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208
Analytical Heat Transfer
Therefore, the law of wall velocity profile can be obtained as
1
+
+
+ C
u = ln y
(10.40)
κ
From the experimental data curve fitting for y + ≥ 30,
κ = 0.4 or 0.41
+
u = 2.5 ln y
+
+ 5.0
(10.41)
Or
+
u = 2.44 ln y
+
+ 5.5
(10.42)
Then, consider the buffer zone between the laminar sublayer and the
turbulence region, 5 ≤ y + ≤ 30, the velocity profile can be obtained as
+
u = 5 ln y
+
− 3.05
(10.43)
It is important to mention that the above three-region velocity profile has
been validated and can be applied for turbulent flow over a flat plate or in a
tube with air, water, or oil as the working fluid. In addition, before showing
the law of wall for temperature profile, it is interesting to point out, just like
for the laminar boundary-layer case, the law of wall for temperature profile
is identical to the law of wall for velocity profile if Prandtl number is unity
(Pr = 1) by replacing the above dimensionless velocity with the appropriate dimensionless temperature with the same dimensionless y-direction wall
coordinate. The effect of Prandtl number on the law of wall for temperature
profile will be discussed in Section 10.3.
10.3 Turbulent Flow Heat Transfer
Consider a fully turbulent flow in a circular tube with uniform wall heat flux
""
(q = C) as the thermal BC [4–6], as sketched in Figure 10.7. From the energy
w
equation,
∂T
∂T
1 ∂
∂T
∂ 2 T
u
+ v
=
r(α + ε H )
+ α
(10.44)
∂x
∂r
r ∂r
∂r
∂x 2
y = R − r
r = R − y
dr = −dy