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

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

HYDRAULIC MODELING OF OPEN CHANNEL FLOWS OVER AN ARBITRARY 3-D SURFACE AND ITS APPLICATIONS IN AMENITY HYDRAULIC ENGINEERING TRAN NGOC ANH August, 2006 Acknowledgements The research work presented in this manuscript was conducted in River System Engineering Laboratory, Department of Urban Management, Kyoto University, Kyoto, Japan First of all, I would like to convey my deepest gratitude and sincere thanks to Professor Dr Takashi Hosoda who suggested me this research topic, and provided guidance, constant and kind advices, encouragement throughout the research, and above all, giving me a chance to study and work at a World-leading university as Kyoto University I also wish to thank Dr Shinchiro Onda for his kind assistance, useful advices especially in the first days of my research life in Kyoto His efforts were helping me to put the first stones to build up my background in the field of computational fluid dynamics My special thanks should go to Professor Toda Keiichi and Associate Professor Gotoh Hitoshi for their valuable commences and discussions that improved much this manuscript I am very very grateful to my best foreign friend, Prosper Mgaya from Tanzania, for all of his helps, discussions and strong encouragements since October, 2003 In addition, my heartfelt gratitude is extended to all of my Vietnamese friends in Japan, Kansai Football Club members, who helped me forget the seduced life in Vietnam, particular Nguyen Hoang Long, Le Huy Chuan and Le Minh Nhat Last but not least, the most deserving of my gratitude is to my wife, Ha Thanh An, and my family, parents and younger brother This work might not be completed without their constant support and encouragement I am feeling lucky because my wife, my parents and my younger brother are always by my side, and this work is therefore dedicated to them iii Abstract Two-dimensional (2D) description of the flow is commonly sufficient to analyze successfully the flows in most of open channels when the width-to-depth ratio is large and the vertical variation of the mean-flow quantities is not significant Based on coordinate criteria, the depth-averaged models can be classified into two groups namely: the depth-averaged models in Cartesian coordinate system and the depth-averaged models in generalized curvilinear coordinate system The basic assumption in deriving these models is that the vertical pressure distribution is hydrostatic; consequently, they possess the advantage of reduction in computational cost while maintaining the accuracy when applied to flow in a channel with linear or almost linear bottom/bed But indeed, in many cases, water flows over very irregular bed surfaces such as flows over stepped chute, cascade, spillway, etc and the alike In such cases, these models can not reproduce the effects of the bottom topography (e.g., centrifugal force due to bottom curvature) In this study therefore, a depth-averaged model for the open channel flows over an arbitrary 3D surface in a generalized curvilinear coordinate system was proposed This model is the inception for a new class of the depth-averaged models, which was classified by the criterion of coordinate system In conventional depth-averaged models, the coordinate systems are set based on the horizontal plane, then the equations are obtained by integration of the 3D flow equations over the depth from the bottom to free surface with respect to vertical axis In contrary the depth-averaged equations derived in this study are derived via integration processes over the depth with respect to the axis iv perpendicular to the bottom The pressure distribution along this axis is derived from one of the momentum equations as a combination of hydrostatic pressure and the effect of centrifugal force caused by the bottom curvature This implies that the developed model can therefore be applied for the flow over highly curved surface Thereafter the model was applied to simulate flows in several hydraulic structures this included: (i) flow into a vertical intake with air-core vortex and (ii) flows over a circular surface The water surface profile of flows into vertical intake was analyzed by using 1D steady equations system and the calculated results were compared with an existing empirical formula The comparison showed that the model can estimate accurately the critical submergence of the intake without any limitation of Froude number, a problem that most of existing models cannot escape The 2D unsteady (equations) model was also applied to simulate the water surface profile into vertical intake In this regard, the model showed its applicability in computing the flow into intake with air-entrainment The model was also applied to investigate the flow over bottom surface with highly curvature (i.e., flows over circular surface) A hydraulic experiment was conducted in laboratory to verify the calculated results For relatively small discharge the flow remained stable (i.e., no flow fluctuations of the water surface were observed) The model showed good agreement with the observations for both steady and unsteady calculations When discharge is increased, the water surface at the circular vicinity and its downstream becomes unstable (i.e., flow flactuations were observed) In this case, the model could reproduce the fluctuations in term of the period of the oscillation, but some discrepancies could be still observed in terms of the oscillation’s amplitude In order to increase the range of applicability of the model into a general terrain, the model was refined by using an arbitrary axis not always perpendicular to the bottom surface The mathematical equation set has been derived and some simple examples of v dam-break flows in horizontal and slopping channels were presented to verify the model The model’s results showed the good agreement with the conventional model’s one vi Preface The depth-averaged model has a wide range of applicability in hydraulic engineering, especially in flow applications having the depth much smaller compare to the flow width In this approach the vertical variation is negligible and the hydraulic variables are averaged integrating from bed channel to the free surface with respect to vertical axis In deriving the governing equations, the merely pure hydrostatic pressure is assumed that is not really valid in case of flows over highly curved bed and cannot describe the consequences of bed curvature Therefore, this work is devoted to derive a new generation of depth-averaged equations in a body-fitted generalized curvilinear coordinate system attached to an arbitrary 3D bottom surface which can take into account of bottom curvature effects This manuscript is presented as a monograph that includes the contents of the following published and/or accepted journal and conference papers: 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) 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) Anh T N and Hosoda T.: Steady free surface profile of flows with air-core vortex at vii vertical intake XXXI IAHR Congress, Seoul, Korea, pp 601-612 (paper A13-1), 11~16 September, 2005 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 Anh T N and Hosoda T.: Free surface profile analysis of flows with air-core vortex Journal of Applied Mechanics, JSCE, Vol 7, pp 1061-1068, 2004 viii Table of contents Acknowledgment iii Abstract iv Preface vii List of Figures xi List of Tables xv Chapter INTRODUCTION 1.1 Classification of depth-averaged modeling 1.2 Depth-averaged model in curvilinear coordinates 1.3 Objectives of study 1.4 Scope of study 1.5 References Chapter LITERATURE REVIEW 2.1 Depth-average modeling 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 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 STEADY ANALYSIS OF WATER SURFACE PROFILE OF FLOWS WITH AIR-CORE VORTEX AT VERTICAL INTAKE 30 4.1 Introduction 30 4.2 Governing equation 35 4.3 Results and discussions 47 4.4 Summary 54 ix 4.4 References 54 Chapter 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 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 MODEL REFINEMENT 100 7.1 Preliminary 100 7.2 Non-orthogonal coordinate system 101 7.3 Application 105 Chapter CONCLUSIONS 111 x List of Figures Chapter Figure 2.1 Definition sketch for variables used in depth-averaged model…… Figure 2.2 Definition of terms in curvilinear system………………………… 11 Figure 2.3 Definition sketch by Sivakumaran et al (1983)…………………… 14 Chapter Figure 3.1 Definition sketch for new generalized coordinate system………… 21 Figure 3.2 Kinetic boundary condition at water surface……………………… 23 Chapter Figure 4.1 An example of free surface air-vortex………………………………31 Figure 4.2 Various stages of development of air-entraining vortex: S1>S2>S3>S4 (Jain et al, 1978)……………………………………31 Figure 4.3 The inflow to and circulation round a closed path in a flow field (Townson 1991)…………………………………… 33 Figure 4.4 The concept of simple Rankine vortex that including two parts: free vortex in outer zone and forced vortex in inner zone (Townson 1991)………………………………………33 Figure 4.5 Definition of coordinate components……………………………….36 Figure 4.6 An example of computed water surface profile with quasi-normal depth line and critical depth line…………………… 45 Figure 4.7 The effect of circulation on water surface profile and discharge at the intake with same water head………………………48 Figure 4.8 Variation of intake discharge with circulation (a=0.025m, b=10-5 m2, water head=0.5m)………………………….49 Figure 4.9 Different water surface profiles with different values of circulation while maintaining the constant intake discharge…… 49 Figure 4.10 Changing of water surface profile with different shape of the intake 51 xi 6.7 References Azmad Z 2005 Flow measument using free overfall in inverted semi-circular channel Flow Meas Instrum., Vol 16, pp 21-26 Davis, A C., Ellett, B G S and Jacob, R P 1998 Flow measurement in sloping channels with rectangular free overfall J Hydr Engrg., ASCE, Vol 124 (7), pp 760-763 Dey, S 1998 End depth in circular channels J Hydr Engrg., ASCE, Vol 124 (8), pp 856-862 Dey, S 2002 Free overfall in circular channels with flat base: a method of open channel flow measurement Flow Meas Instrum., Vol 13, pp 209-221 Dey, S 2003 Free overfall in inverted semicircular channels J Hydr Engrg., ASCE, Vol 129(6), pp 438-447 Dey S 2005 End depth in U-shaped channels: a simplified approach J Hydr Engrg., ASCE, Vol 131(6), pp 513-516 Guo, Y 2005 Numerical modeling of free overfall J Hydr Engrg., ASCE, Vol 131(2), pp 134-138 Pal M and Goel A 2006 Prediction of the end-depth ratio and discharge in semi-circular and circular shaped channels using support vector machines Flow Meas Instrum., Vol 17, pp 49-57 Rouse, H 1936 Discharge characteristic of the free overfall Civ Engrg ASCE, Vol 6(4), pp 257-260 99 Chapter MODEL REFINEMENT 7.1 Preliminary In Chapter 3, a 2D depth averaged model was introduced based on a generalized curvilinear coordinate system, in which the coordinate axis ζ was always perpendicular to the bottom surface Therefore, the orthogonal conditions in Equation 3.10 and 3.11 were applied to obtain the simplified momentum equations as described in (3.33) and (3.34) Nevertheless, because axis ζ is always perpendicular to the bottom surface thus, for a concave bed there is a possibility that ζ will intersect some where above the surface To avoid this difficulty, the above model should be applied in a convex topography as in Chapter 4-6 or we limit the applicability of the model in concave surfaces with a shallow depth therefore with these limitations the condition for ζ not to intersect within the computational domain is satisfied as shown in following figure (Figure 7.1) 100 ζ ξ Figure 7.1 Illustration for limitation of the model in concave topography In order to eliminate these limitations from the previuos model, we are now at the stage to derive a more general equation set that will not use the condition for ζ to be perpendicular to the bottom surface, (i.e ζ could be an arbitrary axis) 7.2 Non-orthogonal coordinate system Coordinate system The new curvilinear coordinate system is set on the bottom surface similar with the previous application but the ζ -axis is arbitrary (as shown in Figure 7.2) The kinematic boundary condition is also derived at the free surface as in (3.23): Ws = ∂h ∂h ∂h + Vs +U s ∂η ∂ξ ∂t (7.1) Depth-average equations: The transformed RAN equations in a new curvilinear coordinates are integrated through the depth from bottom to water surface with respect to ζ -axis, without applying the perpendicular relations, we obtain: Continuity equation 101 ∂ N ∂h ∂ M + + =0 J ∂t ∂ξ J ∂η J (7.2) Momentum equations: ∂ ⎛M ⎜ ∂t ⎜ J ⎝ = − ⎞ ∂ ⎛ UM ⎟+ ⎜ ⎟ ∂ξ ⎜ J ⎝ ⎠ ( ⎞ ∂ ⎛ VM ⎞ ξ ξ ξ ξ 2 ⎟+ ⎟ ⎜ ⎟ ∂η ⎜ J ⎟ + J h M Γ0 ξξ + MNΓ0 ξη + NMΓ0ηξ + N Γ0ηη ⎝ ⎠ ⎠ 2 h ξ ξ x0 + ξ y0 + ξ z ∂ G − J0 J0 ∂ξ h p ∫ρ ξ x 0η x + ξ y 0η y + ξ z 0η z ∂ h p τξ dζ − b J0 ρJ ∂η ∫ ρ dζ − ( ξ x 0ζ x + ξ y 0ζ y + ξ z 0ζ z J0 p ⎜ ⎟ )⎛ ρ ⎞ ⎜ ⎟ ⎝ ⎠b ) h h ∂ τ ξξ ∂ τ ξη dζ + dζ ∂ξ J ∫ ρ ∂η J ∫ ρ 0 + (7.3) and ∂⎛N ⎜ ∂t ⎜ J ⎝ ( ⎞ ∂ ⎛ UN ⎞ ∂ ⎛ VN ⎞ η η η η 2 ⎟+ ⎟ ∂ξ ⎜ J ⎟ + ∂η ⎜ J ⎟ + J h M Γ0 ξξ + MNΓ0 ξη + NMΓ0ηξ + N Γ0ηη ⎜ ⎟ ⎜ ⎟ ⎠ ⎝ ⎠ ⎝ ⎠ 2 h η η x0 + η y0 + η z ∂ = G − ∂η J0 J0 − h p ∫ρ dζ − ( η x 0ζ x + η y 0ζ y + η z 0ζ z J0 p ⎜ ⎟ )⎛ ρ ⎞ ⎜ ⎟ ⎝ ⎠b + ξ x 0η x + ξ y 0η y + ξ z 0η z ∂ ∂ξ J0 h p ∫ρ dζ − h h ∂ τ ηη ∂ τ ηξ dζ + dζ ∂ξ J ∫ ρ ∂η J ∫ ρ 0 ) η τb ρJ (7.4) In order to estimate the pressure distribution along the ζ -axis, the momentum equation in ζ -direction is recast with the assumptions of homogeneity in ζ and shear stress are negligible: ( ) ζ Gζ 2 ∂ ⎛ p⎞ ζ ζ ζ ⎜ ⎟ U Γξξ + UVΓξη + VUΓηξ + V Γηη = − ζx +ζ y +ζz J J J ∂ζ ⎜ ρ ⎟ ⎝ ⎠ ( − ) ∂ ⎛ p⎞ ∂ ⎛ p⎞ ⎜ ⎟ ⎜ ⎟ − ζ xξ x + ζ yξ y + ζ zξ z ζ xη x + ζ yη y + ζ zη z ⎜ρ⎟ J J ∂ξ ⎜ ρ ⎟ ∂η ⎝ ⎠ ⎝ ⎠ ( ) ( ) (7.5) or (ζ x +ζ y +ζz p ⎜ ⎟ ) ∂∂ζ ⎛ ρ ⎞ + (ζ η ⎜ ⎟ ⎝ ⎠ x x + ζ yη y + ζ zη z 102 p ⎜ ⎟ ) ∂∂ ⎛ ρ ⎞ + (ζ ⎜ ⎟ η ⎝ ⎠ x ξ x + ζ yξ y + ζ zξ z ) ∂ ∂ξ ⎛ p⎞ ⎜ ⎟ ⎜ρ⎟ ⎝ ⎠ ( ζ ζ ζ ζ = G ζ − U Γξξ + UVΓξη + VUΓηξ + V Γηη ) (7.6) To solve equation (7.6) by iteration method, the first estimation of pressure term is given as: (ζ 2 + ζ y0 + ζ z0 x0 ) ∂ ∂ζ ⎛ p⎞ ⎜ ⎟ ⎜ρ⎟ ⎝ ⎠ (0 ) ( ζ ζ ζ ζ = G ζ − U Γξξ + UVΓξη + VUΓηξ + V Γηη ) (7.7) hence, taking integration from ζ to h gives p (0 ) ρ = (ζ x0 + ζ y0 + ζ z0 ) [G − (U ζ )] ζ ζ ζ ζ Γξξ + UVΓξη + VUΓηξ + V Γηη (h − ζ ) = f p(0 ) (h − ζ ) Consequently, ∂ ∂ξ ∂f (0 ) ∂f p(0 ) ⎛ p (0 ) ⎞ ∂ ∂h (0 ) ⎜ ⎟ = (h − ζ ) p + f p(0 ) fp h −ζ ⋅ = ⎜ ρ ⎟ ∂ξ ∂ξ ∂ξ ∂ξ ⎝ ⎠ ( ) (7.8) and similarly, ∂f p(0 ) ∂f (0 ) ∂ ∂h ∂ ⎛ p (0 ) ⎞ ⎜ ⎟ = (h − ζ ) p + f p(0 ) f p(0 )h − ζ ⋅ = ∂η ∂η ∂η ∂η ∂η ⎜ ρ ⎟ ⎝ ⎠ ( ) (7.9) Substituting Equations (7.8) and (7.9) into (7.7) and taking integration we attain the equation for the new estimation of pressure distribution as follows (ζ ⎧∂ ∂f p(0 ) ⎪ f p(0 ) h (h − ζ ) − ∂ξ ⎪ ∂ξ ⎩ x ξ x + ζ y ξ y + ζ z ξ z )0 ⎨ (ζ η x [ x ( ) (0 ) ⎧ ∂f p ⎪ ∂ f p(0 ) h (h − ζ ) − + ζ yη y + ζ zη z ⎨ ∂η ⎪ ∂η ⎩ ) ( ( ) )] ⎛ h ζ ⎞⎫ ⎟⎪ ⎜ ⎜ − ⎟⎬ + ⎠⎪ ⎝ ⎭ ⎫ ⎛ h ζ ⎞⎪ 2 ⎜ ⎟ ⎜ − ⎟⎬ − ζ x + ζ y + ζ z ⎝ ⎠⎪ ⎭ ζ ζ ζ ζ = G ζ − U Γξξ + UVΓξη + VUΓηξ + V Γηη (h − ζ ) as a result, 103 ( ) p (1) ρ (ζ x +ζ y +ζz ) p (1) [ )] ( ζ ζ ζ ζ = − G ζ − U Γξξ + UVΓξη + VUΓηξ + V Γηη (h − ζ ) ρ ⎧∂ ∂f p(0 ) ⎪ (0 ) f p h (h − ζ ) − + ζ xξ x + ζ yξ y + ζ zξ z ⎨ ∂ξ ⎪ ∂ξ ⎩ ⎛ h ζ ⎞⎫ ⎟⎪ ⎜ ⎜ − ⎟⎬ ⎠⎪ ⎝ ⎭ ⎧ ∂ ∂f p(0 ) ⎪ (0 ) f p h (h − ζ ) − + ζ xη x + ζ yη y + ζ zη z ⎨ ∂η ⎪ ∂η ⎩ ⎛ h ζ ⎞⎫ ⎜ ⎟⎪ ⎜ − ⎟⎬ ⎝ ⎠⎪ ⎭ ( ( ) ( ) ( ) ) (7.10) Taking integration Equation (7.10) gives (ζ x +ζ y +ζz (1) ) ∫ pρ h [( ) ζ ζ ζ ζ dζ = U Γξξ + UVΓξη + VUΓηξ + V Γηη − G ζ ] h2 (0 ) ⎧∂ h ∂f p h ⎫ ⎪ ⎪ − + ζ xξ x + ζ yξ y + ζ z ξ z ⎨ f p(0 ) h ⎬ ∂ξ ⎪ ⎪ ∂ξ ⎩ ⎭ ( ( ) ) ⎧ ∂ ∂f p(0 ) h ⎫ ⎪ ⎪ (0 ) h fp h + ζ xη x + ζ yη y + ζ zη z ⎨ − ⎬ ∂η ⎪ ⎪ ∂η ⎩ ⎭ ( ( ) ) (7.11) Equation (7.11) describe the pressure distribution along the ζ -axis It is easily seen that, two last terms are added in the pressure distribution and if ζ normal to the bottom surface, the distribution reduced as the first term - a term representative for the hydrostatic pressure and effects of centrifugal force due to bottom curvature shown in Chapter The pressure at the bottom is then obtained: ⎛ p⎞ ⎜ ⎟ = ⎜ρ⎟ 2 ⎝ ⎠b ζ x + ζ y + ζ z ( ) [(U ) ] ζ ζ ζ ζ Γξξ + UVΓξη + VUΓηξ + V Γηη − G ζ h + (0 ) ⎧ ∂ h ∂f p ⎫ ⎪ ⎪ (0 ) fp h − + ζ x ξ x + ζ y ξ y + ζ z ξ z ⎨h ⎬ ∂ξ ⎪ ⎪ ∂ξ ⎩ ⎭ ( ) ( ) (0 ) ⎧ ∂ h ∂f p ⎫ ⎪ ⎪ (0 ) + ζ xη x + ζ yη y + ζ zη z ⎨h fp h − ⎬ ∂η ⎪ ⎪ ∂η ⎩ ⎭ ( ) ( ) 104 (7.12) The contravariant components of gravitational vector are derived similar as in Chapter 4: G ξ = − gJ (xη yζ − xζ yη ) (7.13a) G η = − gJ (xξ yζ − xζ yξ ) (7.13b) G ξ = − gJ (xξ yη − xη yξ ) (7.13c) 7.3 Application In order to verify the validity and the applicability of the new general equations, in this first version of manuscript, the dam-break flows in a horizontal and sloping channel bed are investigated In this application, the 1-D flow is considered using the new coordinate system as described in Figure 7.2 The parameters α and α define the slope of the channel and the angle of the ζ -axis with the channel bottom (also means the angle between two new axes) By changing the value α , it is easily to set the slope of channel and α = defines for the case of horizontal bed z ζ α2 α1 x ξ Figure 7.2 Definition sketch of new generalized coordinate system 105 In this research, the second and third terms in equation (7.6) are neglected, it gives (ζ x0 + ζ y 02 + ζ z 02 p ) ∂∂ ⎛ ρ ⎞ = G − (U ⎜ ⎟ ζ ⎝ ⎠ ζ ζ ζ Γζ + UV Γζ + VU Γηξ + V Γηη ξξ ξη ) (7.14) Consider 1D flow (i.e V = ) and the calculation gives Γζ = then pressure ξξ distribution in (7.14) becomes hydrostatic: ⎛ p⎞ ∂ ⎛ p⎞ Gζ ⇒⎜ ⎟= Gζ ( h − ζ ) ⎜ ⎟= 2 2 ∂ζ ⎝ ρ ⎠ ζ x + ζ y + ζ z ρ ⎠ ζ x0 + ζ y 02 + ζ z 02 ⎝ ( ) ( ) hence, the pressure at the bottom is obtained: ⎛ p⎞ Gζ h = ⎜ ⎟ ζ x02 + ζ y 02 + ζ z 02 ⎝ ρ ⎠b ( (7.15) ) and h ⎛ p⎞ ∫ ⎜ ρ ⎟ d ζ = (ζ ⎝ ⎠ x0 + ζ y0 + ζ z0 ) Gζ h (7.16) Substituting (7.15-7.16) into (7.3) and neglecting the effect of internal stresses gives ∂ ⎛M ⎜ ∂t ⎝ J − ⎞ ∂ ⎛ UM ⎟+ ⎜ ⎠ ∂ξ ⎝ J 2 ξ ⎞ M Γ 0ξξ h ξ ξ x0 + ξ y0 + ξ z0 ∂ G − + = ⎟ J0 h J0 J0 ∂ξ ⎠ h p ∫ ρ dζ τξ ⎛ p⎞ ξ x 0ζ x + ξ y 0ζ y + ξ z 0ζ z ⎜ ⎟ − b J0 ⎝ ρ ⎠b ρ J ( ) or 2 ξ ⎡ ⎤ h ξ ξ x0 + ξ y0 + ξ z0 ∂ ⎢ ∂ ⎛ M ⎞ ∂ ⎛ UM ⎞ U hΓ 0ξξ G − Gζ h ⎥ = ⎜ ⎟+ ⎜ ⎟+ ⎥ ∂t ⎝ J ⎠ ∂ξ ⎝ J ⎠ ∂ξ ⎢ ζ x + ζ y + ζ z J0 J0 J0 ⎣ ⎦ ( − ( ) τξ ξ x 0ζ x + ξ y 0ζ y + ξ z 0ζ z Gζ h − b J0 ρ J0 ζ x02 + ζ y 02 + ζ z 02 ( ) ) (7.17) Equation (7.17) is solved in plugging with continuity equation (7.2) and the geometric 106 variables are calculated from the mesh depicted in Figure 7.2, to investigate the dam-break flow in horizontal and slopping channels Figure 7.3 and 7.4 show the dam-break flow water surface at different times in a dried and wetted bed for horizontal channel The bores are visually observed, and they were compared with the conventional depth-averaged model (i.e., Kuipers and Vreugdenhill equations) The good agreement was achieved in the bore of both cases: dried and wetted bed , but there are still some bias at the upstream of the dam, that might be caused by the difference in boundary conditions at upstream end due to the difference of the grid It infers that, the new model can be applied in such cases that the third axis is not necessary always perpendicular to the bottom surface It also was emphasized in Figure 7.5-7.6, which show the dam-break flows in channels with slop 30 107 2.0 T = 0.00s T = 0.20s T = 0.40s T = 0.60s z (m) Channel bed 0.0 2.0 Figure 7.3 4.0 x (m) 6.0 8.0 Calculated water surface profile at different time steps of dam break flow in a dried-bed horizontal channel: Hini = 1.0m ; α = 30 ; α = 60 2.0 T = 0.00s T = 0.20s T = 0.40s T = 0.60s z (m) Channel bed 0.0 2.0 Figure 7.4 4.0 x (m) 6.0 8.0 Calculated water surface profile at different time steps of dam break flow in a wetted-bed horizontal channel: Hiniup = 1.0m; Hinidown = 0.5m ; α = 0 ; α = 60 108 1.5 Initial depth Conventional depth-averaged model New model z (m) 1.0 0.5 0.0 10 12 14 16 18 x (m) Figure 7.5 Comparison of water surface profile for dried horizontal channel at different times: T = 0.4s; 1.0s; and 1.6s; Initial depth 1.5 Conventional depth-averaged model New model z (m) 1.0 0.5 0.0 10 12 14 16 18 x (m) Figure 7.6 Comparison of water surface profile for wetted horizontal channel at different times: T = 0.4s; 0.6s; and 0.8s; 109 6.0 T = 0.0s T = 0.2s T = 0.4s 4.0 z (m) T = 0.6s Channel bed 2.0 0.0 2.0 Figure 7.7 4.0 6.0 x (m) 8.0 10.0 Calculated water surface profile at different time steps of dam break flow in a dried-bed sloping channel: Hini = 1.0m ; α = 30 ; α = 60 6.0 T = 0.0s T = 0.2s T = 0.4s 4.0 T = 0.6s z (m) Channel bed 2.0 0.0 2.0 Figure 7.8 4.0 x (m) 6.0 8.0 Calculated water surface profile at different time steps of dam break flow in a dried-bed horizontal channel: Hiniup = 1.0m; Hinidown = 0.5m ; α = 30 ; α = 60 110 Chapter CONCLUSIONS In this research, a depth-averaged model of open channel flows in a generalized curvilinear coordinate system over an arbitrary 3D surface was developed This model is the inception for a new class of the depth-averaged models classified by the criteria of coordinate system This work focused on the derivation of governing equations with examples of application in hydraulic amenity The major results and contributions of this research are summarized as follows Mathematical model - The original RAN equations were firstly transformed into the new generalized curvilinear coordinate system, in which two axes lay on an arbitrary surface and the other perpendicular to that surface The depth-averaged continuity and momentum equation were derived by integrating the transformed RAN equation and substituting the kinematic boundary condition at the free surface - The pressure distribution was obtained by integrating the momentum equation along the perpendicular axis as a combination of hydrostatic pressure and the effect of 111 centrifugal force caused by bottom curvature Flow with air-core vortex at a vertical intake - Firstly, 1D analytical analysis was applied to calculate the water surface profile of the flows with air-core vortex at a vertical intake 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 fomula 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 - Secondly, 2D unsteady analysis was applied to simulate the water surface profile and also showed the applicability of the model in computing the flow into intake with air-entrainment Flows over circular surface - A hydraulic experiment was conducted in the laboratory to measure the surface profile of the flows over circular surface It was pointed out that: with small discharge, the free surface remained stable but some oscillations occurred when the discharge was increased The intensity of the oscillation was observed to increase toward the downstream direction of the flow - Both 1D steady and unsteady analysis were applied to estimate the water surface profile of the flows with small discharge Good agreements in comparison with measured data were achieved for both approaches Results from 1D unsteady model demonstrate the fluctuation at the region with circular bottom with large discharge Spectrum analysis was applied to investigate the characteristics of the oscillation The comparison with observed data showed the fairly good agreement in term of 112 frequency or period but some discrepancies in magnitude of fluctuation were still observed 2D simulation was conducted, and also could reproduce the phenomena as well Model refinement - In order to apply the model for the case with highly concave topography, the new depth-averaged equations were derived without using the perpendicular conditions The improved model therefore has the ability in more general geometry of the bottom surface - A simple example was presented to verify the model with dam-break flows in horizontal and sloping channels 113 ... contents of the following published and/ or accepted journal and conference papers: Anh T N and Hosoda T.: Depth-Averaged model of open channel flows over an arbitrary 3D surface and its applications. .. 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... analysis of water surface profile Journal of Hydraulic Engineering, ASCE (accepted on May 12, 2006) Anh T N and Hosoda T.: Oscillation induced by the centrifugal force in open channel flows over