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

We present direct numerical simulations of the spatial development of norma l mode perturbations to boundary layers with Falkner-Skan velocity profiles . Values of the pressure gradient parameter considered range from very smal l, i.e. nearly hat-plate conditions, to relatively large values correspondi ng to incipient separation. In almost all cases, we find that the most effe ctive perturbation is one composed of a plane wave and a pair of oblique wa ves inclined at equal and opposite angles to the primary how 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 s hare a common critical layer, thus facilitating a strong interaction. The oblique waves initially undergo a parametric type of subharmonic resona nce, but in accordance with recent analyses of non-equilibrium critical lay ers, the system subsequently becomes fully coupled. From that point on, the amplification of all modes, including the plane wave, substantially exceed s the predictions of linear stability theory. Good agreement is obtained wi th the experimental small pressure gradient results of Corks & Gruber (1996 ). Our growth rates are slightly larger owing to slight differences in init ial conditions (e.g the angle of inclination of the oblique waves). The spectral element method was used to discretize the Navier-Stokes equati ons and the preconditioned conjugate gradient method was used to solve the resulting system Of algebraic equations. At the inflow boundary, Orr-Sommer feld modes were employed to provide the initial forcing, whereas the buffer domain technique was used at the outflow boundary to prevent convective wa ve reflection or upstream propagation of spurious information.