A Study of the Vortex Sheet Method and its Rate of Convergence

The subject of this study is Chorin's vortex sheet method, which is used to solve the Prandtl boundary layer equations and to impose the no-slip boundary condition in the random vortex method solution of the Navier–Stokes equations. This is a particle method in which the particles carry concentrations of vorticity and undergo a random walk to approximate the diffusion of vorticity in the boundary layer. During the random walk, particles are created at the boundary in order to satisfy the no-slip boundary condition. It is proved that in each of the $L^1 $, $L^2 $, and $L^\infty $ norms the random walk and particle creation, taken together, provide a consistent approximation to the heat equation, subject to the no-slip boundary condition. Furthermore, it is shown that the truncation error is entirely due to the failure to satisfy the no-slip boundary condition exactly. It is demonstrated numerically that the method converges when it is used to model Blasius flow, and rates of convergence are established in terms of the computational parameters. The numerical study reveals that errors grow when the sheet length tends to zero much faster than the maximum sheet strength. The effectiveness of second-order time discretization, sheet tagging, and an alternative particle-creation algorithm are also examined.