x ⎞ t = 0.265 ⎜⎜ c 1 2 c ⎟⎟ qL Sc ⎝ ⎠

0.735

(11.29)

11.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (11.10) and (11.11) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎤ ⎡⎛ ncW 0.0526 ⎞ ⎟ Ac = 2.295⎢⎜⎜ ( q x Q ) + ⎥ L c u 12 ⎟ ⎠ ⎦ ⎣⎝ S c

0.735

(11.30)

Substituting Eq. (11.1) into Eq. (11.28), and solving it gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp: 12

0.735 ⎧⎪ ⎡ ncW 0.0526 (qL xc + Qu ) ⎤ ⎫⎪ 2 − W + ⎨W + 4.592 z ⎢ ⎥ ⎬ S c1 2 ⎪⎩ ⎣ ⎦ ⎪⎭ yc = z

(11.31)

Substituting Eqs. (11.10) and (11.11) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎤ ⎡⎛ n W 0.0526 ⎞ Ap = 2.295⎢⎜⎜ c 1 2 ⎟⎟(q L L p + Qu )⎥ ⎠ ⎦ ⎣⎝ S c

0.735

(11.32)

Substituting Ac = Ap, and yc = yp into Eq. (11.1), and then substituting it into Eq. (11.32), and solving it gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

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0.735 ⎧ ⎫ ⎡ ncW 0.0526 (q L L p + Qu )⎤ ⎪ 2 ⎪ − W + ⎨W + 4.592 z ⎢ ⎥ ⎬ 12 Sc ⎢⎣ ⎥⎦ ⎪⎩ ⎪⎭ yp = z

12

(11.33)

from Eq. (4.51), the distance Lp is:

Lp =

Q p − Qu

(11.34)

qL

For a channel with zero upstream inflow (Qu = 0), Eqs. (11.30)-(11.34) reduce to:

⎛ ncW 0.0526 qL xc ⎞ ⎟⎟ Ac = 2.295⎜⎜ 12 S c ⎝ ⎠

0.735

(11.35)

0.735 ⎡ ⎛ ncW 0.0526 qL xc ⎞ ⎤ 2 ⎟⎟ ⎥ − W + ⎢W + 4.592 z⎜⎜ S c1 2 ⎢⎣ ⎝ ⎠ ⎥⎦ yc = z

12

(11.36)

which are valid for 0 ≤ xc ≤ Lp,

⎛ ncW 0.0526 qL L p ⎞ ⎟ Ap = 2.295⎜ 12 ⎜ ⎟ S c ⎝ ⎠

0.735

⎡ ⎛ ncW 0.0526 q L L p 2 ⎢ − W + W + 4.592 z ⎜ ⎜ ⎢ S c1 2 ⎝ ⎣ yp = z

(11.37)

⎞ ⎟ ⎟ ⎠

0.735 1 2

⎤ ⎥ ⎥ ⎦

(11.38)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp qL

(11.39)

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145

11.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (11.10) and (11.11) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow:

td =

0.735 ⎤ ⎛ S c1 2 ⎞ ⎡ ⎛ ncW 0.0526Qu ⎞ ⎟ ⎜ ⎟ ⎢ ⎥ + Lc qL + Qu − 0.323⎜⎜ 2 . 295 60 q t L q 0.0526 ⎟ ⎜ ⎟ S c1 2 ⎥⎦ ⎝ ncW ⎠ ⎢⎣ ⎝ ⎠

⎛ S q 26.54⎜⎜ ⎝ ncW

12 c L 0.0526

⎞⎡ ⎛nW Q ⎞ ⎟⎟ ⎢2.295⎜⎜ c 1 2 u ⎟⎟ Sc ⎠ ⎢⎣ ⎝ ⎠ 0.0526

0.735

⎤ + 60qL t q ⎥ ⎥⎦

1.360

0.360

(11.40)

For a channel with zero upstream inflow (Qu = 0), Eq. (11.38) reduces to:

⎛ S c1 2 ⎞ 0.360 1.360 ⎟q t q Lc − 84.62⎜⎜ 0.0526 ⎟ L n W ⎝ c ⎠ td = ⎛ S c1 2 ⎞ ⎟(q t )0.360 115.1⎜⎜ 0.0526 ⎟ L q ⎝ ncW ⎠

(11.41)

11.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

11.12.1. Partial Equilibrium Discharge Substituting Eqs. (11.10) and (11.11) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎛ S c1 2 Q p = 0.323⎜⎜ 0.0526 ⎝ n cW

⎞⎡ ⎛ ncW 0.0526 Qu ⎟⎟ ⎢2.295⎜⎜ S c1 2 ⎠ ⎢⎣ ⎝

⎞ ⎟⎟ ⎠

0.735

⎤ + 60q L t q ⎥ ⎥⎦

1.360

For a channel with zero upstream inflow (Qu = 0), Eq. (11.42) reduces to:

(11.42)

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⎛ S c1 2 Q p = 84.62⎜⎜ 0.0526 ⎝ n cW

⎞ ⎟(q L t q )1.360 ⎟ ⎠

(11.43)

Equations (11.42) and (11.43) are valid for tq ≤ t ≤ (tq + td).

11.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc

(11.44)

For a channel with zero upstream inflow (Qu = 0), Eq. (11.44) reduces to:

Q e = q L Lc

(11.45)

Equations (11.44) and (11.45) are valid for tt ≤ t ≤ tq.

11.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (11.10) and (11.11) into Eq. (4.71) gives the equation for the equilibrium flow area profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lp.

⎤ ⎡⎛ ncW 0.0526 ⎞ ⎟ Ac = 2.295⎢⎜⎜ ( q x Q ) + ⎥ L c u 12 ⎟ ⎠ ⎦ ⎣⎝ S c

0.735

(11.46)

Substituting Eq. (11.1) into Eq. (11.46), and solving it gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lp: 12

0.735 ⎫⎪ ⎧⎪ ⎡ ncW 0.0526 (qL xc + Qu )⎤ 2 − W + ⎨W + 4.592 z ⎢ ⎬ ⎥ S c1 2 ⎪⎭ ⎪⎩ ⎣ ⎦ yc = z

(11.47)

For a channel with zero upstream inflow (Qu = 0), Eqs. (11.46) and (11.47) reduce to:

Kinematic-Wave Rainfall-Runoff Formulas

⎛ n W 0.0526 qL xc ⎞ ⎟⎟ Ac = 2.295⎜⎜ c Sc1 2 ⎠ ⎝

147

0.735

(11.48)

0.735 ⎡ ⎛ ncW 0.0526 q L xc ⎞ ⎤ 2 ⎟⎟ ⎥ − W + ⎢W + 4.592 z ⎜⎜ S c1 2 ⎢⎣ ⎝ ⎠ ⎥⎦ yc = z

12

(11.49)

which are valid for 0 ≤ xc ≤ Lc.

11.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (11.10) and (11.11) into Eq. (4.78) gives the equilibrium detention storage for a channel with upstream inflow:

⎛ n W 0.0526 ⎞ Dec = 1.322⎜⎜ c 1 2 ⎟⎟ ⎝ Sc ⎠

0.735

⎡ (Qu + qL Lc )1.735 − Qu1.735 ⎤ ⎢ ⎥ qL ⎣ ⎦

(11.50)

For a channel with zero upstream inflow (i.e. Qu = 0), Eq. (11.50) reduces to:

⎛ n W 0.0526 q ⎞ Dec = 1.322⎜⎜ c 1 2 L ⎟⎟ Sc ⎝ ⎠

0.735

L1c.735

(11.51)

11.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (11.10) and (11.11) into Eq. (4.91) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc.

⎛ S 1 2 A 0.360 xc = 26.36⎜⎜ c 0c.0526 ⎝ n cW

⎡ ⎛ S c1 2 Ac1.360 ⎜⎜ 0 323 . ⎢ ncW 0.0526 ⎞ ⎝ ⎢ ⎟⎟(t − t q ) + ⎢ qL ⎠ ⎢ ⎢⎣

⎤ ⎞ ⎟⎟ − Qu ⎥ ⎠ ⎥ ⎥ ⎥ ⎥⎦

(11.52)

Substituting Eq. (11.1) into Eq. (11.52) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

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⎡S1 2 x c = 26.36 ⎢ c ⎢⎣

(

⎧ ⎡ S 1 2 0.5 zy 2 + Wy c c ⎪ . 0 323 ⎢ c 0.0526 0.360 2 n W ⎤ ⎪ c 0.5 zy c + Wy c ⎪ ⎣⎢ ⎥ (t − t q ) + ⎨ 0.0526 qL ncW ⎥⎦ ⎪ ⎪ ⎪⎩

(

)

)

1.360

⎫ ⎤ ⎥ − Qu ⎪ ⎪⎪ ⎦⎥ ⎬ (11.53) ⎪ ⎪ ⎪⎭

From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

⎛ ncW 0.0526 Qu Ac = 2.295⎜⎜ S c1 2 ⎝

⎞ ⎟ ⎟ ⎠

0.735

(11.54)

Substituting Eq. (11.1) into Eq. (11.54) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf:

⎡ ⎛ ncW 0.0526 Qu 2 ⎢ − W + W + 4.592 z ⎜⎜ S c1 2 ⎢⎣ ⎝ yc = z

⎞ ⎟ ⎟ ⎠

0.735 1 2

⎤ ⎥ ⎥⎦

(11.55)

Substituting Eqs. (11.10) and (11.11) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq:

⎛ S c1 2 ⎞ ⎟ L f = 35.56⎜⎜ 0.0526 ⎟ ⎝ ncW ⎠

0.735

Qu0.265 (t − t q )

(11.56)

For a channel with zero upstream inflow (Qu = 0), Eqs. (11.52) and (11.53) reduce to:

⎛S A xc = 26.357⎜⎜ ⎝ n cW 12 c

⎡S1 2 x c = 26.357 ⎢ c ⎣⎢

0.360 c 0.0526

⎡ ⎛ S c1 2 Ac1.360 ⎢ 0.323⎜⎜ 0.0526 ⎞ ⎝ n cW ⎟⎟(t − t q ) + ⎢ ⎢ qL ⎠ ⎢ ⎢⎣

⎞⎤ ⎟⎟ ⎥ ⎠⎥ ⎥ ⎥ ⎥⎦

(

⎧ ⎡ S 1 2 0.5 zy 2 + Wy c c ⎪ . 0 323 ⎢ c 0.0526 0.360 2 n W ⎤ ⎢⎣ c 0.5 zy c + Wy c ⎪⎪ ⎥ (t − t q ) + ⎨ 0.0526 qL n cW ⎪ ⎦⎥ ⎪ ⎩⎪

(

)

(11.57)

)

1.360

⎤⎫ ⎥⎪ ⎥⎦ ⎪⎪ (11.58) ⎬ ⎪ ⎪ ⎭⎪

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149

which are valid for 0 ≤ xc ≤ Lc.

11.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (11.10) and (11.11) into Eq. (4.101) gives the equation for the falling phase of a hydrograph for a channel with upstream inflow:

0.0281 ⎛ n W 0.0526 ⎞ t = 0.265 ⎜⎜ c 1 2 ⎟⎟ Qc ⎝ Sc ⎠

0.735

⎡ ⎛ Qc − Qu ⎢ Lc − ⎜⎜ ⎝ qL ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

(11.59)

For a channel with zero upstream inflow (Qu = 0), Eq. (11.59) reduces to:

0.0281 ⎛ n W 0.0526 t = 0.265 ⎜⎜ c 1 2 Qc ⎝ Sc

⎞ ⎟⎟ ⎠

0.735

⎡ ⎛ Qc ⎞⎤ ⎢ Lc − ⎜⎜ ⎟⎟⎥ + t q ⎝ q L ⎠⎦ ⎣

(11.60)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 151-161

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 12

12. WORKING FORMULAS FOR FLOW IN TRIANGULAR CHANNEL In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a triangular channel are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

12.1. KINEMATIC WAVE PARAMETERS For flow in a triangular channel as shown in figure 12.1, the flow area Ac, and the wetted perimeter P, are related to the flow depth, yc and the reciprocal of channel side slope, z as follows:

Ac = zyc2

(12.1)

(

P = 2 yc 1 + z 2

)

12

(12.2)

Substituting Eqs. (12.1) and (12.2) and A = Ac into Eq. (3.2) gives:

R=

(

zy c

2 1+ z2

)

12

Substituting Eq. (12.1) into Eq. (12.3) gives:

(12.3)

Tommy S.W. Wong

152

1 z

Figure 12.1. Cross-section of Triangular Channel.

R=

1 ⎛ zAc ⎞ ⎜ ⎟ 2 ⎝1+ z 2 ⎠

12

(12.4)

Substituting Eq. (12.4) and Q = Q c , S =S c, A = Ac, n = nc into Eq. (3.1) gives:

⎛ S1 2 Qc = 0.630⎜⎜ c ⎝ nc

13

⎞⎛ z ⎞ ⎟⎟⎜ Ac4 3 2 ⎟ ⎠⎝ 1 + z ⎠

(12.5)

A comparison of Eq. (12.5) with Eq. (4.7) gives the kinematic wave parameters (Wong 2008b):

⎛ S c1 2 ⎝ nc

α c = 0.630⎜⎜ βc =

⎞⎛ z ⎞ ⎟⎟⎜ 2 ⎟ ⎠⎝ 1 + z ⎠

13

(12.6)

4 3

(12.7)

12.2. FLOW DEPTH Substituting Eq. (12.1) into Eq. (12.5) gives:

⎛ S c1 2 Qc = 0.630⎜⎜ ⎝ nc

⎞⎛ z 5 ⎟⎟⎜⎜ 2 ⎠⎝ 1 + z

13

⎞ ⎟⎟ y c8 3 ⎠

(12.8)

Rearranging Eq. (12.8) gives the equation for the flow depth for a channel with and without upstream inflow:

Kinematic-Wave Rainfall-Runoff Formulas

⎛n Q ⎞ yc = 1.190⎜⎜ c 1 2c ⎟⎟ ⎝ Sc ⎠

38

153

18

⎛1+ z2 ⎞ ⎜⎜ 5 ⎟⎟ ⎝ z ⎠

(12.9)

12.3. FLOW VELOCITY Substituting Eqs. (12.6) and (12.7) into Eq. (4.12) gives the equation of flow velocity along the equilibrium profile for a channel with upstream inflow:

⎛ S1 2 v = 0.707⎜⎜ c ⎝ nc

⎞ ⎟ ⎟ ⎠

34

⎡ z (Qu + q L xc )⎤ ⎢ 1+ z2 ⎥ ⎣ ⎦

14

(12.10)

For a channel with zero upstream inflow zero upstream inflow (Qu = 0), Eq. (12.10) reduces to: 34

14

⎛ S 1 2 ⎞ ⎛ zq x ⎞ v = 0.707⎜⎜ c ⎟⎟ ⎜ L 2c ⎟ ⎝ nc ⎠ ⎝ 1 + z ⎠

(12.11)

12.4. AVERAGE FLOW VELOCITY Substituting Eqs. (12.6) and (12.7) into Eq. (4.15) gives the equation of average flow velocity for a channel with upstream inflow: 34

14

⎛ S1 2 ⎞ ⎛ z ⎞ 0.530⎜⎜ c ⎟⎟ ⎜ ⎟ qL Lc nc ⎠ ⎝ 1 + z 2 ⎠ ⎝ vav = (Qu + qL Lc )3 4 − Qu 3 4

(12.12)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.12) reduces to: 34

14

⎛ S 1 2 ⎞ ⎛ zq L ⎞ vav = 0.530⎜⎜ c ⎟⎟ ⎜ L 2c ⎟ ⎝ nc ⎠ ⎝ 1 + z ⎠

(12.13)

12.5. KINEMATIC WAVE CELERITY Substituting Eqs. (12.6) and (12.7) into Eq. (4.29) gives the equation of wave celerity along the equilibrium profile for a channel with upstream inflow:

Tommy S.W. Wong

154

⎛ S1 2 c k = 0.943⎜⎜ c ⎝ nc

⎞ ⎟ ⎟ ⎠

⎡ z (Qu + q L xc ) ⎤ ⎢ 1+ z2 ⎥ ⎣ ⎦

34

14

(12.14)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.14) reduces to:

⎛ S c1 2 ck = 0.943⎜⎜ ⎝ nc

⎞ ⎟ ⎟ ⎠

34

⎡ zq L xc ⎤ ⎢1 + z 2 ⎥ ⎣ ⎦

14

(12.15)

12.6. AVERAGE WAVE CELERITY Substituting Eqs. (12.6) and (12.7) into Eq. (4.32) gives the equation of wave celerity for a channel with upstream inflow:

34

14

⎛ S1 2 ⎞ ⎛ z ⎞ ⎟ qL Lc 0.707⎜⎜ c ⎟⎟ ⎜⎜ nc ⎠ ⎝ 1 + z 2 ⎟⎠ ⎝ cav = (Qu + qL Lc )3 4 − Qu 3 4

(12.16)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.16) reduces to: 34

14

⎛ S 1 2 ⎞ ⎛ zq L ⎞ cav = 0.707⎜⎜ c ⎟⎟ ⎜ L 2c ⎟ ⎝ nc ⎠ ⎝ 1 + z ⎠

(12.17)

12.7. TIME OF TRAVEL Substituting Eqs. (12.6) and (12.7) into Eq. (4.35) gives the equation of time of travel for a channel with upstream inflow:

⎛ n ⎞ tt = 0.0236⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

34

⎛ 1 + z 2 ⎞ ⎡ (Qu + q L Lc )3 4 − Qu3 4 ⎤ ⎜⎜ ⎟⎟ ⎢ ⎥ qL ⎝ z ⎠ ⎣ ⎦ 14

(12.18)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.18) reduces to:

⎛n L ⎞ tt = 0.0236⎜⎜ c1 2c ⎟⎟ ⎝ Sc ⎠

34

14

⎛1+ z2 ⎞ ⎜⎜ ⎟⎟ ⎝ zqL ⎠

(12.19)

Kinematic-Wave Rainfall-Runoff Formulas

155

12.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (12.6) and (12.7) into Eq. (4.45) gives the equation for the rising phase of the hydrograph for a channel with upstream inflow: 13

⎛ S 1 2 ⎞⎛ z ⎞ Qc = 0.630⎜⎜ c ⎟⎟⎜ 2 ⎟ ⎝ nc ⎠⎝ 1 + z ⎠

⎡ ⎛n Q ⎢1.414⎜⎜ c 1 2u ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

14

⎛1+ z2 ⎞ ⎜⎜ ⎟⎟ z ⎝ ⎠

⎤ + 60qL t ⎥ ⎥⎦

43

(12.20)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.20) reduces to:

⎛ S1 2 Qc = 148.0⎜⎜ c ⎝ nc

⎞⎛ z ⎞ ⎟⎟⎜ 2 ⎟ ⎠⎝ 1 + z ⎠

13

(q L t )4 3

(12.21)

Equations (12.20) and (12.21) are valid for t ≤ tt.

12.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (12.6) and (12.7) into Eq. (4.47) gives the equation for the forward characteristic of the rising phase for a channel with upstream inflow:

⎛ n ⎞ t = 0.0236⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

34

⎛ 1 + z 2 ⎞ ⎡ (Qu + qL xc )3 4 − Qu3 4 ⎤ ⎜⎜ ⎟⎟ ⎢ ⎥ qL ⎝ z ⎠ ⎣ ⎦ 14

(12.22)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.22) reduces to:

⎛n x t = 0.0236⎜⎜ c1 2c ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

⎛1+ z 2 ⎜⎜ ⎝ zq L

⎞ ⎟⎟ ⎠

14

(12.23)

12.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (12.6) and (12.7) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 ⎤ ⎟⎟ (Qu + qL xc )⎥ Ac = 1.414 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ ⎢⎣⎝ S c ⎠⎝ z ⎠ ⎥⎦

34

(12.24)

Tommy S.W. Wong

156

Substituting Eq. (12.1) into Eq. (12.24) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 ⎤ yc = 1.189 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ 5 ⎟⎟ (Qu + q L xc )⎥ ⎢⎣⎝ S c ⎠⎝ z ⎠ ⎥⎦

38

(12.25)

Substituting Eqs. (12.6) and (12.7) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 ⎤ ⎟⎟ (Qu + q L L p )⎥ Ap = 1.414 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ ⎢⎣⎝ S c ⎠⎝ z ⎠ ⎥⎦

34

(12.26)

Substituting Ac = Ap, and yc = yp into Eq. (12.1) and then substituting it into Eq. (12.26) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 ⎤ y p = 1.189 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ 5 ⎟⎟ (Qu + q L L p )⎥ ⎢⎣⎝ S c ⎠⎝ z ⎠ ⎥⎦

38

(12.27)

From Eq. (4.51), the distance Lp is:

Lp =

Q p − Qu

(12.28)

qL

For a channel with zero upstream inflow (Qu = 0), Eqs. (12.24)-(12.28) reduce to: 13 34

⎡n q x Ac = 1.414 ⎢ c 1L2 c ⎢⎣ S c

⎛1+ z2 ⎞ ⎜⎜ ⎟⎟ ⎝ z ⎠

⎡n q x yc = 1.189 ⎢ c 1L2 c ⎢⎣ S c

⎛1+ z2 ⎞ ⎜⎜ 5 ⎟⎟ ⎝ z ⎠

which are valid for 0 ≤ xc ≤ Lp,

⎤ ⎥ ⎥⎦

(12.29)

13 38

⎤ ⎥ ⎥⎦

(12.30)

Kinematic-Wave Rainfall-Runoff Formulas

157

13 34

⎡ nc q L L p Ap = 1.414 ⎢ 1 2 ⎢⎣ S c

⎛1+ z2 ⎞ ⎜⎜ ⎟⎟ ⎝ z ⎠

⎡ nc q L L p y p = 1.189 ⎢ 1 2 ⎢⎣ S c

⎛1+ z2 ⎞ ⎜⎜ 5 ⎟⎟ ⎝ z ⎠

⎤ ⎥ ⎥⎦

(12.31)

13 38

⎤ ⎥ ⎥⎦

(12.32)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp

(12.33)

qL

12.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (12.6) and (12.7) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow: 13

td =

⎛ S 1 2 ⎞⎛ z ⎞ Lc qL + Qu − 0.630⎜⎜ c ⎟⎟⎜ 2 ⎟ ⎝ nc ⎠⎝ 1 + z ⎠

⎡ ⎛n Q ⎢1.414⎜⎜ c 1 2u ⎢⎣ ⎝ Sc

⎛n Q ⎛ S q ⎞⎛ z ⎞ ⎡ ⎟⎟⎜ ⎢1.414⎜⎜ c 1 2u 50.40⎜⎜ 2 ⎟ ⎝ Sc ⎝ nc ⎠⎝ 1 + z ⎠ ⎢⎣ 12 c L

13

⎞ ⎟⎟ ⎠

34

⎞ ⎟⎟ ⎠

34

14 ⎤ ⎛1+ z2 ⎞ ⎜⎜ ⎟⎟ + 60qL t q ⎥ ⎥⎦ ⎝ z ⎠

⎛1+ z ⎜⎜ ⎝ z

2

14

⎞ ⎟⎟ ⎠

⎤ + 60qL tq ⎥ ⎥⎦

13

43

(12.34)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.34) reduces to: 13

⎛ S 1 2 ⎞⎛ z ⎞ 1 3 4 3 Lc − 148.0⎜⎜ c ⎟⎟⎜ qL tq 2 ⎟ n 1 z + ⎝ ⎠ ⎝ c ⎠ td = 13 12 ⎛ S c ⎞⎛ z ⎞ ⎟⎟⎜ (q L t q )1 3 197.3⎜⎜ 2 ⎟ ⎝ nc ⎠⎝ 1 + z ⎠

(12.35)

12.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

Tommy S.W. Wong

158

12.12.1. Partial Equilibrium Discharge Substituting Eqs. (12.6) and (12.7) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎛ S c1 2 Q p = 0.630⎜⎜ ⎝ nc

13 ⎞⎛ z ⎞ ⎡ ⎛ n c Qu ⎟⎟⎜ ⎜⎜ 1 2 ⎢ 1 . 414 ⎟ 2 1 z + ⎝ ⎠ ⎢ ⎠ ⎝ Sc ⎣

⎞ ⎟⎟ ⎠

34

⎛1+ z2 ⎜⎜ ⎝ z

⎞ ⎟⎟ ⎠

14

⎤ + 60q L t q ⎥ ⎥⎦

43

(12.36)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.36) reduces to:

⎛ S c1 2 Q p = 148.0⎜⎜ ⎝ nc

⎞⎛ z ⎞ ⎟⎟⎜ 2 ⎟ ⎠⎝ 1 + z ⎠

13

(q t )

43

(12.37)

L q

Equations (12.36) and (12.37) are valid for tq ≤ t ≤ (tq + td).

12.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc

(12.38)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.38) reduces to:

Q e = q L Lc

(12.39)

Equations (12.38) and (12.39) are valid for tt ≤ t ≤ tq.

12.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (12.6) and (12.7) into Eq. (4.71) gives the equation for the equilibrium flow area profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lp.

⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 ⎤ ⎟⎟ (Qu + qL xc )⎥ Ac = 1.414 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ ⎢⎣⎝ S c ⎠⎝ z ⎠ ⎥⎦

34

(12.40)

Substituting Eq. (12.1) into Eq. (12.40) gives the equation for the equilibrium water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lp:

Kinematic-Wave Rainfall-Runoff Formulas

⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 ⎤ yc = 1.189 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ 5 ⎟⎟ (Qu + q L xc )⎥ ⎢⎣⎝ S c ⎠⎝ z ⎠ ⎥⎦

159

38

(12.41)

For a channel with zero upstream inflow (Qu = 0), Eqs. (12.40) and (12.41) reduce to: 13 34

⎡n q x Ac = 1.414 ⎢ c 1L2 c ⎢⎣ S c

⎛1+ z2 ⎞ ⎜⎜ ⎟⎟ ⎝ z ⎠

⎡n q x yc = 1.189 ⎢ c 1L2 c ⎢⎣ S c

⎛1+ z2 ⎞ ⎜⎜ 5 ⎟⎟ ⎝ z ⎠

⎤ ⎥ ⎥⎦

(12.42)

13 38

⎤ ⎥ ⎥⎦

(12.43)

which are valid for 0 ≤ xc ≤ Lc.

12.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (12.6) and (12.7) into Eq. (4.78) gives the equilibrium detention storage for a channel with upstream inflow:

⎛ n Dec = 0.808⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎛1+ z2 ⎜⎜ ⎝ z

⎞ ⎟⎟ ⎠

14

⎡ (Qu + q L Lc )7 4 − Qu7 4 ⎤ ⎥ ⎢ qL ⎥⎦ ⎢⎣

(12.44)

For a channel with zero upstream inflow (i.e. Qu = 0), Eq. (12.44) reduces to:

⎛ n ⎞ Dec = 0.808⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

34

14

⎛1+ z2 ⎞ 3 4 7 4 ⎜⎜ ⎟⎟ q L L ⎝ z ⎠

(12.45)

12.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (12.6) and (12.7) into Eq. (4.91) gives the equation for the falling phase of the flow area profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

Tommy S.W. Wong

160

13 ⎤ ⎡ ⎛ S c1 2 ⎞⎛ zAc4 ⎞ ⎜ ⎟ ⎜ ⎟ 0 630 . Q − ⎢ 1 3 u ⎥ ⎜ n ⎟⎜ 1 + z 2 ⎟ ⎛ Sc1 2 ⎞⎛ zAc ⎞ ⎥ ⎢ c ⎠⎝ ⎠ ⎝ ⎟⎟⎜⎜ ⎟ (t − t q ) + ⎢ xc = 50.40⎜⎜ 2 ⎟ ⎥ qL ⎝ nc ⎠⎝ 1 + z ⎠ ⎥ ⎢ ⎥⎦ ⎢⎣

(12.46)

Substituting Eq. (12.1) into Eq. (12.46) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc: 13 ⎤ ⎡ ⎛ S c1 2 ⎞⎛ z 5 yc8 ⎞ 1 3 ⎜ ⎟ ⎜ ⎟ 0 630 ⎢ − . Q 2 u ⎥ ⎜ n ⎟⎜ 1 + z 2 ⎟ ⎛ S c1 2 ⎞ ⎡ zyc ⎤ ⎥ ⎢ ⎠ c ⎠⎝ ⎝ ⎟⎟ ⎢ (t − tq ) + ⎢ xc = 50.40⎜⎜ 2 ⎥ ⎥ qL ⎝ nc ⎠ ⎢⎣ 1 + z ⎥⎦ ⎥ ⎢ ⎥⎦ ⎢⎣

( )

(12.47)

From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

⎛n Q Ac = 1.414⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

⎛1+ z 2 ⎜⎜ ⎝ z

⎞ ⎟⎟ ⎠

14

(12.48)

Substituting Eq. (12.1) into Eq. (12.48) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf:

⎛n Q y c = 1.190⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟ ⎟ ⎠

38

⎛1+ z 2 ⎜⎜ 5 ⎝ z

⎞ ⎟⎟ ⎠

18

(12.49)

Substituting Eqs. (12.6) and (12.7) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq: 34

⎡⎛ S c1 2 ⎞⎛ z ⎞1 3 ⎤ ⎟⎟⎜ L f = 56.57 ⎢⎜⎜ Qu1 4 (t − t q ) 2 ⎟ ⎥ ⎣⎢⎝ nc ⎠⎝ 1 + z ⎠ ⎦⎥

(12.50)

For a channel with zero upstream inflow (Qu = 0), Eqs. (12.46) and (12.47) reduce to:

Kinematic-Wave Rainfall-Runoff Formulas

161

13 ⎡ ⎛ S c1 2 ⎞⎛ zAc4 ⎞ ⎤ ⎟⎜ ⎟ ⎥ ⎢ 0.630⎜⎜ 13 nc ⎟⎠⎜⎝ 1 + z 2 ⎟⎠ ⎥ ⎛ S c1 2 ⎞⎛ zAc ⎞ ⎢ ⎝ ⎟⎟⎜⎜ ⎟ (t − t q ) + ⎢ xc = 50.40⎜⎜ 2 ⎟ ⎥ 1 n z qL + ⎠ ⎝ c ⎠⎝ ⎥ ⎢ ⎥⎦ ⎢⎣

(12.51)

⎡ ⎛ S c1 2 ⎞⎛ z 5 y c8 ⎟⎜ ⎢ 0.630⎜⎜ 2 13 nc ⎟⎠⎜⎝ 1 + z 2 ⎛ S c1 2 ⎞ ⎡ zy c ⎤ ⎢ ⎝ ⎟⎟ ⎢ xc = 50.40⎜⎜ ⎥ (t − t q ) + ⎢ 2 n qL 1 z + ⎥⎦ ⎢ ⎝ c ⎠⎣ ⎢ ⎢ ⎣

(12.52)

( )

13 ⎞ ⎤ ⎟⎟ ⎥ ⎠ ⎥ ⎥ ⎥ ⎥ ⎦

which are valid for 0 ≤ xc ≤ Lc.

12.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (12.6) and (12.7) into Eq. (4.101) gives the equation for the falling phase of a hydrograph for a channel with upstream inflow:

⎛ n t = 0.0177⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎛1 + z 2 ⎜⎜ ⎝ zQc

14

⎞ ⎡ ⎛ Q − Qu ⎟⎟ ⎢ Lc − ⎜⎜ c ⎝ qL ⎠ ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

(12.53)

For a channel with zero upstream inflow (Qu = 0), Eq. (12.53) reduces to:

⎛ n t = 0.0177⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎛1 + z 2 ⎜⎜ ⎝ zQc

14

⎞ ⎡ ⎛ Q ⎞⎤ ⎟⎟ ⎢ Lc − ⎜⎜ c ⎟⎟⎥ + t q ⎝ q L ⎠⎦ ⎠ ⎣

(12.54)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 163-174

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

Chapter 13

13. WORKING FORMULAS FOR FLOW IN VERTICAL CURB CHANNEL In this Chapter, based on the Manning’s equation, the kinematic wave parameters for flow in a vertical curb channel are derived. By applying these parameters to the general formulas in Chapter 4, working formulas for the flow depth; flow velocity; average flow velocity; wave celerity; average wave celerity; time of travel; rising, equilibrium and falling phases of hydrograph; forward characteristic; rising, equilibrium and falling phases of water surface profiles; duration of partial equilibrium discharge; and equilibrium detention storage are also derived.

13.1. KINEMATIC WAVE PARAMETERS For flow in a vertical curb channel as shown in figure 13.1, the flow area Ac, and the wetted perimeter P, are related to the flow depth, yc and the reciprocal of channel side slope, z as follows:

Ac = 0.5 zyc2

[

(13.1)

(

P = yc 1 + 1 + z 2

)

12

]

(13.2)

Substituting Eqs. (13.1) and (13.2) and A = Ac into Eq. (3.2) gives:

R=

0.5 zyc

(

1+ 1+ z2

)

12

Substituting Eq. (13.1) into Eq. (13.3) gives:

(13.3)

Tommy S.W. Wong

164

1 z

Figure 13.1. Cross-section of Vertical Curb Channel.

R=

0.707 z1 2 Ac1 2

(

1+ 1+ z2

(13.4)

)

12

Substituting Eq. (13.3) and Q = Q c , S =S c, A = Ac, n = nc into Eq. (3.1) gives:

⎛ S1 2 Qc = 0.794⎜⎜ c ⎝ nc

⎞⎧⎪ z ⎟⎟⎨ ⎠⎪⎩ 1 + 1 + z 2

[

(

13

)

12

⎫⎪ Ac4 3 2 ⎬ ⎪⎭

]

(13.5)

A comparison of Eq. (13.5) with Eq. (4.7), gives the kinematic wave parameters (Wong 2008b):

⎛ S 1 2 ⎞⎧⎪ z α c = 0.794⎜⎜ c ⎟⎟⎨ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2

[

βc =

(

13

)

12

⎫⎪ 2⎬ ⎪⎭

]

4 3

(13.6)

(13.7)

13.2. FLOW DEPTH Substituting Eqs. (13.1) into Eq. (13.5) gives:

⎛ S 1 2 ⎞⎧⎪ z5 Qc = 0.315⎜⎜ c ⎟⎟⎨ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2

[

(

13

)

12

⎫⎪ yc8 3 2⎬ ⎪⎭

]

(13.8)

Kinematic-Wave Rainfall-Runoff Formulas

165

Rearranging Eq. (13.8) gives the equation for the flow depth for a channel with and without upstream inflow:

⎛n Q ⎞ yc = 1.542⎜⎜ c 1 2c ⎟⎟ ⎝ Sc ⎠

38

[

(

⎧ 1+ 1+ z 2 ⎪ ⎨ z5 ⎪⎩

] ⎫⎪⎬

)

18

12 2

(13.9)

⎪⎭

13.3. FLOW VELOCITY Substituting Eqs. (13.6) and (13.7) into Eq. (4.12) gives the equation of flow velocity along the equilibrium profile for a channel with upstream inflow:

⎛ S c1 2 v = 0.841⎜⎜ ⎝ nc

⎞ ⎟⎟ ⎠

34

⎧⎪ z ⎨ ⎪⎩ 1 + 1 + z 2

[

(

)

12

⎫⎪ 2 ⎬ ⎪⎭

14

(Qu + q L xc )1 4

]

(13.10)

For a channel with zero upstream inflow zero upstream inflow (Qu = 0), Eq. (13.10) reduces to:

⎛ S1 2 v = 0.841⎜⎜ c ⎝ nc

⎞ ⎟⎟ ⎠

34

⎧⎪ zq L xc ⎨ ⎪⎩ 1 + 1 + z 2

[

(

14

)

12

⎫⎪ 2 ⎬ ⎪⎭

]

(13.11)

13.4. AVERAGE FLOW VELOCITY Substituting Eqs. (13.6) and (13.7) into Eq. (4.15) gives the equation of average flow velocity for a channel with upstream inflow: 14

⎫⎪ ⎛ S ⎞ ⎧⎪ z ⎟⎟ ⎨ 0.631⎜⎜ ⎬ qL Lc 2 12 ⎝ nc ⎠ ⎪⎩ 1 + 1 + z 2 ⎪⎭ vav = 34 34 (Qu + qL Lc ) − Qu 12 c

34

[

(

]

)

(13.12)

For a channel with zero upstream inflow (Qu = 0), Eq. (13.12) reduces to:

⎛ S c1 2 vav = 0.631⎜⎜ ⎝ nc

⎞ ⎟⎟ ⎠

34

⎧⎪ zq L Lc ⎨ 12 ⎪⎩ 1 + 1 + z 2

[

(

)

14

⎫⎪ 2 ⎬ ⎪⎭

]

(13.13)

Tommy S.W. Wong

166

13.5. KINEMATIC WAVE CELERITY Substituting Eqs. (13.6) and (13.7) into Eq. (4.29) gives the equation of wave celerity along the equilibrium profile for a channel with upstream inflow:

⎛ S1 2 c k = 1.122⎜⎜ c ⎝ nc

⎞ ⎟⎟ ⎠

34

⎧⎪ z ⎨ ⎪⎩ 1 + 1 + z 2

[

(

14

)

12

⎫⎪ 2 ⎬ ⎪⎭

(Qu +q L xc )1 4

]

(13.14)

For a channel with zero upstream inflow (Qu = 0), Eq. (13.14) reduces to:

⎛ S1 2 ck = 1.122⎜⎜ c ⎝ nc

⎞ ⎟⎟ ⎠

34

⎧⎪ zq L xc ⎨ 12 ⎪⎩ 1 + 1 + z 2

[

(

)

14

⎫⎪ 2 ⎬ ⎪⎭

]

(13.15)

13.6. AVERAGE WAVE CELERITY Substituting Eqs. (13.6) and (13.7) into Eq. (4.32) gives the equation of wave celerity for a channel with upstream inflow:

cav =

14

⎧⎪ ⎫⎪ z ⎨ ⎬ qL Lc 12 2 ⎪⎩ 1 + 1 + z 2 ⎪⎭ (Qu + qL Lc )3 4 − Qu 3 4

⎛ S1 2 ⎞ 0.841⎜⎜ c ⎟⎟ ⎝ nc ⎠

34

[

(

]

)

(13.16)

For a channel with zero upstream inflow (Qu = 0), Eq. (13.16) reduces to:

⎛ S1 2 cav = 0.841⎜⎜ c ⎝ nc

⎞ ⎟⎟ ⎠

34

⎧⎪ zq L Lc ⎨ 12 ⎪⎩ 1 + 1 + z 2

[

(

)

14

⎫⎪ 2 ⎬ ⎪⎭

]

(13.17)

13.7. TIME OF TRAVEL Substituting Eqs. (13.6) and (13.7) into Eq. (4.35) gives the equation of time of travel for a channel with upstream inflow:

⎛ n t t = 0.0198⎜⎜ 1c2 ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

)

]

12 2

14

⎫ ⎡ (Q + q L )3 4 − Q 3 4 ⎤ ⎪ u L c u ⎥ ⎬ ⎢ qL ⎪⎭ ⎣⎢ ⎦⎥

(13.18)

Kinematic-Wave Rainfall-Runoff Formulas

167

For a channel with zero upstream inflow (Qu = 0), Eq. (13.18) reduces to:

⎛n L t t = 0.0198⎜⎜ c1 2c ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

] ⎫⎪⎬

)

⎧ 1+ 1+ z2 ⎪ ⎨ zq L ⎪⎩

12 2

14

(13.19)

⎪⎭

13.8. HYDROGRAPH – RISING PHASE Substituting Eqs. (13.6) and (13.7) into Eq. (4.45) gives the equation for the rising phase of the hydrograph for a channel with upstream inflow:

[

]

43

14 34 ⎞ ⎛ 12 2⎫ ⎟ ⎜ ⎛ nc Qu ⎞ ⎧⎪ 1 + 1 + z 2 ⎪ ⎜ ⎟ . q t ⎬ + 60 L ⎟ (13.20) ⎜1 189⎜ 1 2 ⎟ ⎨ 12 z ⎝ S c ⎠ ⎪⎩ ⎟ ⎜ ⎪⎭ ⎠ ⎝ For a channel with zero upstream inflow (Qu = 0), Eq. (13.20) reduces to:

⎛ S ⎞⎧⎪ z ⎟⎟⎨ Qc = 0.794⎜⎜ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2 12 c

⎛ S1 2 Qc = 186.5⎜⎜ c ⎝ nc

[

(

13

⎫⎪ 2⎬ ⎪⎭

]

)

(

)

13

⎞⎧⎪ z ⎟⎟⎨ 12 ⎠⎪⎩ 1 + (1 + z 2 )

⎫⎪ 2 ⎬ ⎪⎭

[

(q L t )4 3

]

(13.21)

Equations (13.20) and (13.21) are valid for t ≤ tt.

13.9. FORWARD CHARACTERISTIC - RISING PHASE Substituting Eqs. (13.6) and (13.7) into Eq.(4.47) gives the equation for the forward characteristic of the rising phase for a channel with upstream inflow:

⎛ n t = 0.0198⎜⎜ 1c2 ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

] ⎫⎪⎬

)

12 2

14

⎡ (Qu + q L xc )3 4 − Qu3 4 ⎤ ⎢ ⎥ qL ⎥⎦ ⎪⎭ ⎢⎣

(13.22)

For a channel with zero upstream inflow (Qu = 0), Eq. (13.22) reduces to:

⎛n x t = 0.0198⎜⎜ c1 2c ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

)

] ⎫⎪⎬

12 2

⎪⎭

14

(13.23)

Tommy S.W. Wong

168

13.10. WATER SURFACE PROFILE – RISING PHASE Substituting Eqs. (13.6) and (13.7) into Eq. (4.49) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

[

⎛ ⎜ ⎛ n ⎞⎧⎪ 1 + 1 + z 2 Ac = 1.189⎜ ⎜⎜ 1c2 ⎟⎟⎨ z ⎜ ⎝ Sc ⎠⎪⎩ ⎝

(

)

]

12 2

⎞ ⎟ (Qu + qL xc )⎟ ⎟ ⎠

13

⎫ ⎪ ⎬ ⎪⎭

34

(13.24)

Substituting Eq. (13.1) into Eq. (13.24) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lp:

[

⎛ ⎜ ⎛ n ⎞⎧⎪ 1 + 1 + z 2 yc = 1.542⎜ ⎜⎜ 1c2 ⎟⎟⎨ z5 ⎜ ⎝ S c ⎠⎪⎩ ⎝

(

]

)

12 2

13

⎫ ⎪ ⎬ ⎪⎭

⎞ ⎟ (Qu + qL xc )⎟ ⎟ ⎠

38

(13.25)

Substituting Eqs. (13.6) and (13.7) into Eq. (4.50) gives the equation for the rising phase of the flow area profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

[

⎛ ⎜ ⎛ nc ⎞⎧⎪ 1 + 1 + z 2 Ap = 1.189⎜ ⎜⎜ 1 2 ⎟⎟⎨ z ⎜ ⎝ S c ⎠⎪⎩ ⎝

(

)

]

12 2

13

⎫ ⎪ ⎬ ⎪⎭

⎞ (Qu + qL Lp )⎟⎟ ⎟ ⎠

34

(13.26)

Substituting Ac = Ap, and yc = yp into Eq. (13.1) and then substituting it into Eq. (13.26) gives the equation for the rising phase of the water surface profile for a channel with upstream inflow, which is valid for Lp ≤ xc ≤ Lc:

[

⎛ ⎜ ⎛ n ⎞⎧⎪ 1 + 1 + z 2 y p = 1.542⎜ ⎜⎜ 1c2 ⎟⎟⎨ z5 ⎜ ⎝ S c ⎠⎪⎩ ⎝

(

)

]

12 2

13

⎫ ⎪ ⎬ ⎪⎭

⎞ (Qu + qL Lp )⎟⎟ ⎟ ⎠

38

(13.27)

From Eq. (4.51), the distance Lp is:

Lp =

Q p − Qu qL

(13.28)

For a channel with zero upstream inflow (Qu = 0), Eqs. (13.24)-(13.28) reduce to:

Kinematic-Wave Rainfall-Runoff Formulas

[

(

)

]

[

(

)

]

⎛ ⎜n q x Ac = 1.189⎜ c 1L2 c ⎜ Sc ⎝

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

⎛ ⎜n q x yc = 1.542⎜ c 1L2 c ⎜ Sc ⎝

⎧ 1+ 1+ z2 ⎪ ⎨ z5 ⎪⎩

13

⎫ ⎪ ⎬ ⎪⎭

12 2

13

⎫ ⎪ ⎬ ⎪⎭

12 2

⎞ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎠

169

34

(13.29)

38

(13.30)

which are valid for 0 ≤ xc ≤ Lp,

[

(

)

]

[

(

)

]

⎛ ⎜ nc qL L p Ap = 1.189⎜ 1 2 ⎜ Sc ⎝

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

⎛ ⎜ nc q L L p y p = 1.542⎜ 1 2 ⎜ Sc ⎝

⎧ 1+ 1+ z2 ⎪ ⎨ z5 ⎪⎩

12 2

12 2

13

⎫ ⎪ ⎬ ⎪⎭

13

⎫ ⎪ ⎬ ⎪⎭

⎞ ⎟ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎟ ⎠

34

(13.31)

38

(13.32)

which are valid for Lp ≤ xc ≤ Lc, and

Lp =

Qp

(13.33)

qL

13.11. DURATION OF PARTIAL EQUILIBRIUM DISCHARGE Substituting Eqs. (13.6) and (13.7) into Eq. (4.63) gives the equation for the duration of partial equilibrium discharge for a channel with upstream inflow:

td =

⎛ S 1 2 ⎞⎧⎪ z Lc qL + Qu − 0.794⎜⎜ c ⎟⎟⎨ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2

[

(

⎡ ⎛ S 1 2 q ⎞ ⎧⎪ z 63.52⎜⎜ c L ⎟⎟ ⎢⎢⎨ n c ⎝ ⎠ ⎢⎪⎩ 1 + 1 + z 2 ⎣

[

(

)

12

13

)

12

⎫⎪ 2⎬ ⎪⎭

]

⎛ ⎜ ⎛ nc Qu ⎜1.189⎜⎜ 1 2 ⎝ Sc ⎜ ⎝

⎫⎪⎛⎜ ⎛n Q 1.189⎜⎜ c 1 2u 2 ⎬⎜ ⎝ Sc ⎪⎭⎜ ⎝

]

⎞ ⎟⎟ ⎠

34

[

⎞ ⎟⎟ ⎠

34

[

(

(

⎧⎪ 1 + 1 + z 2 ⎨ z ⎪⎩

)

12

]

⎞ ⎟ + 60 qL t q ⎟ ⎟ ⎠ 13 2 14 ⎤ ⎞ ⎫⎪ ⎟⎥ ⎬ + 60 qL t q ⎟⎥ ⎟⎥ ⎪⎭ ⎠⎦

⎧⎪ 1 + 1 + z 2 ⎨ z ⎪⎩

)

12 2

14

⎫⎪ ⎬ ⎪⎭

]

For a channel with zero upstream inflow (Qu = 0), Eq. (13.34) reduces to:

43

(13.34)

Tommy S.W. Wong

170

td =

⎛ S 1 2 ⎞⎧⎪ z Lc − 186.5⎜⎜ c ⎟⎟⎨ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2

13

⎫⎪ 13 43 ⎬ qL tq 12 2 ⎪⎭ 13 ⎫⎪ qL tq z ⎛ S c1 2 ⎞⎧⎪ ⎟⎟⎨ 248.7⎜⎜ ⎬ 12 2 ⎝ nc ⎠⎪⎩ 1 + 1 + z 2 ⎪⎭

[

(

[

]

)

(

(13.35)

]

)

13.12. HYDROGRAPH - EQUILIBRIUM PHASE As shown in figures 4.2 and 4.3, the hydrograph may reach partial equilibrium or equilibrium depending on the duration of lateral inflow tq. If tq < tt, the hydrograph reaches partial equilibrium with a constant discharge Qp. If tq ≥ tt, the hydrograph reaches equilibrium with a constant discharge Qe.

13.12.1. Partial Equilibrium Discharge Substituting Eqs. (13.6) and (13.7) into Eq. (4.66) gives the equation for the partial equilibrium discharge for a channel with upstream inflow:

⎛ S 1 2 ⎞⎧⎪ z Qp = 0.794⎜⎜ c ⎟⎟⎨ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2

[

(

13

)

12

⎫⎪ 2⎬ ⎪⎭

]

⎛ ⎜ ⎛ ncQu ⎜1.189⎜⎜ 1 2 ⎝ Sc ⎜ ⎝

⎞ ⎟⎟ ⎠

34

[

(

⎧ 1+ 1+ z 2 ⎪ ⎨ z ⎪⎩

)

]

12 2

43

14 ⎞ ⎫ ⎟ ⎪ (13.36) 60 q t + ⎬ L q⎟ ⎟ ⎪⎭ ⎠

For a channel with zero upstream inflow (Qu = 0), Eq. (13.36) reduces to:

⎛ S1 2 Q p = 186.5⎜⎜ c ⎝ nc

⎞⎧⎪ z ⎟⎟⎨ ⎠⎪⎩ 1 + 1 + z 2

[

(

13

)

12

⎫⎪ 2 ⎬ ⎪⎭

]

(q t )

43

L q

(13.37)

Equations (13.36) and (13.37) are valid for tq ≤ t ≤ (tq + td).

13.12.2. Equilibrium Discharge From Eq. (4.69), the equation for the equilibrium discharge for a channel with upstream inflow is:

Qe = Qu + q L Lc

(13.38)

Kinematic-Wave Rainfall-Runoff Formulas

171

For a channel with zero upstream inflow (Qu = 0), Eq. (13.38) reduces to:

Q e = q L Lc

(13.39)

Equations (13.38) and (13.39) are valid for tt ≤ t ≤ tq.

13.13. WATER SURFACE PROFILE - EQUILIBRIUM PHASE Substituting Eqs. (13.6) and (13.7) into Eq. (4.71) gives the equation for the equilibrium water surface profile for a channel with upstream inflow between –Lu ≤ xc ≤ Lp.

[

⎛ ⎜ ⎛ nc ⎞⎧⎪ 1 + 1 + z 2 Ac = 1.189⎜ ⎜⎜ 1 2 ⎟⎟⎨ z ⎜ ⎝ Sc ⎠⎪⎩ ⎝

(

]

)

13

⎫ ⎪ ⎬ ⎪⎭

12 2

⎞ ⎟ (Qu + qL xc )⎟ ⎟ ⎠

34

(13.40)

Substituting Eq. (13.1) into Eq. (13.40) gives the equation for the equilibrium water surface profile for a channel with upstream inflow, which is valid for –Lu ≤ xc ≤ Lp:

[

⎛ ⎜ ⎛ n ⎞⎧⎪ 1 + 1 + z 2 yc = 1.542⎜ ⎜⎜ 1c2 ⎟⎟⎨ z5 ⎜ ⎝ S c ⎠⎪⎩ ⎝

(

]

)

12 2

13

⎫ ⎪ ⎬ ⎪⎭

⎞ ⎟ (Qu + qL xc )⎟ ⎟ ⎠

38

(13.41)

For a channel with zero upstream inflow (Qu = 0), Eqs. (13.40) and (13.41) reduce to:

[

(

)

]

[

(

)

]

⎛ ⎜n q x Ac = 1.189⎜ c 1L2 c ⎜ Sc ⎝

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

⎛ ⎜n q x yc = 1.542⎜ c 1L2 c ⎜ Sc ⎝

⎧ 1+ 1+ z2 ⎪ ⎨ z5 ⎪⎩

which are valid for 0 ≤ xc ≤ Lc.

12 2

12 2

13 34

⎫ ⎪ ⎬ ⎪⎭

⎞ ⎟ ⎟ ⎟ ⎠

(13.42)

13 38

⎫ ⎪ ⎬ ⎪⎭

⎞ ⎟ ⎟ ⎟ ⎠

(13.43)

Tommy S.W. Wong

172

13.14. EQUILIBRIUM DETENTION STORAGE Substituting Eqs. (13.6) and (13.7) into Eq. (4.78) gives the equilibrium detention storage for a channel with upstream inflow:

⎛ n Dec = 0.679⎜⎜ 1c2 ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

] ⎫⎪⎬

)

14

12 2

⎡ (Qu + q L Lc )7 4 − Qu7 4 ⎤ ⎢ ⎥ qL ⎥⎦ ⎪⎭ ⎢⎣

(13.44)

For a channel with zero upstream inflow (i.e. Qu = 0), Eq. (13.44) reduces to:

⎛ n ⎞ Dec = 0.679⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

)

] ⎫⎪⎬

12 2

14

⎪⎭

qL3 4 L7c 4

(13.45)

13.15. WATER SURFACE PROFILE – FALLING PHASE Substituting Eqs. (13.6) and (13.7) into Eq. (4.91) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc.

⎛ S1 2 xc = 63.52⎜⎜ c ⎝ nc

⎞⎧⎪ zAc ⎟⎨ ⎟ ⎠⎪⎩ 1 + 1 + z 2

[

(

)

12

⎛ ⎛ S c1 2 ⎜ 13 ⎜ 0.794⎜⎜ ⎫ ⎝ nc ⎜ ⎪ ( ) t t − + ⎜ q 2 ⎬ ⎪⎭ ⎜ ⎜ ⎜ ⎝

]

⎞⎧⎪ zAc4 ⎟⎨ ⎟ ⎠⎪⎩ 1 + 1 + z 2 qL

[

(

)

12

⎫ ⎪ 2 ⎬ ⎪⎭

]

13

⎞ ⎟ − Qu ⎟ ⎟ (13.46) ⎟ ⎟ ⎟ ⎟ ⎠

Substituting Eq. (13.1) into Eq. (13.46) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for Lf ≤ xc ≤ Lc:

⎛ ⎛ S c1 2 ⎞⎧⎪ ⎜ z 5 yc8 ⎟ ⎜ . 0 315 ⎜ 23 ⎜ n ⎟⎨ ⎞ ⎝ c ⎠⎪⎩ 1 + 1 + z 2 ⎛ S c1 2 ⎞⎛⎜ zyc ⎜ ⎟ (t − t q ) + ⎟⎟ xc = 50.42⎜⎜ ⎜ 2 12 ⎟ ⎜ qL ⎝ nc ⎠⎝ 1 + 1 + z ⎠ ⎜ ⎜ ⎜ ⎝

(

)

[

(

)

12

13 ⎞ ⎫⎪ ⎟ Q − u ⎟ 2⎬ ⎪⎭ ⎟ (13.47) ⎟ ⎟ ⎟ ⎟ ⎠

]

From Eq. (4.92), the equation for the flow area profile between 0 ≤ xc ≤ Lf is:

Kinematic-Wave Rainfall-Runoff Formulas

⎛n Q Ac = 1.189⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

173

] ⎫⎪⎬

)

14

12 2

(13.48)

⎪⎭

Substituting Eq. (13.1) into Eq. (13.48) gives the equation for the falling phase of the water surface profile for a channel with upstream inflow, which is valid for 0 ≤ xc ≤ Lf:

⎛n Q y c = 1.542⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟ ⎟ ⎠

38

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z5 ⎪⎩

)

] ⎫⎪⎬

12 2

18

(13.49)

⎪⎭

Substituting Eqs. (13.6) and (13.7) into Eq. (4.94) gives the equation for the distance Lf for a channel with upstream inflow, which is valid for t ≥ tq:

⎛ 12 ⎧ ⎜ ⎛ S ⎞⎪ z L f = 67.29⎜ ⎜⎜ c ⎟⎟⎨ ⎜ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2 ⎝

[

(

13 34

)

12

⎫⎪ 2⎬ ⎪⎭

]

⎞ ⎟ 14 ⎟ Qu (t − t q ) ⎟ ⎠

(13.50)

For a channel with zero upstream inflow (Qu = 0), Eqs. (13.46) and (13.47) reduce to:

⎛ S 1 2 ⎞⎧⎪ zAc xc = 63.52⎜⎜ c ⎟⎟⎨ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2

[

⎛ S1 2 x c = 50.42⎜⎜ c ⎝ nc

(

)

⎞⎛⎜ zy c ⎟ ⎟⎜ 2 ⎠⎝ 1 + 1 + z

(

which are valid for 0 ≤ xc ≤ Lc.

12

)

⎛ ⎛ Sc1 2 ⎞⎧⎪ ⎜ zAc4 ⎜ ⎟ 0 794 . 13 ⎜ ⎜ n ⎟⎨ ⎫⎪ ⎝ c ⎠⎪⎩ 1 + 1 + z 2 ⎜ ( ) − + t t q 2⎬ ⎜ qL ⎪⎭ ⎜ ⎜ ⎜ ⎝

[

]

12

⎛ ⎛ S c1 2 ⎜ ⎜ 0 315 . ⎜ 23 ⎜ n ⎞ ⎝ c ⎜ ⎟ (t − t q ) + ⎜ ⎟ ⎜ ⎠ ⎜ ⎜ ⎝

(

13

)

12

⎞⎧⎪ z 5 y c8 ⎟⎨ ⎟ ⎠⎪⎩ 1 + 1 + z 2 qL

[

(

)

⎫⎪ 2⎬ ⎪⎭

]

12

⎫ ⎪ 2 ⎬ ⎪⎭

]

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ 13

(13.51)

⎞ ⎟ ⎟ ⎟ (13.52) ⎟ ⎟ ⎟ ⎟ ⎠

Tommy S.W. Wong

174

13.16. HYDROGRAPH - FALLING PHASE Substituting Eqs. (13.6) and (13.7) into Eq. (4.101) gives the equation for the falling phase of a hydrograph for a channel with upstream inflow:

⎛ n ⎞ t = 0.0149⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ zQc ⎪⎩

)

] ⎫⎪⎬

12 2

14

⎡ ⎛ Q c − Qu ⎢ Lc − ⎜⎜ ⎝ qL ⎪⎭ ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

(13.53)

For a channel with zero upstream inflow (Qu = 0), Eq. (13.53) reduces to:

⎛ n t = 0.0149⎜⎜ 1c2 ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ zQc ⎪⎩

)

] ⎫⎪⎬

12 2

14

⎡ ⎛ Qc ⎞ ⎤ ⎢ Lc − ⎜⎜ ⎟⎟⎥ + t q ⎝ q L ⎠⎦ ⎭⎪ ⎣

(13.54)

In: Kinematic-Wave Rainfall-Runoff Formulas Editor: Tommy S.W. Wong, pp. 175-231

ISBN: 978-1-60692-705-2 © 2009 Nova Science Publishers, Inc.

APPENDICES APPENDIX A. APPLICABILITY OF KINEMATIC WAVE THEORY A.1. Flow on Overland Plane The applicability of the kinematic wave theory to overland flow situations with sufficient accuracy as compared to the solution from the Saint Venant equations have been investigated by several researchers (Woolhiser and Liggett 1967, Ponce et al. 1978, Morris and Woolhiser 1980). For overland flow, the applicability of the theory can be defined by the Morris and Woolhiser (1980) criterion:

KFe2 ≥ 5

(A.1)

where K = kinematic flow number, and Fe = Froude number at the end of the plane at equilibrium. The parameter KFe can be related to the physical characteristics of an overland plane and the rainfall intensity as follows (Wong 2005b):

⎡ S 1.3 L0.4 ⎤ KFe2 = 8586 ⎢ 0.6o o 0.6 ⎥ ⎢⎣ no (C r i ) ⎥⎦

(A.2)

where So = slope of the overland plane, Lo = length of the overland plane, and no = Manning’s roughness coefficient of the overland surface, Cr = runoff coefficient, and i = rainfall intensity. Substituting Equation (A.2) into Equation (A.1) gives:

⎡ S o1.3 L0o.4 ⎤ ≥ 0.000582 ⎢ 0.6 0.6 ⎥ ⎢⎣ no (C r i ) ⎥⎦

(A.3)

In general, the theory is applicable to overland flow situations where the backwater effect is not significant (Overton and Meadows 1976).

Tommy S.W. Wong

176

A.2. Flow in Open Channel The applicability of the kinematic wave theory to open channel flow situations with sufficient accuracy as compared to the solution from the diffusive wave equations can be defined by the Ponce et al. (1978) criterion:

τ=

Tw S c v s > 1.383 ys

(A.4)

where τ = dimensionless wave period. Tw = wave period that can be taken as twice the timeof-rise of the flood wave (Ponce 1991), Sc = channel bed slope, vs = steady-state, uniform, mean flow velocity in the channel, and ys = steady-state, uniform, flow depth in the channel. As a rule of thumb, the American Society of Civil Engineers (1996, 1997) simplified the criterion to:

S c > 0.002

(A.5)

In general, the theory is applicable to most open channel flow situations where backwater effect is not significant (Overton and Meadows 1976).

Kinematic-Wave Rainfall-Runoff Formulas

APPENDIX B. GENERAL FORMULAS FOR FLOW ON OVERLAND PLANE B.1. Flow Depth

⎛ q y o = ⎜⎜ ⎝αo

⎞ ⎟⎟ ⎠

1 βo

B.2. Flow Velocity

[

v = α o (qu + C r ixo )

]

β o −1 1 β o

B.3. Average Flow Velocity vav =

α o 1 β Cr iLo o

[

β o (qu + Cr iLo )1 β − qu 1 β o

o

]

B.4. Kinematic Wave Celerity

[

c k = β o α o (qu + C r ixo )

]

β o −1 1 β o

B.5. Average Wave Celerity cav =

α o1 β Cr iLo o

(qu + Cr iLo )1 β

o

− qu

1 βo

B.6. Time of Concentration

to =

1

α o1 β

o

⎡ (qu + C r iLo )1 β o − qu 1 β o ⎢ Cr i ⎢⎣

⎤ ⎥ ⎥⎦

177

Tommy S.W. Wong

178

B.7. Design Discharge

(

bβ o

)

⎡ aC 1 b ⎤ b + β o −bβ o r Qd Ao = ⎢ 1 βo ⎥ ⎢⎣ (Lo α o ) ⎥⎦

B.8. Hydrograph – Rising Phase ⎤ ⎡⎛ q ⎞1 β o q = α o ⎢⎜⎜ u ⎟⎟ + Cr it ⎥ ⎥⎦ ⎢⎣⎝ α o ⎠

βo

for t ≤ to

B.9. Forward Characteristic – Rising Phase

t=

1

α o1 β

o

⎡ (qu + Cr ixo )1 β o − qu1 β o ⎤ ⎢ ⎥ Cr i ⎣ ⎦

B.10. Water Surface Profile – Rising Phase

⎛ q + Cr ixo ⎞ ⎟⎟ yo = ⎜⎜ u αo ⎝ ⎠

1 βo

for 0 ≤ xo ≤ Lp

⎛ q + Cr iL p ⎞ ⎟⎟ y p = ⎜⎜ u α o ⎝ ⎠ for Lp ≤ xo ≤ Lo

Lp =

q p − qu Cr i

1 βo

Kinematic-Wave Rainfall-Runoff Formulas

B.11. Duration of Partial Equilibrium Discharge

td =

⎡⎛ q ⎞1 βo ⎤ Cr iLo + qu − α o ⎢⎜ u ⎟ + Cr itr ⎥ ⎢⎣⎝ α o ⎠ ⎥⎦ ⎡⎛ q ⎞1 βo ⎤ α o β oCr i ⎢⎜ u ⎟ + Cr itr ⎥ ⎢⎣⎝ α o ⎠ ⎥⎦

βo

βo −1

B.12. Hydrograph – Equilibrium Phase B.12.1. Partial Equilibrium Discharge

⎡⎛ q q p = α o ⎢⎜⎜ u ⎢⎣⎝ α o

⎞ ⎟⎟ ⎠

1 βo

⎤ + C r it r ⎥ ⎥⎦

βo

for tr ≤ t ≤ (tr + td)

B.12.1. Equilibrium Discharge

qe = qu + Cr iLo for to ≤ t ≤ tr

B.13. Water Surface Profile – Equilibrium Phase

⎛ q + Cr ixo ⎞ ⎟⎟ yo = ⎜⎜ u α o ⎝ ⎠

1 βo

for 0 ≤ xo ≤ Lo

B.14. Equilibrium Detention Storage Deo =

[

βo (qu + C r iLo )(1+ β 1/ β (1 + β o )α o C r i o

o

) / βo

− qu

(1+ β o ) / β o

]

179

Tommy S.W. Wong

180

B.15. Water Surface Profile – Falling Phase

⎛q y o = ⎜⎜ u ⎝ αo

⎞ ⎟⎟ ⎠

1 βo

for 0 ≤ xo ≤ Lf β o −1

xo = α o β o yo

⎛ α o yoβ o − qu (t − tr ) + ⎜⎜ Cr i ⎝

for Lf ≤ xo ≤ Lo

L f = α o1 β o β o qu( β o −1) β o (t − t r )

B.16. Hydrograph – Falling Phase ⎛ q − qu ⎞ ⎟ Lo − ⎜ ⎜ Cr i ⎟ ⎝ ⎠ +t t= r 1 β o [1− (1 β o )] β oα o q for t ≥ tr

⎞ ⎟⎟ ⎠

Kinematic-Wave Rainfall-Runoff Formulas

APPENDIX C. GENERAL FORMULAS FOR FLOW IN OPEN CHANNEL C.1. Flow Area 1/ β c

⎛Q ⎞ Ac = ⎜⎜ c ⎟⎟ ⎝ αc ⎠

C.2. Flow Velocity

[

v = α c (Qu + q L x c )

]

β c −1 1 β c

C.3. Average Flow Velocity vav =

α c 1 β q L Lc c

[

β c (Qu + q L Lc )1 β − Qu 1 β c

c

]

C.4. Kinematic Wave Celerity

[

c k = β c α c (Qu + q L xc )

]

β c −1 1 / β c

C.5. Average Wave Celerity cav =

α c1 / β q L Lc c

(Qu + qL Lc )1 / β

c

− Qu

1/ βc

C.6. Time of Travel

tt =

1

α c1 / β

c

⎡ (Qu + q L Lc )1 / β c − Qu 1 / β c ⎢ qL ⎢⎣

⎤ ⎥ ⎥⎦

181

Tommy S.W. Wong

182

C.7. Hydrograph – Rising Phase ⎡⎛ Q Qc = α c ⎢⎜⎜ u ⎢⎣⎝ α c

1/ βc

⎤ + q Lt ⎥ ⎥⎦

⎞ ⎟⎟ ⎠

βc

for t ≤ tt

C.8. Forward Characteristic – Rising Phase

t=

1

α c1 / β

c

⎡ (Qu + q L xc )1 / β c − Qu 1 / β c ⎤ ⎢ ⎥ qL ⎥⎦ ⎣⎢

C.9. Flow Area Profile – Rising Phase 1/ β c

⎛ Q + q L xc ⎞ ⎟⎟ Ac = ⎜⎜ u α c ⎝ ⎠ for 0 ≤ xc ≤ Lp

1/ β c

⎛ Qu + q L L p ⎞ ⎟⎟ Ap = ⎜⎜ αc ⎝ ⎠ for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

C.10. Duration of Partial Equilibrium Discharge ⎤ ⎡⎛ Q ⎞1/ β c Lc qL + Qu − α c ⎢⎜⎜ u ⎟⎟ + q L t q ⎥ ⎥⎦ ⎢⎣⎝ α c ⎠ td = β c −1 ⎤ ⎡⎛ Q ⎞1/ β c u α c β c qL ⎢⎜⎜ ⎟⎟ + q Lt q ⎥ ⎥⎦ ⎢⎣⎝ α c ⎠

βc

Kinematic-Wave Rainfall-Runoff Formulas

C.11. Hydrograph – Equilibrium Phase C.11.1. Partial Equilibrium Discharge

⎡⎛ Q Q p = α c ⎢⎜⎜ u ⎢⎣⎝ α c

⎞ ⎟⎟ ⎠

1 / βc

⎤ + qLtq ⎥ ⎥⎦

βc

for tq ≤ t ≤ (tq + td)

C.11.2. Equilibrium Discharge

Qe = Qu + q L Lc

for tt ≤ t ≤ tq

C.12. Flow Area Profile – Equilibrium Phase 1/ β c

⎛ Q + q L xc ⎞ ⎟⎟ Ac = ⎜⎜ u αc ⎝ ⎠ for 0 ≤ xc ≤ Lc

C.13. Equilibrium Detention Storage

⎡ (Qu + qL Lc )(1+ β ) / β − Qu (1+ β βc Dec = (1 + β c )α c1 / β ⎢⎣ qL c

c

c

C.14. Flow Area Profile – Falling Phase

⎛Q Ac = ⎜⎜ u ⎝ αc

⎞ ⎟⎟ ⎠

1 βc

for 0 ≤ xc ≤ Lf

[

xc = α c β c Ac for Lf ≤ xc ≤ Lc

β c −1

]

⎛ α c Ac β c − Qu (t − t q ) + ⎜⎜ qL ⎝

⎞ ⎟ ⎟ ⎠

c

)/ βc

⎤ ⎥ ⎦

183

Tommy S.W. Wong

184

L f = α c1 β c β c Qu( β c −1) β c (t − t q )

C.15. Hydrograph – Falling Phase

⎛ Q − Qu ⎞ ⎟ Lc − ⎜⎜ c qL ⎟⎠ ⎝ t = 1 / βc + tq α c β c Qc( βc −1) / βc for t ≥ tq

Kinematic-Wave Rainfall-Runoff Formulas

APPENDIX D. KINEMATIC WAVE PARAMETERS D.1. Overland Plane

S o1 2 αo = no

βo =

5 3

D.2. Circular Channel ⎛ S c1 2 D1 / 6 ⎞ ⎟⎟ ⎝ nc ⎠

α c = 0.501⎜⎜

βc =

5 4

D.3. Parabolic Channel ⎛ S c1 2 ⎞ ⎟ 29 ⎟ n H ⎝ c ⎠

α c = 0.493⎜⎜

13 9 D.4. Rectangular (Deep) Channel

βc =

⎛ S c1 2W 2 3 ⎞ ⎟⎟ ⎝ nc ⎠

α c = 0.630⎜⎜

βc = 1

185

Tommy S.W. Wong

186

D.5. Rectangular (Square) Channel

⎛ S c1 2 ⎞ ⎟ ⎟ n c ⎠ ⎝

α c = 0.481⎜⎜ βc =

4 3

D.6. Rectangular (Wide) Channel ⎛ S1 2 ⎞

α c = ⎜⎜ c 2 3 ⎟⎟ ⎝ ncW ⎠

βc =

5 3

D.7. Trapezoidal Channel with Equal Side Slopes ⎛

S c1 2 ⎞ ⎟ 0.0909 ⎟ ⎝ ncW ⎠

α c = 0.340⎜⎜

β c = 1.379

D.8. Trapezoidal Channel with One Side Vertical ⎛

S c1 2 ⎞ ⎟ 0.0526 ⎟ ⎝ ncW ⎠

α c = 0.323⎜⎜

β c = 1.360

Kinematic-Wave Rainfall-Runoff Formulas

D.9. Triangular Channel

⎛ S c1 2 ⎝ nc

α c = 0.630⎜⎜ βc =

⎞⎛ z ⎞ ⎟⎟⎜ 2 ⎟ ⎠⎝ 1 + z ⎠

13

4 3

D.10. Vertical Curb Channel

⎛ S 1 2 ⎞⎧⎪ z α c = 0.794⎜⎜ c ⎟⎟⎨ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2

[

βc =

4 3

(

13

)

12

⎫⎪ 2⎬ ⎪⎭

]

187

Tommy S.W. Wong

188

APPENDIX E. WORKING FORMULAS FOR FLOW DEPTH E.1. Overland Plane

⎛n q⎞ yo = ⎜⎜ o1 2 ⎟⎟ ⎝ So ⎠

35

E.2. Circular Channel 3

2

⎛ nQ ⎞ ⎛y ⎞ ⎛y ⎞ ⎛y ⎞ − 1.195⎜ c ⎟ + 1.801⎜ c ⎟ + 0.397⎜ c ⎟ = 2.213⎜⎜ 1 2c c8 3 ⎟⎟ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝ Sc D ⎠

E.3. Parabolic Channel ⎛ nQ y c = 0.721⎜⎜ 1 2c c1 2 ⎝ Sc H

⎞ ⎟ ⎟ ⎠

6 13

E.4. Rectangular (Deep) Channel ⎛ nQ ⎞ yc = 1.587⎜⎜ 1 2c c5 3 ⎟⎟ ⎝ Sc W ⎠

E.5. Rectangular (Square) Channel

⎛nQ yc = 1.316⎜⎜ c 1 2c ⎝ Sc

⎞ ⎟⎟ ⎠

38

E.6. Rectangular (Wide) Channel

⎛ nQ yc = ⎜⎜ 1c 2 c ⎝ Sc W

⎞ ⎟⎟ ⎠

35

45

Kinematic-Wave Rainfall-Runoff Formulas

E.7. Trapezoidal Channel with Equal Side Slopes ⎡ ⎛ ncW 0.0909 Qc 2 ⎢ − W + W + 8.748 z ⎜⎜ S c1 2 ⎢⎣ ⎝ yc = 2z

⎞ ⎟ ⎟ ⎠

0.725 1 2

⎤ ⎥ ⎥⎦

E.8. Trapezoidal Channel with One Side Vertical ⎡ ⎛ n W 0.0526 Q − W + ⎢W 2 + 4.592 z ⎜⎜ c 1 2 c Sc ⎢⎣ ⎝ yc = z

E.9. Triangular Channel

⎛n Q ⎞ yc = 1.190⎜⎜ c 1 2c ⎟⎟ ⎝ Sc ⎠

38

18

⎛1+ z2 ⎞ ⎜⎜ 5 ⎟⎟ ⎝ z ⎠

E.10. Vertical Curb Channel ⎛n Q ⎞ yc = 1.542⎜⎜ c 1 2c ⎟⎟ ⎝ Sc ⎠

38

[

(

⎧ 1+ 1+ z 2 ⎪ ⎨ z5 ⎪⎩

)

] ⎫⎪⎬

12 2

18

⎪⎭

⎞ ⎟ ⎟ ⎠

0.735 1 2

⎤ ⎥ ⎥⎦

189

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190

APPENDIX F. WORKING FORMULAS FOR FLOW VELOCITY F.1. Overland Plane

⎛ S1 2 ⎞ v = 0.00238⎜⎜ o ⎟⎟ ⎝ no ⎠

35

(3.6 x10 q 6

u

+ Cr ixo

F.2. Circular Channel

⎛ S 1 2 D1 6 ⎞ ⎟⎟ v = 0.575⎜⎜ c ⎝ nc ⎠

45

(Qu + qL xc )1 5

F.3. Parabolic Channel ⎛ S1 2 v = 0.613⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟ ⎟ ⎠

9 13

(Qu + q L xc )4 13

F.4. Rectangular (Deep) Channel ⎛ S c1 2W 2 3 ⎞ ⎟ v = 0.630⎜⎜ ⎟ ⎠ ⎝ nc

F.5. Rectangular (Square) Channel

⎛ S1 2 ⎞ v = 0.578⎜⎜ c ⎟⎟ ⎝ nc ⎠

34

(Qu + qL xc )1 4

F.6. Rectangular (Wide) Channel

⎛ S1 2 ⎞ v = ⎜⎜ c 2 3 ⎟⎟ ⎝ ncW ⎠

35

(Qu + qL xc )2 5

)

25

Kinematic-Wave Rainfall-Runoff Formulas

F.7. Trapezoidal Channel with Equal Side Slopes

⎡ ⎤ ⎛ S c1 2 ⎞ ⎟(Qu + qL xc )0.379 ⎥ v = ⎢0.340⎜⎜ 0.0909 ⎟ ⎝ ncW ⎠ ⎣ ⎦

0.725

F.8. Trapezoidal Channel with One Side Vertical

⎡⎛ S c1 2 ⎞ ⎤ ⎟(Qu + qL xc )0.360 ⎥ v = 0.436⎢⎜⎜ 0.0526 ⎟ ⎠ ⎣⎝ ncW ⎦

0.735

F.9. Triangular Channel ⎛ S c1 2 v = 0.707⎜⎜ ⎝ nc

⎞ ⎟ ⎟ ⎠

34

⎡ z (Qu + q L xc ) ⎤ ⎢ 1+ z2 ⎥ ⎣ ⎦

14

F.10. Vertical Curb Channel

⎛ S c1 2 ⎞ ⎟⎟ v = 0.841⎜⎜ n ⎝ c ⎠

34

⎧⎪ z ⎨ ⎪⎩ 1 + 1 + z 2

[

(

)

12

⎫⎪ 2 ⎬ ⎪⎭

]

14

(Qu + q L xc )1 4

191

Tommy S.W. Wong

192

APPENDIX G. WORKING FORMULAS FOR AVERAGE FLOW VELOCITY G.1. Overland Plane

⎛ S1 2 ⎞ vav = 0.00143⎜⎜ o ⎟⎟ ⎝ no ⎠

35

⎡ Cr iLo ⎢ 35 6 6 ⎢⎣ 3.6 × 10 qu + Cr iLo − 3.6 × 10 qu

(

G.2. Circular Channel 45

⎛ S c1 2 D1 6 ⎞ ⎟⎟ qL Lc 0.460⎜⎜ n c ⎝ ⎠ vav = 45 (Qu + qL Lc ) − Qu 4 5

G.3. Parabolic Channel 9 13

⎛ S1 2 ⎞ 0.424⎜⎜ c 2 9 ⎟⎟ q L Lc ⎝ nc H ⎠ v av = (Qu + q L Lc )9 13 − Qu9 13

G.4. Rectangular (Deep) Channel ⎛ S c1 2W 2 3 ⎞ ⎟⎟ vav = 0.630⎜⎜ n c ⎠ ⎝

G.5. Rectangular (Square) Channel 34

⎛ S1 2 ⎞ 0.433⎜⎜ c ⎟⎟ q L Lc ⎝ nc ⎠ vav = (Qu + qL Lc )3 4 − Qu 3 4

)

(

)

35

⎤ ⎥ ⎥⎦

Kinematic-Wave Rainfall-Runoff Formulas

G.6. Rectangular (Wide) Channel 35

⎛ S1 2 ⎞ 0.600⎜⎜ c 2 3 ⎟⎟ q L Lc ⎝ nW ⎠ vav = (Qu + qL Lc )3 5 − Qu 3 5

G.7. Trapezoidal Channel with Equal Side Slopes 0.725

⎛ S c1 2 ⎞ ⎟ qL Lc 0.332⎜⎜ ncW 0.0909 ⎟⎠ ⎝ vav = (Qu + qL Lc )0.725 − Qu 0.725

G.8. Trapezoidal Channel with One Side Vertical 0.735

⎛ S c1 2 ⎞ ⎟ q L Lc 0.321⎜⎜ ncW 0.0526 ⎟⎠ ⎝ vav = (Qu + qL Lc )0.735 − Qu 0.735

G.9. Triangular Channel 34

14

⎛ S1 2 ⎞ ⎛ z ⎞ 0.530⎜⎜ c ⎟⎟ ⎜ ⎟ qL Lc nc ⎠ ⎝ 1 + z 2 ⎠ ⎝ vav = (Qu + qL Lc )3 4 − Qu 3 4

G.10. Vertical Curb Channel

vav =

14

⎧⎪ ⎫⎪ z ⎨ ⎬ qL Lc 12 2 ⎪⎩ 1 + 1 + z 2 ⎪⎭ 34 34 (Qu + qL Lc ) − Qu

⎛ S c1 2 ⎞ ⎟⎟ 0.631⎜⎜ n ⎝ c ⎠

34

[

(

)

]

193

Tommy S.W. Wong

194

APPENDIX H. WORKING FORMULAS FOR KINEMATIC WAVE CELERITY H.1. Overland Plane

⎛ S1 2 ⎞ ck = 0.00397⎜⎜ o ⎟⎟ ⎝ no ⎠

35

(3.6 ×10 q 6

u

+ Cr ixo

H.2. Circular Channel

⎛ S 1 2 D1 6 ⎞ ⎟⎟ ck = 0.719⎜⎜ c ⎝ nc ⎠

45

(Qu + q L xc )1 5

H.3. Parabolic Channel ⎛ S1 2 c k = 0.885⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟ ⎟ ⎠

9 13

(Qu + q L xc )4 13

H.4. Rectangular (Deep) Channel ⎛ S 1 2W 2 3 ⎞ ⎟⎟ ck = 0.630⎜⎜ c ⎝ nc ⎠

H.5. Rectangular (Square) Channel

⎛ S1 2 ⎞ ck = 0.770⎜⎜ c ⎟⎟ ⎝ nc ⎠

34

(Qu + qL xc )1 4

H.6. Rectangular (Wide) Channel

⎛ S c1 2 ⎞ ⎟ ck = 1.667⎜⎜ 23 ⎟ n W ⎝ c ⎠

35

(Qu + q L xc )2 5

)

25

Kinematic-Wave Rainfall-Runoff Formulas

H.7. Trapezoidal Channel with Equal Side Slopes

⎡⎛ S c1 2 ⎞ 0.379 ⎤ ⎟ ck = 0.630 ⎢⎜⎜ + ( Q q x ) ⎥ u L c 0.0909 ⎟ ⎠ ⎣⎝ ncW ⎦

0.725

H.8. Trapezoidal Channel with One Side Vertical

⎡⎛ S c1 2 ⎞ ⎤ ⎟(Qu + q L xc )0.360 ⎥ ck = 0.593⎢⎜⎜ 0.0526 ⎟ ⎠ ⎣⎝ ncW ⎦

0.735

H.9. Triangular Channel ⎛ S1 2 c k = 0.943⎜⎜ c ⎝ nc

⎞ ⎟ ⎟ ⎠

34

⎡ z (Qu + q L xc ) ⎤ ⎢ 1+ z2 ⎥ ⎣ ⎦

14

H.10. Vertical Curb Channel

⎛ S1 2 c k = 1.122⎜⎜ c ⎝ nc

⎞ ⎟⎟ ⎠

34

⎧⎪ z ⎨ ⎪⎩ 1 + 1 + z 2

[

(

14

)

12

⎫⎪ 2 ⎬ ⎪⎭

]

(Qu + q L xc )1 4

195

Tommy S.W. Wong

196

APPENDIX I. WORKING FORMULAS FOR AVERAGE WAVE CELERITY I.1. Overland Plane

⎛ S o1 2 ⎞ ⎟ c av = 0.00238⎜⎜ ⎟ n ⎝ o ⎠

35

⎡ C r iLo ⎢ 35 6 ⎢ 3.6 × 10 q u + C r i Lo − 3.6 × 10 6 q u ⎣

(

I.2. Circular Channel 45

⎛ S c1 2 D1 6 ⎞ ⎟⎟ qL Lc 0.575⎜⎜ n c ⎝ ⎠ cav = 45 + (Qu qL Lc ) − Qu 4 5

I.3. Parabolic Channel 9 13

⎛ S c1 2 ⎞ ⎟ q L Lc 0.613⎜⎜ nc H 2 9 ⎟⎠ ⎝ c av = (Qu + q L Lc )9 13 − Qu9 13

I.4. Rectangular (Deep) Channel ⎛ S c1 2W 2 3 ⎞ ⎟⎟ cav = 0.630⎜⎜ n c ⎝ ⎠

I.5. Rectangular (Square) Channel 34

⎛ S1 2 ⎞ 0.578⎜⎜ c ⎟⎟ q L Lc ⎝ nc ⎠ cav = (Qu + qL Lc )3 4 − Qu 3 4

)

(

⎤ ⎥ 35 ⎥ ⎦

)

Kinematic-Wave Rainfall-Runoff Formulas

I.6. Rectangular (Wide) Channel 35

⎛ S c1 2 ⎞ ⎜⎜ ⎟ q L Lc ncW 2 3 ⎟⎠ ⎝ cav = (Qu + qL Lc )3 5 − Qu 3 5

I.7. Trapezoidal Channel with Equal Side Slopes 0.725

⎛ S c1 2 ⎞ ⎟ qL Lc 0.457⎜⎜ ncW 0.0909 ⎟⎠ ⎝ cav = (Qu + qL Lc )0.725 − Qu 0.725

I.8. Trapezoidal Channel with One Side Vertical 0.735

⎛ S c1 2 ⎞ ⎟ q L Lc 0.436⎜⎜ ncW 0.0526 ⎟⎠ ⎝ cav = (Qu + qL Lc )0.735 − Qu 0.735

I.9. Triangular Channel 34

14

⎛ S c1 2 ⎞ ⎛ z ⎞ ⎟⎟ ⎜⎜ ⎟ qL Lc 0.707⎜⎜ n 1 + z 2 ⎟⎠ ⎝ c ⎠ ⎝ cav = (Qu + qL Lc )3 4 − Qu 3 4

I.10. Vertical Curb Channel

cav =

14

⎧⎪ ⎫⎪ z ⎨ ⎬ qL Lc 12 2 ⎪⎩ 1 + 1 + z 2 ⎪⎭ 34 34 (Qu + qL Lc ) − Qu

⎛ S1 2 ⎞ 0.841⎜⎜ c ⎟⎟ ⎝ nc ⎠

34

[

(

)

]

197

Tommy S.W. Wong

198

APPENDIX J. WORKING FORMULAS FOR TIME OF CONCENTRATION AND TIME OF TRAVEL J.1. Overland Plane

⎛ n ⎞ to = 6.988⎜⎜ 1o2 ⎟⎟ ⎝ So ⎠

35

(

)

(

⎡ 3.6 ×106 q + C iL 3 5 − 3.6 × 106 q u r o u ⎢ Cr i ⎢⎣

J.2. Circular Channel ⎛ nc ⎞ ⎡ (Qu + qL Lc )4 5 − Qu4 5 ⎤ tt = 0.0290⎜⎜ 1 2 1 6 ⎟⎟ ⎢ ⎥ qL ⎝ Sc D ⎠ ⎣ ⎦ 45

J.3. Parabolic Channel

⎛ nc H 2 9 t t = 0.0272⎜⎜ 1 2 ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

⎡ (Qu + q L Lc )9 13 − Qu9 13 ⎤ ⎢ ⎥ qL ⎣⎢ ⎦⎥

J.4. Rectangular (Deep) Channel ⎛ nL ⎞ tt = 0.0265⎜⎜ 1 2c c2 3 ⎟⎟ ⎝ Sc W ⎠

J.5. Rectangular (Square) Channel

⎛ n t t = 0.0289⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎡ (Qu + q L Lc )3 4 − Qu3 4 ⎤ ⎢ ⎥ qL ⎢⎣ ⎥⎦

J.6. Rectangular (Wide) Channel ⎛ n W 2 3 ⎞ ⎡ (Q + qL Lc )3 5 − Qu3 5 ⎤ tt = 0.0167⎜⎜ c 1 2 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc ⎠ ⎣ ⎦ 35

)

35

⎤ ⎥ ⎥⎦

Kinematic-Wave Rainfall-Runoff Formulas

J.7. Trapezoidal Channel with Equal Side Slopes ⎛ ncW 0.0909 ⎞ ⎟⎟ tt = 0.0364⎜⎜ 12 S c ⎝ ⎠

0.725

⎡ (Qu + qL Lc )0.725 − Qu0.725 ⎤ ⎢ ⎥ qL ⎣ ⎦

J.8. Trapezoidal Channel with One Side Vertical ⎛ ncW 0.0526 ⎞ ⎟⎟ tt = 0.0382⎜⎜ 12 S c ⎝ ⎠

0.735

⎡ (Qu + qL Lc )0.735 − Qu0.735 ⎤ ⎢ ⎥ qL ⎣ ⎦

J.9. Triangular Channel ⎛ n ⎞ tt = 0.0236⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

34

⎛ 1 + z 2 ⎞ ⎡ (Qu + qL Lc )3 4 − Qu3 4 ⎤ ⎜⎜ ⎟⎟ ⎢ ⎥ z qL ⎝ ⎠ ⎣ ⎦ 14

J.10. Vertical Curb Channel ⎛ n t t = 0.0198⎜⎜ 1c2 ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

)

]

12 2

14

⎫ ⎡ (Q + q L )3 4 − Q 3 4 ⎤ ⎪ u L c u ⎥ ⎬ ⎢ qL ⎪⎭ ⎣⎢ ⎦⎥

199

Tommy S.W. Wong

200

APPENDIX K. WORKING FORMULAS FOR HYDROGRAPH – RISING PHASE K.1. Overland Plane S1 2 q= o no

⎡⎛ n q ⎞ 3 5 Cr it ⎤ ⎥ ⎢⎜⎜ o1 2u ⎟⎟ + 60 × 103 ⎥ ⎢⎣⎝ S o ⎠ ⎦

53

for t ≤ to

K.2. Circular Channel 45 ⎤ ⎛ S c1 2 D1 6 ⎞ ⎡ ⎛ nc Qu ⎞ ⎟⎟ ⎢1.738⎜⎜ 1 2 1 6 ⎟⎟ + 60q Lt ⎥ Qc = 0.501⎜⎜ ⎥⎦ ⎝ nc ⎠ ⎢⎣ ⎝ Sc D ⎠

54

for t ≤ tt

K.3. Parabolic Channel

⎛ S 1 2 ⎞⎡ ⎛ n H 2 9Q Qc = 0.493⎜⎜ c 2 9 ⎟⎟ ⎢1.639⎜⎜ c 1 2 u ⎝ nc H ⎠ ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

⎤ + 60q L t ⎥ ⎥⎦

for t ≤ tt

K.4. Rectangular (Deep) Channel

⎤ ⎛ S 1 2W 2 3 ⎞ ⎡ ⎛ nQ ⎞ ⎟⎟ ⎢1.587⎜⎜ 1 2c u2 3 ⎟⎟ + 60qL t ⎥ Qc = 0.630⎜⎜ c ⎝ nc ⎠ ⎣ ⎝ Sc W ⎠ ⎦ for t ≤ tt

13 9

Kinematic-Wave Rainfall-Runoff Formulas

K.5. Rectangular (Square) Channel ⎛ S c1 2 ⎞ ⎡ ⎛n Q ⎟⎟ ⎢1.731⎜⎜ c 1 2u Qc = 0.481⎜⎜ ⎝ nc ⎠ ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎤ + 60qL t ⎥ ⎥⎦

43

for t ≤ tt

K.6. Rectangular (Wide) Channel 35 ⎤ ⎛ S c1 2 ⎞ ⎡⎛ nW 2 3Qu ⎞ ⎜ ⎟ ⎢ ⎥ ⎟ + q t Qc = ⎜⎜ 60 L 12 23 ⎟ ⎜ ⎟ ⎥ ⎝ ncW ⎠ ⎢⎣⎝ S c ⎠ ⎦

53

for t ≤ tt

K.7. Trapezoidal Channel with Equal Side Slopes ⎛ ncW 0.0909Qu ⎛ S c1 2 ⎞ ⎡ ⎟ ⎜ ⎢ Qc = 0.340⎜ 2.186⎜⎜ 0.0909 ⎟ n W S c1 2 ⎢ ⎝ ⎠⎣ ⎝ c

⎞ ⎟⎟ ⎠

0.725

⎤ + 60q Lt ⎥ ⎥⎦

1.379

for t ≤ tt

K.8. Trapezoidal Channel with One Side Vertical ⎛ ncW 0.0526Qu ⎛ S c1 2 ⎞ ⎡ ⎜⎜ ⎟ ⎢ Qc = 0.323⎜⎜ 2 . 295 0.0526 ⎟ S c1 2 ⎝ ⎠ ⎢⎣ ⎝ ncW for t ≤ tt

⎞ ⎟⎟ ⎠

0.735

⎤ + 60qL t ⎥ ⎥⎦

1.360

201

Tommy S.W. Wong

202

K.9. Triangular Channel 13

⎛ S 1 2 ⎞⎛ z ⎞ Qc = 0.630⎜⎜ c ⎟⎟⎜ 2 ⎟ ⎝ nc ⎠⎝ 1 + z ⎠

⎡ ⎛n Q ⎢1.414⎜⎜ c 1 2u ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

14

⎛ 1+ z 2 ⎞ ⎜⎜ ⎟⎟ ⎝ z ⎠

⎤ + 60q Lt ⎥ ⎥⎦

43

for t ≤ tt

K.10. Vertical Curb Channel ⎛ S ⎞⎧⎪ z ⎟⎟⎨ Qc = 0.794⎜⎜ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2 12 c

for t ≤ tt

[

(

13

)

12

⎫⎪ 2⎬ ⎪⎭

]

⎛ ⎜ ⎛ nc Qu ⎜1.189⎜⎜ 1 2 ⎝ Sc ⎜ ⎝

⎞ ⎟⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

)

]

12 2

14 ⎞ ⎫ ⎟ ⎪ ⎬ + 60qL t ⎟ ⎟ ⎪⎭ ⎠

43

Kinematic-Wave Rainfall-Runoff Formulas

203

APPENDIX L. WORKING FORMULA FOR FORWARD CHARACTERISTIC – RISING PHASE L.1. Overland Plane

⎛ n ⎞ t = 6.988⎜⎜ 1o2 ⎟⎟ ⎝ So ⎠

35

(

)

(

⎡ 3.6 × 106 q + C ix 3 5 − 3.6 × 106 q u r o u ⎢ Cr i ⎢⎣

L.2. Circular Channel ⎛ nc ⎞ ⎡ (Qu + q L xc )4 5 − Qu4 5 ⎤ t = 0.0290⎜⎜ 1 2 1 6 ⎟⎟ ⎢ ⎥ qL ⎝ Sc D ⎠ ⎣ ⎦ 45

L.3. Parabolic Channel

⎛ nc H 2 9 t = 0.0272⎜⎜ 1 2 ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

⎡ (q L xc + Qu )9 13 − Qu9 13 ⎤ ⎢ ⎥ qL ⎣⎢ ⎦⎥

L.4. Rectangular (Deep) Channel ⎛ nx ⎞ t = 0.0265⎜⎜ 1 2c c2 3 ⎟⎟ ⎝ Sc W ⎠

L.5. Rectangular (Square) Channel

⎛ n t = 0.0289⎜⎜ 1c2 ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎡ (Qu + q L xc )3 4 − Qu3 4 ⎤ ⎢ ⎥ qL ⎢⎣ ⎥⎦

L.6. Rectangular (Wide) Channel ⎛ n W 2 3 ⎞ ⎡ (Q + qL xc )3 5 − Qu3 5 ⎤ t = 0.0167⎜⎜ c 1 2 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc ⎠ ⎣ ⎦ 35

)

35

⎤ ⎥ ⎥⎦

Tommy S.W. Wong

204

L.7. Trapezoidal Channel with Equal Side Slopes ⎛ ncW 0.0909 ⎞ ⎟⎟ t = 0.0364⎜⎜ 12 S c ⎝ ⎠

⎡ (Qu + qL xc )0.725 − Qu0.725 ⎤ ⎢ ⎥ qL ⎣ ⎦

0.725

L.8. Trapezoidal Channel with One Side Vertical ⎛ ncW 0.0526 ⎞ ⎟⎟ t = 0.0383⎜⎜ 12 S c ⎝ ⎠

0.735

⎡ (Qu + q L xc )0.735 − Qu0.735 ⎤ ⎢ ⎥ qL ⎣ ⎦

L.9. Triangular Channel ⎛ n ⎞ t = 0.0236⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

34

⎛ 1 + z 2 ⎞ ⎡ (Qu + qL xc )3 4 − Qu3 4 ⎤ ⎜⎜ ⎟⎟ ⎢ ⎥ qL ⎝ z ⎠ ⎣ ⎦ 14

L.10. Vertical Curb Channel ⎛ n t = 0.0198⎜⎜ 1c2 ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

)

]

12 2

14

⎫ ⎪ ⎬ ⎪⎭

⎡ (Qu + q L xc )3 4 − Qu3 4 ⎤ ⎢ ⎥ qL ⎢⎣ ⎥⎦

Kinematic-Wave Rainfall-Runoff Formulas

205

APPENDIX M. WORKING FORMULAS FOR WATER SURFACE PROFILE – RISING PHASE M.1. Overland Plane ⎡⎛ n y o = 0.116 × 10 ⎢⎜⎜ 1o2 ⎣⎢⎝ S o −3

⎤ ⎞ ⎟ 3.6 × 10 6 qu + C r ixo ⎥ ⎟ ⎥⎦ ⎠

(

)

35

for 0 ≤ xo ≤ Lp

⎡⎛ n y p = 0.116 × 10 ⎢⎜⎜ 1o2 ⎣⎢⎝ S o −3

⎤ ⎞ ⎟ 3.6 × 10 6 qu + C r iL p ⎥ ⎟ ⎥⎦ ⎠

(

)

35

for Lp ≤ xo ≤ Lo

⎛ q p − qu L p = 3.6 × 10 6 ⎜⎜ ⎝ Cr i

⎞ ⎟⎟ ⎠

M.2. Circular Channel 3 2 ⎡⎛ n ⎤ ⎞ ⎛ yc ⎞ ⎛ yc ⎞ ⎛y ⎞ − 1.195⎜ ⎟ + 1.801⎜ ⎟ + 0.397⎜ c ⎟ = 2.213⎢⎜⎜ 1 2 c 8 3 ⎟⎟(Qu + q L xc )⎥ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎣⎝ S c D ⎠ ⎦

45

for 0 ≤ xc ≤ Lp 3

⎛ yp ⎞ ⎛ yp − 1.195⎜⎜ ⎟⎟ + 1.801⎜⎜ ⎝ D⎠ ⎝ D for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

2

⎞ ⎛y ⎟⎟ + 0.397⎜⎜ p ⎠ ⎝ D

⎡⎛ n ⎤ ⎞ ⎞ ⎟⎟ = 2.213⎢⎜⎜ 1 2 c 8 3 ⎟⎟(Qu + q L L p )⎥ ⎢⎣⎝ S c D ⎠ ⎥⎦ ⎠

45

Tommy S.W. Wong

206

M.3. Parabolic Channel ⎡⎛ n y c = 0.721⎢⎜⎜ 1 2 c 1 2 ⎢⎣⎝ S c H

⎤ ⎞ ⎟(Qu + q L xc )⎥ ⎟ ⎥⎦ ⎠

6 13

for 0 ≤ xc ≤ Lp

⎡⎛ n y c = 0.721⎢⎜⎜ 1 2 c 1 2 ⎢⎣⎝ S c H

⎤ ⎞ ⎟(Qu + q L L p )⎥ ⎟ ⎥⎦ ⎠

for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

M.4. Rectangular (Deep) Channel ⎛ n ⎞ yc = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟(Qu + q L xc ) ⎝ Sc W ⎠ for 0 ≤ xc ≤ Lp

⎛ n ⎞ y p = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟(Qu + qL L p ) ⎝ Sc W ⎠ for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

6 13

Kinematic-Wave Rainfall-Runoff Formulas

M.5. Rectangular (Square) Channel

⎡⎛ n ⎞ ⎤ yc = 1.316 ⎢⎜⎜ 1c2 ⎟⎟(Qu + q L xc )⎥ ⎣⎝ S c ⎠ ⎦

38

for 0 ≤ xc ≤ Lp

⎡⎛ n ⎞ ⎤ y p = 1.316⎢⎜⎜ 1c2 ⎟⎟(Qu + qL L p )⎥ ⎣⎝ S c ⎠ ⎦

38

for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

M.6. Rectangular (Wide) Channel

⎡⎛ n yc = ⎢⎜⎜ 1 2c ⎣⎝ S c W

⎤ ⎞ ⎟⎟(Qu + q L xc )⎥ ⎠ ⎦

35

for 0 ≤ xc ≤ Lp

⎡⎛ n y p = ⎢⎜⎜ 1 2c ⎣⎝ S c W for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

⎤ ⎞ ⎟⎟(Qu + qL L p )⎥ ⎠ ⎦

35

207

Tommy S.W. Wong

208

M.7. Trapezoidal Channel with Equal Side Slopes 12

0.725 ⎧⎪ ⎫⎪ ⎡ ncW 0.0909 (q L x c + Qu ) ⎤ 2 − W + ⎨W + 8.748 z ⎢ ⎬ ⎥ S c1 2 ⎪⎩ ⎪⎭ ⎣ ⎦ yc = 2z

for 0 ≤ xc ≤ Lp 0.725 ⎧ ⎫ ⎡ ncW 0.0909 (q L L p + Qu )⎤ ⎪ 2 ⎪ − W + ⎨W + 8.748 z ⎢ ⎥ ⎬ 12 Sc ⎥⎦ ⎢⎣ ⎪⎩ ⎪⎭ yp = 2z

12

for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

M.8. Trapezoidal Channel with One Side Vertical 12

0.735 ⎧⎪ ⎡ ncW 0.0526 (qL xc + Qu ) ⎤ ⎫⎪ 2 − W + ⎨W + 4.592 z ⎢ ⎥ ⎬ S c1 2 ⎪⎩ ⎣ ⎦ ⎪⎭ yc = z

for 0 ≤ xc ≤ Lp 0.735 ⎧ ⎫ ⎡ ncW 0.0526 (q L L p + Qu )⎤ ⎪ 2 ⎪ − W + ⎨W + 4.592 z ⎢ ⎥ ⎬ 12 Sc ⎥⎦ ⎢⎣ ⎪⎩ ⎪⎭ yp = z

12

for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

Kinematic-Wave Rainfall-Runoff Formulas

M.9. Triangular Channel ⎤ ⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 yc = 1.189 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ 5 ⎟⎟ (Qu + q L xc )⎥ ⎥⎦ ⎢⎣⎝ S c ⎠⎝ z ⎠

38

for 0 ≤ xc ≤ Lp

⎤ ⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 y p = 1.189 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ 5 ⎟⎟ (Qu + q L L p )⎥ ⎥⎦ ⎢⎣⎝ S c ⎠⎝ z ⎠

38

for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

M.10. Vertical Curb Channel

[

(

[

(

⎛ ⎜ ⎛ n ⎞⎧⎪ 1 + 1 + z 2 yc = 1.542⎜ ⎜⎜ 1c2 ⎟⎟⎨ z5 ⎜ ⎝ S c ⎠⎪⎩ ⎝

)

]

)

]

12 2

13

⎫ ⎪ ⎬ ⎪⎭

⎞ ⎟ (Qu + qL xc )⎟ ⎟ ⎠

38

for 0 ≤ xc ≤ Lp

⎛ ⎜ ⎛ n ⎞⎧⎪ 1 + 1 + z 2 y p = 1.542⎜ ⎜⎜ 1c2 ⎟⎟⎨ z5 ⎜ ⎝ Sc ⎠⎪⎩ ⎝ for Lp ≤ xc ≤ Lc

Lp =

Q p − Qu qL

12 2

13

⎫ ⎪ ⎬ ⎪⎭

⎞ (Qu + qL Lp )⎟⎟ ⎟ ⎠

38

209

Tommy S.W. Wong

210

APPENDIX N. WORKING FORMULAS FOR DURATION OF PARTIAL EQUILIBRIUM DISCHARGE N.1. Overland Plane 53 35 ⎧ 12 ⎡ ⎤ ⎫ ⎛ ⎞ S n q C iL C it r o o o u r r ⎪ ⎥ ⎪ ⎢⎜⎜ 1 2 ⎟⎟ + + qu − 3 ⎪ 3.6 × 106 60 10 n S × ⎥⎦ ⎪⎪ ⎢ o ⎪ ⎣⎝ o ⎠ t d = 36 × 103 ⎨ ⎬ 23 35 12 ⎤ ⎡ ⎪ ⎪ S o Cr i ⎛ no qu ⎞ Cr it r ⎥ ⎢⎜⎜ 1 2 ⎟⎟ + ⎪ ⎪ 60 ×103 ⎥ no ⎢⎝ S o ⎠ ⎪⎩ ⎪⎭ ⎦ ⎣

N.2. Circular Channel

td =

⎤ + 60q L t q ⎥ ⎥⎦ 14 45 ⎤ ⎛ S c1 2 D1 6 qL ⎞ ⎡ ⎛ nc Qu ⎞ ⎟⎟ ⎢1.738⎜⎜ 1 2 1 6 ⎟⎟ + 60q Lt q ⎥ 37.58⎜⎜ nc ⎥⎦ ⎝ ⎠ ⎢⎣ ⎝ Sc D ⎠

⎛ S 1 2 D1 6 ⎞ ⎡ ⎛ nQ ⎞ ⎟⎟ ⎢1.738⎜⎜ 1 c2 u1 6 ⎟⎟ Lc qL + Qu − 0.501⎜⎜ c ⎝ nc ⎠ ⎢⎣ ⎝ Sc D ⎠

45

54

N.3. Parabolic Channel

td =

⎤ + 60q Lt q ⎥ ⎥⎦ 4 9 9 13 ⎤ ⎛ S c1 2 qL ⎞ ⎡ ⎛ nc H 2 9Qu ⎞ ⎟ ⎜ ⎟ ⎢ + 42.73⎜⎜ 1 . 632 60 q t L q⎥ 29 ⎟ ⎜ S1 2 ⎟ ⎥⎦ c ⎝ nc H ⎠ ⎢⎣ ⎝ ⎠

⎛ S 1 2 ⎞⎡ ⎛ n H 2 9Q Lc q L + Qu − 0.493⎜⎜ c 2 9 ⎟⎟ ⎢1.632⎜⎜ c 1 2 u ⎝ nc H ⎠ ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

N.4. Rectangular (Deep) Channel

⎤ ⎛ S 1 2W 2 3 ⎞ ⎡ ⎛ nQ ⎞ ⎟⎟ ⎢1.587⎜⎜ 1 2c u2 3 ⎟⎟ + 60q Lt q ⎥ Lc q L + Qu − 0.630⎜⎜ c ⎝ nc ⎠ ⎣ ⎝ Sc W ⎠ ⎦ td = 12 23 ⎛ S W qL ⎞ ⎟⎟ 37.80⎜⎜ c nc ⎝ ⎠

13 9

Kinematic-Wave Rainfall-Runoff Formulas

211

N.5. Rectangular (Square) Channel

td =

⎤ + 60qL t q ⎥ ⎥⎦ 1/ 3 34 ⎤ ⎛ S c1 2 qL ⎞ ⎡ ⎛ nc Qu ⎞ ⎟⎟ ⎢1.731⎜⎜ 1 2 ⎟⎟ + 60qL t q ⎥ 38.48⎜⎜ ⎥⎦ ⎝ nc ⎠ ⎢⎣ ⎝ Sc ⎠

⎛ Sc1 2 ⎞ ⎡ ⎛n Q ⎟⎟ ⎢1.731⎜⎜ c 1 2u Lc qL + Qu − 0.481⎜⎜ ⎝ nc ⎠ ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

43

N.6. Rectangular (Wide) Channel

td =

⎤ + 60qL tq ⎥ ⎥⎦ 23 35 ⎤ ⎛ S c1 2 qL ⎞ ⎡⎛ ncW 2 3Qu ⎞ ⎟ ⎢⎜ ⎟⎟ + 60q Lt q ⎥ 100.0⎜⎜ 23 ⎟ ⎜ 12 n W S ⎥⎦ ⎢ c ⎝ c ⎠ ⎣⎝ ⎠

⎛ S 1 2 ⎞ ⎡⎛ n W 2 3Q Lc qL + Qu − ⎜⎜ c 2 3 ⎟⎟ ⎢⎜⎜ c 1 2 u ⎝ ncW ⎠ ⎢⎣⎝ S c

⎞ ⎟⎟ ⎠

35

53

N.7. Trapezoidal Channel with Equal Side Slopes

td =

⎤ + 60qL tq ⎥ ⎥⎦ 0 . 379 0.725 ⎤ ⎛ Sc1 2 qL ⎞ ⎡ ⎛ ncW 0.0909Qu ⎞ ⎟ ⎜ ⎟ ⎢ 28.13⎜⎜ 2 . 186 60 q t + L q⎥ 0.0909 ⎟ ⎜ ⎟ S c1 2 ⎥⎦ ⎝ ncW ⎠ ⎢⎣ ⎝ ⎠

⎛ Sc1 2 ⎞ ⎡ ⎛ ncW 0.0909Qu ⎟ ⎜⎜ ⎢ 2 186 Lc qL + Qu − 0.340⎜⎜ . 0.0909 ⎟ S c1 2 ⎝ ncW ⎠ ⎢⎣ ⎝

⎞ ⎟⎟ ⎠

0.725

1.379

N.8. Trapezoidal Channel with One Side Vertical

td =

0.735 ⎤ ⎛ S c1 2 ⎞ ⎡ ⎛ ncW 0.0526Qu ⎞ ⎟ ⎜ ⎟ ⎥ ⎢ 2 295 60 + Lc q L + Qu − 0.323⎜⎜ . q t L q 0.0526 ⎟ ⎜ ⎟ S c1 2 ⎥⎦ ⎝ ncW ⎠ ⎢⎣ ⎝ ⎠ 0.735 ⎤ ⎛ S c1 2 qL ⎞ ⎡ ⎛ ncW 0.0526Qu ⎞ ⎟ ⎜ ⎟ ⎢ 26.54⎜⎜ 2 . 295 60 q t + L q⎥ 0.0526 ⎟ ⎜ ⎟ S c1 2 ⎥⎦ ⎝ ncW ⎠ ⎢⎣ ⎝ ⎠

0.360

1.360

Tommy S.W. Wong

212

N.9. Triangular Channel 13

td =

⎛ S c1 2 ⎞⎛ z ⎞ ⎟⎟⎜ Lc q L + Qu − 0.630⎜⎜ 2 ⎟ n ⎝ c ⎠⎝ 1 + z ⎠ 13

⎛ S 1 2 q ⎞⎛ z ⎞ 50.40⎜⎜ c L ⎟⎟⎜ 2 ⎟ ⎝ nc ⎠⎝ 1 + z ⎠

⎡ ⎛n Q ⎢1.414⎜⎜ c 1 2u ⎢⎣ ⎝ Sc

⎡ ⎛n Q ⎢1.414⎜⎜ c 1 2u ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎞ ⎟⎟ ⎠

34

14 ⎤ ⎛1+ z2 ⎞ ⎜⎜ ⎟⎟ + 60q Lt q ⎥ ⎥⎦ ⎝ z ⎠

14 ⎤ ⎛1+ z2 ⎞ ⎜⎜ ⎟⎟ + 60qLt q ⎥ ⎥⎦ ⎝ z ⎠

43

13

N.10. Vertical Curb Channel

td =

[

]

14 ⎛ 34 ⎞ 12 2 ⎫ ⎜ ⎟ ⎛ ncQu ⎞ ⎧⎪ 1+ 1+ z 2 ⎪ ⎬ + 60qLtq ⎟ ⎜1.189⎜⎜ 1 2 ⎟⎟ ⎨ z ⎝ Sc ⎠ ⎪⎩ ⎜ ⎟ ⎪⎭ ⎝ ⎠ 13 14 2 ⎡ 34 ⎞⎤ 12 ⎫ ⎫⎪⎛⎜ ⎟⎥ ⎛ Sc1 2qL ⎞⎢⎧⎪ ⎛ ncQu ⎞ ⎧⎪ 1 + 1+ z 2 z ⎪ ⎜ ⎟ ⎜ ⎟ 63.52⎜ 1.189⎜ 1 2 ⎟ ⎨ ⎨ ⎬ + 60qLtq ⎟⎥ 2 ⎬⎜ ⎟ ⎢ 1 2 2 z ⎝ nc ⎠⎢⎪⎩ 1+ 1+ z ⎝ Sc ⎠ ⎪⎩ ⎟⎥ ⎪⎭⎜ ⎪⎭ ⎝ ⎠⎦ ⎣ 13

⎫⎪ ⎛ S ⎞⎧⎪ z ⎟⎟⎨ Lc qL + Qu − 0.794⎜⎜ ⎬ 12 2 ⎝ nc ⎠⎪⎩ 1+ 1+ z 2 ⎪⎭ 12 c

[

[

(

(

)

)

]

]

[

(

(

)

)

]

43

Kinematic-Wave Rainfall-Runoff Formulas

APPENDIX O. WORKING FORMULAS FOR HYDROGRAPH – EQUILIBRIUM PHASE

O.1. Overland Plane O.1.1. Partial Equilibrium Discharge

S1 2 qp = o no

⎡⎛ n q ⎞3 5 Cr it r ⎤ ⎥ ⎢⎜⎜ o1 2u ⎟⎟ + 60 ×103 ⎥ ⎢⎣⎝ S o ⎠ ⎦

53

for tr ≤ t ≤ (tr + td)

O.1.2. Equilibrium Discharge

qe = qu +

Cr iLo 3.6 × 106

for to ≤ t ≤ tr

O.2.Circular Channel O.2.1. Partial Equilibrium Discharge 45 ⎤ ⎛ S c1 2 D 1 6 ⎞ ⎡ ⎛ nc Qu ⎞ ⎟⎟ ⎢1.738⎜⎜ 1 2 1 6 ⎟⎟ + 60q L t q ⎥ Q p = 0.501⎜⎜ ⎥⎦ ⎝ nc ⎠ ⎢⎣ ⎝ Sc D ⎠

for tq ≤ t ≤ (tq + td)

O.2.2. Equilibrium Discharge

Qe = Qu + q L Lc

for tt ≤ t ≤ tq

54

213

Tommy S.W. Wong

214

O.3. Parabolic Channel O.3.1. Partial Equilibrium Discharge

⎛ S c1 2 Q p = 0.493⎜⎜ 29 ⎝ nc H

⎞⎡ ⎛ nc H 2 9 Qu ⎟⎟ ⎢1.639⎜⎜ 12 ⎠ ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

⎤ + 60q L t q ⎥ ⎥⎦

for tq ≤ t ≤ (tq + td)

O.3.2. Equilibrium Discharge

Qe = Qu + q L Lc

for tt ≤ t ≤ tq

O.4. Rectangular (Deep) Channel O.4.1. Partial Equilibrium Discharge

⎤ ⎛ S 1 2W 2 3 ⎞ ⎡ ⎛ nQ ⎞ ⎟⎟ ⎢1.587⎜⎜ 1 2c u2 3 ⎟⎟ + 60q L t q ⎥ Q p = 0.630⎜⎜ c ⎥⎦ ⎝ nc ⎠ ⎢⎣ ⎝ Sc W ⎠ for tq ≤ t ≤ (tq + td)

O.4.2. Equilibrium Discharge

Qe = Qu + q L Lc for tt ≤ t ≤ tq

13 9

Kinematic-Wave Rainfall-Runoff Formulas

O.5. Rectangular (Square) Channel O.5.1. Partial Equilibrium Discharge

⎛ S c1 2 Q p = 0.481⎜⎜ ⎝ nc

⎞⎡ ⎛n Q ⎟⎟ ⎢1.731⎜⎜ c 1 2u ⎠ ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎤ + 60q L t q ⎥ ⎥⎦

for tq ≤ t ≤ (tq + td)

O.5.2. Equilibrium Discharge

Qe = Qu + q L Lc

for tt ≤ t ≤ tq

O.6. Rectangular (Wide) Channel O.6.1. Partial Equilibrium Discharge

⎛ S 1 2 ⎞ ⎡⎛ nW 2 3 Qu Q p = ⎜⎜ c 2 3 ⎟⎟ ⎢⎜⎜ 12 ⎝ ncW ⎠ ⎢⎣⎝ S c for tq ≤ t ≤ (tq + td)

O.6.2. Equilibrium Discharge

Qe = Qu + q L Lc for tt ≤ t ≤ tq

⎞ ⎟ ⎟ ⎠

35

⎤ + 60q L t q ⎥ ⎥ ⎦

53

43

215

Tommy S.W. Wong

216

O.7. Trapezoidal Channel with Equal Side Slopes O.7.1. Partial Equilibrium Discharge

⎛ S c1 2 ⎞ ⎡ ⎛ ncW 0.0909 Qu ⎟ ⎢2.186⎜⎜ Q p = 0.340⎜⎜ 0.0909 ⎟ S c1 2 ⎝ n cW ⎠ ⎢⎣ ⎝

⎞ ⎟⎟ ⎠

0.725

⎤ + 60q L t q ⎥ ⎥⎦

1.379

for tq ≤ t ≤ (tq + td)

O.7.2. Equilibrium Discharge

Qe = Qu + q L Lc

for tt ≤ t ≤ tq

O.8. Trapezoidal Channel with One Side Vertical O.8.1. Partial Equilibrium Discharge

⎛ S c1 2 ⎞ ⎡ ⎛ ncW 0.0526 Qu ⎜ ⎟ ⎢2.295⎜⎜ Q p = 0.323⎜ 0.0526 ⎟ n W S c1 2 ⎝ c ⎠ ⎢⎣ ⎝ for tq ≤ t ≤ (tq + td)

O.8.2. Equilibrium Discharge

Qe = Qu + q L Lc

for tt ≤ t ≤ tq

⎞ ⎟⎟ ⎠

0.735

⎤ + 60q L t q ⎥ ⎥⎦

1.360

Kinematic-Wave Rainfall-Runoff Formulas

217

O.9. Triangular Channel O.9.1. Partial Equilibrium Discharge

⎛ S c1 2 ⎞⎛ z ⎞ ⎟⎟⎜ Q p = 0.630⎜⎜ 2 ⎟ ⎝ n c ⎠⎝ 1 + z ⎠

13

⎡ ⎛n Q ⎢1.414⎜⎜ c 1 2u ⎢⎣ ⎝ Sc

⎞ ⎟⎟ ⎠

34

⎛1+ z 2 ⎜⎜ ⎝ z

⎞ ⎟⎟ ⎠

14

⎤ + 60q L t q ⎥ ⎥⎦

43

for tq ≤ t ≤ (tq + td)

O.9.2. Equilibrium Discharge

Qe = Qu + q L Lc

for tt ≤ t ≤ tq

O.10. Vertical Curb Channel O.10.1. Partial Equilibrium Discharge

⎛ S ⎞⎧⎪ z ⎟⎟⎨ Qp = 0.794⎜⎜ ⎝ nc ⎠⎪⎩ 1 + 1 + z 2 12 c

[

(

for tq ≤ t ≤ (tq + td)

O.10.2. Equilibrium Discharge

Qe = Qu + q L Lc for tt ≤ t ≤ tq

13

)

12

⎫⎪ 2⎬ ⎪⎭

]

⎛ ⎜ ⎛ nc Qu ⎜1.189⎜⎜ 1 2 ⎝ Sc ⎜ ⎝

⎞ ⎟⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

)

]

12 2

14 ⎞ ⎫ ⎟ ⎪ ⎬ + 60qLtq ⎟ ⎟ ⎪⎭ ⎠

43

Tommy S.W. Wong

218

APPENDIX P. WORKING FORMULAS FOR WATER SURFACE PROFILE – EQUILIBRIUM PHASE P.1. Overland Plane

⎤ ⎡⎛ n ⎞ yo = 0.116 ×10 ⎢⎜⎜ 1o2 ⎟⎟(3.6 ×106 qu + Cr ixo )⎥ ⎣⎝ So ⎠ ⎦

35

−3

for 0 ≤ xo ≤ Lo

P.2. Circular Channel 3 2 ⎡⎛ n ⎤ ⎞ ⎛ yc ⎞ ⎛ yc ⎞ ⎛y ⎞ − 1.195⎜ ⎟ + 1.801⎜ ⎟ + 0.397⎜ c ⎟ = 2.213⎢⎜⎜ 1 2 c 8 3 ⎟⎟(Qu + q L xc )⎥ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎣⎝ S c D ⎠ ⎦

for 0 ≤ xc ≤ Lc

P.3. Parabolic Channel ⎡⎛ n y c = 0.721⎢⎜⎜ 1 2 c 1 2 ⎣⎢⎝ S c H

⎤ ⎞ ⎟(Qu + q L xc )⎥ ⎟ ⎠ ⎦⎥

for 0 ≤ xc ≤ Lc

P.4. Rectangular (Deep) Channel ⎛ n ⎞ yc = 1.587⎜⎜ 1 2 c 5 3 ⎟⎟(Qu + q L xc ) ⎝ Sc W ⎠ for 0 ≤ xc ≤ Lc

6 13

45

Kinematic-Wave Rainfall-Runoff Formulas

P.5. Rectangular (Square) Channel

⎤ ⎡⎛ n ⎞ yc = 1.316⎢⎜⎜ 1c2 ⎟⎟(Qu + q L xc )⎥ ⎣⎝ S c ⎠ ⎦

38

for 0 ≤ xc ≤ Lc

P.6. Rectangular (Wide) Channel

⎡⎛ n yc = ⎢⎜⎜ 1 2c ⎣⎝ S c W

⎤ ⎞ ⎟⎟(Qu + q L xc )⎥ ⎠ ⎦

35

for 0 ≤ xc ≤ Lc

P.7. Trapezoidal Channel with Equal Side Slopes 12

0.725 ⎧⎪ ⎫⎪ ⎡ ncW 0.0909 (q L x c + Qu ) ⎤ 2 − W + ⎨W + 8.748 z ⎢ ⎬ ⎥ S c1 2 ⎪⎩ ⎪⎭ ⎣ ⎦ yc = 2z

for 0 ≤ xc ≤ Lc

P.8. Trapezoidal Channel with One Side Vertical 12

0.735 ⎧⎪ ⎡ ncW 0.0526 (qL xc + Qu ) ⎤ ⎫⎪ 2 − W + ⎨W + 4.592 z ⎢ ⎥ ⎬ S c1 2 ⎪⎩ ⎣ ⎦ ⎪⎭ yc = z

for 0 ≤ xc ≤ Lc

219

Tommy S.W. Wong

220

P.9. Triangular Channel ⎤ ⎡⎛ n ⎞⎛ 1 + z 2 ⎞1 3 yc = 1.189 ⎢⎜⎜ 1c2 ⎟⎟⎜⎜ 5 ⎟⎟ (Qu + q L xc )⎥ ⎥⎦ ⎢⎣⎝ S c ⎠⎝ z ⎠

38

for 0 ≤ xc ≤ Lc

P.10. Vertical Curb Channel

[

⎛ ⎜ ⎛ nc ⎞⎧⎪ 1 + 1 + z 2 yc = 1.542⎜ ⎜⎜ 1 2 ⎟⎟⎨ z5 ⎜ ⎝ S c ⎠⎪⎩ ⎝ for 0 ≤ xc ≤ Lc

(

)

]

12 2

13

⎫ ⎪ ⎬ ⎪⎭

⎞ ⎟ (Qu + qL xc )⎟ ⎟ ⎠

38

Kinematic-Wave Rainfall-Runoff Formulas

221

APPENDIX Q. WORKING FORMULAS FOR EQUILIBRIUM DETENTION STORAGE Q.1. Overland Plane 72.8 × 10 −6 Deo = Cr i

⎛ no ⎜ 12 ⎜S ⎝ o

⎞ ⎟ ⎟ ⎠

35

[(3.6 ×10 q 6

u

+ C r iLo

)

8/5

Q.2. Circular Channel ⎛ n ⎞ ⎡ (Q + qL Lc )9 5 − Qu9 5 ⎤ Dec = 0.966⎜⎜ 1 2 c 1 6 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc D ⎠ ⎣ ⎦ 45

Q.3. Parabolic Channel

⎛n H29 Dec = 0.964⎜⎜ c 1 2 ⎝ Sc

⎞ ⎟⎟ ⎠

9 13

⎡ (Qu + q L Lc )22 13 − Qu22 13 ⎤ ⎥ ⎢ qL ⎢⎣ ⎥⎦

Q.4. Rectangular (Deep) Channel

⎛ n ⎞ ⎡ (Q + qL Lc )2 − Qu2 ⎤ Dec = 0.794⎜⎜ 1 2 c 2 3 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc W ⎠ ⎣ ⎦

Q.5. Rectangular (Square) Channel

⎛ n ⎞ Dec = 0.989⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

3/ 4

⎡ (Qu + q L Lc )7 / 4 − Qu7 / 4 ⎤ ⎢ ⎥ qL ⎢⎣ ⎥⎦

Q.6. Rectangular (Wide) Channel ⎛ n W 2 3 ⎞ ⎡ (Q + qL Lc )8 5 − Qu8 5 ⎤ Dec = 0.625⎜⎜ c 1 2 ⎟⎟ ⎢ u ⎥ qL ⎝ Sc ⎠ ⎣ ⎦ 35

(

− 3.6 × 10 6 qu

)

8/5

]

Tommy S.W. Wong

222

Q.7. Trapezoidal Channel with Equal Side Slopes ⎛ ncW 0.0909 ⎞ ⎟⎟ Dec = 1.268⎜⎜ 12 S c ⎝ ⎠

0.725

⎡ (Qu + q L Lc )1.725 − Qu1.725 ⎤ ⎢ ⎥ qL ⎣ ⎦

Q.8. Trapezoidal Channel with One Side Vertical ⎛ ncW 0.0526 ⎞ ⎟⎟ Dec = 1.322⎜⎜ 12 S c ⎝ ⎠

0.735

⎡ (Qu + q L Lc )1.735 − Qu1.735 ⎤ ⎢ ⎥ qL ⎣ ⎦

Q.9. Triangular Channel

⎛ n ⎞ Dec = 0.808⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠

34

⎛ 1 + z 2 ⎞ ⎡ (Qu + q L Lc )7 4 − Qu7 4 ⎤ ⎜⎜ ⎟⎟ ⎢ ⎥ qL ⎝ z ⎠ ⎢⎣ ⎦⎥ 14

Q.10. Vertical Curb Channel ⎛ n Dec = 0.679⎜⎜ 1c2 ⎝ Sc

⎞ ⎟ ⎟ ⎠

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z ⎪⎩

)

] ⎫⎪⎬

12 2

14

⎡ (Qu + q L Lc )7 4 − Qu7 4 ⎤ ⎢ ⎥ qL ⎪⎭ ⎣⎢ ⎦⎥

Kinematic-Wave Rainfall-Runoff Formulas

223

APPENDIX R. WORKING FORMULA FOR WATER SURFACE PROFILE – FALLING PHASE R.1. Overland Plane

⎤ ⎡⎛ n ⎞ yo = 0.116 × 10 ⎢⎜⎜ 1o2 ⎟⎟(3.6 × 106 qu )⎥ ⎣⎝ S o ⎠ ⎦

35

−3

for 0 ≤ xo ≤ Lf

⎡ ⎛ ⎛ S o1 2 yo 5 3 ⎞ ⎜⎜ ⎟ − qu ⎢ ⎜ n ⎟ ⎜ ⎛ S o1 2 yo 2 3 ⎞ ⎢ o ⎠ ⎟(t − t r ) + 3.6 × 106 ⎜ ⎝ xo = 100.0⎜⎜ ⎢ ⎟ n C i o r ⎝ ⎠ ⎜ ⎢ ⎜ ⎢⎣ ⎝

⎞⎤ ⎟⎥ ⎟⎥ ⎟⎥ ⎟⎥ ⎟⎥ ⎠⎦

for Lf ≤ xo ≤ Lo

⎛ S o1 2 ⎞ ⎟⎟ L f = 0.238⎜⎜ ⎝ no ⎠

35

(3.6 ×10 q ) (t − t ) 25

6

u

r

R.2. Circular Channel 3

2

⎛ nQ ⎞ ⎛y ⎞ ⎛y ⎞ ⎛y ⎞ − 1.195⎜ c ⎟ + 1.801⎜ c ⎟ + 0.397⎜ c ⎟ = 2.213⎜⎜ 1 2c u8 3 ⎟⎟ ⎝D⎠ ⎝D⎠ ⎝D⎠ ⎝ Sc D ⎠ for 0 ≤ xc ≤ Lf

45

Tommy S.W. Wong

224

3 2 ⎛ S c1 2 D 2 3 ⎞ ⎡ yc ⎞ yc ⎞ ⎛ y ⎞⎤ ⎛ ⎛ ⎟⎟ ⎢− 1.195⎜ ⎟ + 1.801⎜ ⎟ + 0.397⎜ c ⎟⎥ xc = 35.38⎜⎜ ⎝ D ⎠⎥⎦ ⎝D⎠ ⎝D⎠ ⎝ nc ⎠ ⎢⎣

14

(t − t ) q

54 3 2 ⎛⎧ ⎛ S c1 2 D 8 3 ⎞ ⎡ yc ⎞ yc ⎞ yc ⎞⎤ ⎫⎪ ⎜⎪ ⎛ ⎛ ⎛ ⎟⎟ ⎢− 1.195⎜ ⎟ + 1.801⎜ ⎟ + 0.397⎜ ⎟⎥ ⎬ − Qu ⎜ ⎨0.370⎜⎜ n ⎝ D ⎠⎥⎦ ⎪ ⎝D⎠ ⎝D⎠ c ⎝ ⎠ ⎢⎣ ⎜ ⎪⎩ ⎭ +⎜ qL ⎜ ⎜⎜ ⎝

for Lf ≤ xc ≤ Lc 45

⎛ S 1 2 D1 6 ⎞ ⎟⎟ Qu1 5 (t − t q ) L f = 43.15⎜⎜ c ⎝ nc ⎠

R.3. Parabolic Channel ⎛ nQ y c = 0.721⎜⎜ 1 2c u1 2 ⎝ Sc H

⎞ ⎟ ⎟ ⎠

6 13

for 0 ≤ xc ≤ Lf 6 ⎧⎡ ⎛ S 1 2 H 1 2 y 13 c 2.033⎜⎜ c ⎪ ⎢ nc ⎛ S c1 2 y c2 3 ⎞ ⎪⎢ ⎝ ⎟⎟(t − t q ) + ⎨ ⎣ xc = 66.08⎜⎜ qL ⎝ nc ⎠ ⎪ ⎪ ⎩

for Lf ≤ xc ≤ Lc

⎛ S1 2 L f = 53.12⎜⎜ c 2 9 ⎝ nc H

⎞ ⎟ ⎟ ⎠

9 13

Qu4 13 (t − t q )

⎫ ⎞⎤ ⎟⎟⎥ − Qu ⎪ ⎪ ⎠⎦⎥ ⎬ ⎪ ⎪ ⎭

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎟ ⎠

Kinematic-Wave Rainfall-Runoff Formulas

R.4. Rectangular (Deep) Channel ⎛ nQ ⎞ y c = 1.587⎜⎜ 1 2c u5 3 ⎟⎟ ⎝ Sc W ⎠ for 0 ≤ xc ≤ Lf

⎧⎡ ⎫ ⎛ S c1 2W 5 3 yc ⎞⎤ ⎟⎟⎥ − Qu ⎪ ⎜⎜ 0 630 . ⎪ ⎢ nc ⎛ S 1 2W 2 3 ⎞ ⎪ ⎪ ⎠⎦ ⎝ ⎟⎟(t − tq ) + ⎨ ⎣ xc = 37.80⎜⎜ c ⎬ qL ⎝ nc ⎠ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ for Lf ≤ xc ≤ Lc

⎛ S c1 2W 2 3 ⎞ ⎟⎟(t − t q ) L f = 37.80⎜⎜ n c ⎝ ⎠

R.5. Rectangular (Square) Channel ⎛n Q y c = 1.316⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟ ⎟ ⎠

38

for 0 ≤ xc ≤ Lf

⎫ ⎧⎡ ⎛ S c1 2 yc 8 3 ⎞⎤ ⎜ ⎟ . Q 0 481 − ⎪ ⎢ u ⎪ ⎜ n ⎟⎥ ⎛ S c1 2 yc 2 3 ⎞ ⎥⎦ ⎪ ⎪ ⎢⎣ c ⎝ ⎠ ⎟(t − t q ) + ⎨ xc = 38.48⎜⎜ ⎬ ⎟ qL ⎝ nc ⎠ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ for Lf ≤ xc ≤ Lc 34

⎛ S1 2 ⎞ L f = 46.21⎜⎜ c ⎟⎟ Qu1 4 (t − t q ) ⎝ nc ⎠

225

Tommy S.W. Wong

226

R.6. Rectangular (Wide) Channel ⎛nQ y c = ⎜⎜ 1c 2 u ⎝ Sc W

⎞ ⎟ ⎟ ⎠

35

for 0 ≤ xc ≤ Lf

⎤ ⎡ ⎛ S c1 2Wyc5 3 ⎞ ⎟⎟ − Qu ⎥ ⎢ ⎜⎜ 12 23 nc ⎛S y ⎞ ⎠ ⎥ xc = 100.0⎜⎜ c c ⎟⎟(t − t q ) + ⎢ ⎝ ⎥ ⎢ q n c L ⎝ ⎠ ⎥ ⎢ ⎦⎥ ⎣⎢ for Lf ≤ xc ≤ Lc

35

⎛ S1 2 ⎞ 25 L f = 100.0⎜⎜ c 2 3 ⎟⎟ Qu (t − t q ) n W ⎝ c ⎠

R.7. Trapezoidal Channel with Equal Side Slopes ⎡ ⎛ n W 0.0909 Q − W + ⎢W 2 + 8.748 z ⎜⎜ c 1 2 u Sc ⎢⎣ ⎝ yc = 2z

⎞ ⎟ ⎟ ⎠

0.725 1 2

⎤ ⎥ ⎥⎦

for 0 ≤ xc ≤ Lf

⎧ ⎫ ⎡ S 1 2 (zy 2 + Wy )1.379 ⎤ c c c − . Q 0 340 ⎪ ⎢ ⎥ u ⎪ 0.0909 n W ⎡ S 1 2 (zy 2 + Wy )0.379 ⎤ ⎪ ⎪ c ⎢ ⎥ ⎣ ⎦ c c xc = 28.14⎢ c ⎥ (t − t q ) + ⎨ ⎬ 0.0909 ncW qL ⎢⎣ ⎥⎦ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ for Lf ≤ xc ≤ Lc

⎛ S c1 2 ⎞ ⎟ L f = 27.84⎜⎜ 0.0909 ⎟ ⎝ ncW ⎠

0.725

Qu0.275 (t − tq )

Kinematic-Wave Rainfall-Runoff Formulas

227

R.8. Trapezoidal Channel with One Side Vertical ⎡ ⎛ ncW 0.0526 Qu 2 ⎢ − W + W + 4.592 z ⎜⎜ S c1 2 ⎢⎣ ⎝ yc = z

⎞ ⎟ ⎟ ⎠

0.735 1 2

⎤ ⎥ ⎥⎦

for 0 ≤ xc ≤ Lf

⎡S1 2 xc = 26.36 ⎢ c ⎢⎣

(

⎧ ⎡ S 1 2 0.5 zy 2 + Wy c c ⎪ . 0 323 ⎢ c 0.0526 0.360 2 n W ⎤ ⎢⎣ c 0.5 zy c + Wy c ⎪⎪ ⎥ (t − t q ) + ⎨ 0.0526 qL n cW ⎥⎦ ⎪ ⎪ ⎪⎩

(

)

for Lf ≤ xc ≤ Lc

⎛ S c1 2 ⎞ ⎟ L f = 35.56⎜⎜ 0.0526 ⎟ ⎝ ncW ⎠

0.735

Qu0.265 (t − tq )

R.9. Triangular Channel ⎛n Q y c = 1.190⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟ ⎟ ⎠

38

⎛1+ z 2 ⎜⎜ 5 ⎝ z

⎞ ⎟⎟ ⎠

18

for 0 ≤ xc ≤ Lf 13 ⎤ ⎡ ⎛ S c1 2 ⎞⎛ z 5 yc8 ⎞ 13 ⎜ ⎟ ⎜ ⎟ 0 630 . Q ⎢ − 2 u ⎥ ⎜ n ⎟⎜ 1 + z 2 ⎟ ⎛ S c1 2 ⎞ ⎡ zyc ⎤ ⎥ ⎢ ⎠ c ⎠⎝ ⎝ ⎟⎟ ⎢ (t − tq ) + ⎢ xc = 50.40⎜⎜ 2 ⎥ ⎥ qL ⎝ nc ⎠ ⎢⎣ 1 + z ⎥⎦ ⎥ ⎢ ⎥⎦ ⎢⎣

( )

for Lf ≤ xc ≤ Lc

)

1.360

⎫ ⎤ ⎥ − Qu ⎪ ⎥⎦ ⎪⎪ ⎬ ⎪ ⎪ ⎪⎭

Tommy S.W. Wong

228

34

⎡⎛ S 1 2 ⎞⎛ z ⎞1 3 ⎤ L f = 56.57 ⎢⎜⎜ c ⎟⎟⎜ Qu1 4 (t − t q ) 2 ⎟ ⎥ ⎢⎣⎝ nc ⎠⎝ 1 + z ⎠ ⎥⎦

R.10. Vertical Curb Channel ⎛n Q y c = 1.542⎜⎜ c 1 2u ⎝ Sc

⎞ ⎟ ⎟ ⎠

38

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ z5 ⎪⎩

)

] ⎫⎪⎬

12 2

18

⎪⎭

for 0 ≤ xc ≤ Lf

⎛ ⎛ Sc1 2 ⎞⎧⎪ ⎜ z 5 yc8 ⎜ ⎟ . 0 315 ⎜ 23 ⎜ n ⎟⎨ ⎞ ⎝ c ⎠⎪⎩ 1 + 1 + z 2 ⎛ Sc1 2 ⎞⎛⎜ zyc ⎜ ⎟ (t − tq ) + ⎟⎟ xc = 50.42⎜⎜ ⎜ 2 12 ⎟ qL ⎝ nc ⎠⎜⎝ 1 + 1 + z ⎠ ⎜ ⎜ ⎜ ⎝

(

[

)

for Lf ≤ xc ≤ Lc

⎛ 12 ⎧ ⎜ ⎛ S ⎞⎪ z L f = 67.29⎜ ⎜⎜ c ⎟⎟⎨ 12 ⎜ ⎝ nc ⎠⎪⎩ 1 + (1 + z 2 ) ⎝

[

13 34

⎫⎪ 2⎬ ⎪⎭

]

⎞ ⎟ 14 ⎟ Qu (t − t q ) ⎟ ⎠

(

13

)

12

⎫⎪ − Qu 2⎬ ⎪⎭

]

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

Kinematic-Wave Rainfall-Runoff Formulas

APPENDIX S. WORKING FORMULA FOR HYDROGRAPH – FALLING PHASE S.1. Overland Plane

⎛ n ⎞ ⎡ C iL − 3.6 × 106 (q − qu )⎤ t = 0.0100⎜⎜ 1o2 ⎟⎟ ⎢ r o ⎥ + tr Cr iq 2 5 ⎦ ⎝ So ⎠ ⎣ 35

for t ≥ tr

S.2. Circular Channel 45

⎛ 0.0232 ⎞⎛ n ⎞ ⎡ ⎛ Q − Qu t = ⎜⎜ 1 5 ⎟⎟⎜⎜ 1 2 c 1 6 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎝ qL ⎝ Qc ⎠⎝ S c D ⎠ ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

for t ≥ tq

S.3. Parabolic Channel

0.0188 ⎛ nc H 2 9 ⎜ t= Qc4 13 ⎜⎝ S c1 2

⎞ ⎟⎟ ⎠

9 13

⎡ ⎛ Qc − Qu ⎢ Lc − ⎜⎜ ⎝ qL ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

for t ≥ tq

S.4. Rectangular (Deep) Channel

⎛ n ⎞⎡ ⎛ Q − Qu t = 0.0265⎜⎜ 1 2 c 2 3 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎝ qL ⎝ Sc W ⎠⎣ for t ≥ tq

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

229

Tommy S.W. Wong

230

S.5. Rectangular (Square) Channel 34

⎛ 0.0216 ⎞⎛ n ⎞ ⎡ ⎛ Q − Qu ⎞⎤ ⎟⎟⎥ + tq t = ⎜⎜ 1 4 ⎟⎟⎜⎜ 1c2 ⎟⎟ ⎢ Lc − ⎜⎜ c Q S q L ⎝ ⎠⎦ ⎝ c ⎠⎝ c ⎠ ⎣ for t ≥ tq

S.6. Rectangular (Wide) Channel 35

⎛ 0.0100 ⎞⎛ ncW 2 3 ⎞ ⎡ ⎛ Q − Qu ⎞⎤ ⎟⎜ 1 2 ⎟⎟ ⎢ Lc − ⎜⎜ c ⎟⎟⎥ + t q t = ⎜⎜ 2 5 ⎟⎜ q Q S L ⎠⎦ ⎝ ⎝ c ⎠⎝ c ⎠ ⎣ for t ≥ tq

S.7. Trapezoidal Channel with Equal Side Slopes 0.0264 ⎛ n W 0.0909 ⎞ t = 0.275 ⎜⎜ c 1 2 ⎟⎟ Qc ⎝ Sc ⎠

0.725

⎡ ⎛ Qc − Qu ⎞⎤ ⎟⎟⎥ + t q ⎢ Lc − ⎜⎜ q L ⎠⎦ ⎝ ⎣

for t ≥ tq

S.8. Trapezoidal Channel with One Side Vertical 0.0281 ⎛ n W 0.0526 ⎞ t = 0.265 ⎜⎜ c 1 2 ⎟⎟ Qc ⎝ Sc ⎠

0.735

⎡ ⎛ Qc − Qu ⎢ Lc − ⎜⎜ ⎝ qL ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

for t ≥ tq

S.9. Triangular Channel

⎛ n t = 0.0177⎜⎜ 1c2 ⎝ Sc for t ≥ tq

⎞ ⎟⎟ ⎠

34

⎛1 + z 2 ⎜⎜ ⎝ zQc

14

⎞ ⎡ ⎛ Q − Qu ⎟⎟ ⎢ Lc − ⎜⎜ c ⎝ qL ⎠ ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

Kinematic-Wave Rainfall-Runoff Formulas

S.10. Vertical Curb Channel ⎛ n ⎞ t = 0.0149⎜⎜ 1c2 ⎟⎟ ⎝ Sc ⎠ for t ≥ tq

34

[

(

⎧ 1+ 1+ z2 ⎪ ⎨ zQc ⎪⎩

)

] ⎫⎪⎬

12 2

14

⎡ ⎛ Q c − Qu ⎢ Lc − ⎜⎜ ⎝ qL ⎭⎪ ⎣

⎞⎤ ⎟⎟⎥ + t q ⎠⎦

231

REFERENCES American Society of Civil Engineers. (1992). “Design and construction of urban stormwater management systems.” ASCE Manuals and Reports of Engineering Practice No. 77, American Society of Civil Engineers and Water Environment Federation, New York, USA. American Society of Civil Engineers. (1996). "River hydraulics." Technical Engineering and Design Guides as adapted from the U. S. Army Corps of Engineers No. 18, American Society of Civil Engineers Press, New York, USA. American Society of Civil Engineers. (1997). "Flood-runoff analysis." Technical Engineering and Design Guides as adapted from the U. S. Army Corps of Engineers No. 19, American Society of Civil Engineers Press, New York, USA. Arcement, G. J., and Schneider, V. R. (1989). "Guide for selecting Manning's roughness coefficients for natural channels and flood plains," U. S. Geological Survey Water-Supply Paper 2339, U. S. Government Printing Office, Washington, D. C., USA. Brady, D. K. (1983). "Kinematic wave parameters for parabolic stream channels." Proceedings of 8th Australasian Fluid Mechanics Conference, University of Newcastle, N.S.W., Australia, pp. 19-22. Chen, C. N., and Evans, R. R. (1977). "Application of kinematic wave method to predict overland peak flows." Proceedings of International Symposium on Urban Hydrology, Hydraulics and Sediment Control, University of Kentucky, Lexington, Kentucky, USA, pp. 113-118. Chow, V. T. (1959). Open channel hydraulics. McGraw-Hill, New York, USA. Chow, V. T., Maidment, D. R., and Mays, L. W. (1988). "Applied Hydrology," McGraw-Hill, New York, USA. DeVries, J. J., and MacArthur, R. C. (1979). "Introduction and application of kinematic wave routing techniques using HEC-1," Training Document No. 10, U. S. Army Corps of Engineers, Hydrologic Engineering Centre, Davis, California, USA. Engman, E. T. (1986). "Roughness coefficients for routing surface runoff." Journal of Irrigation and Drainage Engineering, ASCE, Vol. 112, No. 1, pp. 39-53. Harley, B. M., Perkins, F. E., and Eagleson, P. S. (1970). “A modular distributed model of catchment dynamics.” Report No. 133, Ralph M. Parsons Laboratory for Water Resources and Hydrodynamics, Department of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA.

234

References

Henderson, F. M., and Wooding, R.A. (1964). "Overland flow and groundwater flow from a steady rainfall of finite duration," Journal of Geophysical Research, Vol. 69, No. 8, pp. 1531-1540. Jan, J. T. (1979). Engineering Mathematics Handbook, McGraw-Hill, New York, USA. Lighthill, M. J., and Whitham, G. B. (1955). "On kinematic waves: flood movement in long rivers." Proceedings of Royal Society (London) Series A, Vol. 229, pp. 281-316. Morris, E. M., and Woolhiser, D. A. (1980). "Unsteady one-dimensional flow over a plane: partial equilibrium and recession hydrographs." Water Resources Research, Vol. 16, No. 2, pp. 355-360. Overton, D. E., and Meadows, M. E. (1976). Stormwater Modeling, Academic Press, New York, USA. Ponce V. P. (1991). "The kinematic wave controversy." Journal of Hydraulic Engineering, ASCE, Vol. 117, No. 6, pp. 511-525. Ponce, V. P., Li, R. M., and Simons, D. B. (1978). "Applicability of kinematic and diffusion models." Journal of Hydraulics Division, ASCE, Vol. 104, No. 3, pp. 353-360. Stephenson, D. (1981). Stormwater Hydrology and Drainage. Elsevier, New York, USA. Wong, T. S. W. (1992). An Introduction to Kinematic Wave Method for Storm Drainage Design, Hillview Publications, Singapore. Wong, T. S. W. (1995). "Time of concentration formulae for planes with upstream inflow." Hydrological Sciences Journal, Vol. 40, No. 5, pp. 663-666. Wong, T. S. W. (1996). "Influence of upstream inflow on wave celerity and time to equilibrium on an overland plane." Hydrological Sciences Journal, Vol. 41, No. 2, pp. 195-205. Wong, T. S. W. (2001). "Formulas for time of travel in channel with upstream inflow.” Journal of Hydrologic Engineering, ASCE, Vol. 6, No. 5, pp. 416-422. Wong, T. S. W. (2002). "Generalised formula for time of travel in rectangular channel,” Journal of Hydrologic Engineering, ASCE, Vol. 7, No. 6, pp. 445-448. Wong, T. S. W. (2003). “Comparison of celerity-based with velocity-based time-ofconcentration of overland plane and time-of-travel in channel with upstream inflow.” Advances in Water Resources, Vol. 26, No. 11, pp. 1171-1175. Wong, T. S. W. (2005a). “Influence of loss model on design discharge of homogenous plane.” Journal of Irrigation and Drainage Engineering, ASCE, Vol. 131, No. 2, pp. 210-217. Wong, T. S. W. (2005b). “Kinematic wave method for storm drainage design,” in Water Encyclopedia: Surface and Agricultural Water, (ed. by J. H. Lehr and J. Keeley), John Wiley, Ostrander, Ohio, USA, pp. 242-245. Wong, T. S. W. (2008a). “Discussion of ‘Storm-water predictions by dimensionless unit hydrograph’ by James C. Y. Guo.” Journal of Irrigation and Drainage Engineering, ASCE, Vol. 134, No. 2, p. 269. Wong, T. S. W. (2008b) "Effect of channel shape on time of travel and equilibrium detention storage in channel,” Journal of Hydrologic Engineering, ASCE, Vol. 13, No. 3, pp. 189196. Wong, T. S. W. and Chen, C. N. (1989). "Use of kinematic wave method to assess effects of urban development on flood peak changes," New Directions for Surface Water Modeling: Proceedings of the Baltimore Symposium, Maryland, USA, (ed. by M. L. Kavvas),

References

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International Association Hydrological Sciences Publication No. 181, Wallingford, UK, pp. 93-102. Wong, T. S. W., and Li, Y. (2000). "Determination of equilibrium detention storage for a series of planes." Hydrological Sciences Journal, Vol. 45, No. 5, pp. 787-790. Wong, T. S. W., and Zhou, M. C. (2003). "Kinematic wave parameters and time of travel in circular channel revisited.” Advances in Water Resources, Vol. 26, No. 4, pp. 417-425. Wong, T. S. W., and Zhou, M. C. (2006). “Kinematic wave parameters for trapezoidal and rectangular channels.” Journal of Hydrologic Engineering, ASCE, Vol. 11, No. 2, pp. 173-183. Wooding, R. A. (1965). "A hydraulic model for the catchment-stream problem, I. kinematicwave theory." Journal of Hydrology, Vol. 3, pp. 254-267. Woolhiser, D. A., and Liggett, J. A. (1967). "Unsteady one-dimensional flow over a plane the rising hydrograph." Water Resources Research, Vol. 3, No. 3, pp. 753-771.

INDEX

Celerity

Circular channel, 63-76

kinematic wave, 7-9, 31-32, 43-45,

average flow velocity, 68

68, 82, 93, 103, 115, 129, 141,

average wave celerity, 68-69

153-154, 166, 177, 181, 194-195

duration of partial equilibrium

average wave, 9, 32, 45, 68-69, 82,

discharge, 71-72

94, 103-104, 116, 129, 141-142,

equilibrium detention storage, 74

154, 166, 177, 181, 196-197

equilibrium discharge, 73

Channel, 39-174

flow depth, 67

circular, 63-76

flow velocity, 67-68

deep rectangular, 91-100

forward characteristic – rising phase,

parabolic, 77-89 rectangular (deep), 91-100 rectangular (square), 101-111

70 hydrograph – equilibrium phase, 7273

rectangular (wide), 113-123

hydrograph – falling phase, 75-76

square, 101-111

hydrograph – rising phase, 69

trapezoidal with equal side slopes,

kinematic wave celerity, 68

125-136 trapezoidal with one side vertical, 137-149 triangular, 151-161 vertical curb, 163-174 wide rectangular, 113-123

kinematic wave parameters, 63-66 partial equilibrium discharge, 72 time of travel, 69 water surface profile – equilibrium phase, 73 water surface profile – falling phase, 74-75

Index

238 water surface profile – rising phase, 70-71

Discharge design, 10-12, 33 equilibrium, 20, 36, 53-54, 73, 86-87,

Deep rectangular channel, 91-100

97, 108, 120, 133-134, 146, 158,

average flow velocity, 93 average wave celerity, 94

170-171, 179, 183 partial equilibrium, 19-20, 35-36, 53,

duration of partial equilibrium

72, 86, 97, 107, 119-120, 133,

discharge, 96 equilibrium detention storage, 98 equilibrium discharge, 97

145-146, 158, 170, 179, 183 Duration partial equilibrium discharge, 17-19,

flow depth, 93

35, 51-53, 71-72, 85-86, 96, 106-

flow velocity, 93

107, 119, 132-133, 145, 157, 169-

forward characteristic – rising phase,

170, 179, 182, 210-212

94-95 hydrograph – equilibrium phase, 97

rainfall, 10-11 Dynamic wave equations, 4-5, 40-41

hydrograph – falling phase, 100 hydrograph – rising phase, 94

Equilibrium detention storage, 21-24, 36-

kinematic wave celerity, 93

37, 54-57, 74, 87-88, 98, 109, 121,

kinematic wave parameters, 91-92

134-135, 147, 159, 172, 221-222

partial equilibrium discharge, 97

flow area approach, 55-56

time of travel, 94

hydrograph approach, 22-24, 56-57

water surface profile – equilibrium

water surface approach, 21-22

phase, 97-98 water surface profile – falling phase, 98-100 water surface profile – rising phase, 95-96 Design

Flow area, 41-42 average velocity, 6-7, 31, 42-43, 68, 81-82, 93, 103, 115, 128-129, 141, 153, 165, 177, 181, 192-193 conditions, 3-4, 39-40

discharge, 10-12, 33

depth, 6, 30-31, 81, 93, 102, 114, 127-

rainfall intensity, 11

128, 140, 152-153, 164-165, 177,

Detention storage equilibrium, 21-24, 36-37, 54-57, 74, 87-88, 98, 109, 121, 134-135, 147, 159, 172, 221-222

188-189 open channel, 39-174 overland, 3-38

Index velocity, 6, 31, 42, 67-68, 81, 93, 102103, 114-115, 128, 140, 153, 165, 177, 181, 190-191 Flow area profile, 49-51, 54, 58-61 equilibrium phase, 54 falling phase, 58-61 rising phase, 49-51 Flow velocity, 6, 31, 42, 67-68, 81, 93, 102-103, 114-115, 128, 140, 153, 165,

239 equilibrium discharge, 20, 53-54, 179, 183 flow area, 41-42, 181 flow area profile – equilibrium phase, 54, 183 flow area profile – falling phase, 5861, 183-184 flow area profile – rising phase, 4951, 182

177, 181, 190-191

flow depth, 6, 177

average, 6-7, 31, 42-43, 68, 81-82, 93,

flow velocity, 6, 42, 177, 181

103, 115, 128-129, 141, 153, 165, 177, 181, 192-193 Formulas, 3-174, 177-184, 188-231 General, 3-28, 39-62, 177-184 Working, 29-38, 63-174, 188-231 Forward characteristic rising phase, 14-15, 33-34, 48-49, 70, 83-84, 94-95, 105, 117, 130-131,

forward characteristic – rising phase, 14-15, 48-49, 178, 182 hydrograph – equilibrium phase, 1920, 53-54, 179, 183 hydrograph – falling phase, 28, 61-62, 180, 184 hydrograph – rising phase, 12-14, 4648, 178, 182

142-143, 155, 167, 178, 182, 203-

inflection line, 27-28, 60-61

204

kinematic wave celerity, 7-9, 43-45,

Froude number, 175

177, 181 kinematic wave equations, 5, 41

General formulas, 3-28, 39-62, 177-184 average flow velocity, 6-7, 42-43, 177, 181 average wave celerity, 9, 45, 177, 181

open channel, 39-62, 181-184 overland plane, 3-28, 177-180 partial equilibrium discharge, 19-20, 53, 179, 183

design discharge, 10-12, 178

time of concentration, 9-10, 177

duration of partial equilibrium

time of travel, 45-46, 181

discharge, 17-19, 51-53, 179, 182 dynamic wave equations, 4-5, 40-41 equilibrium detention storage, 21-24, 54-57, 179, 183

water surface profile – equilibrium phase, 20-21, 179 water surface profile – falling phase, 24-28, 180

Index

240 water surface profile – rising phase, 15-17, 178

137-139, 151-152, 163-164, 185187 Kinematic wave theory

Hydrograph

applicability, 175-176

duration of partial equilibrium discharge, 17-19, 35, 51-53, 71-

Manning’s roughness coefficient, 2

72, 85-86, 96, 106-107, 119, 132-

channel surface, 2

133, 145, 157, 169-170, 179, 182,

overland surface, 2

210-212 equilibrium phase, 19-20, 35-36, 5354, 72-73, 86-87, 97, 107-108, 119-120, 133-134, 145-146, 157158, 170-171, 179, 183, 213-217 falling phase, 28, 38, 61-62, 75-76, 89, 100, 110-111, 122-123, 136,

Open channel flow, 39-174, 176, 181-231 applicability of kinematic wave theory, 176 average flow velocity, 42-43, 68, 8182, 93, 103, 115, 128-129, 141, 153, 165, 181, 192-193

149, 161, 174, 180, 184, 229-231

average wave celerity, 45, 68-69, 82,

rising phase, 12-14, 33, 46-48, 69, 83,

94, 103-104, 116, 129, 141-142,

94,104, 116-117, 130, 142, 155, 167, 178, 182, 200-202

154, 166, 181, 196-197 duration of partial equilibrium discharge, 51-53, 71-72, 85-86,

Inflection line, 27-28, 60-61

96, 106-107, 119, 132-133, 145, 157, 169-170, 182, 210-212

Kinematic flow number, 175

dynamic wave equations, 40-41

Kinematic wave

equilibrium detention storage, 54-57,

average celerity, 9,32, 45, 68-69, 82, 94, 103-104, 116, 103-104, 129, 141-142, 154, 166, 177, 181, 196197 celerity, 7-9, 31-32, 43-45, 68, 82, 93,

74, 87-88, 98, 109, 121, 134-135, 147, 159, 172, 183, 221-222 equilibrium discharge, 53-54, 73, 8687, 97, 108, 120, 133-134, 146, 158, 170-171

103, 115, 129, 141, 153-154, 166,

flow area, 41-42, 181

177, 181, 194-195

flow area profile – equilibrium phase,

equations, 5, 41 parameters, 29-30, 63-66, 77-80, 9192, 101-102, 113-114, 125-127,

54, 183 flow area profile – falling phase, 5861, 183-184

Index flow area profile – rising phase, 4951, 182 flow conditions, 39-40 flow depth, 67, 81, 93, 114, 127-128, 140, 152-153, 164-165, 188-189 flow velocity, 42, 67-68, 81, 93, 102103, 114-115, 128, 140, 153, 165, 181, 190-191 forward characteristic – rising phase, 48-49, 70, 83-84, 94-95, 105, 117, 130-131, 142-143, 155, 167, 182, 203-204

241 partial equilibrium discharge, 53, 72, 86, 97, 107, 119-120, 133, 145146, 158, 170 time of travel, 45-46, 69, 83, 94, 104, 116, 130, 142, 154, 166-167, 181, 198-199 water surface profile – equilibrium phase, 73, 87, 97-98, 108, 120121, 134, 146-147, 158-159, 171, 218-220 water surface profile – falling phase, 74-75, 88-89, 98-100, 109-110,

general formulas, 39-62, 181-184

121-122, 135-136, 147-149, 159-

hydrograph – equilibrium phase, 53-

161, 172-173, 223-228

54, 72-73, 86-87, 97, 107-108,

water surface profile – rising phase,

119-120, 133-134, 145-146, 157-

70-71, 84-85, 95-96, 105-106,

158, 170-171, 183, 213-217

117-118, 131-132, 143-144, 155-

hydrograph – falling phase, 61-62, 7576, 89, 100, 110-111, 122-123, 136, 149, 161, 174, 184, 229-231 hydrograph – rising phase, 46-48, 69,

157, 168-169, 205-209 working formulas 63-174 Overland flow/plane, 3-38, 175, 177-180, 185, 188, 190, 192, 194, 196, 198,

83, 94, 104, 116-117, 130, 142,

200, 203, 205, 210, 213, 218, 221,

155, 167, 182, 200-202

223, 229

inflection line, 60-61 kinematic wave celerity, 43-45, 68, 82, 93, 103, 115, 129, 141, 153154, 166, 177, 181, 194-195

applicability of kinematic wave theory, 175 average flow velocity, 6-7, 31, 177, 192

kinematic wave equations, 41

average wave celerity, 9, 32, 177, 196

kinematic wave parameters, 63-66,

design discharge, 10-12, 33, 178

77-80, 91-92, 101-102, 113-114, 125-127, 137-139, 151-152, 163164, 185-187

duration of partial equilibrium discharge, 17-19, 35, 179, 210 dynamic wave equations, 4-5 equilibrium detention storage, 21-24, 36-37, 179, 221

Index

242 equilibrium discharge, 20, 36

average flow velocity, 81-82

flow conditions, 3-5

average wave celerity, 82

flow depth, 6, 30-31, 177, 188

duration of partial equilibrium

flow velocity, 6, 31, 177, 190 forward characteristic – rising phase, 14-15, 33-34, 178, 203

discharge, 85-86 equilibrium detention storage, 87-88 equilibrium discharge, 86-87

Froude number, 175

flow depth, 81

general formulas, 3-28, 177-180

flow velocity, 81

hydrograph – equilibrium phase, 19-

forward characteristic – rising phase,

20, 35-36, 179, 213 hydrograph – falling phase, 28, 38, 180, 229 hydrograph – rising phase, 12-14, 33, 178, 200

83-84 hydrograph – equilibrium phase, 8687 hydrograph – falling phase, 89 hydrograph – rising phase, 83

inflection line, 27-28,

kinematic wave celerity, 82

kinematic flow number, 175

kinematic wave parameters, 77-80

kinematic wave celerity, 7-9, 31-32,

partial equilibrium discharge, 86

177, 194 kinematic wave equations, 5 kinematic wave parameters, 29-30, 185 partial equilibrium discharge, 19-20, 35-36 time of concentration, 9-10, 32, 177, 198 water surface profile – equilibrium phase, 20-21, 36-37, 179, 218 water surface profile – falling phase,

time of travel, 83 water surface profile – equilibrium phase, 87 water surface profile – falling phase, 88-89 water surface profile – rising phase, 84-85 Partial equilibrium discharge duration, 17-19, 35, 51-53, 71-72, 8586, 96, 106-107, 119, 132-133, 145, 157, 169-170, 210-212

24-28, 37, 180, 223 water surface profile – rising phase, 15-17, 34, 178, 205 working formulas, 29-38, 63-174

Rainfall intensity design, 11 duration relationship, 10-11 Rectangular (deep) channel, 91-100

Parabolic channel, 77-89

average flow velocity, 93

Index average wave celerity, 94 duration of partial equilibrium discharge, 96

243 hydrograph – equilibrium phase, 107108 hydrograph – falling phase, 110-111

equilibrium detention storage, 98

hydrograph – rising phase, 104

equilibrium discharge, 97

kinematic wave celerity, 103

flow depth, 93

kinematic wave parameters, 101-102

flow velocity, 93

partial equilibrium discharge, 107

forward characteristic – rising phase,

time of travel, 104

94-95 hydrograph – equilibrium phase, 97 hydrograph – falling phase, 100 hydrograph – rising phase, 94 kinematic wave celerity, 93 kinematic wave parameters, 91-92 partial equilibrium discharge, 97

water surface profile – equilibrium phase, 108 water surface profile – falling phase, 109-110 water surface profile – rising phase, 105-106 Rectangular (wide) channel, 113-123

time of travel, 94

average flow velocity, 115

water surface profile – equilibrium

average wave celerity, 116

phase, 97-98 water surface profile – falling phase, 98-100 water surface profile – rising phase, 95-96 Rectangular (square) channel, 101-112 average flow velocity, 103 average wave celerity, 103-104 duration of partial equilibrium discharge, 106-107

duration of partial equilibrium discharge, 119 equilibrium detention storage, 121 equilibrium discharge, 120 flow depth, 114 flow velocity, 114-115 forward characteristic – rising phase, 117 hydrograph – equilibrium phase, 119120

equilibrium detention storage, 109

hydrograph – falling phase, 122-123

equilibrium discharge, 108

hydrograph – rising phase, 116-117

flow depth, 102

kinematic wave celerity, 115

flow velocity, 102-103

kinematic wave parameters, 113-114

forward characteristic – rising phase,

partial equilibrium discharge, 119-120

105

time of travel, 116

Index

244 water surface profile – equilibrium phase, 120-121 water surface profile – falling phase, 121-122 water surface profile – rising phase, 117-118 Runoff coefficient, 2

Time of concentration, 9-10, 32, 177, 198 Time of travel, 45-46, 69, 83, 94, 116, 130, 142, 154, 166-167, 198-199 Trapezoidal channel with equal side slopes, 125-136 average flow velocity, 128-129 average wave celerity, 129 duration of partial equilibrium

Square channel, 101-112 average flow velocity, 103 average wave celerity, 103-104 duration of partial equilibrium discharge, 106-107

discharge, 132-133 equilibrium detention storage, 134135 equilibrium discharge, 133-134 flow depth, 127-128

equilibrium detention storage, 109

flow velocity, 128

equilibrium discharge, 108

forward characteristic – rising phase,

flow depth, 102 flow velocity, 102-103 forward characteristic – rising phase, 105 hydrograph – equilibrium phase, 107108

130-131 hydrograph – equilibrium phase, 133134 hydrograph – falling phase, 136 hydrograph – rising phase, 130 kinematic wave celerity, 129

hydrograph – falling phase, 110-111

kinematic wave parameters, 125-127

hydrograph – rising phase, 104

partial equilibrium discharge, 133

kinematic wave celerity, 103

time of travel, 130

kinematic wave parameters, 101-102

water surface profile – equilibrium

partial equilibrium discharge, 107 time of travel, 104 water surface profile – equilibrium phase, 108 water surface profile – falling phase, 109-110 water surface profile – rising phase, 105-106

phase, 134 water surface profile – falling phase, 135-136 water surface profile – rising phase, 131-132 Trapezoidal channel with one side vertical, 137-149 average flow velocity, 141 average wave celerity, 141-142

Index duration of partial equilibrium discharge, 145

245 hydrograph – equilibrium phase, 157158

equilibrium detention storage, 147

hydrograph – falling phase, 161

equilibrium discharge, 146

hydrograph – rising phase, 155

flow depth, 140

kinematic wave celerity, 153-154

flow velocity, 140

kinematic wave parameters, 151-152

forward characteristic – rising phase,

partial equilibrium discharge, 158

142-143 hydrograph – equilibrium phase, 145146 hydrograph – falling phase, 149 hydrograph – rising phase, 142 kinematic wave celerity, 141 kinematic wave parameters, 137-139

time of travel, 154 water surface profile – equilibrium phase, 158-159 water surface profile – falling phase, 159-161 water surface profile – rising phase, 155-157

partial equilibrium discharge, 145-146 time of travel, 142 water surface profile – equilibrium phase, 146-147 water surface profile – falling phase, 147-149 water surface profile – rising phase, 143-144 Triangular channel, 151-161

Velocity, 6, 31, 42, 67-68, 81, 93, 102103, 114-115, 128, 140, 153, 165, 177, 181, 190-191 average, 6-7, 31, 42-43, 68, 81-82, 93, 103, 115, 128-129, 141, 153, 165, 177, 181, 192-193 Vertical curb channel, 163-174 average flow velocity, 165

average flow velocity, 153

average wave celerity, 166

average wave celerity, 154

duration of partial equilibrium

duration of partial equilibrium discharge, 157

discharge, 169-170 equilibrium detention storage, 172

equilibrium detention storage, 159

equilibrium discharge, 170-171

equilibrium discharge, 158

flow depth, 164-165

flow depth, 152-153

flow velocity, 165

flow velocity, 153

forward characteristic – rising phase,

forward characteristic – rising phase, 155

167 hydrograph – equilibrium phase, 170171

Index

246 hydrograph – falling phase, 174

average, 9, 32, 45, 68-69, 82, 94, 103-

hydrograph – rising phase, 167

104, 116, 129, 141-142, 154, 166,

kinematic wave celerity, 166

196-197

kinematic wave parameters, 163-164

kinematic, 7-9, 31-32, 43-45, 68, 82,

partial equilibrium discharge, 170

93, 115, 129, 141, 153-154, 166,

time of travel, 166-167

194-195

water surface profile – equilibrium phase, 171 water surface profile – falling phase, 172-173 water surface profile – rising phase,

Wave period, 176 Wide rectangular channel, 113-123 average flow velocity, 115 average wave celerity, 116 duration of partial equilibrium

168-169 Water surface profile, 15-17, 20-21, 24-

discharge, 119 equilibrium detention storage, 121

28, 34, 36-37, 70-71, 73-75, 84-85,

equilibrium discharge, 120

87-89, 95-100, 117-118, 120-122,

flow depth, 114

131-132, 134-136, 143-144, 146-149,

flow velocity, 114-115

155-161, 168-169, 171-173, 205-209,

forward characteristic – rising phase,

218-220, 223-228 equilibrium phase, 20-21, 36, 73, 87,

117 hydrograph – equilibrium phase, 119-

97-98, 108, 120-121, 134, 146147, 158-159, 171, 179, 218-220 falling phase, 24-28, 37, 74-75, 88-89,

120 hydrograph – falling phase, 122-123 hydrograph – rising phase, 116-117

98-100, 109-110, 121-122, 135-

kinematic wave celerity, 115

136, 147-149, 159-161, 172-173,

kinematic wave parameters, 113-114

180, 223-228

partial equilibrium discharge, 119-120

rising phase, 15-17, 34, 70-71, 84-85, 95-96, 105-106, 117-118, 131-

time of travel, 116 water surface profile – equilibrium

132, 143-144, 155-157, 168-169, 178, 205-209

phase, 120-121 water surface profile – falling phase,

Wave celerity, 7-9, 31-32, 43-45, 68-69, 82, 93-94, 103-104, 115-116, 129,

121-122 water surface profile – rising phase,

141-142, 153-154, 166, 194-197

117-118 Working formulas, 29-38, 63-174, 188231

Index average flow velocity, 31, 68, 81-82,

247 hydrograph – falling phase, 38, 75-76,

93, 103, 115, 128-129, 141, 153,

89, 100, 110-111, 122-123, 136,

165, 192-193

149, 161, 174, 229-231

average wave celerity, 32, 68-69, 82,

hydrograph – rising phase, 33, 69, 83,

94, 103-104, 116, 129, 141-142,

94,104, 116-117, 130, 142, 155,

154, 166, 196-197

167, 200-202

circular channel, 63-76

kinematic wave celerity, 31-32, 63-

deep rectangular channel, 91-100

66, 82, 93, 103, 115, 129, 141,

design discharge, 33

153-154, 166,194-195

duration of partial equilibrium

kinematic wave parameters, 29-30,

discharge, 35, 71-72, 85-86, 96,

63-66, 77-80, 91-92, 101-102,

106-107, 119, 132-133, 145, 157,

113-114, 125-127, 137-139, 151-

169-170, 210-212

152, 163-164, 185-187

equilibrium detention storage, 36-37,

overland plane, 29-38, 175, 185, 188,

74, 87-88, 98, 109, 121, 134-135,

190, 192, 194, 196, 198, 200,

147, 159, 172, 221-222

203, 205, 210, 213, 218, 221,

equilibrium discharge, 36, 73, 86-87,

223, 229

97, 108, 120, 133-134, 146, 158,

parabolic channel, 77-89

170-171, 213-217

partial equilibrium discharge, 35-36,

flow depth, 30-31, 81, 93, 114, 127128, 140, 152-153, 164-165, 188189 flow velocity, 31, 67-68, 81, 93, 114115, 128, 140, 153, 165, 190-191 forward characteristic – rising phase,

72, 86, 97, 107, 119-120, 133, 145-146, 158, 170, 213-217 rectangular (deep) channel, 91-100 rectangular (square) channel, 101-111 rectangular (wide) channel, 113-123 square channel, 101-111

33-34, 70, 83-84, 94-95, 117,

time of concentration, 32, 198

130-131, 142-143, 155, 167, 203-

time of travel, 69, 83, 94, 116, 130,

204 hydrograph – equilibrium phase, 3536, 72-73, 86-87, 97, 107-108, 119-120, 133-134, 145-146, 157158, 170-171, 213-217

142, 154, 166-167, 198-199 trapezoidal channel with equal side slopes, 125-136 trapezoidal channel with one side vertical, 137-149 triangular channel, 151-161 vertical curb channel, 163-174

Index

248 water surface profile – equilibrium

water surface profile – rising phase,

phase, 36, 73, 87, 97-98, 120-121,

34, 70-71, 84-85, 95-96, 117-118,

134, 146-147, 158-159, 171, 218-

131-132, 143-144, 155-157, 168-

220

169, 205-209

water surface profile – falling phase, 37, 74-75, 88-89, 98-100, 121122, 135-136, 147-149, 159-161, 172-173, 223-228

wide rectangular channel, 113-123