Critical layer and weakly nonlinear evolution of unstable quasi-monochromatic perturbations in shear flows

Abstract: By Howard’s semicircle theorem, in a plane-parallel shear flow, the real part of the phase velocity $c$ of an unstable perturbation $\sim f(z)\exp[ik(x-ct)]$ is between the minimum and maximum of the flow velocity $V_x = U(z)$ and coincides with $U$ on a critical level $z=z_c$ so that $\mathrm{Re} \, c = U(z_c)$. In a narrow neighborhood of this level, — so called critical layer (CL), — liquid particles are in phase resonance with the wave and intensively interact with it. In the framework of an idealized statement of the problem taking no account of dissipation (viscosity), unsteadiness, and nonlinearity, the perturbation eigenfunction $f(z)$ is singular on the critical level. Taking into consideration any one of these factors makes the solution regular, but the relative magnitude of the perturbation inside the CL remains large. For this reason, it is the CL that makes the leading-order contribution into nonlinear interactions, and this fact simplifies the study of a weakly nonlinear evolution of an unstable perturbation.

Each of these factors specifies the length scale associated with it, namely,

(i) viscous $L_ν = (k^3 \mathrm{Re})^{-1/3} = O(ν^{1/3})$,

(ii) unsteady $L_t = |(kU'_c A)^{-1} d|A|/dt| = O(γ)$,

(iii) nonlinear $L_N \sim |A/U'_c|^δ$,

where $\mathrm{Re}$ is Reynolds number, $A(t)$ is the perturbation amplitude, $δ$ depends on the behavior of $f(z)$ in the neighborhood of the critical level, and the prime denotes the derivative in $z$. The greatest of these scales determines not only the width of the CL, but also the behavior of the solution inside it. Therefore, it is appropriate to distinguish between viscous, unsteady, and nonlinear CLs, taking into account that the CL kind may change in the process of evolution in accordance with the scale ratio.

As a result, only a limited number of basic scenarios of evolution do exist. The realization of one scenario or another, or some sequence of them depends mainly on the degree of supercriticality of the basic unstable flow and on the nature of the singularity at the critical level.

mathematical physicsanalysis of PDEsclassical analysis and ODEsdynamical systemsnumerical analysisexactly solvable and integrable systemsfluid dynamics

Audience: researchers in the topic

Mathematical models and integration methods