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

CHAPTER 3. CHANNEL FLOW 46
obtaining y ∗ is
y ∗ can be related to y + by
y ∗ = y√ k
ν
Likewise, 〈U〉 can be normalised by k.
(3.53)
y ∗ = y +√ k + (3.54)
U ∗ = 〈U〉
√ k
U ∗ =
U +
√ k +
(3.55)
(3.56)
y + is plotted against y ∗ for various values of Reτ in Figure 3.8. U + and U ∗ at
various Reynold’s numbers are plotted in Figure 3.9. It can be seen that U +
and U ∗ can differ significantly. This may be caused by the fact that k + is not
increasing monotonically. An interesting feature, visible in Figure 3.9, is that
U ∗ approaches a constant as y → 0. This happens because, as y → 0, U +
becomes proportional to y + and k + becomes proportional to (y + ) 2 . Thus,
U +
√ k + = U ∗ approaches a constant.
Another choice of variable against which to normalise y and U is 〈v 2 〉. 〈v 2 〉
has the same units as k. Since most of the local variability of k results from
turbulent fluctuations in the mean flow direction (〈u 2 〉), local nondimension-
alisation based on 〈v 2 〉 may be expected to offer smoother profiles. We can
define
y ∗
v 2 = y 〈v 2 〉
ν
U ∗ 〈U〉
v2 = =
〈v2 〉
= y +
〈v2 〉 +
U +
〈v 2 〉 +
(3.57)
(3.58)