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 stress-free 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.

Revised version: 29 pages, 3 figures, 2 tables. Added simple example to Section 1 and complexity estimates to Section 4. Improved discussion in Section 4

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• Optimal cooling of an internally heated disc
• Rigorous scaling laws for internally heated convection at infinite Prandtl number
• Internal heating profiles for which downward conduction is impossible
• Bounds on buoyancy driven flows with Navier-slip conditions on rough boundaries
• Internally heated convection with rotation: bounds on heat transport
• Variational bounds and nonlinear stability of an active nematic suspension
• Bounds on dissipation in three-dimensional planar shear flows: reduction to two-dimensional problems