Calculated Surface Velocity Coeffiecients for Prismatic Open Channels by Three-Dimensional Hydraulic Modeling

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1 Utah State University All Graduate Theses and Dissertations Graduate Studies Calculated Surface Velocity Coeffiecients for Prismatic Open Channels by Three-Dimensional Hydraulic Modeling Nat Marjang Utah State University Follow this and additional works at: Part of the Civil Engineering Commons Recommended Citation Marjang, Nat, "Calculated Surface Velocity Coeffiecients for Prismatic Open Channels by Three-Dimensional Hydraulic Modeling" (008). All Graduate Theses and Dissertations This Dissertation is brought to you for free and open access by the Graduate Studies at It has been accepted for inclusion in All Graduate Theses and Dissertations by an authorized administrator of For more information, please contact

2 CALCULATED SURFACE VELOCITY COEFFICIENTS FOR PRISMATIC OPEN CHANNELS BY THREE-DIMENSIONAL HYDRAULIC MODELING by Nat Marjang A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in Irrigation Engineering Approved: Gary P. Merkley Major Professor Wynn R. Walker Committee Member Christopher M.U. Neale Committee Member Gilberto E. Urroz Committee Member Blake P. Tullis Committee Member Byron R. Burnham Dean of Graduate Studies UTAH STATE UNIVERSITY Logan, Utah 008

3 ABSTRACT ii Calculated Surface Velocity Coefficients for Prismatic Open Channels by Three-Dimensional Hydraulic Modeling by Nat Marjang, Doctor of Philosophy Utah State University, 008 Major Professor: Dr. Gary P. Merkley Department: Biological & Irrigation Engineering A turbulence model was developed for computing surface velocity coefficients and discharge under steady, uniform flow conditions for rectangular and compound open-channel cross sections. Reynolds-Average Navier-Stokes (RANS) equations, Reynolds stress equations, and kinetic energy and dissipation equations were applied in the model using the finite-volume method with the SIMPLER algorithm. The models show graphical results of the velocity distributions in the longitudinal bed slope direction, secondary velocities, pressure, turbulence kinetic energy, and kinetic energy dissipation rate across the cross section. Also, the surface velocity coefficients were computed at increments of one-eighth of the base width from the vertical walls to the center of the cross section, and the submergence depth of the floating object from zero to 30 cm, with a 5-cm depth increment. Four different sets of Reynolds stress equations (one set by Boussinesq hypothesis and three sets of algebraic stress model) were used to calculate the results. Only one version of the algebraic stress model was successful in predicting the depression of the maximum streamwise velocity below the water surface. The model was calibrated and verified using laboratory data collected at Utah State University. Calculated discharges from the turbulence model had very

4 good agreement with the laboratory data. The surface velocity coefficients from model results iii were generally lower than the results from the laboratory data, but higher than the values published by the United States Bureau of Reclamation. Standard cross sections of rectangular and compound cross sections were defined to simulate the model results and model sensitivity to parameter changes. The model results were summarized to show the relationship between surface velocity coefficient and channel characteristics compared with the published values by the USBR. For rectangular cross sections, the coefficients from the model are higher than the published USBR values. But the coefficients from the model and USBR are in very close agreement for the tested compound cross sections. The published coefficients by the USBR are a function of only average water depth. However, the model results show that the coefficients are also related to channel size, surface roughness height, float submergence depth, and lateral location of the float object. These factors should be included in the determination of the surface velocity coefficient to improve the discharge estimations from the application of the float method. (348 pages)

5 ACKNOWLEDGMENTS iv I would like to express my deep gratitude to Dr. Gary P. Merkley who gave me the great opportunity to be a Ph.D. student at USU, provided moral and technical support when I was frustrated, gave invaluable directions and suggestions while I was carrying this research, and contributed too many other things to be listed here. His invaluable encouragement made the final dissertation possible. I am deeply grateful to Dr. Gilberto E. Urroz who supported this research through the loan of many useful documents and books, and who provided numerous helpful comments. I am sincerely grateful to Drs. Wynn R. Walker, Christopher M.U. Neale, and Blake P. Tullis for their help and for serving on my graduate committee. To my wife, thanks for your good food, for emotional support, and for always encouraging me. I am especially thankful to my parents who never stopped believing I would finish this dissertation. Lastly, I wish to thank Thai government and Faculty of Engineering, Kasetsart University, Thailand for providing my scholarship and support. Nat Marjang

6 CONTENTS v Page ABSTRACT ACKNOWLEDGEMENTS LIST OF TABLES LIST OF FIGURES NOTATION ii iv ix xi xviii CHAPTER INTRODUCTION Background Objectives 4 Benefits of the Study 5 LITERATURE REVIEW 6 Velocity Distribution 6 Secondary Currents 8 Governing Equations The RANS Equations 4 Reynolds Stress Equation 6 Turbulence Modeling 6 Algebraic Models 7 One-Equation Models 5 Two-Equation Models 7 Reynolds Stress Models 3 Algebraic Stress Models 38 Boundary Conditions 4 Wall Boundary Conditions 4 Vertical Plane of Symmetry 43 Free-Surface Boundary Conditions 44 3 TURBULENCE MODEL EQUATIONS 46 Continuity Equation 46 Momentum Equations 47 Reynolds Stress Equations 47 Boussinesq Hypothesis 47

7 vi Algebraic Stress Model 48 Algebraic Stress Model 49 Algebraic Stress Model 3 50 Reynolds Stress Model 5 Kinetic Energy and Dissipation Equations 54 k-ε Model based on the Boussinesq Hypothesis 54 k-ε Model for Reynolds Stress Model 54 Auxiliary Equations 55 Boundary Equations 56 First Log-law Equations 57 Second Log-law Equations 58 Vertical Wall Boundary 58 Horizontal Floor Boundary 59 Free Water Surface 60 Distance from the Walls SOLUTION OF EQUATIONS FINITE DIFFERENCE METHOD 6 Algebraic Equations 6 Newton-Raphson Method 67 Grid Spacing and Management 68 Spiral Solutions Approach 69 Multiple Solutions 70 Problems SOLUTION OF EQUATIONS FINITE VOLUME METHOD 80 Introduction 80 Staggered Grid 84 Equations Forms 88 Continuity (Mass Balance) Equation 88 General Form of Equations 88 Momentum Equation in the x-direction 90 Momentum Equation in the y-direction 9 Momentum Equation in the z-direction 93 The k Equation of the k-ε Model Based on the Boussinesq Hypothesis 94 The ε Equation of the k-ε Model Based on the Boussinesq Hypothesis 95 The ε Equation of the k-ε Model for the Reynolds Stress Model 96 Reynolds Stress Model 96 Momentum Equation with Reynolds Stress Equations 99 SIMPLER Algorithm 05 Pressure Equation 05 Pressure Correction Equation 07

8 vii Differencing Scheme 0 Partial Differentials of Velocity Components at Scalar Nodes Solving the Equations 3 Convergence Criteria 7 Convergence Criterion by Comparing with the Values from the Last Iteration 7 Convergence Criterion by Comparing with a Specific Value SOFTWARE DEVELOPMENT History of Model Development -D Laminar Flow 3 -D Laminar Flow 3 3-D Uniform Flow Model with Momentum Equations and Continuity Equation 4 3-D Uniform Flow Model with the Boussinesq Hypothesis 5 Model Design 30 Input Data 3 Chezy Equations 3 Boundary Distance 35 Grid Generation 36 RMS Distance 37 Array Initialization 37 p - v - w Loop 38 Pressure Correction 39 u -k-ε Loops 39 User Interface and Graphical Display 39 Model Validation 47 Modeling Pitfalls and Solutions FINITE-VOLUME SIMULATION RESULTS 55 Introduction 55 Rectangular Cross Section 55 Simulation Cases 55 SVC as a Function of Channel Aspect Ratio 56 SVC as a Function of Channel Width 58 SVC as a Function of Longitudinal Bed Slope 60 SVC as a Function of Surface Roughness 6 SVC as a Function of Lateral Float Location 63 SVC as a Function of Float Submergence Depth below the Water Surface 64 SVC as a Function of Model Parameters 67 Compound Cross Sections 73 Simulation Cases 73 SVC as a Function of Channel Width 74

9 viii SVC as a Function of Longitudinal Bed Slope 76 SVC as a Function of Surface Roughness 76 SVC as a Function of Lateral Float Location 78 SVC as a Function of Float Submergence Depth below the Water Surface SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS 83 Summary 83 Conclusions 85 Model Validity 85 Applicability of SVC Values 86 Modeling Pitfalls and Solutions 88 Recommendations for Future Research 90 LITERATURE CITED 9 APPENDICES 97 APPENDIX A: STRAIN RATE TENSOR 98 APPENDIX B: FINITE DIFFERENCE EQUATIONS 0 APPENDIX C: FUNCTION VECTOR AND PARTIAL DERIVATIVES 08 APPENDIX D: USER INTERFACE AND GRAPHICAL RESULTS 9 Turbulence Model with the Boussinesq Hypothesis 30 Rectangular Cross Sections 30 Compound Cross Sections 34 Turbulence Model with ASM 38 Rectangular Cross Sections 38 Compound Cross Sections 4 Turbulence Model with ASM 46 Rectangular Cross Sections 46 Compound Cross Sections 50 Turbulence Model with ASM3 54 Rectangular Cross Sections 54 Compound Cross Sections 58 APPENDIX E: LABORATORY DATA 6 APPENDIX F: EXAMPLES OF CALCULATION RESULTS 85 VITA 33

10 LIST OF TABLES ix Table Page Turbulence Model (k-ε) Constants 30 Empirical Constants for the Various Turbulence Models 37 3 Number of Unknown Values, and Number of Non-zero Jacobian Matrix Values, for Each of the Turbulence Models 67 4 Turbulence Model Input Data 33 5 Adjusted Turbulence Model Parameters 49 6 ASM3 Model Results and Measured Laboratory Data for a -ft Rectangular Flume Width and a 5-cm Water Depth 5 7 Model Results and Laboratory Data for a 3-ft Rectangular Flume Width and a 0- cm Water Depth 53 8 SVC Coefficients Based on Cross-section Aspect Ratio (H/B) 87 9 SVC Coefficients Based on Surface Roughness Height (k s ) 87 0 Coefficients for the USBR-published SVCs in Rectangular Cross Sections 88 D. Input Data Text File for the Turbulence Model with the Boussinesq Hypothesis for Rectangular Cross Sections 3 D. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with the Boussinesq Hypothesis for Rectangular Cross Sections 3 D.3 Input Data Text File for the Turbulence Model with the Boussinesq Hypothesis for Compound Cross Sections 34 D.4 Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with the Boussinesq Hypothesis for Compound Cross Sections 36 D.5 Input Data Text File for the Turbulence Model with ASM for Rectangular Cross Sections 38 D.6 Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM for Rectangular Cross Sections 40 D.7 Input Data Text File of the Turbulence Model with ASM for Compound Cross Sections 4 D.8 Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM for Compound Cross Sections 44 D.9 Input Data Text File for the Turbulence Model with ASM for Rectangular Cross Sections 46 D.0 Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM for Rectangular Cross Sections 48

11 x D. Input Data Text File for the Turbulence Model with ASM for Compound Cross Sections 50 D. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM for Compound Cross Sections 5 D.3 Input Data Text File for the Turbulence Model with ASM3 for Rectangular Cross Sections 54 D.4 Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM3 for Rectangular Cross Sections 56 D.5 Input Data Text File for the Turbulence Model with ASM3 for Compound Cross Sections 58 D.6 Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM3 for Compound Cross Sections 60 E. Surface Velocity Coefficients of the Large Balls at Various Submergences 63 E. Surface Velocity Coefficients of the Ping-Pong Balls at Various Submergences 65 E.3 Surface Velocity Coefficients of the Papers at Various Diameters 67 E.4 Surface Velocity Coefficients for the -ft flume 69 E.5 Surface Velocity Coefficients for the 3-ft flume 76

12 LIST OF FIGURES xi Figure Page Application of the float method to measure surface velocity in an open channel (curves represent isovels) 3 Contours of constant velocity in a rectangular open-channel cross section and centerline velocity profile 7 3 Measured secondary currents and division line (solid line) in rectangular channels (Yang and Lim 997) 9 4 Calculation results of secondary currents from SSG model (Speziale and Gatski 997) 0 5 Definition sketch for calculating average distance of point P from solid wall surfaces 4 6 Flowchart of the turbulence model solution using the finite difference method 63 7 Definition of variables for the finite difference scheme, applied at node (k, j) in the channel cross section 65 8 Boundary of each type of finite difference scheme 66 9 Spiral movement of calculation point 70 0 Multiple solutions problem 7 Pattern of the initial guesses of v - w 76 Flow chart of the turbulence model as developed by Kra (00) 8 3 Computational cell configuration examples in a finite-volume grid at a cross section 85 4 The finite-volume computational cell configuration for the u (also for all scalar variables) 86 5 The finite-volume computational cell configuration for v 86 6 The finite-volume computational cell configuration for w 87 7 The finite-volume computational cell configurations for all three computational cell types (shaded area for u and scalars, sparse dot line for w and denser dot line for v ) Example of a 4 x 3 control volume 3 9 Incorrect directions of v - w at the two top corner regions of the cross section 6 0 Flowchart of the turbulence model solution using the finite volume method 3 Definition sketch for calculating average distance of point P from solid wall surfaces and from the water surface 37 First screen and file sub-menus of the turbulence model 40 3 Input data window for the turbulence model with ASM3 for a rectangular cross section 4

13 xii 4 Input data screen for turbulence model with ASM3 for compound cross section 4 5 Warning window for input data range violations 43 6 Warning window for incorrect string input data 43 7 Input screen for secondary-velocity arrow attributes 43 8 Sample arrowhead styles for graphical display of calculated secondary velocities 44 9 Example of the graphical display for a rectangular cross section Example of the graphical display for a symmetrical compound cross section 45 3 The simulation menu 46 3 Measured and calculated relationships between discharge and longitudinal bed slope (S o ) for a -ft rectangular flume width with a 0-cm water depth from the laboratory Comparison of surface velocity coefficient, λ, values from the ASM3 turbulence model and hydraulic laboratory data for a -ft rectangular flume width and a 5-cm water depth Comparison of calculated discharge from the ASM3 turbulence model, and measured discharge from a hydraulic laboratory for a -ft rectangular flume width and a 5-cm water depth Comparison of the SVC from ASM3, USBR, and laboratory data for a 3-ft rectangular flume width and a 0-cm water depth 5 36 Comparison of discharge from ASM3, the Chezy equation, and measured laboratory data for a 3-ft rectangular flume width and a 0-cm water depth 5 37 Relationship between calculated SVC and channel aspect ratio (H/B) for rectangular cross sections Relationship between calculated discharge and channel aspect ratio (H/B) for rectangular cross sections Relationship between calculated SVC and channel width (B) for rectangular cross sections Relationship between calculated discharge and channel width (B) for rectangular cross sections 59 4 Relationship between calculated SVC and longitudinal bed slope (S o ) for rectangular cross sections 60 4 Relationship between calculated discharge and longitudinal bed slope (S o ) for rectangular cross sections 6 43 Relationship between calculated SVC and surface roughness height (k s ) for rectangular cross sections 6 44 Relationship between calculated discharge and surface roughness height (k s ) for rectangular cross sections 6 45 Relationship between calculated SVC and lateral surface location for rectangular cross sections 63

14 xiii 46 Relationship between calculated SVC/SVC at center and lateral surface location, where B/ is the center of the cross section and B/8 is one eight of the base width from a vertical wall for rectangular cross sections Relationship between calculated SVC and float submergence depth at the channel centerline for rectangular cross sections Relationship between calculated SVC and float submergence depth at distance 3B/8 from a vertical wall for rectangular cross sections Relationship between calculated SVC and float submergence depth at distance B/4 from a vertical wall for rectangular cross sections Relationship between calculated SVC and float submergence depth at distance B/8 from a vertical wall for rectangular cross sections 66 5 The relationship between calculated SVC and α for a rectangular cross section 68 5 The relationship between calculated SVC and the model parameter α for a rectangular cross section The relationship between calculated SVC and β for a rectangular cross section The relationship between calculated SVC and β for a rectangular cross section The relationship between calculated SVC and c for a rectangular cross section The relationship between calculated SVC and c for a rectangular cross section The relationship between calculated SVC and c 3 for a rectangular cross section 7 58 The relationship between calculated SVC and σ k for a rectangular cross section 7 59 The relationship between calculated SVC and σ ε for a rectangular cross section 7 60 The relationship between calculated SVC and c ε for a rectangular cross section 7 6 The relationship between calculated SVC and c ε for a rectangular cross section 73 6 Relationship between calculated SVC and channel width (T) for compound cross sections Relationship between calculated discharge and channel width (T) for compound cross sections Relationship between calculated SVC and longitudinal bed slope (S o ) for compound cross sections Relationship between calculated discharge and longitudinal bed slope (S o ) for compound cross sections Relationship between calculated SVC and surface roughness height (k s ) for compound cross sections Relationship between calculated discharge and surface roughness height (k s ) for compound cross sections Relationship between calculated SVC and lateral surface location for compound cross sections 79

15 xiv 69 Relationship between calculated SVC/SVC at the center and other lateral surface locations, where T/ is the center of the cross section and T/8 is one-eight of the top width from a vertical wall for a compound cross sections Relationship between calculated SVC and float submergence depth at the channel centerline for compound cross sections 8 7 Relationship between calculated SVC and float submergence depth at distance 3T/8 from a vertical wall for compound cross sections 8 7 Relationship between calculated SVC and float submergence depth at distance T/4 from a vertical wall for compound cross sections 8 73 Relationship between calculated SVC and float submergence depth at distance T/8 from a vertical wall for compound cross sections 8 A. Deformation of a fluid element in the x-y plane 99 D. Input data screen for turbulence model with Boussinesq hypothesis for rectangular cross sections 30 D. Graphical results from the turbulence model with the Boussinesq hypothesis for rectangular cross sections 3 D.3 Input data screen for the turbulence model with the Boussinesq hypothesis for compound cross sections 34 D.4 Graphical results from the turbulence model with the Boussinesq hypothesis for compound cross sections 36 D.5 Input data screen for the turbulence model with ASM for rectangular cross sections 38 D.6 Graphical results from the turbulence model with ASM for rectangular cross sections 40 D.7 Input data screen for the turbulence model with ASM for compound cross sections 4 D.8 Graphical results from the turbulence model with ASM for compound cross sections 44 D.9 Input data screen for the turbulence model with ASM for rectangular cross sections 46 D.0 Graphical results from the turbulence model with ASM for rectangular cross sections 48 D. Input data screen for the turbulence model with ASM for compound cross sections 50 D. Graphical results from the turbulence model with ASM for compound cross sections 5 D.3 Input data screen for the turbulence model with ASM3 for rectangular cross sections 54 D.4 Graphical results from the turbulence model with ASM3 for rectangular cross sections 56 D.5 Input data screen for the turbulence model with ASM3 for compound cross sections 58 D.6 Graphical results from the turbulence model with ASM3 for compound cross sections 60 E. Surface velocity coefficients of the large balls for a 0-cm water depth 64

16 xv E. Surface velocity coefficients of the large balls for a 5-cm water depth 64 E.3 Surface velocity coefficients of the large balls for a 0-cm water depth 65 E.4 Surface velocity coefficients of the ping-pong balls for a 0-cm water depth 66 E.5 Surface velocity coefficients of the ping-pong balls for a 5-cm water depth 66 E.6 Surface velocity coefficients of the ping-pong balls for a 0-cm water depth 67 E.7 Surface velocity coefficients of the papers for a 0-cm water depth 68 E.8 Surface velocity coefficients of the papers for a 5-cm water depth 68 E.9 Surface velocity coefficients of the papers for a 0-cm water depth 69 E.0 Relationship between SVC and longitudinal bed slope (S o ) for -ft flume with a 5- cm water depth 7 E. Relationship between discharge and longitudinal bed slope (S o ) for -ft flume with a 5-cm water depth 7 E. Relationship between SVC and longitudinal bed slope (S o ) for -ft flume with a 0- cm water depth 7 E.3 Relationship between discharge and longitudinal bed slope (S o ) for -ft flume with a 0-cm water depth 7 E.4 Relationship between SVC and longitudinal bed slope (S o ) for -ft flume with a 5- cm water depth 73 E.5 Relationship between discharge and longitudinal bed slope (S o ) for -ft flume with a 5-cm water depth 73 E.6 Relationship between SVC and longitudinal bed slope (S o ) for -ft flume with a 0- cm water depth 74 E.7 Relationship between discharge and longitudinal bed slope (S o ) for -ft flume with a 0-cm water depth 74 E.8 Relationship between SVC and longitudinal bed slope (S o ) for -ft flume with a 5- cm water depth 75 E.9 Relationship between discharge and longitudinal bed slope (S o ) for -ft flume with a 5-cm water depth 75 E.0 Relationship between SVC and longitudinal bed slope (S o ) for 3-ft flume with a 5- cm water depth 77 E. Relationship between discharge and longitudinal bed slope (S o ) for 3-ft flume with a 5-cm water depth 77 E. Relationship between SVC and longitudinal bed slope (S o ) for 3-ft flume with a 0- cm water depth 78 E.3 Relationship between discharge and longitudinal bed slope (S o ) for 3-ft flume with a 0-cm water depth 78 E.4 Relationship between SVC and longitudinal bed slope (S o ) for 3-ft flume with a 5- cm water depth 79

17 xvi E.5 Relationship between discharge and longitudinal bed slope (S o ) for 3-ft flume with a 5-cm water depth 79 E.6 Relationship between SVC and longitudinal bed slope (S o ) for 3-ft flume with a 0- cm water depth 80 E.7 Relationship between discharge and longitudinal bed slope (S o ) for 3-ft flume with a 0-cm water depth 80 E.8 Relationship between SVC and longitudinal bed slope (S o ) for 3-ft flume with a 5- cm water depth 8 E.9 Relationship between discharge and longitudinal bed slope (S o ) for 3-ft flume with a 5-cm water depth 8 E.30 Relationship between SVC and longitudinal bed slope (S o ) for 3-ft flume with a 30- cm water depth 8 E.3 Relationship between discharge and longitudinal bed slope (S o ) for 3-ft flume with a 30-cm water depth 8 E.3 Relationship between SVC and longitudinal bed slope (S o ) for 3-ft flume with a 35- cm water depth 83 E.33 Relationship between discharge and longitudinal bed slope (S o ) for 3-ft flume with a 35-cm water depth 83 E.34 Relationship between SVC and longitudinal bed slope (S o ) for 3-ft flume with a 40- cm water depth 84 E.35 Relationship between discharge and longitudinal bed slope (S o ) for 3-ft flume with a 40-cm water depth 84 F. Plots of u -contour lines and secondary velocities for rectangular cross sections with H/B = 0.5 to.00, B =.0 m, S o = 0, and k s = 5 m 86 F. Plots of u -contour lines and secondary velocities for rectangular cross sections with H/B = 0.5, B = 0.5 to 5.0 m, S o = 0, and k s = 5 m 97 F.3 Plots of u -contour lines and secondary velocities for rectangular cross sections with H/B = 0.5, B =.0 m, S o = 003 to, and k s = 5 m 300 F.4 Plots of u -contour lines and secondary velocities for rectangular cross sections with H/B = 0.5, B =.0 m., S o = 0, and k s = to 0.0 m 307 F.5 Plots of u -contour lines and secondary velocities for compound cross sections with T = 0.5 to 5.0 m, H C = 0.5 m, S o = 0, and k s = 5 m 30 F.6 Plots of u -contour lines and secondary velocities for compound cross sections with T =.0 m, H C = 0.5 m, S o = 003 to, and k s = 5 m 33 F.7 Plots of u -contour lines and secondary velocities for compound cross sections with T =.0 m, H C = 0.5 m, S o = 0, and k s = to 0.0 m 30

18 NOTATION xvii The following symbols are used in this dissertation: Γ = diffusion coefficient (m /s); Π ij = pressure strain tensor (m /s 3 ); Π ij = non-linear turbulent part of pressure strain tensor (m /s 3 ); Π ij = mean-strain (or rapid) part of pressure strain tensor (m /s 3 ); Π ijw = wall-reflection terms (m /s 3 ); Π ijw = first term of wall-reflection terms (m /s 3 ); Π ijw = second term of wall-reflection terms (m /s 3 ); Ω ij = mean-rotation tensor (s - ); Φ(Δy), Φ(Δz) = first-order truncation errors; α = turbulence model constant; β, * β, β o, * β o = turbulence model constant; χ k = turbulence model parameter; χ ω = the vortex-stretching parameter; δ = boundary layer thickness (m); δ ij = the Kronecker delta (δ ij = for i = j, and δ ij = 0 for i j); Δt = time interval (s); Δx = distance between opposite faces of a control volume in x-direction (m); Δy sn = distance between opposite faces of a control volume in y-direction (m); Δy PN = distance between center of one control volume and its north neighbor (m); Δy SP = distance between center of one control volume and its south neighbor (m); Δz we = distance between opposite faces of a control volume in z-direction (m); Δz PE = distance between center of one control volume and its east neighbor (m); Δz WP = distance between center of one control volume and its west neighbor (m); ε = kinetic energy dissipation rate per unit mass (W/kg or m /s 3 ); ε f = kinetic energy dissipation rate per unit mass at water surface (W/kg or m /s 3 ); ε ij = Reynolds stress dissipation tensor per unit mass (W/kg or m /s 3 ); ε w = wall kinetic energy dissipation rate per unit mass (W/kg or m /s 3 ); γ, γ, γ, γ 3 = coefficient and exponent of the power law;

19 γ = turbulence model constant; κ = von Kármán constant; = turbulence scale length (m); λ = surface velocity coefficient; λ USBR = surface velocity coefficient published by the USBR; μ = molecular dynamic viscosity (N-s/m ); μ t = isotropic turbulent (eddy) viscosity (N-s/m ); μ t xy = anisotropic turbulent (eddy) viscosity for τ xy (N-s/m ); xviii μ t xz = anisotropic turbulent (eddy) viscosity for τ xz (N-s/m ); ν = kinematic molecular viscosity (m /s); ν t = kinematic turbulent (eddy) viscosity (m /s); ρ = fluid density (kg/m 3 ); σ, * σ = turbulence model constant; σ k, σ ε = k and ε Schmidt numbers; τ = shear stress (N/m ); τ ij = Reynolds stress tensor (N/m ); τ t = generic turbulent stress (N/m ); τ w = wall shear stress (N/m ); υ = turbulence velocity scale (m/s); ω = rate of dissipation of energy in unit volume and time, (/s); ξ = angle of longitudinal bed slope (radian); ψ = any of the variables; a = acceleration in the direction of the force (m/s ); a ψ P = coefficient of value at point P for variable ψ, in discretized equation; A = channel cross-sectional area (m ); A n, A s, A e, A w = north, south, east, and west control volume face areas (m ); A + = van Driest damping constant; b ψ = volumetric source term for variable ψ; B = rectangular channel width (m); B C = compound section main channel width (m); B L = compound section left side channel width (m);

20 B R = compound section right side channel width (m); B s = wall roughness parameter; c ε,c ε,c ε = turbulence model constants; c μ = isotropic turbulent viscosity coefficient; c μ, xy c μ = anisotropic turbulent viscosity coefficients; xz c, c, c 3, c D = turbulence model constants; c s* = turbulence model constants; c s c s = ratio of dy between node above water surface and the first node below water surface to dy between inner nodes = ratio of dy between the first and second nodes below water surface to dy between inner nodes c S = turbulence empirical constants; C, C, C, C 3, C 4 = coefficient and exponent of the power law; C = Chézy uniform flow equation roughness coefficient (m / /s); C D = turbulence model constants; C l = a dimensionless integration constant related to the thickness of the viscous sublayer C l = Coles wake-strength parameter; C m = friction velocity coefficient; CV = control volume; d 50 = median grain size diameter of the bed material (m); d 90 = the grain size diameter larger than 90% by weight of the bed material (m); d 99 = depth, from water surface at which u 0.99umax >= (m); D = the distance of the location of the depressed origin below the top of roughness elements (m) Dif v = molecular diffusion term of diffusive transport term (N/m/s); Dif ij = turbulent diffusion tensor of diffusive transport term (N/m/s); D ij = turbulent diffusion tensor (m /s 3 ); D ψ i = diffusion term for variable ψ in the i-direction; E r = wall roughness parameter; f = unspecified function; f = wall-damping function; f β = turbulence model constants; f = stress model parameter; xix

21 xx f = stress model parameter; F = force (N); F i = body force per unit mass of water in x i direction (N/kg); F r = Froude number; g = ratio of weight to mass (N/kg or m/s ); h a = root-mean-square reciprocal distances from water surface (m); H = total flow depth for rectangular cross section (m); H C = flow depth in main channel for compound cross section (m); H LR = flow depth in left and right channel for compound cross section (m); H r = ripple height (m); j = index from to 3; k = turbulence kinetic energy per unit mass (specific turbulence kinetic energy) (J/kg or m /s ); k f = turbulence kinetic energy per unit mass at water surface (J/kg or m /s ); k s = equivalent sand roughness height of channel wall (m); uk k + s = τ s ν, dimensionless equivalent sand roughness; k w = wall turbulence kinetic energy per unit mass (J/kg or m /s ); K b = roughness parameter; l = represents the terms (x, y, z) in each coordinate direction; l = dissipation length (m); L = characteristic length (m); m = mass (kg); n = Manning roughness; ny = number of grid nodes in the y-direction; nz = number of grid nodes in the z-direction; ˆn = unit normal vector to the control volume surface; p = pressure (N/m ); * p = guess for pressure (N/m ); p = average pressure (N/m ); p = pressure correction (N/m ); p = pressure fluctuation (N/m ); P = control volume center; Pe = dimensionless Peclet number;

22 xxi P ij = Reynolds stress production tensor (W/kg or m /s 3 ); P k = Production of turbulence kinetic energy (W/kg or m /s 3 ); P kb = Production of turbulence kinetic energy using Boussinesq hypothesis (W/kg or m /s 3 ); P ll = Rate of production of turbulence energy (W/kg or m /s 3 ); Q = discharge (m 3 /s); R = radius of pipe or half width of closed duct or full depth of open channel (m); Re = Reynolds number; R h = hydraulic radius (m); S = distance to the boundary segment that occupies the angle differential, dθ (m); S ψ = source term for variable ψ; S f = friction slope (m/m); S ij = angular rate of strain tensor (s - ); S ij = mean-strain-rate tensor (s - ); S ij = frame-indifferent derivative of S ij (s - ); S o = longitudinal bed slope (m/m); t = time (s); t ij = viscous stress tensor (N/m ); T = channel top width (m); u = velocity in the x-direction (m/s); u + = uu *, dimensionless velocity; u τ = τ w ρ, friction velocity (m/s); av u τ = grhs f, estimate of average u τ (m/s); u i = fluctuating velocity component in x i direction (m/s); u i = instantaneous velocity or velocity component in x i direction (m/s); u j = velocity component in the x j direction (m/s); u = average velocity in longitudinal (x-) direction (m/s); u avg = average u over channel cross section (m/s); u cv = u at the node representing its control volume (m/s); u i = average velocity in the x i direction (m/s); u max = maximum u (m/s);

23 u mean = mean of the bulk velocity in the longitudinal direction (m/s); u sm = average u at the middle of the water surface (m/s); u = velocity vector (m/s); v = velocity in the y-direction (m/s); v = velocity correction in the y-direction (m/s); v = average vertical (y-direction) component of secondary velocity (m/s); * v = guess for velocity in the y-direction (m/s); ˆv = pseudo-velocity in the y-direction (m/s); V = velocity (m); V = volume (m 3 ); w = velocity in the z-direction (m/s); v = velocity correction in the z-direction (m/s); w = average horizontal (z-direction) component of secondary velocity (m/s); * w = guess for velocity in the z-direction (m/s); ŵ = pseudo-velocity in the z-direction (m/s); x n = normal distance from wall (m); y = distance from wall in the y-direction (m); y + = dimensionless distance from wall in the y-direction; xxii y a = root-mean-square reciprocal distances from the solid walls (m); y n = uniform flow normal depth (m); y o = certain distance from wall (m); y w = average distance from solid wall (m); -D = two-dimensional; 3-D = three-dimensional; ASM = Algebraic Stress Model; CFD = Computational Fluid Dynamics; ID = SSG model with isotropic dissipation; IPM = Isotropisation-of-Production model; QIM = Quasi-Isotropic model; RANS = Reynolds-Averaged Navier-Stokes equations; SIMPLE = Semi-Implicit Method for Pressure-Linked Equations; SIMPLER = Revised Semi-Implicit Method for Pressure-Linked Equations;

24 SSG model = Speziale pressure strain model; SVC = surface velocity coefficient; and, USBR = United State Bureau of Reclamation. xxiii SUBSCRIPTS e, w, n, s = control volume surfaces in compass directions: E, W, N, S; E, W, N, S = centers of neighboring control volumes in compass directions; i = row; and, j = column.

25 CHAPTER INTRODUCTION Background Water management is a widely applied term in irrigation engineering and covers a number of design, operational, and maintenance aspects of irrigation systems. In one definition, among many others, water management is the knowledge of water user requirements and the application of such knowledge in the field. But in any definition, water management inherently involves the concept of measurement, because as with most resources, if it cannot be measured, it probably is not managed nearly as well as it could be. In fact, many would argue that management does not exist in the absence of quantitative measurements of a resource, be it financial, human, hydraulic, hydrologic, or other. Dozens of researchers have worked steadily over the past several decades on improved methods for estimating agricultural crop water requirements, or evapotranspiration. Research continues to this day, with a number of annual meetings of professionals where new and improved evapotranspiration techniques and calibrations are presented and discussed. The state of the technology is now at the point in which accurate hourly estimations of crop water use can be calculated, to as much as three significant digits, and irrigation scheduling programs for agricultural crops abound via the Internet and other information sources. The vast amount of knowledge about estimating crop water requirements continues to grow. But how useful are accurate estimations of requirements unless it is also possible to know how much water is applied to agricultural crops? What good is it to have precise recipe instructions in a cookbook without measuring cups and spoons to effectively apply the recipe in practice? Thus, the other side of the equation, so to speak, is knowledge about how much water is actually being applied through irrigation systems to agricultural areas. Estimations of water

26 application are usually much less accurate than predictions of crop water use, and in fact, are often nonexistent. This is partly due to a lack of water measurement capability, mostly in open channels, but also seen in pressurized pipe water delivery systems. Many irrigation canals have a single flow measurement structure, such as a calibrated flume, at the upstream end, but often with no other measurement structures downstream of that point. Many of the existing flow measurement structures have fallen into disrepair of one kind or another, requiring recalibration or even nullifying their application to flow measurement. Open channels are very prevalent in irrigation systems, both for conveying water from the source to the irrigated areas, and for distributing the water within the irrigated area. The simplest method to estimate flow rate in an open channel (other than simply guessing the flow rate by eye ) is the so-called float method, in which a floating object is placed in the channel. The time for the floating object to travel a specified distance (e.g. 0 m) is timed with a stopwatch, the distance is divided by the elapsed time, and the surface velocity is obtained. This is usually repeated in three or more trials to arrive at an average elapsed time for the specified distance (Fig. ). Then, a coefficient is applied to the surface velocity in an attempt to estimate the average flow velocity of the cross section, and this is multiplied by the area of the channel cross section to yield a volumetric flow rate (e.g. lps, m 3 /s, cfs). This is a quick and simple flow measurement method, requiring almost no equipment (other than a tape measure and stop watch), but is highly inaccurate because of the uncertainty in the value of the coefficient which converts the surface velocity to average velocity. If this uncertainty is reduced, the float method could yield sufficiently accurate estimates of openchannel flow rate, perhaps up to three significant digits. Then a technician could quickly and easily determine flow rate values based on the float method, and water measurement could be feasible where it currently is not. Such a capability would go far in quantifying and subsequently

27 3 Fig.. Application of the float method to measure surface velocity in an open channel (curves represent isovels) improving the management of water resources, especially in the area of agricultural irrigation, which is the largest water user in most arid and semiarid areas of the world. Currently, surface velocity discharge coefficients are developed from velocity profile measurements, or from vertical velocity distribution assumptions. The governing equations for calculating the velocity distribution in open channels have been known for many years, particularly with regard to laminar flow. However, the flow regime in most practical applications of open-channel flow is turbulent, which involves more complicated hydraulic equations. The main hindrance to the application of these equations to practical situations has been the unavailability of an appropriate mathematical turbulence model. In recent years, many turbulence models have been developed for simulating turbulence flow situations. The performance of many turbulence flow methods has been published, but detailed definitions of the methods are not included in publications. Nevertheless, with a valid mathematical turbulence model, surface velocity coefficients for the float method can be developed from a calculated velocity distribution in a channel cross section.

28 4 The USBR (997) published a table of surface velocity coefficients for application of the float method to discharge measurement in open channels. The coefficients are a unique function of the average depth of water in the channel at the measurement location. However, other hydraulic factors such as longitudinal bed slope, channel wall roughness, channel uniformity, and channel cross-sectional shape, among others, could possibly affect the true value of the coefficient. The float method, using the USBR (997) coefficients, sometimes yields a discharge within 5% of the current metering or calibrated measurement flume results, but more often than not, the discharge difference might be 5 to 0%, or more. Thus, based on field experience, it is concluded that more factors must be taken into account than just the average water depth in determining the surface velocity coefficient. Objectives The main objective of the proposed research was to develop the capability to theoretically calibrate the float method in different prismatic open channels, allowing the application of the float method with significantly greater accuracy than that which was previously available. The specific objectives of this research were:. To develop an enhanced three-dimensional mathematical model for calculated velocity profiles in steady-state, uniform open-channel flow;. To apply the model to the analysis of surface velocity coefficients as used in the float method for estimating open-channel discharge; and, 3. To provide guidelines for determining surface velocity coefficients for the application of the float method of open-channel discharge estimation.

29 Benefits of the Study 5 Nowadays, sophisticated methods can be implemented on digital computers, such that a hydraulic model will be able to conveniently provide surface velocity coefficients for a wide range of conditions in which the float method might be appropriate. Along with the surface velocity measured by the float method, the velocity distribution and the discharge at the cross section can be calculated. With such a mathematical model for determining surface velocity coefficient values, more expensive flow measurement methods, such as flumes, weirs, and orifices may not be required for flow measurement in many cases.

30 CHAPTER 6 LITERATURE REVIEW This literature review presents relevant fluid mechanics modeling methodologies and approaches in chronological order based on the date of publication. The primary emphasis of this chapter concerns the Reynolds-Averaged Navier-Stokes (RANS) equations. The concept of velocity distribution in a cross section of turbulent open-channel flow is presented first. Then, the basic governing equations and the development of the methodologies for calculating the velocity distribution in a channel cross section are reviewed, along with the turbulence modeling associated with these methodologies. Turbulence models are a required part of the RANS equations and greatly increase the difficulty of the modeling effort, compared to the basic Navier-Stokes equations which pertain only to laminar flow conditions. There are four main categories of turbulence models: Algebraic ( Zero-Equation ) models, One-Equation models, Two-Equation models, and Stress- Transport models. These are presented in order of increasing complexity and completeness. Each of these turbulence models will be discussed in detail in this chapter and in Chapter 3. Velocity Distribution The main difference in the velocity distribution at a cross section between laminar and turbulent flow is the location of the maximum velocity along the longitudinal bed slope (xdirection, or streamwise direction). For most turbulent flow situations, the maximum streamwise velocity occurs somewhat below the water surface, while in laminar flow, the maximum velocity occurs at the water surface. However, this distinction may not hold true for turbulent flow with shallow water depth and high velocities, in which case the maximum velocity may be near or at the water surface (French 985). By observation, the maximum velocity occurs below water surface when the aspect ratio B/H is less than about 5 (Nezu 005). Laboratory and field data

31 7 from many researchers show that the maximum streamwise velocity is found at about 5 to 5% of the water depth below the water surface. Typical isovel (contour lines of velocity in the x- direction) and velocity profiles at the center line of a rectangular cross section are shown in Fig.. C L y Velocity contour Lines u Velocity distribution at center line Fig.. Contours of constant velocity in a rectangular open-channel cross section and centerline velocity profile The lowering of maximum velocity to a point below the water surface with turbulent flow is due to the movement of water in the transverse directions, which is referred to as secondary flow. The bulk of the secondary flow generally rotates in a clockwise direction on the left side of the cross section, and in a counterclockwise direction on the right side. Henderson (966) and Schlichting (987) observed the lowering of maximum velocity and suggested that the circulation of the secondary current causes this phenomenon. This means that small amounts of water momentum in the longitudinal direction are transferred from both sides of the cross section to the center by secondary flow. Consequently, water momentum along the main flow is reduced and has the effect of slowing down the main (streamwise) flow. The location of maximum velocity which falls below the water surface is also correlated with the aspect ratio (depth to width) of the cross section. The relation is that the depth of the maximum velocity location is proportional to the cross-sectional aspect ratio (Schlichting 987; Montes 998).

32 8 Secondary currents only occur in turbulent flow. The secondary currents are also known as turbulence-driven secondary motions (Naot and Rodi 98). However, this secondary current is a completely different phenomenon from that of turbulent fluid movement. Turbulent movement has high-frequency velocity fluctuations with low velocity magnitudes. The average value of the fluctuating velocity is equal to zero comparing with the mean flow direction. In contrast with the secondary current, the average value is not equal to zero and it also does not change rapidly (Brenkert 960). In the case of straight channels with non-buoyant flow, the secondary current is the result of the gradients of the inequality between the normal turbulent stresses in the cross section (Brundrett and Baines 964; Gessner and Jones 965; Perkins 970). This means that the motion of secondary flow cannot be predicted by laminar theory, which is based on an isotropic eddy viscosity (Naot and Rodi 98). Secondary Currents Three examples of secondary flow measurements in rectangular channels are shown in Fig. 3 (Yang and Lim 997). The example pictures show secondary currents in half of the cross sections. There are two main loops of the secondary current: upper loop rotated in clockwise direction, and lower loop rotated in counterclockwise direction. The pattern of the secondary current is also slightly depending on the aspect ratio between water depth and channel width (H/B). The solid line in the figure is so called division line which represents a line of zero Reynolds shear stress in a cross section. This means no energy transfer across this line.

33 9 (a) H/B = (b) H/B = 0.5 (c) H/B = 0.7 Fig. 3. Measured secondary currents and division line (solid line) in rectangular channels (Yang and Lim 997)

34 The first turbulence model which had the capability to calculate the secondary current 0 was developed for channel flow by Launder and Ying (973). After that, many turbulence models had been developed by many researchers such as Naot et al. (974), Tatchell (975), Chiu et al. (978), Gessner and Emery (977), Chiu et al. (978), and Neti and Eichhorn (979). Speziale and Gatski (997) show the calculation results of secondary currents based on the Speziale et al. (99) pressure strain model (SSG model). Figure 4 shows two different patterns of secondary currents depending on model component: SSG model with the new algebraic anisotropic dissipation rate model (ADRM), and SSG model with isotropic dissipation (ID) (calculated by Mompean et al. 996). (a) SSG model with ADRM (b) SSG model with ID Fig. 4. Calculation results of secondary currents from SSG model (Speziale and Gatski 997) However, the first person who successfully developed a turbulence model which could simulate the dip of the maximum velocity below water surface is Reece (976). His model gives good calibration result compared with measurement data from Nikuradse (93). Another successful turbulence model was developed by Naot and Rodi (98). They derived algebraic stress equations to use in the model which can simulate the depression of maximum velocity and

35 calculate both main flow and secondary current in open channel. Goring et al. (997) developed a turbulence model based on the Navier-Stokes equations. The model can simulate the distribution of velocity in a natural river cross section; however, it cannot predict the depression of maximum velocity. Kra (00) developed a turbulence model based on the Naot and Rodi (98) equations and successes to simulate the depression of maximum velocity and secondary currents flow. However, his model has some unexplained and unexpected results in which the depression (below the water surface) of the maximum stream-wise velocity does not occur, in some cases. Governing Equations A fundamental law of nature is the law of conservation of mass which states that the mass in a close system or control volume must remain constant, and that mass cannot be created or destroyed. Following is a version of this law in equation form: d ρ dv = 0 dt (.) V where ρ is the fluid density (kg/m³); t is time (s); and V is the volume (m 3 ) over which the integration is performed. The Reynolds Transport Theorem, which asserts that the control volume does not move with the fluid, can be used in the application of the conservation of energy law. The law of conservation of mass for an infinitesimal control volume, V, in a water stream can be applied by the summation of the rate of change of mass in a control volume, where net mass flux over the control surface, S, is zero: V ρ dv t + ρu nˆ ds = 0 S (.) where u is the velocity vector (m/s); and, ˆn is the unit normal vector at the control volume surface. In Cartesian coordinates, this equation is simplified to:

36 ( ρu) ( ρv) ( ρw) ρ = 0 t x y z (.3) which can be expressed in an abbreviated form as: ( ρu j ) ρ + = 0 t x j (.4) where u j is the velocity component in the x j direction (m/s); and, j is an index from to 3. For incompressible flow, density is constant, whereby the equation for conservation of mass (Eq..4) is reduced to: u x j j = 0 (.5) To prevent ambiguity when using the equation of mass conservation in other equations where the subscript j is used to repeat the terms for each direction, the subscript j in the equation of mass conservation (Eq..5) is changed to k, as follows: u v w uk + + = x y z x k 0 (.6) The general equation of motion for the unbalanced forces acting on the water body is Newton s second law which applied to the water flow in open channel. This well known equation can be found in general fluid mechanic text books which is: F = ma (.7) where F is the sum of the forces or net force (N); m is mass (kg); and, a is acceleration in the same direction of the net force (m/s ). Another basic equation is the conservation of momentum. This equation describes the balance of the inertia force, extraneous forces, pressure forces, and viscous forces on an infinitesimal fluid element which is expressed as:

37 Dui p tij ρ = ρf i + Dt x i xj extraneous inertia forces pressure viscous forces forces 3 (.8) where F i is the component of extraneous force per unit mass of water in the x i direction (N/kg); p is pressure (N/m ); and t ij is the viscous stress tensor (N/m ). Equation.8 can be written in words as: inertia equals gravity forces minus pressure forces, plus viscous forces. The viscous stress tensor is defined by: t ij δij = μ Sij 3 u x k k (.9) where μ is molecular dynamic viscosity (N s/m ); the differential ( u / x ) term is the volumetric dilation rate; δ ij is the Kronecker delta (δ ij = for i = j, and δ ij = 0 for i j); and, S ij is the angular rate-of-strain tensor (s - ), defined as: k k S ij u u i j = + x j x i (.0) which is symmetric for isotropic liquids, as is the case for plain water, such that S ij = S ji. The angular rate-of-strain tensor is described in detail in Appendix A. Finally, the three Navier-Stokes equations are derived by substituting Eqs..0 and.9 into Eq..8, and using Eq..5 for an incompressible fluid which can be written in compressed form as: u u p u ρ ρ ρ μ t x x x i i i + uj = Fi + j i j (.) where u i is the instantaneous velocity (m/s) for i = x, y, z directions; and, j is also a subscript for each of the three dimensional axes in Cartesian coordinates.

38 The RANS Equations 4 For turbulent flow which is the most interesting flow scheme, the mean and fluctuating parts should be separated by using time-averaging approach (Reynolds 895), otherwise, the Navier-Stokes equations could not be applied for developing a turbulence model. ui = ui + u i (.) where u i is the instantaneous velocity, u i is mean flow; and u i is fluctuation component. All terms have units of m/s. There are many forms of Reynolds averaging which are involving an integral or a summation, but only three forms of Reynolds averaging are commonly used in turbulence modeling: () the time average; () the spatial average; and, (3) the ensemble average. For stationary turbulent flows in which the average of the turbulent flows does not vary with time, such as with steady flow in an open channel, the most appropriate form of the Reynolds average is the time averaging of the turbulent fluctuations in the steady flow (Wilcox 006). The time average of ui may be defined by: u t+δt u i ( x =, t ) dt t t i i Δ t Δ (.3) where Δt is the time interval (s); and, x i is the distance in i direction. The time interval should be very long compared with the maximum period of the turbulent fluctuations, but short when compared with the period of the variation in mean motion with time (which, by definition, is zero for steady-state flow). By applying time-averaging approach, the continuity (conservation of mass) equation (Eq..4) becomes: ( ρuj ρ u j) ρ + + ' = 0 t x j (.4) For incompressible flow, such as water, continuity equation will reduce to be:

39 u k x k = 0 5 (.5) Also taking the time-averaging approach applies to the momentum equation (Eq..8) resulting in the Reynolds-Average Navier-Stokes (RANS) equations. The RANS equations for compressible flow are: p ( ρui + ρ ui) + ( ρuu i j + uiρ uj) = ρfi + ( tij ujρ ui ρuu i j ρ uu i j) t x x x j i j (.6) where t ij is the viscous stress tensor (N/m ), which is defined as follows: t ij u u i j u k = μ + δij x j x i 3 xk (.7) For incompressible flow, the RANS equations are reduced to a simplified form of Eq..6, as shown below: u + = + + t x j xi xj xj p ui ( uu i j) Fi ( uu i j) i ρ ρ μ ρ (.8) or, Du p u i i ρ = ρfi + μ ρuu i j Dt xi x j x j (.9) The term ρ uu i j is generally referred to as the Reynolds stress, τ ij, which physically represents the transport of x i -momentum of a fluctuating fluid particle through an area normal to the x j -axis. It is analogous to stress on a fluid element, and it exists only for turbulent flow. u u p u τ ρ ρ ρ μ t x x x x i i i ij + uj = Fi + + j i j j (.0)

40 Reynolds Stress Equation 6 To close (or complete) the RANS system of equations, additional equations must be introduced. The Reynolds stress equation can be established by taking the moment of the Navier- Stokes equation (Wilcox 006). This procedure is done by taking the time average of the product of the Navier-Stokes equation and the fluctuating velocity. The result is: τij τij uj ui u u i j u u i j + ul = τil τ jl + μ + p + t xl xl xl xl x l xj x i τ ij + ν + ρuu i ju l xl xl (.) The equation can be rearranged as: τ ij t τ + u = Diff uu ρp + ρε ρπ ( ) ij l i j ij ij ij xl (.) Each term of the equation and the new unknowns generated from this equation, which will require additional equations, is discussed below. Turbulence Modeling Turbulence models have been studied and developed by many researchers for several decades. Many turbulence theories have been developed, and attempts have been made to apply them in mathematical models. However, none of the turbulence theories are complete (Kra 00). Only in laminar flow have the Navier-Stokes equations been successfully applied to obtain an exact solution because in such cases the flow regime is simple and the non-linear terms in the Navier-Stokes equations can be ignored. For turbulent flow, the Navier-Stokes equations cannot be directly applied because of the random fluctuations of the various flow properties of turbulence itself. Reynolds (895) suggested that all quantities could be expressed as the summation of mean and fluctuating parts. By using the time-averaging approach, nine additional Reynolds shear stresses are added as unknown quantities and this makes the number of unknown

41 7 greater than the number of equations. To close the system of equations, more equations must be established. The effects of turbulence on the mean flow are simulated by solving the closure problem through the application of turbulence models (Rodi 984). Algebraic Models Algebraic models are the simplest kind of turbulence model, and are based on the mixinglength hypothesis. In 877 the concept of eddy viscosity was introduced by Boussinesq. After that, in 895, Reynolds published the results of his turbulence research. The research has proven to be an important part of all subsequent developments. In the beginning of twentieth century, Prandtl (904) introduced the term boundary layer, and after that, in 95, he introduced the mixing length, based on dimensional analysis and an analogy to the mean free path in gas dynamics. After that, many researchers, including von Karman (93), used the mixing-length concept as a basis for turbulence models. In 956, Van Driest used a viscous damping correction to improve the mixing-length model (Wilcox 006). Mixing-Length Hypothesis. The standard log-law is a set of equations that can be used to predict the velocity distribution along a certain distance normal to a solid boundary, or a wall. Thus, in this research, a wall refers to any solid boundary, including vertical walls and horizontal floor surfaces. The standard log-law is useful in hydraulic modeling, but it cannot be applied to cover all of the near-wall region, down to the wall itself (Buschmann 005). To close the gap between the log-law layer and the wall, Prandtl (95) proposed a mixing-length approach as a simplified model of turbulent fluid motion (Wilcox 006). The turbulent (eddy) viscosity is equal to the square of mixing length,, multiplied by the absolute value of velocity gradient perpendicular to the mean flow direction. The turbulent eddy viscosity, μ t, is:

42 μ = ρ t u y 8 (.3) where u is the mean velocity in the flow direction (m/s); and, y is distance orthogonal to the mean flow direction. The turbulent stress with Prandtl's mixing length hypothesis will become: τ du du dy dy t = ρ (.4) Rodi (984) proposed a turbulent viscosity equation for general flow regime as: u u u μ = ρ (.5) t i i j + x x j j xi For one-dimensional uniform flow in a cross section without secondary current, Goring et al. (997) modified the equation for turbulent viscosity to be: μ u u t = ρ + y (.6) z For 3-D thin shear flows, Tannehill et al. (997) interpreted Prandtl s formula for threedimensional flow in the thin shear layer as: μ u w t = ρ + y y (.7) von Kármán (93) (see Schlichting and Gersten 000) used a similarity hypothesis to developed a turbulence model which the mixing length can be calculated from: du d u = κ (.8) dy dy where κ is the empirical von Kármán constant, usually about 0.40 to 0.4. Nikuradse (93) measured pipe flow and duct data to calibrate the mixing length. It is interesting that his results show that the mixing length is independent of the Reynolds number for Re > 0 5 (Schlichting 987).

43 9 4 yw y = R w (.9) R R where y w is the distance from wall (m), and R is pipe radius, half-width of closed duct, or open channel depth (m) (Rodi 984; Schlichting 987). The distribution of mixing length close to the wall approaches the equation proposed by Prandtl (Schlichting 987). = κ y w (.30) In the region close to the wall, the turbulent fluctuations are depleted and equal to zero (Weigand 005). Van Driest (956) introduced the Van Driest damping term into the mixing length equation (Eq..30) in order to enable the equation to be applied throughout the calculation region. The mixing length equation becomes: where, + + ( y / A ) = κ yw e (.3) y yu ν + w τ = (.3) τ u w τ = (.33) ρ where y + is dimensionless distance from the wall; A + is the Van Driest damping constant, which is equal to 6; y w is the normal distance from wall (m); u τ is friction velocity (m/s); and, τ w is wall shear stress (N/m ). Empirical Velocity Laws. The velocity profile for a turbulent boundary layer at a solid surface (e.g. wall or floor) can be defined using three adjacent regions by using the relationship between dimensionless velocity, u +, and dimensionless distance from wall in the y-direction, y +. These three regions are known in the technical literature as the viscous sublayer (at the solid surface), the log layer, and the defect layer. In the viscous sublayer, the viscous force is predominant, while the turbulent shear stresses play a major role in the defect layer. The log

44 0 layer is an overlap region between the viscous and defect layers. However, there is small region between the viscous and log layers. This layer is referred as the buffer region and has y + between 5 and 30 (Hanifi 999). Law of the Wall. The streamwise velocity in turbulent flows near a solid wall can be empirically determined by the law of the wall. This law asserts that the average fluid velocity in the boundary layer varies logarithmically with distance from the wall surface, and is characterized by the log layer at a boundary: ( ) + + u = f y (.34) The dimensionless velocity, u +, and dimensionless normal distance from the wall, y +, are defined as: + uy ( ) τ w + uy τ u = ; uτ = ; y = (.35) u ρ ν τ The laminar sub-layer, or viscous sub-layer, which has the approximate range of y + < 5, is found in the region closest to the wall (solid surface). In this region, the shear stress equals the wall shear stress, which is constant for steady flow. Thus, the dimensionless velocity is directly proportional to the dimensionless distance from the wall. Equation.34 changes to the following form in the laminar sub-layer: u = y (.36) + + (Kay and Nedderman 985; Versteeg and Malalasekera 007) and is supported by laboratory measurements. Further from the wall, beyond the viscous sub-layer, is what has been termed the loglaw layer, or the log layer. This layer is from the edge of the viscous sub-layer to a dimensionless distance from the wall of 500 to 750 (y + = 500 to 750), according to laboratory measurements. Alfrink and Rijn (983) used uk τ s ν = 60 to define the boundary (or threshold)

45 of fully turbulent flow. Kirkgöz (989) defined the range of the log-law layer as 50 to 80 uy τ ν 00 to 600. The wall is still the major influence on the layer, there is not much effect from viscous stress, and the velocity distribution is governed by the log-law. In this layer, the effects of viscosity and turbulent flow are both important (Versteeg and Malalasekera 007). Thus, the law of the wall (Eq..34) can be written in the following form: + u = lny + + Cl (.37) κ where C l is a dimensionless integration constant related to the thickness of the viscous sublayer; for smooth surfaces, C l 5.0 (or, more generally, in the range of 4.9 to 7.0). Kirkgöz (989) reported that the best value of C l was 5.5. For surfaces with wall roughness elements of average height k s (m), the value of C l is a function of the dimensionless roughness height, k + s, where in which, + + Cl = 8.5 ln ks, for ks >> (.38) κ k uk ν + τ s s = (.39) Liu (00) suggested the following k s values according to different types of experiments: k s 0.0, concrete bottom = ( -0 ) d50, flat sand bed ( ) H r, bed with sand ripples (.40) where d 50 is the median grain size diameter of the bed material (m); and, H r = ripple height (m). Also, the equivalent roughness, k s can be calculated from the Manning roughness, n, using the Strickler law: ks ( 4.04n) 6 = (.4)

46 Then, the law of the wall in the viscous sub-layer with a completely rough wall is: y u = ln κ ks (.4) Equation.37 can be rewritten in the following form: u = ln ( E y ) (.43) κ r + + where E r is a wall roughness parameter. For a smooth wall, E r is equal to 9.0. In general, E r can be calculated by: E r ( κ B ) exp s = (.44) + k s where B s is a roughness parameter, which can be estimated by Krishnappan and Lau (986) as follows: ( ) + ( ln ) exp + 0.7( ln ) exp 0.7( ln ) Bs = + ks ks + k s (.45) Finally, the farthest region from the wall is the defect layer, which has the range y + > 500 to y + > 750. The edge of defect layer depends on the shape of the cross section. The beginning of the defect layer is the end of the boundary layer for flow over a flat plate, the center of the pipe (for flow in a full pipe), or the free surface in the case of an open channel. A new item can be added to the law of the wall as it is approaching the free stream, and is called the law of the wake : + + Cl π y u = lny + Cl + sin κ κ δ (.46) where C l is Coles wake-strength parameter (Coles and Hirst 969), usually 0.6 for constant pressure; and, δ is the of the boundary layer thickness (m).

47 Darcy (857), von Kármán (93), and Clauser (956) found that the velocity-defect 3 law is well-correlated with the velocity distribution in this region. The velocity-defect law can be called the Clauser defect law : u u y = f δ max u τ (.47) where f is an unspecified function (Versteeg and Malalasekera 007; Wilcox 000). From Eqs..35 and.37, velocity from a certain distance from the wall can be found by the Prandtl-von Kármán universal velocity distribution law as: u y τ u = ln κ y o (.48) where u τ is friction velocity; and, y o Cmν for hydraulicallysmooth walls, Cm =/9 = uτ (.49) Ck for hydraulicallyrough walls, C=/30 m s m where C m is an empirical coefficient (French 985); k s is the equivalent sand roughness height of the wall (m). The roughness height has a relation with the grain size distribution of the wall surface material as: ks = 3d (.50) 90 where d 90 is the grain size diameter that is larger than 90% by weight of the bed material (Schlichting 987; Olsen 000). The bed shear velocity is estimated by: u = τ grhs (.5) f where R h is hydraulic radius (m); and, S f is friction slope (m/m) (Chow 973). For turbulent flow with smooth wall surfaces, the origin of the logarithmic velocity profile starts at the wall surface. However, the profile origin shifts beneath the top of the roughness wall material in the case of a

48 4 rough wall surface. Smart (999) found that the velocity profile matched the measured data with the depressed distance equal to: D= 0.k s (.5) Power Law. There is yet another alternative law (or, more specifically, approach) to mathematically describe the velocity distribution in the log-law layer: this is the power-law layer. Sarma et al. (983) devised an experiment for calibrating the parameters of the power law. The cross section is divided into three regions: wall region, corner region, and outer region. The power law of wall was applied to the region for both inner region of the bed and the sidewall, and it assumes that the velocity gradient is dependent on molecular viscosity within the range of arbitrarily high Reynolds numbers (Barenblatt and Prostokishin 993), in contrast with the loglaw which considers the velocity gradient to be independent of molecular viscosity. Barenblatt (993) postulated the general form of the power-law as: u ( ) γ = C y (.53) + + where C and γ are the coefficient and exponent of the power law, which were found to be /7 and 8.3, respectively. The experimental data show that there is no significant effect of the Froude number on the equation (Sarma et al. 983). However, Barenblatt and Prostokishin (993) suggested that C and γ are functions of the Reynolds number: C3 C4 C( Re) = Cln Re+ C + + (.54) ln Re ( ) ( ) ( ln Re) γ γ γ 3 γ ( Re ) = (.55) 3 ln Re ln Re ln Re where C, C, C 3, C 4, γ, γ, and γ 3 are constant parameters; and, Re is the Reynolds number (ratio of inertial to viscous forces). Many different values of the parameters have been developed by other researchers. But the proposed parameter values are quite different, such as C = 0.577, C =

49 .50, γ =.5, C 3 = C 4 = γ = γ 3 = 0 (Barenblatt and Prostokishin 993), and C = , C = , γ =.085, γ = 6.535, C 3 = C 4 = γ 3 = 0 (Zagarola et al. 997). Subsequently, Balachandar et al. (00) found that for open-channel flows, the parameters C and γ have no significant relationship with the Reynolds number, and are constant at and 0.55, respectively. Also, the effect of surface roughness was included in the power law as follows: ( ) + + γ + u = C y Δ u (.56) The values of Δu + for the three rough surfaces (wire mesh, sand d 50 = 0.58 mm, and sand d 50 =.0 mm) are between 8 and, with an average of 0 for wire mesh (nominal diameter of 0.6 mm, with wires spaced 7.5 mm on centers), between 4 and 7 with an average of 6 for sand surfaces with median diameter of. mm, and between 4.5 and 6.0 with an average of 5.5 for sand surfaces with median diameter of 0.58 mm. The velocity distribution in the outer region of the bed (including the inner region of the sidewall) is represented by (Sarma et al. 983): u u = α y H max Kb u τ (.57) where K b = 9.6 for flat plate, or 6.3 for a rough, wide open channel; α is the depression of the maximum velocity; y is distance from wall in the y-direction (m), H is flow depth (m); and, u max is the maximum velocity (m/s). One-Equation Models Prandtl (945) proposed that the turbulent kinetic energy per unit mass, k, is based on velocity scale: uu j j k = ( uu + vv + ww ) = (.58)

50 The Reynolds stress equations (Eq..) can be shown to lead to the transport equation for 6 turbulent kinetic energy: k k τ ij u i + u j = ε + t x ρ x x unsteady term or storage term convection j j j production of turbulent kinetic energy dissipation rate of turbulent kinetic energy k uuu i i j pu j ν x j ρ (.59) Molecular Turbulence Pressure Diffusion Transport Diffusion where ε is energy dissipation rate (principally to sensible heat) per unit mass (W/kg), which is defined as: ui ui ε ν = xk xk (.60) Mansour et al. (988) used Direct Numerical Simulation (DNS) to simulate the equation and suggested that the last two terms of Eq..59 be called Turbulence Transport and Pressure Diffusion. These values were small for simple flows and could be approximated as: uuu i i j pu j ν t k + = ρ σ x k j (.6) Then, the turbulent kinetic energy equation can be written as: k k τ ij u i νt k + u ν j = ε + + t xj ρ xj x σ j k x j unsteady term convection production of turbulent kinetic energy dissipation rate of turbulent kinetic energy molecular diffusion (.6) Prandtl (945) postulated an equation for calculating the value of energy dissipation based on the mixing-length model: ε = C D k 3 (.63) Later, Emmons (954) and Glushko (965) suggested the range of C D from 0.07 to For this condition, the turbulent kinetic energy equation is the so-called Prandtl s One-Equation model, which is as follows:

51 7 3 k k τ ij u i k ν t k + uj = CD + ν + t xj ρ x j xj σk xj (.64) and the kinematic eddy viscosity is: k (.65) vt = k = CD ε where the Reynolds-stress tensor can be estimated using the Boussinesq approximation as: τ ij = ρν T Sij ρkδij (.66) 3 Two-Equation Models k-ε Turbulence Models. The k-ε model has been the most commonly applied turbulence model by researchers over the last several years. Many researchers have developed k-ε models, but the first well-known version was developed by Jones and Launder (97). After that, Launder and Sharma (974) proposed the closure coefficients for k-ε model which is referred as the Standard k-ε model (Wilcox 006). The quotient k/ε represents a characteristic of turbulence decay time (s). Turbulence velocity scale, υ, and turbulence length scale,, relate to k and ε by: 3/ k υ = k, = c μ (.67) ε where k is turbulent kinetic energy per unit mass (J/kg); ε is kinetic energy dissipation rate per unit mass (W/kg); and c μ is turbulent viscosity coefficient which is equal to The eddy viscosity is calculated as: k μt = cμρυ = ρcμ (.68) ε The exact ε equation can be derived by taking the moment of the Navier-Stokes equation: u i ν Ν ( ui ) = 0 x x j j (.69)

52 Then, the exact definition of ε is: 8 ε ε u i u u i j u k u k u i ui + uj = ν + νu k t xj x j xk xk xi x j xj xk xj Production of Dissipation u i u i u k u i u i ν ν xk xm xm xk xm xk xm Dissipation of Dissipation ε u i u u i ν p j + ν νu j x j xj xm xm ρ xm x m Molecular Diffusion of Dissipation and Turbulent Transport of Dissipation (.70) The terms on the right-hand-side of the equation are Production of Dissipation, Dissipation of Dissipation, and the sum of Molecular Diffusion of Dissipation and Turbulent Transport of Dissipation, respectively, as seen in Eq..70. The exact equations of k and ε (Eqs..59 and.70) are complicated and have many new unknowns. By using closure coefficients, the standard modeled k equation is: k k τ ij u i ν t k + u ν j = ε + + t xj ρ xj x σ j k x j unsteady term convection term production of turbulent kinetic energy dissipation rate of turbulent kinetic energy diffusion of turbulent kinetic energy (.7) For the k-ε turbulence models, the Reynolds stresses are approximated by employing the Boussinesq hypothesis: τ μ u u i j ij = t + k ij xj x i 3 ρ δ (.7) where σ k is the k Schmidt number, which is the effective ratio of the diffusion of turbulence momentum to the diffusion of turbulent kinetic energy. Emmons (954) and Glushko (965) suggested a value of σ k =.0. The variable ν t is called kinematic turbulent viscosity (m /s), which is related to turbulence kinetic energy, k, and rate of energy dissipation, ε, by:

53 9 k ν t = c μ (.73) ε where c μ is the isotropic turbulence viscosity coefficient, and ranges from 0.07 to By parameterizing some terms as function of large-eddy scales, the standard approximate ε equation is: ε ε τ ij ε ui ε ν t ε + uj = cε c ν ε + + t x x j ρ k xj k x σ j ε j unsteady term convection production of dissipation of diffusion of dissipation term dissipation dissipation (.74) where c ε, and c ε are constants (.44, and.9, respectively); and, σ ε is the ε Schmidt number, equal to.30. Another version of the simplified form of the dissipation rate equation was proposed by CHAM (007): τ jj k ε ε jj ε j j ρ ε j ε ε ε ε ε + uj = c + c P c t x x x k k (.75) Launder et al. (975) used a different ε equation from standard equation to make Reynolds-Stress model. ε ε k τ jk ε ε ε + uj = cε + c ε Pkb cε t xj xj ε ρ xk k k (.76) where c ε, c ε, and c ε are constants which have been used in many models, as shown in Table.

54 Table. Turbulence Model (k-ε) Constants 30 Empirical Constants Isotropisation-of-Production Model (IPM) Quasi-Isotropic Model (QIM) Pressure-Strain Model Launder et al. (975) Younis (984) Launder et al. (975) Speziale et al. (99) c ε c ε c ε k-ω Turbulence Models. The first two-equation model of turbulence was the k-ω model, which was proposed by Kolmogorov (94). The two parameters for the model were turbulence kinetic energy per unit mass, k, and the rate of dissipation of energy in unit volume and time, ω. After the initial development, many versions of the k-ω turbulence model were devised with some improvements by Saffman (970), Launder and Spalding (97), Wilcox and Alber (97), Saffman and Wilcox (974), Saiy (974), Wilcox and Traci (976), Spalding (979), Wilcox and Rubesin (980), Wilcox (988), Speziale et al. (990), Menter (99), and Peng et al. (997). Subsequently, the model was improved (Wilcox 988) to be able to simulate the flow characteristics more accurately. The kinematic turbulent viscosity, ν t, is given by the ratio of turbulence kinetic energy, k, and rate of dissipation of energy in unit volume and time, ω: ν = k ω (.77) t and the equation of turbulence kinetic energy, k is: k k τ ij u i * * k + uj = β kω+ ( ν + σ νt ) t x x j ρ x j x j j (.78) where σ * is constants and equal to 0.5. The parameter * β is calculated from the equations: β β o f β f β * * = *, *, χk 0 = + 680χk, χ 0 k > + 400χk χ ω, k 3 k x j ω x j (.79)

55 where β is constant, equal to The equation of rate of dissipation of energy in unit volume and time, ω, is: * o 3 ω ω τ ij ω ui ω + u ( ν σνt ) j = α βω + + t x x j ρ k xj x j j (.80) where α, σ are constants, equal to 0.5, and 0.50, respectively; and, the parameter β is calculated from the following equations: β = β f o β, f β + 70χω = (.8) + 80χ ω where β o is constant, equal to 0.07; and, χ ω is the vortex-stretching parameter, which is defined by: The mean-rotation tensor (s - ), χ ω ΩΩ ij jkski (.8) * ( βω) 3 o Ω ij, and the mean-strain-rate tensor (s - ), S ij, are defined as: ui Ω ij = x j u j, x i S ij u u i j = + x j x i (.83) Then, the kinetic energy dissipation per unit mass, ε, can be calculated from: * ε = βωk (.84) As for the k-ε turbulence models, the Reynolds stresses which will be used for k-ω turbulence models are approximated by employing the Boussinesq hypothesis (Eq..7). Reynolds Stress Models Reynolds stress models, or differential stress models, are not based on the Boussinesq approximation (or, perhaps more accurately, the Boussinesq assumption). This model is expected to correct some deficiencies of the Boussinesq approximation by including the convection,

56 production, and diffusion of the Reynolds shear stresses; body force terms; and no equality of 3 normal stresses in case of disappearance of the mean stain rate (Wilcox 006). The general stress transport equation is: τ ij τ ij + u = Diff uu ρp + ρε ρπ t xl Rate of Change ( ) l i j ij ij ij Convection Diffusive Transport Stress Viscous Pressure Production Destruction Strain (.85) where the subscript l represents the terms (x, y, z) in each coordinate direction. The first term on the right-hand-side is the diffusive transport term, which consists of the molecular diffusion, Dif v (N/m/s), and turbulent diffusion Dif ij (N/m/s) terms: These two terms are defined as follows: Dif ν + Dif ij Diff ( u iu j) = molecular turbulent (.86) x diffusion diffusion l term term Dif ν τij ρuu i j = ν = ν x x l l (.87) Dif = ρu u u + p u δ + p u δ (.88) ij i j l i jl j il where ν is kinematic molecular viscosity (m /s). In 970, Daly and Harlow suggested a simplified form of the diffusive transport term: k k τ ij Diff ( u ) = ( iu j ρcsulul uiu j) = csτll xl ε xl ρ xl ε xl (.89) where c S is an empirical constant which has a value between 0. and 0.; k is turbulent kinetic energy per unit mass (J/kg); and, ε is the kinetic energy dissipation rate (W/kg). Thus, the term k/ε has units of seconds. However, Launder et al. (975) suggested that the first of the turbulent diffusion terms (Eq..88) is the most dominant of the two and can be used to represent diffusion in the Reynolds stress model. The turbulent velocity fluctuation diffusion can be approximated from:

57 33 k uu j l uu uu l i i j ρuu i ju l = cs* ρ uu i k + u ju k + uu l k ε xk xk xk (.90) where c s* is a constant, equal to 0. (Launder et al. 975), and then: k uu j l uu uu l i i j Diff ( u iu j) = cs* ρ u iuk + u ju k + u luk xl ε xk xk xk (.9) The P ij term on the right-hand-side of the Reynolds stress model is the stress production or generation term, which represents the production (m /s 3 ) of Reynolds stresses from the mean flow: u u i j u u i j Pij = u ju l + uu i l = τ jl + τil xl xl ρ xl xl (.9) The following term on the right-hand-side of the Reynolds stress model is the viscous destruction or dissipation term, which is the dissipation (W/kg) of the Reynolds stresses through the smaller eddies within the large eddies: u u i j εij = ν x x l l (.93) The viscous destruction term represents the dissipative conversion of kinetic energy to heat. With the assumption of isotropic dissipative motions, the viscous destruction term can be estimated by (Launder et al. 975): δij ε = ε (.94) 3 ij Finally, the last term is the pressure-strain term, which is defined as: p u u i j Π ij = + ρ x j x i (.95) The pressure-strain term includes a non-linear turbulent part, Π ij (m /s 3 ), and a meanstrain (or rapid) part, Π ij (m /s 3 ), as seen in Eq..96:

58 34 Π ij = Π ij + Π ij (.96) Non-linear Mean-strain turbulent part (or Rapid) part The simplest model of the pressure-strain term was introduced by Rotta (95) and Launder et al. (975), and is called the Isotropisation-of-Production model (IPM): ε Π ij = c uu i j δijk 3 k (.97) Π ij = c Pij δijpll 3 (.98) where P ll is defined by: ul u l Pll = uu l k + uu l k xk xk (.99) with l, k = representing the x-direction; l, k = representing the y-direction; and, l, k = 3 representing the z-direction. The nonlinear turbulent part involves the fluctuating quantities and another rapid part arises from the presence of the mean rate-of-strain. The parameters c and c are empirical constants. Based on laboratory measurements, Rotta (95) proposed that the value of c should be.4. However, in 96, Rotta obtained better computational results by using c =.8. In 975, Launder et al. (975) suggested that c should be equal to.8. The constant c was set equal to 0.60 by most researchers. Later, the new constants of c = 3.00 and c = 0.30 were suggested by Younis (984), who made significant improvements to the turbulence model. Launder et al. (975) also proposed the complete equations of the presence of the mean rate of strain in the compact form of Π ij, which provides improved simulation results. The new term makes the model somewhat more complex and is called as Quasi-Isotropic model (QIM): α u u i j ij Pij ij ll ij ij ll 3 δ P β k D P xj x γ i 3 δ Π = + (.00)

59 where P ij is stress production term, defined above; and, 35 ul u l Dij = uu i l + u ju l x j x i (.0) The empirical coefficients α, β, and γ are defined as follows: ( c + ) ( ) ( ) 8 30c 8c α =, β =, γ = (.0) 55 Because of the rapid spatial variation of the mean velocity gradient near a solid boundary, the pressure-strain terms proposed by Rotta (95) are not accurate for approximating the influences of mean strain on the pressure-strain term. Both IPM and QIM require wall-reflection terms in the pressure-strain model. Π ij = Π ij + Π ij + Π ijw (.03) Non-linear Mean-strain Wall-reflection turbulent part (or Rapid) part term The wall-reflection terms include two terms: Π ijw and Π ijw (m /s 3 ). The first term was adopted from Shir (973), and is given as: 3 3 ε Π ˆ ˆ ˆ ˆ ˆ ˆ ijw = cw u ku mnknmδ ij u kun i knj u ku jnkni f k (.04) where ˆn is the unit vector normal to a wall. The second term is adopted from Gibson and Launder (978): 3 3 Π ˆ ˆ ˆ ˆ ˆ ˆ ij w = cw Πkmnk nmδij Πkinknj Πkj nk ni f (.05) where f is the wall-damping function, which is calculated from: f cw = x n (.06) and x n is the normal distance from a wall (m). At the near-wall grid points: x n c w = (.07)

60 in which is the turbulence length scale (m), defined by: 36 c k ε 3/ D = (.08) where c D is an empirical constant, equal to 3/4 c μ * ; and, c μ * is the effective eddy-viscosity coefficient, equal to Launder et al. (975) also proposed the near-wall correction term in a simple format: ε Π =Π +Π = 0.5 uu kδ P D k 3 ( ) ijw ijw ij w i j ij ij ij k ε x 3/ n (.09) or, ε τ ij k Π ijw = 0.05( Pij Dij ) kδ ij k ρ 3 ε x 3/ n (.0) However, Wilcox (006) referred to Launder et al. (975) with the negative of Eq..09; that is: ε τ ij Π = kδ 0.05 P D k ρ 3 ( ) ijw ij ij ij k ε x 3/ n (.) Later, the pressure-strain model was improved by Speziale et al. (99). This model does not require the wall-reflection terms: Π = ( cε + c P ) b + c ε b b b b ij S kk ij ik ij mn mn / + c c ( b b ) ks + c k b S + b S b S 5 3 3S mn mn ij 4 ik jk jk ik mn mn ( ik jk bjk ik ) + ck b Ω + Ω δij 3 δij 3 (.) where b ij uu i j δij =, k 3 S ij u u i j = +, x j x i ui Ω ij = x j u j x i (.3) The empirical constants are summarized in Table. Finally, the Reynolds stress equations are approximated from:

61 τij τij u u i j + ul = τ jl + τil + ρδijε ρ Π ij +Π ij +Πijw t xl xl xl 3 pressure-strain term diffusion term generation term k τ ij csτ ll ρ xl ε xl dissipation term ( ) 37 (.4) or, using a different form of the diffusion term, Table. Empirical Constants for the Various Turbulence Models Empirical Constants Isotropisation-of-Production Model (IPM) Quasi-Isotropic Model (QIM) Pressure-Strain Model Launder et al. (975) Younis (984) Launder et al. (975) Speziale et al. (99) Diffusion Transport Model c S Pressure-Strain Model c c S c c c 3S c c Wall-Reflection Model c w c w τij τij u u i j + ul = τ jl + τil + ρδijε ρ Π ij +Π ij +Πijw t xl xl xl 3 pressure-strain term generation term dissipation term c k τ τ τ s* jl li ij τik + τ jk + τlk ρ xl ε xk xk xk diffusion term ( ) (.5) Algebraic Stress Models

62 Algebraic stress models (ASM) are known as k-ε-a models. These models were 38 developed by simplifying the differential stress models to be algebraic stress equations, and assuming certain flow conditions. The easiest way is by removing the turbulent convection and diffusion terms. This assumption can be acceptable in the case of high shear flow which has relatively small convective and diffusive terms. An additional condition is that the two terms (convection and diffusion) are equal in magnitude, which occurs where local turbulence equilibrium exists. With these assumptions, Eq..4 reduces to: i j 0 τ u u x τ = + + l xl 3 ρδ ε ρ Π +Π +Π ( ) jl il ij ij ij ijw (.6) Using the simplest form of pressure-strain model in terms of the wall-reflection terms: ε 0 = ρp + ρδ ε ρ c ij uu δ k c P δ P 3 ij i j 3 ij k ij 3 ij ll (.7) where τ ij ε Pij = δε ij c + δijk + c Pij δijpll 3 ρ 3 k 3 (.8) And, because of P k is defined by: 3 3 u n u n Pk = u nu m = u nu m xm n= m= x (.9) m and, Pll u u v v Pk = = uu + uv ' + uv + vv x y x y (.0) Eq..8 becomes: ε τ ij Pij = δε ij c + δijk + c Pij δijpk 3 k ρ 3 3 (.) or,

63 τ ρ k c ij ( c ) Pij c ijpk ijk c 3 δ 3 ρ = + ε c δ 39 (.) where c and c are adjusted to compensate for the loss in approximation, simplifying the Reynolds stress equations to be algebraic stress equations, and are equal to.3 and 0.4, respectively. A better way to develop ASM is to keep all terms (including convective and diffusion terms), and assume that the summation of the convective and diffusion terms in the Reynolds stress equations is proportional to the summation of the convective and diffusion terms of from the turbulent kinetic energy equation (Eq..6) (Rodi 97). With this assumption, Eq..4 is simplified to be the following form: cd k τij = ρδijk + ρ Pij δijpk P (.3) k 3 ε 3 c + ε where c and c D are constant, and adjusted to compensate for the loss in approximation, equal to. and 0.55, respectively Naot and Rodi (98) were successful in deriving algebraic expressions for the Reynolds stress equations. The model was developed based on these expressions and was able to simulate the depression of the maximum velocity from the water surface in rectangular open channels. These models assume local equilibrium between the production rate of turbulence and dissipation rate. Naot et al. (993) applied the equations on compound rectangular open channels. Their equations are as follows: τ u xy = μ t xy y (.4a) u τxz = μ t xz z (.4b)

64 40 k β β yy u u τ = ρ α + c + τxy τxz + μ v t c+ c3 3 ε y z y (.4c) k β β yy zz u u τ τ = ρ α + c+ τxz τxy + c3 + μ w t c 3 ε z y k z (.4d) β k u u v w τ yz = τxz + τxy + μt + ( c +.5c3) ε y z z y (.4e) The isotropic turbulent viscosity, μ t, is: k = ε μt ρc μ (.5) where c μ = 0.09; and, the anisotropic turbulence viscosities are defined as: c μ μ μ t = c xy μxy t = t ( c+.5c3)( c+ c3) (.6) c +.5c μ = c μ = μ 3 txz μxz t t c + c3 (.7) The constants, α, β, c, and c 3, are expressed as functions of the distance from a solid wall, f, and open surface, f : α = f ; β = f ; c = f; c3 = 0.0 f (.8) where l f =, and ya f l = ha (.9) in which l is the dissipation length, which is defined as: c l = κ 3/4 3/ μ k ε (.30) where y a and h a are the root-mean-square reciprocal distances from the solid walls and water surface, respectively. The term y a and h a (Eq..9) are defined as:

65 4 y a / =, and y ha ( H y) / = l (.3) and, θ d y θ = θ θ (.3) S θ and S is the distance to the boundary segment that occupies the angle differential, dθ. S ( θ ) dθ S ( θ ) P S θ Fig. 5. Definition sketch for calculating average distance of point P from solid wall surfaces form: Speziale (987) and Pezzinga (994) used the turbulent stress equations in the following μt μt τ ij = μtsij + 4CD SimSmj SmnSmnδij + 4CE S ij S mnδij ρδijk ρk 3 ρk 3 3 (.33) where c μ = 0.09; σ k =.0; σ ε =.3; c ε =.44; c ε =.9; C D = C E =.67; and, S ij is the frameindifferent derivative of S ij : S S u u u S S S (.34) ij i j ij = + ij kj ki t xk xk Blanckaert and Sam adapted the Reynolds stress equation from Speziale to take the following form (Ninokata et al. 00): k k τ ij = μtsij + CDcμρ 4 SimSmj SmnSmnδij + CEcμρ S ij S mnδij ρδijk ε 3 ε 3 3 (.35)

66 Boundary Conditions 4 Wall Boundary Conditions In the region of the dimensionless distance from wall, y +, in the range between 30 and 00, the Reynolds stresses are almost constant. Then, the convection and diffusion of the Reynolds stress terms could be considered negligible, whereby local equilibrium is dominant (Rodi 984). This leads to the conclusion that the stress production term is equal to the dissipation term plus the pressure-strain term, as follows: ε ij = Pij + Π ij (.36) Viscous Stress Pressure Destruction Production Strain The wall boundary conditions for k and ε could be set by using the hypothesis of local equilibrium of the turbulence near the wall. In this case, the production rate and the dissipation rate are equal (P = ε). Another hypothesis is that the wall shear stress is equal to the turbulent shear stress, as follows: k w uτ = (.37) c μ 3 uτ ε w = (.38) κ y w The velocity component normal to the vertical plane is set equal to zero: The Reynolds shear stress at wall is equal to wall shear stress. w = 0 (.39) τ w τ ij = τ w or uu i j = (.40) ρ And, the normal derivative of the turbulent quantity k is zero (Ferziger and Perić 996). k = 0 z (.4)

67 The three velocity components in the x-, y-, and z-directions near the walls, which are 43 used as boundary conditions, can be calculated using the so-called law-of-wall. Vertical Plane of Symmetry The boundary conditions for the vertical plane of symmetry are that the normal derivatives of the velocity components parallel to the plane are set to zero: u v = 0, = 0 z z (.4) The velocity component normal to the vertical plane is also zero: w = 0 (.43) And, the normal derivatives of the turbulent quantities k and ε are set to zero: k ε = 0, = 0 z z (.44) Finally, the normal derivatives of the Reynolds Stress tensors are equal to zero. τ ij z = 0 (.45)

68 Free-Surface Boundary Conditions 44 The boundary conditions for the free water surface are almost the same as the boundary conditions for the vertical plane of symmetry because the free surface is, in fact, a plane of symmetry. The normal derivatives of the velocity components parallel to the plane are set to zero: u w = 0, = 0 y y (.46) The velocity component normal to the free surface is also set to zero: v = 0 (.47) And, the normal derivative of the turbulence kinetic energy k is set to zero: k = 0 y (.48) However, it is noteworthy that Kra (00) chose to set the kinetic energy per unit mass to zero at the water surface: k = 0 (.49) The kinetic energy dissipation at the free surface is determined using an equation from Naot and Rodi (98): ε f 3/4 3/ cμ k = + κ 0.07H dw (.50) where κ = 0.4 is the von Kármán constant; H is the local water depth; and, d w is the distance to the nearest wall (or floor). Later, Lau and Krishnappan (98) suggested that the kinetic energy dissipation at the free surface could be calculated from: ε f cf = κ y f k f c μ 3/ (.5)

69 where c f is an empirical constant, equal to 0.64; y f is the distance from the free surface to the 45 nearest computational grid point (m); and, k f is the turbulence kinetic energy at the free surface (m /s ). Finally, the equation of Naot and Rodi was adapted as follows (Ninokata et al. 00): ε f 3/4 3/ cμ k = κ H dw (.5) Rodi (984) suggested equations to account for the effects of wind shear: k f uτ f = (.53) c μ ε f 3/4 3/ cμ kf = u τ f κ dw + ah kf c μ (.54) where a is an empirical constant, equal to 0.07 (Hossain 980). Versteeg and Malalasekera (007) suggested that the turbulence kinetic energy per unit mass, k, and kinetic energy dissipation per unit mass, ε, for the k-ε model are both equal to zero at the free surface; thus, k f = 0, ε = 0 (.55) f Additionally, Versteeg and Malalasekera (007) suggest that all Reynolds stress tensors for Reynolds stress equation models are equal to zero: for all values of i and j. τ = 0 (.56) ij

70 CHAPTER 3 46 TURBULENCE MODEL EQUATIONS This chapter provides a detailed description about the equations used in developing the various turbulence models as applied in this research. The turbulence equations from the literature review were presented in condensed form, but they must be expanded before implementing them in a mathematical model and generate simulation results. Thus, this chapter will show the expanded form of the equations for steady, uniform flow in the x (streamwise) direction. All partial derivatives with time as the independent variable disappear from the equations because this research was limited to steady flow conditions. Also, all partial derivatives with respect to x vanish from the equations under uniform-flow conditions. Using mathematical symbols, all / t and / x terms in the governing equations are equal to zero. There are three major groups of equations for turbulence models: () continuity and momentum equations; () stress equations, which have four sets in algebraic form and one set in partial differential form; and, (3) turbulence equations, which are k-ε equations. The four stress equation sets in algebraic form are: () Boussinesq hypothesis () algebraic stress model (ASM); (3) algebraic stress model (ASM), and (4) algebraic stress model 3 (ASM3). The stress equation set in partial differential form is Reynolds stress model (RSM). As will be seen below, the ASM, ASM, and ASM3 implementations are different versions of the algebraic stress model, and they do not involve any differential terms. The RSM stress model is somewhat more complex than the ASM models. Continuity Equation The condensed form of the continuity (conservation of mass) equation for steady, incompressible flow is:

71 47 u x k k = 0 (3.) And, the expanded form of this equation in Cartesian coordinates is: u v w + + = x y z 0 (3.) Momentum Equations The condensed form of the momentum conservation equation is: Du = p + u i i ρ ρfi μ ρuu i j Dt xi x j x j (3.3) And, the expanded form for steady, incompressible flow is: u u u u τ xy τ ρv + ρw = ρgsin( ξ) + μ + μ + + y z y y z z y z v v p v v τ yy τ ρv + ρw = ρgcos( ξ) + μ + μ + + y z y y y z z y z w w p w w τ yz τ zz ρv + ρw = + μ + μ + + y z z y y z z y z xz yz (3.4) (3.5) (3.6) where ξ ( S ) = tan o (3.7) Reynolds Stress Equations Boussinesq Hypothesis The Reynolds shear stresses are simply approximated using the Boussinesq hypothesis. These equations can be easily applied to the turbulence model. The condensed form is: u u j ρkδ i ij τij = ρuu i j = μt + xj xi 3 (3.8) And, the expanded form is:

72 τ xy u = μ t y 48 (3.9) u τxz = μ t z v ρk τ yy = μ t y 3 (3.0) (3.) τ μ v w yz = t + z y w ρk τzz = μ t z 3 (3.) (3.3) There is no τ xx equation because τ xx x in the momentum equation in the x-direction is absent due to the uniform flow assumption. By definition, τ yx, τ zx, and τ zy are equal to τ xy, τ xz, and τ yz, respectively. Algebraic Stress Model form is: ASM is the simplest form of the algebraic stress model (Eq..). The condensed τ ρ k c ij ( c ) Pij c ijpk ijk c 3 δ 3 ρ = + ε c δ (3.4) And, the expanded form is: k c τ yy = ρ ( c ) P cpk + ρ k cε 3 3 c k c τzz = ρ ( c ) P33 cpk + ρ k cε 3 3 c (3.5) (3.6) k = P (3.7) τxy ρ c ε ( c )

73 τxz ρ c ε ( c ) 3 49 k = P (3.8) k = P (3.9) τ yz ρ c ε ( c ) 3 Algebraic Stress Model ASM is based on the retention of all terms (including the convective and diffusion terms), and assuming that the summation of the convective and diffusion terms in the Reynolds stress equations is proportional to the summation of the convective and diffusion terms from the turbulent kinetic energy equation (Rodi 97). The condensed form is: cd k τij = ρδijk + ρ Pij δijpk P (3.0) k 3 ε 3 c + ε And, the expanded form is: c k (3.) + D τ yy = ρk + ρ P Pk Pk 3 ε 3 c ε c k (3.) + D τzz = ρk + ρ P33 Pk Pk 3 ε 3 c ε c k P P D τxy = ρ k ε c + ε (3.3) c k P P D τxz = ρ 3 k ε c + ε (3.4) c k P P D τ yz = ρ 3 k ε c + ε (3.5) Algebraic Stress Model 3

74 50 ASM3 is based on an assumption of local equilibrium between the rate of turbulence production and the dissipation rate (Naot et al. 993). There is no condensed form for this version of the algebraic stress model. The ASM3 equations are written as follows: k β β yy u u τ = ρ α + c + τxy τxz + μ v t c+ c3 3 ε y z y (3.6) k β β yy zz u u τ τ = ρ α + c+ τxz τxy + c3 + μ w t c 3 ε z y k z (3.7) u τxy = μ t xy y (3.8) τ u xz = μ t xz z (3.9) β k u u v w τ yz = τxz + τxy + μt + ( c+.5c3) ε y z z y (3.30) where α = f ; β = f ; c = f ; c = 0.0 f (3.3) 3 l f =, and ya f l = ha (3.3) c l = κ ε 3/4 3/ μ k, y a / =, y ha ( H y) / = l (3.33) θ d y θ = θ θ (3.34) S θ μ txy = c ( c +.5c )( c + c ) 3 3 μ t (3.35) μ txz c+.5c3 = μt c + c 3 (3.36)

75 k μt = ρc μ ε 5 (3.37) Reynolds Stress Model Reynolds stress models (RSM), or differential stress models, are not based on the Boussinesq approximation. This model is expected to correct some deficiencies of the Boussinesq approximation by including the convection, production, and diffusion terms representing the Reynolds shear stresses, inclusion of the body-force terms; and, the assertion of inequality of normal stresses in case of disappearance of the mean stain rate (Wilcox 006). The RSM is based on the Launder-Reece-Rodi model (975) and the diffusion term from Daly and Harlow (970). The condensed form includes these equations: τ ij τ ij w v + w = ρp + ρδ ε ρ Π +Π +Π Diff uu y z 3 ( ) ( ) ij ij ij ij ij i j (3.38) τ ij ε Π ij = c + δijk ρ 3 k (3.39) α u u i j ij Pij ij k ij ij k 3 δ P β k D P xj x γ i 3 δ Π = + w ε τ ij k Π ij = 0.05( Pij Dij ) kδij k ρ 3 ε x k τ ij Diff ( u iu j) = csτ ll ρ xl ε xl 3/ n (3.40) (3.4) (3.4) And, the expanded form for the main equations is: w ( ) ρ ρε ρ( ) τxx τxx v + w = Diff uu P + Π +Π +Π y z 3 (3.43)

76 τ τ v + w = Diff vv P + Π +Π +Π y z 3 ( ) ρ ρε ρ( ) yy yy w w ( ) ρ 33 ρε ρ( ) τzz τzz v + w = Diff ww P + Π +Π +Π y z 3 τ τ v + w = Diff uv P Π +Π +Π y z τ y ( ) ρ ρ( ) xy xy w xz xz v w Diff uw P w ( ) ρ 3 ρ( ) τ + = Π +Π +Π z τ τ v + w = Diff vw P Π +Π +Π y z ( ) ρ 3 ρ( ) yz yz w 5 (3.44) (3.45) (3.46) (3.47) (3.48) The expanded form for the non-linear turbulent part of the pressure strain terms is: ε τ xx Π = c + k k ρ 3 (3.49) ε τ yy Π = c + k k ρ 3 (3.50) ε τzz Π 33 = c + k k ρ 3 (3.5) ε τ Π = c xy (3.5) k ρ ε τ Π 3 = c xz (3.53) k ρ ε τ Π 3 = c yz (3.54) k ρ And, the expanded form for the mean-strain (or rapid) part of the pressure strain terms is: Π = α P Pk γ D Pk 3 3 (3.55) v Π = α P Pk βk γ D Pk 3 y 3 (3.56)

77 w Π 33 = α P33 Pk βk γ D33 Pk 3 z 3 u Π = α P βk γd y 53 (3.57) (3.58) u Π = α P βk γd z (3.59) α v w P β Π = k + D z y γ (3.60) The expanded form for the wall-reflection terms of the pressure strain terms is: 3/ w k ε τ xx Π = 0.05( P D) k εxn k ρ 3 3/ w k ε τ yy Π = 0.05( P D ) 0.5 k εx + n k ρ 3 3/ w k ε τzz Π 33 = 0.05( P33 D33 ) k εxn k ρ 3 3/ w k ε τ xy Π = 0.05( P D ) 0.5 ε xn k ρ 3/ w k ε τ xz Π 3 = 0.05( P3 D3 ) 0.5 ε xn k ρ 3/ w k ε τ yz Π 3 = 0.05( P3 D3 ) 0.5 ε xn k ρ (3.6) (3.6) (3.63) (3.64) (3.65) (3.66) Finally, the expanded form for the diffusion terms is: Diff ( u u ) = cs k τ xx cs k τ xx τ yy + τzz y ρ ε y z ρ ε z Diff ( v v ) = cs k τ yy cs k τ yy τ yy + τzz y ρ ε y z ρ ε z (3.67) (3.68)

78 54 Diff ( w w ) = cs k τ zz cs k τ zz τ yy + τzz y ρ ε y z ρ ε z Diff ( u v ) = cs k τ xy cs k τ xy τ yy + τzz y ρ ε y z ρ ε z Diff ( u w ) = cs k τ xz cs k τ xz τ yy + τzz y ρ ε y z ρ ε z Diff ( v w ) = cs k τ yz cs k τ yz τ yy + τzz y ρ ε y z ρ ε z (3.69) (3.70) (3.7) (3.7) Kinetic Energy and Dissipation Equations k-ε Model based on the Boussinesq Hypothesis These equations are applied with the stress equation in Boussinesq hypothesis ASM, ASM, and ASM3. The condensed form is: k k μt k ρ + ρuj = ρp μ kb ρε + + t xj x σ j k x j (3.73) ε ε ε ε μt ε ρ + ρuj = ρcε Pkb ρc μ ε + + t x x j k k x σ j ε j (3.74) And, the expanded form is: k k μ k μ k v + w = P t t ρ ρ ρ kb ρε μ μ y z y σk y z σk z (3.75) ε ε ε ε μ t ε μ t ε ρv + ρw = ρcε Pkb ρcε + μ+ + μ+ y z k k y σε y z σε z (3.76) k-ε Model for Reynolds Stress Model These equations are based on Launder et al. (975) and applied in the Reynolds stress model. The condensed form is:

79 55 τ ii k = (3.77) ρ ε ε k ε ε ε ρ + ρuj = cε τ jk + ρc ε Pkb ρcε t xj xj ε xk k k (3.78) And, the expanded form is: k = τ + τ + τ ρ ( xx yy zz) (3.79) ε ε k ε k ε ρv + ρw = cετ yy + cετzz y z y ε y z ε z k ε k ε ε ε + cετ yz + ε cτ yz + ρc ε Pkb ρcε y ε z z ε y k k (3.80) Auxiliary Equations Several auxiliary equations are required to complete the model, such that the number of equations equals the number of variables. The condensed form of these equations is: D u u l l ij = τil + τ jl ρ x j xi (3.8) u u i j ij = τ jl + τil ρ xl xl P (3.8) P k τ nm u = ρ x n m (3.83) P τ ij u i u u i j =, where τ = μ + ρkδ (Boussinesq Hypothesis) ρ x j xj x i 3 kb ij t ij (3.84) And, the expanded form is: D = 0 (3.85) D u v w = τxy + τ yy + τ yz ρ y y y (3.86)

80 D 33 u v w = τxz + τ yz + τzz ρ z z z 56 (3.87) D u v w = τxx + τxy + τxz ρ y y y (3.88) D 3 u v w = τxx τxy τxz ρ + + z z z (3.89) D 3 u u v v w w = τxy + τxz + τ yy + τ yz + + τzz ρ z y z y z y (3.90) P u = τxy + τ ρ y xz u z (3.9) P v = τ yy + τ ρ y yz v z (3.9) P 33 w w = τ yz + τzz ρ y z (3.93) P v u u v = τxy + τ yy + τ yz + τxz ρ y y z z (3.94) P 3 w u w u = τxz + τ yz + τxy + τzz ρ z y y z (3.95) P 3 v w w v = τ yz + + τ yy + τzz ρ y z y z (3.96) P v w u u v w = τ + τ + τ + τ + τ + ρ y z y z z y k yy zz xy xz yz (3.97) P kb u u v w v w ν = t y z y z z y (3.98)

81 Boundary Equations 57 Two types of log-law (logarithmic) equations are used to develop the turbulence models. The first equation is used in the finite difference solution approach, and the second equation (which is much simpler than the first one) is used in the finite volume method. However, the results from both equations are exactly the same. First Log-law Equations uτ ywuτ u = ln Eru κ ν (3.99) E ru τ u xz τ = (3.00) ρ ( κ B ) exp su = (3.0) + k + ( ln ) exp + 0.7( ln ) exp 0.7( ln ) su ( ) Bsu = + ksu ksu + k su It follows from Eqs to 3.03 that: k uk ν (3.0) + τ s su = (3.03) k τ s xz k τ s xz ln exp 0.7 ln + ν ρ ν ρ τ xz y w u = ln exp κ κ ρ k s k τ s xz exp 0.7 ln ν ρ (3.04) Using the same procedure, the log-law equation for v is:

82 v = κ τ yz y ln ρ k w s k τ s yz k τ s yz ln exp 0.7 ln + ν ρ ν ρ exp κ k τ s yz exp 0.7 ln ν ρ 58 (3.05) And, the log-law equation for w is: w = κ τ yz y ln ρ k w s k τ s yz k τ s yz ln exp 0.7 ln + ν ρ ν ρ exp κ k τ s yz exp 0.7 ln ν ρ (3.06) Second Log-law Equations + u = lny + + Cl (3.07) κ yu ln w τ u = uτ + Cl κ ν + + Cl = 8.5 ln ks, for ks >> κ (3.08) (3.09) k uk ν + τ s s = (3.0) Then, y ln w 8.5, + u = uτ + for k >> s κ ks (3.)

83 Vertical Wall Boundary 59 y ln w u = uτ κ k s (3.) v = 0 (3.3) w = 0 (3.4) u k = (3.5) c τ / μ 3 uτ ε = (3.6) κ y w τ = 0 (3.7) yy τ = 0 (3.8) zz τ = 0 (3.9) xy τ xz = ρ (3.0) u τ τ yz = ρ (3.) v τ Horizontal Floor Boundary y ln w u = uτ κ k s (3.) v = 0 (3.3) y ln w w= wτ κ k s (3.4) u k = (3.5) c τ / μ 3 uτ ε = (3.6) κ y w

84 60 τ = 0 (3.7) yy τ = 0 (3.8) zz τ xy = ρ (3.9) u τ τ = 0 (3.30) xz τ yz = ρ (3.3) w τ Free Water Surface u = 0 y w = 0 y k = 0 y (3.3) (3.33) (3.34) v = 0 (3.35) ε f 3/4 3/ cμ k = + κ 0.07H dw (3.36) τ = 0 (3.37) ij Distance from the Walls During the calculation process, the computational nodes in the channel cross section are represented by a small area around the node. Also, some turbulent equations, especially for velocity in the longitudinal direction, cannot be applied to the node closest to a wall. A certain distance from the wall to the first row and first column must be determined to ensure that the equations, which are based on the log-law (logarithmic velocity profile), are correctly applied.

85 6 Based on published laboratory data (Wilcox 006), the average friction velocity can be estimated by: avg uτ = ghs f (3.38) Then, the distance from a wall is defined as follows: y w = + y μ avg ρu τ (3.39) After this, the computational grid points can be generated based on the desired grid density, and the cross-sectional shape and size.

86 CHAPTER 4 6 SOLUTION OF EQUATIONS FINITE DIFFERENCE METHOD The first version of the turbulence model was developed by applying the finite difference method to discretize the equations and convert them to algebraic form. In this research, only the turbulence model for a rectangular channel cross section was applied to this method. The calculations start by initializing the initial guesses for each unknown. The systems of equations are then solved iteratively by the Newton-Raphson method at a point in the computational grid. After the convergence criteria at the point are met, the model records the solution and moves to the next grid point and solves the set of equations again, and so on, until all nodes in the cross section are solved. If this calculation is the first iteration, the solution will be compared with the initial guesses; otherwise it is compared with the values from the previous Newton-Raphson iteration. The model stops running when the overall convergence criteria are met. The calculation processes are shown in Fig. 6. Algebraic Equations Most of the governing equations in a turbulence model are in the form of partial differential equations (PDEs). The Microsoft Visual C#.NET 005 programming language was used to develop the model. The finite difference technique, based on the Taylor series, was used to transform the PDEs to non-linear algebraic equations. Both forward and backward differences are used for transforming the equations depending on the location of the grid. The first-order forward and backward differences are: j+ ψ ψ ψ = +Φ Δ y Δy ( y) (4.)

87 63 Select Model (ASM, ASM, ASM3, RSM) void DefineConstants() Define constants for each model which do not change during calculations void AllocateArrayMemory() Allocate memory for arrays void Chezy() Calculate discharge (Q) from uniform flow conditions void BoundaryDistance() Calculate distance from boundaries using "Law of the Wall" void RmsDistance() Calculate Root-mean-square Distance from boundaries void InitializeArray() Initialize all arrays and estimate variable values void BoundaryCondition() Set boundary conditions for hard surfaces and free water surface iter++ No Converged iter < MaxIter iter = 0 No Stop Yes Boolean DoCalculations() Performs hydraulic calculations for a channel cross section i = 0 i < MaxNR Yes No Stop Boolean GaussJordan(...) Perform Gauss-Jordan elimination with full pivoting Yes Boolean Converged(...) Check the convergence of the overall solution No iz < nz iz = 0 void CopyArray(...) Copy Array to the temporary Array for checking the difference in results of the next iteration void DefineConstants(...) Calculate values of constants which change for every point in the cross section Converged Yes No void UpdateVariables(...) Updates all variable values Stop Yes iy = 0 iz++ No void LoadMatrix(...) Loads the Jacobian matrix of Newton- Raphson method Send values depending on position of point Boolean ConvergedNR(...) Check Newton-Raphson solution convergence iy < ny Yes Check position of the calculating point LoadMatrix.cs Call methods corresponding to grid position, Equation_BB_BB.cs, Equation_BB_CC.cs,..., Equation_FF_FF.cs Converged Yes iy++ No i++ void WriteResultsToFile() Write input data and results to file Fig. 6. Flowchart of the turbulence model solution using the finite difference method

88 k+ ψ ψ ψ = +Φ Δ z Δz ( z) j ψ ψ ψ = +Φ Δ y Δy ( y) k ψ ψ ψ = +Φ Δ z Δz ( z) 64 (4.) (4.3) (4.4) where Ψ represents any of the variables; and, Φ(Δy) and Φ(Δz) are first-order truncation errors. The second-order forward, backward, and central difference for y-direction and z- direction used in this research are: j j+ j+ ψ ψ ψ + ψ = +Φ Δ y Δy ( y) k k+ k+ ψ ψ ψ + ψ = +Φ Δ z Δz j j + ( z) ψ ψ ψ ψ = +Φ Δ y Δy k k + ( y) ψ ψ ψ ψ = +Φ Δ z Δz j+ j ψ ψ ψ + ψ = +Φ Δ y Δy ( z) ( y ) k+ k ψ ψ ψ + ψ = +Φ Δ z Δz ( z ) (4.5) (4.6) (4.7) (4.8) (4.9) (4.0) where Φ(Δy ) and Φ(Δz ) are second-order truncation error. All variables are stored in a matrix representing each of the computational nodes. In contrast with the staggered-grid method used in the finite-volume method (see Chapter 5), the secondary velocities, which are the velocities in the y- and z-directions, are separately located around the representative points of the rest variables. The staggered-grid method will be discussed in detail in the next chapter. The definition of the symbols for the finite difference equations are shown in Fig. 7.

89 65 j+ ψ j+ j+ ψ j+ k ψ k ψ ψ k+ ψ ψ k+ j j ψ j- k- Δy k- j ψ Δz k k+ k+ j- Fig. 7. Definition of variables for the finite difference scheme, applied at node (k, j) in the channel cross section For a first-order differential term, a central ( nd -order) difference gives more accurate results than either forward or backward ( st -order) differences. However, a central difference cannot be applied on the first-order PDE terms because the pressure variable (p) in the momentum equations in the y and z-directions will be cancelled from the finite difference equations. Only p k-, p k+, p j-, and p j- values, which are not variables in the equations, remain in the discrete equations at the grid point of interest. Various types of finite difference are used on different parts of the cross section depending on the availability of data from surrounding nodes, as shown in Fig. 8. The four-letter symbol in Fig. 8 shows variation of finite difference schemes applied on the nodes in the cross section. The letters F, B, C in the symbols represent forward, backward, and central differences, respectively. The first two letters in the symbol are used for first-order difference in the z- and y- directions, respectively. And, the last two letters in the symbol are used for second-order difference in the z- and y- directions, respectively. An example of the finite difference equations for the case of first-order forward differences in the z-direction, backward differences in the y-direction, and second-order central

90 66 differences for both the z- and y- directions (FB_CC) are shown in Appendix B. However, it is noted that he Appendix B shows only one of possible cases. The other sets of finite difference equations were derived during the model development, but are not shown in this dissertation because they would require more than one hundred additional pages, including the equations for the Newton-Raphson method, to present in their entirety. Vertical Wall Boundary FF_FF FB_FB BF_BF BB_BB Vertical Wall Boundary Fig. 8. Boundary of each type of finite difference scheme It is acknowledged that a vertical line of symmetry must exist at the center of this simple rectangular cross section, for the specified conditions. Thus, it is expected that the solution for the left side of the cross section should result in the mirror image of the solution for the right side, in terms of all variables. Nevertheless, the solution was set up for the entire cross section in anticipation of testing the model for non-symmetrical cross-sectional geometries and conditions. For example, there might be varying hydraulic roughness values between the two vertical walls, a curve in the channel alignment, and so forth.

91 Newton-Raphson Method 67 Each region in the cross section in each turbulence model has different set of equations. Each equation set must have the number of equations, n, equal to the number of unknowns to close (complete) the system of equations, thereby enabling a unique overall solution. The Newton-Raphson method is utilized for solving the system of equations by reducing the system of non-linear algebraic equations, through replacement of the differential terms by finitedifference expressions, to linear equations (Merkley 997). This method starts by moving all terms of each non-linear equation to one side to combine which other terms on the other side, thereby setting each of the equations to zero. Function vectors, F, F, F 3,, F n, are introduced to represent each equation. This means that the solution entails a search for each variable such that the function values are equal to zero, by definition. The numbers of unknowns for each turbulence model are shown below: Table 3. Number of Unknown Values, and Number of Non-zero Jacobian Matrix Values, for Each of the Turbulence Models Turbulence Model ASM ASM ASM3 RSM Number of variables (n) Number of non-zero elements Partial derivatives, F i x, with respect to all unknowns, x i, for all equations are i required. Appendix C shows the function vectors and partial derivatives for the case of firstorder forward differences in the z-direction, backward differences in the y-direction, and secondorder central differences in both the z- and y- directions (FB_CC). Now, the initial guesses of all unknowns at the computational grid point in question, and the surrounding grid points, are used to determine the values of all function vectors and partial derivatives. To arrive at the solution to the set of linear equations, these values are arranged in matrix form as:

92 F F F F... x x xn x δ x F n F F F F... x x xn x δ x F n =. Fn Fn Fn F n... δ x x n Fn x xn xn Fn Fn Fn F n... δ xn Fn x x x x n n Corrections Jacobian matrix Function vector 68 (4.) The system of linear equations is solved by Gauss-Jordan elimination (Jennings 977) with full pivoting. The solution of these equations yields correction values for each of the unknowns. The variables are corrected by these adjustment values as follows: iter + iter i = i i x x δ x (4.) where i =,,3,,n; iter xi is the current value of the variable (equal to the initial guess for the first iteration); x is the updated value; and, δ x is the correction value. After all variables have iter+ i been corrected, all function vectors and function derivatives are recalculated and solved by Gauss-Jordan elimination again. This process continues until the absolute adjustment values are less than an acceptable value. At the end of this process, the converged values of all variables are recorded at the computational node in question. Then, the model applies the Newton-Raphson method to the next node in the sequence, and so on until all nodes in the computational grid have been processed. Grid Spacing and Management Grid spacing in the y- and z-directions are determined by the cross-sectional width, B, normal (uniform-flow) depth, H, longitudinal bed slope, S o, desired dimensionless distance from the walls, y +, the number of nodes in the z-direction, nz, and the number of nodes in the y-

93 69 direction, ny. First, the distance from the walls to the first row and first columns is calculated by Eq. 3.39, which is: y w = + y μ avg ρu τ (4.3) Then, the grid spacing in the z-direction is: dz B y nz = w (4.4) and the grid spacing in the y-direction is: dy H y ny = w (4.5) Spiral Solutions Approach The ways to move the calculating point from current node to the next node have been tried by many options such as finish from left to right of a row and move to the next consecutive row, or finish from top to bottom of a column and move to the next consecutive column. The problem from these calculation movements is that the result tends to have asymmetry problems and leads to numerical divergence. Spiral movement of calculating node was introduced due to these problems. The calculation starts from the node at the center of the cross section and moves to the next node in clockwise direction as shown in Fig. 9. The technique ameliorates asymmetry and divergence problems. However, these problems do still exist because there are many factors leading to these problems.

94 70 calculation direction computational nodes law of wall region Fig. 9. Spiral movement of calculation point Multiple Solutions A significant problem in developing the turbulence model solution using the finite difference method is that more than one solution can be obtained from the same set of equations, channel characteristics, and number of computational nodes, but different initial guesses. For example, the RSM version of the stress equations has 6 unknowns at each computational node. This means all 6 equations must be satisfied by a set of values when the solutions converge. Interestingly, there are more than one set of solutions that can satisfy all 6 equations. Figure 0 shows the multiple solutions at a point in the middle left part of a cross section from RSM with a channel width 5.0 m, normal depth of 3.0 m, longitudinal bed slope of 0, roughness height of 4 m, and a 0 x 0 computational node grid.

95 7 u, m/s Case Case Case Newton-Raphson Iterations v, m/s Case Case Case Newton-Raphson Iterations (a) u (b) v Case Case Case 3,630,65,60,65 w, m/s p, N/m,60,605,600,595,590 Case Case Case Newton-Raphson Iterations, Newton-Raphson Iterations (c) w (d) p τ xx, N/m Case Case Case Newton-Raphson Iterations τ yy, N/m Case Case Case Newton-Raphson Iterations (e) τ xx (f) τ yy Fig. 0. Multiple solutions problem

96 7 τ zz, N/m Case Case Case Newton-Raphson Iterations τ xy, N/m Case Case Case Newton-Raphson Iterations (g) τ zz (h) τ xy τ xz, N/m Case Case Case 3 τ yz, N/m Case Case Case Newton-Raphson Iterations Newton-Raphson Iterations (i) τ xz (j) τ yz k, J/kg Case Case Case 3 ε, W/kg Case Case Case Newton-Raphson Iterations Newton-Raphson Iterations (k) k (l) ε Fig. 0. (Continued from above)

97 Case Case Case Case Case Case 3 D, W/kg D 33, W/kg Newton-Raphson Iterations Newton-Raphson Iterations (m) D (n) D 33 D, W/kg Case Case Case Newton-Raphson Iterations D 3, W/kg Case Case Case Newton-Raphson Iterations (o) D (p) D 3 D 3, W/kg Case Case Case Newton-Raphson Iterations P, W/kg Case Case Case Newton-Raphson Iterations (q) D 3 (r) P Fig. 0. (Continued from above)

98 74 P, W/kg 0.0 Case Case Case 3 P 33, W/kg Case Case Case Newton-Raphson Iterations Newton-Raphson Iterations (s) P (t) P 33 P, W/kg Case Case Case Newton-Raphson Iterations P 3, W/kg Case Case Case Newton-Raphson Iterations (u) P (v) P 3 P 3, W/kg Case Case Case 3 P k, W/kg Case Case Case Newton-Raphson Iterations Newton-Raphson Iterations (w) P 3 (x) P k Fig. 0. (Continued from above)

99 75 P kb, W/kg Case Case Case 3 μ t, Ns/m Case Case Case Newton-Raphson Iterations Newton-Raphson Iterations (y) P kb (z) μ t Fig. 0. (Continued from above) These three cases are the solutions resulting from different initial guesses at a computational node, but have the same initial guesses at the surrounding nodes. The solutions from cases and are close, but do not result in exactly the same number. This problem occurs in many, but not all, cases when the initial guesses are changed. Thus, many different initial guesses can sometimes yield the same exact solution. However, this problem leads to a lack of confidence in the solution approach. The obvious question is, Which solution is correct? In many cases, this question can be answered by considering the direction and magnitude of the secondary currents and the pressure values, which should not deviate significantly from hydrostatic pressures. Problems The model code has verified for every calculation process to make sure that the model will not give a wrong solution due to coding errors. The results from all calculation steps have been compared with the manual calculation results computed in a spreadsheet application. After testing the model for several months with many calculation techniques, the model still had many

100 problems, especially, divergence problem at the nodes at water surface. The lists of 76 methodologies which have been used and the problems which come out from the model when developing are explained below. First, it is difficult to attain convergence at a node using Gauss-Jordan elimination because the initial guesses for all variables must be very close to the final solution. Changing the method for looping in the DoCalculations (Fig. 6) function from looping with Gauss-Jordan until convergence at a node and moving to the next node, to performing Gauss-Jordan one time for a node, then moving to the next node and repeating until the solution converges for all variables at all nodes can cure this problem. The pattern and magnitude of the initial guesses of v and w have been carefully generated. The pattern of secondary currents is imitated from the data from the literature review, which has a clockwise direction loop on the left side and counterclockwise direction loop on the right side, as shown in Fig.. The maximum velocity of the secondary currents is set equal to 4 to 9% of the average u. Fig.. Pattern of the initial guesses of v - w

101 By using very high v, and w, 60 to 70% of average u, the model seems to converge 77 readily. However, some authors claim that they should be about to 3% of the normal u. Appropriate initial guesses can help improve convergence, but it is apparent that problems remain with this solution approach. The next problem is that the k values at boundary nodes always become negative at some point in the solution process. This leads to numerical divergence because the k variable is found inside a square root term, which means the argument cannot be negative without dealing with imaginary numbers. A good initial guess for the k variable can help prevent this problem. This looks like the initial guess of k should vary node by node and it might vary case by case. But in practice it is difficult to provide a set of suitable initial k values (initial guesses for k). The sign of the initial guesses of D ij and P ij also affects the convergence of the solutions. That means the sign (+ or -) of the initial guesses of the variables should depend on the position of the node. Many patterns of D ij and P ij were tried in the model during this research. For RSM, after trying to obtain solutions for different nodes in different quadrants, the relationship between the signs of v, and w and other variables (τ xx, τ yy,τ zz, τ xy, τ xz, τ yz, k, ε, D, D 33, D, D 3, D 3, P, P, P 33, P, P 3, P 3, P k, P kb, μ t ) are established. However, when the signs of the variables depending on the signs of v, and w are applied on the initial guesses, the results cannot be obtained as easily as previously thought. After trying to solve the equations by starting at the central node, it is found that the values at the surrounding nodes which have been changed by previous correction have some effect on the calculating node. It appears that the solution yielded variables outside of the acceptable ranges because of the difference between adjusted and unadjusted values at the surrounding nodes. By using the values from the last iteration for all surrounding nodes, the model performs better than when using some current values from the surrounding nodes, and some from the last iteration.

102 A very good initial guess values and distribution pattern of u has been tried on the 78 model. To obtain a reasonable pattern of u, a new methodology to determine initial values of u has been introduced. A location in the grid of n computational nodes has been chosen such that it corresponds to the expected location of u max, and does not fall on any of the nodes. The distance of each computational node from the assumed u max location is calculated by: ( ) ( ) i = m i + m i d x x y y (4.6) where (x m, y m ) is the location of assumed u max in the grid; and, (x i, y i ) is the location of a given computational node. Coefficients are calculated by: ci = d i (4.7) for i = to n. The summation of the coefficients is: = n c c (4.8) sum i= i and all coefficients will be adjusted by: nci ci = c sum (4.9) The maximum and minimum coefficient values (c max and c min ) are determined, then specify the desired range for the coefficients, r (for example, r = 0.). A coefficient, c α, is calculated based on these values. r c = α c c (4.0) max min Then, each coefficient will be modified by: ci = cα ci + (4.) Finally, the initial guess of the u are set as a function of the total flow rate, Q, and the total cross-sectional area, A.

103 ( ) 79 u = c Q A (4.) i i From this initial guess of u, the solution converges at all interior (non-boundary) nodes. However, some nodes have negative (less than atmospheric) pressure, which is impossible. Some of the u values at the interior nodes look strange, such as too high or too low. The model stops running when it moves to calculate at the first node at the water surface. Many sets of water surface boundary conditions were tried but the divergence problem at the water surface nodes could not be solved. Finally, this version of the turbulence model (almost 5 thousand lines of C#.NET code) was abandoned.

104 CHAPTER 5 80 SOLUTION OF EQUATIONS FINITE VOLUME METHOD Introduction The finite volume (FV) method has been used to solve the governing equations in most of the well-established computational fluid dynamics (CFD) applications, such as CFX/ANSYS, FLUENT, PHOENICS and STAR-CD (Versteeg and Malalasekera 007). It has been said that there are three essential properties of exact solutions contributing to a successful CFD model: conservativeness, boundedness, and transportiveness. Conservativeness means the conservation of fluid properties in a local control volume, and in the entire problem domain, is satisfied. For a fluid quantity, the total rate of energy production and dissipation in a control volume must be equal to the net flux across the control volume boundary multiply by surface area. Boundedness refers to the requirement that the solution must be in range of maximum and minimum values of the boundary values in the absence of an internal source and or sink in the problem domain. Transportiveness asserts that the solution must be affected by both convection and diffusion mechanisms. If diffusion is predominant, the solution at a certain distance around a source or sink should be more or less equal in all directions. In contrast with pure convection, the solution at a point will be determined only by the values from upstream points. All these three essential properties can be satisfied by application of the finite volume method. A turbulence model developed by Kra (00) is also based on the finite volume method. He used algebraic stress equations developed by Naot et al. (993) (ASM3) with the k-ε model for coding within the MATLAB software. A forward staggered grid method was applied for grid generation. There is no graphical user interface for input data. The solutions are presented through contour lines of the surface velocity coefficient, λ, calculated from:

105 where uavg u 8 avg λ = (5.) ucv is the value of average u of the cross section, u cv is the u at the node representing its control volume. The model diagram of his model is shown in Fig.. listed here: Kra s (00) research has some application restrictions and unexpected results which are The graph of velocity coefficient (λ) and water depth had some unexplained spikes. The initial pressure value at each node in the cross section and pressure at boundary nodes were not hydrostatic. These values were set to be zero. This means the pressure distribution from the model output was not correct and could not be compared with the real pressure. The gravity source term in the momentum equation in the y-direction was ignored: v v v v τ yy τ yz p ρv + ρw = μ + μ + + ρgcos( So ) y z y y z z y z y Convective terms Diffusive terms Source terms (5.) Wall and surface proximity correction functions, f and f, in the algebraic stress equation (ASM3) were completely ignored (f = 0, and f = 0): = = 0 f l ya, f l ha = = 0 (5.3) α = f = , f β = = 0.09 (5.4) c = f =.50, c3 = 0.0 f = 0 (5.5) μ c μ μ μ c +.5c μ μ txy 3 = t = t, t = xz ( c+.5c3)( c+ c3) c+ c3 t = t (5.6) u u τxy = μt = μ xy t y y (5.7)

106 8 Read input data, Calculate discharge by Chezy equation, Grid construction, Initialize arrays cmax=00, kmax=0, Nu=0, u_tol=0-3, c= c cmax Yes kmx=, k= No c cmax Yes Nuo=Nu, n_u=0, k= k <= kmax k <= kmx Yes Calculate v near wall by "law-of-wall" Calculate p by FV method k_vw= No k = k+ Yes n_uo=n_u Calculate u near wall by "law-of-wall" nu=0, k= k <= 50 Yes nuo=nu Calculate u by FV method nu=norm(u) No nuo-nu /nu<u_tol No No No k_vw <= 500 Yes nvo=nv, nwo=nw Calculate v by FV method Calculate w by FV method nv=norm(v), nw=norm(w) (nwo-nw)/nw < & nvo-nv /nv < Yes k_ms= No c=c+ Yes n_u=norm(u) n_uo-n_u /n_u<u_tol No q=0, n_keo=.0, n_ke=.0 No No k_ms <= 00 Yes n_mso=n_ms, Calculate pcor by FV method Correct v and w Calculate mass source by using continuity equation for each FV n_ms=norm(ms) Yes (n_keo-n_ke)/ n_keo >ke_tol & q<50 & n_ke<0 Yes q=q+, n_keo=n_ke Calculate k by FV method n_ke=norm(ke) Calculate by FV method No (n_ms/dx)/(rho*q)<0-3 No k_ms>000 & n_ms<nms_min Yes nms_min=n_ms, vtemp=v, wtemp=w No k_ms=k_ms+ Yes No k=k+ Nu=norm(u) (Nuo-Nu)/Nu <u_tol nms_min<n_ms No k=k+ Yes v=vtemp, w=wtemp Yes No c== Write Output ud=u (store D results) Yes Fig.. Flow chart of the turbulence model as developed by Kra (00)

107 u u τxz = μt = μ xz t z z 83 (5.8) k τ yy = c + c 3 β β u u v ρ α + c + τxy τxz + μt 3 ε y z y (5.9) k β β u u τ yy w τzz = ρ α + c + τxz τxy + c3 c 3 + μ t ε z y k z (5.0) τ = yz β ( c+.5c3 ) k u τ u v w xz τ xy μ + + t + ε y z z y (5.) The convergent criteria is determined by norm( u ) as: ( ) / ui nk norm( u ) = < (5.) k = where nk is number of nodes in the cross section. The MATLAB code has dimensional inconsistencies. Kra (00) named a procedure u -pressure corrections, but the corrections are not related to pressure: Qc=Qu_tot(uGrid); while(abs((qc-q)/q)>u_tol) Qc=Qu_tot(uGrid); Du=B*yn/(Dz_u(4,4)*Dy_u(4,4)); dpdxc=rx()*(q-qc)/du+(-rx())*dpdxc; dpdx=dpdx-dpdxc; for I=bu+:tu u(i,lu+:ru-)=u(i,lu+:ru-)+du*dpdxc; end end << Dimensionless << unit [L 3 /T] << unit [L 3 /T] << unit [L/T]+[L 3 /T] This code is used for adjusting the values of u to make the total discharge from the model (Q c ) to be close to the value from the Chezy equation. Also, this will make the value of norm( u ) in the second and third iterations to be very close together and make the equations converge. Without these simplifications, the solution will not converge.

108 In this research, rectangular and compound cross sections have been considered to be 84 applied with five types of Reynolds stress equations: Boussinesq hypothesis, ASM, ASM, ASM3, and RSM. Staggered Grid The staggered-grid method has been proposed to increase the accuracy of the pressure gradient term in the momentum equations. In this research, the scalar variables, including the u are located at the ordinary nodes, but the secondary velocity components are stored at the nodes on the control volume faces surrounding the ordinary nodes. This is called a staggered grid node arrangement, and it has advantages related to the solution of the equations. A backward staggered grid is used in this research. Figure 3 shows a sample rectangular cross section with computational nodes and cells. The spacing of the nodes is uniform except near solid walls and free water surface boundaries. Figures 4 to 6 show the control volumes for velocity ( u ) in the x-direction, velocity ( v ) in the y-direction, and velocity ( w ) in the z-direction. As previously mentioned, the control volume for u is also used for all scalar variables such as pressure, p, pressure correction value, p, turbulent kinetic energy k, energy dissipation rate, ε, and Reynolds stresses, τ ij. For three-dimensional flow, the control volumes can be set up as cubes with a total of six faces. These six cube faces might be labeled as east, west, north, south, top, and bottom (Versteeg and Malalasekera 007). However, all gradients with respect to the streamwise direction, x, are zero for uniform open-channel flow. This means that the top and bottom faces can be ignored in the calculations, but the velocities ( u, v, and w ) in all three coordinate directions must be calculated to arrive at a realistic solution of the governing hydraulic equations (Kra 00).

109 85 y w y w dz dz dz dz dz dz y w y w J=ny- j=ny- J=ny- j=ny- w P φ P v P csdy csdy J=ny-3 j=ny-3 w P dy v P dy J=3 j=3 J= j= J= j= J=0 φ P v P w P φ P dy y w y w p,p,v,k, -control volume w-control volume u-control volume (a) Rectangular cross section y w y w J=ny- j=ny- J=ny- B l B c y w y w y w y w dz l dz l dz c dz c dz c dz r dz r B r y w y w csdy j=ny- J=ny-3 j=ny-3 φ P w P yw yw csdyt dyt Hlr Hc nyi_b v P nyi_b dyb J=3 j=3 nzi_l nzi_r dyb ny_b J= j= J= j= J=0 yw yw I=0 i= I= i= I= i=3 I=3 i=4 I=4 (b) Compound cross section i=nz-4 I=nz-4 i=nz-3 I=nz-3 i=nz- I=nz- i=nz- I=nz- Fig. 3. Computational cell configuration examples in a finite-volume grid at a cross section

110 86 Δz we φ N Δy PN Δy sn φw φ P φe Δy SP φ S Δz WP ΔzPE Fig. 4. The finite-volume computational cell configuration for the u (also for all scalar variables) Δz we j+ N v N J n Δy PN Δy sn j W vw w P vp e E ve J- s Δy SP j- S I- i I i+ I+ v S Δz WP ΔzPE Fig. 5. The finite-volume computational cell configuration for v

111 87 Δz we J+ w N N j+ n Δy PN Δy sn J w W W w wp P e we E j s Δy SP J- w S i- I- i I i+ Δz WP S ΔzPE Fig. 6. The finite-volume computational cell configuration for w wp φp v P Fig. 7. The finite-volume computational cell configurations for all three computational cell types (shaded area for u and scalars, sparse dot line for w and denser dot line for v ).

112 Equations Forms 88 Continuity (Mass Balance) Equation Mass balance is represented by the following (where the derivative term with respect to x is zero for uniform flow), with the assumption that water is incompressible (constant density, ρ): ρ v w + = 0 y ρ z (5.3) Dividing through by density, ρ: v w + = 0 y z (5.4) And, integrating over the u control volumes: ρva ρva+ ρwa ρwa = 0 n n s s e e w w (5.5) where the control volume face areas are: An = As =ΔzweΔ x (5.6) Ae = Aw =ΔysnΔ x (5.7) In which Δ zwe is the width of the control volume (north and south faces); and, Δ ysn is the height of the control volume (east and west faces). Let F = ρv Δz F = ρv Δz n n we, s s we, e e sn F = ρw Δ y, Fw = ρwwδysn (5.8) Then, according to the integrated continuity equation (Eq. 5.5), Δ x cancels, and, ( F F ) ( F F ) 0 + = (5.9) n s e w General Form of Equations All equations represented each variable must be integrated over their control volumes and rearranged into a standard equation form. The standard equation form for the finite volume method is:

113 89 ψ ψ ψ ψ ψ ψ ρv + ρw = Γ y + Γ z + S y z y y z z ψ (5.0) where ψ Γ y and ψ Γ z are diffusion coefficients; ψ represents any of the dependent variables; and S ψ represents the source term. Next, integrate over the control volume: ρvn ρvs ρwe ρww A A A A ψ ψ ψ Γy Γy Γz = A A + A Δy Δy Δz ( ψ + ψ ) ( ψ + ψ ) + ( ψ + ψ ) ( ψ + ψ ) n N P s P S e E P w P W ( ψ ψ ) ( ψ ψ ) ( ψ ψ ) n N P s P S e E P PN SP PE ψ Γz Aw P W +ΔV S Δz WP ( ψ ψ ) ψ ( ψ ) (5.) Define the F and D terms as follows: ψ = ρ = ( ρ ), +, F ρva ( ρva), F v A va ψ n n n I j = = (5.) s s s I j ψ = ρ = ( ρ ) +,, F ρwa ( ρwa), F w A wa ψ e e e i J = = (5.3) w w w ij D Γ A Γ A ψ ψ ψ y n y I, j+ n = = ΔyPN ΔyPN, Γ A Γ A ψ Ds = = Δy Δy ψ ψ y s y I, j SP SP (5.4) D Γ A ψ ψ ψ Γz Ae z i+, J e = = ΔzPE ΔzPE, D Γ A ψ ψ z w w = = ΔzWP Γ ψ z Δz A i, J WP (5.5) ( ) b =Δ V S (5.6) ψ ψ ψ Then, ψ ψ ψ ψ Fn Fs Fe Fw ( ψ N + ψ P) ( ψ P + ψs) + ( ψ E + ψ P) ( ψ P + ψw ) ψ ψ ψ ψ = D D + D D + b ( ψ ψ ) ( ψ ψ ) ( ψ ψ ) ( ψ ψ ) n N P s P S e E P w P W ψ (5.7) Next, perform a series of algebraic manipulations: ψ ψ ψ ψ ψ F w ψ F e ψ F s ψ F n Dw + De + + Ds + Dn + ψ P ψ ψ ψ ψ ψ F w ψ F e ψ F s ψ F n = Dw + ψw + De ψ E + Ds + ψs + Dn ψ N + b ψ (5.8)

114 90 ψ ψ ψ ψ F w F e F s F n D + + D + D + + D + F F + F F ( ) ( ) ψ ψ ψ ψ ψ ψ ψ ψ w e s n e w n s P ψ ψ ψ ψ ψ F w ψ F e ψ F s ψ F n = Dw + ψw + De ψ E + Ds + ψs + Dn ψ N + b ψ ψ (5.9) ψ ψ ψ ψ Apply Eq. 5.9 (continuity), ( F F ) ( F F ) 0 terms in parentheses. Then: + =, and use coefficients to replace the n s e w a ψ = a ψ + a ψ + a ψ + a ψ + b (5.30) ψ ψ ψ ψ ψ P P W W E E S S N N ψ where P represents the node represented control volume; W represents the west face of the control volume; E is the east face, S is the south face; N is the north face; and, a represents the equation coefficients at the node and its faces: a = a + a + a + a (5.3) ψ ψ ψ ψ ψ P W E S N a ψ W ψ ψ Fw = Dw +, a ψ E ψ ψ Fe = De, a ψ S ψ ψ Fs = Ds +, a ψ N ψ ψ Fn = Dn (5.3) Momentum Equation in the x-direction The general momentum equation in the x-direction is: u u u u τ xy τ ρv + ρw = ρgsin( ξ) + μ + μ + + y z y y z z y z xz (5.33) Each term in Eq has units of N/m 3. This equation will be applied to the u control volume (Fig. 4). Then, the general finite-volume equation is obtained: au = a u + au + au + au + b (5.34) u u u u u P P W W E E S S N N u where the parameters for the finite volume method are: a = a + a + a + a (5.35) u u u u u P W E S N Γ =Γ = (5.36) u u y z μ = ρ = ( ρ ), +, F u ρva ( ρva), F v A va u n n n I j = = (5.37) s s s I j

115 = ρ = ( ρ ) +,, F u ρwa ( ρwa), F w A wa u e e e i J w w w ij 9 = = (5.38) D u n μ A = = Δy Δy μ An I, j+ PN PN, D u s μ A Δy s = = SP μ A Δy I, j SP, (5.39) D u e μ A = = Δz Δz μ Ae i+, J PE PE, D u w μ A Δz w = = WP μ A Δz ij, WP (5.40) Using central differences for the shear stress terms, ( τ ) ( τ ) ( τ ) ( τ ) bu =Δ VI, Jρgsin( ξ) +Δ VI, J + Δysn xy n xy s xz e xz w Δz we (5.4) in which ( τ xy ) n Δ V =Δy Δz Δ x (5.4) IJ, sn we ( τ ) + xy ( τ ) I, J+ xy I, J =, ( τ xy ) ( τxy ) + ( τxy ) I, J I, J = (5.43) s and ( τ ) xz e ( τxz ) + ( τ ) I, J xz I+, J =, ( τ ) xz w ( τ ) ( τ ) xz + I, J xz I, J = (5.44) Momentum Equation in the y-direction This equation is similar to the momentum equation in the x-direction, but with an additional term representing the pressure gradient: v v p v v τ yy τ ρv + ρw = ρgcos( ξ) + μ + μ + + y z y y y z z y z yz (5.45) where the pressure gradient term is negative: p < 0 y (5.46) (because y is positive in the upward direction, so that p decreases as y increases) with an absolute magnitude near that of ρg cos(ξ), whereby,

116 p ρg cos( ξ) 0 y 9 (5.47) Nevertheless, the velocities in the y-direction, v, are generally of small magnitude, but are not equal to zero. This equation will be applied to v control volume (Fig. 5). The general finite-volume equation is obtained: ( ) av = a v + av + av + av + A p p + b (5.48) v v v v v P P W W E E S S N N n s n v where the parameters for the finite volume method are: a = a + a + a + a (5.49) v v v v v P W E S N Γ =Γ = (5.50) v v y z μ vi, j+ v v I, j+ Fn = ρva n n = ρ AI, J, v v I, j + vi, j Fs = ρvsas = ρ AI, J (5.5) v wi+, J + wi+, J Fe = ρwa e e = ρ Ai+, j, v wij, + wij, Fw = ρwa w w = ρ Ai, j (5.5) D v n μ A Δy n = = PN μ A Δy I, J PN, D v s μ A Δy s = = SP μ A Δy IJ, SP (5.53) D v e μ A = = Δz Δz μ Ae i+, j PE PE, D v w μ A Δz w = = WP μ A Δz i, j WP (5.54) p p s = I, J, pn pi, J = (5.55) τ yy τ yz bv =ΔVI, j ρgcos( ξ) + + y z (5.56) Or, using central differencing for the τ terms, ( τ yy ) ( τ yy ) ( τ yz ) ( τ yz ) bv = Δ VI, jρgcos( ξ) +Δ VI, j + Δysn n s e w Δz we (5.57) in which

117 ( τ yz ) ( τ yz ) ( τ ) = yy ( τ yy ), ( τ ) ( ) n I, J yy τ s yy I, J ( τ yz ) + ( τ yz ) + ( τ yz ) + ( τ yz ) 93 = (5.58) I, J I+, J I, J I+, J = (5.59) e ( τ yz ) + ( τ yz ) + ( τ yz ) + ( τ yz ) 4 I, J I, J I, J I, J = (5.60) w 4 Momentum Equation in the z-direction The momentum equation in the z-direction is similar to that for the y-direction, except without the ρg cos(ξ) term: τ τ ρv ρ μ w w p w w yz zz + w = y z z y z y z (5.6) This equation will be applied to the w control volume (Fig. 6). The general finitevolume equation is obtained: ( ) a w = a w + a w + a w + a w + A p p + b (5.6) w w w w w P P W W E E S S N N e w e w which the parameters for the finite volume method are: a = a + a + a + a (5.63) w w w w w P W E S N Γ =Γ = (5.64) w w y z μ v w I, j+ + vi, j+ Fn = ρva n n = ρ Ai, j+, vi, j+ v w I, j Fs = ρva s s = ρ Ai, j (5.65) w wi+, J + wi, J e ρ e e ρ I, J F = wa = A, w wij, + wi, J Fw = ρwwaw = ρ AI, J (5.66) D w n μ A Δy n = = PN μ A Δy I, J PN, D w s μ A Δy s = = SP μ A Δy i, j SP (5.67) D w e μ A Δz e = = PE μ A Δz I, J PE, D w w μ A Δz w = = WP μ A Δz I, J WP (5.68)

118 p w pi, J 94 =, pe = pi, J (5.69) b τ τ zz y z yz w =Δ Vi, J + (5.70) or, b w ( τ ) ( τ ) ( τ ) ( τ ) =Δ Vi, J + Δysn yz n yz s zz e zz w Δz we (5.7) in which ( τ yz ) ( τ yz ) ( τ ) = zz ( τzz ), ( e I, τ ) ( ) J zz τzz, = (5.7) w I J ( τ yz ) + ( τ yz ) + ( τ yz ) + ( τ yz ) I, J+ I, J+ I, J I, J = (5.73) n ( τ yz ) + ( τ yz ) + ( τ yz ) + ( τ yz ) 4 I, J I, J I, J I, J = (5.74) s 4 The k Equation of the k-ε Model Based on the Boussinesq Hypothesis The equation for turbulent kinetic energy, k is: k k μ k μ k t t ρv + ρw = ρpkb ρε + μ + + μ + y z y σk y z σk z (5.75) This equation will be applied to the u /scalar control volume (Fig. 4). To prevent negative values of k in the calculation process, the negative term on the left-hand side will be moved to combine with the coefficient of the calculating node. Thus, the general finite-volume equation is obtained: ak = a k + ak + ak + ak + b (5.76) k k k k k P P W W E E S S N N k in which the parameters for the finite volume method are:

119 ( ) ΔV, ρ ε k k k k k IJ, I J ap = aw + ae + as + an + (5.77) k P 95 μ Γ =Γ = μ + (5.78) k k t y z σ k ( ), b = ρ P Δ V (5.79) k kb IJ, I J It is noted here that all F and D terms use the same formula as the momentum equation in the x-direction (scalar control volume). The ε Equation of the k-ε Model Based on the Boussinesq Hypothesis The equation for is turbulent energy dissipation rate, ε, is as follows: ε ε ε ε μ t ε μ t ε ρv + ρw = cερ Pkb ρcε + μ+ + μ+ y z k k y σε y z σε z (5.80) where each term has units of kg/m/s 4, or N/m /s. This equation will be applied to the u /scalar control volume (Fig. 4). To prevent negative values of ε in the calculation process, the negative term on the left-hand side is moved to combine with the coefficient of the calculating node. The general finite-volume equation is obtained: a εu = a ε + a ε + a ε + a ε + b (5.8) ε ε ε ε ε P P W W E E S S N N ε in which the parameters for the finite volume method are: ( ε ) ( ), k a = a + a + a + a + ρc Δ V (5.8) ε ε ε ε ε IJ P W E S N ε I, J IJ, μ ε ε t Γ y =Γ z = μ + (5.83) σ ε ( ε ) ( k ) ( ) b = c ρ P Δ V (5.84) ε IJ, ε kb IJ, I, J IJ,

120 96 Again, all F and D terms use the same formula as the momentum equation in the x- direction (scalar control volume). The ε Equation of the k-ε Model for the Reynolds Stress Model The equation for ε (kinetic energy dissipation per unit mass) is as follows: ε ε k ε k ε ρv + ρw = cετ yy + cετzz y z y ε y z ε z k ε k ε ε ε cε τ yz τ yz ρc ε Pkb ρcε y ε + + z z ε y k k (5.85) This equation has the same condition as the ε equation based on the Boussinesq hypothesis. The general finite-volume equation is obtained: a εu = a ε + a ε + a ε + a ε + b (5.86) ε ε ε ε ε P P W W E E S S N N ε in which the parameters for the finite volume method are: ( ε ) ( ), k a = a + a + a + a + ρc Δ V (5.87) ε ε ε ε ε IJ P W E S N ε I, J IJ, k ετ ε Γ ε = y c, ε yy c z zz ( ε ) ( k ) Γ = k ετ (5.88) ε ( ) b = c ρ P Δ V (5.89) ε IJ, ε kb IJ, I, J IJ, Reynolds Stress Model All Reynolds stress model equations are applied to the u /scalar control volume (Fig. 4). All F and D terms use the same formula as the momentum equation in the x-direction (scalar control volume). τ xx equations:

121 τ xx τxx cs k τ xx cs k τ xx v + w = τ yy + τzz y z y ρ ε y z ρ ε z w ρp + ρε ρ( Π +Π +Π) 3 97 (5.90) Then, c ρ k ε Γ τ xx = S y τ, τ yy xx S z Γ = c k τ zz ρ ε (5.9) bτ =ΔV ρp ρε ρ xx + Π +Π +Π 3 w ( ) I, j (5.9) τ yy equations: τ yy τ yy cs k τ yy cs k τ yy v + w = τ yy + τzz y z y ρ ε y z ρ ε z w ρp + ρε ρ( Π +Π +Π ) 3 (5.93) Then, c ρ k ε Γ τ yy = S y τ, τ yy S yy z Γ = c k τ zz ρ ε (5.94) bτ =ΔV ρp ρε ρ yy + Π +Π +Π 3 w ( ) I, j (5.95) τ zz equations: τ zz τzz cs k τ zz cs k τ zz v + w = τ yy + τzz y z y ρ ε y z ρ ε z w ρp33 + ρε ρ( Π 33 +Π 33 +Π33 ) 3 (5.96) Then, c ρ k ε Γ τ zz = S y τ, τ yy zz S z Γ = c k τ zz ρ ε (5.97) bτ =ΔV ρp ρε ρ zz + Π +Π +Π 3 w ( ) I, j (5.98)

122 τ xy equations: τxy τxy cs k τxy cs k τxy v + w = τ yy + τzz y z y ρ ε y z ρ ε z w ( ) ρp ρ Π +Π +Π 98 (5.99) Then, xy c ρ k ε Γ τ xy = S y τ, τ xy S yy z Γ = c k τ zz ρ ε (5.00) ( ( )) w bτ =ΔV ρp ρ Π +Π +Π (5.0) I, j τ xz equations: τ xz τxz cs k τ xz cs k τ xz v + w = τ yy + τzz y z y ρ ε y z ρ ε z w ( ) ρp ρ Π +Π +Π (5.0) Then, xz c ρ k ε Γ τ xz = S y τ, τ yy xz S z Γ = c k τ zz ρ ε (5.03) ( ( )) w bτ =ΔV ρp ρ Π +Π +Π (5.04) I, j τ yz equations: τ yz τ yz cs k τ yz cs k τ yz v + w = τ yy + τzz y z y ρ ε y z ρ ε z w ( ) ρp ρ Π +Π +Π (5.05) Then, yz c ρ k ε Γ τ yz = S y τ, τ yz S yy z Γ = c k τ zz ρ ε (5.06) ( ( )) w bτ =ΔVI, j ρp3 ρ Π 3 +Π 3 +Π 3 (5.07)

123 Momentum Equation with Reynolds Stress Equations 99 During the development of the finite volume method solution, the model had problems with mathematical convergence. In an attempt to ameliorate these difficulties, alternate forms of the momentum equations were developed. The Reynolds stress terms in the momentum equations were substituted by the Reynolds stress equations from the Boussinesq hypothesis, ASM, ASM, and ASM3. This strategy was successful in solving the problem. The new momentum equations and finite volume coefficients are shown below. Momentum Equations + Boussinesq Hypothesis. In the x-direction, u u v w gsin( ) ( t) u ρ ρ ρ ξ μ μ ( μ μt) u + = y z y y z z (5.08) Then, Γ =Γ = μ + μ (5.09) u u y z t ( ρ ( ξ )) bu =Δ VI, J gsin (5.0) In the y-direction, v v p v v ρv + ρw = ρgcos( ξ) + ( μ+ μt) + ( μ+ μt) y z y y y z z ρ k w + μt 3 y z y (5.) Then, v Γ = μ + μ, y t Γ = μ + μ (5.) v z t ρ k w bv =ΔVI, j ρgcos( ξ) + μt 3 y z y (5.3) In the z-direction,

124 w w p w w ρv + ρw = + ( μ + μt) + ( μ + μt) y z z y y z z v ρ k + μt y z 3 z 00 (5.4) Then, w Γ = μ + μ, Γ = μ + μ (5.5) w y t z t b w ρ k =ΔVi, J 3 z (5.6) Momentum equations + ASM. In the x-direction, u u c k u c k u ρv + ρw = ρgsin( ξ) + μ+ τ yy + μ+ τzz y z y c ε y z c ε z c k v u v c y y z z + τxy + τ yz + τxz ε c k w u w c z z y y + τxz + τ yz + τxy ε (5.7) Then, c k u u Γ y = μ + τ yy, z μ c ε c c k Γ = + τ zz (5.8) ε c k v u v ρg sin( ξ) + τ + τ + τ xy yz xz c y ε y z z u =ΔVI, J c k w u w + τxz + τ yz + τ xy c z ε z y y b (5.9) In the y-direction,

125 v v p k v ρv + ρw = ρgcos( ξ) + μ+ c τ yy y z y y c 3 ε y c k v + μ+ τzz z c ε z c c k w u u 3 v w c y c z y z c z y + ρk τzz + τxy + τxz + τ yz + τ yz 3 ε c k v w w z y z y + τ yz + + τ yy c ε 0 (5.0) Then, v k Γ y = μ + c τ c 3 ε c k Γ = + τ zz (5.) ε v yy, z μ c c k v w w ρg cos( ξ) + τ yz + + τ yy c z ε y z y bv =ΔVI, j (5.) c c k w u u 3 v w + ρk τzz + τxy + τxz + τ yz + τ yz 3 c y c ε z y z c z y In the z-direction, w w p c k w k w ρv + ρw = + μ+ τ yy + μ+ c τzz y z z y c ε y z c 3 ε z c k v w v τ τzz c y y z z + yz + + ε c c k v u u v 3 w + τ yz z c y + ρk τ yy + τxy + τxz + τ yz 3 c z c ε y y z (5.3) Then, c k k Γ = μ + c τ zz ε w w Γ y = μ + τ yy, z c ε c 3 (5.4)

126 b w 0 c k v w v τ yz + + τzz c y ε y z z =ΔVi, J (5.5) c c k v u u v 3 w + ρk τ yy + τxy + τxz + τ yz + τ yz 3 c z c ε y y z z c y Momentum equations + ASM. In the x-direction, u u cdk u ρv + ρw = ρgsin( ξ) + μ+ τ yy y z y cε Pk ε y + ck D u ck D v u v + μ+ τzz + τxy + τ yz + τxz z cε Pk ε z + y cε + Pk ε y z z ck w u w z c + P z y y D + τxz + τ yz + τxy ε k ε (5.6) Then, u ck D Γ y = μ + τ yy, cε + P ε k u ck D Γ z = μ + τ zz cε + P ε k (5.7) ck v u v ρg sin ξ + τ + τ + τ ( ) D xy yz xz y cε + Pk ε y z z u =ΔVI, J ck D w u w + τxz + τ yz + τ xy z cε + Pk ε z y y b (5.8) In the y-direction, v v p 4 cdk v ρv + ρw = ρgcos( ξ) + μ+ τ yy y z y y 3 cε Pk ε y + ck D v + μ+ τzz z cε Pk ε z + ck D w u u w v + ρk τzz + τxy + τxz + τ yz 3 y cε + Pk ε z y z y z ck τ v w τ w + + y z y D + yz yy z cε + Pk ε (5.9) Then,

127 Γ = μ + v y 4 3 ck D τ yy, cε + P ε k v ck D Γ z = μ + τ zz cε + P ε k 03 (5.30) ck D v w w ρg cos( ξ) + τ yz + + τ yy z cε Pk ε y z y + bv =ΔVI, j ck D w u u w v + ρk τzz + τxy + τxz + τ yz 3 y cε + Pk ε z y z y z (5.3) In the z-direction, w w p c k w y z z y c + P y D ρv + ρw = + μ+ τ yy ε k ε 4 ck D w ck D v w v + μ+ τzz + τ yz + + τzz z 3 cε Pk ε z + y cε + Pk ε y z z ck D v u u v w + ρk τ yy + τxy + τxz + τ yz 3 z cε + Pk ε y y z z y (5.3) Then, w ck D Γ y = μ + τ yy, cε + P ε k Γ = μ + w z 4 3 ck D τ zz cε + P ε k (5.33) b w ck D v w v τ yz + + τzz y cε Pk ε y z z + =ΔVi, J ck D v u u v w + ρk τ yy + τxy + τxz + τ yz 3 z cε + Pk ε y y z z y (5.34) Momentum equations + ASM3. In the x-direction, u u u u ρv + ρw = ρgsin( ξ) + ( μ+ μt ) ( μ μ ) xy + + txz y z y y z z (5.35) Then, Γ = μ + μ, u y txy Γ = μ + μ (5.36) u z txz ( ρ ( ξ )) bu =Δ VI, J gsin (5.37) In the y-direction,

128 04 v v p v v ρv + ρw = ρgcos( ξ) + ( μ+ μt) + ( μ+ μt) y z y y y z z k β β u u + ρ α + c + μt μ xy txz y c+ c3 3 ε y z k u u w z c+ c3 y z μt z y β ( ) ( μt μ ) xy txz.5 ε (5.38) Then, v Γ = μ + μ, y t Γ = μ + μ (5.39) v z t b v =ΔV β k u u w ρg cos( ξ) + ( ) ( μt + μ xy txz ) + μt z c.5c3 ε y z + z y k β β u u + ρ α + c + μt μ xy txz y c+ c3 3 ε y z I, j (5.40) In the z-direction, w w p w w ρv + ρw = + ( μ+ μt) + ( μ+ μt) y z z y y z z k u u v + μt + μt + μt y c c3 y z + y z β ( ) ( ) xy xz.5 ε c3 β ρ + c k 3 c c α c3 v + + μt z c c 3 β u u z c y + μ t xz μ txy c+ c3 ε z y (5.4) Then, w Γ = μ + μ, Γ = μ + μ (5.4) w y t z t

129 b β k u u v ( ) ( μt + μ ) xy t + μ xz t y c.5c3 ε y z + y z c β ρ + α + c k c3 v + + z c c 3 β u u μt z c y + μ t xz μ t xy c+ c3 ε z y 3 w =ΔVi, J 3 c+ c3 05 (5.43) All F and D terms use the same formula as the momentum equations in each direction. SIMPLER Algorithm Pressure Equation It can see that there is no specific equation for pressure. The SIMPLER algorithm (Patankar 980) was proposed for developing equations to calculate pressure value. The pressure equation is solved at the u /scalar control volumes (Fig. 4) because pressure is a scalar value. From the momentum equation in the y- and z-directions: v v av P P = anbvnb + pa s s pa n n + bv (5.44) w w ap wp = anbwnb + pw Aw pe Ae + bw (5.45) where v v v v v a v nb nb = a v W W + a v E E + a v a v S S + N N (5.46) w w w w w a w nb nb = a w W W + a w E E + a w a w S S + N N (5.47) Let, Then, vˆ P a v + b =, wˆ v nb nb v v ap P a w + b = (5.48) w nb nb w w ap A A v = v + p p, s n ˆ P P s v n v ap ap A A v = vˆ + p p (5.49) I, J I, J I, j I, j v I, J v I, J ai, j ai, j

130 A A w = w + p p, w e ˆ P P w w e w ap ap I, J I, J ij, ij, w I, J w IJ, aij, aij, 06 A A w = wˆ + p p (5.50) A A v = vˆ + p p (5.5) I, J I, J+ I, j+ I, j+ v I, J v I, J+ ai, j+ ai, j+ A A w = wˆ + p p (5.5) I, J I+, J i+, J i+, J w I, J w I+, J ai+, J ai+, J From the continuity equation (Eq. 5.5), ( ρ va ) ( ρ va ) ( ρ wa ) ( ρ wa ) + = 0 I, j+ I, j i+, J i, J (5.53) A A A A ρ vˆ + p p A ρ vˆ + p p A I, J I, J+ I, J I, J I, j+ v I, J v I, J+ I, j+ I, j v I, J v I, J I, j ai, j+ ai, j+ ai, j ai, j A A A A + ρ wˆ + p p A ρ wˆ + p I, J I+, J I, J I, J i+, J w I, J w I+, J i, J i, J w I, J ai, J a + w + i+, J ai, J ai, J p A I, J i, J = 0 (5.54) A A A A ρ A + ρ A + ρ A + ρ A p a a a a A A A = ρ A p + ρ A p + ρ A p a a a IJ, IJ, IJ, IJ, w ij, w i+, J v I, j v I, j+ IJ, i, J i+, J I, j I, j+ I, J I+, J I, J w ij, I, J w i+, J I+, J v I, j IJ, ij, i+, J I, j A + ρ A ˆ ˆ I, j+ pij, + + ρwij, AiJ, ρwi+, JA ˆ ˆ i+, J+ ρvi, jai, j ρvi, j+ AI, j+ a IJ, + v I, j+ (5.55) For parallel grid lines, d A A A = = = = (5.56) p I, J I, J i, J w di, J w w w aij, aij, aij, d A A A = = = = (5.57) p I, J I+, J i+, J e di+, J w w w ai+, J ai+, J ai+, J d A A A = = = = (5.58) p I, J I, J I, j s di, j v v v ai, j ai, j ai, j A A A d = = = = (5.59) p I, J I, J+ I, j+ n di, j+ v v v ai, j+ ai, j+ ai, j+

131 Then, ( ), ρd A + ρda + ρda + ρda p = ρd A p + ρda p+ ρda p+ ρda p p p p p p p p p w w e e s s n n I J w w w e e e s s s n n n + ρwa ˆ ρwa ˆ + ρva ˆ ρva ˆ p p p p w w e e s s n n 07 (5.60) or, a p = a p + a p + a p + a p + b (5.6) p p p p p P I, J W w E e S s N n p where, a = a + a + a + a (5.6) p p p p p P W E S N a = ρd A, p p W w w a = ρd A, p p E e e a = ρd A, p p S s s a = ρd A (5.63) p p N n n The source term for the pressure equation is defined as: b = ρwˆ A ρwˆ A + ρvˆ A ρvˆ A (5.64) p i, J i, J i+, J i+, J I, j I, j I, j+ I, j+ or, p p p p b = ρwˆ A ρwˆ A + ρvˆ A ρvˆ A (5.65) p w w e e s s n n Pressure Correction Equation Let * v, * w, and * p be guesses for the velocities and pressure at a point in the channel cross section. The values of the guesses are specified for the first iteration, but in subsequent iterations of the solution procedure they are set equal to the values from the previous iteration. Then, v * v * * * av P P = anbvnb + pa s s pa n n + bv (5.66) w * w * * * ap wp = anbwnb + pw Aw pe Ae + bw (5.67) * Also, let v, w, and p represent velocity and pressure corrections to the respective guesses ( v, * w, and * p ). Then, * v = v + v, * w= w + w (5.68)

132 Subtract Eqs and 5.45 by Eqs and 5.67, respectively: or, 08 * p = p + p (5.69) * v * * * ( ) = ( ) + ( ) ( ) v ap vp vp anb vnb vnb ps ps As pn pn An (5.70) * w * * * ( ) = ( ) + ( ) ( ) w ap wp wp anb wnb wnb pw pw Aw pe pe Ae (5.7) v v av P P = anbv nb + pa s s pa n n (5.7) w w ap w P = anbw nb + p w Aw p e Ae (5.73) Let, v w anbv nb = anbw nb = 0 (5.74) Then, A A v = p p, or s n P v s v n ap ap A A v = p p (5.75) I, J I, J I, j v I, J v I, J ai, j ai, j A A w = p p, or w e P w w w e ap ap A A w = p p (5.76) I, J I, J ij, w I, J w IJ, aij, aij, Similarly, A A v = p p (5.77) I, J I, J+ I, j+ v I, J v I, J+ ai, j+ ai, j+ A A w = p p (5.78) I, J I+, J i+, J w I, J w I+, J ai+, J ai+, J From Eq. 5.68, A A v = v + p p (5.79) * I, J I, J+ I, j+ I, j+ v I, J v I, J+ ai, j+ ai, j+ A A v = v + p p (5.80) * I, J I, J I, j I, j v I, J v I, J ai, j ai, j

133 * I, J I+, J i+, J i+, J w I, J w I+, J ai+, J ai+, J 09 A A w = w + p p (5.8) A A w = w + p p (5.8) * I, J I, J ij, ij, w I, J w IJ, aij, aij, Again, from the continuity equation (Eq. 5.5), ( ρ va ) ( ρ va ) ( ρ wa ) ( ρ wa ) + = 0 I, j+ I, j i+, J i, J (5.83) A * I, J A I, J A * I, J A + I, J ρ vi, j+ + p I, J p v v I, J+ AI, j+ ρ vi, j+ p v I, J p v I, J AI, j ai, j a I, j ai, j a + + I, j (5.84) A * I, J A I+, J A * I, J A I, J + ρ wi+, J + p w I, J p w I+, J Ai+, J ρ wi, J + p w I, J ai+, J a,, 0 w p I J A i J= i+, J ai, J a ij, A A ρv A + ρ A p ρ A p * IJ, IJ, + I, j+ I, j+ v I, j+ I, J v I, j+ I, J+ ai, j+ ai, j+ A A ρv A ρ A p + ρ A p * IJ, IJ, I, j I, j v I, j I, J v I, j I, J ai, j ai, j A A + ρw A + ρ A p ρ A p * IJ, I+, J i+, J i+, J w i+, J I, J w i+, J I+, J ai+, J ai+, J ρw A * ij, ij, A ρ a A A p + ρ A p = 0 I, J I, J w ij, I, J w ij, IJ, ij, aij, (5.85) AIJ, AIJ, AIJ, A IJ, ρ A w ij, + ρ A w i+, J + ρ A v I, j + ρ A v I, j+ p IJ, aij, ai, J ai, j a + I, j+ A A A = ρ A p + ρ A p + ρ A p a a a I, J I+, J I, J w ij, I, J w i+, J I+, J v I, j IJ, ij, i+, J I, j A + ρ A p + ρw A ρw a A + ρv A ρv A IJ, + * * * * v I, j+ I, J+ i, J i, J i+, J i+, J I, j I, j I, j+ I, j+ I, j+ (5.86) and, a p = a p + a p + a p + a p + b (5.87) p p p p p P P W w E e S s N n p where, a = a + a + a + a (5.88) p p p p p P W E S N

134 0 a = ρd A, p p W w w a = ρd A, p p E e e a = ρd A, p p S s s a = ρd A (5.89) p p N n n which are the same as in the pressure equation. The source term for the pressure correction equation is: b = ρw A ρw A + ρv A ρv A (5.90) * * * * p i, J i, J i+, J i+, J I, j I, j I, j+ I, j+ or, b = ρw A ρw A + ρv A ρv A (5.9) * p * p * p * p p w w e e s s n n Differencing Scheme Patankar (980) proposed the power-law differencing scheme which is more accurate and is a better representation of the exact solution than central, upwind, and hybrid differencing schemes. The patterns of the coefficients in the finite volume equation are: ( ( ) ) max 0, 0. 5 max [, 0 ] a = D Pe + F W w w w ( ( ) ) max 0, 0. 5 max [, 0 ] ae = De Pee + Fe ( ( ) ) max 0, 0. 5 max [,0 ] as = Ds Pes + Fs ( ( ) ) max 0, 0. 5 max [, 0 ] a = D Pe + F N n n n (5.9) (5.93) (5.94) (5.95) where the max function is applied in the model to preclude negative values; and P e is the dimensionless Peclet number, which represents the relative strengths of the convection and diffusion terms: F Pe = (5.96) D

135 Partial Differentials of Velocity Components at Scalar Nodes For nodes (I = to I = nz-) and (J = to J = ny-3): ( ) ( ) u u u u u u u u n u + + = s = = y Δy Δy Δy I, J I, J+ I, J I, J I, J+ I, J sn sn sn (5.97) For nodes (I = to I = nz-) and (J = ): ( ) u u u u 0 u u n u + + = s = = y Δy Δy Δy I, J I, J+ I, J I, J+ sn sn sn (5.98) For nodes (I = to I = nz-) and (J = ny-): ( ) ( ) u u u u u u u u n u + + = s = = y Δy Δy Δy I, J I, J+ I, J I, J I, J I, J sn sn sn (5.99) For nodes (I = to I = nz-3) and (J = to J = ny-): ( ) ( ) u u u u u u u u e u + + = w = = z Δz Δz Δz I, J I+, J I, J I, J I +, J I, J we we we (5.00) For nodes (I = ) and (J = to J = ny-): ( ) u u u u 0 u u e u + + = w = = z Δz Δz Δz I, J I+, J I, J I+, J we we we (5.0) For nodes (I = nz-) and (J = to J = ny-): ( ) u u 0 u u u u e u + + = w = = z Δz Δz Δz I, J I, J I, J I, J we we we (5.0) For nodes (I = to I = nz-) and (J = to J = ny-): v v v y Δy Δy v = n s = sn I, j+ vi, j sn (5.03) For nodes (I = to I = nz-3) and (J = to J = ny-):

136 ( vi, j+ + vi+, j+ + vi, j + vi+, j) ( vi, j+ + vi, j+ + vi, j + vi, j) v ve v = w = z Δz 4Δz we v + v v v = 4Δz I+, j+ I+, j I, j+ I, j we we (5.04) For nodes (I = ) and (J = to J = ny-): ( ) v v v v v v 0 v v v v e v = w = = z Δz 4Δz 4Δz I, j+ I+, j+ I, j I+, j I, j+ I+, j+ I, j I+, j we we we (5.05) For nodes (I = nz-) and (J = to J = ny-): ( ) v v 0 v v v v v v v v e v = w = = z Δz 4Δz 4Δz I, j+ I, j+ I, j I, j I, j+ I, j+ I, j I, j we we we (5.06) For nodes (I = to I = nz-) and (J = to J = ny-): w w w w z Δz Δz = e w = we i+, J wi, J we (5.07) For nodes (I = to I = nz-) and (J = to J = ny-3): ( wij, + + wi+, J+ + wij, + wi+, J) ( wij, + wi+, J+ wij, + wi+, J ) w wn w = s = y Δy 4Δy sn w + w w w = 4Δy ij, + i+, J+ ij, i+, J sn sn (5.08) For nodes (I = to I = nz-) and (J = ): ( ) w w w w w w 0 w w w w n w = s = = y Δy 4Δy 4Δy ij, + i+, J+ ij, i+, J i, J+ i+, J+ i, J i+, J sn sn sn (5.09) For nodes (I = to I = nz-) and (J = ny-): ( wij, + + wi+, J+ + wij, + wi+, J) ( wij, + wi+, J+ wij, + wi+, J ) w wn w = s = y Δy 4Δy sn sn w From = 0 y at the water surface, w = ij, w, + ij, wi+, J+ = wi+, J, and

137 ( ) ( ) w w + w w + w + w + w w + w w w = = y 4Δy 4Δy i, J i+, J i, J i+, J i, J i+, J ij, i+, J ij, i+, J sn sn 3 (5.0) Solving the Equations The solution to the system of equations by the finite volume method is not the same as that used with the finite difference method. The set of equations is solved for all control volume nodes equation-by-equation. For any given equation, every control volume node has its own equation in the form of Eq. 5.30, which is: a ψ = a ψ + a ψ + a ψ + a ψ + b (5.) ψ ψ ψ ψ ψ P P W W E E S S N N ψ This means the number of finite volume equations must be equal to the number of control volume nodes. There are two strategies to solve these finite volume equations: to solve all equations simultaneously or to solve equations column-by-column (or row-by-row). For example, a cross section with 4 x 3 = nodes (as shown in Fig. 8) has the finite volume equations for each node as: ψ 0,4 ψ,4 ψ,4 ψ 3,4 ψ 4,4 ψ 5,4 ψ 0,3 ψ,3 ψ,3 ψ 3,3 ψ 4,3 ψ 5,3 ψ 0, ψ, ψ, ψ 3, ψ 4, ψ 5, ψ 0, ψ, ψ, ψ 3, ψ 4, ψ 5, ψ 0,0 ψ,0 ψ,0 ψ 3,0 ψ 4,0 ψ 5,0 =Boundary nodes =Calculating nodes Fig. 8. Example of a 4 x 3 control volume a ψ a ψ a ψ = b + a ψ + a ψ (5.) ψ ψ ψ ψ ψ,, P,,, E,,, N,,, ψ,, W 0,,, S,0

138 ψ ψ ψ ψ ψ,, P,,, W,,, E 3,,, N,,, ψ,, S,0 4 a ψ a ψ a ψ a ψ = b + a ψ (5.3) a ψ a ψ a ψ a ψ = b + a ψ (5.4) ψ ψ ψ ψ ψ 3,, P 3, 3,, W, 3,, E 4, 3,, N 3, 3,, ψ 3,, S 3,0 a ψ a ψ a ψ = b + a ψ + a ψ (5.5) ψ ψ ψ ψ ψ 4,, P 4, 4,, W 3, 4,, N 4, 4,, ψ 4,, E 5, 4,, S 4,0 a ψ a ψ a ψ a ψ = b + a ψ (5.6) ψ ψ ψ ψ ψ,, P,,, E,,, S,,, N,3,, ψ,, W 0, a ψ a ψ a ψ a ψ a ψ = b (5.7) ψ ψ ψ ψ ψ,, P,,, W,,, E 3,,, S,,, N,3,, ψ a ψ a ψ a ψ a ψ a ψ = b (5.8) ψ ψ ψ ψ ψ 3,, P 3, 3,, W, 3,, E 4, 3,, S 3, 3,, N 3,3 3,, ψ a ψ a ψ a ψ a ψ = b + a ψ (5.9) ψ ψ ψ ψ ψ 4,, P 4,, P 4,, W 3, 4,, S 4, 4,, N 4,3 4,, ψ 4,, E 5, a ψ a ψ a ψ = b + a ψ + a ψ (5.0) ψ ψ ψ ψ ψ,3, P,3,3, E,3,3, S,,3, ψ,3, W 0,3,3, N,4 a ψ a ψ a ψ a ψ = b + a ψ (5.) ψ ψ ψ ψ ψ,3, P,3,3, W,3,3, E 3,3,3, S,,3, ψ,3, N,4 a ψ a ψ a ψ a ψ = b + a ψ (5.) ψ ψ ψ ψ ψ 3,3, P 3,3 3,3, W,3 3,3, E 4,3 3,3, S 3, 3,3, ψ 3,3, N 3,4 a ψ a ψ a ψ = b + a ψ + a ψ (5.3) ψ ψ ψ ψ ψ 4,3, P 4,3 4,3, W 3,3 4,3, S 4, 4,3, ψ 4,3, E 5,3 4,3, N 4,4 The terms on the left-hand side of the finite volume equations are unknown values, and on the right-hand side are known values. To solve all finite volume equations simultaneously, the equations are arranged in matrix form as: where AX = B (5.4)

139 5 a a a A = ψ ψ ψ,, P,, E,, N ψ ψ ψ ψ a,, W a,, P a,, E a,, N ψ ψ ψ ψ a3,, W a3,, P a3,, E a3,, N ψ ψ ψ a4,, W a4,, P a4,, N ψ ψ ψ ψ a,, S a,, P a,, E a,, N ψ ψ ψ ψ ψ a,, S a,, W a,, P a,, E a,, N ψ ψ ψ ψ ψ a3,, S a3,, W a3,, P a3,, E a3,, N ψ ψ ψ ψ a4,, S a4,, W a4,, P a4,, N ψ ψ ψ a,3, S a,3, P a,3, E ψ ψ ψ ψ a,3, S a,3, W a,3, P a,3, E ψ ψ ψ ψ a3,3, S a3,3, W a3,3, P a3,3, E ψ ψ ψ a4,3, S a4,3, W a4,3, P (5.5) X ψ, ψ, ψ 3, ψ 4, ψ, ψ, = ψ, 3, ψ 4, ψ,3 ψ,3 ψ 3,3 ψ 4,3 b + a ψ + a b a a b,, ψ B = b b a a b a a ψ ψ ψ,, ψ,, W 0,,, S,0 ψ b,, ψ + a,, Sψ,0 ψ b3,, ψ + a3,, Sψ 3,0 ψ ψ 4,, ψ + 4,, Eψ5, + 4,, Sψ4,0 ψ b,, ψ + a,, Wψ 0, 3,, ψ ψ b4,, ψ + a4,, Eψ 5, ψ ψ,3, ψ +,3, Wψ0,3 +,3, Nψ,4 ψ b,3, ψ + a,3, Nψ,4 ψ b3,3, ψ + a3,3, Nψ 3,4 ψ ψ 4,3, ψ + 4,3, Eψ5,3 + 4,3, Nψ4,4 (5.6) Due to the banded nature of the matrix, in the computer code, the matrix A can be compacted to minimize memory usage in a form as:

140 6 a a a a a a a a a a a a A = a a a, 3,, 3,, a a a a a a a a a a a a a a a a a a ψ ψ ψ,, P,, E,, N ψ ψ ψ ψ a,, W a,, P a,, E a,, N ψ ψ ψ ψ a3,, W a3,, P a3,, E a3,, N ψ ψ ψ a4,, W a4,, P a4,, N ψ ψ ψ ψ,, S,, P,, E,, N ψ ψ ψ ψ ψ,, S,, W,, P,, E,, N ψ ψ ψ ψ ψ 3,, S 3,, W 3, P a E a N ψ ψ ψ ψ 4,, S 4,, W 4,, P 4,, N ψ ψ ψ,3, S,3, P,3, E ψ ψ ψ ψ,3, S,3, W,3, P,3, E ψ ψ ψ ψ 3,3, S 3,3, W 3,3, P 3,3, E ψ ψ ψ 4,3, S 4,3, W 4,3, P (5.7) Gauss-Jordan elimination can be applied to solve the linear equations. By this means, the values of a variable at all nodes are solved and the model can continue to solve the next equation of calculation step. The solutions could converge very fast with very good initial guesses of boundary values. However, divergence problem is still a big issue of this way of solving equation. After extensive testing of the turbulence model, solving the finite volume equations column-by-column makes the model more robust because the initial guesses of the calculating nodes and boundary nodes affect on the results. However, the results after solving all columns are not necessarily those of the final solution. The final solution will be the input values for the next iteration, and so on, until the solution converges. From the example above, the finite volume equations for the first column can be rearranged as: a ψ a ψ = b + a ψ + a ψ + a ψ (5.8) ψ ψ ψ ψ ψ,, P,,, N,,, ψ,, W 0,,, E,,, S,0 a ψ a ψ a ψ = b + a ψ + a ψ (5.9) ψ ψ ψ ψ ψ,, P,,, S,,, N,3,, ψ,, W 0,,, E, a ψ a ψ = b + a ψ + a ψ + a ψ (5.30) ψ ψ ψ ψ ψ,3, P,3,3, S,,3, ψ,3, W 0,3,3, E,3,3, N,4

141 7 Also, the terms on the left-hand side of the equations are unknown values, and the terms on right hand side are known values. The finite volume equations can be arranged in matrix form as: a a ψ b + a ψ + a ψ + a ψ = + + ψ ψ ψ ψ ψ,, P,, N,,, ψ,, W 0,,, E,,, S,0 ψ ψ ψ ψ ψ a,, S a,, P a,, N ψ, b,, ψ a,, Wψ0, a,, Eψ, ψ ψ ψ ψ ψ a,3, S a,3, P ψ,3 b,3, ψ + a,3, Wψ0,3 + a,3, Eψ,3 + a,3, Nψ,4 (5.3) In the computer code, the Jacobian matrix can store value in a condensed form to minimize memory usage as: a a A= a a a a ψ ψ,, P,, N ψ ψ ψ,, S,, P,, N ψ ψ,3, S a,3, P (5.3) When the finite volume equations of considering column are solved, this procedure is applied to the next column until all columns in the cross section have been processed. Convergence Criteria There are two distinct sets of convergence criteria used in this research which depend on the calculation steps: convergence criterion by comparing the solution with the value from the last iteration, and convergence criterion by comparing with a specific value. Convergence Criterion by Comparing with the Values from the Last Iteration By solving the finite volume equations column-by-column until all nodes have been processed, the solution will be used as the input data for the same finite volume equations. This solution procedure is repeated until the convergence criterion is satisfied by comparing the solution node-by-node with the values from the last iteration. The convergence criterion for this iteration will be applied to all nodes as:

142 8 ψ ψ k k i, j i, j k ψ i, j *00 MaxAlw (5.33) k where ψ i, j is the solution at the node (i, j) from existing iteration; k ψ i, j is the solution from the previous iteration; and MaxAlw is the maximum allowable percentage change in each iteration. This criterion is applied to the equations of pressure, momentum in the x-, y-, and z-directions, turbulence kinetic energy, kinetic energy dissipation rate, and Reynolds stress. Convergence Criterion by Comparing with a Specific Value The above criterion is not applied to the pressure correction equations because the allowable change in value is dependent upon the magnitude of the variable value. The pressure correction values are not used to correct only the pressure values, but also correct the magnitudes and directions of the secondary velocities to satisfy the mass balance at each control volume. This means that if Eq is used as convergence criterion for the pressure correction equations, the mass balance error at each control volume will depend on the magnitude of the pressure at each node. Ideally, the iteration on pressure correction equations should continue until the pressure correction values at all nodes is equal to zero. However, the process is time consuming, and the iterations should stop when all pressure correction values are less than an acceptable value. k ψ i, j MaxAlwval (5.34) where MaxAlw val is the maximum allowable magnitude. The main equations for pressure, momentum in the y-direction, momentum in the z- direction, pressure correction, momentum in the x-direction, turbulence kinetic energy, kinetic energy dissipation rate, and Reynolds stresses are all solved in sequence by the step as described above. When all of the main equations are processed during simulation iteration, the solution at

143 this point may not satisfy all of the equations. This is because when solving an equation at a 9 control volume, all variables except the main variable of the equation are taken to be constants. Thus, the model must iterate to solve all of the equations in an iterative fashion until the model converges. A general convergent criterion, as presented above, cannot be applied to determine when to stop iterating on the main equations because of the small value fluctuation at the nodes from iteration to iteration. This fluctuation is caused by solving the variables by series of the main equations, but not solving all main equations at the same time. For example, after solving the momentum equations in the y- and the z-directions, the secondary velocity, v and w, satisfy these two main equations, but does not necessarily satisfy the continuity equation in the y-z plane. Then the calculation processes move to the next step: solving the pressure correction equations to make the secondary flows satisfy the continuity equation, but which might not satisfy the momentum equations. The iterations continue until all main equations are satisfied. The mass source term is introduced to be an indicator for the convergence criterion. The term mass source is defined as: k mass source = u A u A + v A v A + w A w A (5.35) k u u n n s s e e w w where A u is the cross-sectional area of a control volume. The units of each term in Eq is m 3. Naot et al. (993) set up a convergence criterion for their model by checking the summation of absolute values of mass sources from all grid cells. nk mass source MaxAlwMassSource (5.36) k = where nk is the number of control volume in the cross section, and MaxAlw MassSource is the maximum allowable value for total absolute mass source, m 3. The model is considered to have converged when the summation value is less than a specified target value. In this research, this criterion is used as one of two criteria to determine when the overall calculation process

144 0 converges, and the solution is obtained. However, this was only one of two convergence criteria because the summation of absolute values of mass sources depends on the size of the channel cross section. After testing the model with many cases, an indicator to consider convergence of the solution is changing of the total flow along the longitudinal bed slope. Equation 5.33 is applied for the total flow and used as an option to determine the convergence of model. To make sure that the model converges with this criterion, the solution must satisfy this criterion at least five consecutive iterations before stopping the calculations and accepting the results. Both criteria are used in the model, but the model converges if the solution meets only one of the criteria.

145 CHAPTER 6 SOFTWARE DEVELOPMENT History of Model Development After discovering the significant shortcomings of the finite difference method for solving the governing equations, it was decided to test the finite volume method to develop a different version of the turbulence model. There were many questions and problems arising during the software development using the finite volume method. The major questions were: What is the most appropriate arrangement of the governing equations? The momentum equations should combine with the Reynolds stress equations, or should be solved separately. After extensive finite volume model testing, it was discovered that mathematical convergence of the solution is strongly affected by the equation arrangement. In what order should the finite volume equations be solved? For example, perhaps the model should start by solving the momentum equations, then the Reynolds stress equations, and finally the kinetic energy and dissipation equations. Or, maybe it would be best to solve for all scalar variables, including u, then solve for secondary velocities, and finally apply the pressure correction. These are only two examples of so many cases that were tried in different versions of the turbulence model using the finite volume method. It was found that the order of solving the equations has an effect on the results, and on the ability to achieve mathematical convergence. Should the initial guesses be specified as simply one single value per variable for all nodes or spatial variation throughout the channel cross section? How should the node spacing be determined? Should the distance from node-tonode, node-to-water surface, and node-to-solid walls be the same, or should they be a

146 function of the variables and or parameters? Slight changes in the node-to-node and node-to-boundary separation distances were observed to have an effect on both the mathematical convergence and the direction of secondary flow rotation. Should the model attempt to converge each governing equation while looping throughout all equations, or converge each of the equations simultaneously? The question arose as to whether it is best to iterate for convergence of each equation until its solution converges, or is it preferable to iteration through all equations simultaneously in the same calculation loop until the over-all solution converges? Or, perhaps group some of the equations and iterate until they converge, and then iterate on the next equation group? Are the boundary areas of the finite volume method for each equation different? Do the boundary areas depend on the boundary condition of each variable and the location of the boundary values? All of these major questions are important because if they are not dealt with satisfactorily, mathematical divergence and model failure can occur. The development of a successful model is a very difficult task because all of these questions must be handled in an appropriate manner. In this research, many methods were tried to solve the governing equations. After working continuously for many months to solve all the equations and still experiencing problems of solution divergence, the model was reverted to the simplest flow scheme, which is that of one-dimensional laminar flow. From this humble starting point, greater modeling complexity was incrementally added until arriving at the final successful version.

147 3 -D Laminar Flow The simplest version of the model was developed in a Microsoft TM Excel spreadsheet, with one equation (the momentum equation in the x-direction), and with no turbulence component. Thus, this version involved a solution to a simplified form of the Navier-Stokes equations. Montes (998) referred to Straub et al. (956) who claimed that the Reynolds number used to determine the transition between laminar and turbulent flow regimes is the same for both circular pipes and open channels. The Reynolds number for an open channel is: ρvl Re = μ (6.) where V is velocity (m/s); and, L is characteristic length (m). For an open channel (Rubin and Atkinson 00), the characteristic length is defined as: L= 4R h (6.) where R h is hydraulic radius (m). Thus, the longitudinal bed slope was set to 6.0(0) -, which makes the Reynolds number less than,000, providing laminar flow conditions. Also, the value of y + must be very small to prevent overlapping between wall regions from two vertical walls and a floor. The velocity profile appears to be correct and the average velocity from the model is close to the average velocity from the Chezy equation. In this model the discharge computed by the Chezy equation is used only for comparing the results and for setting up initial guesses for u, but it does not have any effect on the calculation process. The model response is good when the any of the variables are changed. The calculated values of u change, as they should, when changing longitudinal bed slope and or dynamic viscosity. -D Laminar Flow By using the same criteria as for the -D laminar flow model, the -D model was successfully developed with the momentum equation in the x-direction using the Visual C#.NET

148 programming language. The vertical velocity profile at the center of the cross section was 4 compared with the output from the -D model, and was found to have a good relationship. The maximum velocity is lower than that of the -D model because of the effects from the vertical walls. The location of maximum velocity is at the water surface. 3-D Uniform Flow Model with Momentum Equations and Continuity Equation Two momentum equations in the y- and the z-directions and the SIMPLE algorithm (Patankar and Spalding 97) were added to the model to calculate u, v, w, and p at each computational node in the channel cross section. This model successfully solved the three momentum equations and the continuity equation. The difference between the SIMPLE and SIMPLER algorithms is that the SIMPLE algorithm does not need to solve for pressure values using the finite volume method as in the SIMPLER algorithm, rather, the pressure values are adjusted by the pressure correction terms, which also used to correct the velocity components in the y and the z-directions in both algorithms. At this stage of development, the SIMPLE algorithm gave better results than the SIMPLER algorithm by considering the direction and magnitude of v and w after several finite volume iterations of the momentum equations in the y- and z-directions. The SIMPLE method is the key to satisfying the continuity equation in each control volume. The key concept of the SIMPLE algorithm is to join the variable which has no equation, pressure, to the equation which has no variable itself, continuity equation. The main steps of this model are: Attempt to converge the values of u from the momentum equation in the x-direction. Converge v and w by iterating within both control volumes at the same time until obtaining convergence from both of them, thereby satisfying the momentum equation in the y- and the z-directions. However, the values of v and w might not satisfy the continuity equation.

149 5 Calculate pressure correction values and then correct the values of p, v, and w, which are corrected to satisfy the continuity equation, but they might not satisfy the momentum equations. Repeat all steps again until all equations are satisfied and the solution converges. However, in this stage of development, the continuity equation is satisfied for each of the internal control volumes, except the control volumes adjacent to the walls, floor, and water surface, even if the continuity equation was directly applied to these control volumes. The finite volume method for pressure correction was not applied to cover the boundary nodes due to a lack in knowledge at that time during the model development. 3-D Uniform Flow Model with the Boussinesq Hypothesis The complete governing equations were applied to the previous model based on the Boussinesq hypothesis. The model runs without NAN ( Not a Number ) or division by zero, but still cannot achieve overall solution convergence. From the literature review, the boundary conditions at the water surface for k have two alternatives which are: k = 0 or k y = 0. After testing the model, the boundary condition k = 0 provides for a robust model, but another boundary condition, k y = 0, can occasionally lead to model failure. The model that was developed in Visual C#.NET as a console application was migrated to a Visual C#. NET Windows application to be able to show the graphical results of the distributions of u, k, ε, and p, and the directions of v - w (secondary flow velocities) at each node in the computational grid. The ability to view graphical results during a simulation can help to identify abnormal model output, such as when the directions of v-w at the two top corner nodes of the cross section are wrong because the directions of v - w point out from the center of the cross section every each iteration as shown in Fig. 9.

150 6 It was thought that this problem was caused by the continuity equation which is applied by SIMPLE algorithm. However, that adjustment was false because the problem was from the way to apply boundary condition k=0 at water surface. This boundary condition was applied by setting the value of k at the nodes nearest the water surface to be equal to zero, but this was not correct. The k values at the nodes adjacent to the water surface should be calculated using the control volume method which has values of k equal to zero at the water surface, but not at nodes adjacent to the water surface. With this change, the program gave simulation results in reasonable range of variable values, but it still could not achieve overall solution convergence. Fig. 9. Incorrect directions of v-w at the two top corner regions of the cross section At this point, the program still had two problems. First, the variables v - w were not symmetrical along the center line of the cross section after 3 or 4 iterations. Second, the directions of v at the middle of the channel cross section alternated in sign, iteration-by-iteration. The first problem, that of solution asymmetry, was solved by extension of the finite volume method for w to cover the whole cross section instead of using the law-of-wall to calculate w near the walls. After effecting these two changes, the program appeared to produce symmetrical

151 velocity profile results up to 5 iterations. From iteration 6 to, the program results 7 manifested asymmetry at only a few points in the cross section, but this symmetry disappeared after iteration 3. The values of k at the second nodes from the vertical walls and floor were too low, compared with the values of k at the surrounding nodes, the first and third nodes from the vertical walls and floor. Then, the location to store the boundary conditions values for both k and ε near the vertical walls, floor, and water surface were changed to be at vertical walls, at floor, and at water surface. This resulted in an acceptable distribution and reasonable range of k and ε values at near-boundary nodes. Another problem was that the pressure values, p, could not be corrected by the SIMPLE algorithm. The model had the following three calculation steps: () calculate u, k, and ε iteratively until obtaining convergence; () calculate v and w using the momentum equations until convergence; and, (3) correct v and w, but not p (the pressure values are not corrected because they lead to divergent problems of v and w velocities), until the continuity equation is satisfied. The divergence problem of v and w due to the correction of pressure values was solved by rearranging the step to calculate variables. The new calculations steps included: () calculate v and w using the momentum equations until attaining convergence; () correct v, w, and p until satisfying the continuity equation; and, (3) calculate u, k, and ε until each of the values converges. Through the application of these calculation steps, the high values of u occurred at the location which had highest values of v-w. However, the asymmetric velocity profile problem after 0 iterations, and the alternating directions (from one iteration to the next) of v - w still existed. In yet another version of the model, the SIMPLER algorithm was used instead of the SIMPLE algorithm. However, the problem about pressure values still existed in which the values

152 8 were gradually increasing with each succeeding iteration. The respective directions of v and w appeared to be correct, but still had a problem about switching directions between two different flow patterns throughout cross section. Another problem was that the directions of v at the nodes close to both vertical walls contrasted with the directions at the next inner nodes. This problem was solved by increasing the y + value to make a larger space between the walls and the nodes adjacent to the walls. However, this would cause divergent problem if the log-law was applied at the first grid column from walls for v and the first row from floor for w. Then both log-law used to calculate v and w close to the walls and floor were replaced by finite volume method. The k-ε model using the Boussinesq hypothesis was successfully developed. The distance from the water surface to the first row of nodes has a great effect on divergence. A few unexpected eddy motions will occur at the middle of cross section if the first row of nodes from the water surface is too close to the water surface. Also, the distance from the first row and column to water surface, floor, and vertical walls must have enough separation space to make the directions of secondary flow to be in the proper direction. Finally, the model with the Boussinesq hypothesis can give good direction of the secondary current, and good distribution and values of u and scalar variables, but the location of the maximum u value is calculated to exist at the water surface in the center of the channel. This means the model with Boussinesq hypothesis fails to predict the correct location of the maximum u, which by laboratory experiments (French 985) is known to be below the water surface for a fully developed velocity distribution. ASM and ASM were used as options for the Reynolds stress equations in the model. These two models also give good results but, as with the Boussinesq hypothesis, still could not correctly predict the depression of maximum u below the water surface. Subsequently, ASM3 was applied to the model. After coding the model for ASM3, the model failed at the first or

153 9 second iteration. The results showed that there was something wrong with output for the pressure control volume. It took too many iterations (more than 0 iterations) to achieve convergence for pressure values, which should take only about to 3 iterations to converge, based on the results from the version of the model using the Boussinesq hypothesis, ASM, and ASM. The pressure values at the upper two corners are dropped too low and the inner values were too high. The initial guesses of u were changed from using single value for the all nodes to use output of u from the -D model. The initial guess of u appeared to be correct, but it did not help prevent the program from mathematical divergence. After checking step by step, the values of the source term of the momentum equation in the y-direction was very high at some points in the vicinity of the top corners of the cross section. The calculated value of every variable was printed and checked by hand, step-by-step. Finally, it was seen that the initial guesses of k values were too high compared to the values of ε. Thus, the initial guesses of k values were changed from 0., which did not cause any problems when using the Boussinesq hypothesis, ASM, and ASM, to be, and the same initial guess (0.) was used for ε. After these changes the solution converged at every iteration, but the direction of the secondary currents were obviously incorrect. After adjusting some factors on the equations for calculating the parameters α, β, c, and c 3 for ASM3, the model was able to calculate the directions of v and w velocities in the way as they should be. Also, the numbers of grid nodes in the y- and the z- directions were adjusted then the model can give very good results and the location of maximum u is below water surface. The model was tested by applying the finite volume method on all u -control volumes, including the control volume closest to the floor and walls. That means both finite volume method and the law-of-wall were applied at these control volumes. This was helpful in preventing solution divergence caused from the wrong pattern of k (k values at both bottom corners of the cross section are much lower than inner values). However, the convergence

154 30 difficulty appears when applying the model to rectangular channel cross sections wider than 3.5 m, and with aspect ratios of H/B = 0.5. Then the model has to switch back to use the finite volume method outside the boundary nodes for u -control volume. However, the model failed due to an obviously incorrect k pattern that was found to causes from model iteration which is explained later in this chapter. The next option for the Reynolds stress equations was RSM, which has a relatively complicated form. The Reynolds stress terms are partial differential equations and cannot be easily combined with the momentum equations. This means that the Reynolds stress equations must be solved equation-by-equation using the finite volume method. After working on the RSM version for many months and trying many alternative numerical configurations, the solution never converged, so it was abandoned in this research due to time constraints. Model Design A turbulence model has been successfully developed in this research. Two cross-section types were used: rectangular and compound cross sections. The turbulence model comprised three main equation groups: () continuity and momentum equations; () Reynolds stress equations; and, (3) turbulence equations. There are four options for the Reynolds stress equation sets: () Boussinesq hypothesis; () ASM; (3) ASM; and (4) ASM3. All options for the turbulence model have the same main calculation process. The flowchart of calculation process of the turbulence model by finite volume method is shown in Fig. 0. The details of each calculation step are described below. Input Data The input data for the turbulence model are shown in Table 4. The required input data depend on the model options and the types of the cross section. The term base data in the table mean the data required for all model options and cross-section types. Each model option and

155 3 cross-section type has additional required data. The program variables MaxIter_u, MaxIter_k, MaxIter_e, MaxIter_v, MaxIter_w, MaxIter_p are always equal to one (unity) because the variables u, k, and ε must be calculated in the same loop, which is control by the MaxIter_uke variable. The variables p, v, and w are also calculated in the MaxIter_pvw loop. These program variables were required for testing during model development, but they still appear on the input data screen. Chezy Equations The Chezy equation is used for calculating the uniform-flow discharge, which is used to calculate the initial guess for u, which was the same value at each computational node. This is in contrast to the initial u guesses in the finite difference model for which a complex algorithm was used to set up different values at each node to help ensure convergence of the solution. In the finite volume model, the discharge from the Chezy equation was used only for providing the initial guess for u, but was not used in the turbulence model calculation processes. The Chezy equation is: QChezy = AC RhSo (6.3)

156 3 Read input data AllocateArrayMemory : Allocate memory for arrays Chezy : Calculate discharge (Q) for uniform flow conditions BoundaryDistance : Calculate distance from boundaries using "Law of the Wall" GridGeneration : Generate grid data by the means of Backward staggered grid ASM3? N InitializeArray : Initialize all arrays and estimate variable values Y RmsDistance : Calculate Root-mean-square Distance from nodes to boundaries DoCalculations : Performs hydraulic calculations for a channel cross section pcv = vcv = wcv = p cv = ucv = kcv = ecv = F pcv = T & vcv = T & wcv = T? N pcv = T? wcv = T? N Calculate w by FV method NAN? Y Y Stop NAN? N u converged? N ucv = F Y Y Stop ucv = T N Calculate p by FV method NAN? Y N p converged? Y N pcv = F Stop pcv = T Y Y N w converged? Y N wcv = F Calculate p by FV method NAN? Y wcv = T Stop kcv = T? N Calculate k by FV method NAN? Y N k converged? Y Stop kcv = T Y N vcv = T? Y N Correct v, w N Y N kcv = F N Calculate v by FV method NAN? Y N v converged? Y N vcv = F = ( n + n- )/, = u, k, Stop vcv = T Y p' converged? Y ucv = T & kcv = T & ecv = T? N ucv = T? N Calculate u by FV method Y ecv = T? N Calculate by FV method NAN? N converged? & kcv = T? N ecv = F Y Y Stop ecv = T Write output to text file Draw output picture & save to picture file Solution converged? Y Write output summary to text file Stop Fig. 0. Flowchart of the turbulence model solution using the finite volume method

157 Table 4. Turbulence Model Input Data 33 Description Unit Program Variable Name Default Value Range in model Base Data Longitudinal bed slope (S o ) - So 0 00 to Ratio of weight to mass (g) m/s g to 9.8 Water density (ρ) kg/m 3 rho ,00 Molecular dynamic viscosity (μ) N-s/m mu 59 9 to 794 The isotropic turbulence viscosity coefficient - Cm to 0.0 (c μ ) Von Karman constant (κ) - kappa to 0.4 k-schmidt number (σ k ) - sigmak to.50 ε-schmidt number (σ ε ) - sigmae to.60 k-ε model constants, c ε - Ce to.80 k-ε model constants, c ε - Ce.9.40 to.40 Distance along the x-direction (dx) m dx to.00 Ratio of dy ny- to dy (c s ) - dy_surface to.00 Ratio of dy ny- to dy (c s ) - dy_surface to.00 Dimensionless distance from wall in the y-zdirections - y_plus 50 to 35 (y+) Number of nodes in the z-direction (nz) - nz 8 8 to 00 Number of nodes in the y-direction (ny) - ny 7 8 to 00 Maximum allowable correction value % MaxAlw 0.05 to Maximum allowable absolute value of pressure correction N/m MaxAlw_val to 000 Maximum allowable absolute value of % MaxAlw_All 0.05 to changing in last 5-iteration of discharge Maximum allowable of summed absolute values of mass sources m 3 MaxAlw_Mass Source to 000 Maximum overall iteration loops MaxIter 00 0 to 0,000 Maximum u k ε FV loop loops MaxIter_uke to 00,000 Maximum u FV loop loops MaxIter_u 0 to 00,000 Maximum k FV loop loops MaxIter_k 0 to 00,000 Maximum ε FV loop loops MaxIter_e 0 to 00,000 Maximum p v w FV loop loops MaxIter_pvw to 00,000 Maximum v FV loop loops MaxIter_v 0 to 00,000 Maximum w FV loop loops MaxIter_w 0 to 00,000 Maximum p FV loop loops MaxIter_p 0 to 00,000 Maximum p cor FV loop loops MaxIter_pcor to Relaxation for u - relax_u 0.7 to.00 Relaxation for k - relax_k 0.75 to.00 Relaxation for ε - relax_e 0.7 to.00 Table 4. (Continued from above)

158 Description Unit Program Default Range in model Variable Name Value Relaxation for v - relax_v 0.8 to.00 Relaxation for w - relax_w 0.8 to.00 Relaxation for p - relax_p 0.7 to.00 Relaxation for p cor - relax_pcor 0.85 to.00 Additional Data for Cross-Section Type - Rectangular Channel Channel width (B) m. B 0.05 to Uniform-flow water depth (H) m. H to Wall roughness height of floor (k s ) floor m. ks_f 4 to Wall roughness height of left wall (k s ) left m. ks_l 4 to Wall roughness height of right wall (k s ) right m. ks_r 4 to Compound Section Channel - Main channel width (B C ) m. B_c to Left side channel width (B L ) m. B_l to Right side channel width (B R ) m. B_r to Uniform-flow water depth in main channel m. H_c to (H C ) Left and right channel depth (H LR ) m. H_lr to Wall roughness height of main channel (k s ) M m. ks_m 4 to Wall roughness height of left channel (k s ) L m. ks_l 4 to Wall roughness height of right channel (k s ) R m. ks_r 4 to Additional (Changed) Data for Each Reynolds Stress Equation Set - Boussinesq Hypothesis No additional parameter - ASM Stress model constants, c - c.65 to 5.00 Stress model constants, c - c 0.55 to.50 - ASM Stress model constants, c - c to 5.00 Stress model constants, c D - cd 0.65 to.50 - ASM3 Stress model parameter, α - alpha to Stress model parameter, α - alpha to 0.50 Stress model parameter, β - beta to 0.50 Stress model parameter, β - beta to Stress model parameter, c - c to.800 Stress model parameter, c - c to Stress model parameter, c 3 - c to Stress model parameter, c 3 - c to 0.50 k-schmidt number (σ k ) - sigmak to.0 ε-schmidt number (σ ε ) - sigmae..0 to.40 k-ε model constants, c ε - Ce to.60 k-ε model constants, c ε - Ce.8.70 to.00 34

159 35 where A is cross-sectional area (m ); C is the Chezy coefficient (m / /s); and, R h is the hydraulic radius (m). For a rectangular channel cross section: and, A= BH (6.4) R = B+ H (6.5) h For a compound section: ( ) A= B H + B + B H (6.6) C C L R LR and, R = B + B + B + H (6.7) h C L R C The Chezy coefficient for turbulent rough flow (R e > 0,000, and k s > 0) can be estimated by (Jain 00): Rh C = 8glog 0 ( k s ) avg (6.8) where k s is the average roughness height from walls and floor (m). For a rectangular section: ( k ) s avg ( ) + ( ) + ( ) B ks H k floor s H k left s right = (6.9) R h And, for a compound section: ( k ) s avg ( BC + ( HC HLR ))( ks ) + ( BL + HLR )( ks ) + ( BR + HLR )( ks ) M L R = (6.0) R h Boundary Distance This function is used for calculating the average distance from the vertical walls to the first inner column and the floor to the first inner row. First, the average friction velocity at the walls and floor is calculated from Eq..5, which is: u = τ grhs (6.) f

160 Then the average distance from walls to the first inner node can be calculated from 36 adapted Eq..35, which is: y wall + μ y = (6.) ρ u τ This distance affects the direction of v at the nodes close to the vertical walls. The distance should not be too small compared to the distances between the inner nodes. Grid Generation A backward staggered grid was used in this research, as shown in Fig. 3. The parameters c s and c s are used to adjust the distance of the dy value between nodes above the water surface and the first node below the water surface, and the dy distance between the first and second rows of nodes below the water surface. These two distances have a significant effect on the calculated rotation direction of the secondary flow. The distance from the first column and row to the walls and floor are set to be equal to y wall according to Eq. 6.. The distance from the second row and column to the next inner row and column is equal to y wall, the object of which is to gradually vary the distance from the wall to the inner nodes. The inner grid spacing is dependent on the cross-sectional characteristics as described above, and the number of nodes set by nz and ny for the horizontal and vertical rows, respectively. For a compound section shape, the respective widths of the side channels do not need to be equal. Logarithmic grid spacing was tested in the model but it tended to cause problems rather than help obtain better solutions. Thus, a uniform grid spacing was ultimately adopted, as described above.

161 RMS Distance 37 The root-mean square (RMS) reciprocal distance is used for ASM3 options to provide a characteristic length between a point in the cross section and the surrounding solid surfaces and water surface. There are two types of RMS distances used in the formula: () RMS distances considering node-to-solid wall and floor surfaces; and, () distances from nodes to the water surface. Figure 5 shows the sketch for calculating the average distance of point P from solid wall surfaces for a rectangular cross section. For compound cross sections, the effective surfaces of point P are only those surfaces that the point P can reach by a straight line without obstruction by any solid surface, as shown in the example in Fig.. Fig.. Definition sketch for calculating average distance of point P from solid wall surfaces and from the water surface Array Initialization Each of the variables uses a unique value for its initial guess, before beginning iterations. The following equations and constants were found to provide acceptable initial guesses such that the systems of equations ultimately converged on a reasonable solution. The initial guess for u is: u initial QChezy = (6.3) A The coefficient ½ in Eq. 6.3 was found to help reduce the required finite volume iterations and help lead to solution convergence, even though at first glance it would seem to be unnecessary.

162 38 Part of the explanation for this is that the final calculated discharge tended to be less than Q Chezy (as noted above, the Chezy C value was used only to provide initial guesses, and not during finite volume calculations). The initial guesses for v and w are taken to be zero: v initial = w = 0 (6.4) initial The final calculated values of the velocities in the y and z directions are generally non-zero, but zero values for initial guesses was adequate to obtain a solution to the equations. Thus, none of the three velocity variables required sophisticated initial guess algorithms to obtain solutions to the equations using the finite volume method. For p, the hydrostatic pressure is used as an initial guess: p = ρ gy cos( S ) (6.5) initial where y is the distance from the water surface to the grid point in question. The initial guesses for k and ε are: o k = (6.6) initial ε = 0. (6.7) initial p-v - w Loop This loop processes three finite volume equation sets for p, v, and w. The loop continues iterating until each of these variables converge upon a stable and reasonable value. If one variable converges, the loop calculations continue to determine the rest of the variables, but ceases to calculate the variable that has already converged. Thus, the loop continues until all three variables converge.

163 Pressure Correction 39 This loop is used for correcting v and w to satisfy the continuity equation. The loop corrects the secondary velocities until the correction values for each are less than a specified threshold value. u -k-ε Loops This loop handles three finite volume equation sets for u, k, and ε. The loop works in the same way as the p - v - w loop except that if k converges before ε converges, the finite volume equations for k must continue to be applied until ε converges. However, if ε converges before k, the program stops applying the finite volume equations for ε and continues on the rest of the variables. This mechanism prevents an obviously erroneous k pattern in the channel cross section that leads to model failure (whereby pressure values become NAN in the next iteration). User Interface and Graphical Display The Microsoft TM Windows-based model has user interface that permits the user to interactively change the input data, and save the input data and simulation results to files. The first screen and sub-menus are shown in Fig.. (a) First screen

164 40 (b) File new sub menu (c) File open sub menu (d) File save sub menu Fig.. First screen and file sub-menus of the turbulence model First of all, the user must open the input data screen. There are two ways to open the input data screen: () using the menu File>New>[Select Model] will open the input screen with the default data; and, () using the menu File>Open>[Select Model] will open the screen with the data from the input file. The input screens for the turbulence model with ASM3 for a rectangular cross section and a compound cross section are shown in Figs. 3 and 4, respectively. Appendix D shows the input screen for the turbulence model with the Boussinesq hypothesis, ASM, and ASM for a rectangular cross section and a compound cross section. Each input screen has some different parameters depending on the Reynolds stress equations.

165 When the user enters numbers that are out of range (as specified in Table 4), the model will 4 display a warning in a pop-up window. For example, if the user enters 55.0 for channel width and leaves the input box, the model will display a warning as shown in Fig. 5. Also, if the user enters an alpha-numeric string instead of an exclusively numeric value, the model will display a warning as shown in Fig. 6. For both of these error types, after the user clicks the OK button, the model will revert to the previous value. Fig. 3. Input data window for the turbulence model with ASM3 for a rectangular cross section The model displays the secondary velocities using an arrow symbol for each computational grid point in the channel cross section. The length and color of these arrows are dependent on the magnitude of the secondary velocity. The user can set the attributes of the

166 4 arrow head, arrow width, minimum length of the arrow, which represents the smallest secondary velocity magnitude, and the maximum length of the arrow, which represents the maximum secondary velocity magnitude. The input screen for arrow attributes is shown in Fig. 7. Figure 8 shows examples of the arrow according to sample attribute values. Fig. 4. Input data screen for turbulence model with ASM3 for compound cross section

167 43 Fig. 5. Warning window for input data range violations Fig. 6. Warning window for incorrect string input data Fig. 7. Input screen for secondary-velocity arrow attributes

168 44 Fig. 8. Sample arrowhead styles for graphical display of calculated secondary velocities The file name extension of the input data file is dependent on the type of model. The extension names ke_bous, ke_asm, ke_asm, and ke_asm3 are for the Boussinesq hypothesis, ASM, ASM, and ASM3, respectively. The input data are stored in a text file with one line per value. There are three data types stored in the input text file which are model input data, arrow parameters, and screen size, respectively. The model input data are stored in the order shown on the model input screens from top to bottom and left to right. Also, the arrow attributes are stored in order from top to bottom. Lastly, the screen size data are stored as window width and height in pixels. Examples of input text files are found in Appendix D. When the input data are ready, the user should view the example of the output screen for a rectangular cross section or a compound cross section as shown in Figs. 9 and 30, respectively. The custom code to display the dynamically-updated graphical results for major variables was written in the Microsoft Visual C#.NET programming language. The calculated value of u in the cross section is displayed in a blue color; the darker the color, the higher the value of u. The small green area in the cross-sectional display area (Fig. 9) represents the crosssectional region for which u is at least 99.9% of the maximum value of u in the cross section, and the surrounding blue grid represents the region of 99.0 to 99.9% of the maximum u value. The arrows show the direction of the average secondary velocities, v and w, at each node. The

169 45 u 99%of u max LN(k) Secondary velocity u 99.9%of u max u distribution LN(ε) pressure Fig. 9. Example of the graphical display for a rectangular cross section Fig. 30. Example of the graphical display for a symmetrical compound cross section length and color of the arrows are indicative of the resultant velocity magnitude. Red represents the highest resultant secondary velocity and yellow represents the lowest secondary velocity. The green rectangular region at the upper right (Figs. 9 and 30) represents the distribution of the logarithm of turbulent kinetic energy per unit mass, k. The purple area below it

170 46 represents the distribution of the logarithm of kinetic energy dissipation rate, ε. Finally, the skyblue area at the lower right represents the pressure distribution in the cross section. The color intensity is greater for larger variable values, so these three rectangular regions give a visual indication of the distribution of the respective values in the cross section during a simulation. The exact ranges of each parameter and the surface velocity coefficients are shown at the bottom of the screen. All of these graphics and numerical values update automatically with each iteration of the numerical solution process during a simulation. The cross-sectional dimensions shown in the example screen depend on the size of the main window (which has an adjustable size). The user should adjust the main window size to provide a representative cross-sectional shape, one that compares well with the cross-section size and shape from the input data. After the user enters the data and sets the proper window size, the sub-menu in main menu Simulation will be active and can be used to start iterative calculations for the turbulence model, as shown in Fig. 3. Fig. 3. The simulation menu The model will update the graphical results after every iteration. Also, the graphical results of every iteration will be saved to bitmap image files named Output_XXXX.bmp, where XXXX is the iteration number. Also, solutions of the main variables which are u, v, w, p, k, and ε are written to a text file named 3D_Output_XXXX.txt. The numerical information upon which the graphic results are based will be saved to a text file named SumOutput.txt.

171 47 The surface velocity coefficients are calculated at every one-eighth of the top width from the vertical walls to the middle of the cross section, and the submergence depth of the floating object (for simulated application of the float method) from zero to 30 cm, which has a 5-cm increment. The information included in the graphical results will show the surface velocity coefficients at only four points in the cross section, with no submergence depth. Surface velocity coefficients at other submergence depths will be stored in the SumOutput.txt file. All output files are stored in the same directory (folder) as the input data file. If the user does not save or open an input data file before running the model, the output files will be stores in the X:\ directory, where X is the drive name corresponding to the location of the turbulence program itself. Examples of the graphical results and output text files for all four Reynolds stress equations are shown in Appendix D. Model Validation Laboratory data for the measurement of surface velocity coefficients in rectangular channel cross sections were collected and analyzed by Vásquez (004) and Vinukollu (005) at the Utah Water Research Laboratory and the Civil and Environmental Engineering hydraulics laboratory, at Utah State University. Three rectangular flume widths were used in their research: -, 3-, and 4-ft flumes. The discharge and water depth data from the laboratories were checked in this research in an attempt to verify that the flow conditions were uniform flow, as shown in Appendix E. However, only some of the laboratory data were found to be useful for turbulence model calibration and validation. Many data from both of the above-mentioned laboratory research results had problems due to the gradually-varied flow effect from the tail gate at the downstream end of the rectangular flumes, as shown in Fig. 3. In the figure, the discharge does not increase with increasing longitudinal bed slopes when the water depth is held constant. This problem also occurred with other water depths except for the data from the -ft flume with a 5-cm

172 water depth, and from 3-ft flume with a 0-cm water depth. Thus, only some of the reported 48 laboratory results were used in this research Q Lab Q Chezy Eq. (ks=0 to 7) ks=0 Discharge (Q), m 3 /s ks=05 ks= ks= ks=3 ks=5 ks= Longitudinal Bed Slope, S o Fig. 3. Measured and calculated relationships between discharge and longitudinal bed slope (S o ) for a -ft rectangular flume width with a 0-cm water depth from the laboratory Some of the model parameters used in this study are different than the values used by other researchers, where appropriate, as a result of efforts to enhance model convergence and lead to the generation of feasible simulation results. The model parameters used in this research, and those suggested by other researchers, are shown in Table 5. As stated above in this chapter, the turbulence models with Boussinesq hypothesis, ASM, and ASM stress models fail to predict the depression of maximum u below the water surface. Only the ASM3 model can simulate this depression phenomenon. Thus, the laboratory data (Vásquez 004; Vinukollu 005) were compared with the results from the ASM3 model.

173 49 Figures 33 and 34 show the comparison of SVC and discharge between the model results and laboratory data for the -ft flume width and 5-cm water depth. The average values from the ASM3 turbulence model results and the laboratory data in Figs. 33 and 34 are given in Table 6. Table 5. Adjusted Turbulence Model Parameters Model Turbulence Model and Stress Equations Parameters Boussinesq Hypothesis ASM ASM ASM3 This Study Others This Study Others This Study Others This Study Others c c c D α α β β c c c c σ k σ ε C ε C ε In Fig. 33, many SVC values from the model are from different numbers of grid nodes. The numbers of nodes affect the solution of SVC but have a very small effect on calculated discharge. The average SVC values from model fall between the laboratory data and SVC by USBR (997) for the low longitudinal bed slopes, but SVCs from the ASM3 turbulence model show good agreement with the laboratory data at relatively high longitudinal bed slopes. In Fig. 34 the discharges from the model have very good relationship with the measured discharge from the laboratory data. Figures 35 and 36 show the comparison of SVC and discharge between the model results and laboratory data for a 3-ft rectangular flume width and a 0-cm water depth.

174 50.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR λ Lab λ, d 99 /H Longitudinal Bed Slope, S o Fig. 33. Comparison of surface velocity coefficient, λ, values from the ASM3 turbulence model and hydraulic laboratory data for a -ft rectangular flume width and a 5-cm water depth Q Lab Q model (Q model ) avg Q Chezy Eq. (ks=0 to 7) ks=0 Discharge (Q), m 3 /s ks=05 ks= ks= ks=3 ks=5 ks= Longitudinal Bed Slope, S o Fig. 34. Comparison of calculated discharge from the ASM3 turbulence model, and measured discharge from a hydraulic laboratory for a -ft rectangular flume width and a 5-cm water depth

175 Table 6. ASM3 Model Results and Measured Laboratory Data for a -ft Rectangular Flume Width and a 5-cm Water Depth 5 B H S o Model Laboratory Data (m) (m) # Run SVC (λ) Q (m 3 /s) d 99 /H SVC (λ) Q (m 3 /s) Notes: B is base width; H is water depth; S o is longitudinal bed slope; SVC is surface velocity coefficient based on the velocities at the center of the water surface and the average cross-sectional streamwise velocity; and, Q is discharge. The average values of the model results and the laboratory data shown in Figs. 35 and 36 are given in Table 7. In Fig. 35, the number of nodes does not have a significant effect on the calculated SVC. The average SVC values from model fall between those of the laboratory data and the SVC as published by the USBR (ibid) for the full range of longitudinal bed slopes. In Fig. 36, the discharges from the model are seen to have very good relationship with the measured discharge from the laboratory data.

176 5.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR λ Lab λ, d 99 /H Longitudinal Bed Slope, S o Fig. 35. Comparison of the SVC from ASM3, USBR, and laboratory data for a 3-ft rectangular flume width and a 0-cm water depth 0.40 Q Lab Q model (Q model ) avg Q Chezy Eq. (ks=0 to 7) 0.0 ks=0 Discharge (Q), m 3 /s ks=05 ks= ks= ks=3 ks=5 ks= Longitudinal Bed Slope, S o Fig. 36. Comparison of discharge from ASM3, the Chezy equation, and measured laboratory data for a 3-ft rectangular flume width and a 0-cm water depth

177 Table 7. Model Results and Laboratory Data for a 3-ft Rectangular Flume Width and a 0-cm Water Depth B H S o Model Laboratory Data (m) (m) # Run SVC (λ) Q (m 3 /s) d 99 /H SVC (λ) Q (m 3 /s) Modeling Pitfalls and Solutions There are many tactics used in the model to arrive at a reasonable solution. The model will not yield a solution if some of the input data are not suitable for the flow situation. Also, some analysis/protective functions must be added to the program code to help prevent model failure (numerical divergence). Several of the modeling pitfalls and their solutions from this research are listed here. The model fails because the calculated pressure becomes an invalid value (e.g. NAN, or not a number ), and the simulation is unable to continue. Most of the problems are due to an unsuitable number of computational nodes in the y- and z- directions, ny and nz. The proportion of inner grid spacing in the y-direction to the z-direction should be near or close to the proportion of water depth to cross-sectional width (the aspect ratio). The model sometimes failed because the pattern of turbulence kinetic energy, k, goes wrong. The incorrect pattern of k is that the minimum k value occurs at both left and right bottom corners of the rectangular cross section. If this pattern is manifested, the model will fail within a few iterations after that point. This incorrect pattern can be

178 prevented by continuing to calculate the finite volume equations for ε until the k 54 value converges. The model has been coded to prevent this problem. In some cases, the secondary current rotates in the wrong direction (i.e. the main eddy at the left half of a rectangular cross section rotates in a counterclockwise direction, and vice versa). This problem may occur due to many factors. The user should try to: o Increase the distance from the solid walls to the first node by increasing y +. In the model, the distance from the second and third rows from the floor, and the second and the third columns from the vertical walls, are equal to two times y wall. These distances should be close to the grid spacing of the inner nodes. o Increase the distance from the water surface to the first row of computational nodes below the water surface by increasing the model parameter c s, and also by adjusting c s for the proper space between the second and the third rows from the water surface. o Adjust the number of nodes in both directions.

179 CHAPTER 7 55 FINITE-VOLUME SIMULATION RESULTS Introduction This chapter presents the calculation results from the turbulence model with the ASM3 stress equations using the finite volume method. The model results are summarized in the terms of the relationship between surface velocity coefficients (SVC, or λ) and the cross-sectional characteristics. Also, the simulated effects of float locations and submergence depths on the surface velocity coefficient are shown for different cross-section sizes. Finally, the relationships between surface velocity coefficients and stress model parameters are shown to illustrate how the model responds to changes in each parameter. Rectangular Cross Section Simulation Cases A standard rectangular cross section and channel characteristics were specified for calculating the model results in this research. The standard cross section has an aspect ratio (H/B) of 0.50, a base width (B) of.00 m, a longitudinal bed slope (S o ) of 0, and a roughness height (k s ) equal to 5 m. All but one of these standard parameters were held constant during each simulation. Each simulation case had one parameter change within a specific range of values. The range of cross-section aspect ratio (H/B) variations was from 0.5 to.00, representing relatively wide, shallow cross sections to relatively narrow, deep cross sections. Base widths varied from 0.5 to 5.0 m, for a constant aspect ratio, thereby representing a variety of cross-section sizes. The variation in longitudinal bed slope from 003 to were used to represent channels with mild to relatively steep slopes, although in all cases the Froude number was less than.0. The wall roughness heights varied from to 0.0 m to represent channel wall and floor surfaces

180 from smooth to relatively rough surfaces. Examples of model results ( u contour lines and 56 secondary velocities, v - w ) for all of these cases are shown in Appendix F. The effects of float locations (relative position with respect to the center of the channel cross section), and float submergence depths, to the surface velocity coefficient for each crosssection size were analyzed from the results of surface velocity coefficients for base widths from 0.5 to 5.0 m. The simulated submergence depths of the float object were from zero (small, light floats) to 30 cm (large and or relatively dense floats), and the lateral locations of the float object were every one-eighth of the base width from the vertical walls to the center of the cross section. Lastly, model parameters were tested on the standard cross section to see the variation of surface velocity coefficients due to changes in these model parameters. The surface velocity coefficients for all cases except the cases for changes in float location and submergence depth are the values at the center of the cross section with zero float submergence depth. SVC as a Function of Channel Aspect Ratio The variation in the calculated SVC, λ, with channel aspect ratio, H/B, is shown in Fig. 37. A total of 34 simulations were used to develop this relationship. The calculated SVC for each aspect ratio was affected by the number of computational nodes (i.e. grid density) used in the finite-volume method. In Fig. 37, the calculated SVCs are plotted together with the SVC as defined by the USBR (997). The USBR SVC increases with water depth regardless of the channel width. At low aspect ratios, between 0.5 and 0.40, the SVC values calculated from the model developed in this research and those from the USBR are in good agreement. For aspect ratios greater than 0.40, the model-calculated SVCs are higher than those suggested by the USBR (997). The average depth of the region for which u is at least 99% of the maximum calculated value of u, d 99, also varies with the aspect ratio. Figure 38 shows the relationship between

181 57.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR d λ, d 99 /H d 99 (m) Channel Aspect Ratio (H/B) Fig. 37. Relationship between calculated SVC and channel aspect ratio (H/B) for rectangular cross sections 0.70 Q model Q model avg. Q chezy 0.60 Discharge (Q), m 3 /s Channel Aspect Ratio (H/B) Fig. 38. Relationship between calculated discharge and channel aspect ratio (H/B) for rectangular cross sections

182 58 calculated discharge, Q, and channel aspect ratio. The graph shows that the discharge increases with increasing aspect ratio, and is not significantly affected by the number of nodes used in the finite-volume method. The calculated discharges from the model have the same trend, but are slightly lower than the calculated values from the Chezy equation. SVC as a Function of Channel Width Figure 39 shows the relationship between calculated SVC and channel width developed from the calculation results of 9 simulations using the finite-volume method. In the graph, the calculated SVCs and the those given by the USBR (997) have nearly the same trend but the calculated SVCs are a little bit higher than the SVCs from the USBR (997). The number of nodes used in the finite-volume method does not have much effect on the calculated SVC for this relationship compared to the relationship between calculated SVCs and channel aspect ratio. The relative depth of the region for which u is at least 99% of the maximum value of u, d 99 /H, also has the same trend with respect to the SVC, which increases with channel width from 0.50 to.50 m, and remains almost constant for channel widths from.50 to 5.00 m. Figure 40 shows the relationship between calculated discharge, Q, and channel width. The graph shows that discharge increases with increasing channel width, and the discharge experiences some effect from the number of nodes used in the finite-volume method when the channel width is greater than 3.0 m. The calculated discharges from the model are close to the calculated values from the Chezy equation for the full range of cross sections. For a cross-sectional width of 3.50 to 4.00 m, the discharge from the model is lower than that from the Chezy equation.

183 59.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR d λ, d 99 /H d 99 (m) Channel Width, B (m) Fig. 39. Relationship between calculated SVC and channel width (B) for rectangular cross sections 9.00 Q model Q model avg. Q chezy Discharge (Q), m 3 /s Channel Width, B (m) Fig. 40. Relationship between calculated discharge and channel width (B) for rectangular cross sections

184 SVC as a Function of Longitudinal Bed Slope 60 The relationship between calculated SVC and longitudinal bed slope (S o ) is shown in Fig. 4. The relationship was developed from the calculation results of 40 simulations. From the figure, it is seen that the calculated SVCs are almost constant. This means the longitudinal bed slope does not significantly affect the calculated SVCs; this agrees with the SVC as suggested by the USBR (997), which is not a function of longitudinal bed slope. The number of computational nodes used in the finite-volume method affects the calculated SVCs and the relative depths, d 99 /H, only when the longitudinal bed slope is less than 0. The d 99 /H values for slopes less than 0 are divided into two groups due to the number of nodes used in the finite-volume method. However, the discharges for slopes less than 03 are not affected by the number of nodes, as shown in Fig. 4. For slopes less than 05, the d 99 /H values decrease with increasing slope. Finally, for slopes greater than 05, the d 99 /H values are.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR λ, d 99 /H Longitudinal Bed Slope, S o Fig. 4. Relationship between calculated SVC and longitudinal bed slope (S o ) for rectangular cross sections

185 Q model Q model avg. Q chezy Discharge (Q), m 3 /s Longitudinal Bed Slope, S o Fig. 4. Relationship between calculated discharge and longitudinal bed slope (S o ) for rectangular cross sections almost constant. The calculated discharges from the model have the same trend, but are lower than the calculated values from the Chezy equation. SVC as a Function of Surface Roughness Figure 43 shows the relationship between calculated SVC and surface roughness height (k s ) developed from the calculation results of 3 simulations. The USBR (997) does not suggest how the SVC relates to the surface roughness height, or to any other measure of roughness. Thus, the SVC as suggested by the USBR (997) is constant for all of these cases. However, the figure shows that the calculated SVC decreases slightly with increasing surface roughness height. The d 99 /H values are almost constant, indicating that the surface roughness height does not affect the location of the maximum u in the channel cross section. The discharge also decreases with increasing surface roughness height as seen in Fig. 44. Neither calculated SVC nor discharge is significantly affected by the number of nodes used in the

186 6.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR λ, d 99 /H Surface Roughness Height, k s (m) Fig. 43. Relationship between calculated SVC and surface roughness height (k s ) for rectangular cross sections 0.6 Q model Q model avg. Q chezy Discharge (Q), m 3 /s Surface Roughness Height, k s (m) Fig. 44. Relationship between calculated discharge and surface roughness height (k s ) for rectangular cross sections

187 63 finite-volume method. The surface roughness height affects the discharge calculated from the Chezy equation more than the discharge from the model. The higher the surface roughness height, the better the agreement between discharge from the model and from the Chezy equation. SVC as a Function of Lateral Float Location Figure 45 shows the relationship between calculated SVC and location of a simulated float object on the water surface. It is seen that the calculated SVC changes with lateral location on the water surface, as would be expected based on published laboratory data, and indeed, based on intuition. The lowest SVC values occur at the center of the rectangular cross section, where the largest stream-wise velocity occurs, and SCV increases with decreasing distance from the nearest vertical wall. The effect of distance from a vertical wall on SVC is high for smaller cross sections, and low for larger cross sections, as shown in Fig B=0.50 B=.00 B=.50 B=.00 B=.50 B=3.00 B=3.50 B=4.00 B=4.50 B= SVC, λ B/8 B/4 3B/8 B/ Relative Distance from Wall, z/b Fig. 45. Relationship between calculated SVC and lateral surface location for rectangular cross sections

188 64.00 B=0.50 B=.00 B=.50 B=.00 B=.50 B=3.00 B=3.50 B=4.00 B=4.50 B= SVC / SVC at center, λ/λ x=b/ B/8 B/4 3B/8 B/ Relative Distance from Wall, z/b Fig. 46. Relationship between calculated SVC/SVC at center and lateral surface location, where B/ is the center of the cross section and B/8 is one eight of the base width from a vertical wall for rectangular cross sections SVC as a Function of Float Submergence Depth below the Water Surface From the laboratory results by Vásquez (004), as shown in Appendix E, the submergence depth of a float object affects the SVC. The laboratory results show that SVC decreases with increasing submergence depth. The relationship between calculated SVC and float submergence depth in the channel in Figs. 47 to 50 also show the same effect on calculated SVC. This is because the average u along the submergence depth increases with increasing submergence depth, up to a certain point. Thus, SVC decreases with increasing submergence depth. This happens if the submergence depth is not deeper than the location of maximum u. However, the relationship of calculated SVC with float submergence depth for a base width of 0.50 m (where submergence depth is greater than 0.0 m), is much different from that of other channel widths, as seen in Fig. 47. This is because the submergence depth is greater than the

189 B=0.50 B=.00 B=.50 B=.00 B=.50 B=3.00 B=3.50 B=4.00 B=4.50 B= Submerged SVC, λ sub Submergence Depth, y (m) Fig. 47. Relationship between calculated SVC and float submergence depth at the channel centerline for rectangular cross sections.05 B=0.50 B=.00 B=.50 B=.00 B=.50 B=3.00 B=3.50 B=4.00 B=4.50 B= Submerged SVC, λ sub Submergence Depth, y (m) Fig. 48. Relationship between calculated SVC and float submergence depth at distance 3B/8 from a vertical wall for rectangular cross sections

190 66.0 B=0.50 B=.00 B=.50 B=.00 B=.50 B=3.00 B=3.50 B=4.00 B=4.50 B= Submerged SVC, λ sub Submergence Depth, y (m) Fig. 49. Relationship between calculated SVC and float submergence depth at distance B/4 from a vertical wall for rectangular cross sections.45 B=0.50 B=.00 B=.50 B=.00 B=.50 B=3.00 B=3.50 B=4.00 B=4.50 B= Submerged SVC, λ sub Submergence Depth, y (m) Fig. 50. Relationship between calculated SVC and float submergence depth at distance B/8 from a vertical wall for rectangular cross sections

191 67 location of maximum u, so the average velocity along the submergence depth decreases with the increasing submergence depth beyond the location of maximum u. In the other words, SVC increases with increasing submergence depth when the submergence depth exceeds the location of maximum u along a vertical line in the channel cross section. The effect of float submergence depth is high for smaller cross sections and low for larger sections. Also, the Figs. 47 to 50 show that the smaller the distance of the float from a vertical wall, the higher the effect of submergence depth on the calculated SVC. SVC as a Function of Model Parameters The turbulence model with ASM3 stress equations is tested by changing the model parameters on the standard cross section to determine the sensitivity of surface velocity coefficients to these parameters. Figures 5 to 6 show the variation of calculated SVC, location of maximum u, and discharge on the model parameters α, α, β, β, c, c, c 3, σ k, σ ε, c ε, and c ε. In summary, the calculated SVC: Is nearly constant with increasing values of parameters α, α, c, c, and almost the entire range of β ; Increases with increases in β, σ ε, and c ε ; Slightly increases with increasing values of c 3 ; Decreases with increases in σ k ; and, Slightly decreases with increasing c ε. All eleven model parameters affect the location of maximum u in the same way as the effect of model parameters on calculated SVC. Also, from the graphs, the calculated discharge: Is nearly constant with changes in α, α, β, c, c, and c 3 ; Slightly increases with increasing values of σ ε ;

192 68.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H α Discharge, Q (m 3 /s) Fig. 5. The relationship between calculated SVC and α for a rectangular cross section.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H α Discharge, Q (m 3 /s) Fig. 5. The relationship between calculated SVC and the model parameter α for a rectangular cross section

193 69.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H β Discharge, Q (m 3 /s) Fig. 53. The relationship between calculated SVC and β for a rectangular cross section.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H Discharge, Q (m 3 /s) β Fig. 54. The relationship between calculated SVC and β for a rectangular cross section

194 70.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H Discharge, Q (m 3 /s) c Fig. 55. The relationship between calculated SVC and c for a rectangular cross section.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H Discharge, Q (m 3 /s) c Fig. 56. The relationship between calculated SVC and c for a rectangular cross section

195 7.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H c Discharge, Q (m 3 /s) Fig. 57. The relationship between calculated SVC and c 3 for a rectangular cross section.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H Discharge, Q (m 3 /s) σ k Fig. 58. The relationship between calculated SVC and σ k for a rectangular cross section

196 7.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H Discharge, Q (m 3 /s) σ ε Fig. 59. The relationship between calculated SVC and σ ε for a rectangular cross section.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H Discharge, Q (m 3 /s) Fig. 60. The relationship between calculated SVC and c ε for a rectangular cross section c ε

197 73.00 λ d 99.9 /H d 99 /H Q λ, d 99 /H, d 99.9 /H Discharge, Q (m 3 /s) c ε Fig. 6. The relationship between calculated SVC and c ε for a rectangular cross section Rapidly increases with increases in c ε ; Slightly decreases with increasing values of β ; Decreases with increasing σ k ; and, Rapidly decreases with increasing c ε. Compound Cross Sections Simulation Cases A standard compound cross section and channel characteristics were used to investigate the ability to handle non-rectangular sections, and to learn about special modifications to the model to deal with them successfully. In this research, the compound section (Fig. 30) has a central rectangular section which abruptly widens at about half of the water depth. The standard compound cross section, as defined herein, has a channel top width (T) of.00 m, a main channel base width (B C ) of ½T, left and right channel top widths (B L, B R ) of 0.5T, a main channel water depth (H C ) of ½T, and left and right channel water depths (H LR ) of 0.5T. The

198 74 standard longitudinal bed slope (S o ) is 0, and the roughness height (k s ) is 5 m. All but one of these standard parameters were held constant during each simulation using the compound cross section. Each simulation case had one parameter change within a specific range of values. Top widths varied from 0.5 to 5.0 m, with a constant aspect ratio, thereby representing a variety of cross-section sizes. The variation in longitudinal bed slope was from 003 to, and in all cases the Froude number was less than.0 (subcritical flow). The wall roughness heights varied from to 0.0 m to represent channel wall and floor surfaces from smooth to relatively rough surfaces. Examples of model results ( u contour lines, and secondary velocities, v - w ) for all of these cases are shown in Appendix F. SVC as a Function of Channel Width Figure 6 shows the relationship between calculated SVC and channel top width developed from the calculation results of simulations using the finite-volume method. In the graph, the calculated SVCs and the those given by the USBR (997) are in good agreement. The numbers of nodes used in the finite-volume method have almost no effect on the calculated SVC for this relationship. The relative depth of the region for which u is at least 99% of the maximum value of u, d 99 /H, increases with the channel top width over the entire range. Figure 63 shows the relationship between calculated discharge, Q, and channel top width. The graph shows that discharge increases with increasing channel top width, and the discharge experiences some effect from the number of computational nodes used in the finite-volume method when the channel top width is greater than 3.5 m. The calculated discharges from the model are lower than the corresponding values from the Chezy equation. The difference in discharge between the model and the Chezy equation increases with increasing channel top width.

199 75.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR d λ, d 99 /H d 99 (m) Channel Top Width, T (m) Fig. 6. Relationship between calculated SVC and channel width (T) for compound cross sections 6.00 Q model Q model avg. Q chezy 5.00 Discharge (Q), m 3 /s Channel Top Width, T (m) Fig. 63. Relationship between calculated discharge and channel width (T) for compound cross sections

200 SVC as a Function of Longitudinal Bed Slope 76 The relationship between calculated SVC and longitudinal bed slope (S o ) is shown in Fig. 64. The relationship was developed from the calculation results of 78 simulations. From the figure it is seen that the calculated SVCs are almost constant, and are close to the values from the USBR (997). This means that the longitudinal bed slope does not significantly affect the calculated SVCs; and this agrees with the SVC as suggested by the USBR (997), which is not a function of longitudinal bed slope. The number of computational nodes used in the finite-volume method has almost no effect on the calculated SVCs, but the relative depths, d 99 /H, are affected by the number of nodes used in the finite-volume method. However, the discharges for slopes less than 03 are not affected by the number of nodes, as shown in Fig. 65. For slopes less than or equal to 045, the d 99 /H ratio is almost constant. Finally, for slopes greater than 045, the d 99 /H ratio decreases with increasing longitudinal bed slope. The calculated discharges from the model have the same trend but are lower than the calculated values from the Chezy equation. SVC as a Function of Surface Roughness Figure 66 shows the relationship between calculated SVC and surface roughness height (k s ) developed from the calculation results of 99 simulations. The USBR (997) does not suggest how the SVC relates to the surface roughness height, or to any other measure of roughness. Thus, the SVC as suggested by the USBR (997) is constant for all of these cases. However, the figure shows that the calculated SVC decreases slightly with increasing surface roughness height as was the case for a simple rectangular cross section. Calculated values from the model and previously published values from the USBR (997) are nearly the same. The d 99 /H values are almost constant for surface roughness heights less than 5 m, and slightly decrease with increasing surface roughness height from 5 to 0.0 m. The discharge also decreases with

201 77.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR λ, d 99 /H Longitudinal Bed Slope, S o Fig. 64. Relationship between calculated SVC and longitudinal bed slope (S o ) for compound cross sections 0.5 Q model Q model avg. Q chezy 0.0 Discharge (Q), m 3 /s Longitudinal Bed Slope, S o Fig. 65. Relationship between calculated discharge and longitudinal bed slope (S o ) for compound cross sections

202 78.00 λ d 99 /H λ avg (d 99 /H) avg λ USBR λ, d 99 /H Surface Roughness Height, k s (m) Fig. 66. Relationship between calculated SVC and surface roughness height (k s ) for compound cross sections increasing surface roughness height as seen in Fig. 67. Neither calculated SVC nor discharge is significantly affected by the number of nodes used in the finite-volume method. The surface roughness height affects the discharges calculated by the Chezy equation more than the discharges calculated by the model. The discharges from the model are much lower than the corresponding values from the Chezy equation. SVC as a Function of Lateral Float Location Figure 68 shows the relationship between calculated SVC and location of a simulated float object on the water surface. It is seen that the calculated SVC changes with lateral location on the water surface, as was the case in the simple rectangular cross section. The lowest SVC values occur at the center of the compound cross section, where the largest stream-wise velocity occurs, and SCV increases with decreasing distance from the nearest vertical wall. The effect of

203 Q model Q model avg. Q chezy Discharge (Q), m 3 /s Surface Roughness Height, k s (m) Fig. 67. Relationship between calculated discharge and surface roughness height (k s ) for compound cross sections.40 T=0.50 T=.00 T=.50 T=.00 T=.50 T=3.00 T=3.50 T=4.00 T=4.50 T= SVC, λ T/8 T/4 3T/8 T/ Relative Distance from Wall, z/t Fig. 68. Relationship between calculated SVC and lateral surface location for compound channels

204 80 distance from a vertical wall on SVC is high for smaller cross sections, and low for larger cross sections, as shown in Fig. 69. SVC as a Function of Float Submergence Depth below the Water Surface The relationship between calculated SVC and float submergence depth in the channel in Figs. 70 to 73 also show a submergence effect on the calculated SVC, as was the case for the rectangular cross section, in which the SVC decreases with increasing submergence depth. The reasons for this are described above for the rectangular cross sections. The effect of float submergence depth is high for smaller cross sections and low for larger sections. Also, Figs. 70 to 73 show that the smaller the distance of the float from a vertical wall, the higher the effect of submergence depth on the calculated SVC..00 T=0.50 T=.00 T=.50 T=.00 T=.50 T=3.00 T=3.50 T=4.00 T=4.50 T= SVC / SVC at center, λ/λ x=b/ T/8 T/4 3T/8 T/ Relative Distance from Wall, z/t Fig. 69. Relationship between calculated SVC/SVC at the center and other lateral surface locations, where T/ is the center of the cross section and T/8 is one-eight of the top width from a vertical wall for compound cross sections

205 T=0.50 T=.00 T=.50 T=.00 T=.50 T=3.00 T=3.50 T=4.00 T=4.50 T= Submerged SVC, λ sub Submergence Depth, y (m) Fig. 70. Relationship between calculated SVC and float submergence depth at the channel centerline for compound cross sections 0.95 T=0.50 T=.00 T=.50 T=.00 T=.50 T=3.00 T=3.50 T=4.00 T=4.50 T= Submerged SVC, λ sub Submergence Depth, y (m) Fig. 7. Relationship between calculated SVC and float submergence depth at distance 3T/8 from a vertical wall for compound cross sections

206 8.5 T=0.50 T=.00 T=.50 T=.00 T=.50 T=3.00 T=3.50 T=4.00 T=4.50 T= Submerged SVC, λ sub Submergence Depth, y (m) Fig. 7. Relationship between calculated SVC and float submergence depth at distance T/4 from a vertical wall for compound cross sections.45 T=0.50 T=.00 T=.50 T=.00 T=.50 T=3.00 T=3.50 T=4.00 T=4.50 T= Submerged SVC, λ sub Submergence Depth, y (m) Fig. 73. Relationship between calculated SVC and float submergence depth at distance T/8 from a vertical wall for compound cross sections

207 CHAPTER 8 83 SUMMARY, CONCLUSIONS, AND RECOMMENDATIONS Summary A period of almost five years, from 003 to 008, was dedicated to this research. The first two years were spent performing a detailed and meticulous check of the equation forms given by Kra (00), and reviewing many dozens of books and journal articles. Kra (00) developed a turbulence model for a rectangular cross section using algebraic stress equations developed by Naot et al. (993) (ASM3) with the k-ε model, and was coded within the MATLAB commercial software application. However, that model included various application restrictions and unexpected results. After gathering the knowledge about turbulence modeling and understanding the Reynolds-Average Navier-Stokes equations and the closure equations, Reynolds stress equations, and kinetic energy and dissipation equations, both in condensed and expanded forms, a turbulence model was developed using the finite difference method in the third year. There are three major groups of equations for turbulence models: () continuity and momentum equations; () stress equations, which have four sets in algebraic form; and, (3) turbulence equations, which are the k-ε equations. The four stress equation sets in algebraic form are: () Boussinesq hypothesis; () algebraic stress model (ASM); (3) algebraic stress model (ASM); and, (4) algebraic stress model 3 (ASM3). The partial differential equations were converted to an algebraic form. There are more than,000 algebraic equations in the finite difference model. The systems of equations are then solved iteratively by the Newton-Raphson method at a point in the computational grid. However, more than one solution can be obtained from the same set of equations, channel characteristics, and number of computational nodes, but with different initial guesses. After testing the model for

208 several months with many calculation techniques, the model still had numerous problems, 84 especially, solution divergence problems at water surface nodes. Finally, this version of the turbulence model (with almost 5 thousand lines of C#.NET code) was abandoned at the middle of the fourth year. After discovering the significant shortcomings of the finite difference method for solving the governing equations, it was decided to test the finite volume method to develop a different version of the turbulence model. In this research, several methods were tried to solve the governing equations. After working continuously for many months to solve all the equations and still experiencing problems of solution divergence, the model was reverted to the simplest flow scheme, which is that of one-dimensional laminar flow. From this humble starting point, greater modeling complexity was incrementally added until arriving at the final successful version. The successful version of turbulence model can resolve all restrictions in the turbulence model by Kra (00). Two types of cross section are used in this research: rectangular and compound cross sections. Four sets of the Reynolds stress equations, Boussinesq hypothesis, ASM, ASM, and ASM3, are options used to compare the solutions. The Microsoft TM Windows-based model has a user interface that permits the user to interactively change the input data, and save the input data and simulation results to files. The custom code to display the dynamically-updated graphical results for major variables was written in the Microsoft Visual C#.NET programming language. The turbulence models with Boussinesq hypothesis, ASM, and ASM stress models fail to predict the depression of maximum u below the water surface. Only the ASM3 model can simulate this depression phenomenon. Thus, the ASM3 model was used during model calibration and verification and for calculating the model results for different channel characteristics.

209 Conclusions 85 Model Validity Laboratory data for the measurement of surface velocity coefficients in rectangular channel cross sections were collected and analyzed by Vásquez (004) and Vinukollu (005) at the Utah Water Research Laboratory and the Civil and Environmental Engineering hydraulics laboratory, at Utah State University. However, only some of the laboratory data were found to be useful for turbulence model calibration and validation. Many data from both of the abovementioned laboratory research results had problems due to the gradually-varied flow effect from the tail gate at the downstream end of the rectangular flumes except for the data from the -ft flume with a 5-cm water depth, and from 3-ft flume with a 0-cm water depth. These two data sets were used for turbulence model calibration and validation. Some of model parameters have been changed from the values used from other researchers to make the model convergence and provide good results. The model parameters used in this research and other researchers are shown in Table 8. For the -ft flume with a 5-cm water depth data, the average Surface Velocity Coefficient (SVC) values from model fall between those from the laboratory data and the SVC as given by the USBR (997) for low longitudinal bed slopes, but SVCs from the ASM3 turbulence model show good agreement with the laboratory data at relatively high longitudinal bed slopes. For 3-ft flume with a 0-cm water depth data, the average SVC values from model fall between those of the laboratory data and the SVC as published by the USBR (997) for the full range of longitudinal bed slopes. However, the discharges from the model for both data sets have very good relationship with the measured discharge from the laboratory data.

210 Applicability of SVC Values 86 The SVC suggested by USBR (997) is only a function of average water depth. However, the results from the model for both rectangular and compound cross sections show that the SVC varies with water depth, cross-section size and varies slightly with surface roughness height but does not have relationship with longitudinal bed slope. This means the SVC from USBR (997) may not be suitable for all flow conditions. The USBR coefficient must be adjusted by some factors to cover the influence from channel size and surface roughness height. Also, the simulation results from both rectangular and compound cross sections show that SVC varies with the lateral location of the float object on the water surface, and on the float submergence depth. The lowest SVC values occur at the center of the rectangular cross section, where the largest streamwise velocity occurs, and SVC increases with decreasing distance from the nearest vertical wall. The effect of submergence depth on the SVC values is that the SVC decreases with increasing submergence depth. Both effects of distance from a vertical wall and float submergence depth on SVC are high for smaller cross sections and low for larger sections. In conclusion, application of SVC value on the float method for estimating flow in a channel must consider the effects from cross-section sizes, roughness height of the channel walls, location of float object on the water surface, and float submergence depth. Some of these factors have a very large effect on the SVC, such as the location of the float object and submergence depth when the float is close to a wall. Cross-sectional size and water depth have a medium effect on the SVC. The float submergence depth at the middle of the cross section and the surface roughness height have very small effect on SVC. Finally, the SVC is not significantly affected by changes in the longitudinal bed slope over the range of values and cross sections studied in this research. The relationship between calculated SVC and channel width (B) for rectangular cross sections (Fig. 39) is such that the SVC from the model is (approximately) a constant.5 times

211 the corresponding USBR SVCs. The SVC coefficients based on aspect ratio (H/B) were 87 calculated by comparing the SVC from each aspect ratio with the SVC from the standard cross section. And, the SVC coefficients based on aspect ratio are shown in Table 8. Table 8. SVC Coefficients Based on Cross-section Aspect Ratio (H/B) H/B λ model coefficient H/B λ model coefficient H/B λ model coefficient The SVC coefficients based on surface roughness height (k s ) were calculated in the same way as for those which were based on the cross-sectional aspect ratio. The SVC coefficients based on surface roughness height are shown in Table 9. Table 9. SVC Coefficients Based on Surface Roughness Height (k s ) k s (mm) λ model coefficient Finally, based on the model results, the recommended coefficients to be applied to the USBR-published SVC values are shown for rectangular cross sections in Table 0. These coefficients should be multiplied by the USBR values to obtain corrected SVCs.

212 Table 0. Coefficients for the USBR-published SVCs in Rectangular Cross Sections 88 k s (mm) H/B Modeling Pitfalls and Solutions From the literature review, most of the researchers had some restrictions on model application range, grid density, initial guesses, and model convergence. Some special techniques must be added to the code to produce a working model, but these tricks are not reported either in the technical journal articles, or in books on fluid mechanics or computational fluid dynamics. There are many tactics used in the model to arrive at a reasonable solution. The model will not yield a solution if some of the input data are not suitable for the flow situation. Also, some analysis and protective functions must be added to the program code to help prevent model failure (numerical divergence). Several of the modeling pitfalls and their solutions from this research are listed in the following: The model sometimes fails because the calculated pressure becomes an invalid value (e.g. NAN, or not a number ), and the simulation is unable to continue. Most of the problems are due to an unsuitable number of computational nodes in the y- and z-

213 89 directions, ny and nz. The proportion of inner grid spacing in the y-direction to the z- direction should be near or close to the proportion of water depth to cross-sectional width (the aspect ratio) to avoid such problems. The model occasionally fails because the pattern of turbulence kinetic energy, k, becomes infeasible, meaning that the minimum k value occurs at both the left and right bottom corners of a rectangular cross section. If this pattern is manifested, the model will fail within a few subsequent iterations. This incorrect pattern can be prevented by continuing to solve the finite volume equations for ε until the k value converges. The model has been coded to prevent this problem. In some cases, the secondary current rotates in the wrong direction (i.e. the main eddy at the left half of a rectangular cross-section rotates in a counter-clockwise direction, and vice versa). This problem may occur due to many factors. The user should try to: o Increase the distance from the solid walls to the first node by increasing y +. In the model, the distance from the second and third rows from the floor, and the second and the third columns from the vertical walls, are equal to two times y wall. These distances should be close to the grid spacing of the inner nodes; o Increase the distance from the water surface to the first row of computational nodes below the water surface by increasing the model parameter c s, and also by adjusting c s for the proper space between the second and third rows from the water surface; and, o Adjust the number of computational nodes in both the y and z directions.

214 Recommendations for Future Research 90 Research on this topic could be continued indefinitely. Nevertheless, the objectives of the present research have been met successfully, and for the purposes of this dissertation the work is completed. However, it is very likely that the author, and others, will conduct follow-up research in this subject area, given the numerous ideas that have been generated during this work over the past five years. Based on the hydraulic model development in this research, a number of recommendations for the future research can be suggested: Determine the most suitable computational grid density, the distance of the first node from the vertical walls, from the floor surfaces, and from the water surface; Develop turbulence models for others cross-section shapes, such as full or semitrapezoidal, and circular cross sections; Apply the stress equation set in partial differential form using the Reynolds stress model (RSM), apply the k-ω model in the turbulence model, and compare the simulation results; and, Develop a more general version of the turbulence model to be able to calculate gradually-varied, steady-state flow profiles, thereby making it possible to study the effects of back-water profiles on the application of the flow method for flow measurement.

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221 APPENDICES 97

222 APPENDIX A: STRAIN RATE TENSOR 98

223 99 In a general flow field, a fluid element undergoes tensile and shear deformation with the passage of time. For example, in the x-y plane, an initially square side of a fluid element becomes skewed by the angles dα and dβ, as shown in Fig. A for a small change in time, δt. y [u+( u/ y) δy] δt u+( u/ y) δy dα φ u v δy time = t δx vt δ v+( v/ x) δx dβ [v+( v/ x) δx] δt x ut δ Fig. A.. Deformation of a fluid element in the x-y plane When the fluid can be considered incompressible, the volume of the fluid element does not change due to the deformations, nor does the area of the element faces, as shown by the respective gray areas in Fig. A. The rate of shear strain in this figure can be written as: dφ d ( α + β) u v u v = = δyδt+ δxδt = + δt δt δt δy y δx x y x (A.) which is for the x-y plane. Each term in Eq. A. has units of t -, or Hz when time is in seconds.

224 Equation A. is one component of the strain-rate tensor for a fluid element. Other 00 components can be derived in the same way for the other faces of the element. The strain-rate tensor is of rank two and has 3 = 9 components in three-dimensional space. This strain-rate tensor can be written succinctly as follows: S ij u u i j = + x j x i (A.) where, due to symmetry (assuming an isotropic fluid, which is normally the case), S ij = S ji, thereby eliminating three of the nine components, leaving six. Thus, Eq. A. can be expanded as follows: S u u u = + = x x x (A.3) S u v = + y x (A.4) S 3 u w = + z x (A.5) S v v v = + = y y y (A.6) S 3 v w = + z y (A.7) S 33 w w w = + = z z z (A.8) in which, because of symmetry (as noted above), S, S 3, and S 3 are the same as those given by Eqs. A.4, A.5, and A.7, respectively. The components S, S, and S 33 represent the three linear (tensile) strain rates on the fluid element, while the other three are the shear strain rates. The preceding derivation for the strain rate tensor is developed here because it was not found in any of the references.

225 APPENDIX B: FINITE DIFFERENCE EQUATIONS 0

226 0 The following finite-difference equations are for first-order forward differences in the z- direction, first-order backward differences in the y-direction, and second-order central differences in both the z- and y- directions (FB_CC). These equations are used in the model version as described in Chapter 4. There are no partial differential terms in ASM and ASM, so the equations for these two models are not included in this section. The continuity equation is: j k+ v v w w Δy + = 0 Δz (B.) And, the momentum equations are: u u u u u u + u ρv + ρw = ρgsin( ξ) + μ Δy Δz Δy j k+ j+ j ( Δz) ( ) k+ k u u + u τ τ τ τ + μ + + Δy Δz ( Δz) j k+ xy xy xz xz v v v v p p v v + v ρv + ρw = ρgcos( ξ) + μ Δy Δz Δy Δy j k+ j j+ j ( ) k+ k v v + v τ τ τ τ + μ + + Δy Δz ( Δz) j k+ yy yy yz yz w w w w p p w w+ w ρv + ρw = + μ Δy Δz Δz Δy j k+ k+ j+ j ( ) w w w τ τ τ τ + μ + + Δy Δz k+ k j k+ + yz yz zz zz (B.) (B.3) (B.4) The following equations apply to algebraic stress model 3 (ASM3): k β β u u u u v v τ yy = ρ α + c + τxy τxz + μt c+ c3 3 ε Δy Δz Δy j k+ j (B.5) + + = k k j k β β + + u u u u τ yy τzz ρ α c τxz τxy + c3 + μ w w t c 3 ε Δz Δy k Δz (B.6)

227 03 j k u u u + u τxy = μ t, τ = xy xz μ t (B.7) xz Δy Δz j k+ k+ j β k u u u u v v w w yz = xz + xy + t + ( c+.5c3) ε Δy Δz Δz Δy τ τ τ μ (B.8) And, these are the equations for the Reynolds stress model (RSM): j k+ 3/ τ xx τxx τxx τxx k v + w = α 0.05 ρp + ( c ) ρε ( α + γ) ρpk Δy Δz ε xn 3 3 k ε k c τ xx + ρ xn k x / 3/ j+ j j j j τ + xx τxx τxx τxx τxx k k k ε ε τ yy k + ( Δ ) Δy Δ Δ c y y S y ε ρε j j τ xx τ τ yy τ xx yy + k Δy Δy k+ k k+ k+ k+ τ + xx τxx τxx τxx τxx k k k ε ε τ + zz k c ( Δ ) Δ Δ Δ z z z ε z S ρε k+ k+ τ xx τxx τzz τzz + k Δz Δz j k+ 3/ τ yy τ yy τ yy τ yy k v + w = α 0.05 ρp + ( c ) ρε ( α + γ) ρpk Δy Δz ε xn 3 3 n v v k k ε k + βρk + γ ρd c τ + ρ Δy x x k x j 3/ / 3/ yy ε n n n τ τ + τ τ τ τ yy k c ( Δy) Δy Δy S ρε j τ yy τ yy + k Δ y ε Δ y j+ j j j j yy yy yy yy yy k k k ε ε + k+ k k+ k+ k+ τ + yy τ yy τ yy τ yy τ yy k k k ε ε τ zz k + ( Δ ) Δ Δ Δ c S z z z ε z ρε k+ k+ τ yy τ yy τ zz τ zz + k Δ z Δz (B.9) (B.0)

228 04 j k+ 3/ τ zz τzz τzz τzz k v + w = α 0.05 ρp33 + ( c ) ρε ( α + γ) ρpk Δy Δz ε xn 3 3 w w k k ε k + βρk + γ ρd c τ + ρ k+ 3/ / 3/ 33 zz Δz ε xn xn k xn j+ j j j j τzz τzz + τzz τzz τ zz k k k ε ε τ yy k + c ( Δy) Δy S Δy ε Δy ρε j j τ yy τ yy + τzz τzz k Δ Δ y y k+ k k+ k+ k+ τ + zz τzz τzz τzz τzz k k k ε ε τ zz k + ( Δ ) Δz Δ Δ c z z S z ε ρε k+ τzz τ zz + k Δ z j+ k+ 3/ j+ τxy τxy τxy τxy k u u v + w = α 0.05 ρp + βρk Δy Δz ε xn Δy 3/ / k k ε γ 0.05 ρd 0.5 c τxy ε xn xn k j+ j j+ j+ j+ τxy τxy + τxy τxy τxy k k k ε ε τ yy k + c ( Δy) Δy Δy ε Δy S ρε j+ j+ τ xy τxy τ yy τ yy + k Δy Δy k+ k k+ k+ k+ τ + xy τxy τxy τxy τxy k k k ε ε τ zz k + ( Δ ) Δ Δ Δ c S z z z ε z ρε k+ k+ τxy τxy τ zz τ zz + k Δ z Δz (B.) (B.)

229 05 j k+ 3/ k+ τ xz τxz τxz τxz k u u v + w = α 0.05 ρp3 + βρk Δy Δz ε xn Δz 3/ / k k ε γ 0.05 ρd3 0.5 c τxz ε xn xn k j+ j j j j τ + xz τxz τxz τxz τxz k k k ε ε τ yy k + c ( Δy) Δy Δy ε Δy S ρε j j τ xz τ τ yy τ xz yy + k Δy Δy k+ k k+ k+ k+ τ + xz τxz τxz τxz τxz k k k ε ε τ zz k + c ( Δ ) Δ Δ Δ S z z z ε z ρε k+ k+ τ xz τxz τzz τ zz + k Δz Δz τ τ τ τ v w P k j k+ 3/ k+ j yz yz yz yz k v v w w + = α 0.05 ρ 3 + βρ + Δy Δz ε xn Δz Δy 3/ / k k ε γ 0.05 ρd3 0.5 c τ yz ε xn xn k j+ j j j j τ yz τ yz + τ yz τ yz τ yz k k k ε ε τ yy k + c ( Δy) Δy Δy ε Δy S ρε j j τ yz τ yz τ yy τ yy + k Δy Δy k+ k k+ k+ k+ τ + yz τ yz τ yz τ yz τ yz k k k ε ε τ zz k + ( Δ ) Δ Δ Δ c S z z z ε z ρε k+ k+ τ yz τ yz τzz τ zz + k Δ z Δz (B.3) (B.4) Finally, the following are the finite-difference forms of the kinetic energy and dissipation equations based on the Boussinesq hypothesis:

230 06 j k+ j+ j k k k k μ t k k + k ρv + ρw = ρpkb ρε + μ+ Δy Δz σ k ( Δ ) y j j k+ k k k μ + t μt μt k k k + + μ + σ Δ Δ ( Δ ) k y y σk z k+ k+ k k μ t μt + σ k Δz Δz j k+ ε ε ε ε ε ε ρv + ρw = cερ Pkb ρcε Δy Δz k k j+ j + j j μ t ε ε ε ε ε μt μt + μ + + σ ( Δ ) ε Δ Δ y σ ε y y k+ k k k μ t ε ε ε ε ε μt μt + μ + + σ ( Δ ) ε Δz Δz z σ ε (B.5) (B.6) Auxiliary equations: D u u v v w w = τxy + τ yy + τ yz ρ Δy Δy Δy j j j (B.7) D 33 = τ + τ + τ ρ Δz Δz Δz k+ k+ k+ u u v v w w xz yz zz (B.8) D u u v v w w = τxx + τxy + τxz ρ Δy Δy Δy j j j (B.9) D 3 = τ + τ + τ ρ Δz Δz Δz k+ k+ k+ u u v v w w xx xy xz (B.0) D τ τ τ Δz Δy Δz = ρ v v w w w w + τ yz + + τzz Δy Δz Δy k+ j k+ u u u u v v xy + xz + yy 3 j k+ j (B.) P = τ + ρ Δy Δz j k+ u u u u xy τxz (B.)

231 07 P = τ + ρ Δy Δz j k+ v v v v yy τ yz (B.3) P 33 = τ + ρ Δy Δz j k+ w w w w yz τzz (B.4) P j j k+ k+ v v u u u u v v = τxy + τ yy + τ yz + τxz ρ Δy Δy Δz Δz (B.5) P 3 = τ + τ + τ + τ ρ Δz Δy Δy Δz k+ j j k+ w w u u w w u u xz yz xy zz (B.6) P 3 j k+ j k+ v v w w w w v v = τ yz + + τ yy + τzz ρ Δ Δ Δ Δ y z y z (B.7) v v w w u u τ yy + τzz + τxy Δy Δz Δy P = ρ u u v v w w + τxz + τ yz + Δz Δz Δy j k+ j k k+ k+ j (B.8) P kb j k+ j u u u u v v + + μ Δy Δz Δy t = + + ρ k k j w w v v w w Δ Δ Δ z z y (B.9)

232 APPENDIX C: FUNCTION VECTOR AND PARTIAL DERIVATIVES 08

233 09 This appendix presents the function vector and partial derivatives of the finite-difference equations which are used in the Newton-Raphson method, as described in Chapter 4. These equations are for first-order forward difference equations in the z-direction, first-order backward differences in the y-direction, and second-order central differences in both the z- and y- directions. The continuity equations are: j k+ v v w w F = + Δy Δz (C.) F = v Δy, F = w Δz (C.) And, the momentum equations in the x-direction are: j k+ u u u u ( Δy) ( Δz) ( ) F = ρv ρw + ρgsin S Δy Δz j+ j k+ k u u + u u u + u τ τ τ τ + μ + μ + + Δy Δz o j k+ xy xy xz xz (C.3) F = v + w μ μ ρ ρ, u Δy Δz Δy Δz ( ) ( ) j F u u = ρ (C.4) v Δy k+ F u u = ρ, w Δz F τ xy =, Δy F τ xz = Δz (C.5) The momentum equations in the y-direction are: v v v v p p F = ρv ρw ρgcos( So ) Δy Δz Δy j k+ j j+ j k+ k v v + v v v + v τ τ τ τ + μ + μ + + Δy Δz ( Δy) ( Δz) j k+ yy yy yz yz (C.6) j F v v w μ μ = ρ + ρ, v Δy Δz Δy Δz ( ) ( ) w k + F v v = ρ Δz (C.7) F = p Δy, F =, τ Δy yy F τ yz = Δz (C.8)

234 Finally, the momentum equations in the z-direction are: j k+ k+ w w w w p p F = ρv ρw Δy Δz Δz w w w w w w τ τ τ τ + μ Δy Δz j+ j k+ k j k+ + + yz yz zz zz μ ( Δy) ( Δz) 0 (C.9) j F w w = ρ, v Δy k+ F v w w μ μ = ρ ρ (C.0) w Δy Δz Δy Δz ( ) ( ) F = p Δz, F =, τ Δz zz F τ yz = Δy (C.) The τ yy equations for ASM are: k c F = τ yy + ρ ( c ) P cpk + ρ k cε 3 3 c (C.) F τ yy = F ρ c = c P cpk + ρ k cε 3 3 c, ( ) (C.3) F k = ρ ( c ) P c P k ε cε 3 The τ zz equations for ASM are: F P k ρ, c ε, = ( c ) F P k = 3 ck c ρ (C.4) ε k c F F = τzz + ρ ( c ) P33 cpk + ρ k, = cε 3 3 c τ zz F ρ c = ( c ) P33 cpk + ρ k cε 3 3 c F k = c P c P, ρ ( ) 33 k ε cε 3 (C.5) (C.6) The τ xy equations for ASM: F k = ρ ( c ), P c ε 33 F P k = 3 ck c ρ (C.7) ε k F = τxy + ρ ( c ) P (C.8) c ε

235 F τ xy = F k ρ cε, = ( c ) P F ε k c ε, = ρ ( c ) P F P, = ( c ) k ρ (C.9) c ε The τ xz equations for ASM are: k F = τxz + ρ ( c ) P3 (C.0) c ε F τ xz = F k ρ cε, = ( c ) P 3 F ε k c ε, = ρ ( c ) P 3 F P k ρ (C.) c ε, = ( c ) 3 The τ yz equations for ASM are: k F = τ yz + ρ ( c ) P3 (C.) c ε F τ yz = F k ρ cε, = ( c ) P 3 F ε k c ε, = ρ ( c ) P 3 F P k ρ (C.3) c ε, = ( c ) 3 The τ yy equations for ASM are: k c F = τ + ρk + ρ P P yy D k ε c Pk ε (C.4) F τ yy = F c = ρ + ρ P P D, k k 3 3 cε + Pk ε (C.5) F c = ρck D P Pk ε 3 ( + ) cε Pk ε (C.6) F cd = ρk + P c P k ε ( ) ε ε, F P + c = ρck D 3 P k ( cε + P ε) k (C.7) The τ zz equations for ASM are: k c F = τ + ρk + ρ P P zz D 33 k ε c Pk ε (C.8) F τ zz = F c = ρ + ρ P P D, 33 k k 3 3 cε + Pk ε (C.9)

236 c F = ρck D P33 Pk ε 3 ( cε + Pk ε) (C.30) F cd = ρk +, P c ε P ε 33 k P + ( c ) ε F 33 = ρck 3 D P k ( cε + P ) k ε (C.3) The τ xy equations for ASM are: k cd F = τxy + ρ P + ε c P k ε (C.3) F τ xy =, F ρcdp = k c + P ε ε, D ε ( cε + P ε) k F c = ρckp k (C.33) F ρck D = P c + P F = ρc kp ε k ε, D Pk ( cε + Pk ε ) (C.34) The τ xz equations for ASM are: k cd F = τxz + ρ P3 + ε c P k ε (C.35) F τ xz =, F ρcdp3 = k c + P ε ε, D 3 ε ( cε + P ε) k F c = ρckp k (C.36) F ρck D = P c + P F = ρckp 3 ε k ε, D 3 Pk ( cε + Pk ε ) (C.37) The τ yz equations for ASM are: k cd F = τ yz + ρ P3 + ε c P k ε (C.38) F τ yz =, F ρcdp3 = k c + P ε ε, D 3 ε ( cε + P ε) k F c = ρckp k (C.39) F ρck D = P c + P F = ρckp 3 ε k ε, D 3 Pk ( cε + Pk ε ) (C.40)

237 The τ yy equations for ASM3 are: 3 = + k β β F τ yy ρ α + c + τ u u xy τ u u xz + μ v v t c+ c3 3 ε Δy Δz Δy j k+ j (C.4) F β k τ xy τ xz = +, u c + c ε Δy Δz 3 F v = μ t F, = Δy τ yy (C.4) j F βk u u =, τxy ( c+ c3) ε Δy k+ F βk u u = τxz ( c+ c3) ε Δz (C.43) F β β u u u u j k+ = ρ α + c + τxy τxz k c+ c3 3 ε Δy Δz (C.44) j k+ F βk u u u u = ( + τxy τxz, ε c c3) ε Δy Δz The τ zz equations for ASM3 are: F v v = μ Δy t j (C.45) k+ j k+ k β β u u u u τ yy w w F = τzz + ρ α + c+ τxz τxy + c3 + μt c 3 ε Δz Δy k Δz (C.46) F βk τ τ xz xy = +, u cε Δz Δy F μ t F c =, 3 = w Δz τ c yy (C.47) F τ zz =, j F βk u u =, τxy cε Δy τ = cε Δz k+ F βk u u xz (C.48) k+ j F = + β β u u u u ρ α c + τxz τxy k c 3 ε Δz Δy (C.49) F k β u u u u = τ xz τ xy ε c ε Δz Δy k+ j, = μ k+ F w w t Δz (C.50) The τ xy equations for ASM3 are: j u u F = τxy + μ t (C.5) xy Δy

238 4 F u μ t = xy F, =, Δy τ xy F u u = μ Δy txy j (C.5) The τ xz equations for ASM3 are: k + u u F = τxz + μ t (C.53) xz Δz F u = μ t xz F, =, and Δz τ xz μ k+ F u u txz = Δz (C.54) The τ yz equations for ASM3 are: j k+ k+ j β k u u u u v v w w yz xz xy t ( c+.5c3) ε Δy Δz Δz Δy F = τ + τ + τ + μ + (C.55) F β k τ τ xz xy =, u ( c+.5c3) ε Δy Δz F v μ t F μ =, t = Δz w Δy (C.56) k+ F β k u u =, τxy ( c+.5c3) ε Δz j F β k u u = τxz ( c+.5c3) ε Δy (C.57) F τ yz =, F β u u u u j k+ = τxz + τxy k ( c+.5c3) ε Δy Δz (C.58) F β k u u u u, j k+ = τxz + τxy ε ( c+.5c3) ε Δy Δz F v v w w = + μ Δz Δy t k+ j (C.59) The τ xx equations for RSM are:

239 j k+ 3/ τ xx τxx τxx τxx k F = v w + α 0.05 ρp + ( c ) ρε Δy Δz ε xn 3 k ε k ( α + γ) ρpk c τxx + ρ 3 xn k x c ρ S ( Δy) / 3/ j+ j j j j k τ yy k ( τxx τxx + τxx ) + ( τxx τxx ) k + ε ε ε j j + k ( τxx τxx )( τ yy τ yy ) k+ k k+ k+ k+ k c τzz k ( τxx τxx + τxx ) + ( τxx τxx ) k ε S ε ρ( Δz) ε k+ k+ + k ( τxx τxx )( τzz τzz ) n 5 (C.60) F v τ = xx τ Δy j xx, F τ = w τ Δz k + xx xx (C.6) / F v w k ε = c τ Δy Δz x k xx cs j j k j + + τ yy k k ε k( τ yy τ yy ) ρ Δ ε ε ( y) c k τ k k ε k( τ τ ) ρ Δ ε ε S k+ k+ k+ + zz zz zz ( y) ( z) n F cs j+ j j j j k = k ( τxx τxx + τxx ) + ( τxx τxx ) k + ε + k τ ρ Δ ε ε yy F cs k+ k k+ k+ k+ k = k ( τxx τxx + τxx ) + ( τxx τxx ) k ε k τ ρ Δ ε ε zz ( z) F k 0.5 ε k = ρp + + c τ + ρ k ε x x k k 8 x / / / xx n n n c S ε τ τ τ τ τ τ τ τ τ τ ρ( Δy) ε ε j ( ) ( )( ) j+ j j j j yy xx xx xx xx xx xx xx yy yy k+ c + + S k k k ε k+ k+ + τzz τxx τxx τxx ( τxx τxx ) + τxx ρ( Δz) ε τxx τzz τzz ε ( )( ) (C.6) (C.63) (C.64) (C.65)

240 3/ F k τ = 0.05 ρp + ( c ) ρ c ε ε x 3 k j+ j j j j k c τ yy k ( τxx τxx + τxx ) + ( τxx τxx ) k ε S ε ρ( Δy) ε j j j ( )( ) k τxx τxx τ yy τ yy c ρ ( Δz) n k+ k k+ k+ k+ k τzz k ( τxx τxx + τxx ) + ( τxx τxx ) k + ε ε ε k ( + k )( + ) k τxx τxx τzz τzz S xx 6 (C.66) F k = α / P ε xn ρ F, = ( + ) P 3 k α γ ρ (C.67) The τ yy equations for RSM are: j k+ 3/ τ yy τ yy τ yy τ yy k F = v w + α 0.05 ρp + ( c ) ρε ( α + γ) ρpk Δy Δz ε xn 3 3 v v k k ε k + βρk + γ ρd c τ + ρ Δy x x k x ( Δy) j 3/ / 3/ yy ε n n n c S j+ j j j τ yy k ( τ yy τ yy + τ yy ) + ( τ yy τ yy ) k + j k + j ε k τ yy τ ρ ε ε k+ k k+ k+ k+ k c τzz k ( τ yy τ yy + τ yy ) + ( τ yy τ yy ) k ε S ε ρ( Δz) ε k+ k+ + k ( τ yy τ yy )( τzz τzz ) ( yy ) (C.68) j k F τ yy τ yy βρ k F τ + yy τ = +, = v Δy Δy w Δz yy (C.69) / F v w k ε = c τ Δy Δz x k yy cs j+ j j j ( 4 ) ( ) j k j k τ yy τ yy τ yy τ yy τ yy k + ε k( τ yy τ yy ) ρ Δ ε ε ( y) c k k k k( ) ρ Δ ε ε S k+ k+ k+ τ + zz ε τzz τzz ( z) ( z) n F cs k+ k k+ k+ k+ k = k ( τ yy τ yy + τ yy ) + ( τ yy τ yy ) k ε k τ ρ Δ ε ε zz (C.70) (C.7)

241 7 F k v v 0.5 ε k = ρ( D P ) + βρ + + c τ + ρ k ε x y x k k 8 x ( Δz) / j / / yy n Δ n n c S ε τ τ τ τ τ τ τ τ ρ( Δy) ε ε c j ( ) ( ) j+ j j j yy yy yy yy yy yy yy yy ( ) S k+ k k+ + τzz τ yy τ yy τ yy τ yy τ yy ρ ε ( Δz) S 3/ F k τ = 0.05 ρ ( P D ) + ( c ) ρ c ε ε x 3 k n k+ ε + k + k + τ yy τ yy τzz τzz ε ( )( ) j+ j j j j k c τ yy k ( τ yy τ yy + τ yy ) + ( τ yy τ yy ) k ε S ε ρ( Δy) ε j ( ) k τ yy τ yy c ρ τ k ( τ τ + τ ) + ( τ τ ) k + ε ε k ( + k k τ )( + yy τ yy τzz τzz ) k+ k k+ k+ k+ zz yy yy yy yy yy yy k ε (C.7) (C.73) F k = γ / D ε xn ρ, F k = α / P ε xn ρ F = α + γ ρ (C.74), ( ) P 3 k The τ zz equations for RSM are: j k+ 3/ τ zz τzz τzz τzz k F = v w + α 0.05 ρp33 + ( c ) ρε ( α + γ) ρpk Δy Δz ε xn 3 3 w w k k ε k + βρk + γ ρd c τ + ρ c ρ S ( Δy) ( z) k+ 3/ / 3/ 33 zz Δz ε xn xn k xn + τ yy k ( τzz τzz + τzz ) + ( τzz τzz ) k ε j j + k ( τ yy τ yy )( τzz τzz ) + ε j j j j j k ε c S τ τ τ + τ + τ τ + τ τ ρ Δ ε ε k+ k k+ k+ ( ) ( ) k+ k k+ k ε k( ) zz k zz zz zz zz zz zz zz (C.75) F v τ zz τ = Δy j zz, k+ F τ zz τzz βρk = w Δz Δz (C.76) F cs j+ j j j j k = k ( τzz τzz + τzz ) + ( τzz τzz ) k + ε + k τ ρ Δ ε ε yy ( y) (C.77)

242 / F v w k ε = c τ Δy Δz x k zz cs j j k j + + τ yy k k ε k( τ yy τ yy ) ρ Δ ε ε ( y) cs k+ k k+ k+ ( 4 ) ( ) k+ k k+ + + k τzz τzz τzz τzz τzz k ε k( τzz τzz ) ρ Δ ε ε ( z) ( Δz) n F k w w 0.5 ε k = ρ( D P ) + βρ + + c τ + ρ k ε x z x k k 8 x / k+ / / zz n Δ n n c S ε τ τ τ τ τ τ τ τ τ τ ρ( Δy) ε ε c j j+ j j j j yy zz zz zz zz zz yy yy zz zz S k+ k k+ + zz zz zz zz zz ρ ( Δz) ( ) ( )( ) k+ ε + τ τ τ τ τ τ + k ε zz τzz τzz ε 3/ F k τ zz = 0.05 ρ ( P33 D33 ) + ( c ) ρ c ε ε x 3 k n S ( ) ( ) j+ j j j j k c τ yy k ( τzz τzz + τzz ) + ( τzz τzz ) k ε S ε ρ( Δy) ε j j ( )( ) k τ yy τ yy τzz τzz c ρ τ k ( τ τ + τ ) + ( τ τ ) k + ε ε k+ ( ) k τzz τzz k+ k k+ k+ k+ zz zz zz zz zz zz k ε 8 (C.78) (C.79) (C.80) F k = γ ε 3/ D33 xn ρ, F k = α 0.05 ε 3/ P33 xn ρ F, = ( + ) P 3 k α γ ρ (C.8) The τ xy equations for RSM are:

243 τ τ τ τ F = v w + P + k j k+ 3/ j xy xy xy xy k u u α 0.05 ρ βρ Δy Δz ε xn Δy 3/ / k k ε γ 0.05 ρd 0.5 c τxy ε xn xn k c ρ S ( Δy) τ k ( τ τ + τ ) + ( τ τ ) k + ε ε j j + k ( τxy τxy )( τ yy τ yy ) j+ j j j j yy xy xy xy xy xy k ε k+ k k+ k+ k+ k c τzz k ( τxy τxy + τxy ) + ( τxy τxy ) k ε S ε ρ( Δz) ε k+ k+ + k ( τxy τxy )( τzz τzz ) 9 (C.8) F u j βρ k F τ =, xy τ xy =, Δy v Δy F τ = w k + xy Δz τ xy (C.83) F cs j+ j j j j k = k ( τxy τxy + τxy ) + ( τxy τxy ) k + ε + k τ ρ Δ ε ε yy ( y) F cs k+ k k+ k+ k+ k = k ( τxy τxy + τxy ) + ( τxy τxy ) k ε k τ ρ Δ ε ε zz ( Δz) ( z) / F v w k ε = c τ Δy Δz x k xy cs j j k j + + τ yy k k ε k( τ yy τ yy ) ρ Δ ε ε ( y) c k τ k k ε k( τ τ ) ρ Δ ε ε S k+ k+ k+ + zz zz zz ( z) / j+ F k u u 0.5 ε = ρ( D P ) + βρ + + c / τ k ε xn Δy xnk k n c S ε τ τ τ τ τ τ τ τ τ τ ρ( Δy) ε ε c j j+ j j j j yy xy xy xy xy xy xy xy yy yy τ τ τ τ τ τ ε ( ) ( )( ) ( ) S k+ k k+ + zz xy xy xy xy xy ρ k+ ε k + k + + τxy τxy τzz τzz ε xy ( )( ) (C.84) (C.85) (C.86) (C.87)

244 3/ F k τ = 0.05 ρ ( P D ) c ε ε x k j+ j j j j k c τ yy k ( τxy τxy + τxy ) + ( τxy τxy ) k ε S ε ρ( Δy) ε j j ( )( ) k τxy τxy τ yy τ yy c ρ ( Δz) n τ k ( τ τ + τ ) + ( τ τ ) k + ε ε k ( + k k τ )( + xy τxy τzz τzz ) S xy k+ k k+ k+ k+ zz xy xy xy xy xy k ε 0 (C.88) F k = γ / D ε xn ρ, F k = α / P ε xn ρ (C.89) The τ xz equations for RSM are: j k+ 3/ k+ τ xz τxz τxz τxz k u u F = v w + α 0.05 ρp3 + βρk Δy Δz ε xn Δz 3/ / k k ε γ 0.05 ρd3 0.5 c τxz ε xn xn k c ρ S ( Δy) τ k ( τ τ + τ ) + ( τ τ ) k + ε ε j j + k ( τxz τxz )( τ yy τ yy ) j+ j j j j yy xz xz xz xz xz k ε k+ k k+ k+ k+ k c τzz k ( τxz τxz + τxz ) + ( τxz τxz ) k ε S ε ρ( Δz) ε k+ k+ + k ( τxz τxz )( τzz τzz ) (C.90) F u j βρ k F τ =, xz τ xz =, Δz v Δy F τ = w τ Δz k + xz xz (C.9) F cs j+ j j j j k = k ( τxz τxz + τxz ) + ( τxz τxz ) k + ε + k τ ρ Δ ε ε yy ( y) F cs k+ k k+ k+ k+ k = k ( τxz τxz + τxz ) + ( τxz τxz ) k ε k τ ρ Δ ε ε zz ( z) (C.9) (C.93)

245 ( Δz) / F v w k ε = c τ Δy Δz x k xz cs j j k j + + τ yy k k ε k( τ yy τ yy ) ρ Δ ε ε ( y) c k τ k k ε k( τ τ ) ρ Δ ε ε S k+ k+ k+ + zz zz zz ( z) / k+ F k u u 0.5 ε = ρ( D3 P3 ) + βρ + + c / τ k ε xn Δz xnk k n c S ε τ τ τ τ τ τ τ τ τ τ ρ( Δy) ε ε c j j+ j j j j yy xz xz xz xz xz xz xz yy yy ( ) ( )( ) ( ) S k+ k k+ + τzz τxz τxz τxz τxz τxz ρ ε 3/ F k τ = 0.05 ρ ( P3 D3 ) c ε ε x k ( Δz) n S xz k+ ε k + k + + τxz τxz τzz τzz ε xz ( )( ) j+ j j j j k c τ yy k ( τxz τxz + τxz ) + ( τxz τxz ) k ε S ε ρ( Δy) ε j j ( )( ) k τxz τxz τ yy τ yy c ρ τ k ( τ τ + τ ) + ( τ τ ) k + ε ε k ( + k k τ )( + xz τxz τzz τzz ) k+ k k+ k+ k+ zz xz xz xz xz xz k ε (C.94) (C.95) (C.96) F D 3/ k = γ ρ, ε x 3 n 3/ F k = α 0.05 ρ P ε x 3 n (C.97) The τ yz equations for RSM are:

246 j k+ 3/ τ yz τ yz τ yz τ yz k F = v w + α 0.05 ρp Δy Δz ε xn k+ j 3/ / v v w w k k ε + βρk + + γ ρd c τ Δz Δy ε xn xn k c ρ S ( Δy) τ k ( τ τ + τ ) + ( τ τ ) k + ε ε j j + k ( τ yz τ yz )( τ yy τ yy ) j+ j j j j yy yz yz yz yz yz 3 k ε k+ k k+ k+ k+ k c τzz k ( τ yz τ yz + τ yz ) + ( τ yz τ yz ) k ε S ε ρ( Δz) ε k+ k+ + k ( τ yz τ yz )( τzz τzz ) yz (C.98) j k F τ yz τ yz βρ k F τ + yz τ yz βρk =, = + v Δy Δz w Δz Δy (C.99) F cs j+ j j j j k = k ( τ yz τ yz + τ yz ) + ( τ yz τ yz ) k + ε + k τ ρ Δ ε ε yy ( y) F cs k+ k k+ k+ k+ k = k ( τ yz τ yz + τ yz ) + ( τ yz τ yz ) k ε k τ ρ Δ ε ε zz ( Δz) ( z) / F v w k ε = c τ Δy Δz x k yz cs j j k j + + τ yy k k ε k( τ yy τ yy ) ρ Δ ε ε ( y) c k τ ε ( τ τ ) ρ Δ ε ε S k+ k+ k+ + zz k k k zz zz ( z) / k+ j F k v v w w 0.5 ε = ρ( D3 P3 ) + βρ c / τ k ε xn Δz Δy xnk k n c S ε τ τ τ τ τ τ τ τ τ τ ρ( Δy) ε ε c j j+ j j j j yy yz yz yz yz yz yz yz yy yy S k+ k k+ + zz yz yz yz yz yz ρ ( ) ( )( ) k+ ε + + τ τ τ τ ( τ τ ) + k k ε τ yz τ yz τzz τzz ε yz ( )( ) (C.00) (C.0) (C.0) (C.03)

247 3/ F k τ = 0.05 ρ ( P3 D3 ) c ε ε x k j+ j j j j k c τ yy k ( τ yz τ yz + τ yz ) + ( τ yz τ yz ) k ε S ε ρ( Δy) ε j j ( )( ) k τ yz τ yz τ yy τ yy c ρ ( Δz) n τ k ( τ τ + τ ) + ( τ τ ) k + ε ε k ( + k k τ )( + yz τ yz τzz τzz ) S yz k+ k k+ k+ k+ zz yz yz yz yz yz k ε 3 (C.04) F k = γ / D3 ε xn ρ, F k = α / P3 ε xn ρ (C.05) The k equations for k-ε model are: j k+ j+ j k k k k μ t k k + k F = ρv ρw + ρpkb ρε + μ+ Δy Δz σ k ( Δ ) y j j k+ k k k k k μ t μt μt k k k k k μt μt + + μ + + σ Δ Δ ( Δ ) k y y σk Δ Δ z σ k z z (C.06) j F k k = ρ, v Δy k+ F k k = ρ (C.07) w Δz j k+ μ t μt μt μt μt μt μ μ σ k σk σk σk F ρv ρw = k Δy Δz Δy ( Δy) Δz ( Δz) (C.08) F = ρ, ε F P kb = ρ, j+ k F k k k k = + μ σ Δ σ Δ ( y) ( z) t k k (C.09) The ε equations for k-ε model are: j k+ ε ε ε ε ε ε F = ρv ρw + cερ Pkb ρcε Δy Δz k k j+ j + j j μ t ε ε ε ε ε μt μt + μ + + σ ( Δ ) ε Δ Δ y σ ε y y k+ k k k μ t ε ε ε ε ε μt μt + μ + + σ ( Δ ) ε Δz Δz z σ ε (C.0)

248 4 F v j ε ε = ρ, Δy k+ F ε ε = ρ, w Δz F ε ε = cερ + Pkb ρ cε (C.) k k k j F ρv ρw ρc ε Pkb ρcε ε μt μt μt = μ ε Δy Δz k k ( Δy) σ ( Δ ) ε σε y (C.) + k μ t μt μt + μ ( Δz) σ ( Δ ) ε σ ε z F ε = cε ρ, P k kb j+ k F ε ε ε ε = + μ σ Δ σ Δ txy ε ( y) ( z) ε (C.3) The D equations are: u u v v w w F = D + τxy + τ yy + τ yz ρ Δy Δy Δy j j j (C.4) F u τ xy =, ρ Δy F v τ yy =, ρ Δy F τ yz =, w ρ Δy j F v v = τ yy ρ Δy (C.5) j F u u =, τxy ρ Δy j F w w =, τ yz ρ Δy F D = (C.6) The D 33 equations are: F = D33 + τ + τ + τ ρ Δz Δz Δz k+ k+ k+ u u v v w w xz yz zz (C.7) F τ xz =, u ρ Δz F v τ yz =, ρ Δz F τ zz = w ρ Δz (C.8) τ =, ρ Δz k+ F w w zz τ =, ρ Δz k+ F u u xz τ =, ρ Δz k+ F v v yz F D 33 = (C.9) The D equations are: u u v v w w F = D + τxx + τxy + τxz ρ Δy Δy Δy j j j (C.0)

249 5 F u τ xx =, ρ Δy F v τ xy =, ρ Δy F τ xz =, w ρ Δy j F u u = τxx ρ Δy (C.) j F v v =, τxy ρ Δy j F w w =, τxz ρ Δy F D = (C.) The D 3 equations are: F = D3 + τ + τ + τ ρ Δz Δz Δz k+ k+ k+ u u v v w w xx xy xz (C.3) F τ xx =, u ρ Δz F v τ xy =, ρ Δz F τ xz = w ρ Δz (C.4) τ =, ρ Δz k+ F u u xx τ =, ρ Δz k+ F v v xy τ =, ρ Δz k+ F w w xz F D 3 = (C.5) The D 3 equations are: τ τ τ Δz Δy Δz F = D + ρ v v w w w w + τ yz + + τzz Δy Δz Δy k+ j k+ u u u u v v xy + xz + yy 3 j k+ j (C.6) F τ xy τ xz =, u ρ Δz Δy F τ yy τ yz =, v ρ Δz Δy F τ yz τ zz = w ρ Δz Δy (C.7) τ =, ρ Δz k+ F v v yy j F w w =, τzz ρ Δy τ = ρ Δz k+ F u u xy (C.8) j F u u =, τxz ρ Δy j k+ F v v w w = +, τ yz ρ Δy Δz F D 3 = (C.9) The P equations are: F = P + τ + ρ Δy Δz j k+ u u u u xy τxz (C.30) F τ xy τ xz =, u ρ Δy Δz j F u u =, τxy ρ Δy τ =, ρ Δz k+ F u u xz F P = (C.3)

250 The P equations are: 6 F = P + τ + ρ Δy Δz j k+ v v v v yy τ yz (C.3) F τ yy τ yz =, v ρ Δy Δz j F v v =, τ yy ρ Δy τ =, ρ Δz k+ F v v yz F P = (C.33) The P 33 equations are: F = P33 + τ + ρ Δy Δz j k+ w w w w yz τzz (C.34) F τ yz τ zz =, w ρ Δy Δz τ =, ρ Δz k+ F w w zz j F w w =, τ yz ρ Δy F P 33 = (C.35) The P equations are: j j k+ k+ v v u u u u v v F = P + τxy + τ yy + τ yz + τxz ρ Δy Δy Δz Δz (C.36) F τ yy τ yz =, u ρ Δy Δz F τ xy τ xz =, v ρ Δy Δz j F u u = τ yy ρ Δy (C.37) j F v v =, τxy ρ Δy τ =, ρ Δz k+ F v v xz τ =, ρ Δz k+ F u u yz F P = (C.38) The P 3 equations are: F = P3 + τ + τ + τ + τ ρ Δz Δy Δy Δz k+ j j k+ w w u u w w u u xz yz xy zz (C.39) F τ yz τ zz =, u ρ Δy Δz F τ xy τ xz =, w ρ Δy Δz τ = ρ Δz k+ F u u zz (C.40) j F w w =, τxy ρ Δy τ =, ρ Δz k+ F w w xz j F u u =, τ yz ρ Δy F P 3 = (C.4) The P 3 equations are:

251 7 j k+ j k+ v v w w w w v v F = P3 + τ yz + + τ yy + τzz ρ Δ Δ Δ Δ y z y z (C.4) F τ yz τ zz =, v ρ Δy Δz F τ yy τ yz =, w ρ Δy Δz j F w w = τ yy ρ Δy (C.43) τ =, ρ Δz k+ F v v zz j k+ F v v w w = +, τ yz ρ Δy Δz F P 3 = (C.44) The P k equations are: v v w w u u τ yy + τzz + τxy Δy Δz Δy F = P + ρ u u v v w w + τxz + τ yz + Δz Δz Δy j k+ j k k+ k+ j (C.45) F τ xy τ xz =, u ρ Δy Δz F τ yy τ yz =, v ρ Δy Δz F τ yz τ zz = w ρ Δy Δz (C.46) j F v v =, τ yy ρ Δy τ =, ρ Δz k+ F w w zz j F u u = τxy ρ Δy (C.47) τ =, ρ Δz k+ F u u xz F v v w w = + τ yz ρ Δz Δy k+ j F, = P k (C.48) The P kb equations are: j k+ j u u u u v v + + μ Δy Δz Δy t F = Pkb ρ k k j w w v v w w Δ Δ Δ z z y (C.49) F μ t u u u u = + u ρ Δy Δz j k+ ( ) ( ), F μ t v v v v w w = + v ρ ( Δy) ( Δz) ΔyΔz j k+ j (C.50) F μ t w w v v w w = + + w ρ k+ k+ j ( Δz) ΔyΔz ( Δy) F, = P kb (C.5)

252 8 j k+ j u u u u v v + + F Δy Δz Δy = + + μ k k j t ρ w w v v w w Δ Δ Δ z z y (C.5) The μ τ equations are: k F = μt + ρc μ ε (C.53) F k = ρcμ, k ε F k F = ρc μ, = ε ε μ t (C.54) The μ t xy equations are: c F = μ + μ (C.55) txy ( c +.5c )( c + c ) 3 3 t F c = μ + + t ( c.5c )( c c ) 3 3 F, = μ t xy (C.56) The μ t xz equations are: c F = + +.5c μ 3 txz t c+ c μ (C.57) 3 F c+.5c3 F =, = μ c + c μ t 3 t xz (C.58)

253 APPENDIX D: USER INTERFACE AND GRAPHICAL RESULTS 9

254 Turbulence Model with the Boussinesq Hypothesis 30 Rectangular Cross Sections Fig. D.. Input data screen for turbulence model with Boussinesq hypothesis for rectangular cross sections

255 3 Table D.. Input Data Text File for the Turbulence Model with the Boussinesq Hypothesis for Rectangular Cross Sections Text File Data Meaning.00 Channel width (B) 0.50 Uniform-flow water depth (H) 00 Longitudinal bed slope (S o ) 400 Wall roughness height of floor (k s ) floor 400 Wall roughness height of left wall (k s ) left 400 Wall roughness height of right wall (k s ) right 9.80 Ratio of weight to mass (g) 00 Water density (ρ) 59 Molecular dynamic viscosity (μ) 0.09 The isotropic turbulence viscosity coefficient (c μ ) 0.40 Von Karman constant (κ).00 k-schmidt number (σ k ).30 ε-schmidt number (σ ε ).44 k-ε model constants, c ε.9 k-ε model constants, c ε 0.0 Distance along the x-direction (dx) 0.75 Ratio of dy ny- to dy (c s ) 0.90 Ratio of dy ny- to dy (c s ) 5 Dimensionless distance from wall in the y-z-directions (y+) 8 Number of nodes in the z-direction (nz) 7 Number of nodes in the y-direction (ny) Maximum allowable correction value 000 Maximum allowable absolute value of pressure correction Maximum allowable absolute value of changing in last 5-iteration of discharge 000 Maximum allowable of summed absolute values of mass sources 00 Maximum overall iteration 0000 Maximum u k ε FV loop Maximum u FV loop Maximum k FV loop Maximum ε FV loop 0000 Maximum p v w FV loop Maximum v FV loop Maximum w FV loop Maximum p FV loop 0000 Maximum p cor FV loop 0.70 Relaxation for u 0.75 Relaxation for k 0.70 Relaxation for ε 0.80 Relaxation for v 0.80 Relaxation for w 0.70 Relaxation for p 0.85 Relaxation for p cor.0 Arrow length scale to grid space 0.50 Ratio of the shortest arrow to the longest arrow 0.30 Ratio of the inside head to the arrow length 0.40 Ratio of the outside head to the arrow length 0.5 Ratio of the head width to the arrow length 0.05 Ratio of the tail width to the arrow length 58 z-pixel 93 y-pixel

256 3 Fig. D.. Graphical results from the turbulence model with the Boussinesq hypothesis for rectangular cross sections Table D.. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with the Boussinesq Hypothesis for Rectangular Cross Sections Iteration = Acc. abs. Mass sources = 0075 %Q change = -0.09,-0.0,-0.08,-0.0,-0.0 p-fv Converged at loop v-fv Converged at loop 9 w-fv Converged at loop 6 pcor-fv Converged at loop 530 u-fv Converged at loop 8 k-fv Converged at loop 6 ε-fv Converged at loop 7 u range = m/s v range = -3-6 m/s w range = m/s p range = , N/m k range =.334E m/s ε range = 8.60E m/s3 Turbulence Model: ke_boussinesq DiffScheme: Power_Law Q (Chezy) = 0.73 cms. u avg. (Chezy) = m/s Q(i) = cms. (64.07% of Q chezy) u(i) avg = 0.63 m/s Reynold Number = 07, H, 0.38H Submergence depth = m, λ(⅛) =.67, λ(¼) = 0.983, λ(⅜) = 0.807, λ(½) = 0.75

257 Table D.. (Continued from above) 33 Submergence depth = 0.0m, λ(⅛) =.67, λ(¼) = 0.983, λ(⅜) = 0.807, λ(½) = 0.75 Submergence depth = 0.0m, λ(⅛) =.6, λ(¼) = 0.980, λ(⅜) = 0.805, λ(½) = 0.75 Submergence depth = 0.03m, λ(⅛) =.5, λ(¼) = 0.975, λ(⅜) = 0.803, λ(½) = 0.75 Submergence depth = 0.04m, λ(⅛) =.4, λ(¼) = 0.969, λ(⅜) = 0.80, λ(½) = 0.75 Submergence depth = 0.05m, λ(⅛) =.30, λ(¼) = 0.96, λ(⅜) = 0.798, λ(½) = 0.75 Submergence depth = 0.06m, λ(⅛) =.9, λ(¼) = 0.956, λ(⅜) = 0.796, λ(½) = 0.75 Submergence depth = 0.07m, λ(⅛) =.07, λ(¼) = 0.950, λ(⅜) = 0.793, λ(½) = 0.76 Submergence depth = 0.08m, λ(⅛) =.97, λ(¼) = 0.944, λ(⅜) = 0.79, λ(½) = 0.76 Submergence depth = 0.09m, λ(⅛) =.86, λ(¼) = 0.938, λ(⅜) = 0.789, λ(½) = 0.76 Submergence depth = 0.0m, λ(⅛) =.76, λ(¼) = 0.93, λ(⅜) = 0.786, λ(½) = 0.76 Submergence depth = 0.m, λ(⅛) =.66, λ(¼) = 0.96, λ(⅜) = 0.784, λ(½) = 0.76 Submergence depth = 0.m, λ(⅛) =.57, λ(¼) = 0.9, λ(⅜) = 0.78, λ(½) = 0.76 Submergence depth = 0.3m, λ(⅛) =.49, λ(¼) = 0.96, λ(⅜) = 0.78, λ(½) = 0.76 Submergence depth = 0.4m, λ(⅛) =.4, λ(¼) = 0.9, λ(⅜) = 0.779, λ(½) = 0.77 Submergence depth = 0.5m, λ(⅛) =.33, λ(¼) = 0.907, λ(⅜) = 0.777, λ(½) = 0.77 Submergence depth = 0.6m, λ(⅛) =.6, λ(¼) = 0.903, λ(⅜) = 0.776, λ(½) = 0.78 Submergence depth = 0.7m, λ(⅛) =.9, λ(¼) = 0.899, λ(⅜) = 0.775, λ(½) = 0.78 Submergence depth = 0.8m, λ(⅛) =.3, λ(¼) = 0.895, λ(⅜) = 0.774, λ(½) = 0.79 Submergence depth = 0.9m, λ(⅛) =.07, λ(¼) = 0.89, λ(⅜) = 0.773, λ(½) = 0.79 Submergence depth = 0.0m, λ(⅛) =.0, λ(¼) = 0.888, λ(⅜) = 0.77, λ(½) = 0.70 Submergence depth = 0.m, λ(⅛) =.096, λ(¼) = 0.885, λ(⅜) = 0.77, λ(½) = 0.7 Submergence depth = 0.m, λ(⅛) =.09, λ(¼) = 0.883, λ(⅜) = 0.77, λ(½) = 0.7 Submergence depth = 0.3m, λ(⅛) =.086, λ(¼) = 0.880, λ(⅜) = 0.77, λ(½) = 0.73 Submergence depth = 0.4m, λ(⅛) =.08, λ(¼) = 0.878, λ(⅜) = 0.77, λ(½) = 0.74 Submergence depth = 0.5m, λ(⅛) =.078, λ(¼) = 0.876, λ(⅜) = 0.77, λ(½) = 0.76 Submergence depth = 0.6m, λ(⅛) =.075, λ(¼) = 0.874, λ(⅜) = 0.77, λ(½) = 0.77 Submergence depth = 0.7m, λ(⅛) =.07, λ(¼) = 0.873, λ(⅜) = 0.77, λ(½) = 0.79 Submergence depth = 0.8m, λ(⅛) =.069, λ(¼) = 0.87, λ(⅜) = 0.773, λ(½) = Submergence depth = 0.9m, λ(⅛) =.066, λ(¼) = 0.87, λ(⅜) = 0.774, λ(½) = 0.73 Submergence depth = 0.30m, λ(⅛) =.064, λ(¼) = 0.870, λ(⅜) = 0.775, λ(½) = Start: Fri /9/008 9:57:33 AM Stop: Fri /9/008 9:57:50 AM Duration: 00:00:6 Run Successful

258 Compound Cross Sections 34 Fig. D.3. Input data screen for the turbulence model with the Boussinesq hypothesis for compound cross sections Table D.3. Input Data Text File for the Turbulence Model with the Boussinesq Hypothesis for Compound Cross Sections Text File Data Meaning 0.50 Main channel width (B C ) 0.30 Left side channel width (B L ) 0.30 Right side channel width (B R ) 0.50 Uniform-flow water depth in main channel (H C ) 0.30 Left and right channel depth (H LR ) 00 Longitudinal bed slope (S o ) 400 Wall roughness height of main channel (k s ) M 400 Wall roughness height of left channel (k s ) L 400 Wall roughness height of right channel (k s ) R 9.80 Ratio of weight to mass (g) 00 Water density (ρ)

259 Table D.3. (Continued from above) 35 Text File Data Meaning 59 Molecular dynamic viscosity (μ) 0.09 The isotropic turbulence viscosity coefficient (c μ ) 0.40 Von Karman constant (κ).00 k-schmidt number (σ k ).30 ε-schmidt number (σ ε ).44 k-ε model constants, c ε.9 k-ε model constants, c ε 0.0 Distance along the x-direction (dx) 0.75 Ratio of dy ny- to dy (c s ) 0.90 Ratio of dy ny- to dy (c s ) 5 Dimensionless distance from wall in the y-z-directions (y+) 30 Number of nodes in the z-direction (nz) 7 Number of nodes in the y-direction (ny) Maximum allowable correction value 000 Maximum allowable absolute value of pressure correction Maximum allowable absolute value of changing in last 5-iteration of discharge 000 Maximum allowable of summed absolute values of mass sources 00 Maximum overall iteration 0000 Maximum u k ε FV loop Maximum u FV loop Maximum k FV loop Maximum ε FV loop 0000 Maximum p v w FV loop Maximum v FV loop Maximum w FV loop Maximum p FV loop 0000 Maximum p cor FV loop 0.70 Relaxation for u 0.75 Relaxation for k 0.70 Relaxation for ε 0.80 Relaxation for v 0.80 Relaxation for w 0.70 Relaxation for p 0.85 Relaxation for p cor.0 Arrow length scale to grid space 0.50 Ratio of the shortest arrow to the longest arrow 0.30 Ratio of the inside head to the arrow length 0.40 Ratio of the outside head to the arrow length 0.5 Ratio of the head width to the arrow length 0.05 Ratio of the tail width to the arrow length 57 z-pixel 76 y-pixel

260 36 Fig. D.4. Graphical results from the turbulence model with the Boussinesq hypothesis for compound cross sections Table D.4. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with the Boussinesq Hypothesis for Compound Cross Sections Iteration = Acc. abs. Mass sources = 0067 %Q change = 0.03,-3,-0.0,-0.04,-0.05 p-fv Converged at loop v-fv Converged at loop 73 w-fv Converged at loop 08 pcor-fv Converged at loop 569 u-fv Converged at loop 9 k-fv Converged at loop 3 ε-fv Converged at loop 5 u range = m/s v range = -4-4 m/s w range = m/s p range = , N/m k range =.03E m/s ε range = 7.3E m/s3 Turbulence Model: ke_boussinesq DiffScheme: Power_Law Q (Chezy) = cms. u avg. (Chezy) = 0.36 m/s Q(i) = cms. (6.9% of Q chezy) u(i) avg = m/s Reynold Number = 74, H, 0.3H Submergence depth = m, λ(⅛) =.97, λ(¼) = 0.940, λ(⅜) = 0.770, λ(½) = Submergence depth = 0.0m, λ(⅛) =.97, λ(¼) = 0.940, λ(⅜) = 0.770, λ(½) = 0.667

261 Table D.4. (Continued from above) 37 Submergence depth = 0.0m, λ(⅛) =.9, λ(¼) = 0.936, λ(⅜) = 0.768, λ(½) = Submergence depth = 0.03m, λ(⅛) =.83, λ(¼) = 0.930, λ(⅜) = 0.766, λ(½) = Submergence depth = 0.04m, λ(⅛) =.73, λ(¼) = 0.93, λ(⅜) = 0.763, λ(½) = Submergence depth = 0.05m, λ(⅛) =.64, λ(¼) = 0.97, λ(⅜) = 0.760, λ(½) = Submergence depth = 0.06m, λ(⅛) =.56, λ(¼) = 0.9, λ(⅜) = 0.757, λ(½) = Submergence depth = 0.07m, λ(⅛) =.48, λ(¼) = 0.905, λ(⅜) = 0.754, λ(½) = Submergence depth = 0.08m, λ(⅛) =.40, λ(¼) = 0.900, λ(⅜) = 0.75, λ(½) = Submergence depth = 0.09m, λ(⅛) =.34, λ(¼) = 0.896, λ(⅜) = 0.750, λ(½) = Submergence depth = 0.0m, λ(⅛) =.9, λ(¼) = 0.89, λ(⅜) = 0.748, λ(½) = Submergence depth = 0.m, λ(⅛) =.4, λ(¼) = 0.889, λ(⅜) = 0.746, λ(½) = Submergence depth = 0.m, λ(⅛) =., λ(¼) = 0.887, λ(⅜) = 0.744, λ(½) = Submergence depth = 0.3m, λ(⅛) =.9, λ(¼) = 0.885, λ(⅜) = 0.743, λ(½) = Submergence depth = 0.4m, λ(⅛) =.7, λ(¼) = 0.884, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.5m, λ(⅛) =.7, λ(¼) = 0.883, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.6m, λ(⅛) =.8, λ(¼) = 0.883, λ(⅜) = 0.740, λ(½) = Submergence depth = 0.7m, λ(⅛) =.0, λ(¼) = 0.884, λ(⅜) = 0.740, λ(½) = 0.67 Submergence depth = 0.8m, λ(⅛) =.3, λ(¼) = 0.886, λ(⅜) = 0.740, λ(½) = 0.67 Submergence depth = 0.9m, λ(⅛) =.7, λ(¼) = 0.888, λ(⅜) = 0.740, λ(½) = 0.67 Submergence depth = 0.0m, λ(⅛) =.33, λ(¼) = 0.89, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.m, λ(⅛) =.40, λ(¼) = 0.896, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.m, λ(⅛) =.49, λ(¼) = 0.90, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.3m, λ(⅛) =.60, λ(¼) = 0.908, λ(⅜) = 0.744, λ(½) = Submergence depth = 0.4m, λ(⅛) =.7, λ(¼) = 0.96, λ(⅜) = 0.745, λ(½) = Submergence depth = 0.5m, λ(⅛) =.87, λ(¼) = 0.95, λ(⅜) = 0.747, λ(½) = Submergence depth = 0.6m, λ(⅛) =.04, λ(¼) = 0.935, λ(⅜) = 0.749, λ(½) = 0.68 Submergence depth = 0.7m, λ(⅛) =.3, λ(¼) = 0.948, λ(⅜) = 0.75, λ(½) = Submergence depth = 0.8m, λ(⅛) =.46, λ(¼) = 0.963, λ(⅜) = 0.754, λ(½) = Submergence depth = 0.9m, λ(⅛) =.83, λ(¼) = 0.986, λ(⅜) = 0.757, λ(½) = Submergence depth = 0.30m, λ(⅛) =.366, λ(¼) =.03, λ(⅜) = 0.760, λ(½) = Start: Fri /9/008 0:3:09 AM Stop: Fri /9/008 0:3:7 AM Duration: 00:00:7 Run Successful

262 Turbulence Model with ASM 38 Rectangular Cross Sections Fig. D.5. Input data screen for the turbulence model with ASM for rectangular cross sections Table D.5. Input Data Text File for the Turbulence Model with ASM for Rectangular Cross Sections Text File Data Meaning.00 Channel width (B) 0.50 Uniform-flow water depth (H) 00 Longitudinal bed slope (S o ) 400 Wall roughness height of floor (k s ) floor 400 Wall roughness height of left wall (k s ) left 400 Wall roughness height of right wall (k s ) right 9.80 Ratio of weight to mass (g) 00 Water density (ρ) 59 Molecular dynamic viscosity (μ) 0.09 The isotropic turbulence viscosity coefficient (c μ )

263 Table D.5. (Continued from above) 39 Text File Data Meaning 0.40 Von Karman constant (κ).65 Stress model constants, c 0.55 Stress model constants, c.00 k-schmidt number (σ k ).30 ε-schmidt number (σ ε ).44 k-ε model constants, c ε.9 k-ε model constants, c ε 0.0 Distance along the x-direction (dx) 0.75 Ratio of dy ny- to dy (c s ) 0.90 Ratio of dy ny- to dy (c s ) 5 Dimensionless distance from wall in the y-z-directions (y+) 8 Number of nodes in the z-direction (nz) 7 Number of nodes in the y-direction (ny) Maximum allowable correction value 000 Maximum allowable absolute value of pressure correction Maximum allowable absolute value of changing in last 5-iteration of discharge 000 Maximum allowable of summed absolute values of mass sources 00 Maximum overall iteration 0000 Maximum u k ε FV loop Maximum u FV loop Maximum k FV loop Maximum ε FV loop 0000 Maximum p v w FV loop Maximum v FV loop Maximum w FV loop Maximum p FV loop 0000 Maximum p cor FV loop 0.70 Relaxation for u 0.75 Relaxation for k 0.70 Relaxation for ε 0.80 Relaxation for v 0.80 Relaxation for w 0.70 Relaxation for p 0.85 Relaxation for p cor.0 Arrow length scale to grid space 0.50 Ratio of the shortest arrow to the longest arrow 0.30 Ratio of the inside head to the arrow length 0.40 Ratio of the outside head to the arrow length 0.5 Ratio of the head width to the arrow length 0.05 Ratio of the tail width to the arrow length 6 z-pixel 66 y-pixel

264 40 Fig. D.6. Graphical results from the turbulence model with ASM for rectangular cross sections Table D.6. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM for Rectangular Cross Sections Iteration = Acc. abs. Mass sources = 0060 %Q change = -0.03,-0.06,-0.08,-0.08, p-fv Converged at loop v-fv Converged at loop 9 w-fv Converged at loop 507 pcor-fv Converged at loop 585 u-fv Converged at loop 7 k-fv Converged at loop 7 ε-fv Converged at loop 8 u range = m/s v range = m/s w range = -8-8 m/s p range = ,76.84 N/m k range =.569E m/s ε range =.050E-05-0 m/s3 Turbulence Model: ke_asm DiffScheme: Power_Law Q (Chezy) = 0.73 cms. u avg. (Chezy) = m/s Q(i) = cms. (60.57% of Q chezy) u(i) avg = m/s Reynold Number = 0, H, 0.039H Submergence depth = m, λ(⅛) =.063, λ(¼) = 0.88, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.0m, λ(⅛) =.063, λ(¼) = 0.88, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.0m, λ(⅛) =.060, λ(¼) = 0.87, λ(⅜) = 0.73, λ(½) = 0.67

265 Table D.6. (Continued from above) 4 Submergence depth = 0.03m, λ(⅛) =.056, λ(¼) = 0.85, λ(⅜) = 0.73, λ(½) = 0.67 Submergence depth = 0.04m, λ(⅛) =.05, λ(¼) = 0.84, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.05m, λ(⅛) =.047, λ(¼) = 0.8, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.06m, λ(⅛) =.043, λ(¼) = 0.8, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.07m, λ(⅛) =.040, λ(¼) = 0.80, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.08m, λ(⅛) =.036, λ(¼) = 0.809, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.09m, λ(⅛) =.033, λ(¼) = 0.808, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.0m, λ(⅛) =.09, λ(¼) = 0.807, λ(⅜) = 0.75, λ(½) = 0.68 Submergence depth = 0.m, λ(⅛) =.06, λ(¼) = 0.806, λ(⅜) = 0.76, λ(½) = 0.68 Submergence depth = 0.m, λ(⅛) =.04, λ(¼) = 0.806, λ(⅜) = 0.76, λ(½) = Submergence depth = 0.3m, λ(⅛) =.0, λ(¼) = 0.805, λ(⅜) = 0.77, λ(½) = Submergence depth = 0.4m, λ(⅛) =.09, λ(¼) = 0.805, λ(⅜) = 0.78, λ(½) = Submergence depth = 0.5m, λ(⅛) =.06, λ(¼) = 0.805, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.6m, λ(⅛) =.05, λ(¼) = 0.805, λ(⅜) = 0.7, λ(½) = 0.69 Submergence depth = 0.7m, λ(⅛) =.03, λ(¼) = 0.805, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.8m, λ(⅛) =.0, λ(¼) = 0.805, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.9m, λ(⅛) =.00, λ(¼) = 0.806, λ(⅜) = 0.75, λ(½) = Submergence depth = 0.0m, λ(⅛) =.009, λ(¼) = 0.806, λ(⅜) = 0.77, λ(½) = Submergence depth = 0.m, λ(⅛) =.008, λ(¼) = 0.807, λ(⅜) = 0.78, λ(½) = 0.70 Submergence depth = 0.m, λ(⅛) =.008, λ(¼) = 0.808, λ(⅜) = 0.730, λ(½) = Submergence depth = 0.3m, λ(⅛) =.007, λ(¼) = 0.809, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.4m, λ(⅛) =.007, λ(¼) = 0.80, λ(⅜) = 0.734, λ(½) = Submergence depth = 0.5m, λ(⅛) =.007, λ(¼) = 0.8, λ(⅜) = 0.736, λ(½) = 0.7 Submergence depth = 0.6m, λ(⅛) =.008, λ(¼) = 0.83, λ(⅜) = 0.739, λ(½) = 0.74 Submergence depth = 0.7m, λ(⅛) =.008, λ(¼) = 0.85, λ(⅜) = 0.74, λ(½) = 0.77 Submergence depth = 0.8m, λ(⅛) =.009, λ(¼) = 0.87, λ(⅜) = 0.744, λ(½) = 0.70 Submergence depth = 0.9m, λ(⅛) =.00, λ(¼) = 0.89, λ(⅜) = 0.747, λ(½) = 0.73 Submergence depth = 0.30m, λ(⅛) =.0, λ(¼) = 0.8, λ(⅜) = 0.750, λ(½) = 0.77 Start: Fri /9/008 0:00: AM Stop: Fri /9/008 0:00:36 AM Duration: 00:00:4 Run Successful

266 Compound Cross Sections 4 Fig. D.7. Input data screen for the turbulence model with ASM for compound cross sections Table D.7. Input Data Text File of the Turbulence Model with ASM for Compound Cross Sections Text File Data Meaning 0.50 Main channel width (B C ) 0.30 Left side channel width (B L ) 0.30 Right side channel width (B R ) 0.50 Uniform-flow water depth in main channel (H C ) 0.30 Left and right channel depth (H LR ) 00 Longitudinal bed slope (S o ) 400 Wall roughness height of main channel (k s ) M 400 Wall roughness height of left channel (k s ) L 400 Wall roughness height of right channel (k s ) R 9.80 Ratio of weight to mass (g) 00 Water density (ρ) 59 Molecular dynamic viscosity (μ)

267 Table D.7. (Continued from above) 43 Text File Data Meaning 0.09 The isotropic turbulence viscosity coefficient (c μ ) 0.40 Von Karman constant (κ).65 Stress model constants, c 0.55 Stress model constants, c.00 k-schmidt number (σ k ).30 ε-schmidt number (σ ε ).44 k-ε model constants, c ε.9 k-ε model constants, c ε 0.0 Distance along the x-direction (dx) 0.75 Ratio of dy ny- to dy (c s ) 0.90 Ratio of dy ny- to dy (c s ) 5 Dimensionless distance from wall in the y-z-directions (y+) 30 Number of nodes in the z-direction (nz) 7 Number of nodes in the y-direction (ny) Maximum allowable correction value 000 Maximum allowable absolute value of pressure correction Maximum allowable absolute value of changing in last 5-iteration of discharge 000 Maximum allowable of summed absolute values of mass sources 00 Maximum overall iteration 0000 Maximum u k ε FV loop Maximum u FV loop Maximum k FV loop Maximum ε FV loop 0000 Maximum p v w FV loop Maximum v FV loop Maximum w FV loop Maximum p FV loop 0000 Maximum p cor FV loop 0.70 Relaxation for u 0.75 Relaxation for k 0.70 Relaxation for ε 0.80 Relaxation for v 0.80 Relaxation for w 0.70 Relaxation for p 0.85 Relaxation for p cor.0 Arrow length scale to grid space 0.50 Ratio of the shortest arrow to the longest arrow 0.30 Ratio of the inside head to the arrow length 0.40 Ratio of the outside head to the arrow length 0.5 Ratio of the head width to the arrow length 0.05 Ratio of the tail width to the arrow length 57 z-pixel 76 y-pixel

268 44 Fig. D.8. Graphical results from the turbulence model with ASM for compound cross sections Table D.8. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM for Compound Cross Sections Iteration = 3 Acc. abs. Mass sources = 0030 %Q change = 0.09,7,-6,-0.08,-0.04 p-fv Converged at loop v-fv Converged at loop 96 w-fv Converged at loop 50 pcor-fv Converged at loop 635 u-fv Converged at loop 5 k-fv Converged at loop 9 ε-fv Converged at loop 0 u range = m/s v range = -4-5 m/s w range = m/s p range = , N/m k range =.59E m/s ε range = 7.966E m/s3 Turbulence Model: ke_asm DiffScheme: Power_Law Q (Chezy) = cms. u avg. (Chezy) = 0.36 m/s Q(i) = cms. (55.8% of Q chezy) u(i) avg = 0.48 m/s Reynold Number = 67, H, 0.056H Submergence depth = m, λ(⅛) =.089, λ(¼) = 0.875, λ(⅜) = 0.76, λ(½) = Submergence depth = 0.0m, λ(⅛) =.089, λ(¼) = 0.875, λ(⅜) = 0.76, λ(½) = Submergence depth = 0.0m, λ(⅛) =.084, λ(¼) = 0.873, λ(⅜) = 0.76, λ(½) = 0.637

269 Table D.8. (Continued from above) 45 Submergence depth = 0.03m, λ(⅛) =.076, λ(¼) = 0.869, λ(⅜) = 0.75, λ(½) = Submergence depth = 0.04m, λ(⅛) =.068, λ(¼) = 0.864, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.05m, λ(⅛) =.06, λ(¼) = 0.860, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.06m, λ(⅛) =.056, λ(¼) = 0.857, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.07m, λ(⅛) =.053, λ(¼) = 0.855, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.08m, λ(⅛) =.05, λ(¼) = 0.853, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.09m, λ(⅛) =.049, λ(¼) = 0.85, λ(⅜) = 0.709, λ(½) = 0.64 Submergence depth = 0.0m, λ(⅛) =.049, λ(¼) = 0.85, λ(⅜) = 0.709, λ(½) = 0.64 Submergence depth = 0.m, λ(⅛) =.050, λ(¼) = 0.850, λ(⅜) = 0.709, λ(½) = Submergence depth = 0.m, λ(⅛) =.05, λ(¼) = 0.850, λ(⅜) = 0.709, λ(½) = Submergence depth = 0.3m, λ(⅛) =.053, λ(¼) = 0.85, λ(⅜) = 0.709, λ(½) = Submergence depth = 0.4m, λ(⅛) =.057, λ(¼) = 0.85, λ(⅜) = 0.709, λ(½) = Submergence depth = 0.5m, λ(⅛) =.06, λ(¼) = 0.853, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.6m, λ(⅛) =.066, λ(¼) = 0.855, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.7m, λ(⅛) =.07, λ(¼) = 0.858, λ(⅜) = 0.7, λ(½) = 0.65 Submergence depth = 0.8m, λ(⅛) =.079, λ(¼) = 0.86, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.9m, λ(⅛) =.088, λ(¼) = 0.866, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.0m, λ(⅛) =.098, λ(¼) = 0.87, λ(⅜) = 0.75, λ(½) = Submergence depth = 0.m, λ(⅛) =.09, λ(¼) = 0.877, λ(⅜) = 0.77, λ(½) = Submergence depth = 0.m, λ(⅛) =., λ(¼) = 0.884, λ(⅜) = 0.79, λ(½) = 0.66 Submergence depth = 0.3m, λ(⅛) =.38, λ(¼) = 0.89, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.4m, λ(⅛) =.56, λ(¼) = 0.90, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.5m, λ(⅛) =.77, λ(¼) = 0.93, λ(⅜) = 0.76, λ(½) = Submergence depth = 0.6m, λ(⅛) =.0, λ(¼) = 0.96, λ(⅜) = 0.79, λ(½) = 0.67 Submergence depth = 0.7m, λ(⅛) =.7, λ(¼) = 0.94, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.8m, λ(⅛) =.55, λ(¼) = 0.957, λ(⅜) = 0.736, λ(½) = Submergence depth = 0.9m, λ(⅛) =.300, λ(¼) = 0.983, λ(⅜) = 0.740, λ(½) = Submergence depth = 0.30m, λ(⅛) =.395, λ(¼) =.03, λ(⅜) = 0.744, λ(½) = Start: Fri /9/008 0:3:4 AM Stop: Fri /9/008 0:4:33 AM Duration: 00:00:5 Run Successful

270 Turbulence Model with ASM 46 Rectangular Cross Sections Fig. D.9. Input data screen for the turbulence model with ASM for rectangular cross sections Table D.9. Input Data Text File for the Turbulence Model with ASM for Rectangular Cross Sections Text File Data Meaning.00 Channel width (B) 0.50 Uniform-flow water depth (H) 00 Longitudinal bed slope (S o ) 400 Wall roughness height of floor (k s ) floor 400 Wall roughness height of left wall (k s ) left 400 Wall roughness height of right wall (k s ) right 9.80 Ratio of weight to mass (g) 00 Water density (ρ) 59 Molecular dynamic viscosity (μ) 0.09 The isotropic turbulence viscosity coefficient (c μ )

271 Table D.9. (Continued from above) 47 Text File Data Meaning 0.40 Von Karman constant (κ).00 Stress model constants, c 0.65 Stress model constants, c D.00 k-schmidt number (σ k ).30 ε-schmidt number (σ ε ).44 k-ε model constants, c ε.9 k-ε model constants, c ε 0.0 Distance along the x-direction (dx) 0.75 Ratio of dy ny- to dy (c s ) 0.90 Ratio of dy ny- to dy (c s ) 5 Dimensionless distance from wall in the y-z-directions (y+) 8 Number of nodes in the z-direction (nz) 7 Number of nodes in the y-direction (ny) Maximum allowable correction value 000 Maximum allowable absolute value of pressure correction Maximum allowable absolute value of changing in last 5-iteration of discharge 000 Maximum allowable of summed absolute values of mass sources 00 Maximum overall iteration 0000 Maximum u k ε FV loop Maximum u FV loop Maximum k FV loop Maximum ε FV loop 0000 Maximum p v w FV loop Maximum v FV loop Maximum w FV loop Maximum p FV loop 0000 Maximum p cor FV loop 0.70 Relaxation for u 0.75 Relaxation for k 0.70 Relaxation for ε 0.80 Relaxation for v 0.80 Relaxation for w 0.70 Relaxation for p 0.85 Relaxation for p cor.0 Arrow length scale to grid space 0.50 Ratio of the shortest arrow to the longest arrow 0.30 Ratio of the inside head to the arrow length 0.40 Ratio of the outside head to the arrow length 0.5 Ratio of the head width to the arrow length 0.05 Ratio of the tail width to the arrow length 6 z-pixel 66 y-pixel

272 48 Fig. D.0. Graphical results from the turbulence model with ASM for rectangular cross sections Table D.0. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM for Rectangular Cross Sections Iteration = Acc. abs. Mass sources = 0050 %Q change = ,-0.03,-0.034,-0.04, p-fv Converged at loop v-fv Converged at loop 34 w-fv Converged at loop 3 pcor-fv Converged at loop 490 u-fv Converged at loop 9 k-fv Converged at loop 9 ε-fv Converged at loop 0 u range = m/s v range = m/s w range = -7-7 m/s p range = ,76.78 N/m k range = 9.597E-05-3 m/s ε range = 4.90E m/s3 Turbulence Model: ke_asm DiffScheme: Power_Law Q (Chezy) = 0.73 cms. u avg. (Chezy) = m/s Q(i) = cms. (5.6% of Q chezy) u(i) avg = 0.35 m/s Reynold Number = 86, H, 0.073H Submergence depth = m, λ(⅛) =.3, λ(¼) = 0.84, λ(⅜) = 0.7, λ(½) = 0.69 Submergence depth = 0.0m, λ(⅛) =.3, λ(¼) = 0.84, λ(⅜) = 0.7, λ(½) = 0.69 Submergence depth = 0.0m, λ(⅛) =.9, λ(¼) = 0.839, λ(⅜) = 0.7, λ(½) = 0.69

273 Table D.0. (Continued from above) 49 Submergence depth = 0.03m, λ(⅛) =.4, λ(¼) = 0.837, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.04m, λ(⅛) =.07, λ(¼) = 0.834, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.05m, λ(⅛) =.0, λ(¼) = 0.83, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.06m, λ(⅛) =.095, λ(¼) = 0.88, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.07m, λ(⅛) =.090, λ(¼) = 0.86, λ(⅜) = 0.79, λ(½) = Submergence depth = 0.08m, λ(⅛) =.085, λ(¼) = 0.84, λ(⅜) = 0.79, λ(½) = Submergence depth = 0.09m, λ(⅛) =.08, λ(¼) = 0.8, λ(⅜) = 0.79, λ(½) = Submergence depth = 0.0m, λ(⅛) =.077, λ(¼) = 0.8, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.m, λ(⅛) =.073, λ(¼) = 0.89, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.m, λ(⅛) =.070, λ(¼) = 0.88, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.3m, λ(⅛) =.066, λ(¼) = 0.87, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.4m, λ(⅛) =.064, λ(¼) = 0.86, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.5m, λ(⅛) =.06, λ(¼) = 0.86, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.6m, λ(⅛) =.059, λ(¼) = 0.85, λ(⅜) = 0.73, λ(½) = 0.70 Submergence depth = 0.7m, λ(⅛) =.057, λ(¼) = 0.85, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.8m, λ(⅛) =.055, λ(¼) = 0.85, λ(⅜) = 0.75, λ(½) = Submergence depth = 0.9m, λ(⅛) =.053, λ(¼) = 0.85, λ(⅜) = 0.76, λ(½) = Submergence depth = 0.0m, λ(⅛) =.05, λ(¼) = 0.85, λ(⅜) = 0.78, λ(½) = Submergence depth = 0.m, λ(⅛) =.05, λ(¼) = 0.86, λ(⅜) = 0.79, λ(½) = Submergence depth = 0.m, λ(⅛) =.050, λ(¼) = 0.86, λ(⅜) = 0.73, λ(½) = 0.70 Submergence depth = 0.3m, λ(⅛) =.049, λ(¼) = 0.87, λ(⅜) = 0.73, λ(½) = 0.7 Submergence depth = 0.4m, λ(⅛) =.048, λ(¼) = 0.88, λ(⅜) = 0.734, λ(½) = 0.74 Submergence depth = 0.5m, λ(⅛) =.048, λ(¼) = 0.89, λ(⅜) = 0.736, λ(½) = 0.76 Submergence depth = 0.6m, λ(⅛) =.048, λ(¼) = 0.80, λ(⅜) = 0.738, λ(½) = 0.79 Submergence depth = 0.7m, λ(⅛) =.048, λ(¼) = 0.8, λ(⅜) = 0.740, λ(½) = 0.7 Submergence depth = 0.8m, λ(⅛) =.049, λ(¼) = 0.83, λ(⅜) = 0.743, λ(½) = 0.74 Submergence depth = 0.9m, λ(⅛) =.050, λ(¼) = 0.85, λ(⅜) = 0.746, λ(½) = 0.77 Submergence depth = 0.30m, λ(⅛) =.05, λ(¼) = 0.87, λ(⅜) = 0.748, λ(½) = Start: Fri /9/008 0:00:49 AM Stop: Fri /9/008 0:0: AM Duration: 00:00:33 Run Successful

274 Compound Cross Sections 50 Fig. D.. Input data screen for the turbulence model with ASM for compound cross sections Table D.. Input Data Text File for the Turbulence Model with ASM for Compound Cross Sections Text File Data Meaning 0.50 Main channel width (B C ) 0.30 Left side channel width (B L ) 0.30 Right side channel width (B R ) 0.50 Uniform-flow water depth in main channel (H C ) 0.30 Left and right channel depth (H LR ) 00 Longitudinal bed slope (S o ) 400 Wall roughness height of main channel (k s ) M 400 Wall roughness height of left channel (k s ) L 400 Wall roughness height of right channel (k s ) R 9.80 Ratio of weight to mass (g) 00 Water density (ρ)

275 Table D.. (Continued from above) 5 Text File Data Meaning 59 Molecular dynamic viscosity (μ) 0.09 The isotropic turbulence viscosity coefficient (c μ ) 0.40 Von Karman constant (κ).00 Stress model constants, c 0.65 Stress model constants, c D.00 k-schmidt number (σ k ).30 ε-schmidt number (σ ε ).44 k-ε model constants, c ε.9 k-ε model constants, c ε 0.0 Distance along the x-direction (dx) 0.75 Ratio of dy ny- to dy (c s ) 0.90 Ratio of dy ny- to dy (c s ) 5 Dimensionless distance from wall in the y-z-directions (y+) 30 Number of nodes in the z-direction (nz) 7 Number of nodes in the y-direction (ny) Maximum allowable correction value 000 Maximum allowable absolute value of pressure correction Maximum allowable absolute value of changing in last 5-iteration of discharge 000 Maximum allowable of summed absolute values of mass sources 00 Maximum overall iteration 0000 Maximum u k ε FV loop Maximum u FV loop Maximum k FV loop Maximum ε FV loop 0000 Maximum p v w FV loop Maximum v FV loop Maximum w FV loop Maximum p FV loop 0000 Maximum p cor FV loop 0.70 Relaxation for u 0.75 Relaxation for k 0.70 Relaxation for ε 0.80 Relaxation for v 0.80 Relaxation for w 0.70 Relaxation for p 0.85 Relaxation for p cor.0 Arrow length scale to grid space 0.50 Ratio of the shortest arrow to the longest arrow 0.30 Ratio of the inside head to the arrow length 0.40 Ratio of the outside head to the arrow length 0.5 Ratio of the head width to the arrow length 0.05 Ratio of the tail width to the arrow length 57 z-pixel 76 y-pixel

276 5 Fig. D.. Graphical results from the turbulence model with ASM for compound cross sections Table D.. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM for Compound Cross Sections Iteration = 4 Acc. abs. Mass sources = 003 %Q change = ,-0.034,-0.038,-0.04, p-fv Converged at loop v-fv Converged at loop 97 w-fv Converged at loop 59 pcor-fv Converged at loop 65 u-fv Converged at loop 8 k-fv Converged at loop 8 ε-fv Converged at loop 9 u range = m/s v range = -4-8 m/s w range = -8-8 m/s p range = 7.0-4, N/m k range = 8.68E-05-5 m/s ε range = 4.448E-06-3 m/s3 Turbulence Model: ke_asm DiffScheme: Power_Law Q (Chezy) = cms. u avg. (Chezy) = 0.36 m/s Q(i) = cms. (49.75% of Q chezy) u(i) avg = 0. m/s Reynold Number = 59, H, 0.070H Submergence depth = m, λ(⅛) =.44, λ(¼) = 0.866, λ(⅜) = 0.708, λ(½) = 0.65 Submergence depth = 0.0m, λ(⅛) =.44, λ(¼) = 0.866, λ(⅜) = 0.708, λ(½) = 0.65 Submergence depth = 0.0m, λ(⅛) =.40, λ(¼) = 0.863, λ(⅜) = 0.707, λ(½) = 0.65

277 Table D.. (Continued from above) 53 Submergence depth = 0.03m, λ(⅛) =.33, λ(¼) = 0.859, λ(⅜) = 0.705, λ(½) = 0.65 Submergence depth = 0.04m, λ(⅛) =.5, λ(¼) = 0.855, λ(⅜) = 0.704, λ(½) = 0.65 Submergence depth = 0.05m, λ(⅛) =.8, λ(¼) = 0.85, λ(⅜) = 0.703, λ(½) = 0.65 Submergence depth = 0.06m, λ(⅛) =.3, λ(¼) = 0.848, λ(⅜) = 0.70, λ(½) = 0.65 Submergence depth = 0.07m, λ(⅛) =.09, λ(¼) = 0.845, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.08m, λ(⅛) =.06, λ(¼) = 0.843, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.09m, λ(⅛) =.04, λ(¼) = 0.84, λ(⅜) = 0.700, λ(½) = Submergence depth = 0.0m, λ(⅛) =.03, λ(¼) = 0.84, λ(⅜) = 0.700, λ(½) = Submergence depth = 0.m, λ(⅛) =.03, λ(¼) = 0.84, λ(⅜) = 0.700, λ(½) = Submergence depth = 0.m, λ(⅛) =.05, λ(¼) = 0.84, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.3m, λ(⅛) =.07, λ(¼) = 0.843, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.4m, λ(⅛) =.0, λ(¼) = 0.844, λ(⅜) = 0.70, λ(½) = Submergence depth = 0.5m, λ(⅛) =.3, λ(¼) = 0.846, λ(⅜) = 0.703, λ(½) = Submergence depth = 0.6m, λ(⅛) =.8, λ(¼) = 0.849, λ(⅜) = 0.704, λ(½) = Submergence depth = 0.7m, λ(⅛) =.4, λ(¼) = 0.85, λ(⅜) = 0.706, λ(½) = 0.66 Submergence depth = 0.8m, λ(⅛) =.30, λ(¼) = 0.856, λ(⅜) = 0.707, λ(½) = Submergence depth = 0.9m, λ(⅛) =.38, λ(¼) = 0.86, λ(⅜) = 0.709, λ(½) = Submergence depth = 0.0m, λ(⅛) =.46, λ(¼) = 0.866, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.m, λ(⅛) =.56, λ(¼) = 0.87, λ(⅜) = 0.73, λ(½) = Submergence depth = 0.m, λ(⅛) =.66, λ(¼) = 0.879, λ(⅜) = 0.76, λ(½) = Submergence depth = 0.3m, λ(⅛) =.78, λ(¼) = 0.887, λ(⅜) = 0.79, λ(½) = 0.67 Submergence depth = 0.4m, λ(⅛) =.90, λ(¼) = 0.896, λ(⅜) = 0.7, λ(½) = Submergence depth = 0.5m, λ(⅛) =.04, λ(¼) = 0.905, λ(⅜) = 0.75, λ(½) = Submergence depth = 0.6m, λ(⅛) =.9, λ(¼) = 0.97, λ(⅜) = 0.78, λ(½) = Submergence depth = 0.7m, λ(⅛) =.38, λ(¼) = 0.930, λ(⅜) = 0.73, λ(½) = 0.68 Submergence depth = 0.8m, λ(⅛) =.59, λ(¼) = 0.945, λ(⅜) = 0.736, λ(½) = Submergence depth = 0.9m, λ(⅛) =.95, λ(¼) = 0.968, λ(⅜) = 0.74, λ(½) = Submergence depth = 0.30m, λ(⅛) =.377, λ(¼) =.04, λ(⅜) = 0.746, λ(½) = 0.69 Start: Fri /9/008 0:4:50 AM Stop: Fri /9/008 0:5:5 AM Duration: 00:00:5 Run Successful

278 Turbulence Model with ASM3 54 Rectangular Cross Sections Fig. D.3. Input data screen for the turbulence model with ASM3 for rectangular cross sections Table D.3. Input Data Text File for the Turbulence Model with ASM3 for Rectangular Cross Sections Text File Data Meaning.00 Channel width (B) 0.50 Uniform-flow water depth (H) 00 Longitudinal bed slope (S o ) 400 Wall roughness height of floor (k s ) floor 400 Wall roughness height of left wall (k s ) left 400 Wall roughness height of right wall (k s ) right 9.80 Ratio of weight to mass (g) 00 Water density (ρ) 59 Molecular dynamic viscosity (μ)

279 Table D.3. (Continued from above) 55 Text File Data Meaning 0.09 The isotropic turbulence viscosity coefficient (c μ ) 0.40 Von Karman constant (κ) Stress model parameter, α Stress model parameter, α Stress model parameter, β Stress model parameter, β.6400 Stress model parameter, c Stress model parameter, c 00 Stress model parameter, c Stress model parameter, c 3.00 k-schmidt number (σ k ). ε-schmidt number (σ ε ).49 k-ε model constants, c ε.8 k-ε model constants, c ε 0.0 Distance along the x-direction (dx) 0.75 Ratio of dy ny- to dy (c s ) 0.90 Ratio of dy ny- to dy (c s ) 5 Dimensionless distance from wall in the y-z-directions (y+) 8 Number of nodes in the z-direction (nz) 7 Number of nodes in the y-direction (ny) Maximum allowable correction value 000 Maximum allowable absolute value of pressure correction Maximum allowable absolute value of changing in last 5-iteration of discharge 000 Maximum allowable of summed absolute values of mass sources 00 Maximum overall iteration 0000 Maximum u k ε FV loop Maximum u FV loop Maximum k FV loop Maximum ε FV loop 0000 Maximum p v w FV loop Maximum v FV loop Maximum w FV loop Maximum p FV loop 0000 Maximum p cor FV loop 0.70 Relaxation for u 0.75 Relaxation for k 0.70 Relaxation for ε 0.80 Relaxation for v 0.80 Relaxation for w 0.70 Relaxation for p 0.85 Relaxation for p cor.0 Arrow length scale to grid space 0.50 Ratio of the shortest arrow to the longest arrow 0.30 Ratio of the inside head to the arrow length 0.40 Ratio of the outside head to the arrow length 0.5 Ratio of the head width to the arrow length 0.05 Ratio of the tail width to the arrow length 6 z-pixel 66 y-pixel

280 56 Fig. D.4. Graphical results from the turbulence model with ASM3 for rectangular cross sections Table D.4. Example Summary Output Data Text File at the Last Iteration of the Turbulence Model with ASM3 for Rectangular Cross Sections Iteration = 9 Acc. abs. Mass sources = 0078 %Q change = -0.03,-0.03,-0.03,-0.043, p-fv Converged at loop v-fv Converged at loop 70 w-fv Converged at loop 94 pcor-fv Converged at loop 460 u-fv Converged at loop 8 k-fv Converged at loop 3 ε-fv Converged at loop 4 u range = m/s v range = -3-8 m/s w range = m/s p range = ,76.43 N/m k range =.4E m/s ε range = 8.6E m/s3 Turbulence Model: ke_asm3 DiffScheme: Power_Law Q (Chezy) = 0.73 cms. u avg. (Chezy) = m/s Q(i) = cms. (8.74% of Q chezy) u(i) avg = 0.07 m/s Reynold Number = 38, H, 0.44H Submergence depth = m, λ(⅛) =.96, λ(¼) =.05, λ(⅜) = 0.847, λ(½) = Submergence depth = 0.0m, λ(⅛) =.96, λ(¼) =.05, λ(⅜) = 0.847, λ(½) = Submergence depth = 0.0m, λ(⅛) =.89, λ(¼) =.047, λ(⅜) = 0.845, λ(½) = 0.740

281 Table D.4. (Continued from above) 57 Submergence depth = 0.03m, λ(⅛) =.78, λ(¼) =.04, λ(⅜) = 0.843, λ(½) = Submergence depth = 0.04m, λ(⅛) =.67, λ(¼) =.034, λ(⅜) = 0.84, λ(½) = Submergence depth = 0.05m, λ(⅛) =.54, λ(¼) =.06, λ(⅜) = 0.838, λ(½) = Submergence depth = 0.06m, λ(⅛) =.4, λ(¼) =.09, λ(⅜) = 0.835, λ(½) = Submergence depth = 0.07m, λ(⅛) =.8, λ(¼) =.0, λ(⅜) = 0.833, λ(½) = Submergence depth = 0.08m, λ(⅛) =.5, λ(¼) =.003, λ(⅜) = 0.830, λ(½) = Submergence depth = 0.09m, λ(⅛) =.0, λ(¼) = 0.995, λ(⅜) = 0.87, λ(½) = Submergence depth = 0.0m, λ(⅛) =.90, λ(¼) = 0.988, λ(⅜) = 0.84, λ(½) = Submergence depth = 0.m, λ(⅛) =.78, λ(¼) = 0.98, λ(⅜) = 0.8, λ(½) = Submergence depth = 0.m, λ(⅛) =.67, λ(¼) = 0.974, λ(⅜) = 0.89, λ(½) = Submergence depth = 0.3m, λ(⅛) =.57, λ(¼) = 0.967, λ(⅜) = 0.86, λ(½) = Submergence depth = 0.4m, λ(⅛) =.47, λ(¼) = 0.96, λ(⅜) = 0.84, λ(½) = Submergence depth = 0.5m, λ(⅛) =.38, λ(¼) = 0.954, λ(⅜) = 0.8, λ(½) = Submergence depth = 0.6m, λ(⅛) =.9, λ(¼) = 0.948, λ(⅜) = 0.809, λ(½) = Submergence depth = 0.7m, λ(⅛) =., λ(¼) = 0.943, λ(⅜) = 0.807, λ(½) = Submergence depth = 0.8m, λ(⅛) =.3, λ(¼) = 0.937, λ(⅜) = 0.804, λ(½) = Submergence depth = 0.9m, λ(⅛) =.06, λ(¼) = 0.93, λ(⅜) = 0.803, λ(½) = Submergence depth = 0.0m, λ(⅛) =.099, λ(¼) = 0.97, λ(⅜) = 0.80, λ(½) = Submergence depth = 0.m, λ(⅛) =.093, λ(¼) = 0.9, λ(⅜) = 0.799, λ(½) = Submergence depth = 0.m, λ(⅛) =.087, λ(¼) = 0.98, λ(⅜) = 0.798, λ(½) = Submergence depth = 0.3m, λ(⅛) =.08, λ(¼) = 0.94, λ(⅜) = 0.796, λ(½) = Submergence depth = 0.4m, λ(⅛) =.076, λ(¼) = 0.90, λ(⅜) = 0.795, λ(½) = Submergence depth = 0.5m, λ(⅛) =.07, λ(¼) = 0.906, λ(⅜) = 0.794, λ(½) = Submergence depth = 0.6m, λ(⅛) =.066, λ(¼) = 0.903, λ(⅜) = 0.793, λ(½) = Submergence depth = 0.7m, λ(⅛) =.06, λ(¼) = 0.900, λ(⅜) = 0.793, λ(½) = Submergence depth = 0.8m, λ(⅛) =.058, λ(¼) = 0.897, λ(⅜) = 0.79, λ(½) = 0.74 Submergence depth = 0.9m, λ(⅛) =.055, λ(¼) = 0.894, λ(⅜) = 0.79, λ(½) = 0.74 Submergence depth = 0.30m, λ(⅛) =.05, λ(¼) = 0.89, λ(⅜) = 0.79, λ(½) = Start: Fri /9/008 0:0:34 AM Stop: Fri /9/008 0:0:48 AM Duration: 00:00:4 Run Successful

282 Compound Cross Sections 58 Fig. D.5. Input data screen for the turbulence model with ASM3 for compound cross sections Table D.5. Input Data Text File for the Turbulence Model with ASM3 for Compound Cross Sections Text File Data Meaning 0.50 Main channel width (B C ) 0.30 Left side channel width (B L ) 0.30 Right side channel width (B R ) 0.50 Uniform-flow water depth in main channel (H C ) 0.30 Left and right channel depth (H LR )