A time accurate compressible interactive boundary layer procedure for airfoils using the quasi-simultaneous method of Veldman is developed. It couples the high frequency transonic small disturbance equation with the complete set of unsteady compressible boundary equations in Levy-Lees variable form, using a pseudo-time derivative of displacement thickness for enhanced stability. Included is a simple procedure for time accurately updating the viscous wake location. The basis of the interaction is an extension of the asymptotic matching condition of Davis for unsteady compressible interaction. This analysis identifies several possible unsteady transonic separation structures and highlights the importance of the pseudo-time derivative in stabilizing the interaction. The method is applied to oscillating airfoils experiencing light shock-induced stall. Comparisons are made with several standard turbulence models. Shock-induced oscillatory flow about the 18% circular arc airfoil is investigated with this method and found to be modeled quite accurately.

The objective of this study is to obtain a better understanding of the fundamental mechanisms governing the unsteady flow past airfoil leading edges. The approach taken is to study the leading edge in isolation and to use the semi-infinite parabola as a model for the leading-edges of conventional airfoils. Numerical solution methods were developed and implemented for the two-dimensional, unsteady, incompressible Navier-Stokes and boundary-layer equations for arbitrary motion of the parabola. Navier-Stokes solutions of the impulsively-started flow past a stationary parabola and of the flow past a pitching parabola compare well with the corresponding computational results for the NACA0012 airfoil. Navier-Stokes solutions for the pitching leading edge were obtained for chord Reynolds numbers up to half-a-million. The sequence of events leading to the unsteady breakaway of the boundary layer, in both the impulsive and pitchup cases, was qualitatively similar for the range of Reynolds numbers considered;We show using Navier-Stokes simulations that small perturbations in the flow field can lead to the formation of eddies in the boundary layer before flow reversal occurs in the base flow. The cases considered here are impulsive changes in the angle of attack, smooth but rapid variations in the angle of attack and introduction of small-amplitude inviscid vortices in the freestream. This type of eddy creation prior to base-flow reversal is a feature of the high-frequency Rayleigh instability. A study of the Reynolds-number scaling of the wavelength of these instabilities yielded a value reasonably close to that predicted by theory. A linear stability analysis of the boundary layer over the parabola was carried out to map the neutral curve for the Rayleigh instability. Preliminary indications are that the disturbances that lead to the eddies in the Navier-Stokes simulations are being initiated within the linearly unstable region bounded by the neutral curve. The linear stability analysis showed that the Rayleigh instability occurs a little after boundary layer velocity profiles become inflectional but much before flow reversal sets in.

The details of an interacting boundary layer algorithm capable of calculating large scale laminar separation past airfoils at low speeds is given. Rationale behind various convergence acceleration methods is given. It is shown that linear based acceleration methods are limited to 50% savings in convergence rate. A nonlinear extrapolation method is proposed and tested on two simple model problems. Savings exceed the 50% limitation of the previous methods. Boundary layer results for laminar flow past symmetric airfoils at zero incidence are presented as a test of the methods. Leading edge marginal separation results at finite Reynolds numbers are presented. Richardson extrapolation of successive calculations is used to improve accuracy. Results for a zero thickness uncambered plate at angle of attack are presented.