Model Reduction of Turbulent Fluid Flows Using the Supply Rate

TL;DR

This paper introduces a novel model reduction technique for turbulent flow simulations that maintains bounds on turbulent energy production by decomposing the Navier-Stokes equations and applying a balanced truncation-like method.

Contribution

It presents a new approach combining feedback decomposition and Riccati equations to reduce turbulence models while preserving energy bounds, improving upon existing methods.

Findings

The method accurately approximates the supply rate of turbulent energy.

Application to pipe flow demonstrates improved bounds preservation.

Compared to canonical balanced truncation, the new method performs better or equally well.

Abstract

A method for finding reduced-order approximations of turbulent flow models is presented. The method preserves bounds on the production of turbulent energy in the sense of the $\curly{L}_2$ norm of perturbations from a notional laminar profile. This is achieved by decomposing the Navier-Stokes system into a feedback arrangement between the linearised system and the remaining, normally neglected, nonlinear part. The linear system is reduced using a method similar to balanced truncation, but preserving bounds on the supply rate. The method involves balancing two algebraic Riccati equations. The bounds are then used to derive bounds on the turbulent energy production. An example of the application of the procedure to flow through a long straight pipe is presented. Comparison shows that the new method approximates the supply rate at least as well as, or better than, canonical balanced truncation.