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

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wall heat flux can be obtained from a prescribed logarithmic temperature pro-
file. The temperature distribution shape is similar to the velocity distribution
shown in Figure 4.2 when the Prandtl number is unity. As the Prandtl number
increases, the thickness of the molecular dominated regime in the thermal field
increases, whilst the opposite happens when the Prandtl number is less than
unity. The logarithmic temperature profile can be written as:
where
Θ ∗ = 1
χ∗ln
E ′
∗ +
y
Θ ∗ = ρcp(Θwall −Θ)k 1/2
P
qwall
(4.14)
(4.15)
In the above equation, E ′ ∗ is a function of molecular Prandtl number and the
universal constant χ ∗ isχ ∗ = χC 1/4
µ ≡ 0.25. An alternative form of writing this
logarithmic temperature profile is:
Θ ∗ = Prt
U + + P
C 1/4
µ
(4.16)
P is called the pee-function and it is derived by Jayatilleke [57] from pipe flow
measurements.
P = 9.24
Pr
Prt
0.75
−1 1+0.28exp
−0.007 Pr
Prt
(4.17)
In the case of prescribed wall heat flux, wall temperature can be calculated by:
Θwall =
qwallΘ ∗
ρcpk 1/2
P
+ΘP
(4.18)
In the case of prescribed wall temperature, wall heat flux can be obtained from:
qwall = ρcp(Θwall −Θ)k 1/2
P
Θ∗ (4.19)
The above standard wall function has only been used in combination with
the high-Reynolds-number k-ε model, whilst the more advanced schemes
have used the AWF, described in the following section.