The initial-value problem for a modeled boundary layer

Abstract

The stability of a basic parallel flow of a viscous incompressible fluid has been traditionally investigated by linearizing the Navier–Stokes equations and using the method of normal modes in order to determine the temporal behavior. In this paper we adopt a method introduced by Lord Kelvin and used much in recent years, solving the equations instead as an initial-value problem and thereby the full dynamics, both the early transient and the long time asymptotic, can be determined directly. The basic flow investigated is a boundary-layer modeled by a piecewise linear profile. Various initial distributions of velocity and vorticity are used as a basis of fundamental solutions to represent more general initial disturbances of the basic flow. The simplicity of the model permits the problem to be solved by use of a moving-coordinate system and of matched asymptotics in the limit of small viscosity. The solutions are remarkably explicit, although complicated and substantiate the increasingly common view that linear disturbances, although decaying ultimately according to the linear theory, may grow so much transiently as to excite nonlinear growth and effective instability; in particular, streamwise vortices are found to be strongly amplified. Although not done here, the method can be applied to other parallel shear flows.