Receptivity Problems in the Weakly-Nonlinear Stability Theory at Large Reynolds Numbers

Abstract

Weakly-nonlinear analysis introduced in [1,2] for the case of marginally unstable waves in a strictly parallel basic flow at finite Reynolds numbers has been widely used in more recent theories which deal with nonlinear instabilities in complicated nonparallel and time-dependent flows at, however, asymptotically large values of the typical Reynolds number; see e.g. [3–5]. Our concern in this paper is with the derivation of physically realistic and mathematically consistent initial conditions for the controlling equations of the high-Reynolds-number weakly-nonlinear theory or, in other words, with the nonlinear receptivity problem as contrasts with linear receptivity processes studied within the asymptotic framework in [6–8] and in many subsequent contributions. The analysis will be restricted to spatial instabilities induced by a time-periodic source such as e.g. a vibrating ribbon, a time-dependent injection/suction through the flow boundary, or an external acoustic field interacting with a surface irregularity. The main point we intend to discuss is a fundamental difference in the receptivity properties of lower- and upper-branch neutral modes stemming from the fact that the limiting equations for the lower-branch instability contain typically growing disturbances corresponding to harmonics of the forcing frequency.