Similarity-Type Transformation of Boundary-Layer Equations in Contravariant Form

Abstract

In particular for three-dimensional boundary layers it is worthwhile to consider transformations of the boundary-layer equations which keep the thickness of the boundary layer in the computational space nearly constant. In two-dimensional laminar boundary-layer predictions similarity-type, such as Blasius or Levy-Lees, transformations generally achieve the desired goal. In turbulent boundary layers the advantage of such transformations is not entirely clear since the Blasius or Levy-Lees scaling had been devised for laminar flows, and turbulent boundary layers grow at a faster rate. One of the main features of such transformations is that the flow variables are scaled by the local values at the “edge” of the boundary layer which yields a constant value of one there, and which prevents in particular the transformed velocity profiles to change drastically in the streamwise direction, at least in the absence of a sustained strong adverse pressure gradient. Such transformations have been maintained for three-dimensional problems (see e.g. BLOTTNER and ELLIS [15] or CEBECI et al. [18]) although it is, in general, not possible to scale the crosswise velocity component with its edge value since that one may change its sign.