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1979
Hydrodynamic Drag on Bottom-Mounted
Smooth and Rough Cylinders in Periodic Flow
Sarpkaya, Turgut
Offshore Technology Conference
http://hdl.handle.net/10945/48946
Downloaded from NPS Archive: Calhoun
L
OTC 3761
HYDRODYNAMIC DRAG ON BOTTOM-MOUNTED SMOOTH
AND ROUGH CYLI NDERS INPER I00 ICFUJiI
by Turgut Sarpkaya and Farhad Rajabi,
Naval Postgraduate School
@Copyright 1979. Olfshore Technology Conference
This paper was presented at the 11th Annual OTe in Houston, lex April30-May 3, 1979. The material is subject to correction by the author. Permission to copy is restricted to an abstract of not more than 300 words
ABSTRACT
The in-line and transverse forces acting on smooth
and sand-roughened circular cylinders placed on a plane
boundary (with no gap) in a sinusoidally oscillating
flow in a U-shaped vertical water tunnel have been
measured. The drag and inertia coefficients for the
in-line force have been determined through the use of
the Morisonls equation and the Fourier analysis. The
transverse force has been analyzed in terms of its
maximum and minimum values, root-mEan-square values,
and the time it remains above a fraction of its
maximum. The results have been presented as a function
of the Keulegan-Carpenter number and the frequency
parameter B = D2/vT.
INTRODUCTION
The design of unburied pipelines laid on or near
the ocean bottom requires an understanding of the
external fluid forces acting on them and an appreciation of the complexities stemming from all other
environmental conditions.
Numerous studies have been conducted both in
laboratory and in the field on the determination of
the forces acting on submerged pipelines. l - 10 In spite
of these past efforts, however. there still remains
much to be learned about the fluid forces acting on
smooth and rough cylinders resting on a plane boundary.
Recently j Yamamoto, Nath, and Slotta? and vJri ght and
Yamamoto a investigated the wave forces on cylinders
near a plane boundary for small values of the Reynolds
number and the Keulegan-Carpenter number. It is a
well known fact that for small values of the period
parameter the flow about the cylinder does not quite
separate, the wake-dependent drag force is very small
or negligible, and the in-line as well as the transverse forces are essentially from the wave temporal
acee 1erati on.
Under such ci rcums tances, the appropri-
ate coefficients may be evaluated from the potential
theory. The results of such a study show that? When
the cylinder touches the boundary a net force exists
away from the wall. However, if even a very small gap
exists between the cylinder and the wall, then a large
net force exists toward the wall. The inertial forces
.
alone do not usually give rise to the maximum load
situation for small structures, such as pipelines
located on or near bottom, and the separation effects
become extremely important not only in the determination of the magnitude but also the direction of the
forces.
. Sarpkaya 8 ,g determined the drag, inertia, and
11ft coefficients for circular cylinders placed near a
plane boundary in oscillatory flow at high Reynolds
numbers and Keulegan-Carpenter numbers for various
relative gaps of e/D. He has shown that the drag and
inertia coefficients for the in-line force acting on
a cylinder in the vicinity of a plane wall are increased by the presence of the wall. This increase
is most evident in the range of e/D values smaller
than about 0.5. Both coefficients depend on the
Reynolds number, Keule~an-Carpenter number, and the
relative gap. The effect of the boundary layer or
the penetration depth of the viscous wave is small
provi ded that the boundary' 1ayer remains 1ami nar. For
turbulent oscillatory boundary layers the characteristics of the waJl jetand separation over the cylinder may be Significantly affected. Sarpkaya has also
shown that the transverse force toward the wall is
relatively small and fairly independent of e/D. It
occurs during the periods of low velocity or high
acceleration. The transverse force away from the wall
is quite large and-depen'dent on e/D, particularly for
e/D smaller than about 0.5. The two forces are about
90 degrees out of phase.
The present investigation was undertaken for the
purpose of measuring the in-line and transverse forces
on smooth and rough cylinders resting, with no gap,
o~ a plane boundary in a fluid oscillating strictly
slnuso1dally and for the purpose of assessing the
applicability of Morison's equation. It is hoped that
the results obtained from highly idealized tests will
provide the designer not only with a rough estimate
of the forces involved but also with some understandin
of the physical mechanisms with which he can assess
the favorable or unfavorable influences of the ocean
environment such as the variation of the amplitude and
frequency of the waves, orientation of the waves and
currents, and the scour and deposition of sediment
around the pipes .
References and illustrations at end of paper.
219.
of several methods previously described. ll The data
were recorded in analog form and then digitized and
processed through the use of an HP-9845 computerdigitizer system. .
EXPERIMENTAL EQUIPMENT AND PROCEDURES
The equipment used to generate the harmonically
oscillating flow has been extensively used at this
facility over the past six years.ll ,12 Only the
The in-line and transverse forces acting on each
cylinder were measured and analyzed in terms of the
appropriate lift coefficients and the drag and inertia
coefficients through the use of the Fourier analysis.
salient features, most recent modifications, as well
as the adaptation for this work, are briefly described
in the following.
The length of the U-shaped water tunnel has been
increased from 30 ft to 35 ft and its height from 16
FORCE COEFFICIENTS AND THE GOVERNING PARAMETERS
ft to 22 ft. Previously, a butterfly valve arrangement at the top of one of the legs was used to initiate
Data reduction with the forces in-line with the
the oscillations. During the past two years, the
direction of oscillation is based on Morison1s equatunnel has been modified so that oscillations can be
tion and the Fourier analysis.
generated and maintained indefinitely at the desired
The in-line force, which consists of the inertia
amplitude. For this purpose the output of a 2 Hp fan
was connected to the top of one of the legs of the
force Fi and the drag force Fd, is assumed to be
tunnel with a large pipe. A small butterfly valve,
expressible by
placed in a special housing between the top of the
tunnel and the supply line, oscillated harmonically at
F = 0.5C dPLDIUlu + 0.25WPC mLD 2dU/dt
(1)
a frequency equal to the natural frequency of the
oscillations in the tunnel. The oscillation of the
For an oscillating flow represented by U = -Umcose,
valve was perfectly synchronized with that of the flow with 8 = 2wt/T, the Fourier averages of Cd and em are
through the use of a feedback control system. The
given by
2w
output of a pressure transducer (sensing the instan(2)
taneous acceleration of the flow) was connected to an
-(3/4)
FCOS8/(pLDU~)d8
electronic speed-control unit coupled to a DC motor
a
oscillating the valve plate. The circuit maintained
and
the period of oscillations of the valve plate within
2w
0.001 seconds. The amplitude of oscillations was
varied by constricting or enlarging an orifice at the
(3)
(2K/w3)
FSin8/(pLDU~)d8
exit of the fan. The flow oscillated at a given amplio
tude as long as desired.
f
f
Two circular cylinders of diameters 0 = 5 inches
and 0 = 6.50 in. were used. Each cylinder was mounted
in the test section, with its axis normal to the flow.
A clearance of 1/8 inch was provided between the plane
boundary (bottom of the tunnel) and the bottom of the
cylinder. The gap was sealed with a very thin plastic
wrapping sheet (1/100 inch) attached both to the plane
boundary and the bottom of the' cylinder. The width of
the plastic sheet was slightly larger than 1/8 inch so
as to allow the cylinder and hence the force transducer
to deflect freely in any direction.
Furthermore, the
gaps between the tunnel walls and the cylinders ends
were sealed with foamy material (about 3/16 inch thick)
The tunnel side of the foam was lubricated so as to
eliminate or minimize friction.
Data reduction for the transverse or lift forces
;s based on the following expressions:
(maximum transverse force away from the
wall in a cycle)/(0.5pLDU~)
(4)
C
Lmin
=
(minimum transverse force)/(0.5pLDU~ (5)
In addition, the root-mean-square value of the lift
coefficient and the fraction of the time the lift force
remains above a specified fraction of its maximum are
calculated.
The above force coefficients may be shown to
depend primarily on K, Re, and k/O where K = UmT/D and
Two identical force transducers, one at each end
Re = UmD/v. Evidently, additional parameters are
of the test cylinder, were used to measure the instanneeded to describe the characteristics of the oscillatory boundary layer of the on-coming flow on the plate.
taneous in-line and transverse forces. A special
housing was built for each gage so that it can be
One such parameter is the depth of penetration of the
mounted on the tunnel window and rotated to measure
laminar viscous wave in oscillatory flow over an
infinite plate. The exact solution of the equation of
either the in-line or the transverse force alone. The
calibration of the gages was accomplished by hanging
motion shows that the amplitude of the viscous wave
loads in the middle of the cylinder after setting both decreases rapidly with increasing n = In/2v y and
gages to sense only the transverse force. The calibra- becomes practically negligible at about n = 10. Thus,
tion was checked for horizontally-applied loads through for the case under consideration where n = 2TI/T,
T = 6.000 seconds, and v = 0.00001 ft2/sec., one has
the use of a simple pulley and weight system.
y = 0.5 inches. Evidently, n may also be expressed in
Experiments were first conducted with 5 and 6.5
terms of the cylinder diameter 0 as
inch smooth cylinders. Then each cylinder was coated
(6)
with sand. For this purpose, sand was sieved to the
y /0 = n/vT /w02
desired size and applied on the cylinder surface after
For the values cited above and for Re = 300,000 and
the latter was first coated with a thin layer of airK = 100, one has S = Re/K = D2/v T = 3,000, and y/D =
drying epoxy paint. The relative roughness of each
0.1. Thus, the depth of penetration of the viscous
cylinder was kiD = 1/100.
wave is relatively small (about 0.5 inch for a 5 inch
cylinder) for the case under consideration provided
The displacement, velocity, and acceleration of
that the boundary layer remains laminar.
the fluid in the tunnel were obtained through the use
220
It is on the basis of the foregoing that the force
coefficients will be regarded as functions of the
Reynolds number Re (or the frequency parameter a),
Keulegan-Carpenter number K, and the relative roughness
kiD and the boundary-layer effects will be ignored.
Thus, one has
(7)
It appears for the purposes of Eq. (7) that the
Reynolds number is not the most suitable parameter
involving viscosity. The primary reason for this is
= fi(K, a, kiD)
(8)
somewhat surprising in view of the fact that one would
have anticipated earlier separation and greater span-
in which S is called the Ufrequency parameter~1 8,12
wise correlation with rough cylinders, leading to
larger lift forces.
From the standpoint of dimensional analysis,
either the Reynolds number or the frequency parameter
can be used as an independent vari ab 1e. Evi dently,
S ;s constant for a series of experiments conducted
with a cylinder of diameter 0 in water of uniform and
constant temperature since T is kept constant in a
U-shaped asci 11 ati ng flow tunne 1. Then the vari ati on
The ratio of the lift-force frequency to the flowoscillation frequency (f = liT) is shown in Figs. 11
and 12 for the smooth an~ rough cyl i nders for two
values of the frequency parameter.
Clearly, the lift
force has twice the flow-oscillation frequency for all
values of K a"d Re. This shows that it is the separation of flow over the cylinder at each half cycle of
flow and not the subsequent shedding of vortices that
determines the fluctuations of the lift force for the
type of flow-cylinder combination considered herein.
of a force coefficient with K may be plotted for
constant values of 8.
Subsequently, one can easily
recover the Reynolds number from Re
and 8 as a function of K for two values of the frequency parameter a. The "maximum ,value of CL at about K =
7 is slightly smaller than that corresponding to the
smooth cylinder. Otherwise, there is very little
difference between the lift coefficients for the smooth
and rough cylinders. In general, the effect of rouhness is in the order of magnitude of the effect of the
Reynolds number.
The root-mean-square values of the lift coefficient is shown in Figs. 9 and 10 for the smooth and
rough cylinders, respectively. For large values of K
the rms value of CL for the rough cylinder is slightly
smaller than that for the smooth cylinder. This is
that Urn appears in both K and Re. Thus, replacing Re
by a = Re/K = 02NT in Eq. (7), one has
Ci(a coefficient)
The maximum and the minimum values of the lift
coefficient for ,rough cylinders are shown in Figs. 7
= aK and connect
the points, on each B = constant curve, representing
a given Reynolds number.
Fi na lly, Fi gs. 13 and 14 show for one smooth and
rough cylinder how much time P(t), expressed as a
fraction of the total time, the lift force spends
above a fraction x of the maximum force. Clearly, the
DISCUSSIONS OF RESULTS
The drag and inertia coefficients for smooth
cylinders are shown in Figs. 1 and 2 for two values
lift force remains large for a large fraction of
time and then decreases to almost zero. The general
of the frequency parameter. Clearly, Cd can reach
very high values and is a function of the Reynolds
shape of the curves for both smooth and rough cylinder remains practically independent of K. The only
significant difference between the smooth and rough
cylinder P(x) values is that P(x), following a rapid
number for a given value of K. The inertia coefficient does not appear to depend on Re and increases
with increasing K. For very small values of K where
the separation effects are negligible, em approaches
remains practically constant over a large fracits theoretical potential flow value of 3.29. Clearly, rise,
tion
of
time. The variation of P(x) with P(t) for the
no generalizations can b,e made regarding the relative
theoretical and experimental values of em' For a
cylinder sufficiently away from a boundary, Cm is
smooth cylinder is more gradual. Evidently, these are
related in a complex wayan the occurrence and motion
within the range of Reynolds numbers encountered.
CONCLUSIONS
Evidently, it is not possible to explain the complex
variations of Cd and Cm with K since it is largely
determined by separation effects.
on smooth and rough cylinders resting on a wall were
of the separation points and the correlation of the
vortices along the cylinder.
always smaller than its theoretical value of 2. In
the present case (e/D = 0), the theoretical value of
Cm is smaller than the experimental value, at least
Although the i n-l i ne and transverse forces acti ng
measured for sinusoidally oscillating flow rather than
for a wavy-type fluid motion, the results presented
The drag and inertia coefficients for the rough
cylinders (kiD
=
herein enable a number of conclusions concerning the
11100) are shown in Figs. 3 and 4.
latter type of flow to be drawn.
The effect of roughness on Cd is quite significant.
Once again, the inertia coefficient is very little
Morison's equation may be used to predict the
affected either by the Reynolds number or by roughness. in-line
force acting on such cylinders subjected to
local
wave
motion through the use of the FourierThe maximum and the minimum values of the lift
averaged coefficients.
coefficient (lift force is always away from the wall)
are shown in Figs. 5 and 6 for the smooth cylinders.
The drag coefficient for bottom-mounted cylinders
The lift coefficient reaches very high values at
can acquire very large values. The effect of roughness
relatively small K values. The potential flow value
may be quite significant. The inertia coefficient
for the lift force away from the wall when elD = 0
at very low values of K is nearly identical to that
is given by von MUller 3 as CL = rr(rr 2+3)/9 = 4.493.
Clearly, the experimental values for the smooth cylin- obtained from the potential theory and increases with
ders are relatively larger than the theoretical value, increasing K.
at about K = 7. The effect of separation at this
The transverse forces, like the in-line forces,
value of K is such as to increase the lift. It is
strongly influenced by flow separation. The 1ift
expected that the lift coefficient will reduce to 4.5 are
force is always away from the wall (e/D = 0) and the
as K approaches zero.
221
on the Ocean Bed, II Paper No. OTC-1383, Houston,
lift coefficient decreases with increasing K and Re.
Texas, [1971].
The lift coefficient for very small values of K (no
separation) approaches that predicted theoretically.
5.
der occurs at smaller K values than that for a cylinder away from the wall. In general the effects of
separation even at small K values are quite profound
and the potential flow values of the inertia coefficien
and the lift
coeffici~nt
6.
Evidently,
the potential flow theory gives no clues about the drag
force and the calculations must be based on the
results obtained experimentally.
7.
"wave Forces on Cylinders Near Plane Boundary ,"
calculations and help to plan and interpret the results
8.
This work was supported by a grant from the
National Science Foundation and forms part of a larger
investigation into basic studies of time-dependent
flows wHh relevance to industrial applications. The
authors wish to express their appreciation to NSF and
9.
Reynol ds Numbers, II OTC-2898, Houston', Texas,
[1977].
10.
REFERENCES
Wright, J. C. and Yamamoto, T.: IIWave Forces on
Cylinders Near Plane Boundaries," Journal of the
Haterway, Port, Coastal and Ocean Division, No.
WW1, Feruary, [1979], pp. 1-13.
Ralston, D. O. and Herbich, J. B.: "The Effects of
Waves and Currents on Submerged Pipelines,1I Coastal
Sarpkaya, T.: IIIn-line and Transverse Forces on
Cylinders Near a Wall in Oscillatory Flow at High
to Mr. Jack McKay for his consistently excellent
assistance with the experiments.
11.
and Ocean Engineering Division Report No. 101C.O.E., Texas A&M University, March, [1968].
2.
Sarpkaya, T.: IIForces on Cylinders Near a Plane
Boundary in a Sinusoidally Oscillating Fluid,"
Journal of Fluids Enqineerinq, Trans. ASME, Vol.
98, Series 1, No.3, September, L1976J, pp. 499505.
ACKNOWLEDGEMENTS
1.
Yamamoto, T., Nath, J. H. and Slotta, L. S.,
Journal of the Waterways, Harbors and Cloastal
Engineering Division, ASCE, Vol. 100, No. WW4,
Nov., [1974J, pp. 345-359.
Relatively idealized results such as the ones
presented herein can form the basis of preliminary
of relatively expensive field measurements.
"Submarine Pipe Lines - A Model Investigation of
the Wave Forces," Report No. EX. 158, Department of
Scientific and Industrial Research, Wallingford,
England, July, [1961].
tend to underestimate the
forces acting on a bottom-mounted cylinder.
Johansson, B. and Reinius, E.: IIWave Forces Acting
on a Pipe at the Bottom .of the Sea," Paper No. 1.7!
IAHR Congress, London, [1963].
It appears that separation over a bottom-mounted cylin-
Sarpkaya, T.: "Vortex Shedding and Resistance in
Harmonic Flow about Smooth and Rough Circular
Cylinders at High Reynolds Numbers," Naval Postgraduate School Technical Report No. NPS-59SL76021
February, [1976], Monterey, California.
Al-Kazily, M. F.: "Forces on Submerged Pipelines
Induced by Water Waves,1I Ph.D. dissertation,
College of Engineering, Univ. of California,
Berkeley, Calif., October, [1972].
3.
12.
Sarpkaya, T.: "The Hydrodynami c Res i stance of
Roughened Cyl i nders in Harmoni c Flow, II Trans. of
the Royal Institution of Naval Architects, Vol.
120, [1978], pp. 41-55.
Johnson, R. E.: IIRegression Model of Wave Forces on
Ocean Outfall s," Journal of the Waterways and
Harbors Division, Proc. ASCE, Vol. 96, No. WWZ,
May, [1970], pp. 284-305.
13.
von M~ller, W.: "Systeme von Doppe1quellen in der
Ebenen Strtlmung, Insbesondere die Strtlmung urn Zwei
Ore; szyl i nder, II Zei tschri ft fur angewandte Mathe-
4.
matik, Vol. 9, No.3, June, [1929J, pp. 200-213.
Priest, M. S.: IIWave Forces on Exposed Pipelines
~-
3.0
2.0
...
1.0
0.9
•
•
•
•
•
•
•••
•
•
• •••
•
.. ..
•
.
f3 = 2840
."
.
• •• •
•
fJ =4100 •• •• •
• ••
..
"
K
0.8L4--~5--~6--L7~8~9~10------~--~20~~--~30~--.~4~0--~5~0--&-~70~A-~100
Fig. 1 - Drag coefficient for the bottom-mounted smooth cylinders.
-
{
I
Cm
.' • . ."
• • '.
• ..
Il = 2840
4 .4 ...
•
.0
• • • • '• . • . • ••
.
",.... 0",
3 -
0
0
..t.:."~
~
.. ~
••••
4800
K
2
4
•
• •
5
6
I
I I
8 9 10
7
40
30
20
50
100
70
Fig. 2 - Inertia coefficient for the bottom-mounted smooth cvl inders.
·C
d
-•
•
•
2 i-
I
•
•
•
7""
61-
. ..
6
C
7
3
..
'.
,
,
,
"
•
~
7
6
4
•
=4800
•• •
• • •
,
,
•
K
• • •
-
kiD
8 9 10
100
70
,...., . • ••
·• .., ' ...
• 2840
•
.
•• 2840
20
30
40
50
Fig. 3 - Drag coefficient for the roughened cylinders.
m
..... ..
/ ' Il
8 9 10
1'1=4800,,
3.29
• •
.. .. 0'
•
'\0
.
q ..
4 i"
• •
•
I
1
4
•
•
• •
•
=1/100
kiD
•
•
.
20
30
= 1/100
•
40
K
I
I
50
•
70
100
Fig. 4 - Inertia coefficient for the I'Dughened cylinders .
••
... • •
......
... •
.
~
f3
A 2840
....
.....
...
•
....
•1·
o
...
....•
4800
·
• • ...
I
( ... (.1
K
o~~~~~~~~~~~~~~~~~~~~~~
10
~
~
~
00
o
ro
Fig. 5 - Maximum lift coefficient for the smooth cylinders.
\
1
4
e l min.
1.5
...
...
•
•
f3
....•
1.0
...
0.5
... 2840
• 4800
....... ...t ....•
A
0
• ••
.~
•• •
...
... ... •...
...
" ... ...
•(
10
0
30
10
•
••
... ... • ... ...
\
. ... ...
K
so
50
40
i::"; q. 6 - Minimum 1ift coefficient for the smooth cylinders.
~
.•
•
.1640
• 4800
0°
I•
e
.
.
I
k/D= 1/100
•
° ••
0
0
...
...
A
K
o~~~~~~~~~--~~~~~~~~~~--~
o
10
20
30
40
50
60
Fig. 7 - Maximum lift coefficient for the roughened cylinders.
1.5
j3
......
1.0
.1640 lk/D =1/100
• 4800
o
.~
0.5
..... -.A..
• .'I. • ...
•
•
•
... 0" 0° ...
•••
o~~~~~~~~~~~~~~~~~~~~~~K~
o
10
211
e •
o·o
30
40
50
Fig. 8 - Minimum lift coefficient for the roughened cylinders.
60
4
3
C lrms
..
•
.
•
~
2840
• 4800
0 ..
• •
..
2
•
..
•
..
.
eo.
K
40
50
30
10
20
Fig. 9 - RMS valve of the lift coefficient for smooth
5
4
70
60
cy';n~ers.
CLrms
•
"
{<
•
.. 2840
• 4800
..
3
••
2
.
•
...
•
.•
I
kID = 1/100
.
"
K
00
60
40
10
20
30
50
Fig. 10 - RMS valve of the lift coefficient for roughened cylinders.
..
.. °ei ............ , I
.
;.. J ........ i
70
t
~
A 2840
•
4900
K
o~~~~~~
[}
10
i-ig. 11
__~~~~~~~~~~~~~~~~~~
20
311
40
50
Lift and flow frequency ratio for smooth cylinders.
60
•• A·.••
A
•
...••
•
...
•
0=28401
k/O=1/100
0=4800
K
0
m
10
0
~;g.
~
~
W
~
12 - Lift and fl.ow frequency ratio for rough cylinders.
100
6.4 10
57
90
80
K=27
70
pix)
60
50
40
30
20
10
10
20
30
40
50
60
70
80
90
100
Fig. 13 - P(x) versus P(t) fo'r smooth cyl inders.
49
50
68
21
40
II = 2140
k/ D = 1/100
30
20
10
0
0
10
30
20
40
50
60
70
80
Fig. 14 - P(x) versus P(t) for roughened cylinders.
90
100