Robust layer-resolving methods for various Prandtl problems

Abstract

In this thesis we deal with four Prandtl boundary layer problems for incompressible laminar flow. When the Reynolds and Prandtl numbers are large the solution of each problem has parabolic boundary layers. For each problem we construct a direct numerical method for computing approximations to the solution of the problem using a piecewise uniform fitted mesh technique appropriate to the parabolic boundary layer. We use the numerical method to approximate the self-similar solution of the Prandtl problem in a finite rectangle excluding the leading edge of the wedge, which is the source of an additional singularity caused by incompatibility of the problem data. We verify that the constructed numerical method is robust in the sense that the computed errors for the components and their derivatives in the discrete maximum norm are parameter uniform. For each problem we construct and apply a special numerical method related to the Blasius technique to compute a reference solution for the error analysis of the components and their derivatives. By means of extensive numerical experiments we show that the constructed direct numerical methods are parameter-uniform.