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

CHAPTER 2. TURBULENCE MODELS 23
way that
The difference between ˜ε and ε is defined as ˆε:
From Equations 2.22 and 2.23, it follows that
˜ε| y=0 = 0 (2.22)
ε = ˜ε + ˆε (2.23)
ˆε| y=0 = ε| y=0
(2.24)
The equation for νt is modified to use ˜ε for convenience. Also, damping
terms exist in the equation for νt and the transport equation for ˜ε. These
damping terms are a function of the turbulent Reynolds number, ˜
Ret. ˜
Ret
is a Reynolds number based on local turbulence quantities (in this case, k
and ˜ε). The tilde is a reminder that, for convenience, the turbulent Reynolds
number used in the low-Reynolds-number standard k-ε model uses ˜ε rather
than ε. ˜ Ret is defined as
˜Ret ≡ k2
˜εν
(2.25)
In the low-Reynolds-number version of the standard k-ε model [33], the trans-
port equation of ˜ε is
D˜ε ν + νt
= ▽ ·
Dt σε
where
▽ ˜ε + Cε1f1
E = 2ννt
˜ε
Pk − Cε2f2
k
∂ 2 〈U〉
∂y 2
2
2 ˜ε
+ E (2.26)
k
(2.27)
f1 = 1 (2.28)
−Ret ˜
f2 = 1 − 0.3e 2
(2.29)