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Journal of Vibration Testing and System Dynamics

C. Steve Suh (editor), Pawel Olejnik (editor),

Xianguo Tuo (editor)

Pawel Olejnik (editor)

Lodz University of Technology, Poland

Email: pawel.olejnik@p.lodz.pl

C. Steve Suh (editor)

Texas A&M University, USA

Email: ssuh@tamu.edu

Xiangguo Tuo (editor)

Sichuan University of Science and Engineering, China

Email: tuoxianguo@suse.edu.cn


A Subgrid Stabilized Method for Lid-driven Cavity Flow at Higher Reynolds Number

Journal of Vcibration Testing and System Dynamics 4(3) (2020) 249--258 | DOI:10.5890/JVTSD.2020.09.002

Yamiao Zhang$^{1}$, Langhuan Lou$^{1}$, Jiazhong Zhang$^{2}$, Yongshen He$^{2}$

$^{1}$ School of Science, Xi’an University of Posts and Telecommunications, Xi’an 710121, China

$^{2}$ School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China

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Abstract

A subgrid stabilized method based on two local Gauss integrations is presented to the numerical simulation of 2-D steady incompressible lid-driven cavity flow at higher Reynolds numbers. The main idea of this method is to use the subgrid model based on two local Gauss integrations as a stabilized term on the finite element discretization. In this method, the discretization equation is solved by Oseen iterative scheme. It is shown that steady flow simulations of the lid-driven cavity problem are computable up to Re = 45000. This maximum Reynolds number has not been reached by other stabilized finite element methods reported. Moreover, the computed vorticity values at the center of the primary vortex agree well with previous analytical solutions in the limit of infinite Reynolds number, and the numerical results for the properties of the primary vortex and the velocity components are also in agreement with the benchmark data in earlier studies.

Acknowledgments

This research was supported by the Research Fund(No.106-205020027), the Key Research and Development Program of Shaanxi Province (No. 2017ZDCXL-GY02-02) and the State Key Laboratory of Compressor and Key Laboratory of Compressor of Anhui Province(No. SKL-YSJ201802).

References

  1. [1]  Franca, L.P. and Frey, S.L. (1992), Stabilized finite element methods: II. The incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 99, 209-233.
  2. [2]  Franca, L.P. and Hughes, T.J.R. (1993), Convergence analyses of Galerkin least-squares methods for symmetric advective-diffusive forms of the Stokes and incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 105, 285-298.
  3. [3]  Brooks, A.N. and Hughes, T.J.R. (1982), Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 32, 199-259.
  4. [4]  Layton, W., Lee, H.K., and Peterson, J. (2002), A defect-correction method for the incompressible Navier-Stokes equations, Applied Mathematics and Computation, 129, 1-19.
  5. [5]  Si, Z. and He, Y. (2011), A defect-correctionmixed finite element method for stationary conduction-convection problems, Mathematical Problems in Engineering, 2011, 1-28.
  6. [6]  Arndt, D., Dallmann, H., and Lube, G. (2015), Local projection fem stabilization for the time-dependent incompressible navier-stokes problem, Numerical Methods for Partial Differential Equations, 31, 1224-1250.
  7. [7]  Guermond, J.L. (1999), Subgrid stabilization of Galerkin approximations of linear contraction semi-groups of class C0 in Hilbert spaces, Numerical Methods for Partial Differential Equations, 2, 131-138.
  8. [8]  Kaya, S., Layton, W., and Riviere, B. (2006), Subgrid stabilized defect correction methods for the Navier-Stokes equations, Siam Journal on Numerical Analysis, 44, 1639-1654.
  9. [9]  John, V. and Kaya, S. (2005), A finite element variational multiscale method for the Navier-Stokes equations, Siam Journal on Scientific Computing, 26, 1485-1503.
  10. [10]  Zheng, H., Hou, Y., Shi, F., and Song, L. (2009), A finite element variational multiscale method for incompressible flows based on two local gauss integrations, Journal of Computational Physics, 228, 5961-5977.
  11. [11]  Zhang, Y., Huang, B., Zhang, J., and Zhang, Z. (2017), A multilevel finite element variational multiscale method for incompressible Navier-Stokes equations based on two local Gauss integrations, Mathematical Problems in Engineering, 2017, 1-13.
  12. [12]  Zhang, Y., Zhang, J., and Zhu, L. (2018), Iterative finite element variational multiscale method for the incompressible Navier-Stokes equations, Journal of Computational and Applied Mathematics, 340, 53-70.
  13. [13]  Layton, W. (2002), A connection between subgrid scale eddy viscosity and mixed methods, Applied Mathematics and Computation, 133, 147-157.
  14. [14]  Guermond, J.L. (1999), Stabilization of Galerkin approximations of transport equations by subgrid modeling, Rairo-Mathematical Modelling and Numerical Analysis, 33, 1293-1316.
  15. [15]  He, Y. and Li, J. (2009), Convergence of three iterative methods based on the finite element discretization for the stationary Navier-Stokes equations, Computer Methods in Applied Mechanics and Engineering, 198, 1351-1359.
  16. [16]  Huang, P.Z. (2014), Iterative methods in penalty finite element discretizations for the steady Navier-Stokes equations, Numerical Methods for Partial Differential Equations, 30, 74-94.
  17. [17]  An, R. (2014), Comparisons of Stokes/Oseen/Newton iteration methods for Navier-Stokes equations with friction boundary conditions, Applied Mathematical Modelling, 38, 5535-5544.
  18. [18]  Xu, H. and He, Y. (2013), Some iterative finite element methods for steady Navier-Stokes equations with different viscosities, Journal of Computational Physics, 232, 136-152.
  19. [19]  He, Y. (2015), Stability and convergence of iterative methods related to viscosities for the 2D/3D steady Navier-Stokes equations, Journal of Mathematical Analysis and Applications, 423, 1129-1149.
  20. [20]  Kaya, S. and Riviere, A. (2006), A two-grid stabilization method for solving the steady-state Navier-Stokes equations, Numerical Methods for Partial Differential Equations, 22, 728-743.
  21. [21]  Kaya, S., Layton, W., and Riviere, B. (2006), Subgrid stabilized defect correction methods for the Navier- Stokes equations, Siam Journal on Numerical Analysis, 44, 1639-1654.
  22. [22]  Botti, L. and Pietro, D. A. (2011), A pressure-correction scheme for convection-dominated incompressible flows with discontinuous velocity and continuous pressure, Journal of Computational Physics, 230, 572-585.
  23. [23]  Hachem, E., Rivaux, B., Kloczko, T., Digonnet, H., and Coupez, T. (2010), Stabilized finite element method for incompressible flows with high Reynolds number, Journal of Computational Physics, 229, 8643-8665.
  24. [24]  Girault, V. and Raviart, P.A. (1986), Finite element method for Navier-Stokes equations: Theory and algorithms Springer-Verlag: New York.
  25. [25]  Taylor, C. and Hood, P. (1973), A numerical solution of the Navier-Stokes equations using the finite element technique, Computers and Fluids, 1, 73-100.
  26. [26]  Ghia, U., Ghia, K.N., and Shin, C.T. (1982), High-Re solutions for incompressible flow using the Navier- Stokes equations and a multigrid method, Journal of Computational Physics, 48, 387-411.
  27. [27]  Erturk, E., Corke, T.C., and Gokcol, C. (2005), Numerical solutions of 2-D steady incompressible driven cavity flow at high Reynolds numbers, International Journal for Numerical Methods in Fluids, 48, 747-774.
  28. [28]  Wahba, E.M. (2012), Steady flow simulations inside a driven cavity up to Reynolds number 35,000, Computers and Fluids, 66, 85-97.
  29. [29]  Wang, D. and Shui, Q. (2016), SUPG finite element method based on penalty function for lid-driven cavity flow up to Re = 27500, Acta Mechanica Sinica, 32, 54-63.
  30. [30]  Burggraf, O. (1965), Analytical and numerical studies of the structure of steady separated flows, Journal of Fluid Mechanics, 24, 113-151.