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.

The impact of static and dynamic roughness elements on flow separation

“…The downstream (or upstream) movement of the local separation point has been seen to depend on the placement, width, height and, in the dynamic case, oscillation frequency of the roughness element. Despite the differences in configurations studied, the choice of parameters for the most downstream shift in the position of local separation is qualitatively similar to the required choice for the greatest increase in the critical parameter Γ c , linked to the angle of attack, for Braun & Kluwick (2004), above which no marginally separated flow solutions exist. The same is true for the most upstream shift of local separation point or decrease in Γ c .…”

“…This correction to the leading order wall shear was obtained in part as the solution to the forced Fisher equation-forced by the form of the obstacle vibrations and the leading order solution-and their main interest in it was as an analysis of the bursting of the small separation bubble as a result of the finite time blow-up of the governing equation. Indeed, it is well known that equation (4.1) is ill-posed and that a numerical solution through time marching will lead to a singularity at some finite time T 0 , when the magnitude of the displacement function A becomes arbitrarily large at some streamwise position (Smith 1982;Braun & Kluwick 2004).…”

“…Equally inspiring is the study by Braun & Kluwick (2004) (henceforth referred to as BK), which is particularly relevant to the study of such dynamic roughness elements and concerns marginal separation. The local separation of a steady two-dimensional flow from the surface of a body is generally taken to occur where the skin friction becomes zero after a length of attached flow (positive skin friction) immediately upstream.…”

“…To maintain consistency between this paper and those of BK (Braun & Kluwick 2002, 2004, we will keep the same notation. A bifurcation takes place for subcritical values of Γ , with two flow regimes possible (Braun & Kluwick 2002); while for supercritical values, as mentioned, no steady solution is possible.…”

Unified description of bifurcation processes associated with laminar boundary‐layer separation

“…In contrast to weakly 3D flows the solution of the eigenvalue problem does not depend on the lateral coordinate which is no longer passive but takes part actively in the time evolution process by entering the function c which by inspection of the problem for A 2 is found to be governed by the general form of Fisher's equation obtained from equ. (1) by the substitution du/dt → ∂u/∂t − ∂ 2 u/∂z 2 where use has been made of the transformed quantity c(Z,T ) ∝ (2u(z, t) − 1) withZ ∝ z,T ∝ t, [4].…”