On the stabilizing mechanism of 2D absolute and global instabilities by 3D streaks

TL;DR

This paper demonstrates that spanwise periodic modulations can stabilize 2D wake instabilities, and this effect can be modeled using a generalized complex Ginzburg-Landau equation with second order sensitivity analysis.

Contribution

It shows that stabilization by spanwise modulations is a general mechanism and can be captured through a second order sensitivity analysis in a simplified model.

Findings

Spanwise modulations stabilize 2D wake instabilities.

Second order sensitivity analysis accurately predicts stabilization effects.

Modulation alters wave diffusion, reducing growth rates.

Abstract

Global and local absolute instabilities of 2D wakes are known to be stabilized by spanwise periodic modulations of the wake profile. The present study shows that this stabilizing effect is of general nature and can be mimicked by enforcing spanwise periodic modulations of the wave advection velocity in the generalized complex Ginzburg-Landau equation. The first order sensitivity of the absolute and global growth rate to the enforced modulation is zero, exactly as in the Navier-Stokes case. We show that a second order sensitivity analysis is effective to quantify and interpret the observed stabilizing effect. The global growth rates predicted by the second order expansion closely match those issued from a direct computation of the eigenvalues. It is shown that, at leading order, the modulation of the wave advection velocity alters the effective wave diffusion coefficient in the dispersion relation and that this variation induces a reduction of the absolute growth rate and the stabilization of the global instability.