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

CHAPTER 3. CHANNEL FLOW 39
3.4.1 Empirical Profile for U +
Reichardt’s log law [58] states that
U + = 1
κ ln 1.0 + 0.4y +
+7.8 1 − exp − y+
+ y
− exp −
11 11
y+
3
(3.34)
This is plotted against experimental and DNS data in Figure 3.1. Reichardt’s
law follows DNS results very closely near the wall. As y + increases, Re-
ichardt’s law approaches the log law, while experimental values of U + tend
to fall above the log law for high y + .
3.4.2 Empirical Profile for − 〈uv〉 +
Based on Reichardt’s law, a profile can be derived for − 〈uv〉 + . This was
originally done by Alexander Davroux and Dominique Laurence at Electricité
de France.
dτ
dy is a constant, τ| y=0 = τw and τ| y=δ = 0. Therefore, the equation for τ (y)
in a channel is
τ (y) = τw
1 − y
δ
This can be inserted into Equation 3.17 to give
− 〈uv〉 = τw
ρ
In nondimensional form, this becomes
1 − y
δ
− 〈uv〉 +
= 1 − y
δ
−
− ν dU
dy
dU +
dy +
(3.35)
(3.36)
(3.37)