Linear stability and receptivity analyses of the Stokes layer produced by an impulsively started plate

Abstract

The stability and the receptivity of the boundary layer produced by the impulsive motion of a flat plate in its plane is studied. The evolution of two-dimensional traveling disturbance waves for this physical situation, known as Stokes’ first problem, is treated by integrating directly the (parabolic in time) linearized Navier–Stokes equation and by a multiple-scale approach. In the asymptotic analysis, the Orr–Sommerfeld equation is found at leading order. Through the compatibility condition for the equation at next order, an O(1) correction to growth rates and frequencies is achieved. Such corrections are found to be very mild. After having established that the leading-order results are adequate when looking at the stability characteristics of the flow for a given (large) time, a receptivity analysis is performed. The adjoint of the parabolic system is obtained, and through its backward-in-time integration, the initial and wall Green’s functions are obtained. These are then compared to the results of the multiple-scale receptivity analysis.