Civil Engineering Reference
In-Depth Information
The Lagrangian coordinate of the fluid particle
y
2
t
0
,
t
()
is
linked to its velocity by
t
(
)
(
)
∫
′
′
yt t
+=
vt tdt
+
2
0
2
0
0
We can show ([HIN 75]) that:
T
T
t
t
1
1
()
(
)
(
)
∫
∫ ∫ ∫
yt
2
=
yt tdt
2
+
=
dtdtdtvt tvt t
′
"
+
'
(
+
")
2
2
0
0
0
2
0
2
0
T
T
0
0
0
0
[4.44]
t
t
(
)
(
)
∫∫
′
′′
=
dt
dt v
"
t
+
t
'
v
t
+
t
20
20
0
0
The Lagrangian correlation is defined by the relation
()
(
)
( )
Rt
vt tvt
′ +
'
()
A
vv
*
2
2
Rt
=
=
[4.45]
22
A
vv
22
2
2
v
v
2
2
The variance of the particle motions is linked to the
Lagrangian correlation. By combining the definition of
Lagrangian correlation with equation [4.44], we obtain
t
t
′′
()
()
∫∫
A
[4.46]
yt
2
=
2
v dtdtR
2
′′
′
*
t
′
2
2
vv
22
0
0
,
such that the correlation
v
2
q
of equation [4.43] is expressed
by means of
In addition, we can int
rod
uce a turbulent viscosity
()
ν
Q
t
dQ
()
[4.47]
vq
=−
ν
t
2
Q
dx
2
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