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

vi
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.


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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

viii
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.

ix

Table of contents

Acknowledgment iii
Abstract iv
Preface vii
List of Figures xi
List of Tables xv
Chapter 1. INTRODUCTION
1
1.1 Classification of depth-averaged modeling 2
1.2 Depth-averaged model in curvilinear coordinates 3
1.3 Objectives of study 4
1.4 Scope of study 5
1.5 References 6
Chapter 2. LITERATURE REVIEW
7

2.1 Depth-average modeling 7
2.2 Depth-average model in generalized curvilinear coordinate system 10
2.3 Effect of bottom curvature 13
2.4 Motivation of study 16
2.5 References 16
Chapter 3. MATHEMATICAL MODEL
20

3.1 Coordinate setting 20
3.2 Kinetic boundary condition at the water surface 23
3.3 Depth-averaged continuity and momentum equations 24
Chapter 4. STEADY ANALYSIS OF WATER SURFACE PROFILE OF FLOWS
WITH AIR-CORE VORTEX AT VERTICAL INTAKE
30

4.1 Introduction 30
4.2 Governing equation 35
4.3 Results and discussions 47
4.4 Summary 54

x
4.4 References 54
Chapter 5. UNSTEADY PLANE-2D ANALYSIS OF FLOWS WITH
AIR-CORE VORTEX
56

5.1 Governing equation 56
5.2 Numerical method 59
5.3 Results and discussions 62
5.4 Summary 64
5.5 References 65
Chapter 6. WATER SURFACE PROFILE ANALYSIS OF FLOWS OVER
CIRCULAR SURFACE
66

6.1 Preliminary 66
6.2. Hydraulic experiment 67
6.3 Steady analysis of water surface profile 74
6.4 Unsteady characteristics of the flows 81
6.5 2D simulation 94
6.6 Summary 94
6.7 References 99
Chapter 7. MODEL REFINEMENT
100

7.1 Preliminary 100
7.2 Non-orthogonal coordinate system 101
7.3 Application 105
Chapter 8. CONCLUSIONS
111



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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
Chapter 3
Figure 3.1 Definition sketch for new generalized coordinate system………… 21
Figure 3.2 Kinetic boundary condition at water surface……………………… 23
Chapter 4
Figure 4.1 An example of free surface air-vortex………………………………31
Figure 4.2 Various stages of development of air-entraining vortex:
S1>S2>S3>S4 (Jain et al, 1978)……………………………………31
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
m
2
, 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


xii
Figure 4.11 The effects of
b
on 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

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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

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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:
mHini 0.1=
;
0
1
30=
α
;
0
2
60=
α
………………………………108
Figure 7.4 Calculated water surface profile at different time steps of
dam break flow in a dried-bed horizontal channel:
mHini 0.1=
;
0
1
0=
α
;
0
2
60=
α
……………………………… 108
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:
mHini 0.1=
;
0
1
30=
α
;
0
2
60=
α
…………………………………110
Figure 7.8 Calculated water surface profile at different time steps of
dam break flow in a dried-bed horizontal channel:
mHinimHini
downup
5.0;0.1 ==
;
0
1
30=
α
;
0
2
60=
α
……………110



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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

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