Geoscience Reference
In-Depth Information
3.07
Re
d
0.168
, for 800
approximated according to
C
D
¼
Re
d
8,000
C
D
1.0,
10
5
C
D
and for 8,000
1.2 (e.g., Rowinski and Kubrak
2002
;
Schlichting and Gersten
2006
). For large cylinder, Reynolds numbers (
Re
d
>
Re
d
4
4
10
5
)
C
D
suddenly decreases to
C
D
0.3 in the so-called supercritical flow regime
as the location of the separation point of the flow around the cylinder changes
(e.g., Nakayama
1999
).
However, the flow structure in a multicylinder array differs substantially from
the flow structure around a single isolated element as wake flow and sheltering
effects caused by upstream cylinders dominate the flow pattern. Petryk (
1969
), as
well as Lindner (
1982
), showed that, for a regular arrangement of cylinders,
C
D
is a
function of the cylinder diameter
d
, the cylinder spacing
a
x
and
a
y
, the approach
velocity of an individual cylinder
u
0i
, and the slope
S
.
Petryk (
1969
) used the wake superposition concept to quantify the flow structure
within a cylinder array. The superposition concept is based on the assumptions that
(1) the channel is straight and has rectangular cross-section; (2) the flow in the channel
is uniform and the flow approaching the cylinders is two-dimensional and fully
developed; (3) the velocity defects caused by upstream cylinders are small; (4) the
turbulence theory of Reichardt (
1941
)) is applicable and can be used to describe the
influence of multiple wake-flows; (5) the cylinder diameters are small compared
to the surrounding flow field; and (6) each wake flow can develop independent of
the wake flow caused by upstream cylinders (Lindner
1982
).
The distribution of the velocity defect
u
d
in a cross-section at a downstream
distance
D
x
of a cylinder can be determined from (Lindner
1982
):
2
z
u
D
x;
max
e
0
:
69
b
1
u
d
;
D
x
ð
z
Þ¼
(2)
=
2
;
D
x
where
z
maximum velocity defect at the center
of the wake (averaged over flow depth), and
b
1/2
¼
¼
transverse coordinate,
u
D
x
,max
¼
one-half of the wake width in
which
u
d,
D
x
¼
u
D
x
,max
/2 (Li and Shen
1973
; Lindner
1982
). According to Petryk
(
1969
), the maximum velocity defect can be estimated according to:
0
1
0
:
7
1
:
5
D
x
C
D
d
1
@
A
u
D
x;
max
¼
0
:
9
u
0
(3)
2
g
D
xS
u
0
1
þ
where
S
¼
bed slope,
g
¼
acceleration due to gravity,
u
0
¼
approach velocity of
the cylinder causing the wake, and
C
D
¼
drag coefficient of the cylinder causing
the wake. The wake width
b
1/2
can be calculated according to:
0
:
41
24
D
x
0
:
7
C
D
d
b
1
=
2
;
D
x
¼
0
:
ð
Þ
(4)
u
0
, the approach velocity
of the cylinders in the second row can be calculated using (
2
)-(
4
). The approach
Starting with the first row of cylinders for which
u
01
¼
