Instability in a viscous flow driven by streamwise vortices

Abstract

Unsteady separation processes at large finite, Reynolds number, Re, are considered, as well as the possible relation to existing descriptions of boundary-layer separation in the limit Re → ∞. The model problem is a fundamental vortex-driven three-dimensional flow, believed to be relevant to bursting near the wall in a turbulent boundary layer. Bursting is known to be associated with streamwise vortex motion, but the vortex/wall interactions that drive the near-wall flow toward breakdown have not yet been fully identified. Here, a simulation of symmetric counter-rotating vortices is used to assess the influence of sustained pumping action on the development of a viscous wall layer. The calculated solutions describe a three-dimensional flow at finite Re that is independent of the streamwise coordinate and consists of a crossflow plane motion, with a developing streamwise flow. The unsteady problem is constructed to mimic a typical cycle in turbulent wall layers and numerical solutions are obtained over a range of Re. Recirculating eddies develop rapidly in the near-wall flow, but these eddies are eventually bisected by alleyways which open up from the external flow region to the wall. At sufficiently high Re, an oscillation was found to develop in the streamwise vorticity field near the alleyways with a concurrent evolution of a local spiky behaviour in the wall shear. Above a critical value of Re, the oscillation grows rapidly in amplitude and eventually penetrates the external flow field, suggesting the onset of an unstable wall-layer breakdown. Local zones of severely retarded streamwise velocity are computed which are reminiscent of the low-speed streaks commonly observed in turbulent boundary layers. A number of other features also bear a resemblance to observed coherent structure in the turbulent wall layer.

Turbulent Marginal Separation and the Turbulent Goldstein Problem

“…In the limitŝ → 0 the stream function in the boundary layer where Y = O(1) is given by ψ ∼ α Ψ 00 (Y ), see Eqs. (16) and (51). We now seek the perturbations there owing to the induced pressure p 10 upstream and downstream ofŝ = 0.…”

“…It rather is a consequence of the behavior of the mixing length, which, related to the two-tiered splitting of the wake regime of the boundary layer as sketched in figure 2, characterizes the outer wake layer. We furthermore mention that, as a result of the limits assumed in (16) and (17), in both the main regions 2 − and 2 + where Y = O(1), see figure 4 on page 11, their perturbations for small values of s are governed by the convective operator in Eq. (8) only.…”

“…where the upper and lower signs refer to the cases s → 0 − and s → 0 + , respectively. The conditions at η = 0 follow from the wake-type boundary conditions (9), and the requirement for η → ∞ reflects the match with the flow regimes 2 − and 2 + , see figure 4 on the next page, where the relations (16) and (17) hold. It can be shown (as will be demonstrated in a separate paper) that in the upstream case the problem (22) has only the obvious solution…”

“…Moreover, it must be emphasized that David Walker was substantially involved in providing the initial steps towards a systematic insight into the very complex turbulent near-wall dynamics by investigating the unsteady Navier-Stokes equations in the high-Reynolds-number limit. 1,[14][15][16][17] Despite the undeniable progress, where much of it must be attributed to David Walker, asymptotic methods have contributed towards an understanding of the fundamental physics of wall-bounded turbulent shear flows, a fully self-consistent theory describing turbulent boundary layer separation in the limit of high Reynolds number is not available at the moment. In particular, the literature lacks a rational description of the, from an engineering point of view, very important case of separation from a smooth surface which is caused by a smooth adverse pressure gradient, imposed by the external flow.…”

Singularity formation for Prandtl’s equations

“…The findings of Brinckman and Walker are probably somehow related to this result. In [7], using the full 3D Navier-Stokes equations, they studied the response of the boundary layer to an array of counter-rotating vortices placed in the outer flow. Their goal was to mimic the mechanism through which vorticity regenerates in the boundary layer.…”