A Pseudo-Spectral Approach for the Incompressible Boundary Layer Equations with Automatic Normal Scaling

Abstract

A pseudo-spectral numerical method for the solution of the incompressible 2D boundary layer equations is presented. The method is based on Fourier expansion in the normal transformed coordinate, similarly to the transformation that leads to the polynomial Tchebichev expansion in finite intervals, but appropriated to semi-infinite intervals, so that no extra parameter is needed for the outer boundary of the layer. A scaling is applied to the normal coordinate but with the innovation that it is based on the computed boundary layer thickness, i.e. not assuming a priori a variation for it. Spectral decay of the expansion coefficients is shown for the similar solution to the family of wedge flows. Also, spectral convergence of the error is shown for the case of a convergent channel (one of the similar "wedge flows"), for which an analytical solution is available. The method pretends to have a good performance also when using very few parameters, so that results with four terms in the Fourier series (it amounts to two independent parameters) are compared with the well known method from von Karman and Pohlhausen.

For 3D problems, the boundary layer equations are solved in a completely general mesh and coordinate system on the surface using the tensorial form of the equations. In order to advance the solution in the streamwise coordinate a mesh-less approximation is used in the coordinates on the surface. This feature allows the treatment of very general geometries. 3D numerical examples include the yawed cylinder and flat plate, and 3D axisymmetric flows like the cone, the sphere and a rotating sphere.