| The focus of this thesis is on mathematical results, both numerical and analytical, for the unsteady viscous and inviscid Prandtl equations. We will present a straightfoward finite-difference method for solutions of the boundary layer equations. The numerical method is used to illustrate results about singularity formation and instability for both the viscous and inviscid unsteady Prandtl equations.; We prove singularity formation for a certain class of smooth initial velocity profiles, independently of the computations. Computations for several problems show that the normal velocity forms a singularity in finite time, in agreement with previous studies using Lagrangian coordinates. We have also explicitly solved the linearization of the inviscid Prandtl equations about a shear flow. The resulting equations are weakly, but not strongly, well-posed, and have a continuous spectrum which is unstable when the unperturbed shear flow profile has a critical point. The question about instability of the pressured Prandtl equations with zero mean in y is also considered. This type of equation arises in studying the interaction between large amplitude voriticity waves and mean flows in the incompressible Euler and Navier-Stokes equations. The vorticity wave is governed by the nonlinear boundary layer equations and the mean flow is governed by the Reynolds averaged Euler equations. This topic of nonlinear boundary layer equations and mean flow interactions is one of our motivations for studying the Prandtl equations. |
