An Improved Rigorous Bound on Heat Transport in Fixed-Flux Rayleigh–Bénard Convection between Free-Slip Boundaries: 𝑁 𝑢 ≤ 0.5500 𝑅^1/3

Jeff Pickhardt

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An Improved Rigorous Bound on Heat Transport in Fixed-Flux Rayleigh–Bénard Convection between Free-Slip Boundaries: 𝑁 𝑢 ≤ 0.5500 𝑅^1/3

Abstract

The background-field method of Doering and Constantin yields rigorous upper bounds on turbulent heat transport, and improving the explicit constants in these bounds for canonical flows is a long-standing open problem. For three-dimensional Rayleigh–Bénard convection between free- slip boundaries driven by an imposed heat flux, the standing record is Fantuzzi’s bound 𝑁 𝑢 ≤ 0.5999 𝑅1/3 (equivalently 𝑁 𝑢 ≤ 0.4646 𝑅𝑎1/2), uniform in the Prandtl number, where 𝑅 is the flux-based Rayleigh number. We prove 𝑁 𝑢 ≤ 0.5500 𝑅^1/3, 𝑁 𝑢 ≤ 0.4079 𝑅𝑎^1/2, uniformly in 𝑃 𝑟 ∈ (0, ∞] and for all 𝑅 ≥ 120 — an 8.3% (respectively 12.2%) improvement of the prefactor. The proof retains Fantuzzi’s variational framework and balance-parameter structure verbatim; the improvement comes entirely from a new boundary-layer estimate for the indefinite term of the spectral constraint. Instead of the pointwise splitting-and-Cauchy–Schwarz chain that produces the coeﬀicient (1 + √2)/√2 ≈ 1.707 on the ‖𝜃′‖ ‖𝑤′‖ pairing, we use an exact fundamental- theorem-of-calculus identity anchored at the midplane (one Cauchy–Schwarz, coeﬀicient 1), plus a correction term that is exactly a Volterra bilinear form whose operator norm 𝜎0 = 0.24921 … we certify rigorously by an exact trace computation: 𝜎0 ≤ (tr 𝑇 4)1/8 = (13/873180)1/8 < 0.2492327, where 𝑇 is an explicit positive integral operator with kernel (1 − max(𝑠, 𝑡))3/3. We also prove a structural ceiling: within this proof architecture the prefactor cannot be pushed below 41/3/3 = 0.52913; our result is within 4% of that ceiling. Closing the remaining gap is stated precisely as an open problem.

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