Weakly nonlinear acoustic instabilities

With the aim of eventually improving numerical solutions of small-scale phenomena, the Hunter-Keller theory of weakly nonlinear high-frequency waves is applied to the study of short wavelength instabilities in inviscid fluids driven by a heat or pressure source. A nonlinear damping effects is found which, for acoustic perturbations of a stationary, homogeneous state, reduces the growth rate to half the linear estimate. This is due primarily to the interactions of the expansion fan and the weak shock generated by the cumulative effect of the nonlinear convective term. For acoustic perturbations driven by an unbalanced heat source, the nonlinear damping actually stabilizes some modes which are unstable according to the linear theory. For the isentropic compression of a spherical shell of material obeying a γ-law equation of state, it is shown that the nonlinear damping again reduces the acoustic growth rate to the half the value predicted by conventional linear stability analyses.