Civil Engineering Reference
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diverge if we suppose that
is constant [AUB 88]. For a
homogeneous channel flow, the equation governing the
instantaneous velocity is obtained after the mean in the
homogeneous planes in the conventional manner, i.e.
<
U
>
∂
U
∂
u
∂
u
∂
u
∂
u
NS
:
i
+
i
U
+
v
δ
+
i
u
−
i
u
i
i
1
j
j
∂∂
t
x
∂
y
∂
x
∂
x
[4.71]
j
j
1
∂
p
2
=−
+ ∇
ν
u
i
ρ∂
x
i
In practice, it is sometimes necessary to select a
truncation
N
{
m
=1
(
m
)
for the base of POD functions. We then
apply a G
ale
rkin projection on a chosen basis. Using the
notation
NS
i
to represent the Fourier transform of the
Navier-Stokes equation [4.71], we take the inner product in
a zone delimited by
(0,
y
)
of the channel
ϕ
y
()
m
dc
⎡
⎤
()
()
[4.72]
()
m
⎦
∫
()*
m
ϕ
,
NS
=
NS
y
ϕ
y dy
⇒
⎢
⎥
⎣
i
i
i
i
dt
0
in order to obtain a system of ordinary differential equations
for the resolved modes
dc
(
m
)
N
{
m
=1
(
m
)
involves a non-resolved part which is modeled by a
Smagorinsky model [AUB 88]. These authors use
experimental data limited to
y
+
<
/
dt
. The truncation
ϕ
40
of a pipe flow at
Re
τ
=
265
. The dynamic equation
dc dt
therefore contains
various terms, including the pseudo-pressure terms and
unresolved mode closure.
()
/
m
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