平面流动高精度算法的精度和瞬态模拟特性研究
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摘要
本文利用高精度分块耦合求解方法，对其瞬态时间精度和非定常时间发展解的跟随性问题进行了研究，同时也对包含边界条件的线法高精度格式的稳定性也作了分析，研究结果表明高精度的分块耦合求解方法可以很好地跟随Taylor问题的时间发展解，与单圆柱绕流和双圆柱绕流的实验结果比较以及他人结果比较符合很好，对升阻力系数在涡脱落时的脉动问题的结果优于他人结果。高精度格式的分块耦合求解方法的时间跟随性问题是受国内外关注的问题，通过本论文研究证实分块耦合求解方法完全可用于求解瞬态流动问题。
另外，本文对压力Poisson方程的差分求解方法做了较多的研究，探讨在Runge-Kutta法中如何实施压力解法，说明对Runge-Kutta法各步全都进行压力求解反而不及单步压力Poisson方程计算。
The paper researches the transient time precision and the following problem of the results developing with time for 2-D high order blocking and matched method, at the same time, anlysing the stability of high order methods including boundary conditions. The results show that the domain decomposition and matched method using high order method can follow the time developing solution of Taylor problem well, the computation result of the flow over a single or two tandem arranging circular cylinder agree the experiment and others' results well, moreover, the panting results of lift and drag coefficient are better than others. The time following problem is worthy of being noticed and the paper prove that DDM can be used to solve transient flow problem.
On the other hand, the paper researches the solution of Poisson equation and studies how to solve the equation in the Runge-Kutta method. The result show it is better when solve the Poisson equation at the third step in Runge-Kutta method.
另外，本文对压力Poisson方程的差分求解方法做了较多的研究，探讨在Runge-Kutta法中如何实施压力解法，说明对Runge-Kutta法各步全都进行压力求解反而不及单步压力Poisson方程计算。
The paper researches the transient time precision and the following problem of the results developing with time for 2-D high order blocking and matched method, at the same time, anlysing the stability of high order methods including boundary conditions. The results show that the domain decomposition and matched method using high order method can follow the time developing solution of Taylor problem well, the computation result of the flow over a single or two tandem arranging circular cylinder agree the experiment and others' results well, moreover, the panting results of lift and drag coefficient are better than others. The time following problem is worthy of being noticed and the paper prove that DDM can be used to solve transient flow problem.
On the other hand, the paper researches the solution of Poisson equation and studies how to solve the equation in the Runge-Kutta method. The result show it is better when solve the Poisson equation at the third step in Runge-Kutta method.
引文
[1] 李光正 绕圆柱非定常周期性涡旋脱落的数值模拟 水动力学研究与进展A辑，2000，15(4)：493-504．
[2] A Borthwick Comparison between two finite-difference schemes for computing the flow around a cylinder. International Journal for Numerical Methods in Fluids, 1986, 6: 275-290.
[3] Keun-Shik Chang and Jong-Youb Sa The effect of buoyancy on vortex shedding in the near wake of a circular cylinder. J. Fluid Mech., 1990, 220:253-266
[4] Tritton D J Ecperiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech., 1959, 6, 547.
[5] Sadatoshi Teneda Experimental investigation of the wakes behind cylinders and Plates at low Reynolds numbers. J. Phys. Soc. Jpn., vol. 11, 1956,3:302-307
[6] Hiroyuli Honji and Sadatoshi Teneda Unsteady flow past a circular cylinder. J. Phys. Soc. Jpn., vol. 27, 1969,6: 1668-1677.
[7] Sadatoshi Teneda Visualization of separating stokes flows. J. Phys. Soc. Jpn., vol. 46,1979,6:1935-1942.
[8] Sadatoshi Teneda Experimental investigation of the wake behind a sphere at low Reynolds numbers. J. Phys. Soc. Jpn., vol. 46, 1979,6:1935-1942.
[9] 马延文、傅德薰 气动计算中德紧致格式与迎风紧致格式 计算数学 第二期，1992．5
[10] Ming Li and Tao Tang A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations. Int. J. Numer. Mthods fluids, 1995, 20:1137-1151
[11] Murli M. Gupta High accuracu solutions of incompressible Navier-Stokes equations. J. Comp. Phys. 1991,93:343-359
[12] 刘宏、傅德薰、马延文 迎风紧致格式与驱动方腔问题德直接数值模拟 中国科学A辑第23卷 第六期，1993．6
[13] S. C. R. Dennis and J. D. Hudson Compact four-order finite-difference approximations to operators of Navier-Stokes type. J. Comp. Phys. 1989, 85:190-416
[14] Weinam E. and Jian-Guo Liu Essentially compact schemes for unsteady viscous incompressible flows. J. Comp. Phys. 1996, 126:122-138
[15] 任安禄、修东滨 二维垂直平板起动流动和漩涡脱落过程德数值模拟95年全国水动力学研讨会论文集
[16] 修东滨、任安禄 高阶紧致格式求解二维粘性不可压缩复杂流场 力学学报 第28卷第三期，1996．5
[17] 任安禄、丁宏等 核反应堆吊篮模型下腔流场的计算 现代力学与科技进步，北京，1997．8
[18] 鲁晓东 粘性不可压缩复杂流场高精度分块耦合求解方法研究 浙江大学硕士学位论文
[19] Carpenter, M.H., Gottlieb, D. and Abarbanel, S. The stability of numerical boundary treatments for compact high-order finite-difference schemes, Journal of omputational Physics,1993, 108:272-295
[20] 任安禄 计算流体力学 浙江大学力学教材 1995．9
[21] Fady M. Najjar and S. P. Vanka Simulations of the unsteady separated flow past a normal flat plate. Int. J. Numer. Methods fluids, 1995, 21:525-547
[22] F. Sotiropoulos and S. Abdallah The discrete continuity equation in primitive variable solutions of incompressible flow. J. Comp. Phys. 1991, 95:212-227
[23] F. Sotiropoulos and S. Abdallah A primitive variable method for the solution of three-dimension incompressible viscous flows. J. Comp. Phys. 1992, 103:336-349
[24] S. Abdallah Numerical solutions for the pressure Poisson equation with Neumann boundary condition using a nonstaggerod grid Ⅰ. J. Comp. Phys. 1987, 20:182-192
[25] S. Abdallah Numerical solutions for the pressure Poisson equation with Numann boundary condition using a nonstaggered grid Ⅱ. J. Comp. Phys. 1987, 70:193-202
[26] Philip M. Gresho and Robert L. Sani On pressure boundary conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods fluids. 1987, 7:1111-1145
[27] A. Segal et al Invariant discretization of the incompressible Navier-Stokes equations in boundary fitted co-ordinates. Int. J. Numer. Methods fluids, 1991, 15:411-426
[28] Alexander G. Churbanov, A. N. Pavlov and P. N. Vabishchevich Operator-splitting methods for the incompressible Navier-Stokes equations on non-staggered grids. Part 1: First-order schemes. Int. J. Numer. Methods Fluids, 1995, 21:617-640
[29] K. M. Smith, W. K. Cope and S. P. Vanka A multigrid procedure for three-dimensional flows on non-orthogonal collocated grids. Int. J. Numer. Methods fluids, 1993, 17:887-904
[30] 朱自强等 应用计算流体力学 北京航空航天大学出版社 1998
[31] 任安禄 郁春伟 卢云 粘性不可压缩流动三维复杂流场分块耦合求解 空气动力学学报，v15(3)，1997．9
[32] 卢云 任安禄 二维粘性不可压缩流场的分块耦合求解方法 空气动力学学报，v12(4)，1994．12
[33] Ren Anlu and DingHong High accurate solution of incompressible viscous flow. The 7th International Symposium on Computational Fluid Dynamics, Sept. 15～19 1997 Beijing China
[34] 任安禄 丁宏 陈耀松 用高阶紧致格式及压力修正法求解粘性不可压缩流场 第11届全国水动力学研讨会暨第4届全国水动力学会议 1997：111-116
[35] 张涵信 无波动、无自由参数的耗散差分格式 空气动力学学报 V．6(2)，1988
[36] 付德薰、马延文 计算空气动力学等等 宇航出版社1994．11
[37] D.X. Fu & Y.W. Ma A High Order Accurate Difference Scheme for Complex Flow Fields J.C.P 134 1997
[38] G.Q. Chen and Z. Gao, Z.F. Yang A Perturbation h~4 Exponential Finite Difference Scheme for the Convective Diffusion Equation. J.C.P.V. 104, 1993
[39] XiaoGang Deng & Hiroshi Maekawa Compact High-Order Accurate Nonlinear Scheme. J. C.P. V. 130, 1997
[40] 邓小刚、毛枚良 高精度耗散紧致格式的边界格式和渐近稳定性分析第六届全国计算流体力学会议
[41] Lele SK. Compact finite difference schemes with spectral-like resolution. J Comp Phys, 1992,103:16～42
[42] Jameson A and Schmidt W. Some recent development in numerical methods for transonic flows. Comp. Methods In App. Mech and Engr. 51(1985) or AIAA paper 81-1259 198
[43] 任安禄，林雪纲，鲁晓东 模型方程线法渐近稳定性分析及非线性影响的思考 第十届全国计算流体力学会议
[44] 任安禄，林雪纲，鲁晓东 模型方程线性稳定性分析和非线性影响的思考 第四届亚洲计算流体力学会议
[45] 任安禄，林雪纲，鲁晓东 The Research for Numerical Stability Analysis of Non-Linear
[46] Dan Givoli Non-reflecting boundary condition. J. Comp. Phys. 1991, 95:212-227
[47] 卢云 复杂流场的分块耦合求解方法 浙江大学硕士学位论文1994．1
[48] S.Abdallah and J.Dreyer Dirichlet and Nuemann boundary conditions for the pressure Poisson equation of incompressible flow. Int. J. Numer. Methods Fluids, 1988, 8:1029-1036
[49] 吕涛等 区域分解算法—偏微分方程数值解新技术科学出版社1992．5第一版
[50] J. F. Thompson, Z. U. A. Warsi and C. W. Mastin Numerical grid generation. Foundation and applications. North-Holland. 1985
[51] 赵峰 周连第 任安禄 通用三维分块帖体网格的耦合生成方法 水动力学研究与进展A辑 第9卷第4期，1994年8月
[52] Kazuyoshi Miki and Toshiyuki Takagi A domain decomposition and overlapping method for the generation of three-dimensional boundary-fitted coordinate systems. J. Comp. Phys. 1984, 53:319-330
[53] Pollard A. and Spalding D. B. The prediction of the three-dimensional turbulent flow field in a flow-splitting tee-junction. Computer Meth. Appl. Mech. Engng 1978, 13:293
[54] Zdravkovich M M. Review of flow interference between two circular culinder in various arrangements[J], ASME Journal of Fluids Engineering, 1977, 99:618-633
[55] Slaoutiand A, Stansby P K. Flow around two circular cylinder by the random-vortex method[J], Journal of Fluids and Structures, 1992,6:641-670
[56] Stansby P K. A numerical study of vortex shedding from one and two circular cylinders[J], Aeronautical Quarterly. 1981, 32:48-71
[57] 刘松 符松 串列双圆柱绕流问题的数值模拟 计算力学学报 第17卷第3期2000．8
[58] 邵立 硕士学位论文 2000年1月
[59] M. B Braza, P. Chassaing, H. Ha Minh Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech, 1986,165:79-130
[2] A Borthwick Comparison between two finite-difference schemes for computing the flow around a cylinder. International Journal for Numerical Methods in Fluids, 1986, 6: 275-290.
[3] Keun-Shik Chang and Jong-Youb Sa The effect of buoyancy on vortex shedding in the near wake of a circular cylinder. J. Fluid Mech., 1990, 220:253-266
[4] Tritton D J Ecperiments on the flow past a circular cylinder at low Reynolds numbers. J. Fluid Mech., 1959, 6, 547.
[5] Sadatoshi Teneda Experimental investigation of the wakes behind cylinders and Plates at low Reynolds numbers. J. Phys. Soc. Jpn., vol. 11, 1956,3:302-307
[6] Hiroyuli Honji and Sadatoshi Teneda Unsteady flow past a circular cylinder. J. Phys. Soc. Jpn., vol. 27, 1969,6: 1668-1677.
[7] Sadatoshi Teneda Visualization of separating stokes flows. J. Phys. Soc. Jpn., vol. 46,1979,6:1935-1942.
[8] Sadatoshi Teneda Experimental investigation of the wake behind a sphere at low Reynolds numbers. J. Phys. Soc. Jpn., vol. 46, 1979,6:1935-1942.
[9] 马延文、傅德薰 气动计算中德紧致格式与迎风紧致格式 计算数学 第二期，1992．5
[10] Ming Li and Tao Tang A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations. Int. J. Numer. Mthods fluids, 1995, 20:1137-1151
[11] Murli M. Gupta High accuracu solutions of incompressible Navier-Stokes equations. J. Comp. Phys. 1991,93:343-359
[12] 刘宏、傅德薰、马延文 迎风紧致格式与驱动方腔问题德直接数值模拟 中国科学A辑第23卷 第六期，1993．6
[13] S. C. R. Dennis and J. D. Hudson Compact four-order finite-difference approximations to operators of Navier-Stokes type. J. Comp. Phys. 1989, 85:190-416
[14] Weinam E. and Jian-Guo Liu Essentially compact schemes for unsteady viscous incompressible flows. J. Comp. Phys. 1996, 126:122-138
[15] 任安禄、修东滨 二维垂直平板起动流动和漩涡脱落过程德数值模拟95年全国水动力学研讨会论文集
[16] 修东滨、任安禄 高阶紧致格式求解二维粘性不可压缩复杂流场 力学学报 第28卷第三期，1996．5
[17] 任安禄、丁宏等 核反应堆吊篮模型下腔流场的计算 现代力学与科技进步，北京，1997．8
[18] 鲁晓东 粘性不可压缩复杂流场高精度分块耦合求解方法研究 浙江大学硕士学位论文
[19] Carpenter, M.H., Gottlieb, D. and Abarbanel, S. The stability of numerical boundary treatments for compact high-order finite-difference schemes, Journal of omputational Physics,1993, 108:272-295
[20] 任安禄 计算流体力学 浙江大学力学教材 1995．9
[21] Fady M. Najjar and S. P. Vanka Simulations of the unsteady separated flow past a normal flat plate. Int. J. Numer. Methods fluids, 1995, 21:525-547
[22] F. Sotiropoulos and S. Abdallah The discrete continuity equation in primitive variable solutions of incompressible flow. J. Comp. Phys. 1991, 95:212-227
[23] F. Sotiropoulos and S. Abdallah A primitive variable method for the solution of three-dimension incompressible viscous flows. J. Comp. Phys. 1992, 103:336-349
[24] S. Abdallah Numerical solutions for the pressure Poisson equation with Neumann boundary condition using a nonstaggerod grid Ⅰ. J. Comp. Phys. 1987, 20:182-192
[25] S. Abdallah Numerical solutions for the pressure Poisson equation with Numann boundary condition using a nonstaggered grid Ⅱ. J. Comp. Phys. 1987, 70:193-202
[26] Philip M. Gresho and Robert L. Sani On pressure boundary conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods fluids. 1987, 7:1111-1145
[27] A. Segal et al Invariant discretization of the incompressible Navier-Stokes equations in boundary fitted co-ordinates. Int. J. Numer. Methods fluids, 1991, 15:411-426
[28] Alexander G. Churbanov, A. N. Pavlov and P. N. Vabishchevich Operator-splitting methods for the incompressible Navier-Stokes equations on non-staggered grids. Part 1: First-order schemes. Int. J. Numer. Methods Fluids, 1995, 21:617-640
[29] K. M. Smith, W. K. Cope and S. P. Vanka A multigrid procedure for three-dimensional flows on non-orthogonal collocated grids. Int. J. Numer. Methods fluids, 1993, 17:887-904
[30] 朱自强等 应用计算流体力学 北京航空航天大学出版社 1998
[31] 任安禄 郁春伟 卢云 粘性不可压缩流动三维复杂流场分块耦合求解 空气动力学学报，v15(3)，1997．9
[32] 卢云 任安禄 二维粘性不可压缩流场的分块耦合求解方法 空气动力学学报，v12(4)，1994．12
[33] Ren Anlu and DingHong High accurate solution of incompressible viscous flow. The 7th International Symposium on Computational Fluid Dynamics, Sept. 15～19 1997 Beijing China
[34] 任安禄 丁宏 陈耀松 用高阶紧致格式及压力修正法求解粘性不可压缩流场 第11届全国水动力学研讨会暨第4届全国水动力学会议 1997：111-116
[35] 张涵信 无波动、无自由参数的耗散差分格式 空气动力学学报 V．6(2)，1988
[36] 付德薰、马延文 计算空气动力学等等 宇航出版社1994．11
[37] D.X. Fu & Y.W. Ma A High Order Accurate Difference Scheme for Complex Flow Fields J.C.P 134 1997
[38] G.Q. Chen and Z. Gao, Z.F. Yang A Perturbation h~4 Exponential Finite Difference Scheme for the Convective Diffusion Equation. J.C.P.V. 104, 1993
[39] XiaoGang Deng & Hiroshi Maekawa Compact High-Order Accurate Nonlinear Scheme. J. C.P. V. 130, 1997
[40] 邓小刚、毛枚良 高精度耗散紧致格式的边界格式和渐近稳定性分析第六届全国计算流体力学会议
[41] Lele SK. Compact finite difference schemes with spectral-like resolution. J Comp Phys, 1992,103:16～42
[42] Jameson A and Schmidt W. Some recent development in numerical methods for transonic flows. Comp. Methods In App. Mech and Engr. 51(1985) or AIAA paper 81-1259 198
[43] 任安禄，林雪纲，鲁晓东 模型方程线法渐近稳定性分析及非线性影响的思考 第十届全国计算流体力学会议
[44] 任安禄，林雪纲，鲁晓东 模型方程线性稳定性分析和非线性影响的思考 第四届亚洲计算流体力学会议
[45] 任安禄，林雪纲，鲁晓东 The Research for Numerical Stability Analysis of Non-Linear
[46] Dan Givoli Non-reflecting boundary condition. J. Comp. Phys. 1991, 95:212-227
[47] 卢云 复杂流场的分块耦合求解方法 浙江大学硕士学位论文1994．1
[48] S.Abdallah and J.Dreyer Dirichlet and Nuemann boundary conditions for the pressure Poisson equation of incompressible flow. Int. J. Numer. Methods Fluids, 1988, 8:1029-1036
[49] 吕涛等 区域分解算法—偏微分方程数值解新技术科学出版社1992．5第一版
[50] J. F. Thompson, Z. U. A. Warsi and C. W. Mastin Numerical grid generation. Foundation and applications. North-Holland. 1985
[51] 赵峰 周连第 任安禄 通用三维分块帖体网格的耦合生成方法 水动力学研究与进展A辑 第9卷第4期，1994年8月
[52] Kazuyoshi Miki and Toshiyuki Takagi A domain decomposition and overlapping method for the generation of three-dimensional boundary-fitted coordinate systems. J. Comp. Phys. 1984, 53:319-330
[53] Pollard A. and Spalding D. B. The prediction of the three-dimensional turbulent flow field in a flow-splitting tee-junction. Computer Meth. Appl. Mech. Engng 1978, 13:293
[54] Zdravkovich M M. Review of flow interference between two circular culinder in various arrangements[J], ASME Journal of Fluids Engineering, 1977, 99:618-633
[55] Slaoutiand A, Stansby P K. Flow around two circular cylinder by the random-vortex method[J], Journal of Fluids and Structures, 1992,6:641-670
[56] Stansby P K. A numerical study of vortex shedding from one and two circular cylinders[J], Aeronautical Quarterly. 1981, 32:48-71
[57] 刘松 符松 串列双圆柱绕流问题的数值模拟 计算力学学报 第17卷第3期2000．8
[58] 邵立 硕士学位论文 2000年1月
[59] M. B Braza, P. Chassaing, H. Ha Minh Numerical study and physical analysis of the pressure and velocity fields in the near wake of a circular cylinder. J. Fluid Mech, 1986,165:79-130