A Reduced Basis Technique for Long-Time Unsteady Turbulent Flows

TL;DR

This paper introduces a novel reduced basis method for long-time simulation of parametrized turbulent flows, incorporating a constrained formulation, an efficient error indicator, and a Greedy algorithm for parameter space coverage.

Contribution

The paper presents a constrained Galerkin formulation, an posteriori error indicator, and a Greedy algorithm for efficient reduced basis construction in turbulent flow simulations.

Findings

Effective reduced basis for long-time turbulent flow simulation

Error indicator correlates with mean flow prediction accuracy

Method demonstrates strong performance on lid-driven cavity flow

Abstract

We present a reduced basis technique for long-time integration of parametrized incompressible turbulent flows. The new contributions are threefold. First, we propose a constrained Galerkin formulation that corrects the standard Galerkin statement by incorporating prior information about the long-time attractor. For explicit and semi-implicit time discretizations, our statement reads as a constrained quadratic programming problem where the objective function is the Euclidean norm of the error in the reduced Galerkin (algebraic) formulation, while the constraints correspond to bounds for the maximum and minimum value of the coefficients of the $N$-term expansion. Second, we propose an \emph{a posteriori} error indicator, which corresponds to the dual norm of the residual associated with the time-averaged momentum equation. We demonstrate that the error indicator is highly-correlated with the error in mean flow prediction, and can be efficiently computed through an offline/online strategy. Third, we propose a Greedy algorithm for the construction of an approximation space/procedure valid over a range of parameters; the Greedy is informed by the \emph{a posteriori} error indicator developed in this paper. We illustrate our approach and we demonstrate its effectiveness by studying the dependence of a two-dimensional turbulent lid-driven cavity flow on the Reynolds number.