Unsteady three-dimensional marginal separation caused by surface-mounted obstacles and/or local suction

Abstract

Earlier investigations of steady two-dimensional marginally separated laminar boundary layers have shown that the non-dimensional wall shear (or equivalently the negative non-dimensional perturbation displacement thickness) is governed by a nonlinear integro-differential equation. This equation contains a single controlling parameter $\Gamma$ characterizing, for example, the angle of attack of a slender airfoil and has the important property that (real) solutions exist up to a critical value $\Gamma_c$ of $\Gamma$ only. Here we investigate three-dimensional unsteady perturbations of an incompressible steady two-dimensional marginally separated laminar boundary layer with special emphasis on the flow behaviour near $\Gamma_c$. Specifically, it is shown that the integro–differential equation which governs these disturbances if $\Gamma_c\,{-}\,\Gamma\,{=}\,O(1)$ reduces to a nonlinear partial differential equation – known as the Fisher equation – as $\Gamma$ approaches the critical value $\Gamma_c$. This in turn leads to a significant simplification of the problem allowing, among other things, a systematic study of devices used in boundary-layer control and an analytical investigation of the conditions leading to the formation of finite-time singularities which have been observed in earlier numerical studies of unsteady two-dimensional and three-dimensional flows in the vicinity of a line of symmetry. Also, it is found that it is possible to construct exact solutions which describe waves of constant form travelling in the spanwise direction. These waves may contain singularities which can be interpreted as vortex sheets. The existence of these solutions strongly suggests that solutions of the Fisher equation which lead to finite-time blow-up may be extended beyond the blow-up time, thereby generating moving singularities which can be interpreted as vortical structures qualitatively similar to those emerging in direct numerical simulations of near critical (i.e. transitional) laminar separation bubbles. This is supported by asymptotic analysis.