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

CHAPTER 4. NUMERICAL IMPLEMENTATION 59
1
1
Pk dy − (ε)p dy
ρ p ρ
must be added to the transport equation of k
in the near-wall cell as a source. k may be factored out of the equation for
(ε) p , and the remaining equation (including the k 1/2 term) may be placed in
the SP source. This amounts to modelling k 3/2
p in the equation for (ε) p as
1/2
kp · k old
p
, where k old
p is the previous iteration value of kp.
The terms in the k equation are modified as follows:
SP = S ′
C
P −
3/4
µ k 1/2
U +
p
P
∆yp
ρ yp
C 1/4
µ k 1/2
p (〈U〉 P ) 2
SU = S ′ U +
U +
P
· yp
∆yp
(4.25)
In treating the transport equation of ε, the value of ε is simply prescribed in
the near-wall cell according to the mixing length hypothesis (Equation 2.47):
εp = C3/4 µ k3/2 κyp
(4.26)
By making SP and SU large, other terms in the transport equation of ε may
be made numerically insignificant in the near-wall cell, allowing a prescribed
value to be specified in SU. The source terms are set as follows:
SP = −G
C
SU = G ·
3/4
µ k3/2
κyp
(4.27)
where G is a very large number, chosen to be several orders of magnitude
greater than any other terms that will appear in the transport equation of ε.