Improved variational principle for bounds on energy dissipation in turbulent shear flow

TL;DR

This paper enhances the variational method for bounding energy dissipation in turbulent shear flows by introducing a balance parameter, resulting in significantly tighter bounds for plane Couette flow.

Contribution

It extends the Doering-Constantin approach by adding a balance parameter, improving the bounds on energy dissipation without extra complexity.

Findings

Bounds for plane Couette flow are reduced by a factor of 27/32.

The new method simplifies obtaining tighter bounds.

The approach is applicable without additional technical difficulties.

Abstract

We extend the Doering-Constantin approach to upper bounds on energy dissipation in turbulent flows by introducing a balance parameter into the variational principle. This parameter governs the relative weight of different contributions to the dissipation rate. Its optimization leads to improved bounds without entailing additional technical difficulties. For plane Couette flow, the high-Re-bounds obtainable with one-dimensional background flows are methodically lowered by a factor of 27/32.