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

Wall Functions 106
After replacing Θ1 from the thermal analytical wall function, expression
of equation 4.30, and substituting for µ from equations 4.24 and 4.25, then
integrating the above equation in the viscous sublayer region, the gradient of
the wall-parallel component of velocity is:
∂U1
µυ
∂y∗ = C1y ∗ +A1 +b(Θwall −Θref)y ∗ +b Prυ
+ C1bµ
Prυ
+ bbµ
µυ
Prυ
+ bbµ
µυ
+ bb 2 µ
− bb 2 Prυ
µ
µυ
y ∗2
−y ∗ υy ∗
Cth1y ∗4
6
Cth1y ∗4
Cth1y ∗5
+ Ath1y ∗3
µυ
+bbµ(Θwall −Θref)
2
12 − Cth1y∗ υ −Ath1
6
12 − Cth1y∗ υ −Ath1
6
Cth1y ∗ υ
12 y∗4
− Cth1y ∗ υy ∗3
6
y ∗4
y ∗3
Cth1y ∗3
6
+ Ath1y ∗2
y ∗2
−y ∗ υy ∗
− Ath1y ∗ υy ∗2
− Ath1y ∗ υ
2
− Ath1y∗ υ
y
2
∗3
− (Cth1y∗ υ −Ath1)y ∗ υ
y
6
∗3
−
2
y ∗2
2
∗2
Ath1yυ y
2
∗2
+ A1bµ(y ∗ −y ∗ υ) (4.55)
where A1 is a constant of integration.
After a second integration and applying the boundary condition that at
y ∗ = 0 : U1 = 0, the wall parallel velocity profile within the zeroµt sublayer is: