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Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer

Published online by Cambridge University Press:  14 October 2011

Luca Brandt ,
Denis Sipp ,
Jan O. Pralits  and
Olivier Marquet
Luca Brandt*
Affiliation:
Linné Flow Centre, KTH Mechanics, S-100 44 Stockholm, Sweden
Denis Sipp
Affiliation:
ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France
Jan O. Pralits
Affiliation:
DIMEC, University of Salerno, Via Ponte don Melillo, 84084 Fisciano (SA), Italy
Olivier Marquet
Affiliation:
ONERA/DAFE, 8 rue des Vertugadins, 92190 Meudon, France
*
†Email address for correspondence: [email protected]
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Abstract

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 modifications on flow stability by considering the sensitivity of the flow non-modal behaviour. 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 the optimal forcing and optimal 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 modifications have a strong impact on the Tollmien–Schlichting (TS) instability mechanism whereas the amplification of streamwise streaks is a very robust process. This is explained 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 is reduced to the region between branch I and branch II for the TS waves.

JFM classification

Boundary Layers: Boundary layer receptivity Flow Control: Control theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 687 , 25 November 2011 , pp. 503 - 528
Copyright
Copyright © Cambridge University Press 2011

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