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velocity on the cylinders in the second row is required to estimate the approach
velocities for the following cylinders. Hence, the approach velocity on the n th
cylinder can be estimated from (Lindner 1982 ):
0
@
1
A
2
2
Z
X
n 1
znz i
b 1
1
B
zz i
e 0 : 69
e 0 : 69
u 0 ;n ¼
u 0 þ
u D x;i; max
b 1
d z
(5)
=
2
; D x
;
i
=
2
; D x
;
i
1
B
where B
¼
flume width and z n , z i ¼
transverse coordinate of the n th and i th cylinder,
respectively.
In an infinite array of cylinders, the superposition of the velocity defect according
to ( 5 ) can, in theory, be carried out infinitely. In order to reduce the computational
effort, Lindner ( 1982 ) defined a threshold criterion so that the velocity defect is only
considered if it is larger than 3% of the undisturbed flow velocity. Applying this
criterion, Lindner ( 1982 ) found that u 0, n is roughly constant for n
20 cylinders.
Using ( 2 )-( 5 ), the depth-averaged velocity can be predicted at any point in the
channel. Thus, it becomes possible to calculate the drag exerted by individual
cylinders from F D i ¼
¼
C D A p u 0, i 2 , where the corresponding drag coefficient
can be determined dependent on the cylinder Reynolds number Re d ,i ¼
0.5
r
.
On the other hand, using the cross-sectional averaged velocity u 0 , the average drag
force of the i th cylinder follows from F D i ¼
u 0, i d /
n
C D ,i A p u 0 2 . Equating both expres-
sions yields the following formulation for the estimation of the drag coefficient
C D ,i ¼
0. 5
r
( u 0, i / u 0 ) 2 C D (e.g., Li and Shen 1973 ). Lindner ( 1982 ) extended this formu-
lation taking into account the blockage ratio according to the approach by Richter
( 1973 ) and taking into account the effect of gravity waves:
u 0 ;i
u 0
2
C D þ D C D
9 d
C D ;i ¼
1
þ
1
:
a y C D
(6)
where D C D ¼
additional drag caused by surface waves around the cylinder. The
additional drag depends on the Froude number of the averaged flow field and the
flow depth in front of and just behind the cylinder (e.g., Lindner 1982 ; Pasche and
Rouv ´ 1985 ).
For cylinder arranged with a constant transverse distance a y , Lindner ( 1982 )
suggested reducing the computational effort by replacing B with a y in ( 5 )and
assuming that the cylinder lines act as imaginary walls. Lindner ( 1982 ) pointed
out that this simplification is valid only for a y / d
10 as below this threshold
the wakes of two neighboring upstream cylinders influence each other, i.e., the
wake-flows cannot develop independently. For the computations with a parallel
cylinder setup, Lindner ( 1982 ) assumed that the cylinders are located in the
center of the interval
>
a y /2 to a y /2 while, for a staggered setup, he assumed
that the cylinders are located at the imaginary side walls. The assumption that
the cylinders lines act as imaginary walls was the reason that blockage effects
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