Classical boundary layer behavior in the vicinity of a curvature jump of the wall contour

Abstract

Laminar flow past a parametrized family of semi-infinite thin bodies with adjustable curvature jump in their contour is investigated according to Prandtl's classical (first order) boundary layer theory. Accounting for the slenderness a perturbation ansatz is used to derive a potential flow solution as an approximation for the part of the surrounding flow, where viscosity effects can be neglected. The assumption of a slightly perturbed oncoming flow is violated in close vicinity of the leading edge stagnation point, where a local solution is found in an appropriately stretched coordinate system. Both solutions are combined in the spirit of matched asymptotic expansions using Van Dyke's matching rule to receive a uniformly valid approximation (compound solution) for the velocity distribution at the surface, which defines the prescribed pressure distribution along the viscous boundary layer in the immediate neighborhood of the solid wall. Finally, the boundary layer equations are derived and numerically solved for increasing strength of the curvature discontinuity at the surface. The break-down of the classical boundary layer theory is indicated by the occurrence of a Goldstein-singularity at the location of the curvature jump independent of its magnitude. This outcome suggests the application of advanced asymptotic concepts to the presented problem to account for the interaction between the boundary layer and the external flow (i.e. triple-deck theory).