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

CHAPTER 4. NUMERICAL IMPLEMENTATION 54
The fully implicit scheme also requires information on φ t−1
p . This does not
appear explicitly in Equation 4.12 because it is included in the source term,
SU.
As an example of the expressions of the coefficients presented in Equation
4.12, the coefficients on 〈U〉 are
AN = (ν + νt) n
∆yn
AS = (ν + νt) s
∆ys
AP = (ν + νt) n
∆yn
= AN + AS
SP = − 1
∆t ∆yp
SU =
〈U〉 t−1
P
∆t
+ (ν + νt) s
∆ys
∆yp −
4.4 Boundary Conditions
1
ρ
∂ 〈P 〉
∆yp
∂x
(4.13)
At y = δ, a symmetry boundary condition is employed. The governing
equations are not solved at y = δ. Values of 〈U〉, k, ˜ε, and ω are copied to
the symmetry plane node from the nearest adjacent node. At this adjacent
node, the coefficients are adjusted as follows for each of 〈U〉, k, ˜ε, and ω:
AN = 0
AP = AS (4.14)
This ensures that all gradients in y are zero at the symmetry plane.
At the wall, velocities are zero. This creates large gradients in the near-