Adjoint Methods for Receptivity Prediction in Nonparallel Flows

The Adjoint Parabolized Stability Equations (APSE) are used to predict the receptivity of boundary layers to a variety of disturbances. The method is validated through comparisons to direct solutions of the linearized Navier–Stokes (LNS) and adjoint Navier–Stokes (ANS) equations and APSE is found to accurately predict receptivity with significant gains in efficiency compared to Navier–Stokes based methods. The APSE are used to document the nonparallel receptivity characteristics of both the Blasius and Falkner–Skan boundary layers for two-dimensional and oblique Tollmein– Schlichting (TS) instabilities. These results are compared to receptivity predictions based on local parallel theory in order to establish the effects of mean boundary layer growth on receptivity. In general, the inclusion of nonparallel effects for the receptivity prediction of TS instabilities is found to be small over a wide range of frequencies, pressure gradients, and wave angles. However, as wave angle is increased, the influence of nonparallel effects does increase as the streamwise wavelength becomes longer compared to the growth of the boundary layer. Likewise, we show that for distributed sources, the use of parallel theory can lead to more significant errors that mainly arise due to errors in the spatial growth rate predicted by parallel theory. Comparing results from Blasius and Falkner–Skan base flows shows that adverse pressure gradients tend to reduce receptivity while favorable pressure gradients lead to an increase in receptivity. Likewise, our investigations for three-dimensional disturbances also show that oblique modes have greater receptivity than two dimensional waves. Particularly high receptivity can be achieved for oblique disturbances in favorable pressure gradient flows, although these disturbances show only weak linear instability. In general there is a trade-off between receptivity and instability – the stronger the instability, typically the weaker the receptivity.