A weakly nonlinear treatment of shear-layer instability in an inviscid compressible fluid, in which a multiple-scales expansion of the compressible Euler equations about the linear neutral threshold yields a Landau-Stuart amplitude equation dA/dT = mu*A - zeta*|A|^2*A for the slowly varying disturbance amplitude. The sign of the real part of the complex Landau coefficient zeta decides whether the instability saturates into a finite-amplitude limit cycle (supercritical Hopf) or is driven away by finite-amplitude disturbances that linear theory calls stable (subcritical Hopf); in a confined shear layer that sign alternates non-monotonically with Mach number, so a flow can pass through several critical Mach numbers. The transferable skill is deriving and reading an amplitude equation, and seeing why linear growth rates alone cannot predict the long-time state of a compressible shear flow.
This paper examines nonlinear effects in hydrodynamic stability by extending linear stability theory for compressible shear flows into the weakly nonlinear regime using the method of multiple scales.…