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

CHAPTER 4. NUMERICAL IMPLEMENTATION 53
The discretised transport equation for k is
kP − k t−1
P
∆t ∆yp =
ν + νt
∂k
∂y
where
σk
n
+Pk∆yp − ε∆yp
Pk = νt
n
〈U〉n − 〈U〉 s
∆yp
and 〈U〉 n and 〈U〉 s are interpolated values.
ν + νt
−
2
σk
s
∂k
∂y s
(4.8)
(4.9)
Discretised transport equations for ˜ε and ω can be found in a similar manner.
Some further equations whose discretised forms are noteworthy are
⎛
⎞2
∂〈U〉 ∂〈U〉
− ∂y
∂y
E = 2ννt ⎝ n
s ⎠
∆yp
⎛
ˆε = 2ν ⎝
√k n
−
∆yp
√k s
⎞
⎠
2
(4.10)
(4.11)
Thus, an algebraic equation may be generated for each node P . The discre-
tised differential equations are expressed in the code in the form
(AP − SP ) φp = ANφN + ASφS + SU
(4.12)
where φ is the unknown parameter from the original differential equation.
AP , AN and AS are coefficients on nodal values. Previous time step infor-
mation is included as a source. The source is split into two terms, SU and
SP φp for reasons of numerical stability. It is advantageous to have a large
coefficient on φp, so negative quantities are sometimes moved from SU into
SP (dividing by φp) to artificially increase this coefficient. At other times,
sources involve a product of φp, and the use of SP is a natural choice.