Adjoint parabolized stability equations for receptivity prediction

Abstract

This paper presents the Adjoint Parabolized Stability Equations (APSE) which are used to predict the receptivity of shear layers to a variety of disturbances. Results from the APSE are first carefully validated against solutions of the Adjoint Navier-Stokes (ANS) equations which demonstrates that APSE is an accurate and effecient means of predicting receptivity. Then APSE is used to document the nonparallel receptivity characteristics of both 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 under all conditions. 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. This is in contrast to the well known effects of pressure gradient on TS instability growth rates. Likewise, our investigations for three-dimensional disturbances also show that oblique modes have greater receptivity than two dimensional waves, again in contrast to the effects of obliquity on instability growth rates. In general there is a trade-off between receptivity and instability -the stronger the instability the weaker the receptivity.

Structural changes of laminar separation bubbles induced by global linear instability

“…The second term in (3.13) imposes the conservation of the bi-linear concomitant (Hill 1995;Dobrinsky & Collis 2000) in the domain, and determines the boundary conditions for the adjoint problem on account of the boundary conditions for the direct problem.…”

On predicting receptivity to surface roughness in a compressible infinite swept wing boundary layer

“…Hill 22 first utilised the ALNS formula in the context of the two-dimensional (2D), parallel Blasius boundary layer, and carried out a thorough parametric study on the optimum conditions for generating TS wave disturbances. Non-parallel receptivity effects of the Blasius flow were examined using adjoint parabolised stability equations (APSE) 23 , while Dobrinsky and Collis 24,25 extended the analysis to include the full spectrum of Falkner-Skan flows. The APSE for a quasi three-dimensional compressible flow were derived by Pralits at al.…”

High-Order Numerical Methods for BiGlobal Flow Instability Analysis and Control

“…In open systems containing boundary layers, adjoint boundary conditions may be devised following the general procedure of expanding the bilinear concomitant in order to capture traveling disturbances Dobrinsky & Collis (2000). When the focus is on global modes concentrated in certain regions of the flow, as the case is, for example, for the global mode of laminar separation bubble (Theofilis (2000); Theofilis et al (2000)) the following procedure may be followed.…”

“…Here the operator N † (q) results from linearization of the convective and viscous terms in the direct and adjoint Navier-Stokes equations and is explicitly stated elsewhere (e.g. Dobrinsky & Collis (2000)). The quantitiesq * =(ũ * ,ṽ * ,w * ) T andp * denote adjoint disturbance velocity components and adjoint disturbance pressure, and j(q * ,q * ) is the bilinear concomitant.…”

“…The derivation of the complex BiGlobal eigenvalue problem governing adjoint perturbations is constructed using the Euler-Lagrange identity Bewley (2001); Dobrinsky & Collis (2000); Giannetti & Luchini (2007); Morse & Feshbach (1953); Pralits & Hanifi (2003),…”