粘性流基于特征线的四阶Runge-Kutta有限元法
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摘要
对于二维不可压缩粘性流,通过沿流线方向的坐标变换,推导了无对流项的二维N-S(Navier-Stokes)方程。采用四阶Runge-Kutta法对N-S方程进行时间离散,并沿流线进行Taylor展开,得到显式的时间离散格式,然后利用Galerkin法对其进行空间离散,得到了高精度的有限元算法。利用本文算法对方腔驱动流和圆柱绕流进行了数值计算,通过对时间步长、网格尺寸和流场区域的计算分析,进一步验证了本文算法相比经典CBS法在时间步长、收敛性、耗散性和计算精度方面更具有优势。
For two-dimensional incompressible viscous flow,the two-dimensional Navier-Stokes(N-S)equation without convection term is derived by the coordinate transformation along the streamline direction.The explicit time discrete format is obtained via introducing the fourth order Runge-Kutta method and the Taylor expansion along the streamline direction,and then the space discretization format is carried out by the Galerkin method.Finally,a high precision finite element algorithm is obtained.This algorithm is applied to simulate flow in a cavity and flow around a circular cylinder.Through analyzing the effects of the different time step sizes,mesh sizes and flow field regions,the algorithm is further validated.Compared with the classical CBS method,it has more advantages in time step,characteristics of convergence and dissipation and accuracy.
For two-dimensional incompressible viscous flow,the two-dimensional Navier-Stokes(N-S)equation without convection term is derived by the coordinate transformation along the streamline direction.The explicit time discrete format is obtained via introducing the fourth order Runge-Kutta method and the Taylor expansion along the streamline direction,and then the space discretization format is carried out by the Galerkin method.Finally,a high precision finite element algorithm is obtained.This algorithm is applied to simulate flow in a cavity and flow around a circular cylinder.Through analyzing the effects of the different time step sizes,mesh sizes and flow field regions,the algorithm is further validated.Compared with the classical CBS method,it has more advantages in time step,characteristics of convergence and dissipation and accuracy.
引文
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[12]田俊武,袁湘江.三维复杂流动的间断有限元方法模拟[J].计算力学学报,2015,32(2):239-242.(TIAN Jun-wu,YUAN Xiang-jiang.Numerical simulation of three dimensional complex flows with RKDGmethod[J].Chinese Journal of Computational Mechanics,2015,32(2):239-242.(in Chinese))
[13]Erturk E,Corke T C,G9kc C.Numerical solutions of2-D steady incompressible driven cavity flow at high Reynolds numbers[J].International Journal for Numerical Methods in Fluids,2005,48:747-774.
[14]时忠民,刘名名,郭晓玲.计算域对圆柱绕流数值模拟结果的影响[J].中国水运,2013,13(7):83-87.(SHIZhong-min,LIU Ming-ming,GUO Xiao-ling.The influence of calculating domain on numerical simulation of flow around a cylinder[J].China Water Transport,2013,13(7):83-87.(in Chinese))
[15]水庆象,王大国.非定常不可压N-S方程的最小二乘算子分裂有限元数值求解[J].计算力学学报,2014,31(5):640-645.(SHUI Qing-xiang,WANG Da-guo.Numerical solution for unsteady incompressible N-Sequations by least-squares-based operator-splitting finite element method[J].Chinese Journal of Computational Mechanics,2014,31(5):640-645.(in Chinese))
[16]包艳.典型钝体结构流致效应分析与计算方法研究[D].上海交通大学,2011.(BAO Yan.The Research on Flow-Induced Responses of Fundamental Bluff Bodies and Its Numerical Sumulation Methods[D].Shanghai Jiao Tong University,2011.(in Chinese))
[2]Hughes T J R,Franca L P,Hulbert G M.A new finite element formulation for computational fluid dynamics,VIII:The Galerkin/least-squares method for advective-diffusive equations[J].Computer Methods in Applied Mechanics and Engineering,1989,73(2):173-189.
[3]O■nate E.A stabilized finite element method for incompressible viscous flows using a finite increment calculus formulation[J].Computer Methods in Applied Mechanics and Engineering,2000,182(314):355-370.
[4]Heinrich J C,Marshall R S,Zienkiewicz O C.Penalty function solution of coupled convective and conductive heat transfer[J].Numerical Methods in Laminar and Turbulent Flow,1978:935-946.
[5]Zienkiewicz O C,Codina R.A general algorithm for compressible and incompressible flow,Part I:The split,characteristic-based scheme[J].International Journal for Numerical Methods in Fluids,1995,20:869-885.
[6]Thomas C G,Nithiarasu P.Influences of element size and variable smoothing on inviscid compressible flow solution[J].International Journal of Numerical Methods for Heat and Fluid Flow,2005,15(5):420-428.
[7]孙旭,张家忠,周志宏,等.不可压缩粘性流动的CBS有限元法[J].计算力学学报,2010,27(5):862-867.(SUN Xu,ZHANG Jia-zhong,ZHOU Zhi-hong,et al.On the application of the CBS finite element method to the incompressible flow[J].Chinese Journal of Computational Mechanics,2010,27(5):862-867.(in Chinese))
[8]Liu C B,Nithiarasu P.The characteristic-based split(CBS)scheme for viscoelastic flow past a circular cylinder[J].International Journal for Numerical Methods in Fluids,2008,57(2):157-176.
[9]Morandi-Cecchi M,Venturin M.Characteristic-based split(CBS)algorithm finite element modelling for shallow waters in the Venice lagoon[J].International Journal for Numerical Methods in Engineering,2006,66(10):1641-1657.
[10]Bao Y,Zhou D,Zhao Y J.A two-step Taylor-characteristic-based Galerkin method for incompressible flows and its application to flow[J].International Journal for Numerical Methods in Fluids,2010,62(11):1181-1208.
[11]Nithiarasu P,Liu C B.An artificial compressibility based characteristic based split(CBS)scheme for steady and unsteady turbulent incompressible flows[J].Computer Methods in Applied Mechanics and Engineering,2006,195(23-24):2961-2982.
[12]田俊武,袁湘江.三维复杂流动的间断有限元方法模拟[J].计算力学学报,2015,32(2):239-242.(TIAN Jun-wu,YUAN Xiang-jiang.Numerical simulation of three dimensional complex flows with RKDGmethod[J].Chinese Journal of Computational Mechanics,2015,32(2):239-242.(in Chinese))
[13]Erturk E,Corke T C,G9kc C.Numerical solutions of2-D steady incompressible driven cavity flow at high Reynolds numbers[J].International Journal for Numerical Methods in Fluids,2005,48:747-774.
[14]时忠民,刘名名,郭晓玲.计算域对圆柱绕流数值模拟结果的影响[J].中国水运,2013,13(7):83-87.(SHIZhong-min,LIU Ming-ming,GUO Xiao-ling.The influence of calculating domain on numerical simulation of flow around a cylinder[J].China Water Transport,2013,13(7):83-87.(in Chinese))
[15]水庆象,王大国.非定常不可压N-S方程的最小二乘算子分裂有限元数值求解[J].计算力学学报,2014,31(5):640-645.(SHUI Qing-xiang,WANG Da-guo.Numerical solution for unsteady incompressible N-Sequations by least-squares-based operator-splitting finite element method[J].Chinese Journal of Computational Mechanics,2014,31(5):640-645.(in Chinese))
[16]包艳.典型钝体结构流致效应分析与计算方法研究[D].上海交通大学,2011.(BAO Yan.The Research on Flow-Induced Responses of Fundamental Bluff Bodies and Its Numerical Sumulation Methods[D].Shanghai Jiao Tong University,2011.(in Chinese))