The UMIST-N Near-Wall Treatment Applied to Periodic Channel Flow

CHAPTER 3. CHANNEL FLOW 41
profile is
k + =
1
Reτ 2
0.07 + 0.05
∗
1600
y +
2
∗ exp − y+
7
+4.5 1 − exp − y+
+ −1
4y
+ 1
20 Reτ
+ 2
y
∗ 1 − exp −
3
Figure 3.4 shows this profile for k + plotted against DNS results.
(3.41)
Table 3.1 offers qualitative descriptions of the contributions of the various
terms in Equation 3.41. The final term ensures that k + = 0 at the wall, but
the shape of the profile in the very near-wall region does not closely match
DNS data. An analysis of the near-wall behaviour of turbulence quantities
will follow.
Table 3.1: Qualitative behaviour of terms in Equation 3.41
Reτ
1600
0.07 + 0.05
4.5 1 − exp
∗ (y + ) 2
∗ exp − y+
7
4y + −1
+ 1
− y+
20
1 − exp −
y +
3
Reτ
2
3.4.4 Empirical Profile for u 2 +
The following formula for 〈u2 〉 +
k +
fits the peak value of k +
helps to roughly capture the curve
at high Reynolds numbers
forces a value of zero at the wall
was adapted from a profile fitted by Alexan-
der Davroux and Dominique Laurence at Electricité de France.