Effect of base flow variation on non-modal stability

Authors: Brandt, L.B., Sipp, , Pralits, J. O., Marquet, 
Document Type: Article
Pubstate: Published
Journal: J. Fluid Mech.
Volume: 687   503-528
Year: 2011


Non-modal analysis determines the potential for energy amplification in stable flows. The latter is quantified in the frequency domain by the singular values of the resolvent operator. The present work extends previous analysis on the effect of base-flow modifi- cations on flow stability by considering the sensitivity of the flow non-modal behavior. Using a variational technique, we derive an analytical expression for the gradient of a singular value with respect to base-flow modifications and show how it depends on the singular vectors of the resolvent operator, also denoted as the optimal forcing and opti- mal response of the flow. As an application, we examine zero-pressure-gradient boundary layers where the different instability mechanisms of wall-bounded shear flows are all at work. The effect of the component-type non-normality of the linearized Navier-Stokes operator, which concentrates the optimal forcing and response on different components, is first studied in the case of a parallel boundary layer. The effect of the convective- type non-normality of the linearized Navier-Stokes operator, which separates the spatial support of the structures of the optimal forcing and response, is studied in the case of a spatially evolving boundary layer. The results clearly indicate that base-flow modi- fications have a strong impact on the Tollmien-Schlichting (TS) instability mechanism whereas the amplification of stream-wise streaks is a very robust process. This is ex- plained by simply examining the expression for the gradient of the resolvent norm. It is shown that the sensitive region of the Lift-up (LU) instability spreads out all over the flat plate and even upstream of it, whereas it reduces to the region comprised between branch I and branch II for the TS waves.