Transport Phenomena in Wall Turbulence - Transport and Coherent Structures in Wall Turbulence - page 69

This relation is obtained using the equation of continuity
in an incompressible flow, where

⎡ ⎤

⎛ ⎞

∂ ∂ u

⎜ ⎜
⎝

i

⎟ ⎟
⎠

u = 0

⎢
⎣

⎥
⎦

k

∂ x ∂ x

⎢ ⎥ k i

Equation [2.19] can therefore be written as

∂ K

k ∂

K

k ∂

U

i

+ U

x k =−

u

i u

∂ t ∂ ∂ x k

[2.28]

⎛ ⎞ ⎠

2 K

∂ p

ρ +

∂

x i − ν ∂

u k
∂

∂ u k

∂

− K u + ν

⎜ ⎟

k

∂ x k ∂ x i ∂ x i x i

⎝

This equation is considerably simplified in a channel flow,
which is homogeneous in the streamwise and spanwise
directions. For example, in this case, the inertial terms
disappear. We obtain

T K 0 = P K + + D K + N K − ε K

uv ∂

U
∂

P K =−

y

∂
∂

T K

=− u i K

x i

[2.29]

2 K

∂

D K = ν

∂ x k ∂ x k

1
ρ

∂
∂

N K =− u k p

x k

= ν ∂

u i ∂ u i

ε K

∂ x k ∂ x k

Each term in this equation has a physical meaning of its
own. The term P K is the kinetic energy production. It is by
way of this term that the turbulence is regenerated. The
terms

T K and N K , respectively, represent the turbulent