On the stability and the numerical solution of the unsteady interactive boundary-layer equation

Abstract

By means of a high-frequency analysis it is shown that a Rayleigh instability is possible within the interactive boundary-layer formulation. This instability reflects the tendency of large-Reynolds-number flows to be unstable. For most, but not all, pressure-displacement relations both Rayleigh's and Fjortoft's theorems hold, although Fjortoft's criterion is not a sufficient condition for instability. However, for two pressure-displacement relations neither theorem could be proved, and for one of these, unstable flows exist which are free of inflexion points. Analytically, the existence of this instability may result in a finite-time singularity, while numerically the presence of Rayleigh modes often leads to accuracy problems which cannot be overcome by simple grid refinement. A test integration resulted in the generation of small grid-dependent eddies. It is suggested that the instability may be a possible cause of the eddy splitting observed in experiments on unsteady flows through distorted channels. This Rayleigh instability is also possible within the ‘inverse’ boundary-layer formulation, but is absent from classical boundary-layer problems.