Inner and outer scales in turbulent Rayleigh–Bénard convection

Abstract

As in other kinds of wall-bounded turbulence, flow and heat transport in turbulent Rayleigh–Bénard convection (RBC) can be divided into an inner layer and an outer layer. This paper refines the traditional inner scales, the Townsend inner scales, by determining the Prandtl number Pr effect, and proposes new scales for the outer layer. Major findings for the inner layer include (i) the mean modified pressure peaks in the inner layer, and the peak location scales with the Townsend inner length scale lν = ν/uinner, where ν is the kinematic viscosity and uinner is the Townsend inner velocity. (ii) The peak value of the mean modified pressure Pmax scales as ΨPρrefuinner2, where ρref is the fluid density and the coefficient ΨP is largely independent of the Reynolds number but is strongly influenced by the Prandtl number. (iii) The thickness of the thermal inner layer scales with a thermal diffusional length scale lα = Ψα α/uinner, where α is the thermal diffusivity and the coefficient Ψα is largely independent of the Reynolds number but is strongly influenced by the Prandtl number. Like passive scalar transport in a pressure-driven turbulent plane Poiseuille flow, the Prandtl number dependence of Ψα (and ΨP) can be approximated by a power law Ψα ∼ ΨP ∼ Prm, where m is a constant of about 0.5. In the outer layer, the vertical component of velocity fluctuation variance at the RBC midplane ⟨ww⟩mp is introduced as a new governing parameter in the scaling of flow and heat transfer. The new outer velocity and temperature scales for turbulent RBC are different from the Deardorff scales, which were developed for convective atmospheric boundary layers. The new outer scales are compared with direct numerical simulation data and experimental measurements.