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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. iii 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 iv 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 v 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. vi 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 vii 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. viii 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 ix 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 x 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 xi

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