Development of a generic three dimensional model for stratified flows

DEVELOPMENT OF A GENERIC THREEDIMENSIONAL MODEL FOR STRATIFIED FLOWS WANG DONGCHAO (M Eng., TIANJIN) A THESIS SUBMITTED FOR THE DEGREE OF MASTER OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2003 ACKNOWLEDGEMENTS I would like to give my sincere thankfulness to my supervisors Assistant Professor Pengzhi Lin and Professor N Jothi Shankar They are always open to suggestions and very flexible in dealing with my specific needs This greatly helps me keep the motivation high and guarantee the best possible outcome of my project I acknowledge all people in Environmental & Hydraulic Division and Civil Engineering Department at the National University of Singapore for their cooperation and efforts throughout my education and research work I would like also to acknowledge the National University of Singapore for its benevolence in providing a full scholarship grant for my study This thesis is dedicated to my family and my friends for their much-needed support and caring during these past months I would not have accomplished this without them i TABLE OF CONTENTS ACKNOWLEDGEMENTS i TABLE OF CONTENTS ii SUMMARY iv NOMENCLATURE vi LIST OF FIGURES viii LIST OF TABLES x Chapter Introduction and Literature Review…… ……………….1 1.1 General Background and Purpose……………………………………………….1 1.2 Stratified Flows……………………………………………………………… 1.3 Sediment Dumping……………………………………………………… ……3 1.4 Review of Stratified Flow Models……………………………………… … 1.5 Objectives and Scope of the Present Study …………………………………… 1.6 Outline of the Present Work…………………………………………………….9 Chapter Mathematical Review of Hydrodynamic Model……… 11 2.1 The Navier-Stokes Equations………………………………………………….11 2.2 Turbulence Modelling………………………………………………………….12 2.2.1 Reynolds Averaged Navier-Stokes Equations…… ……………………12 2.2.2 The Eddy Viscosity Concept……………………… ………………… 15 2.2.3 The Mixing Length Model……………………………………………….16 2.2.4 The Standard k-ε Model…………………………… ………………… 17 2.2.5 Large Eddy Simulation Model (LES)……………… ………………….19 2.2.6 Direct Numerical Simulation (DNS)……………… ………………… 20 2.3 Concluding Remarks…………………………………………………………21 Chapter Development and Formulations of the 3D Hydrodynamic Model…………………………………………………….…22 3.1 Governing Equations in Cartesian Coordinate…………………………… 22 ii 3.2 Governing Equations in σ-Coordinate………………………………………25 3.3 Numerical Approximations…………………………………………………30 3.3.1 Advection Step………………………………………………………… 32 3.3.2 Diffusion Step……………………………………………………………34 3.3.3 Pressure-updating Step………………………….……………………35 3.3.4 Velocity-correction Step…………… ………………………………… 39 3.3.5 Density Tracking Step……………………………………………………39 3.3.6 Free Surface Tracking Step………………………………………………42 3.4 Boundary Conditions in σ-Coordinate …………………………………… 43 3.5 Stability Criterion………………… …………………………………… 46 Chapter Results and Discussions……………………… ……………47 4.1 Additional Convective Effect in a Stratified Flow…………….………….… 47 4.2 Density-driven Flow…………………………………….……………….…….50 4.3 Computation of the Sediment Dumping into Water… ……………………….53 4.3.1 Drift Velocity Assumption…………………………………………….…54 4.3.2 Turbulence Model…… …………… …………………………………54 4.3.3 Computational Conditions in 2-D Cases………………………………55 4.3.4 Calculated Results in 2-D Cases……………………………………… 57 4.3.5 3-D Demonstration with Free Surface………………………………….65 4.4 Future Works on Buoyant Jets and Plumes.…… …………………………….72 4.4.1 A Horizontal Buoyant Jet ……………………………………………….72 4.4.2 Preliminary Results for Jet Centerline Trajectory…… ……………… 74 4.4.3 Mixing of a Buoyant Plume with and without Waves…….…………… 78 Chapter Conclusions and Recommendations………………… ……82 5.1 Conclusions…………………………………………………………………….82 5.2 Recommendations……………………………………………………………84 References…………………………………………………………………85 Appendix …………………………………………………………………90 iii SUMMARY A three-dimensional numerical model has been developed to simulate stratified flows with free surface The model solves the original Navier-Stokes equations (NSE) with a variable fluid density without the employment of Boussinesq approximation The modified NSE are solved in a transformed σ-coordinate system with the use of operatorsplitting method (Lin & Li, 2002), which splits the solution procedure into advection, diffusion and pressure-correction steps Assuming the free surface is the single function of the horizontal plane, a slightly modified σ-coordinate introduced by Blumberg and Mellor (1983) would map the irregular computation domain to a regular computational domain By introducing density variation, a transport equation for fluid density is added, which is solved by the Cubic-Interpolated propagation (CIP) method as developed by Yabe et al (1991) The CIP method has the advantage to capture the moving sharp interface front Without Boussinesq assumption, no restriction is made for the rate of density variation and the model can be used to simulate both continuously stratified flows and layered flows The numerical model is validated against the one-dimensional convection-diffusion problem, which displayed fairly the influence of strong density stratification to the diffusion term Another comparison of the model prediction to an analytical solution was conducted for a horizontal density-gradient flow in a closed basin Excellent agreements are obtained between numerical results and analytical solutions The model is then used to study transport phenomena of dumped sediments into a quiescent water body, which is modeled in this study as a strongly stratified flow A mixing length model is applied to represent the induced turbulence with a well-defined iv length scale For the two-dimensional problem, the numerical results are compared well with experimental data in terms of the mean particle falling velocity and the spreading rate of the sediment cloud for both fine and coarse sediments The present model is further extended to study the dumping of sediments in a 3D environment with the presence of free surface A reasonable explanation is given by the current model on the behaviours of particle dumping The model simulation results reveal an inverse relationship between the rate of spreading of the cloud and the settling velocity, and show that the frontal velocity approaches the settling velocity in the ultimate stage It is found that an annulus-like cloud will be formed for fine sediments whereas a plate-like cloud for coarse sediments The model is proven to be a good tool to simulate strongly stratified free surface flow, in which the conventional Boussinesq assumption may become invalid Keywords: 3-D numerical model, stratified flows, σ-coordinate, free surface, sediment dumping, non-Boussinesq v NOMENCLATURE b nominal width of the cloud C mixing length coefficient CD drag coefficient Cr Courant number Cs d empirical constant in SGS model d50 mean diameter of the particle D0 jet and plume diameter F body force F0 buoyancy flux Fr densimetric Froude number FX momentum flux in x-direction FY momentum flux in y-direction gx g′ diameter of the particle gy gz gravity acceleration in x-, y-, z- direction reduced gravity h water depth k turbulent kinetic energy lm n p mixing length q0 initial volume per unit depth Rc Re dispersion width Rij u strain rate of the mean flow velocity component in x-direction ui velocity component in i-th direction U0 v initial mean jet velocity V velocity scale Vs w settling velocity wd drift velocity Wc falling velocity of particle cloud Normal coordinate pressure Reynolds number velocity component in y-direction velocity component in z-direction vi x* y* z* t * x y σ t Z spatial and temporal coordinates in physical domain spatial and temporal coordinates in computational domain dumping distance Greek Symbols β weighting coefficient Γt turbulent diffusivity δ ij Kronecker delta(1 when i = j ) ∆x, ∆y, ∆z space steps in x-, y-, z- direction ∆ρ mean density difference between particle cloud and ambient fluid ∆ρ0 initial density difference between jet and ambient fluid ε η dissipation rate λ µ horizontal density gradient free surface displacement dynamic molecular viscosity µt ν eddy viscosity νc νt ρ molecular viscosity of the fluid ρs ρa ρ0 σt τ ij ϕ′ ϕ density of the particle Φ steady mean component of a flow property kinematic molecular viscosity kinematic eddy viscosity density density of the ambient fluid initial ambient fluid density turbulent Prandtl/Schmidt number viscous stress tensor time-varying fluctuating component of a flow property time-averaged component of a flow property χ convergence number in diffusion step Ω convergence factor Subscripts x, y, z in x-, y-, z- direction i, j, k grid points indices vii LIST OF FIGURES Figure 1.1 Stability definitions showing related hydrostatic density distributions 03 Figure 1.2 General overview of the disposal of dredged material 04 Figure 3.1 The sigma-transformation in the vertical direction 26 Figure 3.2 Schematic plot of mesh definition σ−coordinate 31 Figure 4.1 Comparisons of numerical velocity profile at t = 1,2,5,10,20 (s) with analytical solution 50 Figure 4.2 Schematic diagram of density-driven flow in a tank 51 Figure 4.3 Comparisons of numerical velocity profile at x=10 km, y=2.5 km with analytical solution for constant horizontal density gradient 53 Figure 4.4 Illustration of sediment dumping 55 Figure 4.5 Variation of falling velocity of cloud normalized by characteristic velocity scale against dumping distance with d50 at 5.0 mm 58 Figure 4.6 Variation of falling velocity of cloud normalized by characteristic velocity scale against dumping distance with d50 at 0.8 mm 59 Figure 4.7 Dispersion width of the dumping clouds normalized by square root of q0 = 10 cm2 60 Figure 4.8 Velocity fields and density contours at intervals of one tenth of density difference between thermal maximum value and ambient water at a cross-section in the middle of ycoordinates for the case d50=5.0 mm with q0 = cm2 at time t= 0.3, 0.7, 1.0, 1.5 (s) 61 Figure 4.9 Velocity fields and density contours at intervals of one tenth of density difference between thermal maximum value and ambient water at a cross-section in the middle of ycoordinates for the case d50=5.0 mm with q0 = 10 cm2 at time 62 viii t= 0.3, 0.7, 1.0, 1.5 (s) Figure 4.10 Velocity fields and density contours at intervals of one tenth of density difference between thermal maximum value and ambient water at a cross-section in the middle of ycoordinates for the case d50=0.8 mm with q0 = cm2 at time t= 0.3, 0.7, 1.0, 1.5 (s) 63 Figure 4.11 Velocity fields and density contours at intervals of one tenth of density difference between thermal maximum value and ambient water at a cross-section in the middle of ycoordinates for the case d50=0.8 mm with q0 = 10 cm2 at time t= 0.3, 0.7, 1.0, 1.5 (s) 64 Figure 4.12 Spatial profiles of surface elevation and density isosurface of sediment cloud in 50% density difference between thermal maximum value and ambient water with d50=0.15 mm at time t = 0.1, 0.3, 0.6, 1.0, 1.5, 2.0, 2.5, 3.0 (s) 68 Spatial profiles of surface elevation and density isosurface of sediment cloud in 50% density difference between thermal maximum value and ambient water with d50=1.3 mm at time t = 0.1, 0.3, 0.6, 1.0, 1.5, 2.0, 2.5, 3.0 (s) 69 Figure 4.14 Velocity fields and density contours at intervals of one tenth of density difference between thermal maximum value and ambient water at a cross-section through the cloud center for the case with d50=0.15 mm at time t = 0.6, 1.0, 2.0, 3.0(s) 70 Figure 4.15 Velocity fields and density contours at intervals of one tenth of density difference between thermal maximum value and ambient water at a cross-section through the cloud center for the case with d50=1.3 mm at time t = 0.6, 1.0, 2.0, 3.0(s) 71 Figure 4.16 Sketch of a horizontally discharged buoyant round jet in a still ambient fluid 73 Figure 4.17 Velocity fields and density contours and comparison of the jet trajectory determined experimentally with the prediction of this study and mathematical model by Davidson for various Froude numbers (a) Fr = 10 , (b) Fr = 15 , (c) Fr = 20 75, 76 Figure 4.18 Surface elevation profiles 3and density isosurface of the jet evolution at ρ = 1004 kg/m with Fr=15 78 Figure 4.19 Density contour of Case A (no wave effect) in a vertical plotting window (x-z plane at y=0.1 m) in the middle of the channel 80 Figure 4.13 ix Chapter3 3.3.1 Advection Step As mentioned before, the entire computational procedure is broken into three major steps The first step is to treat the advection terms in the momentum equations (3.18) to (3.20) Due to the similarity of these three equations, only equation (3.18) is discussed The other two equations can be solved in the same way The finite difference form for the advection step in (3.18) can be represented (Lin & Li, 2002) as, uin, +j 1,k3 − uin, j ,k ∆t n  ∂u ∂u ∂u  + u + v +ω = 0,  ∂y ∂σ i , j ,k  ∂x (3.30) In fact, the above form can be further split into three sub-steps as follows, uin, +j 1,k9 − uin, j ,k ∆t uin, +j ,2k − uin, +j ,1k9 ∆t uin, +j ,3k − uin, +j ,2k ∆t n  ∂u  + u  = 0,  ∂x i , j ,k (3.31) n +1  ∂u  +v  = 0,  ∂y i , j ,k (3.32) n+  ∂u  + ω =   ∂σ i , j ,k (3.33) Since the above three advection sub-steps have almost the same characteristics, they can essentially be solved by the same numerical scheme In this study, the combination of quadratic backward characteristic method and Lax-Wendroff method (e.g Lin & Li, 32 Chapter3 2002) is used to solve the flow advection Without losing generality, only equation (3.31) is solved here For simplicity, only ui , j ,k >0 is considered The negative velocity and equations (3.32) and (3.33) could be solved similarly In order to employ the quadratic backward characteristics method defined with subscript as QC, the advection distance ∆xa is first defined as ∆xa = uin, j , k ∆t Equation (3.31) is then solved as, (u ) n +1/ i , j ,k QC = (∆xi −1 − ∆xa )(−∆xa ) n (∆xi − + ∆xi −1 − ∆xa )(−∆xa ) n ui − 2, j , k + ui −1, j ,k ∆xi − (∆xi − + ∆xi −1 ) (∆xi − )(−∆xi −1 ) (∆xi − + ∆xi −1 − ∆xa )(∆xi −1 − ∆xa ) n + ui , j ,k (∆xi − + ∆xi −1 )∆xi −1 (3.34) The Lax-Wendroff method defined with subscript as LW solves equation (3.31) as, (u ) n +1/ i , j ,k LW = ∆xa (∆xi + ∆xa ) n (∆xi −1 − ∆xa )(−∆xi − ∆xa ) ui −1, j ,k + ui , j , k n ∆xi −1 (∆xi −1 + ∆xi ) ∆xi −1 (−∆xi ) (∆xi −1 − ∆xa )(−∆xa ) n + ui +1, j ,k (∆xi −1 + ∆xi )∆xi (3.35) This study uses the average of the above two methods to achieve the stable and accurate numerical results, i.e., ( uin, +j ,1/k =  uin, +j ,1/k  ) QC ( + uin, +j ,1/k ) LW   (3.36) 33 Chapter3 3.3.2 Diffusion Step The diffusion process is solved after the advection is completed In this step, the density variation needs to be treated properly when stresses are calculated Again, without the loss of generality, we still discuss equation (3.18) only, uin, +j ,2k − uin, +j 1,k3 ∆t n +1/ ∂τ ∂σ ∂τ ∂σ  ∂τ ∂σ ∂τ  ∂τ =  xx + xx * + xy + xy * + xz *  ρ  ∂x ∂σ ∂x ∂y ∂σ ∂y ∂σ ∂z i , j ,k (3.37) All stress terms in the above equations can be calculated using equation (3.28) The central difference method is used to discretize all the partial differentiation terms in the above equation For example, n +1/ (τ xx )i +1/ 2, j ,k − (τ xx )i −1/ 2, j ,k  ∂τ xx  = ,  ∂x  ( ∆xi −1 + ∆xi ) /  i , j , k n +1/ n +1/ (3.38) Where  ui +1, j ,k − ui , j ,k + ∆xi  (τ xx )i+1/ 2, j ,k = (ν + ν t ) ρi+1/ 2, j ,k  n +1/  ui +1/ 2, j , k +1 − ui +1/ 2, j ,k −1  ∂σ    * ∆σ k −1 + ∆σ k  ∂x i +1/ 2, j ,k  n +1/  ui , j ,k − ui −1, j ,k + ∆ x i −1  (3.39) (τ xx )i−1/ 2, j ,k = (ν + ν t ) ρi −1 2, j ,k  n +1/  ui −1/ 2, j ,k +1 − ui −1/ 2, j ,k −1  ∂σ    * ∆σ k −1 + ∆σ k  ∂x i −1/ 2, j ,k  34 n +1/ (3.40) Chapter3 The velocity between nodes is obtained by the linear interpolation The derivatives of σ are calculated based on formula (3.24)-(3.27), which is discreted in center difference scheme The other stress terms in (3.37) could be discretized similarly 3.3.3 Pressure-updating Step Pressure-updating step solves the additional source and sink terms besides the advection and diffusion In NSE, these terms include pressure and gravitational forces The projection method (Lin & Liu, 1998) is used to calculate the pressure and velocity field so that the updated velocity field satisfies the divergence-free condition as imposed by the continuity equation (3.17) The finite difference forms are written as follows, u in, +j 1,k − u in, +j ,2k ∆t n +1  ∂p ∂p ∂σ  =−  + + gx  ρ  ∂x ∂σ ∂x *  i , j ,k (3.41) n +1 vin, +j ,1k − vin, +j ,2k  ∂p ∂p ∂σ   = −  + + gy ρ  ∂y ∂σ ∂y * i , j , k ∆t win, +j 1,k − win, +j ,2k ∆t (3.42) n +1  ∂p ∂σ  =−  + gz  ρ  ∂σ ∂z *  i , j ,k (3.43) and the continuity equation (3.17) is discretized as follows: n +1  ∂u ∂u ∂σ ∂v ∂v ∂σ ∂w ∂σ   +  + + + =0 * * *   ∂x ∂σ ∂x ∂y ∂σ ∂y ∂σ ∂z i , j , k Performing the following operation: 35 (3.44) Chapter3 ∂ (3.41) ∂ (3.41) ∂σ ∂ (3.42) ∂ (3.42) ∂σ ∂ (3.43) ∂σ + + + + , and substituting (3.44) into ∂x ∂σ ∂x* ∂y ∂σ ∂y * ∂σ ∂z * the manipulation, we obtain the modified Poisson pressure equation as follows,  ∂  ∂p  ∂  ∂p   ∂σ   ∂σ   ∂σ   ∂  ∂p  +   +  *  +  *  +  *       ∂x  ρ ∂x  ∂y  ρ ∂y   ∂x   ∂y   ∂z   ∂σ  ρ ∂σ ∂σ  ∂  ∂p  ∂  ∂p  ∂σ  ∂  ∂p  ∂  ∂p  +   + +   + *     ∂x  ∂x  ρ ∂σ  ∂σ  ρ ∂x  ∂y *  ∂y  ρ ∂σ  ∂σ  ρ ∂y  n +1  ∂σ ∂ σ  ∂p   +  * + *   ∂x ∂x ∂y ∂y  ρ ∂σ  i , j , k 2    (3.45) n+2 /  ∂u ∂u ∂σ ∂v ∂v ∂σ ∂w ∂σ   =  + + + + ∆t  ∂x ∂σ ∂x * ∂y ∂σ ∂y * ∂σ ∂z *  i , j ,k For individual term in LHS of Equation (3.45), we wrote the finite difference form with respect to density variation For simplicity again, only the term in x-direction is listed below, m  ∂  ∂p    ∂p  =       ∂x  ρ ∂x   i , j ,k ∆x  ρ ∂x i +1 2, j ,k =   ∂p  −    ρ ∂x i −1 2, j ,k  pi +1, j ,k − pi , j ,k pi , j ,k − pi −1, j ,k   1 −   ρi −1 2, j ,k ∆x  ρi +1 2, j , k ∆xi ∆xi −1  m m (3.46) For impressible fluids, the fluid density is a function of x, y, z, t, during the derivation The Successive-Over-Relaxation (SOR) method is used to solve the above equation for pin, +j ,1k , provided that the proper pressure boundary conditions are given,   (PS + PC ) − DU pim, j+,1k = pim, j , k + Ω  − pim, j , k  , 1/ ∆   36 (3.47) Chapter3 where Ω is the convergence factor which lies between 1.0 and 2.0 and m is the iteration number PS represents the terms resulting from the second-order derivatives in equation (3-45) and PC from the cross differential and other terms, pi +1, j ,k pi −1, j ,k   1 PS = +   ρi −1 2, j ,k ∆xi −1  ∆x  ρi +1 2, j ,k ∆xi m pi , j +1,k pi , j −1,k   1 + +   ρi , j −1 2,k ∆y j −1  ∆y  ρi , j +1 2,k ∆y j + ∆σ PC = m (3.48)  pi , j , k +1 pi , j ,k −1   ∂σ   ∂σ   ∂σ 2  + +     + ρi , j ,k −1 ∆σ k −1   ∂x*   ∂y*   ∂z *    ρi , j ,k +1 ∆σ k 1 ∂σ * ∂x ∆xi −1 + ∆xi ∆σ k −1 + ∆σ k m   1  − +  p i +1, j ,k +1   ρ ρ  i j k i j k + , , , , +      1    pi +1, j ,k −1  + − p i −1, j ,k +1  + + p i −1, j ,k −1  + ρ  ρ  ρ   i +1, j ,k ρ i , j ,k −1   i −1, j ,k ρ i , j ,k +1   i −1, j ,k ρ i , j ,k −1  + m   1 1  ∂σ  p − +  i , j + , k + ρ  ∂y * ∆y j −1 + ∆y j ∆σ k −1 + ∆σ k   i , j +1, k ρ i , j , k +1     1    pi , j +1,k −1  + − p i , j −1,k +1  + + p i , j −1,k −1  + ρ  ρ  ρ   i , j +1,k ρ i , j ,k −1   i , j −1,k ρ i , j ,k +1   i , j −1,k ρ i , j ,k −1   ∂ 2σ  ∆σ k −1 ∆σ k ∂ 2σ     * + *  + p i , j ,k +1 − p i , j ,k −1   ρ i , j ,k +1 ∆σ k ρ i , j ,k −1 ∆σ k −1  ∂x ∂x ∂y ∂y  ∆σ k −1 + ∆σ k  (3.49) where ∆x = ∆xi −1 + ∆xi ∆y = ∆y j −1 + ∆y j 37 ∆σ = ∆σ k −1 + ∆σ k (3.50) Chapter3 DU represents the divergence of the intermediate velocity at n+2/3 time step, which is the right hand side of equation (3.45) The central difference method is used to calculate n+2 /  ∂u  is shown below and the other terms can be DU For simplicity, only the term    ∂x  i , j ,k discretized similarly, n+2 / u u − ui , j , k − ui −1, j , k   ∂u   ∆xi −1 i +1, j , k = + ∆xi i , j , k   ∆xi −1 + ∆xi  ∆xi ∆xi −1  ∂x i , j , k    n+2 / (3.51) The value of / ∆2 is calculated as,      1 1 +   + + 1/∆2 =       x x x y y y ∆ ∆ ∆ ∆ ∆ ∆ ρ ρ ρ ρ   i +1 2, j , k i i −1 , j , k i −1  i , j −1 , k j −1   i , j +1 , k j (3.52)   ∂σ   ∂σ   ∂σ     1    + + +    + ∆σ  ρi , j , k +1 ∆σ k ρ i , j , k −1 ∆σ k −1   ∂x*   ∂y *   ∂z *       ∂ 2σ  ∆σ k −1 ∆σ k ∂ 2σ    + + − + * *    ρ ρ i , j ,k −1 ∆σ k −1  ∂x ∂x ∂y ∂y  ∆σ k −1 + ∆σ k i , j , k +1 ∆σ k  Equation (3.45) is solved by iteration with the initial guess of pressure being the previous time step result The convergence criterion is chosen to be that at any computational node, the difference of the calculated pressure between two immediate iterations is smaller than a small number δ times the pressure itself, i.e., p m +1 − p m ≤ δ Normally, pm the smaller the value of δ is chosen, the more accurate the numerical results are but the computation becomes more expensive The numerical tests show that for most case, when δ = 10 −4 , the numerical solution converges The further reduction of δ will make little 38 Chapter3 difference of numerical solution Therefore, in the following computation, unless otherwise stated, the default value of δ is 10-4 The optimal value of convergence factor Ω in general depends on the choice of δ When δ = 10 −4 , the optimal Ω is found to be 1.4 3.3.4 Velocity-correction Step For updating the velocity field after the calculation of Poisson equation, rewrite the velocity scheme from Equation (3.41) in weighting formula in terms of density variation For simplicity, only horizontal x-direction velocity is listed uin, +j 1,k − uin, +j ,2k ∆t n +1  ∂p ∂p ∂σ  =−  + + gx ρ  ∂x ∂σ ∂x* i , j ,k  ( pi+1, j,k − pi, j ,k ) ∆xi−1 + ( pi, j ,k − pi−1, j ,k ) ∆xi    ∆xi −1 + ∆xi  ρi +1 2, j ,k ∆xi ∆xi −1 ρi −1 2, j , k    ( pi, j ,k +1 − pi, j ,k ) ∆σ k −1 + ( pi, j ,k − pi, j ,k −1 ) ∆σ k  − ∆σ k −1 + ∆σ k  ρi , j , k +1 ∆σ k ρi , j ,k −1 ∆σ k −1  ∂σ + gx ∂x* =− 3.3.5 (3.53)     Density Tracking Step Following velocity-correction step, density needs to be updated owing to the density variation For incompressible flows, we need to solve equation (3.21) in σ-coordinate The similar split operator method is used to treat the advection and diffusion terms in equation (3.21) separately The advection terms in RHS of Equation (3.21) are discretized by the central difference scheme The finite difference form for the advection terms in equation (3.21) can be represented as, 39 Chapter3 ρin, +j ,1k − ρin, j ,k ∆t n  ∂ρ ∂ρ ∂ρ  + u +v +ω =0  ∂y ∂σ i , j ,k  ∂x (3.54) The advection terms in equation (3.54) should be adopted by a high-order numerical scheme in order to maintain high accuracy and numerical stability The CIP method (Cubic-Interpolated propagation) as introduced by Yabe and Aoki (1991) is a general hyperbolic solver It has been shown that the CIP method is able to solve hyperbolic equation like equation (3.54) with small numerical diffusion and dispersion, when it is used to capture front motion In this study, we applied CIP method in stratified flows with sharp density interfaces Compared with other schemes e.g Upwind, Lax-Wendroff and backward quadratic interpolation in the same testing case, CIP method provides the best agreement with analytical profile Watanabe & Saeki (1999, 2002) adopted this method for the simulation of breaking waves, and proved the validity of numerical results from comparisons with analytical and experimental results In the present model, the density transport equation (3.21) is updated by the CIP method as a numerical solver for the hyperbolic equation Let us therefore outline this CIP method, to numerically solve the general advection equation (3.55) as follow, Df ∂f ∂f ∂f ∂f = +u +v + w = Dt ∂t ∂x ∂y ∂z (3.55) In the three-dimensional CIP method, the function f and its space derivatives at an arbitrary point ( x + ∆x, y + ∆y, z + ∆z ) in a rectangular cell are interpolated by a cubicpolynomial and its space derivatives in x-, y-, z-directions, 40 Chapter3 f ( x + ∆x, y + ∆y, z + ∆z ) = ( a1∆x + a2 ∆y + a3∆z + a4 ) ∆x + a5 ∆y + ∂ x f i , j ,k  ∆x + ( a6 ∆y + a7 ∆z + a8 ∆x + a9 ) ∆y + a10 ∆z + ∂ y f i , j ,k  ∆y (3.56) + ( a11∆z + a12 ∆x + a13∆y + a14 ) ∆z + a15 ∆x + ∂ z f i , j ,k  ∆z + a16 ∆x∆y∆z + f i , j ,k ∂f ( x + ∆x, y + ∆y, z + ∆z ) = 3a1∆x + 2a2 ∆x∆y + 2a3∆x∆z + 2a4 ∆x + a5 ∆y ∂x + a6 ∆z + a8 ∆y + a13∆z + a16 ∆y∆z + ∂ x f i , j ,k ∂f ( x + ∆x, y + ∆y, z + ∆z ) ∂y = a2 ∆x + a5 ∆x + 3a7 ∆y + 2a8 ∆x∆y + 2a9 ∆y∆z (3.57) (3.58) + 2a10 ∆y + a11∆z + a14 ∆z + a16 ∆x∆z + ∂ y f i , j ,k ∂f ( x + ∆x, y + ∆y, z + ∆z ) = a3∆x + a6 ∆x + a9 ∆y + a11∆y + 3a12 ∆z (3.59) ∂z + 2a13∆x∆y + 2a14 ∆y∆z + 2a15 ∆z + a16 ∆x∆y + ∂ z f i , j ,k where ∆x , ∆y and ∆z are local coordinates where the origin is set at the grid coordinate x , y and z Here, the remaining coefficients a1 − a16 can be determined form the continuity of f , ∂ x f , ∂ y f and ∂ z f at grid points ( i, j, k + 1) , and that of f at ( i + 1, j + 1, k ) , ( i + 1, j, k ) , ( i, j + 1, k ) ( i, j + 1, k + 1) , ( i + 1, j , k + 1) and and ( i + 1, j + 1, k + 1) Thus, equation (3.56) means that f ( x + ∆x, y + ∆y, z + ∆z ) is advected to the coordinate ( x, y, z ) by the velocities u ( = − ∆x ∆t ) , v ( = − ∆y ∆t ) and w ( = − ∆z ∆t ) during a small time interval ∆t , so provided ∆t is sufficiently small and the flow field is locally steady, at the next time step f can be updated by setting ∆x = −u ∆t , ∆y = −v ∆t and 41 Chapter3 ∆z = − w∆t Since the interpolation by the spatial derivative of f is worked as a restriction condition on the gradient of f , the sharp profile of f is retained through the advection phase 3.3.6 Free Surface Tracking Step Finally, the free surface displacement is updated by solving Equation (3.29) The finite difference form is as follows, ηin, +j ,1k − ηin, j ,k ∆t =− ∆xi −1 + ∆xi FX i +1, j ,k − FX i , j ,k FX i , j ,k − FX i −1, j ,k   + ∆xi  ∆xi −1  ∆xi ∆xi −1   − ∆y j −1 + ∆y j  FYi , j +1,k − FYi , j ,k FY − FYi , j −1,k + ∆y j i , j ,k  ∆y j −1 ∆y j ∆y j −1  n    n (3.60) where FX and FY are the momentum fluxes in the x and y-directions, which can be calculated as,  k max ∂η n  n FX i , j , k = ( h + η )i , j ,k  ∑ uin, +j ,2k +31 ∆σ k + β ∆tg z  ∂x   k =1 FYi , j , k = ( h + η )i , j ,k n  k max n + ∂η n  v tg σ β ∆ + ∆ z  ∑ i , j ,k +1 k  ∂y   k =1 (3.61) where the second term in the bracket of Equation (3.61) is the artificial diffusion term that will not change the leading order solution of Equation (3.60) but can be used to control spurious wiggles due to numerical dispersions (Lin and Li, 2002) This treatment is similar to the Lax-Wendroff method for nonlinear advection equations The coefficient β is the weighting coefficient that lies between and When β = , the scheme is normally too dissipative; when β = , instability could occur locally The numerical tests 42 Chapter3 show that the best results are achieved when β ≈ / In this study, β = 0.65 is used unless otherwise mentioned 3.4 Boundary Conditions in σ-Coordinate Boundary conditions are applied at the end of each computational step discussed above In the section, the details of how to apply boundary conditions are provided Certain approximations are made to simplify the computation All the Dirichlet type of boundary conditions that give the values of the variables will remain the same However, for the Neumann type of boundary condition that involves partial differentiation, modifications are needed in the transformed plane For example, in the σ-coordinate, the boundary condition for pressure on bottom of equation (3.6) is changed to, ∂p = ρDg z ∂σ (3.62) The pressure gradient at the inflow and radiation boundaries and on the vertical wall should also be modified so that ∂p ∂p ∂p ∂σ ∂p ∂p ∂p ∂σ = + = + and ∂y * ∂y ∂σ ∂y * ∂x * ∂x ∂σ ∂x * On bottom: The velocities are all zero based on the no-slip boundary condition However, it is noted that if the potential flow is simulated, not only the kinematic viscosity should be set as zero, but also the no-slip boundary condition should be changed to free-slip boundary condition for consistency When the free-slip boundary condition is applied, the normal gradient of tangential velocity and the normal velocity itself are required to be zero, i.e., ∂ut ∂n = and un = In most cases when the bottom slope is mild, we 43 Chapter3 replace ut with u and v and un with w The boundary condition for pressure in equation (3.62) can be applied directly in the pressure-correction step when the modified Poisson equation is solved On free surface: The zero pressure condition on the free surface can be easily applied when the Poisson equation is solved The continuity of stress condition imposes the zero shear and normal stresses on the free surface when the wind effect is absent To simplify the computation, the mild slope of free surface is further assumed, which is consistent with the premise of non-breaking wave The original stress boundary condition on the free surface (3.8) can then by simplified as follows during the computation, ∂u =0 ∂σ ∂v =0 ∂σ ∂w = ∂σ (3.63) The second-order three-point finite difference form is used to discretize the above equation Here only the formula for u is presented, u i , j ,l +1 = (∆σ l + ∆σ l −1 )2 ∆σ l −1 (2∆σ l + ∆σ l −1 ) u i , j ,l − ∆σ l u i , j ,l −1 ∆σ l −1 (2∆σ l + ∆σ l −1 ) (3.64) The pressure boundary condition is worth more discussion Although the Neumann type of boundary condition for pressure can be derived directly from the governing equation, i.e., (3.10), the application of such condition is complicated In this study, the simpler condition for pressure is used in the actual calculation with the assumption that the vertical acceleration of fluids at these boundaries is negligibly small, ∂p ∂p ∂σ ∂η + = − ρg z * ∂x ∂σ ∂x ∂x ∂p ∂p ∂σ ∂η + = − ρg z * ∂y ∂σ ∂y ∂y 44 (3.65) Chapter3 The backward difference which used the boundary node and its adjacent interior node is used to discretize the spatial differentiation On vertical wall: The model can also treat the interior surface-piercing structure with vertical walls On the vertical wall, the similar condition as that on bottom could be used for velocity For free surface displacement and pressure, the zero normal gradient should be applied in principle However, since the variable is defined at the node that is placed right on the solid surface, the accurate application of such boundary condition numerically is difficult Alternatively, no explicit boundary condition is applied for η At the solid surface, the value of η is calculated using the backward difference method by realizing that the momentum flux on the wall equals zero For the pressure, the same boundary condition as for inflow and radiation boundaries (3.65) is used, assuming that the pressure difference on the wall between two nodes are mainly caused by the difference of free surface displacement at that location 45 Chapter3 3.5 Stability Criterion There are two stability criteria that have to be satisfied to make the scheme stable One is related to the advection process that is characterized by the Courant number restriction, CrI = U I ∆t ≤α , ∆x I (3.66) where I=1,2,3 and UI is the bigger value of maximum particle velocity and wave celerity (when in x-y plane) The value of α is often taken as 0.2 ~ 0.3 to ensure the stability everywhere in the computational domain Another stability restriction is related to the diffusion process Based on the stability analysis, the following condition should be satisfied, ν∆t ∆x I ≤χ, (3.67) where again I=1,2,3 and χ is normally taken as 1/6 in the computation 46 ... flows Although two -dimensional models are Chapter computationally efficient and their results are generally reasonable, most practical applications generally require a full three- dimensional analysis... Reynolds Averaged Navier-Stokes Equations The time averaged form of the Navier-Stokes equations are called the Reynolds Averaged 12 Chapter Navier-Stokes (RANS) equations These models calculate a mean,... turbulent part can be very large and of the same order as the mean Examples are unsteady flow in general, wake flows or flows with large separation For this type of flows, it is more appropriate to