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To: 1.3 Lagrangian methods for diffusion |
One possible approach to handle diffusion in vortex methods is to use a conventional mesh. Since this combines both vortex and mesh based approaches it is called a hybrid method. The basic procedure is the following: First, the vorticity at mesh points is determined from the vortices using some interpolation scheme. Then the diffusion equation is solved on this mesh to obtain diffused vorticity values at the mesh points. After this, the strengths of the existing vortices can be updated using the diffused vorticity values of the mesh points and the mesh can be discarded [92]; alternately, the existing vortices can be removed and new ones created at the mesh points [40,138,139,163].
One of the main reasons for using a mesh is the convenience in evaluating the spatial derivatives of the vorticity or any other flow variable. This is also one of the reasons for using a mesh in the Vortex-In-Cell method (VIC) [56,61,76] and the Particle-In-Cell (PIC) type methods [108,158]; in both these methods the numerical diffusion is a major disadvantage [30]. Similarly, using a mesh for diffusion has the disadvantage of high numerical diffusion due to the interpolations. This is not desirable for high Reynolds number flows or other situations where small scales need to be resolved. It is also difficult to ensure that the interpolated vorticity satisfies the governing equations and conservation laws.
Apart from the above methods, there is another hybrid approach to handle diffusion. In this approach the flow domain is divided into viscous regions adjoining the solid boundaries (usually, the boundary layers) and convection dominated regions outside it. In each of those regions, different formulations of the Navier-Stokes equations [179] can be used; or approximations to the Navier-Stokes equations may instead be used. Any of the above equations can be solved using vortices or a mesh or a combination of both. This opens the way for numerous variations, some of which can be found in [5,52,55,66,103,111,207,216].
The motivation for using these methods is their ability to handle the viscous regions accurately and efficiently [38,65,66,111]. For example, the no-slip boundary condition can be handled accurately [55,103]. Moreover, efficient numerical schemes can be used in various regions; typically, a finite difference or a finite element scheme is chosen.
However, dividing the flow domain into different regions has an inherent difficulty: it may not be easy to formulate appropriate conditions at the boundary between the regions [66,103,111,206]. Some authors consider the boundary layer equations to be simpler to use in the viscous regions instead of the full Navier-Stokes equations [4,216]; the difficulty here is that the boundary layer equations could quickly become invalid. Such is the case whenever unsteady boundary layer separation occurs, as shown by Van Dommelen and Shen [241,243].
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 domain. All these difficulties are eliminated in mesh-free methods; we will discuss such methods in the next section.