Civil Engineering Reference
In-Depth Information
The pressure (rapid or slow) i
n
the Fourier space is given
as a function of Green's function
G k
*
,
y
,
(
)
by
y
′
1
(
)
(
)
(
)
∫
*
′
′
′
[2.77]
p k
,,
y k
=
Gk
,,
yy Fk y
, ,
kdy
x
z
x
z
−
1
The Green's function associated with equation [2.75] is
⎡
*
′
⎤ ⎡
*
⎤
cosh
ky
(
−
1) cosh
ky
(
+
1)
⎣
⎦ ⎣
⎦
(
)
yy Gkyy
′
>
:
*
,
,
′
= −
()
()
*
2 h h
k
k
k
⎡
*
′
⎤ ⎡
*
⎤
cosh
ky
(
+
1) cosh
ky
(
−
1)
(
)
⎣
⎦ ⎣
⎦
′
*
′
yy Gkyy
<
:
,
,
= −
()
()
2 h nh
k
*
k
k
when
k
*
0
and
k
2
k
z
2
=
+
≠
1
(
)
(
)
′
′
′
y
>
y
:
G
0
y
,
y
=
y
−
y
2
1
(
)
(
)
y
′
<
y
:
G
0
y
,
y
′
=
y
−
y
′
2
The Green's function is obviously indepe
nd
ent of the
Reynolds number. Near to the wall,
G k
*
,
y
(
)
corresponding to high wavenumbers
k
*
decreases rapidly
toward the center of the channel, whereas the Green's
function near to the source with low wavenumbers varies
slightly [KIM 89]. In
other
words, the large structures with
small values of
k
*
→±
1,
y
′
k
z
2
influence the pressure far from
the wall, while the effect of the smaller structures is more
local, as might be expected. Similarly, the contribution to the
pressure/velocity
k
2
=
+
gradient
correlation
terms
(
)
, wherein one of the predominant terms is
p
∂
u
i
∂
x
j
+ ∂
u
j
∂
x
i
(
)
, is local in the inner layer
y
+
<
50
, whereas it is
p
∂
u
∂
y
+ ∂
v
∂
x
global and stems from all the points
y
+
100
[KIM 89].
>
Search WWH ::
Custom Search