Nonlinear roll-up of externally excited free shear layers

Abstract

We consider the effects of strong critical-layer nonlinearity on the spatially growing instabilities of a shear layer between two parallel streams. A composite expansion technique is used to obtain a single formula that accounts for both shear-layer spreading and nonlinear critical-layer effects. Nonlinearity causes the instability to saturate well upstream of the linear neutral stability point. It also produces vorticity roll-up that cannot be predicted by linear theory.

Nonlinear development of subsonic modes on compressible mixing layers: a unified strongly nonlinear critical-layer theory

“…Without loss of generality, the origin of the coordinate is taken to be x n . Close to x n , a critical layer emerges, where nonlinear effects may become important first, whilst the unsteady flow outside the critical layer remains linear (Goldstein & Leib 1988). The analysis must therefore be performed for two distinct regions: a linear inviscid outer layer, and a nonlinear viscous inner region, i.e.…”

“…Note that when β T = 1 and M = 0, T ′ c = 0. In this case, the critical-layer nonlinearity is associated with the logarithmic term inũ (1) 1 (see (2.36)), and the dynamics is governed by the strongly nonlinear theory of Goldstein & Leib (1988). If β T = 1 and/or M = O(1), then T ′ c is of O(1) in general, and the criticallayer nonlinearity is associated with the pole in T 0 (see (2.30)), and so nonlinear effects operate in a weakly nonlinear fashion (Goldstein & Leib 1989, Leib 1991.…”

“…Figure 6 shows some of the harmonics, measured by (cf. Goldstein & Leib 1988) 18) for the case of Λ = 0.001. As is illustrated, the first harmonic m = 2 has the most significant magnitude.…”

“…At high Reynolds numbers of practical interest, previous studies have shown that the dominant nonlinear effect controlling the evolution of an instability wave comes from the so-called critical layer. In the incompressible limit (M = 0), the critical-layer dynamics are strongly nonlinear, with nonlinearity being associated with the logarithmic singularity of the velocity fluctuation (Goldstein & Leib 1988). In contrast, in the fully compressible regime (M = O(1)), nonlinearity is associated with a simple-pole singularity in the temperature fluctuation, and enters in a weakly nonlinear fashion (Goldstein & Leib 1989).…”

“…In addition, the non-equilibrium effect associated with the slow modulation of the wave envelope may also appear at leading order. Goldstein & Leib (1988) and Goldstein & Hultgren (1988) showed that a planar mode on an incompressible mixing layer evolves into a nonlinear stage where the critical layer is non-equilibrium, and strongly nonlinear in the sense that nonlinearity enters as a 'coefficient' of the operator governing the dynamics rather than as an inhomogeneous term of a linear system in weakly nonlinear theory (cf. Stuart 1960).…”