Boundary layer structure in turbulent thermal convection and its consequences for the required numerical resolution

Abstract

Results on the Prandtl-Blasius type kinetic and thermal boundary layer thicknesses in turbulent Rayleigh-Bénard convection in a broad range of Prandtl numbers are presented. By solving the laminar Prandtl-Blasius boundary layer equations, we calculate the ratio of the thermal and kinetic boundary layer thicknesses, which depends on the Prandtl number Pr only. It is approximated as 0.588Pr −1/2 for Pr Pr * and as 0.982Pr −1/3 for Pr * Pr, with Pr * ≡ 0.046. Comparison of the Prandtl-Blasius velocity boundary layer thickness with that evaluated in the direct numerical simulations by Stevens, Verzicco, and Lohse (J. Fluid Mech. 643, 495 (2010)) gives very good agreement. Based on the Prandtl-Blasius type considerations, we derive a lower-bound estimate for the minimum number of the computational mesh nodes, required to conduct accurate numerical simulations of moderately high (boundary layer dominated) turbulent Rayleigh-Bénard convection, in the thermal and kinetic boundary layers close to bottom and top plates. It is shown that the number of required nodes within each boundary layer depends on Nu and Pr and grows with the Rayleigh number Ra not slower than ∼ Ra 0.15 . This estimate agrees excellently with empirical results, which were based on the convergence of the Nusselt number in numerical simulations. arXiv:1109.6870v1 [physics.flu-dyn] 30 Sep 2011Boundary layer structure in turbulent thermal convection 2 IntroductionRayleigh-Bénard (RB) convection is the classical system to study properties of thermal convection. In this system a layer of fluid confined between two horizontal plates is heated from below and cooled from above. Thermally driven flows are of utmost importance in industrial applications and in natural phenomena. Examples include the thermal convection in the atmosphere, the ocean, in buildings, in process technology, and in metal-production processes. In the geophysical and astrophysical context one may think of convection in Earth's mantle, in Earth's outer core, and in the outer layer of the Sun. E.g., the random reversals of Earth's or the Sun's magnetic field have been connected with thermal convection.Major progress in the understanding of the Rayleigh-Bénard system has been made over the last decades, see e.g. the recent reviews [1,2]. Meanwhile it has been well established that the general heat transfer properties of the system, i. e. Nu = Nu(Ra, Pr) and Re = Re(Nu, Pr), are well described by the Grossmann-Lohse (GL) theory [3,4,5,6]. In that theory, in order to estimate the thicknesses of the kinetic and thermal boundary layers (BL) and the viscous and thermal dissipation rates, the boundary layer flow is considered to be scalingwise laminar Prandtl-Blasius flow over a plate. We use the conventional definitions: The Rayleigh number is Ra = αgH 3 ∆/νκ with the isobaric thermal expansion coefficient α, the gravitational acceleration g, the height H of the RB system, the temperature difference ∆ between the heated lower plate and the cooled upper plate, and the material constants ν, kinema...

Horizontal structures of velocity and temperature boundary layers in two-dimensional numerical turbulent Rayleigh-Bénard convection

“…Note that comparison of the mean bulk temperature calculated using the Prandtl-Blasius theory with that measured in both liquid and gaseous nonOberbeck-Boussinesq RB convection shows very good agreement [16][17][18] . In addition, the kinematic BL thickness evaluated by solving the laminar Prandtl-Blasius BL equations was found to agree well with that obtained in the direct numerical simulation (DNS) 19 .…”

Intensified heat transfer in modulated rotating Rayleigh–Bénard convection

“…The spatial resolution is taken as ðn r ; n h ; n z Þ ¼ ð385; 193; 385Þ, which was found adequate to also resolve the finer scales of the flow associated with the boundary layers that develop near the two end plates and the side walls (Kunnen et al, 2009;Shishkina et al, 2010). We work in cylindrical coordinates in which case several terms in the governing equations possess a factor 1=r with r the radial coordinate.…”

New perspectives in turbulent Rayleigh-Bénard convection

“…In refs. [27,77,78] new criteria have been suggested which incorporate the strong vertical variation of gradients and thus of the kinetic energy dissipation rate. Based on Blasius theory the following estimates for 10 6 ≤ Ra ≤ 10…”