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To: 1.3.6 Free-Lagrange 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 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. The diffusion velocity is then added to the incompressible velocity field to convect the vortices.
Ogami & Akamatsu [165] applied this method to the one-dimensional Stokes flow of an initially uniform vortex patch. Strickland, Kempka & Wolfe [218] have applied it to several simple one-dimensional problems involving solid walls. Clarke & Tutty [59,227], and Huyer & Grant [112,113] have used this method for the flow over a cylinder and an airfoil. Recently, Ogami & Cheer [164] have extended the diffusion velocity method [165] to compressible flows.
The diffusion velocity method is mesh-free since the vorticity gradients are evaluated as in the SPH method mentioned in the subsection 1.3.4. However, the definition of the diffusion velocity gives rise to inherent problems: special care is needed in regions where the vorticity vanishes and also where vorticity gradients are large. The method also requires a large number of overlapping vortices for accurate simulations due to large variations in the diffusion velocity in different parts of the flow. Kempka and Strickland [116] noticed that the diffusion velocity is not divergence free. They interpret the effect of this nonzero divergence as a change in the size of the vortices. They show that the accuracy of the diffusion process can be improved by modifying the size of the vortices according to the nonzero divergence. However, such modifications are not easy to handle and lead to severe restrictions on the size of the time step [116]. Further, the overlap of the vortices must be carefully monitored similar to PSE and Fishelov's methods.
Finally, the observations of Degond & Mustieles [71] may be noteworthy: The diffusion velocity may become infinite in regions of vanishing vorticity or large vorticity gradients. However, for numerical implementation the diffusion velocity has to be finite; this creates a `diffusion front' and could lead to numerical instability. In fact, their one-dimensional computation of the diffusion of an initially smooth vorticity field shows numerical oscillations in the regions of vanishing vorticity. They also point out that this method may be less accurate and more expensive than other methods for the Navier-Stokes and the heat equations. However, it may be suited to problems in the kinetic theory of plasma physics.