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The above energy integral equation 10.45 can be obtained in the following
three-regions and the Prandtl number effect is shown in Figure 10.8.
y +
1 − (y + /R + )
T
+
=
+
dy
(1/Pr) + (ε H /ν)
0
ε H =
ε m
ν
ν
For the region 0 ≤ y + ≤ 5 with y + ≈ 0 and (ε m /ν) ≈ 0,
y +
T
+
= Pr
+
dy = Pr y
+
0
+
+
u = y , T
+
= Pr y
+
(10.46)
For the region 5 ≤ y + ≤ 30 with u + = 5 ln y + − 3.05 and (y + /R + ) ≈ 0,
ε m
ε H
y +
=
=
− 1
ν
ν
5
y +
1
(1/Pr) + (y + /5) − 1
T
+
− T
+
5
+
dy
=
5
y +
1
1
y +
= 5
d
− 1
+
(1/Pr) + (y + /5) − 1
Pr
5
5
u
+
= u
u*
20
(T −T )
w
T +
⎛ q" ⎞
=
w
⎝ ρC p u* ⎠
5
Pr
If Pr = 1,
u + ≅ T
+
5
30
500
1000
yu*
+
ln y = ν
211
Turbulent Flow Heat Transfer
FIGURE 10.8
Law of wall temperature profile for turbulent boundary layer.
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The above energy integral equation 10.45 can be obtained in the following
three-regions and the Prandtl number effect is shown in Figure 10.8.
y +
1 − (y + /R + )
T
+
=
+
dy
(1/Pr) + (ε H /ν)
0
ε H =
ε m
ν
ν
For the region 0 ≤ y + ≤ 5 with y + ≈ 0 and (ε m /ν) ≈ 0,
y +
T
+
= Pr
+
dy = Pr y
+
0
+
+
u = y , T
+
= Pr y
+
(10.46)
For the region 5 ≤ y + ≤ 30 with u + = 5 ln y + − 3.05 and (y + /R + ) ≈ 0,
ε m
ε H
y +
=
=
− 1
ν
ν
5
y +
1
(1/Pr) + (y + /5) − 1
T
+
− T
+
5
+
dy
=
5
y +
1
1
y +
= 5
d
− 1
+
(1/Pr) + (y + /5) − 1
Pr
5
5
u
+
= u
u*
20
(T −T )
w
T +
⎛ q" ⎞
=
w
⎝ ρC p u* ⎠
5
Pr
If Pr = 1,
u + ≅ T
+
5
30
500
1000
yu*
+
ln y = ν
211
Turbulent Flow Heat Transfer
FIGURE 10.8
Law of wall temperature profile for turbulent boundary layer.