Civil Engineering Reference
In-Depth Information
⎛
⎝
⎞
D
Dt
u
i
u
k
∂
U
i
∂
p
ρ
+
K
=−
x
k
−
K
u
k
⎜
⎟
∂
∂
x
k
⎠
P
K
Τ
K
[2.20]
⎛
⎞
⎛
⎞
∂
∂
∂
x
k
+
∂
u
i
u
k
∂
− ν
∂
u
k
∂
∂
u
i
x
k
+
∂
u
k
∂
+ν
u
k
⎜
⎟
⎜
⎟
x
i
∂
x
i
x
i
∂
x
i
⎝
⎠
⎝
⎠
D
*
*
ε
K
The inertial terms in the turbulent kinetic energy
transport equation are grouped together in the form of the
operator
DDt
on the left-hand side of equation [2.20].
Turbulent production is
re
presented by
P
K
. It cannot exist in
a flow without shear
x
k
, which is essential for the
maintenance of the turbulent state. The quantity
∂
U
i
∂
Τ
K
, which
encompasses the turbulent diffusion terms, comprises
(among other values) the pressure/velocity correlations and
D
*
, which is linked to viscous diffusion. The dissipation per
unit mass is defined by
⎛
⎝
⎞
⎠
= ν
∂
u
k
∂
∂
u
i
x
k
+
∂
u
k
∂
[2.21]
ε
*
⎜
⎟
x
i
∂
x
i
Let us now introduce the symmetrical velocity gradient
tensor
⎛
⎞
⎠
1
2
∂
u
i
x
k
+
∂
u
k
∂
[2.22]
s
ik
=
⎜
⎟
∂
x
i
⎝
It is then possible to show that the dissipation
ε
*
is
identical to
ε
*
s
ik
s
ik
[2.23]
=
2
ν
Consider the equation for the kinetic energy at the wall.
For an established steady-state flow, the inertial term on the
left-hand side of the equation, along with the production and
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