By Shankar Subramaniam
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Additional info for A NEW MESH-FREE VORTEX METHOD
1, we believe that this is due to our brute force approach to finding the redistribution fractions. Our requirement that the redistribution weights are positive was motivated in part by the standard five-point explicit finite difference scheme for the diffusion process. For that finite difference scheme, the transition from a stable to an unstable scheme occurs when one of the fractions becomes negative.
16) at each time step of the computation. For viscous flows, in addition to moving the vortices, the diffusion process must also be represented; we will discuss that in the next section. 4) is Dω = ν ∇2 ω Dt . 17) numerically, we use the viscous splitting algorithm mentioned in the introduction of this chapter. Mathematically, this algorithm is expressed by splitting 33 each time-step into a convection step and a diffusion step as follows (Chorin et al. 20) . 21) In the above equations, xi and Γi are the position and circulation of vortex i respectively; Kδ is the velocity kernel K ∗ φδ ; and ω is the smooth vorticity distribution represented by the vortices Γi .
2). This makes it possible to approximate the trailing exponentials in the two Fourier transforms by a truncated Taylor series. It does turn out to be possible to equate the Fourier transforms using these truncated Taylor series. The detailed derivation is given in Appendix A. The resulting equations are the redistribution equations we were looking for. 7) that are bounded by the neighborhood radius; ξij ≤ R. 11) j Higher-order moment equations, m = 4, . . , M + 1 . 12) 38 From these equations, the redistribution fractions fijn are to be found.
A NEW MESH-FREE VORTEX METHOD by Shankar Subramaniam