A PN×PN-2 Spectral Element Method Based on the Picard Iteration for Steady Incompressible Navier-Stokes Equations

QIU Zhouhua; ZENG Zhong; LIU Hao

doi:10.21656/1000-0887.410289

Volume 42 Issue 2

Feb. 2021

Turn off MathJax

Article Contents

Article Navigation > Applied Mathematics and Mechanics > 2021 > 42(2): 142-150

QIU Zhouhua, ZENG Zhong, LIU Hao. A PN×PN-2 Spectral Element Method Based on the Picard Iteration for Steady Incompressible Navier-Stokes Equations[J]. Applied Mathematics and Mechanics, 2021, 42(2): 142-150. doi: 10.21656/1000-0887.410289

Citation:

QIU Zhouhua, ZENG Zhong, LIU Hao. A P_N×P_N-2 Spectral Element Method Based on the Picard Iteration for Steady Incompressible Navier-Stokes Equations[J]. Applied Mathematics and Mechanics, 2021, 42(2): 142-150. doi: 10.21656/1000-0887.410289

Citation:

PDF( 2165 KB)

A P_N×P_N-2 Spectral Element Method Based on the Picard Iteration for Steady Incompressible Navier-Stokes Equations

doi: 10.21656/1000-0887.410289

QIU Zhouhua^1,2,3,
ZENG Zhong⁴,
LIU Hao^1,2,3

¹Chongqing Southwest Research Institute for Water Transport Engineering, Chongqing Jiaotong University, Chongqing 400016, P.R.China;²Key Laboratory of Transport Industry of Inland Waterway Regulation Engineering(Chongqing Jiaotong University), Chongqing 400074, P.R.China;³Chongqing Xike Waterway Engineering Consulting Center, Chongqing 400016, P.R.China;⁴College of Aerospace Engineering, Chongqing University, Chongqing 400044, P.R.China

Received Date: 2020-09-24
Rev Recd Date: 2020-10-12
Publish Date: 2021-02-01

Abstract

Abstract

A P_N×P_N-2 spectral element method based on the Picard linearized iteration was presented for the solution of 2D steady incompressible Navier-Stokes equations. Through the Picard iteration, the Navier-Stokes equations were converted to a series of Stokes-type equations to be solved with the P_N×P_N-2 spectral element method on the non-staggered grid in each iteration step. In order to eliminate the pseudo pressure mode, the pressure discretization is 2 orders lower than the velocity discretization, and the application of non-staggered grids makes the discretization of the equation convenient and avoids the interpolation error. The Stokes flow, the Kovasznay flow and the lid-driven cavity flow were simulated with the present method. The numerical results show that, the error converges with the spectral accuracy. In addition, avoidance of the pressure oscillation phenomenon indicates the accuracy and reliability of the present method.
- steady Navier-Stokes equations,
- Picard iteration,
- spectral Galerkin method,
- pseudo pressure mode

FullText(HTML)

References(22)

References

[1]	AUTERI F, GUERMOND J L, PAROLINI N. Role of the LBB condition in weak spectral projection methods[J]. Journal of Computational Physics,2001,174(1): 405-420.
[2]	SCHUMACK M R, SCHULTZ W W, BOYD J P. Spectral method solution of the Stokes equations on nonstaggered grids[J]. Journal of Computational Physics,1990,89(2): 30-58.
[3]	DEBLOIS B M. Linearizing convection terms in the Navier-Stokes equations[J]. Computer Methods in Applied Mechanics and Engineering,1997,143(3/4): 289-297.
[4]	REHMAN M U, VUIK C, SEGAL G. Numerical solution techniques for the steady incompressible Navier-Stokes problem[C]// Proceedings of the World Congress on Engineering.London, 2008: 844-849.
[5]	苏铭德, 陈霜立. 定常不可压缩粘性流体流动Navier-Stokes方程的推进迭代法[J]. 计算物理, 1989,6: 321-334.(SU Mingde, CHEN Shuangli. Solution of the N-S equation of the steady incompressible viscous flow with marching-iterative method[J]. Chinese Journal of Computational Physics,1989,6: 321-334.(in Chinese))
[6]	CASARIN M A. Schwarz preconditioners for the spectral element discretization of the steady Stokes and Navier-Stokes equations[J]. Numerische Mathematik,2001,89: 307-339.
[7]	PONTAZA J P, REDDY J N. Spectral/hp least-squares nite element formulation for the Navier-Stokes equations[J]. Journal of Computational Physics,2003,190: 523-549.
[8]	KNOLL D A, KEYES D. Jacobian-free Newton-Krylov methods: a survey of approaches and applications[J]. Journal of Computational Physics,2004,193: 357-397.
[9]	马东军, 柳阳, 孙德军, 等. 高阶谱元区域分解算法求解定常方腔驱动流[J]. 计算力学学报, 2006,23(6): 668-673.(MA Dongjun, LIU Yang, SUN Dejun, et al. Spectral element method with a domain decomposition Stokes solver for steady cavity driven flow[J]. Chinese Journal of Computational Mechanics,2006,23(6): 668-673.(in Chinese))
[10]	ZHANG W, ZHANG C H, XI G. An explicit Chebyshev pseudospectral multigrid method for incompressible Navier-Stokes equations[J]. Computers & Fluids,2010,39(1): 178-188.
[11]	章争荣, 张湘伟. 二维定常不可压缩粘性流动N-S方程的数值流形方法[J]. 计算力学学报, 2010,27(3): 415-421.(ZHANG Zhengrong, ZHANG Xiangwei. Numerical manifold method for steady incompressible viscous 2D flow Navier-Stokes equtions[J]. Chinese Journal of Computational Mechanics,2010,27(3): 415-421.(in Chinese))
[12]	覃燕梅, 冯民富, 罗鲲, 等. Navier-Stokes方程的局部投影稳定化方法[J]. 应用数学和力学, 2010,31(5): 618-630.(QIN Yanmei, FENG Minfu, LUO Kun, et al. Local projection stabilized finite element method for the Navier-Stokes equations[J]. Applied Mathematics and Mechanics,2010,31(5): 618-630.(in Chinese))
[13]	HE Y, ZHANG Y, SHANG Y, et al. Two-level Newton iterative method for the 2D/3D steady Navier-Stokes equations[J]. Numerical Methods for Partial Differential Equations,2012,28: 1620-1642.
[14]	MELCHIOR S A, LEGAT V, DOOREN P V, et al. Analysis of preconditioned iterative solvers for incompressible flow problems[J]. International Journal for Numerical Methods in Fluids,2012,68: 269-286.
[15]	OZCELIKKALE A, SERT C. Least-squares spectral element solution of incompressible Navier-Stokes equations with adaptive refinement[J].Journal of Computational Physics,2012,231: 3755-3769.
[16]	戴海, 潘文峰. 谱元法求解Helmholtz方程透射特征值问题[J]. 应用数学和力学, 2018,39(7): 833-840.(DAI Hai, PAN Wenfeng. A spectral element method for transmission eigenvalue problems of the Helmholtz equation[J]. Applied Mathematics and Mechanics,2018,39(7): 833-840.(in Chinese))
[17]	GHIA U, GHIA K N, SHIN C T. High- Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method[J].Journal of Computational Physics,1982,48(3): 387-411.
[18]	AIDUN C K, TRIANTAFILLOPOULOS N G, BENSON J. Global stability of a lid-driven cavity with through-flow: flow visualization studies[J].Physics of Fluids A: Fluid Dynamics,1991,3(9): 2081-2091.
[19]	ALBENSOEDER S, KUHLMANN H C, RATH H J. Three-dimensional centrifugal-flow instabilities in the lid-driven-cavity problem[J]. Physics of Fluids,2001,13: 121-135.
[20]	THEOFILIS V. Global linear instability[J]. Annual Review of Fluid Mechanics,2011,43: 319-352.
[21]	KOSEFF J R, STREET R L. The lid-driven cavity flow: a synthesis of qualitative and quantitative observations[J]. Journal of Fluids Engineering,1984,106(4): 390-398.
[22]	KOSEFF J R, STREET R L, GRESHO P M, et al. A three-dimensional lid-driven cavity flow: experiment and simulation[C]// International Conference on Numerical Methods in Laminar and Turbulent Flow.Seattle, WA, 1983.