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

CHAPTER 4. NUMERICAL IMPLEMENTATION 62
the subgrid, SU of the main grid near-wall cell node is modified as follows:
SU = S ′
〈U〉s1
U − ν
(4.29)
To obtain the main grid boundary condition on k, volume-weighted subgrid
averages of turbulent production (Pk) and dissipation (ε = ˜ε+ˆε) are required.
These are readily obtained after the subgrid governing equations have been
solved. These values may be integrated with respect to the main grid near-
wall cell volume and then incorporated as a source in the main grid near-wall
cell equation for k:
For the k-ω model
ys1
SU = S ′ U + Pk − ε (4.30)
ε = (ωk)β ∗
(4.31)
Note that, in the above equation for ε, the quotient is averaged. If the
individual terms are averaged separately and then multiplied, the boundary
condition will not be correct. Terms may only be averaged separately where
the net source applied to an equation is a linear combination of those terms.
The boundary condition on ε is obtained in a very similar manner to the
boundary condition on k. Rather than prescribing a value of ε in the main
grid near-wall cell, the terms leading to the production and distruction of ˜ε
are averaged throughout the subgrid, leading to a net production of ˜ε to be
applied to the main grid. (In the main grid node, ˜ε may be taken as being
synonymous with ε, since ˆε → 0 away from solid boundaries.) The modified
source term applied to the main grid near-wall cell is
SU = S ′ U + Cε1
f1
˜ε
Pk − Cε2 f2
k
˜ε 2
− E − Y (4.32)
k