Civil Engineering Reference
In-Depth Information
t
is
b
G
.
Readers can easily verify that relation [4.54] is written as
point
a
G
and whose random initial position at time
−
(
)
(
)
[4.57]
uv
=+
uv
v U
−+
U
v U
−
U
aa
ba
a
b
a
a
a
b
The last term is simply
0
0
1
∂
P
(
)
()
()
()
()
∫
∫
vU U
−
−
v
0
sds
+
ν
v
0
∇
2
Usds
=Θ
a
b
a
1
ρ
∂
x
−
t
−
t
(
)
on the right-hand side of equation
[4.57] illustrates the mechanism of gradient-driven Prandtl
transport, which we can briefly recap while considering the
concept of the mixing length. A fluid particle “remembers”
and recovers the mean momentum of its origin
b
G
at the
poi
nt
a
G
.
Therefore, at point
a
, it induces a fluctuation
The term
v
a
U
b
−
U
a
A
where
A
is the mixing length. The
length scales characteristic of the turbulent structure are of
the same order of magnitude in direction
x
i
, and the order of
magnitude of the fluctuating intensities is therefore
identical. In addition, a particle coming from a low-velocity
zone with
v
uU U
=−∝−
∂∂
u y
a
b
0
causes a local fluctuation
u
0
.
>
<
Consequently,
uv
0
, and by combining these elements, we
are able to construct the Prandtl turbulence closure model
<
2
⎛
⎞
∂
∂
u
−=
uv
A
2
⎜
⎝ ⎠
y
in relation [4.57] represents the
transpor
t of fluid
particles by displacement. On the other
hand,
The term
u
b
v
a
(
)
+
v
a
U
b
−
U
a
(
)
has an entirely different meaning,
representing the correlation of
v
a
with the modification of
the total momentum including the accelerations and
v
a
U
a
−
U
b
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