The background method: Theory and computations

Abstract

The background method is a widely used technique to bound mean properties of turbulent flows rigorously. This work reviews recent advances in the theoretical formulation and numerical implementation of the method. First, we describe how the background method can be formulated systematically within a broader "auxiliary function" framework for bounding mean quantities, and explain how symmetries of the flow and constraints such as maximum principles can be exploited. All ideas are presented in a general setting and are illustrated on Rayleigh-Bénard convection between free-slip isothermal plates. Second, we review a semidefinite programming approach and a timestepping approach to optimizing bounds computationally, revealing that they are related to each other through convex duality and low-rank matrix factorization. Open questions and promising directions for further numerical analysis of the background method are also outlined.

The role of boundary conditions in scaling laws for turbulent heat transport

“…The background field method will be carefully described in Section 3.1.1 and we refer to [20] for a recent review on it. A generalization of the background field method was proposed in [21] to improve the available bounds.…”

Systematic Search For Extreme and Singular Behavior in Some Fundamental Models of Fluid Mechanics

“…When the auxiliary function is fixed and quadratic whereas optimization is performed with respect to the form of a certain "background flow", this bounding framework reduces to the background method originally developed by Doering & Constantin [87] to obtain rigorous a priori bounds on energy dissipation in wall-bounded flows. The background method has been since used, both analytically and computationally, to derive bounds on average quantities in different flows and we refer the reader to [88] for a recent survey of this topic.…”