Abstract

There is evidence to suggest that the boundary-layer equations are not the high-Reynolds number limit of solutions to the Navier-Stokes equations. Numerical calculations by Brinckman and Walker show that at sufficiently high Reynolds number, a short-wavelength instability may appear before the separation time of solutions to the unsteady boundary-layer equations. Using the Kuramoto-Sivashinsky equation as a model for the problem with key similarities and one spatial dimension, we will show that a similar short-wavelength instability can arise before the shock formation time of the kinematic-wave equation. We will then show that this instability can be explained through tracking exponentially-small terms in the asymptotic solution structure, invisible to traditional matched asymptotics approaches. These terms, and their associated Stokes and anti-Stokes lines, can be found by tracking singularities of the kinematic-wave equation in the complex plane.