Roughness as a Route to the Ultimate Regime of Thermal Convection

Abstract

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...

Wall-to-wall optimal transport in two dimensions

“…Heat transfer via fluid advection is a critical component of atmospheric, oceanographic, geophysical, and astrophysical dynamics, as well as being the basis of cooling systems in engineering applications. Numerous studies on how to design systems that achieve enhanced heat transfer by either manipulation of domain geometry or through the discovery of suitable flow structures have recently appeared in the literature Toppaladoddi et al (2017); Alben (2017); Marcotte et al (2018); Motoki et al (2018a,b). A particularly fruitful approach to discovering flow structures was first introduced by Hassanzadeh et al (2014) where it was formulated via an optimal control approach.…”

Wall roughness induces asymptotic ultimate turbulence

“…Today, this issue is often considered as one of the most important open problems in the thermal convection community. In fact, the exponent 1/2 has only been achieved with rough walls [26] as presumably a transient, local effective scaling, which saturates back to smaller exponent at even larger Ra [12,27,28], or in artificial configurations, such as numerical simulations of so-called "homogeneous convective turbulence" [29] with periodic boundary conditions and no boundary layers, or experimental realisations thereof such as in…”

Thermal transfer in Rayleigh–Bénard cell with smooth or rough boundaries

“…Recently, Toppaladoddi et al (2017) and Zhu et al (2017) focused on the influence of the density of roughness structures on the thermal transfer. To do so, they performed several 2D numerical simulations in a RayleighBénard system with sinusoidal roughness on both plates.…”