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

107
+A1y ∗ + by∗2
2
+ bµy ∗2
∗ y
3 − y∗
υ
2
µυU1 = C1
2 y∗2
Prυy
+ bbµ
∗3
µυ
Cth1y ∗2
+ bb 2 Prυy
µ
∗3
Cth1
6µυ
+ bb 2 Prυy
µ
∗3
6µυ
+ bµA1y ∗
∗ y
2 −y∗
υ
(Θwall −Θref)+
[C1 +b(Θwall −Θref)]
20 − (Cth1y∗ υ −2Ath1)
10
12 y∗3 − (3Cth1y∗ υ −2Ath1)
10
(Cth1y∗ υ −4Ath1)y ∗ υ
In the fully turbulent region y ∗ > y ∗ υ
4
Prυy ∗
3µυ
Cth1y ∗
y ∗ − Ath1y ∗ υ
y ∗2
y ∗ +Ath1y ∗2
υ
3
4 +Ath1
(4.56)
the momentum equation is written as:
∂
∂y∗
(µ+µt) ∂U2
∂y∗
= C2 +b(Θ2 −Θref) (4.57)
First the equation for Θ2 from the thermal analytical wall function (equa-
tion 4.38) is substituted into the above equation. Then we assume that the
turbulent viscosity varies linearly across the fully turbulent region, according
to equation 4.20. After a first integration, the wall-parallel velocity gradient
across the fully turbulent region of the near wall control volume is:
µυ
∂U2
=
∂y∗ C2
[1+α(y ∗ −y ∗ υ )]y∗ +
+ b PrυCth2
2µυαt
+ b M
αt
y ∗2
[1+α(y ∗ −y ∗ υ
A2
[1+α(y ∗ −y∗ υ )] +b(R+Θwall −Θref)
[1+α(y ∗−y ∗ y∗
υ )]
)] −b Prυ
µυαt
YT
[1+α(y ∗ −y ∗ υ )] (lnYT −1)−
y ∗
Cth2y ∗ υ
[1+α(y ∗−y ∗ υ )]
bbµRµy ∗
[1+α(y ∗ −y ∗ υ )]
(4.58)
The parameters, M, R and Rµ, are calculated by imposing the appropriate
boundary conditions.
Let
Y = [1+α(y ∗ −y ∗ υ)] (4.59)