Nonlinear evolution of oblique waves on compressible shear layers

Abstract

We consider the effects of critical-layer nonlinearity on spatially growing oblique instability waves on compressible shear layers between two parallel streams. The analysis shows that mean temperature non-uniformities cause nonlinearity to occur at much smaller amplitudes than it does when the flow is isothermal. The nonlinear instability wave growth rate effects are described by an integro-differential equation which bears some resemblance, to the Landau equation in that it involves a cubic-type nonlinearity. The numerical solutions to this equation are worked out and discussed in some detail. We show that inviscid solutions always end in a singularity at a finite downstream distance but that viscosity can eliminate this singularity for certain parameter ranges.

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

“…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.…”

“…The resulting system can be obtained by combining those given in §3 and in the previous subsection. We thus have 14) where the temperature equation (4.12) includes linear inhomogeneous terms involving both T ′ c and T ′′ c , and the vorticity equation (4.13) includes linear inhomogeneous terms involving U ′′′ c and T ′′ c . We note that…”

“…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 this paper, we first consider a weakly compressible regime, corresponding to the distinguished scaling M = O(ǫ 1/4 ), for which the strongly nonlinear structure persists but is affected by compressibility at leading order (where ǫ ≪ 1 measures the magnitude of the instability mode).…”

“…We then assume that T ′ c = O(ǫ 1/2 ) so as to include the compressible nonlinear effect in the strongly nonlinear structure. The final evolution system reduces to that in the incompressible limit, while on the other hand contains the amplitude equation in the weakly nonlinear theory of Goldstein & Leib (1989), in that the latter can be derived in an appropriate limit. Thus our final system is a composite theory uniformly valid for all 0 M O(1).…”

Non-equilibrium, nonlinear critical layers in laminar-turbulent transition

“…In passing, we remind that if the flow is compressible, the temperature perturbation, 0 say, has a singularity of simple pole [11] x/Re = xn + #(Ax) with Ax = 0(1) (3.2)…”