We use highly resolved numerical simulations to study turbulent Rayleigh-Bénard convection in a cell with sinusoidally rough upper and lower surfaces in two dimensions for P r = 1 and Ra = 4 × 10 6 , 3 × 10 9 . By varying the wavelength λ at a fixed amplitude, we find an optimal wavelength λopt for which the Nusselt-Rayleigh scaling relation is N u − 1 ∝ Ra 0.483 maximizing the heat flux. This is consistent with the upper bound of Goluskin and Doering [1] who prove that N u can grow no faster than O(Ra 1/2 ) as Ra → ∞, and thus the concept that roughness facilitates the attainment of the so-called ultimate regime. Our data nearly achieve the largest growth rate permitted by the bound. When λ λopt and λ λopt, the planar case is recovered, demonstrating how controlling the wall geometry manipulates the interaction between the boundary layers and the core flow. Finally, for each Ra we choose the maximum N u among all λ, and thus optimizing over all λ, to find N uopt − 1 = 0.01 × Ra 0.444 .The ubiquity and importance of thermal convection in many natural and man-made settings is well known [2][3][4]. The simplest scenario that has been used to study the fundamental aspects of thermal convection is the Rayleigh-Bénard system [5]. The flow in this system is governed by three non-dimensional parameters: (1) the Rayleigh number Ra = gα∆T H 3 /νκ, which is the ratio of buoyancy to viscous forces, where g is the acceleration due to gravity, α the thermal expansion coefficient of the fluid, ∆T the temperature difference across a layer of fluid of depth H, ν the kinematic viscosity (or momentum diffusivity) and κ the thermal diffusivity; (2) the Prandtl number, P r = ν/κ; and (3) the aspect ratio of the cell, Γ, defined as the ratio of its width to height.The primary aim of the corpus of studies of turbulent Rayleigh-Bénard convection has been to determine the Nusselt number, N u, defined as the ratio of total heat flux to conductive heat flux (Eq. 1), as a function of the three governing parameters, viz., N u = N u(Ra, P r, Γ). For Ra 1 and fixed P r and Γ, this relation is usually sought in the form of a power law: N u = A(P r, Γ)Ra β , where β has a fundamental significance for the mechanisms underlying the transport of heat.The classical theory of Priestley [6], Malkus [7] and Howard [8] is based on the argument that as Ra → ∞ the dimensional heat flux should become independent of the depth of the cell, resulting in β = 1/3. A consequence of this scaling is that the conductive boundary layers (BLs) at the upper and lower surfaces, which are separated by a well mixed interior, do not interact.However, Kraichnan [9] reasoned that for extremely large Ra the BLs undergo a transition leading to the generation of smaller scales near the boundaries that increase the system's efficiency in transporting the heat, predicting that N u ∼ Ra/ (ln Ra). In this, "KraichnanSpiegel" or "ultimate regime" (β = 1/2), it is argued that the heat flux becomes independent of the molecular properties of the fluid [e.g. and hence values of Ra) t...

By tailoring the geometry of the upper boundary in turbulent Rayleigh-Bénard convection we manipulate the boundary layer -interior flow interaction, and examine the heat transport using the Lattice Boltzmann method. For fixed amplitude and varying boundary wavelength λ, we find that the exponent β in the Nusselt-Rayleigh scaling relation, N u − 1 ∝ Ra β , is maximized at λ ≡ λmax ≈ (2π) −1 , but decays to the planar value in both the large (λ ≫ λmax) and small (λ ≪ λmax) wavelength limits. The changes in the exponent originate in the nature of the coupling between the boundary layer and the interior flow. We present a simple scaling argument embodying this coupling, which describes the maximal convective heat flux.

We study the effects of externally imposed shear and buoyancy driven flows on the stability of a solid-liquid interface. By reanalyzing the data of Gilpin et al. [J. Fluid Mech., 99(3), 619 (1980)] we show that the instability of the ice-water interface observed in their experiments was affected by buoyancy effects, and that their velocity measurements are more accurately described by Monin-Obukhov theory. A linear stability analysis of shear and buoyancy driven flow of melt over its solid phase shows that buoyancy is the only destabilizing factor and that the regime of shear flow here, by inhibiting vertical motions and hence the upward heat flux, stabilizes the system. It is also shown that all perturbations to the solid-liquid interface decay at a very modest strength of the shear flow. However, at much larger shear, where flow instabilities coupled with buoyancy might enhance vertical motions, a re-entrant instability may arise.

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