Identifying spatially-localized instability mechanisms using sparse optimization

Abstract

Recent investigations have established the physical relevance of spatially-localized instability mechanisms in fluid dynamics and their potential for technological innovations in flow control. In this letter, we show that the mathematical problem of identifying spatially-localized optimal perturbations that maximize perturbation-energy amplification can be cast as a sparse (cardinality-constrained) optimization problem. Unfortunately, cardinality constrained optimization problems are non-convex and combinatorially hard to solve in general. To make the analysis viable within the context of fluid dynamics problems, we propose an efficient iterative method for computing sub-optimal spatially-localized perturbations. Our approach is based on a generalized Rayleigh quotient iteration algorithm followed by a variational renormalization procedure that reduces the optimality gap in the resulting solution. The approach is demonstrated on a sub-critical plane Poiseuille flow at Re = 4000, which has been a benchmark problem studied in prior investigations on identifying spatially-localized flow structures. Remarkably, we find that a subset of the perturbations identified by our method yield a comparable degree of energy amplification as their global counterparts. We anticipate our proposed analysis tools will facilitate further investigations into spatially-localized flow instabilities, including within the resolvent and input-output analysis frameworks.

Figures (2)

• Figure 1: Transient growth in kinetic energy $G(t=T)$ for a specified time-horizon $T$ for plane Poiseuille flow at $Re = 4000$ with $(\alpha,\beta)=(1,2)$ due to sparse optimal perturbations (red) approach that of the (non-sparse, $k=2N=200$) optimal perturbation as $k$ is increased. At $k=50$, sparse optimal perturbations yield comparable transient growth for all time-horizons, including the maximum transient growth over all time-horizons $G=G_\text{max}$. Associated optimal perturbations and responses are reported in figure \ref{['fig:composite_fields']}.
• Figure 2: Optimal perturbations (left) and associated responses (right) resulting in $G=G_\text{max}$ for $Re=4000$ and $(\alpha,\beta)=(1,2)$. Non-sparse optimal perturbations (black) are composed of wall-normal velocity $v$ and wall-normal vorticity $\eta$ distributed across the wall-normal direction ($y$). Sparse (sub-)optimal perturbations (red) with $k \le 50$ are composed only of wall-normal velocity.