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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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