Hostname: page-component-745bb68f8f-kw2vx Total loading time: 0 Render date: 2025-02-06T16:08:01.922Z Has data issue: false hasContentIssue false
  • English
  • Français

Three Iterative Finite Element Methods for the Stationary Smagorinsky Model

Published online by Cambridge University Press:  28 May 2015

Haiyan Su ,
Pengzhan Huang ,
Juan Wen  and
Xinlong Feng
Haiyan Su*
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, RR., China
Pengzhan Huang*
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, RR., China
Juan Wen*
Affiliation:
Faculty of Science, Xi'an Jiaotong University, Xi'an 710049, RR., China
Xinlong Feng*
Affiliation:
College of Mathematics and System Sciences, Xinjiang University, Urumqi 830046, RR., China
*
Corresponding author. Email Address: yanzi880308xju@sina.com
Corresponding author. Email Address: hpzh007@gmail.com
Corresponding author. Email Address: zhongnanjicuan@163.com
Corresponding author. Email Address: fxlmath@gmail.com
Get access

Abstract

Three iterative stabilised finite element methods based on local Gauss integration are proposed in order to solve the steady two-dimensional Smagorinsky model numerically. The Stokes iterative scheme, the Newton iterative scheme and the Oseen iterative scheme are adopted successively to deal with the nonlinear terms involved. Numerical experiments are carried out to demonstrate their effectiveness. Furthermore, the effect of the parameters Re (the Reynolds number) and δ (the spatial filter radius) on the performance of the iterative numerical results is discussed.

Keywords

65N3065N1276D07Smagorinsky modelstabilised finite element methodlocal Gauss integrationiterative schemelid driven cavity flow
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 4 , Issue 2 , May 2014 , pp. 132 - 151
Copyright
Copyright © Global-Science Press 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Adams, R.A., Fournier, J.J.F., Sobolev Spaces. Academic Press, New York, 1975.Google Scholar
[2]Bochev, P.B., Dohrmann, C.R. & Gunzburger, M.D., Stabilization of low-order mixed finite elements for the Stokes equations, SIAM J. Numer. Anal. 44, 82–101 (2006).Google Scholar
[3]He, Y.N. & Li, J., Convergence of three iterative methods based on the finite element discretisation for the stationary Navier-Stokes equations. Comput. Methods Appl. Mech. Engrg. 198, 1351–1359 (2009).Google Scholar
[4]Feng, X.L., Kim, I., Nam, H. & Sheen, D., Locally stabilized P1-nonconforming quadrilateral and hexahedral finite element methods for the Stokes equations, J. Comput. Appl. Math. 236, 714–727 (2011).Google Scholar
[5]Huang, P.Z., Feng, X.L. & Liu, D.M., Two-level stabilized method based on Newton iteration for the steady Smagorinsky model, Nonlinear Anal. Real World Appl. 14, 1795–1805 (2013).Google Scholar
[6]Borggaard, J. & Iliescu, T., Approximate deconvolution boundaryconditions for large eddy simulation, Appl. Math. Lett. 19, 725–740 (2006).CrossRefGoogle Scholar
[7]Borggaard, J., Lee, H., Roop, J.P. & Son, H., A two-level discretization method for the Smagorinsky model, Multiscale Model. Simul. 7, 599–621 (2008).Google Scholar
[8]Borggaard, J., Iliescu, T. & Roop, J.P, A bounded artificial viscosity large eddy simulation model, SIAM. J. Numer. Anal. 47, 622–645 (2009).Google Scholar
[9]Li, J. & Chen, Z.X., A new local stabilised nonconforming finite method for the Stokes equations, Computing 82, 157–170 (2008).CrossRefGoogle Scholar
[10]Li, J. & He, Y.N., A new stsbilised finite element method based on two local Gauss integrations for the Stokes equations, J. Comput. Appl. Math. 214, 58–65 (2008).Google Scholar
[11]Berselli, L.C., Iliescu, T. & Layton, W.J., Mathematics of Large Eddy Simulation of Turbulent Flows, Scientific Computation. Springer-Verlag, Berlin, 2006.Google Scholar
[12]Su, H.Y., Huang, P.Z. & Feng, X.L., Two-level stabilized nonconforming finite element method for the Stokes equations, Appl. Math. 58, 643–656 (2013).CrossRefGoogle Scholar
[13]Iliescu, T., & Fischer, P.F., Large eddy simulation of turbulent channel flows by the rational LES model, Phys. Fluids. 15, 3036–3047 (2003).CrossRefGoogle Scholar
[14]Girault, V. & Raviart, P.A., The Finite Element Method for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag, Berlin, Heidelberg, 1986.Google Scholar
[15]Dai, X.X. & Cheng, X.L., A two-grid method based on Newton iteration for the Navier-Stokes equations J. Comput. Appl. Math. 220, 566–573 (2008).Google Scholar