On Howard's upper bound for heat transport by turbulent convection

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Abstract
The variational problem introduced by Howard (1963) for the derivation of an upper bound on heat transport by convection in a layer heated from below is analyzed for the case in which the equation of continuity is added as constraint for the velocity field. Howard's conjecture that the maximizing solution of the Euler equations is characterized by a single horizontal wave-number is shown to be true only for a limited range of the Rayleigh number, Ra. A new class of solutions with a multiple boundary-layer structure is derived. The upper bound for the Nusselt number, Nu, given by these solutions is Nu ≤ (Ra/1035)½ in the limit when the Rayleigh number tends to infinity. The comparison of the maximizing solution with experimental observations by Malkus (1954a) and Deardorff & Willis (1967) emphasizes the similarity pointed out by Howard.
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