A Reynolds and Prandtl uniform numerical method for Prandtl's boundary layer problem for flow past a wedge with heat transfer

We consider Prandtl's boundary layer problem for incompressible laminar flow past a wedge with heat transfer. When the Reynolds or Prandtl number is large the solution of this problem has two parabolic boundary layers; one in the velocity components, the other in the temperature component. We construct a direct numerical method for computing approximations to the solution of this problem using a compound piecewise-uniform mesh appropriately fitted to the parabolic boundary layer. Using this numerical method we approximate the self--similar solution of Prandtl's 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. By means of extensive numerical experiments, for a range of values of the Reynolds number, Prandtl number and number of mesh points, we verify that the constructed numerical method is Reynolds and Prandtl uniform, in the sense that the computed errors for the velocity components, their derivatives and the temperature component, in the discrete maximum norm are Reynolds and Prandtl uniform. We use a special numerical method related to the Blasius technique to compute a semi--analytic reference solution with required accuracy with respect to the Reynolds and Prandtl numbers for use in the error analysis.