Multiple-scale approximation of instabilities in unsteady boundary layers

A general procedure is developed to study the stability of unsteady boundary layers using complex-ray theory. The propagation of small disturbances is described by a high-frequency (optical) approximation similar to the one adopted for wave propagation in nonuniform media. The ray trajectories, formally defined as the characteristic lines of the eikonal equation, are described by a system of first-order differential equations. These lines are complex valued and provide the main contribution to the propagation of the wave (its Green’s function). As an application, we present the analysis of the flow past on an oscillating airfoil. The propagation of a harmonic disturbance inside the boundary layer is considered and some numerical transition-prediction results are discussed.