A numerical investigation of resonant interactions in adverse-pressure-gradient boundary layers

Abstract

We present direct numerical simulations of the spatial development of normal mode perturbations to boundary layers with Falkner–Skan velocity profiles. Values of the pressure gradient parameter considered range from very small, i.e. nearly flat-plate conditions, to relatively large values corresponding to incipient separation. In almost all cases, we find that the most effective perturbation is one composed of a plane wave and a pair of oblique waves inclined at equal and opposite angles to the primary flow direction. The frequency of the oblique waves is half that of the fundamental plane wave and because the conditions for resonance are satisfied exactly, all modes share a common critical layer, thus facilitating a strong interaction.The oblique waves initially undergo a parametric type of subharmonic resonance, but in accordance with recent analyses of non-equilibrium critical layers, the system subsequently becomes fully coupled. From that point on, the amplification of all modes, including the plane wave, substantially exceeds the predictions of linear stability theory. Good agreement is obtained with the experimental small pressure gradient results of Corke & Gruber (1996). Our growth rates are slightly larger flowing to slight differences in initial conditions (e.g. the angle of inclination of the oblique waves).The spectral element method was used to discretize the Navier–Stokes equations and the preconditioned conjugate gradient method was used to solve the resulting system of algebraic equations. At the inflow boundary, Orr–Sommerfeld modes were employed to provide the initial forcing, whereas the buffer domain technique was used at the outflow boundary to prevent convective wave reflection or upstream propagation of spurious information.