Spatial evolution of nonlinear acoustic mode instabilities on hypersonic boundary layers

Abstract

We consider the effects of strong critical-layer nonlinearity on the spatial evolution of an initially linear ‘acoustic mode’ instability wave on a hypersonic flat-plate boundary layer. Our analysis shows that nonlinearity, which is initially confined to a thin critical layer, first becomes important when the amplitude of the pressure fluctuations becomes O(1/M4InM2), where M is the free-stream Mach number. The flow outside the critical layer is still determined by linear dynamics and therefore takes the form of a linear instability wave — but with its amplitude completely determined by the flow within the critical layer. The latter flow is determined by a coupled set of nonlinear equations, which we had to solve numerically.

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

“…Vorticity roll-up is much stronger than for β T = 1 shown in figure 7. Closed "islands", similar to those found in Goldstein & Wundrow (1990), soon appear. Further downstream, irregular small-scale motions cause contours to break up, forming tiny islands or excessive "wiggles".…”

“…The perturbed flow can be written as 15) where the perturbation is assumed to be an instability mode, whose overall magnitude is small, i.e. ǫ ≪ 1.…”

Nonlinear Stability of Nonstationary Cross‐Flow Vortices in Compressible Boundary Layers

“…The equations ( We note that the equations are now of a form similar to the evolution equations obtained by Goldstein and Wundrow [40] 17), govern the nonlinear evolution of the cross-flow vortex. Note that for a given basic flow, the constants J, J I are fixed and the only free parameters are Yc and the wave angle ().…”

“…Excellent reviews of the early ideas on nonlinear critical layers are given in [34,35], and more recent work is surveyed in [16,36]. The assumptions and scalings used here lead to a fully nonlinear partial differential system for the evolution of the vortex, as in [1, [37][38][39][40], and not an integro-differential equation of the Hickemell [41] type. The scalings adopted are appropriate for low-frequency long-wavelength cross-flow vortices that have small growth rates.…”

Nonlinear evolution of singular disturbances to a tanh3y mixing layer

“…This work was later extended [10] to include viscosity, and to a more general flow including comparisons with experimental data [17]. Other authors later added additional effects, such as hypersonic flow [13] and stratified flow [19], or studied different base flows, such as the Bickley jet [18] for which there were not one but two neutral modes. In the present study, rather than the traditional hyperbolic tangent, we take as our base flow u o (;y) = tanh 3 y.…”