Logarithmic Temperature Profiles in Turbulent Rayleigh-Bénard Convection

Abstract

We report results for the temperature profiles of turbulent Rayleigh-Bénard convection (RBC) in the interior of a cylindrical sample of aspect ratio Γ≡D/L=0.50 (D and L are the diameter and height, respectively). Both in the classical and in the ultimate state of RBC we find that the temperature varies as A×ln(z/L)+B, where z is the distance from the bottom or top plate. In the classical state, the coefficient A decreases in the radial direction as the distance from the side wall increases. For the ultimate state, the radial dependence of A has not yet been determined. These findings are based on experimental measurements over the Rayleigh-number range 4×10(12)≲Ra≲10(15) for a Prandtl number Pr≃0.8 and on direct numerical simulation at Ra=2×10(12), 2×10(11), and 2×10(10), all for Pr=0.7.

Completing the Mechanical Energy Pathways in Turbulent Rayleigh-Bénard Convection

“…Another important result is that the simulations at Ra > 10 10 revealed the logarithmic temperature profile typical of non-convecting turbulent boundary layers, as reported by Ref. [5]. These observations support the conclusion that dissipation is dominated by the interior turbulence.…”

“…Introduction.-Flow in a horizontal layer of fluid heated uniformly from below and cooled from above-RayleighBénard convection (RBC)-is an idealized problem from which much can be learned about the nature of convective flow and heat transfer. Recent attention has focused on the relative importance of the boundary layer and interior behavior, including flow structures, at very large Rayleigh numbers [1][2][3][4][5][6]. The salient features of the flow can be usefully interpreted in terms of the energy budget, and previous work has considered the kinetic and thermal energy [2,3,7,8].…”

Logarithmic Spatial Variations and Universalf−1Power Spectra of Temperature Fluctuations in Turbulent Rayleigh-Bénard Convection

“…This state is known as "classical" RBC. As Ra increases and exceeds a critical value Ra * which for Pr ≃ 1 is O(10 14 ), the shear stress from the turbulent bulk will become sufficiently large to force the BLs into a turbulent state as well and the system enters the "ultimate" state which is expected to be asymptotic as Ra tends toward infinity [9][10][11][12].Recently, it was found that the time-averaged temperature T (t, z, r) t (z is the vertical and r the radial coordinate), both in the classical and the ultimate state but outside the BLs, varies logarithmically with the distance z/L from the plates when this distance is not too large (say z/L < ∼ 0.1 or so) [13,14]. Similar logarithmic behavior is well known from mean velocity profiles of near-wall turbulence in shear flows, such as pipe, channel, and Taylor-Couette flows [15][16][17]; there it is known as the "Law of the Wall" [18][19][20] (for recent reviews, see [21,22]).…”

“…Recently, it was found that the time-averaged temperature T (t, z, r) t (z is the vertical and r the radial coordinate), both in the classical and the ultimate state but outside the BLs, varies logarithmically with the distance z/L from the plates when this distance is not too large (say z/L < ∼ 0.1 or so) [13,14]. Similar logarithmic behavior is well known from mean velocity profiles of near-wall turbulence in shear flows, such as pipe, channel, and Taylor-Couette flows [15][16][17]; there it is known as the "Law of the Wall" [18][19][20] (for recent reviews, see [21,22]).…”

Convective ripening and initiation of rainfall

“…The upper and lower plates are at temperatures T up and T low respectively. The expectation value of the temperature is close to T av = (T up + T low )/2 (with small logarithmic corrections) [21], except in the vicinity of the upper and lower plates, and at any time most of the gas in the convection cell is at a temperature close to T av . Gas which is in contact with the lower plate of the cell is heated to a higher temperature T av + ∆T (where ∆T ≤ ∆T h /2), and joins a plume of rising gas.…”