Boundary layer theory for turbulent Rayleigh-Benard convection: Temperature boundary layer profiles

We have derived the boundary layer equations for turbulent Rayleigh-Benard convection. We consider a quasi-two-dimensional fluid flow along a semi-infinite horizontal heated plate with the requirement that the horizontal velocity vanishes far away from the plate. The turbulent fluctuations are taken into account by an eddy viscosity ν_t and an eddy thermal diffusivity κ_t. Based on Prandtl's mixing length ideas, we approximate (ν_t/ν)_ξ ≈ k₁ψ and (κ_t/κ)_ξ ≈ k₂ψ where ψ is the dimensionless stream function, ξ is the similarity variable and k₁ and k₂ are constants. For high Prandtl number (Pr), the dimensionless temperature boundary layer profile Θ(ξ) does not depend on ψ and is given by Eqs. (24) and (25) in Shishkina et al., Phys. Rev. Lett. 114, 114302 (2015). For low Pr and high Rayleigh number, Θ(ξ) is obtained by solving the boundary layer equations

(1+k₁g)ψ_ξξξ + (1/4+ 9k₁/8)ψψ_ξξ +(1/2-k₁/4)(ψ_ξ)² = 0

(1+k₂g)Θ_ξξ + [k₂+ Pr(1/4+k₁/8)]ψΘ_ξ = 0

with suitable boundary conditions at ξ=0 and ξ tends to ∞. Here, g_ξ= ψ. Our theoretical results are in good agreement with the direct numerical simulation results.