Second-order sensitivity of parallel shear flows and optimal spanwise-periodic flow modifications

TL;DR

This paper develops a second-order sensitivity analysis for spanwise-periodic flow modifications to optimize flow stabilization, validated with stability calculations and considering transient growth effects for improved control strategies.

Contribution

It introduces a 2nd-order sensitivity operator for predicting eigenvalue changes due to spanwise modifications, enabling optimal flow control design without full eigenvalue computations.

Findings

Second-order sensitivity accurately predicts eigenvalue shifts.

Optimal spanwise wavenumbers for stabilization are identified.

Combined transient growth and stabilization strategies outperform traditional methods.

Abstract

The question of optimal spanwise-periodic modification for the stabilisation of spanwise-invariant flows is addressed. A 2nd-order sensitivity analysis is conducted for the linear temporal stability of parallel flows U0 subject to small-amplitude spanwise-periodic modification e*U1, e<<1. Spanwise-periodic modifications have a quadratic effect on stability, i.e. the 1st-order eigenvalue variation is zero. A 2nd-order sensitivity operator is computed from a 1D calculation, allowing one to predict how eigenvalues are affected by any U1, without actually solving for modified eigenvalues/eigenmodes. Comparisons with full 2D stability calculations in a plane channel flow and in a mixing layer show excellent agreement. Next, optimisation is performed on the 2nd-order sensitivity operator: for each eigenmode streamwise wavenumber and base flow modification spanwise wavenumber b, the most stabilising profiles U1 are computed, together with lower bounds for the variation in leading eigenvalue. These bounds increase like b^-2 as b goes to 0, yielding a large stabilising potential. However, 3D modes with wavenumbers |b0|=b and b/2 are destabilised, thus larger control wavenumbers should be preferred. The modification U1 optimised for the most unstable streamwise wavenumber has a stabilising effect on other streamwise wavenumbers too. Finally, the potential of transient growth to amplify perturbations and stabilise the flow is assessed. Combined optimal perturbations that achieve the best balance between transient linear amplification and flow stabilisation are determined. In the mixing layer with b<1.5, these combined optimal perturbations appear similar to transient growth-only optimal perturbations, and achieve a more efficient overall stabilisation than optimal 1D and 2D modifications computed for stabilisation only. This is consistent with the efficiency of streak-based control strategies.