The methods that have come to be known as the Malkus–Howard–Busse (MHB) and the Constantin–Doering–Hopf (CDH) techniques have, over the past few decades, produced the few rigorous statements available about average properties (e.g. momentum and heat transport) of turbulent flows governed by the Navier–Stokes equation and the heat equation. In this, the first of two papers investigating upper bounds on the heat transport in infinite-Prandtl-number convection, we show that the methods of MHB and CDH yield equivalent optimal bounds: as at a saddle–one from above, and one from below.

We also demonstrate that here, in contrast to earlier applications of the CDH method, the simplest possible, one-parameter, ‘test function’ does not capture the leading-order scaling associated with the fully optimal solution. We explore the consequences of a two-parameter test function in modifying the scaling of the upper bound. In the case of no-slip, the suggestion is that a hierarchy of test functions of increasing complexity is required to yield the correct limiting behaviour.