The thesis is centered around an adaptive method for calculating vorticity dominated flows in two dimensions. We use the vortex method after a general transformation is applied to the flow region because the vortex elements describe the local flow more accurately if the transformation is suitably chosen. A good example of this is boundary layer flow, where vortex sheets serve to represent the vorticity as long as the Prandtl equation holds, but the method is inaccurate in the region where transition from Prandtl to fall Navier-Stokes equations occurs. Physical intuition tells us that the transition should be a natural process and therefore should be smooth, depending fully on the local flow and geometry. It is shown that this can be realized with a good spatial transformation which takes account of the above factors. In the special case of finite area vortex regions and within 1st order accuracy the Biot-Savart law is explicit and equivalent to the elliptic vortex method with the axes ratio and orientation evolving according to the continuous transformation. The transformation is obtained from the real flow by averaging, truncation and satisfying the same boundary condition as the real flow. Some experiments for typical flows are carried out in detail. We also modeled a two-dimensional, dilute fluid-particle system with low Reynolds number flow around cylindrical particles and high Reynolds number with respect to the bulk flow. Full particle methods are used to solve both the fluid and particle phase flows. The vortex method is used for the nearly incompressible fluid phase. The compressible particle phase is taken care of by using Voronoi diagrams suitably. On the microscopic scale the Stokes-Oseen formula is used to represent the forces on particles. Interactions between the two phases lead to fluid vorticity creation from the particles and drift of particles forced by the fluid. Numerical examples compare well with some experimental results.
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