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

CHAPTER 3. CHANNEL FLOW 33
of x and time, hence Pw(x, t). Integrating Equation 3.6 with respect to y
and applying the appropriate boundary conditions yields
ρ v 2 + [〈P 〉 − Pw(x, t)] = 0 (3.9)
Equation 3.9 can be differentiated with respect to x to give
Thus
∂〈P 〉
∂x
∂ 〈P 〉
∂x
is not a function of y.
= ∂
∂x Pw(x, t) (3.10)
In the above analysis, continuity has served to simplify the momentum equa-
tions, and y-momentum has yielded the insight that 〈P 〉 varies only in the x
direction and in time. The remaining x-momentum equation is solved in the
CFD code. Applying the EVM to Equation 3.5, this becomes:
∂ 〈U〉 ∂ 〈P 〉 ∂ ∂ 〈U〉
= −1 + (ν + νt)
∂t ρ ∂x ∂y
∂y
(3.11)
In summary, channel flow is governed by a single momentum equation con-
taining terms for fluid acceleration, shear stress, and a driving pressure gra-
dient. Furthermore, convection does not take place in any channel flow, since
wall-normal velocity is zero throughout and all gradients in the wall-parallel
direction are zero.
3.1.1 The k-ε Model
In channel flow, the transport equations in the k-ε model (Equations 2.16 &
2.33) become
k :
˜ε : ∂ ˜ε
∂t
∂k ∂ ν+νt
= ∂t ∂y σk
∂ ν+νt ∂ ˜ε
˜ε
= + Cε1f1
∂y σε ∂y
k
∂k
∂y
+ Pk − ε (3.12)
˜ε 2
Pk − Cε2f2 + E + Y (3.13)
k