Optimal bounds with semidefinite programming: an application to stress driven shear flows

TL;DR

This paper presents a novel convex optimization-based numerical method using semidefinite programming to compute rigorous bounds on fluid flow dissipation, improving accuracy and applicability over traditional approaches.

Contribution

It introduces a new SDP-based technique for variational problems in fluid flows that accounts for all modes and handles boundary fluxes effectively.

Findings

Achieved over 10 times tighter bounds than previous analytical results.

Bounds become independent of domain aspect ratio as viscosity vanishes.

Demonstrated the method's efficiency in analyzing flow stability.

Abstract

We introduce an innovative numerical technique based on convex optimization to solve a range of infinite dimensional variational problems arising from the application of the background method to fluid flows. In contrast to most existing schemes, we do not consider the Euler--Lagrange equations for the minimizer. Instead, we use series expansions to formulate a finite dimensional semidefinite program (SDP) whose solution converges to that of the original variational problem. Our formulation accounts for the influence of all modes in the expansion, and the feasible set of the SDP corresponds to a subset of the feasible set of the original problem. Moreover, SDPs can be easily formulated when the fluid is subject to imposed boundary fluxes, which pose a challenge for the traditional methods. We apply this technique to compute rigorous and near-optimal upper bounds on the dissipation coefficient for flows driven by a surface stress. We improve previous analytical bounds by more than 10 times, and show that the bounds become independent of the domain aspect ratio in the limit of vanishing viscosity. We also confirm that the dissipation properties of stress driven flows are similar to those of flows subject to a body force localized in a narrow layer near the surface. Finally, we show that SDP relaxations are an efficient method to investigate the energy stability of laminar flows driven by a surface stress.