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

CHAPTER 4. NUMERICAL IMPLEMENTATION 58
source. 6 Since τw contains a 〈U〉 P term, the SP source is used:
SP = S ′ P − C1/4 µ k 1/2
p
U + p
(4.19)
where S ′ P is the value of SP before the log-law term is added (containing only
the time-dependence term).
Also, AS is modified to cut the link to the wall and prevent a second (in-
accurate) accounting of τw via the normal functioning of the x-momentum
equation. The modified terms are
AS = 0
AP = AN (4.20)
This modification is common to all wall function treatments on 〈U〉, k, ε,
and ω.
Average production and dissipation of k in the near-wall cell (
Pk p and
(ε) p ) are calculated from Equations 2.45 and 2.46. In descretised form, these
become
Substituting Equation 4.18,
〈U〉P
Pk = τw
p
yp
(kp)
(ε) p = ρCµ
2
〈U〉P
Pk p
τw
yp
ρC1/4 µ k
= 1/2
p (〈U〉 P ) 2
U +
P
(ε) p = C 3/4
µ k 3/2
· yp
U +
P
yp
(4.21)
(4.22)
(4.23)
(4.24)
6 In general, one would integrate τw with respect to the surface area over which it acts
before applying the result as a source to the x-momentum equation. In channel flow
geometry, τw acts over a surface of unit area.