Civil Engineering Reference
In-Depth Information
Then, equation [4.43] assumes the form
()
2
2
dy
t
t
1
2
()
() ()
()
2
∫
A
*
′
′
[4.48]
ν
t
=
=
y
t v
t
=
v
R
t
dt
Q
2
2
2
v v
dt
22
0
The above relation takes different forms depending on the
value of the diffusion time
t
. For high
t
values (
t
), we
∝
→∞
t
find
∞
()
( )
()
∫
A
[4.49]
ν
t
=∞=
ν
v
2
R
*
t
′
dt
′
= Λ
v
2
Q
Q
2
v v
2
A
v v
22
22
0
where
Λ
A
is the Lagrangian integral length scale. These
preliminary notions will be sufficient for use in the rest of
this topic. For further details - in particular for the
formulation of expression [4.48] for short diffusion times -
readers are invited to consult specialized publications such
as [HIN 75].
vv
22
4.10.2.
Lagrangian description of transport by
energetic events
A different way of analyzing energetic events is to track
the fluid particles and determine their trajectories [BER 89,
BER 90a].
15
The Lagrangian approach is based on the
integral of the Navier-Stokes equation, tracking a particle
along the entire length of its trajectory
s
. Consider such a
particle whose origin is arbitrary but whose trajectory
finishes at a given position
y
+
at a given time
t
= 0. The
integral of the component
u
in the Navier-Stokes equation
along its trajectory, with backtracking in time, is
15 Readers can also consult [BER 90b] for a Lagrangian approach to the
vorticity transport equation.
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