The computation of turbulent natural convection flows - Turbulence ...

Wall Functions 100
temperature were neglected in the main flow transport equations (since tem-
perature difference in the outer flow field will be even smaller than across the
near-wall viscous layer).
4.2.2 Thermal Wall Function When the Near Wall Cell Is Thicker
Than the Viscous Sublayer
First, we convert the simplified temperature equation into y ∗ coordinates:
∂
∂y∗
µ µt
+
Pr Prt
∂Θ
∂y ∗
= µ2 υ
ρ 2 υkP
∂(ρUΘ)
∂x
+ ∂(ρVΘ)
∂y P
In the viscous sub-layer region (y ∗ < y ∗ υ ), where µt = 0, this gives
∂
∂y∗
µ
Pr
After a first integration:
∂Θ
∂y∗
= µ2υ ρ2 υkP ∂(ρUΘ)
∂x
µ ∂Θ1
Pr ∂y∗ = Cth1y ∗ +Ath1
+ ∂(ρVΘ)
∂y P
= Cth (4.26)
= Cth1
(4.27)
(4.28)
for some constant of integration Ath1. Subscript 1 denotes parameters inside
the viscous sublayer region.
After substituting for µ from equations 4.24 and 4.25, in the viscous sub-
layer, the gradient of temperature is:
∂Θ1
∂y
∗ = PrυCth1
µυ
y ∗ [1+bµ(y ∗ −y ∗ PrυAth1
υ )]+ [1+bµ(y
µυ
∗ −y ∗ υ )] (4.29)
Then after a second integration, imposing the boundary condition that at
y ∗ = 0,Θ1 = Θwall, the temperature profile within the viscous sublayer is: