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

CHAPTER 2. TURBULENCE MODELS 21
2.2 The k-ε Model
The high-Reynolds number version of the standard k-ε model of Launder &
Sharma [33] defines turbulent viscosity, νt as
2 k
νt = Cµ
ε
(2.15)
where k is turbulent kinetic energy, and ε is the rate of dissipation of k. Cµ
is a constant given in Table 2.1.
The transport equation for k is based on Equation 2.14 (with 〈Φ〉 = k). The
source term consists of production, Pk and dissipation, ε:
Dk ν + νt
= ▽ ·
▽ k + Pk − ε (2.16)
Dt σk
Pk is
Pk = −aij 〈Sij〉 (2.17)
where the right hand side of the equation is summed over all permutations
of i and j to obtain Pk.
aij is defined as
〈Sij〉 is defined as
aij = u iu j
〈Sij〉 ≡ 1
∂ 〈U i〉
2 ∂xj − 2
3 kδij
+ ∂ U j
∂xi
Applying the EVM (Equation 2.10) to Equation 2.17,
(2.18)
(2.19)
Pk = 2νt 〈Sij〉 〈Sij〉 (2.20)