| Popis: |
In this dissertation, I introduce a vortex method that is generally applicable to any two-dimensional, incompressible flow with or without boundaries. This method is deterministic, accurate, convergent, naturally adaptive, geometry independent and fully localized. For viscous flows, the vorticity distribution of each vortex element must evolve in addition to following a Lagrangian trajectory. My method relies upon an idea called core spreading. Core spreading is inconsistent by itself, but I have corrected it with a deterministic process known as "vortex fission" where one "fat" vortex is replaced by several "thinner" ones. Also, I examine rigorously a method for merging many blobs into one. This process maintains smaller problem sizes thus boosting the efficiency of the vortex method. To prove that this corrected core spreading method will converge uniformly, I adapted a continuous formalism to this grid-free scheme. This convergence theory does not rely on any form of grid. I only examine the linear problem where the flow field is specified, and treat the full nonlinear problem as a perturbation of the linear problem. The estimated rate of convergence is demonstrated to be sharp in several examples. Boundary conditions are approximated indirectly. The boundary is decomposed into a collection of small linear segments. I solve the no-slip and no-normal flow conditions simultaneously by superimposing a potential flow and injecting vorticity from the boundary consistent with the unsteady Rayleigh problem. Finally, the ultimate test for this new method is to simulate the wall jet. The simulations produce a dipole instability along the wall as observed in water tank and wind tunnel experiments and predicted by linear stability analysis. Moreover, the wavelength and height of these simulations agree quantitatively with experimental observations. |
