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To: 1.3.4 Fishelov's method |
A deterministic method to simulate diffusion has been developed by Raviart [180], Choquin & Huberson [45], and Cottet & Mas-Gallic [64]. They use viscous/inviscid splitting of the vorticity equation and then solve the diffusion equation exactly using the fundamental solution of the heat equation. Recently the `Deterministic Particle (or Vortex) Method' has been developed along different lines by Degond & Mas-Gallic [72], and Mas-Gallic & Raviart [147]. The basic ingredients in this approach are: (a) to consider the strength (circulation) of each particle (vortex) as an unknown coefficient that changes with time due to diffusion effects, (b) to approximate the diffusion operator by an integral operator, and (c) to discretize the integral using the particle positions as quadrature points.
In practice, such methods start out with a fixed number of particles, distributed uniformly over the domain, each with a prescribed initial strength [119,170]. Changes in the particle strengths then simulate the diffusion effects through a system of ordinary differential equations [119,170]. These changes can be interpreted as changes in the strengths of particles due to the neighboring particles. For this reason, Winckelmans [247] and Koumoutsakos [119] call the deterministic particle method the `Particle Strength Exchange' method (PSE)
Choquin & Lucquin-Desreux [46] have investigated the accuracy of the method for axisymmetric vorticity distributions in two-dimensions. Huberson, Jollès & Shen [111] applied it to a concentrated vortex in a shear flow. Choquin & Huberson [45] studied the Kelvin-Helmholtz instability of a shear layer. Winckelmans & Leonard [247] have used the PSE scheme to study the fusion of vortex rings. Cottet [67] and Mas-Gallic [148] have extended the deterministic particle method to problems with boundary conditions. Pépin [170] used the local vorticity flux to adjust the strength of the vortices near a boundary. Koumoutsakos, Leonard & Pépin [118] describe the no-slip boundary condition in terms of the vorticity flux based on the fundamental solution of the heat equation. Koumoutsakos & Leonard [117] have applied the scheme to impulsively started and stopped flows around translating and rotating cylinders for Reynolds numbers from 40 to 9500; further computations of the flow over a cylinder include that of Cottet [65,66], Guermond, Huberson & Shen [103], and Huberson, Jollés, & Shen [111]. Shen & Phuoc Loc [206] have simulated the flow over an airfoil using PSE.
However, the PSE method has some disadvantages. The vortex size must be sufficiently large that there is a significant overlap of vortices. This requirement reduces the ability of the method to resolve the smallest scales; for example, the Stokes layer generated by an impulsive change in boundary conditions is numerically diffused over a significant number of vortices away from the boundary. Further, the PSE method requires that the uniformity of the particle distribution be periodically restored [65,66,111,119,170]. The uniformity is restored using a mesh [119,170] and this makes it difficult to handle flows over complicated geometries. For flows with separated vorticity, the mesh would have to be adaptive to be effective, greatly increasing the difficulties. Even then, the interpolation to the mesh introduces significant errors and inaccuracies.