Hydraulic modeling of open channel flows over an arbitrary 3-D surface and its applications in amenity hydraulic engineering - Trần Ngọc Anh

HYDRAULIC MODELING OF OPEN CHANNEL FLOWS OVER AN ARBITRARY 3-D SURFACE AND ITS APPLICATIONS IN AMENITY HYDRAULIC ENGINEERING TRAN NGOC ANH August, 2006 iii 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. iv 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 v 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 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. vii 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 601-612ç (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 xi 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 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 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 xiii 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 xiv 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= ; 01 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= ; 01 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;ç ………………..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= ; 01 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 xv 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 1 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). 2 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.). (6) Turbulence model: a. 0-equation model (eddy viscosity proportional to depth multiplied by friction velocity), b. Depth averaged ε−k model. (7) Single layer model or two layered model: A multi-layered model which has more than two layers is classified as 3-D model. (8) Open channel flow or partially full pressurized flow: 3 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