The background method: theory and computations.

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. This article is part of the theme issue 'Mathematical problems in physical fluid dynamics (part 1)'.