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

CHAPTER 2. TURBULENCE MODELS 19
The RANS equations are
D
Dt
Continuity : ▽ · 〈U〉 = 0 (2.4)
Momentum :
T ransport :
D〈U〉
Dt
D〈Φ〉
Dt
= −1 ▽ 〈P 〉 + ▽ · (ν ▽ 〈U〉)
ρ
∂ 〈uu〉 ∂ 〈vu〉 ∂ 〈wu〉
− − −
∂x ∂y ∂z
ν
= ▽ · ▽ 〈Φ〉 + Sφ
σ
∂ 〈uφ〉 ∂ 〈vφ〉 ∂ 〈wφ〉
− − −
∂x ∂y ∂z
may be referred to as the mean material derivative and is defined by
D
Dt
(2.5)
(2.6)
∂
≡ + 〈U〉 · ▽ (2.7)
∂t
It can be shown [53] that the mean material derivative is related to the
material derivative according to
DΘ
=
Dt
D 〈Θ〉
Dt
for a quantity Θ = 〈Θ〉 + θ.
∂ ∂ ∂
+ 〈uθ〉 + 〈vθ〉 + 〈wθ〉 (2.8)
∂x ∂y ∂z
For convenience, Equation 2.5 may be re-written as
D 〈U i〉 ∂ 〈P 〉
= −1 +
Dt ρ ∂xi ∂
ν
∂xj ∂ 〈U
i〉
−
∂xj ∂
ujui ∂xj
(2.9)
where each side of the equation is a vector in i comprised of terms that are
summed in j.
By decomposing velocities into mean and fluctuating components, RANS
models seek to treat separately the physical details of turbulence. The fluctu-
ating velocity components which are associated with turbulence are therefore