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By Shankar Subramaniam

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To summarize, many of the difficulties of the above approaches are caused by the mesh; in particular, the high numerical diffusion is a major disadvantage. In addition, it may be difficult to generate a mesh for flow around a complicated geometry; and in an external flow, it may be difficult for a mesh to exactly represent an infinite 8 domain. All these difficulties are eliminated in mesh-free methods; we will discuss such methods in the next section. 3 Lagrangian methods for diffusion A number of numerical schemes model diffusion in vortex methods without using a mesh.

The actual form of G is [13, 95], G(x; x′ ) = 1 ln(| x − x′ |) 2π . 17) . 18) for the velocity are known as the Biot-Savart law [109, 126]. The function K is also called the kernel [11], the Biot-Savart Kernel [109] or the velocity kernel [17]. 13). 3. 16), the initial condition and the boundary conditions together describe the evolution of the vorticity. Next we review the conservation laws derived from the vorticity equation. 3 Vorticity conservation laws The vorticity conservation laws can be viewed as constraints on the motion of the vorticity.

It also simplifies the computation of flows about complex geometries. We did not want a partitioning of the domain as in ‘unstructured’ computations. Such a partitioning is still a form of structure that must be regenerated. It brings in complicating aspects such as the geometry of triangles that are not found in the physical flow. Our scheme uses all available vortices within some reasonable distance, rather than a selected subset, to find a suitable discretization for the diffusion of each vortex.

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