On nonlinear effects near the wing-tips of an evolving boundary-layer spot

Abstract

A study is made concerning nonlinear effects arising in the free evolution of three-dimensional disturbances in boundary layers, these disturbances having a spot-like character sufficiently far downstream of the initial disturbance. The theory proposed here takes an inviscid initial-value formulation, typically that involving the three-dimensional unsteady Euler equations; such a formulation is felt to offer hope of considerable analytical progress on the nonlinear side, as well as being suggested by some of the available experimental evidence on turbulent spots and by engineering modelling and previous related theory. As the typical disturbance amplitude is increased, nonlinear effects first enter the reckoning of the large-time large-distance behaviour in edge layers near the spot’s wing-tips which correspond to caustics in linear theory. This behaviour is associated with the major length scales, proportional to (time) 1/2 and to (time), in the evolving spot; within the former scale it is interesting that the three-dimensional Euler flow exhibits a triple-deck-like structure; within the latter scale, in contrast, there are additional time-independent scales in operation. Two possible mechanisms I and II are found for nonlinearity to affect the evolution significantly, near a wing-tip. In I, weakly nonlinear effects make themselves felt in the bulk of the wing-tip flow, accompanied by a near-wall layer where full nonlinearity substantially alters the local vorticity. The analysis then leads to a nonlinear amplitude equation which is the second Painlevé transcendent, whose solution properties are well known. In contrast, mechanism II, which is believed to be the more likely case, has its nonlinearity being mostly due to a three-dimensional mean-flow correction that varies relatively slowly. The resulting nonlinear amplitude equation then has a novel form, solutions for which are obtained computationally and analytically. The further repercussions from the two mechanisms are somewhat different, although each one points to a subsequent flooding of nonlinear effects into the middle of the spot. Mechanism I suggests that strong nonlinearity is produced next by relatively high-amplitude disturbances, whereas the favoured mechanism II indicates instead a strongly nonlinear influence on the mean flow next occurring at input amplitudes that are still relatively low. The additional significance of viscous sublayer bursts is also noted, along with comments on comparisons with experiments and direct numerical simulations. Firm comparisons are felt likely to arise for the next stage implied above, where the middle of a typical spot is affected substantially.