Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T20:09:35.537Z Has data issue: false hasContentIssue false
  • English
  • Français

Solution of Two-Dimensional Stokes Flow Problems Using Improved Singular Boundary Method

Published online by Cambridge University Press:  09 January 2015

Wenzhen Qu  and
Wen Chen
Wenzhen Qu
Affiliation:
Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, China
Wen Chen*
Affiliation:
Department of Engineering Mechanics, College of Mechanics and Materials, Hohai University, Nanjing 210098, China
*
*Email:chenwen@hhu.edu.cn(W. Chen)
Get access

Abstract

In this paper, an improved singular boundary method (SBM), viewed as one kind of modified method of fundamental solution (MFS), is firstly applied for the numerical analysis of two-dimensional (2D) Stokes flow problems. The key issue of the SBM is the determination of the origin intensity factor used to remove the singularity of the fundamental solution and its derivatives. The new contribution of this study is that the origin intensity factors for the velocity, traction and pressure are derived, and based on that, the SBM formulations for 2D Stokes flow problems are presented. Several examples are provided to verify the correctness and robustness of the presented method. The numerical results clearly demonstrate the potentials of the present SBM for solving 2D Stokes flow problems.

Keywords

76D0776M25Singular boundary methodorigin intensity factorStokes flowfundamental solution
Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 7 , Issue 1 , February 2015 , pp. 13 - 30
Copyright
Copyright © Global Science Press Limited 2015 

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]Li, M., Tang, T. and Fornberg, B., A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations, Int. J. Numer. Methods Fluids, 20 (1995), pp. 1137–1151.Google Scholar
[2]Taylor, C. and Hood, P., A numerical solution of the Navier-Stokes equations using the finite element technique, Comput. Fluids, 1 (1973), pp. 73–100.Google Scholar
[3]Power, H. and Wrobel, L. C., Boundary Integral Methods in Fluid Mechanics, Computational Mechanics Publications Southampton, 1995.Google Scholar
[4]Smyrlis, Y. S. and Karageorghis, A., Some aspects of the method of fundamental solutions for certain biharmonic problems, Comput. Model. Eng. Sci., 4 (2003), pp. 535–550.Google Scholar
[5]Ogata, H., A fundamental solution method for three-dimensional Stokes flow problems with obstacles in a planar periodic array, J. Comput. Appl. Math., 189 (2006), pp. 622–634.CrossRefGoogle Scholar
[6]Alves, C. J. S. and Silvestre, A., Density results using Stokeslets and a method of fundamental solutions for the Stokes equations, Eng. Anal. Boundary Elements, 28 (2004), pp. 1245–1252.Google Scholar
[7]Fairweather, G. and Karageorghis, A., The method of fundamental solutions for elliptic boundary value problems, Adv. Comput. Math., 9 (1998), pp. 69–95.Google Scholar
[8]Chen, W. and Gu, Y., An improved formulation of singular boundary method, Adv. Appl. Math. Mech., 4 (2012), pp. 543–558.CrossRefGoogle Scholar
[9]Liu, Y., A new boundary meshfree method with distributed sources, Eng. Anal. Boundary Elements, 34 (2010), pp. 914–919.Google Scholar
[10]Chen, W. and Wang, F. Z., A method of fundamental solutions without fictitious boundary, Eng. Anal. Boundary Elements, 34 (2010), pp. 530–532.CrossRefGoogle Scholar
[11]Gu, Y., Chen, W. and Zhang, C. Z., Singular boundary method for solving plane strain elasto-static problems, Int. J. Solids Structures, 48 (2011), pp. 2549–2556.Google Scholar
[12]Golberg, M. and Chen, C., The method of fundamental solutions for potential, Helmholtz and diffusion problems, Boundary Integral Methods Numer. Math. Aspects, (1998), pp. 103–176.Google Scholar
[13]Young, D., Jane, S., Fan, C., Murugesan, K. and Tsai, C., The method of fundamental solutions for 2D and 3D Stokes problems, J. Comput. Phys., 211 (2006), pp. 1–8.CrossRefGoogle Scholar
[14]Young, D., Chen, C., Fan, C., Murugesan, K. and Tsai, C., The method of fundamental solutions for Stokes flow in a rectangular cavity with cylinders, Euro. J. Mech. B Fluids, 24 (2005), pp. 703–716.Google Scholar
[15]Chen, W. and Fu, Z. J., A novel numerical method for infinite domain potential problems, Chinese Science Bulletin, 55 (2010), pp. 1598–1603.Google Scholar
[16]Tanaka, M., Sladek, V. and Sladek, J., Regularization techniques applied to boundary element methods, Appl. Mech. Rev., 47 (1994), pp. 457.Google Scholar
[17]Young, D. L., Chen, K. H. and Lee, C. W., Novel meshless method for solving the potential problems with arbitrary domain, J. Comput. Phys., 209 (2005), pp. 290–321.Google Scholar
[18]Gu, Y., Chen, W. and Zhang, J., Investigation on near-boundary solutions by singular boundary method, Eng. Anal. Boundary Elements, 36 (2012), pp. 1173–1182.Google Scholar
[19]Liu, Y., A new fast multipole boundary element method for solving 2-D Stokes flow problems based on a dual BIE formulation, Engineering Anal. Boundary Elements, 32 (2008), pp. 139–151.CrossRefGoogle Scholar
[20]Burggraf, O. R., Analytical and numerical studies of the structure of steady separated flows, J. Fluid Mech, 24 (1966), pp. 113–151.Google Scholar
[21]Young, D. L., Jane, S. C., Lin, C. Y., Chiu, C. L. and Chen, K. C., Solutions of 2D and 3D Stokes laws using multiquadrics method, Engineering Anal. Boundary Elements, 28 (2004), pp. 1233–1243. ˇGoogle Scholar
[22]Šarler, B., Solution of potential flow problems by the modified method of fundamental solutions: formulations with the single layer and the double layer fundamental solutions, Engineering Anal. Boundary Elements, (2009), pp. 1374–1382.Google Scholar
[23]Young, D., Chen, K., Chen, J. and Kao, J., A modified method of fundamental solutions with source on the boundary for solving Laplace equations with circular and arbitrary domains, Comput. Model. Eng. Sci., 19 (2007), pp. 197.Google Scholar
[24]Pozrikidis, C., Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, 1992.Google Scholar
[25]Zhang, Y. M., Wen, W. D., Wang, L. M. and Zhao, X. Q., A kind of new nonsingular boundary integral equations for elastic plane problems, Acta Mech., 36 (2004), pp. 311–321.Google Scholar