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

CHAPTER 4. NUMERICAL IMPLEMENTATION 50
Table 4.1: The notation for discretised values
current node subscript P
node above subscript N
node below subscript S
vertex above subscript n
vertex below subscript s
previous time step superscript t − 1
cell height ∆yp
node-to-node dist. ∆yn,s
time step ∆t
4.2 Volume Integral Form
The finite volume discretisation method is convenient because quantities are
conserved within each finite volume and are therefore necessarily conserved
throughout the flow field. In order to employ the finite volume method, it is
first necessary to integrate the differential equations governing the flow with
respect to volume. The volume-integrated forms of the equations are pre-
sented below. Note that, in channel flow geometry, integrating with respect
to volume is equivalent to integrating with respect to y.
〈U〉 :
k :
˜ε :
∂〈U〉
∂k
∂t
∂〈P 〉
∂〈U〉
dy + (ν + νt) ∂x ∂y
ν+νt ∂k
∂t dy = − 1
ρ
dy =
∂ ˜ε dy = ∂t
ω : ∂ω dy = ∂t
− Cε2f2
ν+νt
σk
ν+νt ∂ ˜ε
σω
σε
˜ε 2
(4.1)
∂y + Pkdy − εdy (4.2)
∂y +
˜ε
Cε1f1 Pkdy k
dy + k
Edy + Y dy (4.3)
∂ω
∂y + γ
ω Pkdy − k
βω2dy (4.4)