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

CHAPTER 4. NUMERICAL IMPLEMENTATION 55
wall region. Because CFD uses discretisation, the storage and computational
requirements associated with accurately representing large gradients are high.
This results in a great potential for computation-saving approaches at the
wall. Many methods exist for applying wall boundary conditions to CFD
codes, and these may be model-specific. Boundary conditions on the wall
are discussed below.
4.4.1 Wall Boundaries on k-ε
In the low-reynolds-number standard k-ε model, ˜ε is defined as being equal
to zero at the wall. Furthermore, k = 0 at the wall, arising from the fact that
k represents turbulent kinetic energy and all velocities, including turbulent
fluctuations, are zero at the wall. In the low-Reynolds-number model, these
quantities are simply prescribed to be zero at the wall. Transport equations
are not solved at the wall, but are solved in the wall-adjacent cell. Because
of the large gradients involved, finer grid resolution is required near the wall,
as shown in Figure 4.1.
4.4.2 Wall Boundaries on k-ω
As with the k-ε model, 〈U〉 = 0 and k = 0 at the wall, and transport
equations may be solved up to the wall-adjacent cell. However, ω → ∞ as
y → 0, so it is not numerically possible to specify ω at the wall. Wilcox [89]
notes that the limiting behaviour of ω at the wall is
ω → 6ν
βy 2 as y → 0 (4.15)