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The method of fundamental solutions for solving incompressible Navier-Stokes problems. (English) Zbl 1244.76104

Summary: A novel meshless numerical procedure based on the method of fundamental solutions (MFS) is proposed to solve the primitive variables formulation of the Navier-Stokes equations. The MFS is a meshless method since it is free from the mesh generation and numerical integration. We will transform the Navier-Stokes equations into simple advection-diffusion and Poisson differential operators via the operator-splitting scheme or the so-called projection method, instead of directly using the more complicated fundamental solutions (Stokeslets) of the unsteady Stokes equations. The resultant velocity advection-diffusion equations and the pressure Poisson equation are then calculated by using the MFS together with the Eulerian-Lagrangian method (ELM) and the method of particular solutions (MPS). The proposed meshless numerical scheme is a first attempt to apply the MFS for solving the Navier-Stokes equations in the moderate-Reynolds-number flow regimes. The lid-driven cavity flows at the Reynolds numbers up to 3200 for two-dimensional (2D) and 1000 for three-dimensional (3D) are chosen to validate the present algorithm. Through further simulating the flows in the 2D circular cavity with an eccentric rotating cylinder and in the 3D cube with a fixed sphere inside, we are able to demonstrate the advantages and flexibility of the proposed meshless method in the irregular geometry and multi-dimensional flows, even though very coarse node points are used in this study as compared with other mesh-dependent numerical schemes.

MSC:

76M28 Particle methods and lattice-gas methods
76D05 Navier-Stokes equations for incompressible viscous fluids
65M80 Fundamental solutions, Green’s function methods, etc. for initial value and initial-boundary value problems involving PDEs
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[1] Fairweather, G.; Karageorghis, A., The method of fundamental solutions for elliptic value problems, Adv Comput Math, 9, 69-95 (1998) · Zbl 0922.65074
[2] Golberg, M. A.; Chen, C. S., The method of fundamental solutions for potential, Helmholtz and diffusion problems, (Golberg, M. A., Boundary integral methods: numerical and mathematical aspects (1998), Computational Mechanics Publications: Computational Mechanics Publications Boston), 103-176 · Zbl 0945.65130
[3] Kupradze, V. D.; Aleksidze, M. A., The method of fundamental equations for the approximate solution of certain boundary value problem, Zh Vychisl Mat, 4, 4, 82-126 (1964) · Zbl 0154.17604
[4] Mathon, R.; Johnston, R. L., The approximate solution of elliptic boundary-value problems by fundamental solutions, SIAM J Numer Anal, 14, 638-650 (1977) · Zbl 0368.65058
[5] Bogomolny, A., Fundamental solutions method for elliptic boundary value problems, SIAM J Numer Anal, 22, 644-669 (1985) · Zbl 0579.65121
[6] Karageorghis, A., The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation, Appl Math Let, 14, 837-842 (2001) · Zbl 0984.65111
[7] Karageorghis, A.; Fairweather, G., The method of fundamental solutions for the numerical solution of the biharmonic equation, J Comput Phys, 69, 434-459 (1987) · Zbl 0618.65108
[8] Young, D. L.; Jane, S. J.; Fan, C. M.; Murugesan, K.; Tsai, C. C., The method of fundamental solutions for 2D and 3D Stokes problems, J Comput Phys, 211, 1-8 (2006) · Zbl 1160.76332
[9] Golberg, M. A., The method of fundamental solutions for Poisson’s equations, Eng Anal Bound Elem, 16, 205-213 (1995)
[10] Golberg, M. A.; Muleshkov, A. S.; Chen, C. S.; Cheng, A. H.-D., Polynomial particular solutions for certain partial differential operators, Numer Methods Partial Differential Equation, 19, 1, 112-133 (2002) · Zbl 1019.65096
[11] Young, D. L.; Tsai, C. C.; Murugesan, K.; Fan, C. M.; Chen, C. W., Time-dependent fundamental solutions for homogeneous diffusion problems, Eng Anal Boundary Elem, 29, 1463-1473 (2004) · Zbl 1098.76622
[12] Young, D. L.; Fan, C. M.; Tsai, C. C.; Chen, C. W.; Murugesan, K., Solution of advection-diffusion equation using the Eulerian-Lagrangian method of fundamental solutions, Int Math Forum, 1, 14, 687-706 (2006) · Zbl 1143.65384
[13] Young, D. L.; Fan, C. M.; Hu, S. P.; Atluri, S. N., The Eulerian-Lagrangian method of fundamental solutions for two-dimensional unsteady Burgers’ equations, Eng Anal Boundary Elem, 32, 395-412 (2008) · Zbl 1244.76096
[14] Tsai, C. C.; Young, D. L.; Fan, C. M.; Chen, C. W., MFS with time-dependent fundamental solutions for unsteady Stokes equations, Eng Anal Boundary Elem, 30, 897-908 (2006) · Zbl 1195.76324
[15] Young, D. L.; Chen, C. W.; Fan, C. M., The method of fundamental solutions for low Reynolds number flows with moving rigid body, (Chen, C. S.; Karageorghis, A.; Smyrlis, Y. S., The Method of Fundamental Solutions-A Meshless Method (2008), Dynamic Publishers), 181-206
[16] Chorin, A. J., Numerical solution of the Navier-Stokes equations, Math Comput, 22, 745-762 (1968) · Zbl 0198.50103
[17] Temam, R., Une méthode d’approximation de la solution des équations de Navier-Stokes, Bull Soc Math France, 98, 115-152 (1968) · Zbl 0181.18903
[18] Patankar, S. V.; Spalding, D. B., A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows, Int J Heat Mass Transfer, 15, 1787-1806 (1972) · Zbl 0246.76080
[19] Goldberg, D.; Ruas, V., A numerical study of projection algorithms in the finite element simulation of three-dimensional viscous incompressible flow, Int J Numer Methods Fluids, 30, 233-256 (1999) · Zbl 0945.76045
[20] Brown, D.; Cortez, R.; Minion, M., Accurate projection method for incompressible Navier-Stokes equations, J Comput Phys, 168, 464-499 (2001) · Zbl 1153.76339
[21] Chang, W.; Giraldo, F.; Perot, B., Analysis of an exact fractional step method, J Comput Phys, 180, 183-199 (2002) · Zbl 1130.76394
[22] Duchon, J., Splines minimizing rotation invariant seminorms in Sobolev spaces, (Constructive theory of functions of several variables (1977), Springer: Springer Berlin), 85-110
[23] Golberg, M. A.; Chen, C. S., Discrete projection methods for integral equations (1997), Computational Mechanics Publication: Computational Mechanics Publication Southampton
[24] Burggraf, O. R., Analytical and numerical studies of the structure of steady separated flow, J Fluid Mech, 24, 113-151 (1966)
[25] Ghia, U.; Ghia, K. N.; Shin, C. T., High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method, J Comput Phys, 48, 387-411 (1982) · Zbl 0511.76031
[26] Jiang, B. N.; Lin, T. L.; Provinelli, L. A., Large-scale computation of incompressible viscous flow by least-square finite element method, Compu Methods Appl Mech Eng, 114, 213-231 (1994)
[27] Tsai, C. C.; Lin, Y. C.; Young, D. L.; Atluri, S. N., Investigations on the accuracy and condition number for the method of fundamental solutions, CMES, 16, 2, 103-114 (2006)
[28] Young, D. L.; Chiu, C. L.; Fan, C. M., A hybrid Cartesian/immersed-boundary finite-element method for simulating heat and flow patterns in a two-roll mill, Numer Heat Transfer, B, 51, 251-274 (2007)
[29] Chiu, C. L.; Fan, C. M.; Young, D. L., 3D hybrid Cartesian/immersed-boundary finite-element analysis of heat and flow patterns in a two-roll mill, Int J Heat Mass Transfer, 52, 7-8, 1677-1689 (2009) · Zbl 1157.80314
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