Concerning marginal singularities in the boundary-layer flow on a downstream-moving surface

Abstract

The formation of separation singularities in solutions of the classical boundary-layer equations is studied numerically and analytically for the case of a two-dimensional incompressible steady flow near a solid surface moving in the direction of the main stream. Unlike the previously studied regime of the incipient separation located at the maximum point in the external pressure distribution, the breakdown in this work occurs under an adverse pressure forcing and involves a regular flow field upstream of the Moore-Rott-Sears point with an algebraic non-analyticity downstream. Small deviations from the precisely regular approach to the singular point are shown to result in an exponential amplification of linear disturbances; in the subsequent nonlinear stage the solution terminates in a finite-distance blow-up singularity or, alternatively, continues in a regular fashion across the singular station. The case of asymptotically small slip velocities is considered and a connection with marginal separation on a fixed wall is discussed.

On the nonlinear growth of two-dimensional Tollmien–Schlichting waves in a flat-plate boundary layer

“…and Professor S. I. Chernyshenko (private communication) suggest that x s = 0. If this is the case here, then the singularity for b = b 0s will have the form proposed by Timoshin (1996) for marginal separation in non-periodic boundary-layer flow over a downstream-moving wall. Figure 5(d) is a plot against b 0 of the rate of change of amplitude db 0 /dT calculated from (5.37) and (5.71).…”

“…Its analytic structure will depend on whether or not the position of the singularity, x s , is located at the maximum point in the external pressure distribution, i.e. whether or not x s = 0 (Professor S. N. Brown, private communication 1987;Sychev 1987;Negoda & Sychev 1987;Timoshin 1996). For the time being we note that more detailed calculations for other external velocities U e by S.J.C.…”

“…We do not pursue the details of this new régime here, other than to note that when the separation is small, i.e. marginal, and if x s = 0, then the flow in the Stokes layer will be as given by Timoshin (1996).…”

On-wall and interior separation in a two-fluid boundary layer

“…4a-c for the obstacle shape h(x) = h 0 exp(−x 2 ) ,ū s = 0.15 and several values of h 0 . Note that the solution does not experience the moving-wall singularity of the type described in [24] at the first appearance of flow reversal since the pressure function p + is not specified in advance. However, as the depth of the indentation increases, the shape of the streamlines at the incipient vortex develops a cusp (Fig.…”