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Boussinesq global modes and stability sensitivity, with applications to stratified wakes

Published online by Cambridge University Press:  12 January 2017

Kevin K. Chen Open the ORCID record for Kevin K. Chen [Opens in a new window]  and
Geoffrey R. Spedding
Kevin K. Chen
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
Geoffrey R. Spedding
Affiliation:
Department of Aerospace and Mechanical Engineering, University of Southern California, Los Angeles, CA 90089, USA
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Abstract

For the Boussinesq equations, we present a theory of linear stability sensitivity to base flow density and velocity modifications. Given a steady-state flow with small density variations, the sensitivity of the stability eigenvalues is computed from the direct and adjoint global modes of the linearised Boussinesq equations, similarly to Marquet et al. (J. Fluid Mech., vol. 615, 2008, pp. 221–252). Combinations of the density and velocity components of these modes reveal multiple production and transport mechanisms that contribute to the eigenvalue sensitivity. We present an application of the sensitivity theory to a stably linearly density-stratified flow around a thin plate at a $90^{\circ }$ angle of attack, a Reynolds number of 30 and Froude numbers of 1, 8 and $\infty$. The global mode analysis reveals lightly damped undulations pervading through the entire domain, which are predicted by both inviscid uniform base flow theory and Orr–Sommerfeld theory. The sensitivity to base flow velocity modifications is primarily concentrated just downstream of the bluff body. On the other hand, the sensitivity to base flow density modifications is concentrated in regions both immediately upstream and immediately downstream of the plate. Both sensitivities have a greater upstream presence for lower Froude numbers.

JFM classification

Instability: Instability Geophysical and Geological Flows: Stratified flows Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 812 , 10 February 2017 , pp. 1146 - 1188
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© 2017 Cambridge University Press 

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