Hydraulic modeling of open channel flows over an arbitrary 3-d surface and its applications in amenity hydraulic engineering

Hydraulic modeling of open channel flows over an arbitrary 3-d surface and its applications in amenity hydraulic engineering

HYDRAULIC MODELING OF OPEN CHANNEL FLOWS OVER AN ARBITRARY 3-D SURFACE

AND ITS APPLICATIONS

IN AMENITY HYDRAULIC ENGINEERING

TRAN NGOC ANH

August, 2006

Acknowledgements

The research work presented in this manuscript was conducted in River System Engineering Laboratory, Department of Urban Management, Kyoto University, Kyoto, Japan

First of all, I would like to convey my deepest gratitude and sincere thanks to Professor

Dr Takashi Hosoda who suggested me this research topic, and provided guidance,

constant and kind advices, encouragement throughout the research, and above all, giving me a chance to study and work at a World-leading university as Kyoto University

I also wish to thank Dr Shinchiro Onda for his kind assistance, useful advices especially

in the first days of my research life in Kyoto His efforts were helping me to put the first stones to build up my background in the field of computational fluid dynamics

My special thanks should go to Professor Toda Keiichi and Associate Professor Gotoh

Hitoshi for their valuable commences and discussions that improved much this

manuscript

I am very very grateful to my best foreign friend, Prosper Mgaya from Tanzania, for all

of his helps, discussions and strong encouragements since October, 2003

In addition, my heartfelt gratitude is extended to all of my Vietnamese friends in Japan, Kansai Football Club members, who helped me forget the seduced life in Vietnam, particular Nguyen Hoang Long, Le Huy Chuan and Le Minh Nhat

Last but not least, the most deserving of my gratitude is to my wife, Ha Thanh An, and my family, parents and younger brother This work might not be completed without their constant support and encouragement I am feeling lucky because my wife, my parents and my younger brother are always by my side, and this work is therefore dedicated to them

Abstract

Two-dimensional (2D) description of the flow is commonly sufficient to analyze successfully the flows in most of open channels when the width-to-depth ratio is large and the vertical variation of the mean-flow quantities is not significant Based on coordinate criteria, the depth-averaged models can be classified into two groups namely: the depth-averaged models in Cartesian coordinate system and the depth-averaged models in generalized curvilinear coordinate system The basic assumption in deriving these models is that the vertical pressure distribution is hydrostatic; consequently, they possess the advantage of reduction in computational cost while maintaining the accuracy when applied to flow in a channel with linear or almost linear bottom/bed But indeed, in many cases, water flows over very irregular bed surfaces such as flows over stepped chute, cascade, spillway, etc and the alike In such cases, these models can not reproduce the effects of the bottom topography (e.g., centrifugal force due to bottom curvature)

In this study therefore, a depth-averaged model for the open channel flows over an arbitrary 3D surface in a generalized curvilinear coordinate system was proposed This model is the inception for a new class of the depth-averaged models, which was classified by the criterion of coordinate system In conventional depth-averaged models, the coordinate systems are set based on the horizontal plane, then the equations are obtained by integration of the 3D flow equations over the depth from the bottom to free surface with respect to vertical axis In contrary the depth-averaged equations derived in this study are derived via integration processes over the depth with respect to the axis

perpendicular to the bottom The pressure distribution along this axis is derived from one of the momentum equations as a combination of hydrostatic pressure and the effect of centrifugal force caused by the bottom curvature This implies that the developed model can therefore be applied for the flow over highly curved surface Thereafter the model was applied to simulate flows in several hydraulic structures this included: (i) flow into a vertical intake with air-core vortex and (ii) flows over a circular surface

The water surface profile of flows into vertical intake was analyzed by using 1D steady equations system and the calculated results were compared with an existing empirical formula The comparison showed that the model can estimate accurately the critical submergence of the intake without any limitation of Froude number, a problem that most of existing models cannot escape The 2D unsteady (equations) model was also applied to simulate the water surface profile into vertical intake In this regard, the model showed its applicability in computing the flow into intake with air-entrainment

The model was also applied to investigate the flow over bottom surface with highly curvature (i.e., flows over circular surface) A hydraulic experiment was conducted in laboratory to verify the calculated results For relatively small discharge the flow remained stable (i.e., no flow fluctuations of the water surface were observed) The model showed good agreement with the observations for both steady and unsteady calculations When discharge is increased, the water surface at the circular vicinity and its downstream becomes unstable (i.e., flow flactuations were observed) In this case, the model could reproduce the fluctuations in term of the period of the oscillation, but some discrepancies could be still observed in terms of the oscillation’s amplitude

In order to increase the range of applicability of the model into a general terrain, the model was refined by using an arbitrary axis not always perpendicular to the bottom surface The mathematical equation set has been derived and some simple examples of

dam-break flows in horizontal and slopping channels were presented to verify the model The model’s results showed the good agreement with the conventional model’s one

Preface

The depth-averaged model has a wide range of applicability in hydraulic engineering, especially in flow applications having the depth much smaller compare to the flow width In this approach the vertical variation is negligible and the hydraulic variables are averaged integrating from bed channel to the free surface with respect to vertical axis In deriving the governing equations, the merely pure hydrostatic pressure is assumed that is not really valid in case of flows over highly curved bed and cannot describe the consequences of bed curvature Therefore, this work is devoted to derive a new generation of depth-averaged equations in a body-fitted generalized curvilinear coordinate system attached to an arbitrary 3D bottom surface which can take into account of bottom curvature effects

This manuscript is presented as a monograph that includes the contents of the following published and/or accepted journal and conference papers:

1 Anh T N and Hosoda T.: Depth-Averaged model of open channel flows over an

arbitrary 3D surface and its applications to analysis of water surface profile Journal

of Hydraulic Engineering, ASCE (accepted on May 12, 2006)

2 Anh T N and Hosoda T.: Oscillation induced by the centrifugal force in open

channel flows over circular surface 7th International Conference on

Hydroinformatics (HIC 2006), Nice, France, 4~8 September, 2006 (accepted on April

21, 2006)

3 Anh T N and Hosoda T.: Steady free surface profile of flows with air-core vortex at

vertical intake XXXI IAHR Congress, Seoul, Korea, pp 601-612 (paper A13-1),

11~16 September, 2005

4 Anh T.N and Hosoda T.: Water surface profile analysis of open channel flows over a

circular surface Journal of Applied Mechanics, JSCE, Vol 8, pp 847-854, 2005

5 Anh T N and Hosoda T.: Free surface profile analysis of flows with air-core vortex

Journal of Applied Mechanics, JSCE, Vol 7, pp 1061-1068, 2004

2.2 Depth-average model in generalized curvilinear coordinate system 10

Chapter 4 STEADY ANALYSIS OF WATER SURFACE PROFILE OF FLOWS

WITH AIR-CORE VORTEX AT VERTICAL INTAKE 30

List of Figures

Chapter 2

Figure 2.1 Definition sketch for variables used in depth-averaged model…… 8

Figure 2.2 Definition of terms in curvilinear system……… 11 Figure 2.3 Definition sketch by Sivakumaran et al (1983)……… 14

Figure 4.3 The inflow to and circulation round a closed path

in a flow field (Townson 1991)……… 33

Figure 4.4 The concept of simple Rankine vortex that including

two parts: free vortex in outer zone and forced vortex

in inner zone (Townson 1991)………33

Figure 4.5 Definition of coordinate components……….36

Figure 4.6 An example of computed water surface profile with

quasi-normal depth line and critical depth line……… 45

Figure 4.7 The effect of circulation on water surface profile and

discharge at the intake with same water head………48

Figure 4.8 Variation of intake discharge with circulation

(a=0.025m, b=10-5 m2, water head=0.5m)……….49

Figure 4.9 Different water surface profiles with different values

of circulation while maintaining the constant intake discharge…… 49

Figure 4.10 Changing of water surface profile with different shape of the intake 51

Figure 4.11 The effects of bon discharge (17a) and submergence

(17b) at an intake………51

Figure 4.12 Definition sketch of critical submergence……… 52

Figure 4.13 Comparison of computed critical submergence by the model (Eq 47) and by Odgaard’s equation (51)……….52

Figure 4.14 The variation of critical submergence wit different values of b…53 Chapter 5 Figure 5.1 Definition sketch of the new coordinates………57

Figure 5.2 Illustration of the computational grid……… 59

Figure 5.3 Definition sketch of cell-centered staggered grid in 2D calculation……… 60

Figure 5.4 Illustration for the discretization scheme in momentum equations……… 61

Figure 5.5 Water surface of flow with different discharges at the intake…….63

Figure 5.6 Water surface of flow with different velocity at the outer-zone boundary……… 63

Figure 5.7 Water surface of flow with different shape of the intake………….64

Chapter 6 Figure 6.1 Side view of the experimental facility ………68

Figure 6.2 Experimental site……….68

Figure 6.3 Schematic of sensor connection……… 71

Figure 6.4 Sensor calibration………71

Figure 6.5 Time history of the free surface at four locations in different experiments: a) Exp-1; b) Exp-2; c) Exp-3; d) Exp-4;………72

Figure 6.6 The oscillation density at four locations in circular region……… 73

Figure 6.7 Curvilinear coordinates attached to the bottom……… 75

Figure 6.8 Illustration of computed water surface profile with quasi-normal and critical depth lines………78

Figure 6.9 Steady water surface profile with conditions of Exp-1……….79

Figure 6.10 Steady water surface profile with conditions of Exp-2……….79

Figure 6.11 Steady water surface profile with conditions of Exp-5……….80

Figure 6.12 Steady water surface profile with conditions of Exp-6……….80

Figure 6.13 Illustration of staggered grid……… 81

Figure 6.14 Computed water surface profile in Exp-1……….84

Figure 6.15 Computed water surface profile in Exp-2……….84

Figure 6.16 Computed water surface profile in Exp-5……….85

Figure 6.17 Computed water surface profile in Exp-6……….85

Figure 6.18 Computed water surface profile in Exp-3……….86

Figure 6.19 Computed water surface profile in Exp-4……….86

Figure 6.20 Computed water surface profile in Exp-7……….87

Figure 6.21 Computed water surface profile in Exp-8……….87

Figure 6.22 Power spectrum of water surface displacement at point 3 in Exp-3…88 Figure 6.23 Power spectrum of water surface displacement at point 4 in Exp-3…88 Figure 6.24 Comparison of calculated and experimental results at point 3 in Exp-3………89

Figure 6.25 Comparison of calculated and experimental results at point 4 in Exp-3………89

Figure 6.26 Power spectrum of water surface displacement at point 3 in Exp-4…90 Figure 6.27 Power spectrum of water surface displacement at point 4 in Exp-4…90 Figure 6.28 Comparison of calculated and experimental results at point 3 in Exp-4………91

Figure 6.29 Comparison of calculated and experimental results at point 4 in Exp-4………91

Figure 6.30 Power spectrum of water surface displacement at point 3 in Exp-8…92 Figure 6.31 Power spectrum of water surface displacement at point 4 in Exp-8…92 Figure 6.32 Comparison of calculated and experimental results at point 3 in Exp-8………93

Figure 6.33 Comparison of calculated and experimental results at point 4 in Exp-8………93

Figure 6.34 Carpet plot of water surface in 2D simulation of Exp-1 ……….95

Figure 6.35 Carpet plot of water surface in 2D simulation of Exp-2……… 95

Figure 6.36 Carpet plot of water surface in 2D simulation of Exp-3……… 96

Figure 6.37 Carpet plot of water surface in 2D simulation of Exp-4……… 96

Figure 6.38 Carpet plot of water surface in 2D simulation of Exp-5……… 97

Figure 6.39 Carpet plot of water surface in 2D simulation of Exp-6……… 97

Figure 6.40 Carpet plot of water surface in 2D simulation of Exp-7……… 98

Figure 6.41 Carpet plot of water surface in 2D simulation of Exp-8……… 98

Chapter 7

Figure 7.1 Illustration for limitation of the model in concave topography………101

Figure 7.2 Definition sketch of new generalized coordinate system……….105

Figure 7.3 Calculated water surface profile at different time steps of

dam break flow in a dried-bed sloping channel:

Figure 7.4 Calculated water surface profile at different time steps of

dam break flow in a dried-bed horizontal channel:

Figure 7.5 Comparison of water surface profile for dried horizontal

channel at different times: T = 0.4s; 1.0s; and 1.6s; ……… 109

Figure 7.6 Comparison of water surface profile for wetted horizontal

channel at different times: T = 0.4s; 0.6s; and 0.8s; 109

Figure 7.7 Calculated water surface profile at different time steps of

dam break flow in a dried-bed sloping channel:

Figure 7.8 Calculated water surface profile at different time steps of

dam break flow in a dried-bed horizontal channel:

Hiniup =1.0; down =0.5 ; 01 =30

List of Tables

Table 4.1 Parameters used in the calculations of results in Figure 4.7…… 48

Table 4.2 Parameters used in the calculations of results in Figure 4.9…… 49

Table 4.3 Parameters used in the calculations of results in Figure 4.10……51

Table 6.1 Experiment conditions………73

Chapter 1

INTRODUCTION

The advent of modern computers has had a profound effect in all branches of engineering especially in hydraulics The recent development of numerical methods and the capabilities of modern machines has changed the situation in which many problems were, up to recently, considered unsuited for numerical solution can now be solved without any difficulties (Brebbia and Ferrante 1983)

The most open-channel flows of practical relevance in civil engineering are always strictly three-dimensional (3D); however, this feature is often of secondary importance, especially when the width-to-depth ratio is large and the vertical variation of the mean-flow quantities is not significant due to strong vertical mixing induced by the bottom shear stress Based on these facts, a two-dimensional (2D) description of the flow is sufficient to successfully analyze the flows in most of open channels using the depth-averaged equations of motion The depth averaging process used to derive these equations sacrifices flow details over the vertical dimension for simplicity and substantially reduces computational effort (Steffler and Jin 1993)

1.1 Classification of depth-averaged models:

In spite of the variation of numerical methods applied in solving the governing equations in different practical problems, the depth-averaged models can be classified using several criteria such as:

(1) Time dependence: a Steady,

b Unsteady

(2) Spatial integral or spatial dimension:

a Integrate over a cross-section to get 1-D equations,

b Integrate the 3D equations from bottom to water surface (i.e depth averaged model) to get 2D equations

(3) Pressure distribution: a Hydrostatic pressure,

b Consideration of vertical acceleration (Boussinesq eq.) (5) Velocity distribution and evaluation of bottom shear stresses:

a Uniform velocity distribution or self-similarity of distribution,

b Modeling of local change of velocity distribution (secondary currents caused by stream-line curvature, velocity distribution with irrotational condition, etc.)

a Fully free water surface,

b Co-existence of open channel flows and pressurized flows in underground channels such as sewer networks

(9) Coordinate system:

a Cartesian coordinate set on a horizontal plane,

b (moving) Generalized curvilinear coordinate on a horizontal plane, c Generalized curvilinear coordinate on an arbitrary 3-D surface

Based on their characteristics, one model can be classified as a combination of the above sub-classes as follows: Unsteady 2D model for open channel flows in a curvilinear coordinate system using the assumption of hydrostatic pressure and 0-equation closure for turbulent model

In fact, to select the most appropriate model to solve a practical problem is usually the most important step for the hydraulic engineers that can satisfy the required accuracy and meet the reasonable computational cost

1.2 Depth-averaged models and Coordinate system

The original depth-averaged model was derived in Cartesian coordinate system (Kuipers and Vreugdenhill 1973) that was then applied extensively in various practical problems mostly related with the regular boundaries

To apply the depth-averaged equations to calculate the flows in the natural rivers or meandering channels, the boundary-fitted curvilinear coordinates were introduced The use of curvilinear coordinates can overcome the problems of irregular boundaries consequently it could decrease the computational cost effectively, spreads out widely and increases the range of the applicability of depth-averaged equations

Up to now, most of hydraulic depth-averaged models have been developed based on

either Cartesian coordinates or a generalized curvilinear coordinates that set on a horizontal plane The basic assumption in deriving these models is that the vertical pressure distribution is hydrostatic; hence they possess the advantage of reduction in computational cost while maintaining the accuracy when applied to flow in a channel with linear or almost linear bottom/bed But indeed, in many cases, water flows over very irregular bed surfaces such as flows over stepped chute, cascade, spillway, etc and the alike In such cases, these models can not reproduce the effects of the bottom topography (e.g., centrifugal force due to bottom curvature) For this reason, the present research work is developed at proposing a new class of models that use the generalized curvilinear coordinates based on an arbitrary surface which can conform and reflect better the variation of bed surface; that is classified as class 9(c) in Section 1.1

1.3 Objectives of Study

The objectives of this study are:

- To develop the new class of general depth-averaged equations in a generalized curvilinear coordinate system set based on an arbitrary 3D surface that could be applied to simulate the flows over a complex channel bed’s topography with highly curvatures

- To apply the equations to analysis of water surface profile of flows in Amenity Hydraulic Structures

A general mathematical model based on a new coordinate system attached to 3D arbitrary bottom surface with an axis perpendicular to it, is developed to solve the 2D depth-averaged equations In this approach, the assumption of shallow water is utilized and the internal turbulent stresses are neglected Firstly, the initial 3D equations are transformed into generalized curvilinear coordinate system, then, the continuity and

momentum equations are integrated over the depth from bottom to free water surface with respect to the axis perpendicular to the bottom

The model is then applied to several flows to analyze the free water surface profiles and the results are compared with the experimental data or/and with an existing empirical formula to show the applicability and effectiveness of the model

1.4 Scope of Study

The manuscript consists of eight chapters including the Introduction and the Conclusions Chapter 1 describes the classification of depth-averaged models, difference of Coordinate systems and clarifies the objectives of the study

Chapter 2 reviews the related studies in the past, including studies on depth-averaged models in Cartesian coordinates, generalized curvilinear coordinates and studies on the effect of bottom curvature It analyzes these methods and their limitation in applying in flow over complex bed topography consequently it emphasizes the motivation of the study In Chapter 3, the derivation of new generalized depth-averaged equations from the original Reynold Averaged Navie-Stokes (RAN) equations is described The flow over a vertical intake with the air-core vortex is investigated in Chapter 4 and Chapter 5, in which the steady water surface profiles are examined firstly in Chapter 4, and then Chapter 5 is devoted for the 2D characteristics using the unsteady equations The same model is applied to simulate the steady and unsteady characteristics of flows over circular surface in Chapter 6

Although the new model has shown its applicability in several problems, it has some limitations as well in some convex bottoms; therefore Chapter 7 is dedicated to model refinements This improvement is illustrated by a simple application of dam-break flows in horizontal and sloping channels Finally, the Conclusions - Chapter 8 summarizes all

the contributions of this study

1.5 References

1 Brebia C A and Ferrante A.: Computational Hydraulics Butterworths Press, 1983 2 Kuipers, J and Vreugdenhill, C B 1973 Calculations of two-dimensional horizontal

flow Rep S163, Part 1, Delft Hydraulics Laboratory, Delft, The Netherlands

3 Steffler M P and Jin Y C 1993 Depth averaged and moment equations for moderately shallow free surface flow J Hydr Res 31 (1), 5-17

The equations were derived by integrating the Reynold-Averaged Navier Stokes equations from the bed to free surface with respect to vertical direction in plugging with the kinematic boundary condition at the free surface Therefore, this equations system is often called as the depth-averaged or shallow water equations

Continuity equation:

(2.1) Momentum equation:

Figure 2.1 Definition sketch for variables used in depth-averaged model

bybx τ

τ , : bottom shear stresses

T , , : effective shear stresses defined as follows:

( )uudzu

( )( )uuvvdzv

( )vvdzv

Jin and Steffler (1993) developed a depth-averaged model and included the effects of velocity distribution in depth by using empirical velocity distribution equations They

then applied their model to a 270o bend and compared the results with experimental data for water surface profile and velocity distribution with satisfactory results Steffler and Jin (1993) also used the classical depth-averaged equations for shallow free surface flow extended to treat the problem with non-hydrostatic pressure and non-uniform velocity distributions Khan and Steffler (1996) analyzed the momentum conservation within a hydraulic jump utilized the depth-averaged equations; with the velocity distribution was evaluated using a moment of longitudinal momentum equation, coupled with a simple linear velocity distribution (Steffler and Jin 1993)

Kimura and Hosoda (1997) investigated the properties of flows in open channels with dead zone by mean of depth averaged model in a variable grid system The fairly good agreement was found between the computed results and experimental data More recently, Molls et al (1998) applied the 2D shallow water equations to examine the effect of sidewall friction in analyzing a backwater profile in a straight rectangular channel

2.2 Depth-averaged Model in Generalized Curvilinear Coordinate System

In solving these equations for practical problems, the natural rivers are almost meandering and have the irregular boundaries, a “stair stepped” approximation is commonly used by the conventional finite-difference method and finite volume method (Vreugdenhil and Wijbenga 1982) This can result in either poor resolution due to coarse grid distribution near the boundary or increase computational cost by refining the grid system to achieve a better representation of the physical boundary The boundary-fitted system of generalized curvilinear coordinate technique has been developed to overcome this deficiency that Thompson (1980) was one of the pioneers

Firstly, the boundary-fitted curvilinear coordinates ( )ξ,η are introduced as ξ =ξ( )x,y ,

( )x,y

η= (Figure 2.2), and then the governing equations (2.1-2.3) are transformed

from Cartesian to non-orthogonal curvilinear coordinates, ( ) ( )x,y → ξ,η , as follows: Continuity equation:

Momentum equation:

( ) () ( ) ( uvh)

ξξ

J : Jacobian matrix defined by J= 1(xξyη −xηyξ)

Using the same idea, Younus and Chaudhry (1994) transformed the original depth-averaged equations from the physical coordinates (x,y,t) to the computational coordinates (ξ,η,t) then analyzed (i) the developed uniform flow in a straight rectangular channel, (ii) hydraulic jump in a diverging channel, (iii) supercritical flow in a diverging channel as well as (iv) circular hydraulic jump Molls and Chaudhry (1995)

presented a depth-averaged model in a general coordinate system with an explicit algorithm and constant eddy viscosity This model was used for both subcritical and supercritical flows in a contraction, hydraulic jump, spur dike, and a channel with an 180obend A two-dimensional, vertically averaged, unsteady circulation model using a non-orthogonal boundary-fitted technique was employed in spherical coordinates for predicting sea level, currents in estuarine and shelf water (Muin and Spaulding 1996) Ye and McCorquodale (1997) developed a depth-averaged model in a non-orthogonal curvilinear system with collocated grid arrangement Comparison of their results with available experimental data for side discharge into an open channel, flow in meandering channel, and flow in Parshall flume with supercritical outflow was satisfactory Zhou and Goodwill (1997) presented a depth-averaged model and compared its predictions for a water surface profile along the 180o bend channel The bend-flow was also considered using depth-averaged model in a generalized curvilinear coordinate by Lien et al (1999), Hsieh and Yang (2003), Lackey and Sotiropoulos (2005), etc The similar equations were also employed to investigate the fundamental characteristics of high velocity flows in a sinuous open channel by Hosoda and Nishihama (2002) The calculated results of water surface profile were visualized and compared with the experimental data to verify the

numerical model

In a more recent paper, Zarrati et al (2005) developed a two-dimensional depth-averaged model in a nonorthogonal curvilinear coordinates for prediction of flow pattern in meandering channel with 60o and 90o bend, and also with compound meandering channel In this study, the Cartesian velocity components were selected as the dependent variables to avoid curvature sensitive terms (Christoffel symbols) The comparison showed that the model predicted the water surface profile and velocity distribution well in simple channels, and the predictions of the model in main channel of compound meandering channel were also in general agreement with the experimental results

2.3 Effect of Bottom Curvature

Despite of the efficiency and reasonable accuracy of above conventional depth-averaged models, they are limited in use due to the assumption of moderate slope of the channel bed In all above models, the governing equations in Cartesian coordinate are firstly integrated over the depth from the bottom to the water surface in vertical direction, (i.e

z to zs which respect to a horizontal datum plane), and then, transformed into a 2D generalized curvilinear coordinate The basic assumption in deriving equations mentioned above is that the vertical accelerations of a particle are negligible, (i.e assuming the vertical pressure distribution to be merely hydrostatic) When the channel bottom is almost linear either horizontal or with only a small inclination, these models yield a good solutions for shallow flows, but in complex terrains they can not describe the significant effect of the bottom curvature

Dressler (1978) derived a set of one-dimensional shallow water equations that included the effect of bottom curvature New equations were derived which is the generalization of the nonlinear unsteady shallow-flow partial differential equations, usually called the

Saint-Vernant equations The new equations possess terms containing explicitly the curvature and the derivative of the curvature The resulting streamlines were therefore curved, and the pressure expression contained terms, in addition to the hydrostatic term, which describe the effect of streamline curvatures (Equation 2.15)

n-axis (upward normal to the bed)

Centre of curvature of the bed profile at Pθ

s-axis

x z

0 stu

),

and the velocity components and pressure are given by

Sivakumaran et al (1981, 1983) applied these equations to analyze the steady state shallow flows over curved beds such as spillway These studies were based on the procedure of asymptotic approximation proposed by Friedrichs (1948) and extended by Keller (1948) However, this approach is efficient only when the flow is sufficiently shallow and gradually varied For this reason, these models appear to be inapplicable for the case of highly curved surface

Berger and Carey (1998a, b) generalized the derivation to a two-dimensional surface based on an orthogonal curvilinear co-ordinate system, which improved the Dressler’s equations by including the vorticity features In fact, it is difficult to set an orthogonal curvilinear coordinate on an arbitrary surface

On the other hand, to apply the 2D shallow water equations for prediction of dam-break flows over complex bed topography, Zhou et al (2004) developed a surface gradient method for the accurate treatment of the sources term The ideas were originally proposed by Zhou et al (2001) and extended for treating bed topography involving a vertical step (Zhou et al 2002) However, the technique used in these studies that involved simply

adding the head loss in the source term of the equations is similar to that of suddenly contracted channel This is not applicable in general geometry and did not reflect the effect of the bottom curvature

2.4 Motivation of Study

A huge number of studies have been performed in the past regarding to the conventional depth-averaged model, however some difficulties as mentioned earlier still remain unsolved entirely In order to reach to a compromise that could take the advantages of shallow water models while could represent the effect of bottom curvature; this study is therefore devoted to develop a new general form of depth-averaged equation that can be applied to simulate the flow over complex topography

2.5 References

1 Berger, R C and Carey, G F 1998 Free-surface flow over curved surfaces, Part I:

Perturbation analysis Int J Numer Meth Fluids Vol 28, pp 191-200

2 Berger, R.C and Carey, G F 1998 Free-surface flow over curved surfaces, Part II:

Computational model Int J Numer Meth Fluids Vol 28, pp 201-213

3 Brebia C A and Ferrante A 1983 Computational Hydraulics Butterworths Press

150pp

4 Dressler, R F 1978 New nonlinear shallow-flow equations with curvature J

Hydraul Res., Vol 16, pp 205-222

5 Friedrichs, K.O 1948 Appendix: On the derivation of the shallow water theory

Comm Appl Math., Vol 1, pp 81-87

6 Hosoda T and Nishihama M 2002 Water surface profile of high velocity flows in a

sinuous open channel The 10th International Symposium on Flow Visualization,

August 26-29, Kyoto, Japan

7 Hsieh T and Yang J C 2003 Investigation on the Suitability of Two-Dimensional

Depth-Averaged Models for Bend-Flow Simulation J Hydr Engrg., ASCE, Vol

129(8), pp 597-612

8 Jin Y C and Steffler P M 1993 Predicting flow in curved open channels by

depth-averaged method J Hydr Engrg., ASCE, Vol 119(1), pp 109-124

9 Kalkwijk J P T and De Vriend H J 1980 Computational of the flow in shallow

river bends J Hydraul Res., Vol 18(4), pp 327-342

10 Keller, J B 1948 The solitary wave and periodic waves in shallow water Comm

Appl Math., Vol 1, pp 323-339

11 Khan A A and Steffler P M 1996 Physically based hydraulic jump model for

depth-averaged computations J Hydr Engrg., ASCE, Vol 122(10), pp 540-548

12 Kimura I And Hosoda T 1997 Fundamental properties of flows in open channel

with dead zone J Hydr Engrg., ASCE, Vol 123(2), pp 98-107

13 Kuipers, J and Vreugdenhill, C B 1973 Calculations of two-dimensional horizontal

flow Rep S163, Part 1, Delft Hydraulics Laboratory, Delft, The Netherlands

14 Lackey T C and Sotiropoulos F 2005 Role of artificial dissipation scaling and multigrid acceleration in numerical solutions of the depth-averaged free-surface flow

equations J Hydr Engrg., ASCE, Vol 131(6), pp 476-487

15 Leendertse, J J 1967 Aspects of a computational model for long period water-wave

propagation Memo RM-5294-PR, Rand Corporation

16 Lien H C., Hsieh T Y., Yang J C and Yeh K C 1999 Bend-flow simulation using

2-D depth-averaged model J Hydr Engrg., ASCE, Vol 125(10), pp 1097-1108

17 McGuirk, J J and Rodi, W 1978 A depth-averaged mathematical model for the near

field of side discharge into open-channel flow J Fluid Mech Vol 86(4), pp

761-781

18 Molls T and Chaudhry M H 1995 Depth-averaged open-channel flow model J

Hydr Engrg., ASCE, Vol 121(6), pp 453-465

19 Molls T., Zhao G and Molls F 1998 Friction slope in depth-averaged flow J Hydr

Engrg., ASCE, Vol 124 (1), pp 81-85

20 Muin M and Spaulding M 1996 Two dimensional boundary-fitted circulation model

in spherical coordinates J Hydr Engrg., ASCE, Vol 122(9), pp 512-521

21 Ponce V M and Yabusaki S B 1981 Modeling circulation in depth-averaged flow J

Hydr Div., ASCE, Vol 107(11), pp 1501-1518

22 Sivakumaran N S, Hosking, R J and Tingsanchali, T 1981 Steady shallow flow

over a spillway J Fluid Mech., Vol 111, pp.411-420

23 Sivakumaran N S, Tingsanchali, T and Hosking, R J 1983 Steady shallow flow

over curved bed J Fluid Mech., Vol 128, pp.469-487

24 Steffler M P and Jin Y C 1993 Depth averaged and moment equations for

moderately shallow free surface flow J Hydr Res., Vol 31(1), pp 5-17

25 Thompson J F 1980 Numerical solution of flow problems using body-fitted

coordinate systems Computational fluid dynamics W Kollmann, ed., Hemisphere

Publishing Corp., Bristol, PA

26 Tingsanchali T And Maheswaran S 1990 2-D depth-averaged flow computation

near groyne J Hydr Engrg., ASCE, Vol 116(1), pp 71-86

27 Vreugdenhill C B and Wijbenga J 1982 Computation of flow pattern in rivers J

Hydr Div., ASCE, Vol 108(11), pp 1296-1310

28 Ye J and McCorquodale J A 1997 Depth-averaged hydrodynamic model in

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∂∂

and its inverse

The derivatives of spatial independent variables (xξ,xη,xζ,yξ,yη,yζ,K) are denoted by the following equations:

Jx ηyζz ηzζy

Jx ζyξz ζzξy

Jx ξyηz ξzηy

Jy ηzζx ηxζz

Jy ζzξz ζxξz

Arbitrary surface

Figure 3.1 Definition sketch for new generalized coordinate system

zxxzη ξ ηξ

Jz ηxζy ηyζx

xyyxη ξ ηξ

The transformation relationship between the contravariant velocity vector V(U,V,W)in curvilinear coordinate and its counterpart v(u ,,vw) in Cartesian coordinate system is

jijic v

jic V

The conditions for the axis ζ to be perpendicular to the surface (i.e the axis ζ

perpendicular to both axes ξ and η) are expressed as follows: (x,y,z)⋅grad (x,y,z)=0

3.2 Kinematic boundary condition at the water surface

The free water surface (Figure 3.2) at a time t and t+∆t are 0

−hξ η t

−∆

thus

where

and similarly,

Vt =∆

Wt =∆

3.3 Depth-averaged continuity and momentum equations

The fundamental equations in Cartesian co-ordinates are firstly transformed into an arbitrary generalized curvilinear coordinate system as follows:

Continuity equation:

Momentum equation:

Based on the shallow water condition, the transforming Jacobian is approximately homogenous with respect to ζ -axis and is set equal to the value of Jacobian at the bottom J Other flow quantities such as U and V are uniform in 0 ζ -direction Integrating Equation (3.24) over the depth from the curved bottom to the water surface with respect to ζ -axis and applying the kinetic boundary condition at the water surface (Equation 3.23), the depth-averaged continuity equation is derived:

where M =Uh and N =Vh From now onwards, the subscript zero “0” denotes the

values right on the bottom surface

Substituting the relations described in Equation (3.7) into Equation (3.25), the ξ

component of momentum equation in 3D space is recast

ξ ⎟⎠+ Γ + Γ + Γ + Γ⎞

JG

2

( ξηζηζξηηηηξηξηξξξζξζξξηξηξξξξξξζ

η ⎟⎟⎠+ Γ + Γ + Γ + Γ + Γ + Γ

1 term)

1

1 term)

(pressure 2 2 2

Substituting pressure term in Equation (3.29) into (3.27) and taking integration over the depth from the curved bottom to the water surface with respect to ζ -axis, the depth-averaged momentum equation in ξ direction is obtained:

ξ 2 000 2 00

ζξζξζ