Civil Engineering Reference
In-Depth Information
interests of general applicability, we will assume that
xt
()
is
*
complex. It must also be remembered that the notation
()
indicates the complex conjugate. Equation [4.63] leads to
(
)
[4.65]
*
nm
cc
=
λδ
n
−
m
n
Using the definitions combined with equation [4.65], we
can easily show that
∞
(
)
( )
(
)
∑
() ( )
[4.66]
*
*
R tt
,
=
xt x t
=
λϕ
t
ϕ
t
12
1
2
n
1
2
n
=
1
In particular, we have
∞
2
() ()()
∑
()
*
[4.67]
R tt
,
=
x t x t
=
λϕ
t
nn
n
=
1
4.11.2.
POD and wall turbulence
This section is heavily inspired by the work of Panton
[PAN 01], which offers an excellent overview. We will
consider a homogeneous turbulent channel flow of length
L
x
in the streamwise direction and
L
z
in the spanwise direction.
The velocity vector is then decomposed into a Fourier series
in the homogeneous directions
x
and
z
, and a Karhunen-
Loève series in direction
y
. The component
u
i
is thus
expressed as follows
G
(
)
∑∑∑
(
)
()
(
)
()
mnq
mnq
imx
inz
[4.68]
uxt
,
=
c
t
ϕ
yee
i
mn
q
are essential in POD analysis.
They are determined on the basis of the Fredholm integral
equation of the first kind applied to the finite domain
D
of
the flow
The basic functions
()
ϕ
y
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