一种能准确模拟波浪复杂演化过程的非静力学数值模型
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摘要
选取σ坐标系下的不可压缩Navier-Stokes方程组作为控制方程。将速度变量定义在计算单元的中心位置,同时将非静力学压力定义在计算单元竖直方向界面位置以便于结合Godunov型格式。采用有限体积和有限差分混合方法结合Godunov型格式对控制方程进行空间离散。利用HLL黎曼求解器求解计算单元之间的通量以实现激波捕捉,并采用二阶非线性强稳定性保持龙格库塔(SSP Runge-Kutta)格式进行时间步迭代。基于以上数值方法,文中发展了一种能准确模拟波浪复杂演化过程的非静力学数值模型。将非静力学数值模型用于数值模拟淹没式堤坝上波浪传播、孤立波沿斜坡爬高和海底滑坡诱发海啸三种波浪复杂演化过程,数值计算结果与相应的实验结果较为吻合。
The incompressible Navier-Stokes equations in conservative form written in σ coordinates are chosen as the governing equations. In order to apply a Godunov-type scheme,the velocities are defined at cell centers,and the dynamic pressure is defined at vertically facing cell faces. A combined finite-volume and finite-difference scheme with a Godunov-type method is applied to discretize the governing equations.The HLL Riemann approximation is employed to estimate fluxes at cell faces. The nonlinear Strong Stability-Preserving(SSP) Runge-Kutta scheme is adopted for time stepping. Based on above numerical methods,a non-hydrostatic model for precise simulation of complex surface wave processes is developed. The model is validated using three test cases based on experimental data,respectively periodic wave propagation over a submerged bar,solitary wave run-up along a slope beach and tsunami generation by underwater landslides. The numerical results are in good agreement with the corresponding experimental data.
The incompressible Navier-Stokes equations in conservative form written in σ coordinates are chosen as the governing equations. In order to apply a Godunov-type scheme,the velocities are defined at cell centers,and the dynamic pressure is defined at vertically facing cell faces. A combined finite-volume and finite-difference scheme with a Godunov-type method is applied to discretize the governing equations.The HLL Riemann approximation is employed to estimate fluxes at cell faces. The nonlinear Strong Stability-Preserving(SSP) Runge-Kutta scheme is adopted for time stepping. Based on above numerical methods,a non-hydrostatic model for precise simulation of complex surface wave processes is developed. The model is validated using three test cases based on experimental data,respectively periodic wave propagation over a submerged bar,solitary wave run-up along a slope beach and tsunami generation by underwater landslides. The numerical results are in good agreement with the corresponding experimental data.
引文
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[19] Zijlema M, Stelling G S. Efficient computation of surf zone waves using the nonlinear shallow water equations with nonhydrostatic pressure[J]. Coastal Eng., 2008, 55(10):780-790.
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[21]朱德军,陈永灿,刘昭伟,任华堂.激波捕捉格式求解复杂床面浅水方程的源项处理[J].清华大学学报(自然科学版),2007, 47(6):809-813.Zhu Dejun, Chen Yongchan, Liu Zhaowei, Ren Huatang. Treatment of source terms in shock-capturing schemes for shallow-water equations over complex bed topographies[J]. Journal of Tsinghua University(Science and Technology), 2007,47(6):809-813.
[22] Kazolea M, Delis A I. A well-balanced shock-capturing hybrid finite volume-finite difference numerical scheme for extended 1D Boussinesq models[J]. Applied Numerical Mathematics, 2013, 67:167-186.
[23]喻海军.城市洪涝数值模拟技术研究[D].广州:华南理工大学, 2015.Yu Haijun. Research on numerical simulation technology of urban floods[D]. Ph.D thesis, South China University of Technology, Guangzhou, 2015.
[24] Lin P, Li C W. Aσ-coordinate three-dimensional numerical model for free surface wave propagation[J]. Int. J Numer.Methods Fluids, 2002, 38:1045-1068.
[25] Gottlieb S, Shu C W, Tadmor E. Strong stability-preserving high-order time discretization methods[J]. SIAM Rev., 2001,43(1):89-112.
[26] Liang Q, Marche F. Numerical resolution of well-balanced shallow water equations with complex source terms[J]. Adv.Water Resour., 2009, 32(6):873-884.
[27] Harten A, Lax P, van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[J].SIAM Rev., 1983, 25(1):35-61.
[28] Beji S, Battjes J A. Experimental investigation of wave propagation over a bar[J]. Coastal Eng., 1993, 19(1-2):151-162.
[29] Larsen J, Dancy H. Open boundaries in short-wave simulations-a new approach[J]. Coastal Eng., 1983, 7(3):285-297.
[30] Synolakis C E. The runup of solitary waves[J]. J Fluid Mech., 1987, 185:523-545.
[31] Enet F, Grilli S T. Experimental study of tsunami generation by three dimensional rigid underwater landslides[J]. J Waterway Port Coastal Ocean Eng., 2007, 133(6):442-454.
[2] Wei G, Kirby J T, Grilli S T, Subramanya R. A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves[J]. J Fluid Mech., 1995, 294:71-92.
[3] Gobbi M F, Kirby J T, Wei G. A fully nonlinear Boussinesq model for surface waves. II. Extension to O(kh4)[J]. J Fluid Mech., 2000, 405:181-210.
[4] Lynett P J, Liu P L-F. A two-layer approach to wave modelling[J]. Proc. Roy. Soc. A, 2002, 460(2049):2637-2669.
[5] Kim D H, Lynett P J, Socolofsky S A. A depth-integrated model for weakly dispersive, turbulent, and rotational fluid flows[J]. Ocean Modell., 2009, 27(3-4):198-214.
[6] Kim D H, Lynett P J. Turbulent mixing and scalar transport in shallow and wavy flows[J]. Phys. Fluids, 2011, 23, 016603.
[7] Watanbe Y, Saeki H, Hosking R J. Three-dimensional vortex structures under breaking waves[J]. J Fluid Mech., 2005,545:291-328.
[8] Christensen E D. Large eddy simulation of spilling and plunging breakers[J]. Coastal Eng., 2006, 53(5-6):463-485.
[9] Shi F, Kirby J T, Ma G. Modeling quiescent phase transport of air bubbles induced by breaking waves[J]. Ocean Modell.,2010, 35(1-2):105-117.
[10] Namin M M, Lin B, Falconer R. An implicit numerical algorithm for solving non-hydrostatic free-surface flow problems[J]. Int. J Numer. Methods Fluids, 2001, 35(3):341-356.
[11] Casulli V, Zanolli P. Semi-implicit numerical modeling of non-hydrostatic free-surface flows for environmental problems[J]. Math. Comput. Model., 2002, 36(9-10):1131-1149.
[12] Chen Q, Kirby J T, Dalrymple R A, Shi F, Thornton E B. Boussinesq modeling of longshore currents[J]. J Geophys. Res.,2003, 108, doi:10.1029/2002JC001308.
[13] Lee J W, Terbner M D, Nixon J B, Gill P M. A 3-D non-hydrostatic pressure model for small amplitude free surface flows[J]. Int. J Numer. Methods Fluids, 2006, 50(6):649-672.
[14] Young C C, Wu C H, Liu W C, Kuo J T. A higher-order non-hydrostaticσmodel for simulating nonlinear refractiondiffraction of water waves[J]. Coastal Eng., 2009, 56(9):919-930.
[15] Wu C H, Young C C, Chen Q, Lynett P J. Efficient non-hydrostatic modeling of surface waves from deep to shallow water[J]. J Waterway Port Coastal Ocean Eng., 2010, 136(2):104-118.
[16] Stelling G S, Zijlema M. An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow with application to wave propagation[J]. Int. J Numer. Methods Fluids, 2003, 43:1-23.
[17] Yuan H, Wu C H. An implicit three-dimensional fully non-hydrostatic model for free-surface flows[J]. Int. J Numer.Methods Fluids, 2004, 46:709-733.
[18] Young C C, Wu C H. Aσ-coordinate non-hydrostatic model with embedded Boussinesq-type-like equations for modeling deep-water waves[J]. Int. J Numer. Methods Fluids, 2010, 63:1448-1470.
[19] Zijlema M, Stelling G S. Efficient computation of surf zone waves using the nonlinear shallow water equations with nonhydrostatic pressure[J]. Coastal Eng., 2008, 55(10):780-790.
[20] Alcrudo F, Garcia-Navarro P. A high-resolution Godunov-type scheme in finite volume for 2D shallow-water equations[J]. Int. J Numer. Methods Fluids, 1993, 16(6):489-505.
[21]朱德军,陈永灿,刘昭伟,任华堂.激波捕捉格式求解复杂床面浅水方程的源项处理[J].清华大学学报(自然科学版),2007, 47(6):809-813.Zhu Dejun, Chen Yongchan, Liu Zhaowei, Ren Huatang. Treatment of source terms in shock-capturing schemes for shallow-water equations over complex bed topographies[J]. Journal of Tsinghua University(Science and Technology), 2007,47(6):809-813.
[22] Kazolea M, Delis A I. A well-balanced shock-capturing hybrid finite volume-finite difference numerical scheme for extended 1D Boussinesq models[J]. Applied Numerical Mathematics, 2013, 67:167-186.
[23]喻海军.城市洪涝数值模拟技术研究[D].广州:华南理工大学, 2015.Yu Haijun. Research on numerical simulation technology of urban floods[D]. Ph.D thesis, South China University of Technology, Guangzhou, 2015.
[24] Lin P, Li C W. Aσ-coordinate three-dimensional numerical model for free surface wave propagation[J]. Int. J Numer.Methods Fluids, 2002, 38:1045-1068.
[25] Gottlieb S, Shu C W, Tadmor E. Strong stability-preserving high-order time discretization methods[J]. SIAM Rev., 2001,43(1):89-112.
[26] Liang Q, Marche F. Numerical resolution of well-balanced shallow water equations with complex source terms[J]. Adv.Water Resour., 2009, 32(6):873-884.
[27] Harten A, Lax P, van Leer B. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws[J].SIAM Rev., 1983, 25(1):35-61.
[28] Beji S, Battjes J A. Experimental investigation of wave propagation over a bar[J]. Coastal Eng., 1993, 19(1-2):151-162.
[29] Larsen J, Dancy H. Open boundaries in short-wave simulations-a new approach[J]. Coastal Eng., 1983, 7(3):285-297.
[30] Synolakis C E. The runup of solitary waves[J]. J Fluid Mech., 1987, 185:523-545.
[31] Enet F, Grilli S T. Experimental study of tsunami generation by three dimensional rigid underwater landslides[J]. J Waterway Port Coastal Ocean Eng., 2007, 133(6):442-454.