A New Diffusion Procedure for Vortex Methods

A new method is proposed for simulating diffusion in vortex methods for two-dimensional incompressible flows. The method resolves length scales up to the spacing of the vortices. The grid-free nature of vortex methods is fully retained and the distribution of the vortices can be irregular. It is shown for the Stokes equations that in principle, the method can have any order of accuracy. It also conserves circulation, linear, and angular momentum. The method is based on exchanging a conserved quantity between arbitrary computational points. This suggests that extensions to more general flows may be possible. For the two-dimensional incompressible flows studied, circulation is exchanged between vortices to simulate diffusion. The amounts of circulation exchanged must satisfy a linear system of equations. Based on stability considerations, the exchanged amounts should further be positive. A procedure to find a solution to this problem is formulated using linear programming techniques. To test the method, the decay processes of a single point vortex and of a counterrotating pair of point vortices are computed. Current limitations of the method are discussed.

[1]  P. Koumoutsakos,et al.  Boundary Conditions for Viscous Vortex Methods , 1994 .

[2]  L. V. Dommelen.  Unsteady separation from a Lagrangian point of view , 1987 .

[3]  P. Raviart.  An analysis of particle methods , 1985 .

[4]  Giovanni Russo,et al.  A deterministic vortex method for the Navier-Stokes equations , 1993 .

[5]  Charles S. Peskin,et al.  A Lagrangian fractional step method for the incompressible Navier-Stokes equations on a periodic domain , 1987 .

[6]  Peter S. Bernard.  A deterministic vortex sheet method for boundary layer flow , 1995 .

[7]  Trease,et al.  [Lecture Notes in Physics] Advances in the Free-Lagrange Method Including Contributions on Adaptive Gridding and the Smooth Particle Hydrodynamics Method Volume 395 || Smooth particle hydrodynamics with strength of materials , 1991 .

[8]  Ole H. Hald,et al.  Convergence of Vortex methods for Euler's equations, III , 1987 .

[9]  J. Sethian,et al.  Validation study of vortex methods , 1988 .

[10]  K. W. Morton,et al.  Moving Point, Particle, and Free-Lagrange Methods for Convection-Diffusion Equations , 1991, SIAM J. Sci. Comput..

[11]  Dalia Fishelov,et al.  A new vortex scheme for viscous flow , 1990 .

[12]  J. Strikwerda.  Finite Difference Schemes and Partial Differential Equations , 1989 .

[13]  G. Cottet,et al.  A particle method to solve the Navier-Stokes system , 1990 .

[14]  P. Colella,et al.  A Fast Adaptive Vortex Method in Three Dimensions , 1994 .

[15]  Pierre Degond,et al.  A Deterministic Approximation of Diffusion Equations Using Particles , 1990, SIAM J. Sci. Comput..

[16]  S. Huberson,et al.  Particles simulation of viscous flow , 1989 .

[17]  G. Batchelor,et al.  An Introduction to Fluid Dynamics , 1968 .

[18]  L. L. Van Dommelen,et al.  A new diffusion scheme in vortex methods for three-dimensional incompressible flows , 1996 .

[19]  Elke A. Rundensteiner,et al.  Fast, adaptive summation of point forces in the two-dimensional Poisson equation , 1989 .

[20]  Grégoire Winckelmans,et al.  Contributions to vortex particle methods for the computation of three-dimensional incompressible unsteady flows , 1993 .