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 …

We describe methods for proving upper and lower bounds on infinite-time averages in
deterministic dynamical systems and on stationary expectations in stochastic systems. The …

The calculus of variations is employed to find steady divergence-free velocity fields that
maximize transport of a tracer between two parallel walls held at fixed concentration for one …

We prove the first rigorous bound on the heat transfer for three-dimensional Rayleigh–
Bénard convection of finite-Prandtl-number fluids between free-slip boundaries with an …

To analyse the physics underlying gravity-driven runoff of thin wavy films, a film flow model is
developed, and is solved with computational fluid dynamics. This model is based on the …

The background method has been a successful tool in obtaining strict bounds on global
quantities such as the rate of energy dissipation and heat transfer in turbulent flows …

In this paper, we use the well-known background method to obtain a rigorous lower bound
on the volume flow rate through a helical pipe driven by a pressure differential in the limit of …

We introduce an innovative numerical technique based on convex optimization to solve a
range of infinite-dimensional variational problems arising from the application of the …

Turbulence is ever-present: from flows in engineering, such as a wake behind a submarine,
the boundary layer over an aircraft wing, and the swirl in an internal combustion engine, to …

We introduce a new technique to optimize a linear cost function subject to an affine
homogeneous quadratic integral inequality, ie, the requirement that a homogeneous …