A theory for natural convection turbulent boundary layers next to heated vertical surfaces

The turbulent natural convection boundary layer next to a heated vertical surface is analyzed by classical scaling arguments. It is shown that the fully developed turbulent boundary layer must be treated in two parts: an outer region consisting of most of the boundary layer in which viscous and conduction terms are negligible and an inner region in which the mean convection terms are negligible. The inner layer is identified as a constant heat flux layer. A similarity analysis yields universal profiles for velocity and temperature in the outer and constant heat flux layers. An asymptotic matching of these profiles in an intermediate layer (the buoyant sublayer) as H_δ &z.tbnd gβF ₀δ⁴ α³ → ∞ yields analytical expressions for the buoyant sublayer profiles as (T-T_w) T₁ = K₂( y n)^{- 1 3} + A(pr), U U₁ = K₁( y n) ^{1 3} + B(pr), where K₁, K₂ are universal constants and A(Pr), B(Pr) are universal functions of Prandtl number. Asymptotic heat transfer and friction laws are obtained as Nu_x = C'_H(Pr)H*_{x 1 4}, τ_w grU²₁ = C_f(Pr), where C'_H(Pr) is simply related to A(Pr). Finally, conductive and thermo-viscous sublayers characterized by a linear variation of velocity and temperature are shown to exist at the wall. All predictions are seen to be in excellent agreement with the abundant experimental data.