A NEW MESH-FREE VORTEX METHOD by Shankar Subramaniam

By Shankar Subramaniam

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5 Diffusion velocity method The basic idea of the diffusion velocity method is to handle diffusion as a part of the convection process. To do that an artificial velocity field is defined to represent the diffusion process. Golubkin & Sizykh [89] and Ogami & Akamatsu [165] identify the ‘diffusion velocity’ by absorbing the diffusion term into the convection term in the vorticity equation. Kempka and Strickland [116] derive the same expression for 15 the diffusion velocity in a different way. It turns out that the diffusion velocity is proportional to the ratio of the vorticity gradient to the vorticity.

If there is no acceptable solution, we add new vortices until there is one. Fundamentally, our procedure differs from the usual particle methods by separating the computation of the vorticity into two distinct steps: (a) determination of vortex strengths from the localized conservation laws; (b) reconstruction of the vorticity field by convolution. This separation allows us to achieve any chosen order of accuracy regardless of the geometry of the vortex distribution. However, unlike the particle methods, and other numerical methods, in our scheme an individual vortex strength has no identifiable meaning.

1) 36 where ∆t is the size of the time step and ν the coefficient of kinematic viscosity. A vortex j is part of the neighborhood of vortex i if |xj − xi | ≤ R hv , where R is a chosen constant. 2) √ 12hv in all computa- tions presented in this thesis. 3. The diffusion of a vortex i will be approximated by moving fractions of its circulation towards the other vortices within this neighborhood. We will indicate the fraction moved from vortex i to a vortex j by fijn , where n indicates the time level.

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