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To: 1.3.3 Deterministic particle method |
Another procedure to model diffusion is the core expansion method proposed by Leonard [126]; in two dimensions, the core of a vortex is the characteristic size of that vortex. In the core expansion method, the core of each vortex is allowed to expand according to the diffusion equation. Earlier Kuwahara & Takami [121] have used expanding vortex cores to compute the motion of a vortex sheet in an inviscid fluid. However, their objective was not to model diffusion but to eliminate the large velocities induced by point vortices. For that, they use the velocity field of a diffusing vortex instead of the velocity field of a point vortex; they do point out that the viscosity is an artificial viscosity and that its value must be chosen as small as possible.
The core expansion method has been used to simulate several flows. Rossi [186] has used the method to simulate wall jets. Zhang & Ghoniem [252] have applied the method to buoyancy-driven plumes. Chua [57] and Leonard [124] have used the method to simulate the collision of vortex rings. Meiburg has used the method for simulating diffusion flames [152] and mixing layers [153]. Nagano, Naita & Takata [162] have used the method for flows over rectangular prisms.
The core expansion method is exact for the Stokes equation. However, Greengard [96] has shown that it cannot model convection correctly when applied to the Navier-Stokes equations. The error in convection arises if the vortices become finite in size compared to the length scale of the flow. To reduce the convection error, Rossi [185] proposed the `corrected core spreading vortex method' in which the large vortices are split into smaller ones; however, the number of vortices grows exponentially in time. Also, the number of smaller vortices, their sizes, and the frequency of vortex splitting are critical control parameters that must be chosen apriori; these are sources of uncertainty in a computation.