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

Abstract

This paper is concerned with the nonlinear instability of compressible mixing layers in the regime of small to moderate values of Mach numberM, in which subsonic modes play a dominant role. 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 the nonlinearity being associated with the logarithmic singularity of the velocity fluctuation (Goldstein & Leib,J. Fluid Mech.vol. 191, 1988, p. 481). 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,J. Fluid Mech.vol. 207, 1989, p. 73). In this paper, we first consider a weakly compressible regime, corresponding to the distinguished scalingM=O(ε1/4), for which the strongly nonlinear structure persists but is affected by compressibility at leading order (where ε ≪ 1 measures the magnitude of the instability mode). A strongly nonlinear system governing the development of the vorticity and temperature perturbation is derived. It is further noted that the strength of the pole singularity is controlled byT′c, the mean temperature gradient at the critical level, and for typical base-flow profilesT′cis small even whenM=O(1). By treatingT′cas an independent parameter ofO(ε1/2), we construct a composite strongly nonlinear theory, from which the weakly nonlinear result forM=O(1) can be derived as an appropriate limiting case. Thus the strongly nonlinear formulation is uniformly valid forO(1) Mach numbers. Numerical solutions show that this theory captures the vortex roll-up process, which remains the most prominent feature of compressible mixing-layer transition. The theory offers an effective tool for investigating the nonlinear instability of mixing layers at high Reynolds numbers.