A new transformation and integration scheme for the compressible boundary layer equations, and solution behavior at separation

Abstract

A new coordinate and variable transformation for the two-dimensional boundary layer equations is presented. The normal coordinate is stretched with a scaling length determined by the local solution. The boundary layer thickness is then essentially constant in computational space for the most types of flows, including separation bubbles and rapidly growing turbulent boundary layers. Similarity solutions can be obtained for all wedge flows. Two finite difference schemes are presented: the Shifted Box Scheme and the Double-Shifted Box Scheme. Both schemes are more resistant to streamwise profile oscillations than the standard Keller's Box Scheme. All governing equations, including the turbulence model, are solved simultaneously as a fully coupled system. This is faster and more robust than conventional weak-coupling iteration schemes. The solution scheme implementation presented makes no restriction on one boundary condition. Any point or integral quantity such as edge velocity, wall shear, displacement thickness, or some functional relationship between two or more of such quantities can be prescribed. The behavior of the boundary layer solution near separation is investigated. It is demonstrated that non-unique solutions always exist whenever an adverse pressure gradient is specified. This bifurcation of the solution is responsible for inability of calculations with prescribed pressure or edge velocity to be carried past separation.