Tailoring boundary geometry to optimize heat transport in turbulent convection

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, Nu − 1 ∝ Ra, is maximized at λ ≡ λmax ≈ (2π) , 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. editor’s choice Copyright c © EPLA, 2015 Thermal and compositional convection underlie the behavior of a wide range of systems from planetary and stellar interiors and the motions of Earth’s atmosphere and oceans, to the solidification of multicomponent melts [1–3]. The simplest setting to study thermal convection is in a Rayleigh-Bénard cell [4], wherein the flow is controlled by the Rayleigh number (Ra), which describes the ratio of buoyancy to dissipative forces, the Prandtl number (Pr), which is the ratio of momentum to thermal diffusivities of the fluid, and the aspect ratio (Γ). The key quantity of interest is the vertical heat flux across the cell, expressed in non-dimensional form as the Nusselt number, Nu(Ra, Pr), which describes the ratio of the total heat flux to the heat flux solely due to conduction. For Ra ≫ 1, the function Nu(Ra, Pr) is usually sought in the form of a scaling law: Nu ∼ PrRa . The value β = 1/3 emerges from the classical argument that when Ra ≫ 1 the dimensional heat flux should become independent of the depth of the cell, implying that the boundary layers (BLs) at the upper and lower surfaces do not interact [5–7]. However, the exponents obtained from experiments and numerical simulations range from approximately β = 2/7 [8–12] to β = 1/3 [11–15]. Theories with specific assumptions concerning the structure of the flow [8] and/or the nature of the BLs [16] have been proposed to explain the 2/7 scaling. For extremely large Ra, however, Nu is predicted to become independent of the molecular properties of the fluid, and hence the boundary layers, and heat transport is achieved solely by convection [17,18]. In this so-called “ultimate regime”, Nu ∼ Ra [18]. Finally, we note that a means of examining the various “crossovers” in the Ra-Pr plane has been proposed [19]. Taking a different approach, Howard [20] sought to determine upper bounds on Nu using a variational formulation with incompressibility as one of the constraints on the statistically stationary flow. When a single horizontal wave number dominates the flow, he found an upper bound ofNu−1 = (Ra/248). Kerswell [21] and Hassanzadeh et al. [22] (and references therein) provide a detailed discussion of this approach. A recent variational study of the two-dimensional problem by Whitehead and Doering [23] has shown that the thermal BLs (TBLs) do play a role in limiting the heat flux in the “ultimate regime”, even when there are no momentum boundary layers (free-slip conditions were used). Hence, the nature of the interaction between the BLs and the core flow plays a central role in turbulent Rayleigh-Bénard convection. This interaction can be probed either by manipulating the boundary geometry itself or by introducing inhomogeneous temperature boundary conditions [24]. The former can be achieved by corrugating one or both horizontal