On the nature of singularities inherent, under a given analytic distribution of the external pressure, in solutions of the prandtl equations near the point of separation

Abstract

A comprehensive analysis of a classical two-dimensional boundary layer is developed with the aim of revealing possible types of singularities related to separation. According to the basic assumption, the limit flow regime with a singular point of vanishing skin friction obeys an analytic solution where the frictional intensity approaches zero according to quadratic law rather than varing linearly with distance. Small deviations from the limit regime are described in terms of eigenfunctions, three of which involve singularities. The flow field for all supercritical regimes is continued into a small region centered about the singular characteristic springing from the point of vanishing skin friction in the undisturbed limit solution. The solvability condition for a key boundary-value problem posed in this region as a result of matching the global-scaled solution upstream gives three different types of singularities of the Prandtl equation. According to the weakest-type singularity the skin friction varies as the square root of the cube of the local distance. The next type includes a sudden change in the wall shear stress derivative with respect to the coordinate along the solid surface, it was ruled out of the basic limit solution. Then the famous Landau-Goldstein singularity with the skin friction being proportional to the square root of the local distance evolves. A still more complicated flow pattern may be composed of the singularity with a sudden change in the wall shear stress derivative superseded by the Landau-Goldstein singularity at some small distance downstream. A qualitative comparison between theoretical predictions and experimental data for a circular cylinder in an incompressible stream is made with the emphasis on conceivable explanations for the nonmonotonic behavior of frictional intensities in the transitional range of Reynolds numbers.