Thermal Boundary Layer Equation for Turbulent Rayleigh–Bénard Convection

Abstract

We report a new thermal boundary layer equation for turbulent Rayleigh-Bénard convection for Prandtl number Pr > 1 that takes into account the effect of turbulent fluctuations. These fluctuations are neglected in existing equations, which are based on steady-state and laminar assumptions. Using this new equation, we derive analytically the mean temperature profiles in two limits: (a) Pr 1 and (b) Pr ≫ 1. These two theoretical predictions are in excellent agreement with the results of our direct numerical simulations for Pr = 4.38 (water) and Pr = 2547.9 (glycerol) respectively.PACS numbers: 44.20.+b, 44.25.+f, 47.27.ek, 47.27.te Turbulent Rayleigh-Bénard convection (RBC) [1][2][3][4][5], consisting of a fluid confined between two horizontal plates, heated from below and cooled from above, is a system of great research interest. It is a paradigm system for studying turbulent thermal convection, which is ubiquitous in nature, occurring in the atmosphere and the mantle of the Earth as well as in stars like our Sun. Convective heat transfer is also an important problem in engineering and technological applications. The state of fluid motion in RBC is determined by the Rayleigh number Ra = αg∆H 3 /(κν) and Prandtl number Pr = ν/κ. Here α denotes the isobaric thermal expansion coefficient, ν the kinematic viscosity and κ the thermal diffusivity of the fluid, g the acceleration due to gravity, ∆ the temperature difference between the bottom and top plates, and H the distance between the plates.In turbulent RBC, there are viscous boundary layers (BLs) near all rigid walls and two thermal BLs, one above the bottom plate and one below the top plate. We denote the thicknesses of the viscous and thermal BLs by l and λ respectively. Both viscous and thermal BLs play a critical role in the turbulent heat transfer of the system and in particular λ is inversely proportional to the heat transport. Grossmann and Lohse (GL) [6], [7] developed a scaling theory of how the Reynolds number Re, determined by the mean large-scale circulation velocity U 0 above the viscous BL, and the dimensionless Nusselt number Nu, measuring the heat transport, depend on Ra and Pr for moderate Ra. The theory makes explicit use of the result l/H ∝ Re −1/2 with the proportionality constant depending only on Pr. This result follows from the assumptions that the BLs are laminar and their mean profiles, averaged over time, are described by the Prandtl-Blasius-Pohlhausen (PBP) theory [8-10] for steady-state forced convection above an infinite weakly-heated plate. Although the GL theory gives perfect agreement with the heat transport measurements, the assumption that the BLs are described by PBP theory is not fulfilled. Systematic deviations of the mean velocity and temperature profiles from the PBP predictions have been reported both in experiments and in direct numerical simulations (DNS) [11][12][13][14][15]. These deviations remain even after a dynamical rescaling procedure [16] that takes into account of the time variations of λ is used, and increase ...

Thermal convection in inclined cylindrical containers

“…The resulting dimensionless equations are solved numerically with the code GOLDFISH, as in Shishkina et al (2015) and Shishkina & Wagner (2016). Computational grids of up to (N r , N φ , N z ) = (192, 512, 384) nodes satisfy the resolution requirements for DNS (Shishkina et al 2010).…”

Heat and momentum transport scalings in horizontal convection

“…To derive the limiting scalings, the following assumptions are made with respect to the BL thicknesses: λ θ ∼ l / N u and . As it was derived for 2‐D thermal BLs in Shishkina et al [] (see equations (13) and (14), and explanations there), the latter relation must be fulfilled for the existence of a similarity solution of the thermal BL equation, even if the BLs are strongly fluctuating.…”

Towards parallel-in-time simulations of turbulent Rayleigh-Bénard convection

“…To calculate the velocity and temperature at the surfaces of each finite volume, it uses higher-order discretization schemes in space, up to the fourth order in the case of equidistant meshes. Goldfish has been used to study thermal convective flows in different configurations [152,153,155], in cylindrical and parallelepiped domains. For the time integration, the leapfrog scheme is used for the convective term and the explicit Euler scheme for the viscous term.…”

“…For the cylindrical simulations we use the latest version of RBflow, which is an optimized version of the code used by Stevens et al [165,163]. The second code is Goldfish by Shishkina et al [153,155,152], which is based on a finite-volume approach and uses discretization schemes of the fourth-order in space. Goldfish can be used to study turbulent thermal convection in cylindrical and parallelepiped domains.…”

“…In Chapter 3, we compare four of these codes based on different discretization techniques: (1) a second-order finite difference method (AFID/RBflow [163,165,174,179,180]); (2) a fourth-order finite volume method (Goldfish [153,155,152]); (3) an eighth-order spectral element method (Nek5000 [24,48,75,97,148]); (4) a second-order finite volume method (OpenFOAM [183]). The first two codes are dedicated to RB convection in simple geometries, whereas the latter two codes are designed for general problems also in complex geometries.…”