Luận văn 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 601-612ỗ (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;ỗ ………………..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 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 4 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 5 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 6 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. 7 Chapter 2 LITERATURE REVIEW 2.1 Depth-averaged models Kuipers and Vreugdenhill (1973) developed one of the first mathematical models capable of solving the two-dimensional (2D) depth averaged equations using the finite-difference scheme developed by Leendertse (1967) (Molls and Chaudhry, 1995). Four assumptions were the basis of the equations set used in this study: (i) water is incompressible; (ii) vertical velocities and accelerations are negligible; (iii) wind stresses and geostrophic effects are negligible; and (iv) average values are sufficient to describe flow properties which vary over the flow depth. 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. 8 Continuity equation: ( ) ( ) 0= ∂ ∂ + ∂ ∂ + ∂ ∂ y hv x hu t h (2.1) Momentum equation: y hT hx hT hhx z g y uv x uu t u xyxx bx s ∂ ∂ + ∂ ∂ +− ∂ ∂ −= ∂ ∂ + ∂ ∂ + ∂ ∂ )(1)(11 ρρ τ ρ (2.2) y hT hx hT hhy z g y vv x vu t v yyyx by s ∂ ∂ + ∂ ∂ +− ∂ ∂ −= ∂ ∂ + ∂ ∂ + ∂ ∂ )(1)(11 ρρ τ ρ (2.3) in which vu, : depth-averaged velocities t : time yx, : Cartesian coordinates g : gravitational acceleration h : flow depth sz : water surface elevation ( bs zhz += ) bz : bed elevation z x ẕ h w u Figure 2.1 Definition sketch for variables used in depth-averaged model 9 bybx ττ , : bottom shear stresses yyxyxx TTT ,, : effective shear stresses defined as follows: ( ) dzuuu x u h T s b z z xx ∫ ⎥⎦ ⎤⎢⎣ ⎡ −−− ∂ ∂ = 22'21 ρρρυ (2.4) ( )( ) dzvvuuvu x v y u h TT s b z z yxxy ∫ ⎥⎦ ⎤⎢⎣ ⎡ −−−−⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ == ρρρυ ''1 (2.5) ( ) dzvvv y v h T s b z z yy ∫ ⎥⎦ ⎤⎢⎣ ⎡ −−− ∂ ∂ = 22'21 ρρρυ (2.6) where υ : kinematic viscosity; and ',' vu : turbulent velocity fluctuations (Ponce and Yabusaki 1981). Since then, many researchers have developed and applied the 2D depth-averaged models in variety of hydraulic engineering problems as follows: McGuirk and Rodi (1978) studied the side discharge into an open channel and its re-circulating flow field by using a depth-averaged model. Kalkwijk and De Vriend (1980) calculated flow pattern in a shallow bend, in which the depth gradually approaches to zero toward the banks by mean of a depth-averaged model. Ponce and Yabusaki (1981) applied the shallow water model to simulate the horizontal circulation of the flows in channel-pool configuration. Vreugdenhil and Wijbenga (1982) studied the flow pattern in rivers with a depth-averaged model in Cartesian coordinates and introduced the curved river boundary to the model through steps. Tingsanchali and Mahesawaran (1990) presented a mathematical depth-averaged model for prediction of flow pattern around groynes. 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 10 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 ( )ηξ , are introduced as ( )yx,ξξ = , ( )yx,ηη = (Figure 2.2), and then the governing equations (2.1-2.3) are transformed 11 from Cartesian to non-orthogonal curvilinear coordinates, ( ) ( )ηξ ,, →yx , as follows: Continuity equation: 0=⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ J Vh J Uh J h t ηξ Momentum equation: J z J z J gh J VM J UM J M t bxsxsx ρ τ η η ξ ξ ηξ −⎟⎟⎠ ⎞⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ −=⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ ( ) ( ) ( ) ( )hvu J hu J hvu J hu J yxyx '''''' 22 − ∂ ∂ +− ∂ ∂ +− ∂ ∂ +− ∂ ∂ + η η η η ξ ξ ξ ξ (2.7) J z J z J gh J VN J UN J N t bysysy ρ τ η η ξ ξ ηξ −⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ −=⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ ( ) ( ) ( ) ( )hv J hvu J hv J hvu J yxyx 22 '''''' − ∂ ∂ +− ∂ ∂ +− ∂ ∂ +− ∂ ∂ + η η η η ξ ξ ξ ξ (2.8) where UhM = and VhN = VU , : contravariant velocities that are perpendicular to ξη, curvilinear coordinates, respectively ξ y x ξ η η Figure 2.2 Definition of terms in curvilinear system a) Physical mesh b) Computational mesh a) b) 12 vuU yx ξξ += , vuV yx ηη += (2.9) J : Jacobian matrix defined by ( )ξηηξ yxyxJ −= 1 Using the same idea, Younus and Chaudhry (1994) transformed the original depth-averaged equations from the physical coordinates ( )tyx ,, to the computational coordinates ( )t,,ηξ 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 180o bend. 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 13 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. bz to sz 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 14 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). ( ) 01 = ∂ ∂ + ∂ ∂ − s q t hhκ (2.10) 01 0 = ∂ ∂ + ∂ ∂ s E t u g (2.11) where the flow per unit width q and energy head E are ( ) ( )hutsq κ κ −−≡ 1ln, 0 (2.12) ( ) ( ) 220 1 2 cos, −−+++≡ h g u g p htsE h κ ρ θζ (2.13) n -axis (upward normal to the bed) Centre of curvature of the bed profile at P θ θcosh s -axis x z ),(0 tsu ),( hs q Radius of curvature at κ1=P Figure 2.3 Definition sketch by Sivakumaran et al (1983) 15 and the velocity components and pressure are given by ( ) n u tnsu κ− = 1 ,, 0 (2.14) ( ) ( ) ( ) ( )[ ]2220 1121cos,, −− −−−+−+= nhunhgptnsp h κκρθρ (2.15) where κ is the curvature of the bed, s and n are the respective coordinates along and normal to the bed, t is the time; the height ζ above some datum defines the bed profile, and the angle θ measures bed slope, hp is the constant atmospheric pressure (often set equal to zero) and ρ is the fluid density as shown in Figure 2.3 (Simakuvaran 1983). 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 16 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. 17 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 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 curvilinear collocated grid. J. Hydr. Engrg., ASCE, 123(5), pp. 380-388. 29. Younus M. And Chaudhry M. H. 1994. A depth-averaged ε−k turbulence model for the computation of free-surface flow. J. Hydr. Res., Vol. 32(3), pp. 415-444. 19 30. Zarrati A. R., Tamai N. and Jin Y. C. 2005. Mathematical modeling of meandering channels with generalized depth averaged model. J. Hydr. Engrg., ACSE, Vol. 131(6), pp. 467-475. 31. Zhou J. G., Causon D. M., Mingham C. G. and Ingram D. M. 2004. Numerical prediction of dam-break flow in general geometries with complex bed topography. J. Hydr. Engrg., ACSE, Vol. 130(4), pp. 332-340. 32. Zhou J. G., Causon D. M., Ingram D. M. and Mingham C. G. 2002. Numerical solutions of the shallow water equations with discontinuous bed topography. Int. J. Numer. Methods Fluids, Vol. 38, pp. 769-788. 33. Zhou J. G., Causon D. M., Mingham C. G. and Ingram D. M. 2001. The surface gradient method for the treatment of source terms in the shallow water equations. J. Comput. Phys., Vol. 168, pp. 1-25. 34. Zhou J. G. and Goodwill I. M. 1997. A finite volume method for state 2D shallow water flows. Int. J. Numer. Methods Heat Fluid Flow, Vol. 7(1), pp. 4-23. 20 Chapter 3 MATHEMATICAL MODEL 3.1 Coordinate setting A co-ordinate system composed of three axes ( )ζηξ ,, is used to define the three-dimensional flow domain, where ( )ηξ , are set on a 3D arbitrary curved bottom surface as a body fitted coordinates and ζ is a straight line perpendicular to the surface as shown in Figure 3.1. The surface is expressed mathematically by equation ( ) 0,, =zyxζ , which implies that ( ) constzyx =,,ζ defines a co-ordinate surface above the bottom. In order to introduce the notation to be used, we first consider a transformation from the Cartesian coordinates ( )zyxxi ,, to the curvilinear coordinates ( )ζηξξ ,,i . We define the transformation matrix: j i i j x c ∂ ∂ = ξ (3.1) 21 and its inverse j ii j xc ξ∂ ∂ = (3.2) The derivatives of spatial independent variables ( K,,,,,, ζηξζηξ yyyxxx ) are denoted by the following equations: J x yzzy ζηζη ξ − = (3.3a) J x yzzy ξζξζ η − = (3.3b) J x yzzy ηξηξ ζ − = (3.3c) J y zxxz ζηζη ξ − = (3.4a) J y zxzz ξζξζ η − = (3.4b) Arbitrary surface ξ η ζ Figure 3.1 Definition sketch for new generalized coordinate system 22 J zxxz ηξηξζ −=y (3.4c) J z xyyx ζηζη ξ − = (3.5a) x J z xyy ξζξζ η − = (3.5b) J xyyx ηξηξ ζ − =z (3.5c) where: zyx ,, are the Cartesian co-ordinates and the Jacobian of transformation ( J ) is defined as: zyx zyx zyx J ζζζ ηηη ξξξ = (3.6) The transformation relationship between the contravariant velocity vector ),,( WVUV in curvilinear coordinate and its counterpart ( )wvu ,,v in Cartesian coordinate system is ji j i c vV = , or j i j i c Vv = (3.7) 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: ( ) ( ) 0,, =⋅ x,y,zgradzyxgrad ξζ (3.8) ( ) ( ) 0,, =⋅ x,y,zgradzyxgrad ηζ (3.9) or: 0=++ zzyyxx ξζξζξζ (3.10) 0=++ zzyyxx ηζηζηζ (3.11) And condition for ζ axis to remain a straight line with the scale factor equal 1 is described as 23 1222 =++ ζζζ zyx (3.12) Plugging (3.10, 3.11) with (3.1, 3.2), it yields: 0=++ ζηζηζη zzyyxx (3.13) 0=++ ζξζξζξ zzyyxx (3.14) From the definition in Equations (3.3-3.5) we can derive ( ) ( ) ( )ξηηξζξξζηζζη ζηξ zyzyJzyzyJzyzyJ xxx −=−=−= , , (3.15) ( ) ( ) ( )ξηηξζξξζηζζη ζηξ xzxzJxzxzJxzxzJ xyy −=−=−= , , (3.16) ( ) ( ) ( )ξηηξζξξζηζζη ζηξ yxyxJyxyxJyxyxJ zzz −=−=−= , , (3.17) Using Equations (3.15-3.17), it easily gives 0=⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ JJJ iii ζ ζ η η ξ ξ ( )zyxi ,,= (3.18) 3.2 Kinematic boundary condition at the water surface The free water surface (Figure 3.2) at a time t and tt ∆+ are 0),,( =− th ηξζ 0),,( =∆+∆+∆+−∆+ tth ηηξξζζ Figure 3.2 Kinetic boundary condition at water surface h ζ ξ 24 thus η η ξ ξζ ∂ ∂ ∆ ∆ + ∂ ∂ ∆ ∆ + ∂ ∂ = ∆ ∆ h t h tt h t (3.19) where szsysxszyx Uwvut z t y t x t =++= ∆ ∆ + ∆ ∆ + ∆ ∆ = ∆ ∆ ξξξξξξξ (3.20) and similarly, sVt = ∆ ∆η (3.21) sWt = ∆ ∆ζ ỗ ỗ ỗ (3.22) in which ),,( sss WVU are contravariant components of velocity vector at free water surface. Applying Equations (3.20-22) into (3.19), the kinetic boundary condition at the free surface is obtained ηξ ∂ ∂ + ∂ ∂ + ∂ ∂ = hVhU t hW sss (3.23) 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: 0=⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ J W J V J U ζηξ (3.24) Momentum equation: 25 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ + + + ∂ ∂ + ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ + + + ∂ ∂ + ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ + + + ∂ ∂ + ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ∂ ∂ JpWw JpWv JpWu JpVw JpVv JpVu JpUw JpUv JpUu Jw Jv Ju t z y x z y x z y x ρζ ρζ ρζ ζ ρη ρη ρη η ρξ ρξ ρξ ξ ( ) ( ) ( ) ( ) ( ) ( ) ⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ++ ++ ++ ∂ ∂ + ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ++ ++ ++ ∂ ∂ + ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ = J J J J J J Jg Jg Jg zzzzyyzxx yzzyyyyxx xzzxyyxxx zzzzyyzxx yzzyyyyxx xzzxyyxxx z y x ρτητητη ρτητητη ρτητητη η ρτξτξτξ ρτξτξτξ ρτξτξτξ ξ ( ) ( ) ( ) ⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ++ ++ ++ ∂ ∂ + J J J zzzzyyzxx yzzyyyyxx xzzxyyxxx ρτζτζτζ ρτζτζτζ ρτζτζτζ ζ (3.25) 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 0J . Other flow quantities such as U and V are uniform in 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: 01 000 = ∂ ∂ + ∂ ∂ + ∂ ∂ J N J M t h J ηξ (3.26) where UhM = and VhN = . 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 ( ξηξξξζξξηξξξζηξ Γ+Γ+Γ+Γ+⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ VUUWUVU JJ WU J VU J U J U t 2 2 1 ) ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +=Γ+Γ+Γ+Γ+Γ+ J WWVWUVWV ρ τ ξ ξξξξ ζζ ξ ζη ξ ζξ ξ ηζ ξ ηη term)(pressure - J G 22 26 ( ξηζηζξηηηηξηξηξξξζξζξξηξηξξξξξξζξη ττττττρρτζρτη Γ+Γ+Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + JJJ 1 )ξζζζζξζηζηξζξζξ τττ Γ+Γ+Γ+ (3.27) where ( ) ( ) ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +++⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ++= ρη ηξηξηξ ρξξξξξξξ p J p zzyyxxzzyyxx 1 J 1 term)(pressure ( ) ⎭⎬ ⎫ ⎩⎨ ⎧ ⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +++ JJJ pp J xxx xzzyyxx ζ ζ η η ξ ξρξρζζξζξζξ 1 ⎭⎬ ⎫ ⎩⎨ ⎧ ⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎟⎠ ⎞⎜⎝ ⎛ ∂ ∂ +⎪⎭ ⎪⎬⎫⎪⎩ ⎪⎨⎧ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + JJJ p JJJ p zzz z yyy y ζ ζ η η ξ ξρξ ζ ζ η η ξ ξρξ (3.28) Applying the perpendicular condition (Equations 3.10, 3.11) and relation in Equation (3.18) to Equation (3.28) gives: ( ) ( ) ⎟⎟⎠ ⎞⎜⎜⎝ ⎛ ∂ ∂ +++⎟⎟⎠ ⎞⎜⎜⎝ ⎛ ∂ ∂ ++= ρη ηξηξηξ ρξξξξ p J p zzyyxxzyx 1 J 1 term)(pressure 222 (3.29) 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: ( )ξηηξηξξξηξξξηξ 0200020000 1 Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ NNMMNM hJJ VM J UM J M t 000 000000 00 2 0 2 0 2 0 0 J dp J dp J G J h b h zzyyxx h zyx ρ τζ ρη ηξηξηξζ ρξ ξξξ ξξ − ∂ ∂++ − ∂ ∂++ −= ∫∫ ( ) ∫∫ ∂∂+∂∂+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ++− hh b zzyyxx dJ d J p J 0000 000000 0 111 ζ ρ τ η ζ ρ τ ξρζξζξζξ ξηξξ (3.30) 27 Using the same procedures the η -component of depth-averaged momentum equation is derived: ( )ηηηηηξηξηηξξηξ 0200020000 1 Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ NNMMNM hJJ VN J UN J N t 000 000000 00 2 0 2 0 2 0 0 J dp J dp J G J h b h zzyyxx h zyx ρ τζ ρξ ηξηξηξζ ρη ηηη ηη − ∂ ∂++ − ∂ ∂++ −= ∫∫ ( ) ∫∫ ∂∂+∂∂+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ++− hh b zzyyxx dJ d J p J 0000 000000 0 111 ζ ρ τ η ζ ρ τ ξρζηζηζη ηηηξ (3.31) where ξG , ηG , ξτ b and ητ b are the contravariant components of gravitational vector and shear stress acting on the bottom surface; ijkΓ ỗ is the Riemann-Christoffel symbols: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ∂ ∂ ∂ ∂ −=Γ m i kj m i jk x x ξ ξξ , 1,2,3)(m = (3.32) Substituting the conditions of orthogonality (Equations 3.10 and 3.11) into Equations (3.30, 3.31), and neglecting the effect of Reynold stress gives: ( )ξηηξηξξξηξξξηξ 0200020000 1 Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ NNMMNM hJJ VM J UM J M t 000 000000 00 2 0 2 0 2 0 0 J dp J dp J G J h b h zzyyxx h zyx ρ τζ ρη ηξηξηξζ ρξ ξξξ ξξ − ∂ ∂++ − ∂ ∂++ −= ∫∫ (3.33) ( )ηηηηηξηξηηξξηξ 0200020000 1 Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ NNMMNM hJJ VN J UN J N t 000 000000 00 2 0 2 0 2 0 0 J dp J dp J G J h b h zzyyxx h zyx ρ τζ ρξ ηξηξηξζ ρη ηηη ηη − ∂ ∂++ − ∂ ∂++ −= ∫∫ (3.34) 28 Pressure term in the momentum equation in ζ -direction is ( ) ( ) ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +++⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ++= ρη ζηζηζη ρξζξζξζξ pp zzyyxxzzyyxx J 1 J 1 term)(pressure ( ) ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +++ ρη ζζζ p J zyx 2221 Applying the orthogonal conditions, it is reduced as: ( ) ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ++= ρη ζζζ p J zyx 2221 term)(pressure To estimate the pressure terms in Equations (3.33), (3.34), the momentum equation in ζ -direction of Equation (3.25) is considered. If the acceleration and shear stress components are negligible, the equation can be reduced as: ( ) ρζ ζζζζζ ηη ζ ηξ ζ ξη ζ ξξ p JJ GVVVUUVUU J zzx ∂ ∂++ −=Γ+Γ+Γ+Γ 0 222 00 1 (3.35) therefore, ( )[ ]ζζηηζηξζξηζξξζζζρζ GVVVUUVUU p zzx +Γ+Γ+Γ+Γ− ++ = ∂ ∂ 222 1 ζ is a straight axis consequently, 1222 =++ zzx ζζζ , ( ) ζζηηζηξζξηζξξρζ GVVVUUVUUp +Γ+Γ+Γ+Γ−=∂∂ (3.36) The expression for pressure distribution is obtained by integrating Equation (3.36) with respect to ζ . It follows that: { } 2 2 0 0 000 hGVVUVVUUUdp h ζζ ηη ζ ξη ζ ξη ζ ξξζρ −Γ+Γ+Γ+Γ=∫ (3.37) Equations (3.26, 3.33, 3.34) and (3.37) are the depth-averaged equations in a general form used in calculating water surface profile of flows in the following applications. 29 Notations t : time zyx ,, : Cartesian coordinates ζηξ ,, : generalized curvilinear coordinates K,,,,,, zyxzyx ηηηξξξ : components of transformation matrix J : transformation Jacobian ( )wvu ,, : velocity components in Cartesian coordinate ( )WVU ,, : contravariant components of velocity vector ( )sss WVU ,, : contravariant components of velocity vector at water surface h : water depth in ζ -direction p : pressure ρ : water density g : gravitational acceleration ζηξ GGG ,, : contravariant components of gravitational vector Kyyyxxzxyxx τττττ ,,,, : effective shear stress ηξ ττ bb , : contravariant of shear stress acting on the bottom i jkΓ : Christoffel symbol 30 Chapter 4 STEADY ANALYSIS OF WATER SURFACE PROFILE OF FLOWS WITH AIR-CORE VORTEX AT VERTICAL INTAKE 4.1 Introduction Air-core vortex formation at intakes is a significant hydraulic engineering problem in many situations such as intakes for irrigation, drainage system, hydropower generation in which the water is normally drawn from rivers, channels or reservoir,… as well as in water art-works or sculptures. It occurs typically whenever the submergence is less than a critical value and causes some detrimental effects as reduction in intake discharge, resulting vibrations and noises as well as operational difficulties. Figure 4.1 shows an example of free surface air-core vortex occurred in laboratory. When the flow in a large body of slowly moving water is diverted and locally accelerated or drawn off, any associated vortex tube is extended and its rotation is thereby increased. Higher velocities incur lower pressures and, if a free surface exists, it becomes locally 31 Figure 4.1 An example of free surface air-vortex Figure 4.2 Various stages of development of air-entraining vortex: S1>S2>S3>S4 (Jain et al., 1978) Q Q S1 S2 S4 Q S3 a. No depression on the surface at large submergence b. Formation of a dimple d. Air-entraining vortexc. Air core extends deeper as the vortex becomes stronger 32 depressed. Thus in hydraulic structures, where flow negotiates a vertical shaft, submerged conduit or gated outlet, the vortex may be sufficiently intense to form a hollow core, which could transmit air. This may lead to the undesirable effects already mentioned, and considerable efforts have been made to predict and control such phenomena. Indeed, the deliberated introduction of a vortex at the vertical intake has been described in Jain et al. (1978). Figure 4.2 shows the various stages in the development of an air-entraining vortex when the water depth is decreased gradually. At the first stage, when the submergence is large, there is no depression on the surface (Figure 4.2a), but together with decreasing water depth, a dimple forms as in Figure 4.2b. If the submergence is further decreased the air-core vortex occurs (Figure 4.2c) and Figure 4.2d shows critical condition under which a vortex just tends to entrain air (Jain et al. 1978). Suppose that steady flow occurs in the horizontal yx − plane and that closed path, S , is traced out in the fluid (Figure 4.3). The path encloses an area A by linking adjacent flow particles and subsequently moves with the flow. At a point on the path, one particle may have velocity components nV and tV normal and tangential to the path. For the path to remain unbroken and assuming the flow is incompressible, the sum ∑ ∆SVn is the net inflow into the closed area A and must always be zero. The corresponding sum along the path ∑ ∆SVt is not necessarily zero and is known as the circulation Γ (being defined as positive anticlockwise). The Kenvin’s theorem states that the circulation remains constant with time unless an external shear stress exists along S , as described in Townson (1991). 33 Figure 4.3 The inflow to and circulation round a closed path in a flow field (Townson 1991) Figure 4.4 The concept of simple Rankine vortex including two parts: free vortex in outer zone and forced vortex in inner zone (Townson 1991) S A nV S∆ tV r ∞→r 0→tV H h H∆ free vortex forced vortex ideal fluid real fluid 34 If we consider an infinitely wide reservoir of constant depth is drawn off at a central point with rate q per unit depth. At some distance from the center, a circular vortex tube may be defined with circulation Γ about a vertical axis through the drawn-off point. As the tube contracts to radius r under the influence of drawn-off, the tangential velocity rVt π2/Γ= . The radial velocity is rqVr π2/= , and both clearly accelerate, producing a spiral flow towards the center. Applying the Kelvin’s theorem, the circulation is conserved therefore the velocity at the center is infinite. If the pressures remain hydrostatic, the depression of the free surface is also infinite. But the viscosity present in real fluids prevents this condition arising, and a zone in center of the flow rotates as a solid mass (Townson 1991). This central part is known as a forced vortex and the composite system as Rankine’s vortex (Figure 4.4). Several approaches have been presented in the literature to deal with the problem of determination and prediction of critical submergence serving in design works. These approaches basically can be labeled as analytical models (Jain 1984; Odgaard 1986; Hite and Mih 1994) and physical models (Anwar et al. 1978; Jain et al. 1978; Yildirim and Kocabas 1995; Yildirim and Kocabas 1997; Yildirim and Kocabas 1998). Many analytical attempts have been presented in the literature in order to attain a theoretical view of the far-field velocity; in fact the flow representation has not been defined so far by any comprehensive analytical analysis. The concept of simple Rankine vortex normally used in the basic equations (Odgaard 1986; Hite and Mih 1994), therefore this approach obviously could not be applied for the case of air-entraining vortex and moreover, the Kelvin’s theorem is invalid for the central region. Trivellato et al. (1999) set the water surface equal to the stationary headwater while other experimental works only focused on the critical submergence. Consequently, these approaches could not be used to predict the water surface profile of flow with air-core 35 vortex. In this study, the water surface profile of a steady air core vortex flow into a vertical intake was derived through out a depth-averaged model of open channel flows over the 3-D curvilinear bottom using a generalized and body fitted coordinate system. The assumption of fully free air-core vortex in the new coordinate allowed us to use the Kelvin’s theorem of the conservation of circulation for the whole flow field. The vortex was assumed axisymmetric and steady. The assumption of shallow water and kinetic boundary condition at water surface were also used. The equation describing water surface profile was derived and calculated results were compared to the formula introduced by Orgaard (1986). The application’s results showed us the ability of the model in analyzing the water surface profile and can be improved to simulate the flow structure of an air-core vortex. 4.2 Governing equations Coordinate setting To consider the vortex occurring at a cavity on the bottom surface (Figure 4.5), the position of any point say P , is defined by three coordinates ( )ζηξ ,, where ( )ηξ , define the position of 0P (projection of P ) on the bottom, and ζ is the distance from point P to that bottom surface. Assuming that the shape of bottom surface (i.e at 0=ζ ) has the form of )( 0 0 ar bz − − = , (4.1) where a and b are the coefficients which define the shape of the bottom; 0r is the distance from any point on the bottom to the z -axis 2 0 2 00 yxr += (4.2) 36 Figure 4.5 Definition of coordinate components x r y η a O ξ ζ P P0 r x ad 2= aa− • z l • 37 Based on Figure 4.5, we can derive the relations ηcos00 rx = and ηsin00 ry −= . Then, the bottom surface is expressed by the following equation: 0)(),,( 00000 =+−=Φ bzarzyx (4.3) The unit vector )(n normal to the bottom surface is derived as: k arz ar j r zy arz i r zx arzgrad gradn rrrr 2 0 2 0 0 0 00 2 0 2 00 00 2 0 2 0 )()( 1 )( 1 −+ − + −+ + −+ = Φ Φ = At ζ the relations between ),,( zyx and ),,( ζηξ can be expressed as: ζηηζ 24 0 0 0 00 2 0 2 0 0 )( coscos )( 1 bar br r zx arz xx +− −= −+ += (4.4a) ζηηζ 24 0 0 0 00 2 0 2 0 0 )( sinsin )( 1 bar br r zy arz yy +− +−= −+ += (4.4b) ζζ 24 0 2 0 020 2 0 0 0 )( )( )()( bar ar ar b arz ar zz +− − + − −= −+ − += (4.4c) Then the covariant base vector components on the bottom surface are denoted as follows: k arp bj p i p kzjyixe 2 0111 000 )( sincos − −+−=++= ηη ξξξξ rv (4.5a) jrirkzjyixe ηηηηη ηη cossin 00000 −−=++= rr (4.5b) j bar bi bar bkzjyixe rrr 24 0 24 0 000 )( sin )( cos +− + +− −=++= ηη ζζζζ k bar ar r 24 0 2 0 )( )( +− − + (4.5c) Let’s denote that 2 0 2 0 22 0 2 0 00 2 0 1 )(2)()( )()( barlbarlarr brarbarl p +−+−+− +−+− = (4.6) 38 in which l is defined in Figure 4.5. From Figure 4.5, it could be easily seen that ( ) ( )arr barl r ar bl r zl − +− = − + = − = 00 0 0 0 0 0tanξ (4.7) then ( ) ( ) 2020 00 2 0 00 0 00 )( )()(tan arr brarbarl arr barl rr − +−+− −=⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − +− ∂ ∂ = ∂ ∂ ξ (4.8) From (4.6) we can obtain: ( ) ( ) ( ) ( )2020 2 0 2 0 22 0 2 02 2 2 tan1 cos 1tan arr barlbarlarr − +−+−+− =+== ∂ ∂ ξξξ ξ (4.9) but 0 00 0 tan tan tantan r rr r ∂ ∂ ∂ ∂ = ∂ ∂⇒ ∂ ∂ ⋅ ∂ ∂ = ∂ ∂ ξ ξ ξ ξξ ξ ξ ξ (4.10) Thus, substituting (4.8) and (4.9) into (4.10) gives 100 2 0 2 0 2 0 22 0 2 00 1 )()( )(2)()( pbrarbarl barlbarlarrr −= +−+− +−+−+− −= ∂ ∂ ξ (4.11) From (4.4a): ( ) ( )[ ] ζ ηη 2324 0 3 0 0 cos2 cos bar arb r x +− − += ∂ ∂ (4.12) Using the relations in (4.11), (4.12), it can be obtained: ( ) ( )[ ] ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −⋅⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ +− − += ∂ ∂ = 1 2324 0 3 0 1cos2cos pbar arbxx ζηηξξ ( ) ( )[ ] ζ ηη ⋅ +− − −−= 2324 01 3 0 1 cos2cos barp arb p (4.13) 39 Similarly, ( ) ( )[ ] ζ ηη ξ ⋅ +− − += 2324 01 3 0 1 sin2sin barp arb p y (4.14) ( ) ( ) ( )[ ] ζξ ⋅+− − − − −= 2324 01 0 2 2 01 2 barp arb arp bz (4.15) and ( ) ζ ηηη ⋅ +− +−= 24 0 0 sinsin bar brx (4.16) ( ) ζ ηηη ⋅ +− +−= 24 0 0 coscos bar bry (4.17) 0=ηz (4.18) ( ) 240 cos bar bx +− −= η ζ (4.19) ( ) 240 sin bar by +− = η ζ (4.20) ( ) ( ) 240 2 0 bar ar z +− − =ζ (4.21) and the Jacobian right on the bottom surface is: 2 0 24 00 100 )( )(1 11 000 000 000 ar barr p zzz yyy xxx JJ − +− === = ζηξ ζηξ ζηξ ζ (4.22) The gravitational vector can be expressed in the new coordinate system as follows: ζ ζ η η ξ ξ eGeGeGG ++= (4.23) Substituting the base vector components from (4.5) into (4.23) 40 ( ) ( ) ( ) kzjyixGkzjyixGkzjyixGG ζζζζηηηηξξξξ ++++++++= ( ) ( ) ( ) kzGzGzGjyGyGyGixGxGxG ζζηηξξζζηηξξζζηηξξ ++++++++= but kgG −= consequently, gG G G zzz yyy xxx − =⋅ 0 0 ξ η ξ ζηξ ζηξ ζηξ (4.24) hence, the contravariant components of gravitational vector are derived ( ) ( )( ) 240 2 0 1 bar ar bgpyxyxgJG +− − =−−= ηζζη ξ (4.25) ( ) 0=−−= ξζζξη yxyxgJG (4.26) ( ) ( )( ) 240 2 0 bar ar gyxyxgJG +− − −=−−= ξηηξ ξ (4.27) Using Equations (4.13-4.22) in plugging with (3.15-3.17), it gives: ( ) ( ) 240 4 0 1 cos bar ar px +− − −= ηξ (4.28) ( ) ( ) 240 4 0 1 sin bar ar py +− − = ηξ (4.29) ( ) ( ) 240 2 0 bar arb z +− − −=ξ (4.30) 0 sin rx ηη −= (4.31) 0 cos ry ηη −= (4.32) 41 0=zη (4.33) ( ) 240 cos bar b x +− −= ηζ (4.34) ( ) 240 sin bar b x +− = ηζ (4.35) ( ) ( ) 240 2 0 bar ar z +− − =ζ (4.36) Equation of water surface profile Dealing with the flows at a vertical intake as in Figure 4.5, the depth-averaged equations (3.26, 3.33 and 3.34) are applied. With the assumptions of steady vortex ( 0= ∂ ∂ t ) and axisymmetric flow, i.e. the flow is homogenous in η -direction ( 0= ∂ ∂ η ), those equations become: Continuity equation: 0 0 = J M d d ξ or .00 constQJ M == (4.37) Momentum equations: ξ -direction: ( ) 00 0 2 000 2 00 1 J G J hNNMMNM hJJ UM d d b ρ τ ξ ξξξ ηη ξ ηξ ξ ξη ξ ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ({ ) ⎭⎬ ⎫ −Γ+Γ+Γ+Γ ++ − 2 2 0000 0 2 0 2 0 2 0 hGVVUVVUUU d d J zyx ζζ ηη ζ ξη ζ ξη ζ ξξξ ξξξ (4.38) η -direction: ( ) 00 0 2 000 2 00 1 J G J hNNMMNM hJJ UN d d b ρ τ ξ η ηη ηη η ηξ η ξη η ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ 42 ({ ) ⎭⎬ ⎫ −Γ+Γ+Γ+Γ ++ − 2 2 0000 0 000000 hGVVUVVUUU d d J zzyyxx ζζ ηη ζ ξη ζ ξη ζ ξξξ ηξηξηξ (4.39) Using the definition in Equation (3.32) in plugging with Equations (4.28 – 4.36), the Christoffel symbols are hereafter calculated: ( ) ( )[ ]24001 2 0 1 2 1 0 21 bararp b dr dp p zyx zyx +−− += ∂ ∂ − ∂ ∂ − ∂ ∂ −=Γ ξ ξ ξ ξ ξ ξ ξξξ ξ ξξ (4.40) 000 =Γ=Γ ξ ηξ ξ ξη (4.41) ( ) ( ) 240 4 00 10 bar arr p +− − =Γ ξηη (4.42) 000 =Γ=Γ η ηη η ξξ (4.43) 01 00 1 rp −=Γ=Γ ηηξ η ξη (4.44) ( ) ( ) 2400210 2 bararp b +−− −=Γ ζξξ (4.45) 000 =Γ=Γ ζ ηξ ζ ξη (4.46) ( ) 240 0 0 bar br +− −=Γ ζηη (4.47) ( ) ( ) 240 4 0 2 12 0 2 0 2 0 bar arp zyx +− − =++ ξξξ (4.48) 2 0 2 0 2 0 2 0 1 rzyx =++ ηηη (4.49) ξ and η introduced in Figure (4.5) are obviously perpendicular with each other, thus it is given that 0000000 =++ zzyyxx ξηξηξη 43 Therefore, Equation (4.39) becomes: ( ) 00 0 2 000 2 00 1 J G J hNNMMNM hJJ UN d d b ρ τ ξ η ηη ηη η ηξ η ξη η ξξ −=Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ (4.50) By substituting Equations (4.26, 4.43-4.44) and neglecting the effect of friction in η -direction, Equation (4.50) is reduced as: 0 2 0100 =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ rphJ MN J MV d d ξ (4.51) Applying Equation (4.37) into (4.51) gives ( ) 02 01 00 =⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ −+ rph NQV d dQ ξ or 0 0 01 2 2 r dr V dV rp V d dV −=⇒=ξ (4.52) Integrating Equation (4.52) between two arbitrary points A and B with distance Ar0 and 0Br respectively from z -axis, the following relation is obtained: 2 0 2 0 0 02 B A A B B A B A r r V V r dr V dV =⇒−= ∫∫ (4.53) where AV and BV are the contravariant components of velocity vector at A and B respectively. Based on equation (4.53), the depth averaged tangential velocity component at 0r is defined as: BBAA B A A A B B rvrv r r r v r v VreVv 002 0 2 0 0 0 0 =⇒=⇒== ηr (4.54) Eq. (4.54) is identical to the free vortex condition which allows us to assume the conservation of circulation vr02π=Γ for the whole flow regime, thus Vr 2 02π=Γ and 44 from which: 2 02 r V π Γ = (4.55) The evaluation of the contravariant component of shear stress acting on the bottom is approximated using the following relation: 2 0 2 0 0 2 hr Qf J UefUf bb =⇒== ρ τ ρ τ ξ ξ ξ rrV (4.56) Using the relations described in Equations (4.40-4.49) and (4.45-4.46), Equation (4.38) becomes ρ τ ξ ξ ξξ ξξ 00 0 0 2 0 2 J G J h hJ M hJ M b −=Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ( ) ⎥⎦ ⎤⎢⎣ ⎡ −Γ+Γ ∂ ∂ ++− 222 1 2 0 2 0 2 2 0 2 0 2 0 0 hGNM J zyx ζζ ηη ζ ξξξξξξ (4.57) Substituting (4.37) into (4.57) gives: ρ τ ξ ξ ξξ ξξ 00 0 0 2 0 0 2 0 1 J G J h h JQ hJ Q b−=Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ ( ) [ ]ζξξξξξξ 02020200 2 0 2 Γ ∂ ∂ ++− zyxJ Q ( ) ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ Γ ∂ ∂ ++ Γ − ζ ηηξξξξπ 040 2 0 2 0 2 0 0 2 2 8 rJ zyx ( ) [ ]2202020 02 1 hG J zyx ζ ξξξξ ∂ ∂ +++ Then, using the relations (4.56) and after some manipulation, we obtain the equation for water surface profile in the form of ),( ),( 021 01 rhfp rhf d dh =ξ (4.58) in which 45 -0.1 0.0 0.1 0.2 x (m) -0.2 0.0 0.2 z (m ) Water surface Critical depth line Quasi-normal depth line Bottom Figure 4.6 An example of computed water surface profile withỗ quasi-normal depth line and critical depth line 46 [ ] [ ] [ ] 4 2324 0 2 0 3 00 24 0 2 2 2324 0 0 2 02 01 )( )(2)(3 8 )( )( ),( h barr arrbar b bar arr gbrhf ⎪⎭ ⎪⎬ ⎫ +− −++−Γ − ⎪⎩ ⎪⎨ ⎧ +− − = π [ ] [ ] [ ] 2 2524 0 24 0 2 02 02324 00 3 02 0 )( )()(3 )( )(2 h bar barar bQ barr ar bQ ⎪⎭ ⎪⎬ ⎫ +− −−− + ⎪⎩ ⎪⎨ ⎧ +− − + [ ] 2 0 2124 02 0 0 2 03 0 2 2 2 0 2 0 )( )( .- 1 4)( gb ar bar Qfh r Q h rar r − +− −⎪⎭ ⎪⎬⎫⎪⎩ ⎪⎨⎧ Γ− − + π (4.59) and [ ] [ ] 2 0 3 2124 00 2 2 3 2124 0 2 0 2 0 02 )( 1 4)( )( ),( Qbh barr gh bar arr rhf − +− Γ + +− − = π (4.60) Method of Calculation For convenience of simulating the water surface profile, the transformation of special coordinates from ξ to 0r is needed. Substitute Equation (4.11) into (4.58) it follows that ),( ),( ),( ),(1. 02 01 0021 01 01 0 0 rhf rhf dr dh rhfp rhf dr dh pd dr dr dh d dh −=⇒=−== ξξ (4.61) A common method of analysis including singular point often used in hydraulic engineering is applied. The singular point is the point at which both functions ),( 01 rhf and ),( 02 rhf in Equation (4.58) are equal to zero. The equation 0),( 01 =rhf expresses the quasi-normal depth line whereas 0),( 02 =rhf expresses critical depth line as shown in Figure 4.6. The water surface profile is derived from Equation (4.58) by using the fourth-order Runge-Kutta scheme with the initial slope near the singular point defined by the following equation: 47 S SSSSSS S h f r f h f h f r f h f r f dr dh ∂ ∂ ∂ ∂ ∂ ∂ −⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ ±⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ + ∂ ∂ − = 2 0 12 2 1 0 21 0 2 0 2 4 (4.62) (The subscript “s” denotes the derivatives at singular point) 4.3 Results and discussions Posey and Hsu (1950) and Jain et al. (1978) have reported that a large reduction in intake discharge was due to formation of vortices, especially in case of flow entering the intake with entrapped air. The model herein derived has been applied with different imposed circulations (i.e. different strengths of vortex) and the computed results show reduction of discharge when the circulation increases while keeping the same submergence (water head) (Figure 4.7). It is observed that the larger the circulation the larger the air core of the vortex extends into the intake, and in case of air coming into intake the intake discharge decrease significantly (from sm331031.0 −ì , in case TC01 to sm331006.0 −ì , in case TC04 as in Table 4.1). This phenomenon is in agreement with the previous experiments (Jain et al. 1978; Posey and Hsu 1950). Intake discharge is inversely proportional to circulation as shown in Figure 4.8 where the circle marks are the calculated results and solid line is the trend line calculated by the least squares method. The results also show that for small decrease in submergence, the air core becomes larger (Figure 4.9). Similar results were reported by Anwar et al. (1978). In Figure 4.9, it is noted that when the circulation increases, the submergence also would increase to maintain the same intake discharge (see Table 4.2 for reference). Based on these results, it is evident that the model is capable of representing the effects of circulation on the flow through vertical intake. 48 Table 4.1 Parameters used in the calculations of results in figure 4.7 Case a (m) b (m2) Q0ỗ (10-3m3/s) Γ (m2/s) f TC 01 0.025 10-4 0.310 0.182 0.015 TC 02 0.025 10-4 0.230 0.200 0.015 TC 03 0.025 10-4 0.135 0.225 0.015 TC 04 0.025 10-4 0.060 0.250 0.015 -0.05 0.00 0.05 0.10 x (m) -0.05 0.00 0.05 0.10 0.15 z (m ) +++++++++ + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + TC 01 TC 02 TC 03 TC 04 Bottom Figure 4.7 The effect of circulation on water surface profile andỗ discharge at the intake with same water head 49 Table 4.2 Parameters used in the calculations of results in figure 4.9 Case a (m) b (m2) Q0ỗ (10-3m3/s) Γ (m2/s) f TC 12 0.02 10-4 0.15 0.30 0.015 TC 13 0.02 10-4 0.15 0.25 0.015 TC 14 0.02 10-4 0.15 0.20 0.015 0.3 0.4 0.5 0.6 0 0.001 0.002 0.003 0.004 Γ (m2/s) Q (m 3/ s) Figure 4.8 Variation of intake discharge with circulation (a=0.025m, b=10-5 m2, water head=0.5m) -0.1 0.0 0.1 0.2 x (m) -0.1 0.0 0.1 0.2 z (m ) TC 12 TC 13 TC 14 Bottom Figure 4.9 Different water surface profiles with different values of circulationỗ while maintaining the constant intake discharge 50 The effects of the shape of intake have been examined by changing the value of ''b in equation (4.1). In this case, three simulations were considered with values of 54 10 5 ;10 −− ì=b and 2510 m− while maintaining the radius of intake (i.e value of ''a in Equation 4.1) at m2102 −ì . It can be seen from Figure 4.10 that the water depth increases when the intake entrance becomes sharper (decreasing value of b as shown in Table 4.3). For more clarity, the computation has been done with the same intake and circulation to test the effects of value of b on intake discharge without changing the submergence (Figure 4.11a) and the effects on submergence with same discharge (Figure 4.11b). This phenomenon is consistent with the physical meaning of the coefficient ''b and it also improves the performance of bell-mouth intake. So far we have been able to verify the model qualitatively based on the fact that no previous studies are available which describe the variation in water surface profile at the intake. However, a larger quantity of experimental data has been used by Odgaard (1986) to verify the equation representing the critical submergence, which is defined as the submergence when the tip of air-core vortex just reaches the intake (Figure 4.12), in the absence of surface tension (Eq. 18 in Odgaard, 1986) presented as follows: [ ] 5.05.0)(6.5 2.23 FrN R d S e Γ= (4.63) where circulation function 0)( QSN Γ=Γ , Froude number ( ) 21gdVFr = and Reynold number dQ υ=Re . This equation can be applied for the case of vortex with turbulent core, in which υ is replaced by Γ+ kυ . From calibration the value of k was then estimated to be 5106 −ì≈k . 51 Table 4.3 Parameters used in the calculations of results in figure 4.10 Case a (m) b (m2) Q0ỗ (10-3m3/s) Γ (m2/s) f TC 15 0.025 1.x10-4 0.3 0.225 0.015 TC 16 0.025 5.x10-5 0.3 0.225 0.015 TC 17 0.025 1.x10-5 0.3 0.225 0.015 0.00000 0.00006 0.00012 b (m2) 0.0000 0.0005 0.0010 0.0015 0.0020 Q (m 3/ s) 0.00000 0.00004 0.00008 0.00012 b (m2) 0 5 10 15 20 25 H /d Figure 4.11 The effects of b on discharge (17a) and submergence (17b) at an intake -0.1 0.0 0.1 0.2 -0.10 0.00 0.10 0.20 z (m ) TC 15 TC 16 TC 17 Bottom Figure 4.10 Changing of water surface profile with different shape of the intake 52 x z S d Figure 4.12 Definition sketch of critical submergence 100 101 102 Computed Critical Submergence (S/d) 100 101 102 O dg aa rd 's C rit ic al S ub m er ge nc e (S /d ) Figure 4.13 Comparison of computed critical submergence by the model (Equation 4.58) and by Odgaard’s equation (4.63) 53 For qualitative verification, the model is applied to predict the submergence at the intake, and then comparisons are made with the Odgaard’s equation. A total of 12 simulations were made with 2510 mb −= and the range of intake radius between 01.0 and m2.0 . Comparison of computed critical submergences and those computed using Odgaard’s equation is shown in Figure 4.13. The horizontal coordinate is the computed critical submergences by the present model and vertical coordinate is by Odgaard’s. The perfect fit between the model and Odgaard’s equation is on the 045 line. It can be seen that there exists a fairly good agreement. These comparisons show that the model can be used to predict critical submergence of a vertical intake. 0 0.04 0.08 0.12 0.16 0.2 0.24 d (m) 0 2 4 6 8 10 12 C rit ic al S ub m er ge nc e (S /d ) + + + + + + + + + + x x x x x x x x x x b=0.000005 b=0.00001 + x Figure 4.14 The variation of critical submergence wit different values of b To verify whether the comparisons above are independent of the effect of intake shape, in this case the parameter ''b , the variation of critical submergence with different value of b while both the intake’s diameter and discharge are kept constant are shown in Figure 4.14. It is observed that the critical submergence is not significantly affected by a small 54 change of b with order of 2510 m− . Based on this observation we strongly believe that the effect of ''b on the critical submergence is negligible when ''b is small enough. 4.4 Summary In this application, a theoretical model based on the depth-averaged and momentum equations has been developed to calculate the variation of water depth of flow with air-core vortex at an intake. Hence, the water surface profile can be directly derived from the model. The use of curvilinear coordinate avoids invalidation of the Kelvin’s theorem in the core of vortex, therefore the conservation of circulation can be applied for whole flow. The comparisons of calculated data using the model and an empirical equation show that the proposed model yields reliable results in predicting the critical submergence of the intake without any limitation of Froude number, a problem that most of existing models cannot escape. 4.5 References 1. Anwar O. H., Weller A. J. and Amphlett B. M. 1978. Similarity of free-vortex at horizontal intake. J. Hydr. Res., Vol. 16 (2). pp. 95-105. 2. Hite J. E. and Mih C. W. 1994. Velocity of air core vortices at hydraulic intakes. J. Hydr. Eng. ASCE, Vol. 120(3), pp.284-297. 3. Hosoda T. 2003. Depth averaged model of open channel flows over an arbitrary surface. Proceeding of the International symposium on Shallow flows, Part II, June 16-18, pp. 267-271. 4. Jain A. K., Raju K. G. R. and Garde R. J. 1978. Vortex formation at vertical pipe intake. J. Hydr. Div. ASCE, Vol. 104 (10), pp.1429-1445. 5. Jain S. C. 1984. Tangential vortex-inlet. J. Hydr. Eng. ASCE, Vol. 110(12), 55 pp.1693-1699. 6. Odgaard A. J. 1986. Free-surface air core vortex. J. Hydr. Eng. ASCE, Vol. 112(7), pp.610-620. 7. Posey C. J. and Hsu H. 1950. How the vortex affects orifice discharge. Engineering News Record, Vol. 144, p. 30. 8. 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. 9. Townson J. M. 1991. Free-surface hydraulics. London Unwin Hyman. 10. Trivellato F., Bertolazzi abd Firmani B. 1999. Finite volume modeling of free surface draining vortices. J. Comp. App. Mech., Vol. 103, pp. 175-185. 11. Yıldırım N. and Kocabas F. 1995. Critical submergence for intakes in open channel flow. J. Hydr. Eng., ASCE, Vol. 121(12), pp 900–905. 12. Yıldırım N. and Kocabas F. 1997. Closure to ‘Critical submergence for intakes in open channel flow. J. Hydr. Eng., ASCE, Vol. 123(6), pp 589–590. 13. Yıldırım N. and Kocabas F. 1998. Critical submergence for intakes in still-water reservoir. J. Hydr. Eng., ASCE, Vol. 124(1), pp. 103–104. 56 Chapter 5 UNSTEADY PLANE-2D ANALYSIS OF FLOWS WITH AIR-CORE VORTEX In most of the studies in the past, because of the complexity of 2D equation the variation of the flow quantities in the tangential direction at the same radius from the center of the vortex was neglected with the assumption of homogeneity of flow in circular direction. In this study, with the advantages of the 2D depth-averaged model, it is possible to estimate the variation in 2D flow. 5.1 Governing equations Coordinate setting: In order to apply the depth-averaged equations (3.26, 3.33 & 3.34) to simulate the 2D flows with air-core vortex, a new coordinate system is introduced as in Figure 5.1. Two coordinates ),( ηξ are set on the bottom and the other ( )ζ perpendicular with that surface. Parameters ( )0 1, , ,a b R R define the geometry of the bottom in which ( )1R is 57 radius of the intake, ( )0R refers to the curvature of the edge of the intake entrance, while a and b are radii of the flow approach area and the length of intake, respectively. Figure 5.1 Definition sketch of the new coordinates Governing equations: The original depth-averaged equations are used: Continuity equation: 01 000 = ∂ ∂ + ∂ ∂ + ∂ ∂ J N J M t h J ηξ (5.1) Momentum equations: ξ -direction ( )ξηηξηξξξηξξξηξ 0200020000 1 Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ NNMMNM hJJ VM J UM J M t 000 000000 00 2 0 2 0 2 0 0 J dp J dp J G J h b h zzyyxx h zyx ρ τζ ρη ηξηξηξζ ρξ ξξξ ξξ − ∂ ∂++ − ∂ ∂++ −= ∫∫ (5.2) b 0R 1R a x y z η ξ ζ 58 η -direction ( )ηηηηηξηξηηξξηξ 0200020000 1 Γ+Γ+Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ NNMMNM hJJ VN J UN J N t 000 000000 00 2 0 2 0 2 0 0 J dp J dp J G J h b h zzyyxx h zyx ρ τζ ρξ ηξηξηξζ ρη ηηη ηη − ∂ ∂++ − ∂ ∂++ −= ∫∫ (5.3) In this case, it can easily be seen that the coordinates setting on the bottom surface with two axis ),( ηξ are orthogonal system, therefore it was proved that: 0=Γ=Γ ξηξ ξ ξη 0=Γ=Γ ηηη η ξξ 0=Γ=Γ ζηξ ζ ξη 0000000 =++ zzyyxx ηξηξηξ and 0=ηG As a result, Equations (5.2-5.3) are reduced to ( )ξηηξξξηξ 02020000 1 Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ NM hJJ VM J UM J M t 0 222 0 2 0 2 0 2 0 0 222 J hGNM J G J h bzyx ρ τ ξ ξξξ ξζζ àη ζ ξξ ξ −⎥⎦ ⎤⎢⎣ ⎡ −Γ+Γ ∂ ∂++ −= (5.4) ( )ηηξηξηηξ 000000 1 Γ+Γ+⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ +⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ ∂ ∂ NMMN hJJ VN J UN J N t 0 222 0 2 0 2 0 2 0 222 J hGNM J bzyx ρ τ η ηηη ηζζ àη ζ ξξ −⎥⎦ ⎤⎢⎣ ⎡ −Γ+Γ ∂ ∂++ −= (5.5) These equations together with continuity Equation (5.1) are then applied to calculate 2D 59 water surface profile of flow with air-core vortex at a vertical intake in the following sections. j=2 j=3 i=2 j=1 i=1 Figure 5.2 Illustration of the computational grid 5.2 Numerical method Based on the generalized curvilinear coordinate system set on the bottom surface (Figure 5.1), the computation mesh is generated as shown in Figure 5.2. Then the set of equations (5.1, 5.4-5.5) is discretized by using the cell-centered staggered grid (Prayet and Taylor, 1983), in which the hydraulic variables are defined at different positions as shown in Figure 5.3 (Fletcher, 1991). The Finite Difference Method and the Finite Volume Method are used in dicretization process, in which the First-Upwind scheme is employed for convective terms while the Two-point Forward scheme is utilized for time derivatives. It follows that: Continuity equation: ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ − ∆ + ∆ − + + ++ ++++ + ++ ++ 21,0 21, 21,10 21,121,21 1 21,21 21,210 11 ji ji ji ji n ji n ji ji J M J M t hh J ξ 60 Figure 5.3 Definition sketch of cell-centered staggered grid in the 2D calculation 01 ,210 ,21 1,210 1,21 =⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ − ∆ + + + ++ ++ ji ji ji ji J N J N η (5.6) Momentum equations: ξ -direction ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ − ∆ + ∆ − +− ++−+− ++ +++++ + + + 21,210 21,121,21 21,210 21,21,2121, 1 21, 21,0 11 ji jbiji ji jaiji n ji n ji ji J MU J MU t MM J ξ [ ] =Γ+Γ+⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ − ∆ + + + − + ++ 21.0 2 0 2 21.0,0 ;21,, 1,0 21,1, 11 ji jiji djiji ji cjiji hVhU JJ MV J MV ξ ηη ξ ξξη [ −Γ++ ∆ −−= ++++ ++ + + + 21.21 2 21.21 21.0 2 0 2 0 2 0 21. 21. 21.0 21. 2 1 jiji ji zyx ji b ji ji ji M J G J h ζ ξξ ξ ξ ξξξ ξρ τ 21.21 2 21.2121.21 2 21.2121.21 2 21.21 +−+−+++++−+− Γ−Γ+Γ jijijijijiji NNM ζηη ζ ηη ζ ξξ ]2 21.2121.212 21.2121.21 +−+−++++ +− jijijiji hGhG ζζ (5.7) M, U M, U N, V N, V h i+1i j j+1 61 Figure 5.4 Illustration of the discretization scheme for the momentum equations N, V h i j-1/2 j+1/ h M, U i+1 j M, U M, U M, U M, U h i-1/2 j j+1 h N, V N, V N, V N, V i+1/i 62 η -direction ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ − ∆ + ∆ − +− + ++++ + + + ji jbiji ji jaiji n ji n ji ji J NU J NU t NN J ,0 ,21, ,10 ,21,1,21 1 ,21 ,210 11 ξ [ ] =Γ+Γ+⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ − ∆ + + −+ +−+−+ ++ ++++ ji ji djiji ji cjiji J UVh J NV J NV ,21 021,210 1,2121,21 21,210 ,2121,211 η ηξ η ξηη [ 21,21 2 21,2121,21 2 21,21 ,210 2 0 2 0 2 0 ,21 2 1 −+−+++++ ++ Γ−Γ ++ ∆ −−= jijijiji ji zyx ji b MM J ζ ξξ ζ ξξ η ηηη ηρ τ 21,21 2 21,2121,21 2 21,2121,21 2 21,21 ++++−+−+++++ −Γ−Γ+ jijijijijiji hGNN ηη ζζ ηη ζ ηη ] 21,21 2 21,21 −+−+ + jiji hG ηη ζ (5.8) To close the above equation system, the boundary conditions were given with the discharge at the upstream end and Neumann condition at the downstream boundary. The computational domain is chosen from 1=j to 1+= JYj , therefore the hydraulic variables at 1+= JYj is used for boundary condition when calculating variables at 1=j and vice versa. 5.3 Results and discussions With different discharges ( )0M and tangential velocities ( )0V given at the outer-zone, some results of the surface profile were obtained as shown in Figures (5.5-5.7). Similarly as the phenomenon monitored in Chapter 4, the water surface and submergence of the flow with air-core vortex strongly depend on the flow condition such as discharge at the intake, the circulation (which in this simulation is defined from the velocity at boundary of outer zone) and also the geometry of the intake. When the discharge at the intake is increased, the submergence is also increased as shown in Figure 5.5 with the same circulation. If the discharge at the intake is maintained, when 63 0.0 0.1 0.2 0.3 0.4 0.5 z ( m ) -0.3 -0.1 0.0 0.2 0.3 x (m) -0.30 -0.15 0.00 0.15 30 y (m) 0.0 0.1 0.2 0.3 0.4 0.5 z ( m ) -0.3 -0.1 0.0 0.2 0.3 x (m) -0.30 -0.15 0.00 0.15 30 y (m) Figure 5.5 Water surface of flow with different discharges at the intake a) 0 0 0 10.35; 10.; 0.03 ; 0.05 ;QM V R m R m= = = = b) 0 0 0 10.5; 10.; 0.03 ; 0.05 ;QM V R m R m= = = = 0.0 0.1 0.2 0.3 0.4 0.5 z ( m ) -0.3 -0.1 0.0 0.2 0.3 x (m) -0.30 -0.15 0.00 0.15 30 y (m) 0.0 0.1 0.2 0.3 0.4 0.5 z ( m ) -0.3 -0.1 0.0 0.2 0.3 x (m) -0.30 -0.15 0.00 0.15 30 y (m) Figure 5.6 Water surface of flow with different velocity at the outer-zone boundary a) 0 0 0 110.; 0.5; 0.05 ; 0.05 ;V QM R m R m= = = = b) 0 0 0 115.; 0.5; 0.05 ; 0.05 ;V QM R m R m= = = = a) b) a) b) 64 velocity at boundary varies the water surface also varies. The higher circular velocity ( )0V , the higher submergence at the intake, or in other words, the circular velocity obstruct or impede the water drainage at the intake. Consequently to maintain the same discharge, water head would be increased with increased circular velocity (Figure 5.6). In order to investigate the effect of the intake geometry to water surface, the parameter 0R is used to control the shape of the join between intake and bottom. Figure (5.7) shows that the submergence increases when 0R increases while keeping other parameters constant. 0.0 0.1 0.2 0.3 0.4 0.5 z ( m ) -0.3 -0.1 0.0 0.2 0.3 x (m) -0.30 -0.15 0.00 0.15 30 y (m) 0.0 0.1 0.2 0.3 0.4 0.5 z ( m ) -0.3 -0.1 0.0 0.2 0.3 x (m) -0.30 -0.15 0.00 0.15 30 y (m) Figure 5.7 Water surface of flow with different shape of the intake a) 0 0 0 110.; 0.5; 0.05 ; 0.05 ;V QM R m R m= = = = b) 0 0 0 110.; 0.5; 0.03 ; 0.05 ;V QM R m R m= = = = 5.4 Summary Based on the simulation results, it can be concluded that, not only 1D water profile as described in Chapter 4 can be estimated by using analytical method, but also the 2D unsteady plane water surface can be reasonably simulated using the depth-averaged equations without any difficulty. Consequently, it also validates the broad perspective of the model’s applicability. a) b) 65 5.5 References 2 Peyret R. and Taylor T. D. 1983. Computational methods for fluid flow, Springer Ser. Comput. Phys. (Springer, Berlin, Heidelberg). 3 Fletcher C. A. J. 1991. Computational techniques for fluid dynamics. Springer Ser. Comput. Phys. (Springer Verlag). 66 Chapter 6 WATER SURFACE PROFILE ANALYSIS OF FLOWS OVER CIRCULAR SURFACE 6.1 Preliminary In a chute spillway, the stability of the free surface is strongly dependent on the geometry of the approach channel as well as that of spillway. If there is an abrupt drop of the channel bed, a free overfall or a vertical fully aerated drop may occur. This phenomenon has attracted considerable attentions of investigators (e.g., Davis et al. 1998; Dey 1998, 2002, 2003, 2005; Ahmad 2005; Guo 2005; Pal and Goel 2006) since the pioneering work of Rouse (1936) due to its practical engineering use in simple flow-measuring devices (e.g., Dey 1998; Guo 2005). But their approaches could not be applied in case of an abrupt change of channel’s bed with circular shape, accompanied with small discharge that cannot result in free overfall. In this chapter a mathematical model based on depth-averaged equations with a 67 generalized curvilinear coordinate system attached to the bottom surface is developed from the 2D depth-averaged model described in Chapter 3. The developed model was applied to calculate the water surface profile in open channel with circular surface at the end. To verify the model, an experiment was conducted to measure the water surface profile, where the model results of both steady and unsteady analysis were compared with the observations. 6.2 Hydraulic Experiment 6.2.1 Experimental Setup The experiments were conducted in an open-channel main flume consisted of two straight channels joined perpendicular to each other by a step ‘A’ at the end as described in Figure 6.1. The flume was 135cm long, 10cm wide and 20cm deep (Figure 6.1a) , made of a steel frame with glass in all side walls and bed which allowed the convenience for visual observations. In this study two detachable circular parts with radius of 15cm and 5cm respectively were used (Figure 6.1b). These parts were made such that, it is easily to be attached and detached to and from the main flume at step ‘A’, without altering the other parts of experiment facility to form a circular channel bottom at the joint (Figure 6.1c). These parts (B and C) were purposely constructed to investigate the effect of different curvatures to flows. For simplicity of discussion, from now onward, the curvature with 15cm and 5cm radius will be referred to “large case” and “small case” respectively. The water surface elevations were measured by the ultrasonic sensor Keyence UD-500 accompanied with the receiver and converter Keyence NR-2000 (Figure 6.2). For each case the measurements were focused in the region with the curvature of the bottom at four points 0o (point 1), 30o (point 2), 60o (point 3) and 90o (point 4) to the vertical direction (Figure 6.1d). 68 Figure 6.2 Experimental site Figure 6.1 Side view of the experimental facility (All units are in centimeter) b) Part B and part C R = 15 R = 5 c) Connection of main flume and parts B, C Point 1 Point 2 Point 3 Point 4 d) Sensors’ arrangement 135 100 20 20 20 75 20 20 a) Main flume Inflow Outflow A Sensor UD-500 Circular surface Digitizer NR-2000 69 The ultrasonic sensor Keyence UD-500 measures the distance from the sensor to the surface by calculating the reflected ultrasonic signal. Actually, the sensor is combined both the transmitter and receiver of the ultrasonic signals, and the output is voltage of the direct electric current, therefore it needs a receiver and digitizer to convert the analog signal into digital signal (Keyence NR-2000) for out-putting to the PC. Thus, the sensor is connected with the receiver and converter as in Figure 6.3. 6.2.2 Experimental procedures and results: Before using the sensor and converter, it is required to calibrate the sensor, and establish a correlation line between the voltage and the distance from the sensor to the surface. The calibration process was done by using the sensor to measure the known distances (Figure 6.4) and the correlation equation was found: tt VD *77.4035.52 −= (6.1) where: tD is the distance (in mm ) and tV is the measured voltage (in Voltage V ) at time t . At the first stage of experiment, the distance from the fixed sensor to the bed was measured in advance ( )bD and then the time series of distance from the sensor to the water surface were saved ( )sD , thereafter water depth was easily obtained by subtracting sD from bD . Water was drawn from a constant head tank being filled by a turbine pump to ensure the steady in-flow rate. The flow rate was controlled by a control valve and discharge measurements were carried out at the end of the flume for different flow conditions. At beginning, the discharge was set at very small value, then increased with small intervals. After each increase, the flows were kept for some time to get the equilibrium condition before taking measurements. Upon attainment of steady conditions at the 70 upstream end of the main flume, the water surface level in the region within circular bottom was measured by the sensor at four locations as in Figure 6.1d. The samples of time series results of water depth are shown in Figure 6.5. And the time-averaged values with the flow conditions are listed in Table 1. At the first stage, with relatively small discharge, the flows over the circular part were observed with very slightly fluctuation and considered to be stable (Figure 6.5a &6.5b). In the second stage, when the discharge was continuously increased, the oscillation occurs at the circular region and become significant with increasing flow rate while remaining other parameters as the same. With each subsequent increase of the discharge, the amplitudes of oscillation become larger (Figure 6.5c & 6.5d). The intensities of oscillation can be estimated by Equation (6.2) and the results are shown in Figure 6.6. ∑ ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛ − = n i h hh n 1 2 1 σ (6.2) From this figure, it is observed that, in both small- and large-case, the bigger intensities were observed in downstream end of the flow field, or in other words, the oscillation intensity increases from point 1 to point 4. 71 Figure 6.3 Schematic of sensor connection Figure 6.4 Sensor calibration Sensor UD-500 Digitizer NR - 2000 Transmitter & Receiver PC 72 0 1 2 3 4 5 Time (s) 0 5 10 15 D ep th (m m ) Point 1 Point 3 Point 2 Point 4 a) 0 1 2 3 4 5 Time (s) 0 5 10 15 D ep th (m m ) b) 0 1 2 3 4 5 Time (s) 0 5 10 15 D ep th (m m ) c) 0 1 2 3 4 5 Time (s) 0 5 10 15 D ep th (m m ) d) Figure 6.5 Time history of the free surface at four locations in different experiments: a) Exp1; b) Exp2; c) Exp3; d) Exp4; 73 Table 6.1 Experiment conditions Time averaged water depth (mm) Experiment Radius of circular part Discharge (l/s) Point1 Point2 Point3 Point4 Small-case Exp 1 5 cm 0.1704 6.2 3.8 2.2 1.4 Small-case Exp 2 5 cm 0.2441 8.6 4.8 3.1 1.7 Small-case Exp 3 5 cm 0.6350 11.1 7.8 5.0 3.5 Small-case Exp 4 5 cm 0.7754 14.4 10.5 7.8 5.6 Large-case Exp 5 15 cm 0.4330 10.8 5.1 2.2 1.8 Large-case Exp 6 15 cm 0.5421 12.7 5.7 3.9 3.1 Large-case Exp 7 15 cm 0.8295 15.8 8.6 5.1 3.8 Large-case Exp 8 15 cm 0.9936 18.6 10.8 7.3 5.1 0.00 0.05 0.10 0.15 0.20 0.25 0.30 Point1 Point2 Point3 Point4 σ Exp-1 Exp-2 Exp-3 Exp-4 Exp-5 Exp-6 Exp-7 Exp-8 Figure 6.6 The oscillation density at four locations in circular