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Hydrodynamic coefficients of a harmonically oscillated tower
Hydrodynamic coefficients of a harmonically oscillated tower SUBRATA K. CHAKRABARTI, DENNIS C. COTTER and ALAN R. LIBBY Marine Research Division, Chicago Bridge & Iron Co., Plainfield, Illinois 60544-8929, USA
A tower hinged at the bottom was oscillated mechanically in a sinusoidal fashion in a plane in still water. An instrumented section in the tower measured the inline and transverse forces locally on the tower due to the hydrodynamic effects. These forces are analyzed for the added mass, drag and lift coefficients which are presented as functions of Keulegan-Carpenter and Reynolds number. The lift force frequencies are also investigated. The measured overall reactions on the tower are used to verify the values of the local coefficients. The results presented here are not only applicable to articulated towers but to other moving elements of an offshore structure, e.g. risers, tension-legs, etc. Key Words: Added mass coefficient, drag coefficient, lift coefficient, articulated tower, oscillation, hydrodynamic coefficient.
INTRODUCTION The semi-empirical force model known as the Morison equation and its several modifications have been widely used in designing offshore structures. This approach depends on cfioosing inertia and drag coefficients which are determined experimentally. Generally there have been two different approaches used in determining these coefficients. One is to test members in oscillatory waves either in a laboratory tank or in the ocean. The second approach is to oscillate the member harmonically in an otherwise still fluid or oscillate the fluid past a stationary instrumented member. It can be shown that these last two methods are equivalent. Sarpkaya in his experiments in an U-tube used the method of oscillating the fluid past a horizontal tube 1 or a group of t u b e s ) On the other hand, several other investigations have been made with a member oscillated in still water. Garrison et al. a oscillated a horizontal cylinder, Rains and Chakrabarti 4 and Laird and Warren s oscillated an articulated group of cylinders while Loken et aL 6 oscillated a group of risers. These methods of testing are sometimes preferred over the tests in waves because the tests can be more easily controlled, the uncertainty of the water particle kinematics can be avoided and higher values of Reynolds number can be achieved compared to a wave tank test. However, the direct application of these results to the case of waves past an offshore member may be questioned. On the other hand, these results may be directly applied to cases where added mass and drag coefficients are required for moving members of an offshore structure, e.g., an articulated mooring tower, risers, tension le~. With these applications in mind, a test was designed with a cylinder hinged at the bottom so that it was free to oscillate in a vertical plane and the tube was oscillated mechanically through still water in a harmonic fashion. A short instrumented section was incorporated into the cylinder in order to measure the inline and transverse forces locally. In addition, the angular motion and overall reaction Received February, 1983. Written discussion closes Dec. 1983.
226
Applied Ocean Research, 1983, Vol. 5, No. 4
at the hinge of the cylinder were measured. An articulated tower can be vertical or inclined in its equilibrium position due to steady loads on it. Therefore, the cylinder was tested at vertical (0 degrees) and two inclined (30 and 45 degrees) positions to investigate the effect o f the tower angle on the hydrodynamic coefficients. The derivation of the motion of a tower in waves depends on the knowledge o f a few hydrodynamic coefficients, namely, inertia coefficient, drag coefficient, added mass coefficient and lift coefficients. The purpose of this paper is to present the values of added mass, drag and lift coefficients applicable to an articulated tower. The top of the cylinder was attached to a mechanical forcing device and oscillated in a vertical plane in a pure sinusoidal motion. During all tests, the instrumented section was submerged throughout the oscillation. The elevation of the bottom of the cylinder was adjusted so that the instrumented section of the cylinder was always at the same elevation and the top of the cylinder was always above the water. The angular motion was measured at the base of the cylinder. Since the measured motion is sinusoidal, the velocity and acceleration at any point on the cylinder can be simply obtained by employing the differential equations of the harmonic motion. The twocomponent forces on the instrumented section were measured. The measured inline forces were used to obtain the added mass and drag coefficients. The transverse force on the instrumented section was used to calculate the lift coefficient and the lift force frequencies. These hydrodynamic coefficients are plotted as functions of the corresponding Keulegan-Carpenter parameter and Reynolds number. They are then used to predict the reaction at the bottom joint. The predicted bottom joint load is compared with the measured load.
THEORETICAL BACKGROUND For a fixed tower in waves, theempirical Morison equation for wave forces is written in terms of the local water 0309-1708183/040226-08 $02.00 © 1983 CML Publications
Hydrodynamic coefficients of a harmonically oscillated tower: S. K. Chakrabarti, D. C. Cotter and A. R. Libby particle velocity and acceleration components as
f = CMf~ + Cold
O)
fi=pn--D2fi
(2)
where
4 and
fD = ½pDIulu
(3)
in which f = force per unit length, p = mass density of water, D = tower diameter, u and fi = local water particle velocity and acceleration and CM, CD = inertia and drag force coefficients respectively. This expression for wave forces on a fixed cylinder was modified to describe reaction forces on an oscillating tube in still water as follows: rr
1
f =rr£# + CAP 4D22 +-CDDI212 +
(4)
in which m is the mass of the tube, x, ~ and £ are the tangential displacement, velocity and acceleration of the tube and CA = a d d e d mass coefficient, Co = drag coefficient, k = spring constant due to buoyancy. A linear damping term (which is generally small for small tubes under consideration) may be added to the right hand side of equation (4).
oscillation of the cylinder was measured with a Rotary Variable Differential Transformer (RVDT). The loads in the three orthogonal directions were measured by an X Y Z load cell of 25 lbf. capacity. Slots were provided in the mounting brackets to allow for alignment of the cross-tank axes of the instrumented section in the tower and the X Y Z load cell with the cylinder's axis of rotation. All alignments were performed and confirmed in the dry on a calibration stand. To allow the top of the cylinder to remain above the water for the three different test angle orientations, the articulated joint was mounted on top of a different height pedestal for each of the three test conditions of the tower ((3°, 30 ° and 45 ° to vertical). This allowed the tank water depth to remain constant throughout the testing. The pedestal was bolted to a rectangular steel plate that was weighted with lead for added stability. Three large bolts were attached to the plate that could be used to level the plate.
C. Mechanical forcing device This test plan required that the cylinder be oscillated at a known frequency and amplitude in still water. This oscillation was achieved by a mechanical forcing device which was composed of hydraulic piston, a pendulum, and a Moog electro-hydraulic servo-control device (see Figs. 1 and 2). The hydraulic piston was attached to the pendulum and causes it to move back and forth in a prescribed
DESCRIPTION OF THE MODEL
A. Tower The test structure is a circular cylinder with an outside diameter of 3 in. and length of 9 ft. The test tower contains three sections. The bottom section is 6 ft long and made from 6061-T6 aluminium with a thickness of 0.065 in. The bottom element contains a buoyant section consisting of two PVC plates with O-rings separated by a 0.25 in. ID PVC tube. In this manner a buoyant section is maintained and the wiring from the instrumented section above can be passed through the section to the computer data acquisition system. The centre section of the test tower is the instrumented section for measuring locally applied loads. This load cell has a 3 in. outer diameter, is 12 in. in length and measures loads in two orthogonal directions. The upper section has a length of 2 ft and is composed of the same material and thickness as the lower section and a buoyancy section is also maintained in this section by circular PVC plates with O-rings. In this case, no wiring need be passed through the section and no central tubing is required. A Plexiglas shroud is attached to the top of the tower to insure that the tower never becomes completely submerged during oscillation. An 0.5 in diameter rod which is threaded on one end and smooth on the other end is threaded through the top cap of the test tower. This rod is used for connection to the mechanical drive device.
Hydraulic
C y l i n d e r -
V Hydr. Pump
Wovctck (Freq..Oscillator')
~1 ~
~
I~--
Electro Hydraulic Moog $¢rvo Valve
Electronic Scrvo Cong=roller
XYZ Load C¢)1
-- 1 Foot Test S¢ctlon
B. Articulated ]oint An articulated joint was designed to support the lower end of the cylinder. The support joint allowed the test cylinder unrestricted rotation about one axis only and was oriented so that the cylinder oscillated in a plane parallel to the tank walls. The articulated joint was instrumented such that the cylinder rotation and support loads in three orthogonal directions could be measured. The angular
RVDT ( Rol.ary Variable Dif fcrcntlol "l"r'Qnsf (
Figure 1.
Lood C¢11.
Mechanical drive schematic
Applied Ocean Research, 1983, Vol. 5, No. 4
227
Hydrodynamic coefficients of a harmonically oscillated tower: S. K. Chakrabarti, D. C Cotter and A. R. Libby 1. Tower loads An X Y Z load cell with a capacity of 25 lbf. was attached to the upper restraint. This load cell measured the load in three orthogonal directions. In this case, during measurement one axis was aligned with the cylinder axis. During all tests the load at the articulated joint was measured. This load was also measured by a 25 lbf. X Y Z load cell but in this case one of the axes was always aligned with the vertical. These load cells are calibrated by applying weights in given directions to the load cells and measuring the electrical output signal.
2. Tower motion During the test runs for the mechanically oscillated cylinder, the cylinder was allowed to rotate in one plane about an articulated joint. The oscillation of the cylinder about the axis orthogonal to the vertical was monitored by an RVDT. This instrument is calibrated together with the articulated joint in the dry by moving the joint through known angles and measuring the output from the RVDT.
3. Instrument section
Figure 2.
Photograph o f test set-up
The loads at discrete intervals of the cylinder were determined by the instrumented section. This load cell was designed and developed in-house by CBI and measures the loads on a test section that is 1 ft long. It consists of a 3 in. diameter tube floating on a central core. The 3 in. cylinder is attached to two cantilever beams which deflect as loads are applied to the cylinder. Strain gauges mounted to these cantilever beams output an electrical signal which varies with the bending of the beam and thus the load. This load cell is also calibrated in the dry in a calibration stand. TEST RUNS
motion. The pivot point of the pendulum was attached to a rigid support structure mounted on the wave tank bridge. The bottom of the pendulum was attached to a machined stainless steel rod at the top of the cylinder with an assembly that included a ball-bushing that allowed free longitudinal movement of the assembly along the rod and a pair of radial bearings that allowed the assembly to rotate with the cylinder. The attachment assembly was instrumented with an X Y Z load cell similar to the one supporting the bottom of the cylinder. This load cell measured the cross-tank and down-tank driving loads normal to the cylinder axis and any load induced in the direction of the cylinder axis due to friction in the ball-bushing. The frequency and amplitude of the cylinder oscillation was controlled by the Moog servo-control unit. The Moog received a feedback signal from the RVDT at the bottom of the cylinder and adjusted the hydraulic piston motion as necessary to cause the feedback signal to match a reference signal from a Wavetek generator. The Wavetek is simply an electronic device which outputs a pure sine wave of known frequency and amplitude.
D. Transducers The following measurements were made: the loads in three orthogonal directions at the restraining points of the cylinder, the loads at two orthogonal directions at the specified instrumented section along the axis of the cylinder and the angular displacement of the cylinder.
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The frequency of the mechanical system was varied from 0.1 Hz to 0.8 Hz. At each frequency the amplitude of oscillation varied from 5 to 30 degrees. A total of 132 runs were made during these tests. The average KeuleganCarpenter number varied from 0.40 to 44.65 and the average Reynolds number varied from 2000 to 40000. During all phases of the test low pass filters with a cutoff frequency of 5 Hz were used to separate the desired signal from the electrical noise inherent in the data acquisition system. In addition a numerical filter with a 3 Hz cut-off frequency was used to separate mechanical noise from the signal in the in-line direction. DATA ANALYSIS When the tower is oscillated mechanically, the equation of motion is given by equation (4). Rewriting
fM = -- m2 -- kx + CAfA + CDfD
(5)
where 7r
f A = - - P 4D22;
1
fo=--2PDlYclf¢
in which x = rO, r = distance from the pin to the center of instrumented section, 0 = cylinder oscillation angle (measured, sinusoidal), k = s p r i n g constant from net buoyancy. In equation (5) the left hand side fM, fA andfD are known from the measured values. The velocity ~ = ar0 and 5( = o2rO are obtained by shifting the phase angle by 90 ° of the measured tower oscillation in which o = oscilla-
Hydrodynamic coefficients of a harmonically oscillated tower." S. K. Chakrabarti D. C Cotter and A. R. Libby ting circular frequency. The quantities, Ca and CD are obtained from the least square assumptions. In matrix form, the equations for CA and CO are written as
LfDfA
f~ J Co
IfMfoJ
TOWER ANGLE TO
VERTICAL - DEGREES
(6) ~i
where the summation over i = 1 . . . . . N (N = no. of data points in a cycle) applies to all terms. The solution is obtained by matrix inversion. The lift coefficients in the transverse direction are calculated from the measured transverse force. The coefficients related to the harmonic components of the lift force are obtained by a Fourier series analysis using the equation
BOTTOM:PIN
o
LOA[~ X - D I R E C t I O N
i
Ni
-
POUNDS
i
i
TOTAL EOAD ON ONE FOOT SECTION
DIRECTI:ON - POUNDS
M
fL = E CLmfDm m=l
(7)
where M = no. of Fourier components of lift coefficients, Czm, and fDm are the Fourier components off/;. The most predominant lift frequencies are obtained by taking the Fast Fourier Transform (FFT) of the lift force profile and choosing the frequency of the spectral peak.
POUNDS
ooi TOTAL LOAD ON ONE FOOT SECTION t-DIRECTiON d ........................................ i..................... i................. i..................... i~ ................ !
~O,O0
I
2',00
q',OO
DISCUSSION OF RESULTS The 3 in. diameter cylinder with the instrumental section was oscillated in still water by an electro-mechanical servocontrol device. It was oscillated in still water about a pin joint such that it resembled an inverted pendulum. The angular displacement at the pin was measured in time along with the driving force causing the cylinder oscillation, the inline and transverse loads on the instrumented section and the loads in three directions at the pin, that is, the inline, transverse and vertical loads. Even after the 5 Hz electronic filter was applied, spikes were left in the data. Although the cylinder angle was a smooth sinusoidal function of time, the loads in the inline direction demonstrated high frequency noise. This noise occurred because of servo-control device's tendency to overcompensate for errors between the reference and feedback signals and therefore oscillate about the mean reference value. This 'hunting' had little effect on the cylinder's angular position (the feedback to the drive system) but did cause cylinder accelerations that appeared as spikes in the inline loads, especially where the reference signal was changing most rapidly. The solution was to apply a 3 Hz digital signal filter to the inline loads and motions only after the data had been collected. These spikes did not occur in the transverse direction and hence, these loads were not faltered and thus, allowed any high frequency transverse loads to remain. These high frequency transverse loads are caused not by the mechanical system but by the lift forces and therefore should remain. See Fig. 3 for the filtered data after the 3 Hz digital t'dter was applied. The desired results of the mechanical oscillation tests were estimations of the added mass coefficient (CA), the drag coefficient (CD) and the lift force coefficient (CL). These coefficients are formulated as functions of the Keulegan-Carpenter number (KC) and the Reynolds number (Re) and plotted versus these non-dimensional numbers in Figs. 4-9. Various trends for these coefficients can be found from these plots. Figures 4 - 6 demonstrate the dependence of CA on the Keulegan-Carpenter number through various angles from vertical. The added mass coefficient was found to be non-dependent on the Re
6',OO
TIME -
Figure 3.
8',00
tO,O0
12,00
SECONDS
Sample of filtered test data
=cov
d~
I
Ol o
, ~
F~,ure 4.
o ~o
, is
r~-~
2'o KC
15
~0
i
F~-a i z ~ , !
~o
4'o
15
CA for cylinder at 0 °
I
d~
~o
~
I : cov
[~
l'~i
20
A
o o
i
~o
KC
Figure 5.
CA for cylinder at 30 °
I
I : COV
~9
0
o KC
Figure 6.
1'5
A
is
CA for cylinder at 45 °
Applied Ocean Research, 1983, Vol. 5, No. 4
229
Hydrodynamic coefficients o f a harmonically oscillated tower: S. K. Chakrabarti, D. C Cotter and A. R. Libby
number. In general, the added mass coefficient seems to decrease with angle for low KC numbers, that is, less than eight. In the mid-range the orientation angle of the cylinder seemed to make very little difference in the result. At high KC numbers, that is, greater than 20, added mass coefficients seemed to decrease with angle once more. Curves presented in Figs. 4-6 are mean curves drawn through the data points developed from the test. In general, there was scatter to these points about the mean line shown by the coefficient of variation (COV) at various values of KC in Figs. 4-6. The COV is computed by letting the collection of estimates for Ca at a given KC number be Yi (i = 1,2, 3 , . . . , iV) and the chosen mean value of the estimate be p. Then the random error of the estimate, Yi, is defined by the standard deviation of the estimate about the mean value, random error = Oy =
ai
o
~ "2
0
/
° //.
O.
/ e .
--
l#I/
LEGEND
/ #
~
o
......... - -
{3 - -
- -
~__
lb
1~
Re t ,,
RANGE ,i
"
"
2ODO0
- 30,000
30,000
-
,i
40.000-
0
10,000
2'o
-10,000 - 20,000 40,000 50,000
a'5
:70
:i~
2o
KC
Figure 8. at 30 °
Co versus K C f o r various R e ranges - cylinder
(8)
(Yi - Y) 1
The coefficient of variation is defined as the normalized random error, _
Oy
COV -
o
(9)
--
P The values of COV are shown in Figs. 4-6 at equal intervals of KC = 5. Contrary to the added mass coefficient, the drag coefficient shows a strong dependence on the Re number for the Re number range tested. It was found that there is comparatively little scatter of the data points about the mean line for the drag coefficient versus the Re number. The drag coefficient showed similar scatter when displayed versus the Kuelegan-Carpenter number but less scatter when plotted versus the Reynold's number. Accordingly the data was broken up into Reynold's number regions and plotted versus the Kuelegan-Carpenter number in Figs. 7-9. This reduced the scatter in the data and demonstrates the dependence of the drag coefficient on both of the nondimensional parameters. A computer program was written to compare the theoretical wave loads on the cylinder with the measured loads at the top and bottom, that is, at the driving arm of the mechanical oscillation device and at the restraining point on the pin. In this program, the cylinder is segmented into 20 sections and for each section and for each wave cycle a particular KC number is derived depending on the distance from the pin, the maximum angular velocity of the cylinder, the period of the oscillation, and the diameter of
C o, L E G E N D
el'
Re
=
Re
= 10,OO0-20,OOO
0-10,0OO
R e = 20,OOO-30,OO0 R e = 30,OO0-40,OO0
0 0
Figure 9. at 45 °
%
,b
KC
?s
~o
17~
CD versus K C f o r various R e ranges - cylinder
g
Z rid tl_
~. ~g
I ~L!A L) er-
o
iE
ei"
2E tO.
bJ
oo. LEGEND Re "~r R A N G E ,i ...........
o - -
o
"
~b
0
¢5
O
-
bJ ~E
10,OO0
10.000 - 20,000 20,00030.000 30,000 - 40,000 > 40,000
2'o
~
8¢" 3'o
g~
go
KC
Figure 7. at 0 °
230
CD versus K C f o r various R e ranges
Applied Ocean Research, 1983, Vol. 5, No. 4
cylinder
~.oo
~'.oo e'.oo Ib.oo tb.oo 2b.oo CRLCULRTED MAXIMUH F O R C E -- P O U N D S
Figure lO. Correlation o f m a x i m u m measured pin loads - cylinder at 30 °
theoretical
2b,.Do
and
Hydrodynamic coefficients o f a harmonically oscillated tower: S. K. Chakrabarti, D. C Cotter and A. R. Libby the cylinder. Then for each KC number and for each segment an added mass coefficient and a drag coefficient are determined from the mean curves of these coefficients versus the KC number. The loads from each segment are determined using these two coefficients and the time history of the angular motion at the pin. The loads for each section are then summed up and compared to the driving load and the restraining load at the pin. An example of these correlations is presented in Fig. 10. In this figure, a perfect correlation would lie along the 45 ° line. The bordering lines on each side of this line are the plus or minus 10% lines. It is seen that the correlation results are quite good for the smaller loads but as the loads become larger the theoretical loads are larger than the measured loads. It is felt that this deviation arises because the loads on the upper portion of the cylinder predominate over the lower portion of the cylinder. In those sections above the instrumented section where the drag load is quite high, the KC number is higher than for those cases experienced by the instrumented section because the velocity is larger. In those cases where the KC number required for the segment was greater than that presented by the mean curve, a linear extrapolation was used to develop the value for the drag or added mass coefficients from the mean curves. This could result in considerable error. The dependence of the lift force coefficient (CL) on the KC number is demonstrated in Figs. 11-16. It can be seen in these figures that the KC number range decreased for every increasing angle tested. For these mechanical oscillation tests, the KC number was strictly dependent on the amplitude of the oscillations. The maximum possible oscillations amplitude decreased with increasing angle in order to keep the instrumented section from leaving the water. Therefore, for each increasing angle the range of KC LEGEND ..........
Fundamental 2 ~ Harmonic 3r~ 4t__~
<5
O
5~
"2 ~ t oo. t~c5
c5
o o
5
I0
15
20 KC
25
35
CL versus KC for cylinder at 0 °
Figure 11.
LEGEND
o o
. . . . . . . .
Fundamentals 2 n~-~° H o r m o n i ¢ 4~ 5 I~
O
0
j
30
•
d"
~
o O
Figure 12.
, 5
• 10
....
• 15 ~C
• 20
--
= 25
CL versus KC for cylinder at 30 °
• 30
40
o
LEGEND
o.
.........
Fundamant01 2~
Harmonic
3~ 4L"
6'
a
5~
O'
5
10
15
20
25
KC
Figure 13.
CL versus KC for cylinder at 45 °
number tested decreased. In these figures there are two types of charts which enable the designer to determine the coefficients for the lift forces. Figures 11-13 are the first type of presentation for the lift force coefficients. In order to get this figure, the time history of the transverse load was broken up into its Fourier components. These loads were then divided by the factor p x ~ D / 2 where p is the density of the water, x m is the maximum normal velocity. This term is much like the term for the drag contribution in Morison's equation. Coefficients up to the fifth harmonic are presented because the energy tends to decrease rapidly beyond these harmonics. The second type of chart is presented in Figs. 14-16 which is the frequency ratio plot. In order to make this plot, the FFT of the time history of the transverse load and the angular motion were taken. For each test the frequency that contained the maximum energy of the transverse force and the angular motion data was determined and the ratio of these two frequencies was calculated. This did not mean there was no energy at other frequencies for the transverse load. This is simply the frequency with the maximum energy for each test for the transverse loads on the instrumented section. In most cases other peaks were present; they simply were not as large as the one chosen. In practice this ratio is usually considered an integer value. If the ratio is equal to 1, then the fundamental frequency or first harmonic is dominant. If it is equal to 2, then the second harmonic is dominant, etc. The test data shows that the frequency ratio takes on an integer value for low KC numbers. As the KC number increases the ratios tend to cluster around the integer values but become far more confused. At higher KC numbers, the trend persists but the basic assumption of an integer value relationship seems to begin to break down. The choice of method of determining the lift force is then up to the user and is dependent upon the type of information that can be used. For example, the user would first determine the KC number expected for any particular segment of the structure. Having this KC number he can then go into the second type of chart as in Fig. 14, and determine which frequency is dominant. With this information he can proceed to the first type of chart, Fig. 11 and choose the lift force coefficient for the KC number and harmonic required. Alternately, the user can determine a KC number and go to the first chart and pick out the five coefficients for the first five harmonics of the lift force. The phase angles of the lift force components are arbitrary. Therefore, whether one frequency is used or all five, the phases for these individual lift force components can be chosen from a random number generator. It can be seen in
Applied Ocean Research, 1983, Vol. 5, No. 4
231
Hydrodynamic coefficients o f a harmonically oscillated tower." S. K. Chakrabarti, D. C. Cotter and A. R. Libby Figs. 11-13 that the lift force coefficient for the second harmonic that is, C/,(2) tends to take on the largest value. In addition, the peak value for a particular harmonic tends to occur at higher KC number for increasing number of the harmonic.
50-
40*
30-
1
20-
l
10-
/I
0 0
FREQUENCY RATIO (TRANSVERSE / WAVE)
Figure 14. Dominant lift force frequency versus KC for cylinder at 0 ° 30-
25"
20-
"~15-
1 m
10-
0
I
0
FREQUENCY
RATIO (TRANSVERSE / WAVE)
Figure 15. Dominant lift force frequency versus KC for cylinder at 30 ° 25'
20-
15 ¸
l
u
10'
B 0
-
FREQUENCY
RATIO(TRANSVERSE / W A V E )
Figure 16. Dominant lift force frequency versus KC for cylinder at 45 °
232
Applied Ocean Research, 1983, Vol. 5, No. 4
CONCLUSIONS AND RECOMMENDATIONS The following conclusions may be drawn from the results presented here. 1. It is shown that the added mass coefficient is a function of the non-dimensional Keulegan-Carpenter number (KC) and not of the Reynolds number (Re) for the region of Reynolds number tested. 2. The hydrodynamic drag coefficients of a moving cylinder in still water showed clear dependence on Reynolds number in addition to the Keulegan-Carpenter number. 3. According to Froude's law, the Keulegan-Carpenter number between a prototype and its model is the same. The range of KC numbers attained in these tests will cover a wide range of prototype situations. Thus, as far as KC numbers are concerned, the mean curves of hydrodynamic coefficients presented in this report are directly applicable in a design. On the other hand, the coefficients are found to be at least weak functions of Re number (within the small range of test data). The Reynolds number between a prototype and its model differ significantly. In laboratory tests of this type, the largest Re number that can be generated is of the order of l0 s while many design conditions require an Re of about 1 0 7 . The trend of the coefficients with Re number should be determined from other similar tests (e.g., ocean tests, forced water oscillation, etc.) for extrapolation purposes. 4. Coefficients generated by an analysis of a particular location on the tower are similarly applicable over the entire length of the tower when these coefficients are formulated as functions of the Kuelegan-Carpenter number. The values of the coefficients should be varied over the length of a tower. 5. The drag coefficient shows a marked dependence on the Reynolds number while the added mass coefficient does not. This dependence becomes important at large KC numbers. 6. The lift (or transverse force) coefficients are presented in terms of its first five harmonics (by a Fourier series analysis). These harmonic coefficients are found to be functions (through some scatter) of the KC number. Since the lift force profiles are somewhat random in nature, these time histories of lift force may be reconstructed from the mean curves presented in the report if each harmonic is assigned an arbitrary ,phase angle. Thus, the results can be applied to a numerical analysis program quite easily. 7. The plot of the predominant lift force frequencies vs. KC number can be used to study the eddy-shedding process. Note that the predominant lift force frequency is almost an even multiple of the wave frequency (except at the higher end of the KC values where the flow becomes more turbulent). When the ratio is one, no eddies are shed (but probably a weak eddy formed) and the lift force is found to be regular. For a ratio of two, one eddy is shed alternatively on either side of the tower. For the ratio of three, two eddies are shed, and so on. 8. The largest value of the lift coefficient occurs at the second harmonic of the motion. The value of the KC number at which a particular coefficient and harmonic of
Hydrodynamic coefficients o f a harmonically oscillated tower: S. K. Chakrabarti, D. C Cotter and A. R. Libby the lift force becomes dominant increases with the harmonic. In other words, the higher the KC number, the higher the dominating harmonic of the lift force. 9. The test tower was restrained from lateral (or transverse) motion and, hence, the coefficients presented here are applicable to cylindrical members in a planar motion. If a member is free to oscillate laterally as well as inline, then the application of these coefficients directly without further corroboration may not be justified.
N Re )3 0 p o Oy
= = = =
No. of date points in a cycle Reynolds number Mean value of variables Yi (i = 1 , 2 . . . . . N ) Angular displacement of cylinder from equilibrium position = Water density = Circular frequency = Standard deviation of a variable setyi (i = 1,2 . . . . . N)
REFERENCES LIST OF SYMBOLS Force per unit length of structures
f fo fA,h
= Drag force for drag coefficient equal to 1.0 = Inertia force for inertia force coefficient equal to
JcL fM
=
k
=
m r x
= cA co cL
= = =
D KC M
= = =
1.0 Lift force Total measured force on instrumented section Restoring forces on instrumented section due to buoyancy Mass of instrumented section Distance from pin to centre of instrumented section Linear tangential displacement of instrumented section from equilibrium position Linear velocity of instrumented section Linear acceleration of instrumented section Maximum linear velocity during the cycle Added mass coefficient Drag coefficient Lift force coefficient Diameter of instrumented section Keulegan-Carpenter number No. of Fourier components
1. Sarpkaya, T. In-line and transverse forces on cylinders in oscillating flow at high Reynolds numbers. Proceedings on the Eighth Offshore Technology Conference, Houston, Texas, OTC 2533, 1976, pp. 95-108 2. Sarpkaya, T. Hydrodynamic forces on various multiple tube riser configurations, Proceedings on the Eleventh Offshore Technology Conference, Houston, Texas, OTC 3539, 1979, pp. 1603-1606 3. Garrison, C. J., Field, J. B. and May, M. D. l)rag and inertia forces on a cylinder in periodic flow, Journal o f the Waterway, Port, Coastal and Ocean Div., ASCE, Vol. 103, No. WW2, May, 1977 4. Rains, C. P. and Chakrabarti, S. K. Mechanical excitation of offshore tower model, Journal o f the Waterways, Harbors and Coastal Engineering Div., ASCE, Vol. 98, No. WWl, February, 1972, pp. 35-47 5. Laird, A. D. K. and Warren, R. P. Groups of vertical cylinders oscillating in water, Journal o f the Engineering Mechanics Division, ASCE, Vol. 89, No. EM1, February, 1963, pp. 25-35 6. Loken, A. E., Torset, P. P., Mathiassen, S. and Arnesan, T. Aspects of hydrodynamic loading in design of production risers, Proceedings on Eleventh Offshore Technology Conference, Houston, Texas, OTC 3538, 1979, pp. 1591-1601 7. Bearman, P. W. and Graham, J. M. R. Hydrodynamic forces on cylindrical bodies in oscillatory flow, Proceedings of the Second International Conference on Behaviour of Offshore Structures, London, England, August, 1979, pp. 309-322
Applied Ocean Research, 1983, Vol. 5, No. 4
233
A tower hinged at the bottom was oscillated mechanically in a sinusoidal fashion in a plane in still water. An instrumented section in the tower measured the inline and transverse forces locally on the tower due to the hydrodynamic effects. These forces are analyzed for the added mass, drag and lift coefficients which are presented as functions of Keulegan-Carpenter and Reynolds number. The lift force frequencies are also investigated. The measured overall reactions on the tower are used to verify the values of the local coefficients. The results presented here are not only applicable to articulated towers but to other moving elements of an offshore structure, e.g. risers, tension-legs, etc. Key Words: Added mass coefficient, drag coefficient, lift coefficient, articulated tower, oscillation, hydrodynamic coefficient.
INTRODUCTION The semi-empirical force model known as the Morison equation and its several modifications have been widely used in designing offshore structures. This approach depends on cfioosing inertia and drag coefficients which are determined experimentally. Generally there have been two different approaches used in determining these coefficients. One is to test members in oscillatory waves either in a laboratory tank or in the ocean. The second approach is to oscillate the member harmonically in an otherwise still fluid or oscillate the fluid past a stationary instrumented member. It can be shown that these last two methods are equivalent. Sarpkaya in his experiments in an U-tube used the method of oscillating the fluid past a horizontal tube 1 or a group of t u b e s ) On the other hand, several other investigations have been made with a member oscillated in still water. Garrison et al. a oscillated a horizontal cylinder, Rains and Chakrabarti 4 and Laird and Warren s oscillated an articulated group of cylinders while Loken et aL 6 oscillated a group of risers. These methods of testing are sometimes preferred over the tests in waves because the tests can be more easily controlled, the uncertainty of the water particle kinematics can be avoided and higher values of Reynolds number can be achieved compared to a wave tank test. However, the direct application of these results to the case of waves past an offshore member may be questioned. On the other hand, these results may be directly applied to cases where added mass and drag coefficients are required for moving members of an offshore structure, e.g., an articulated mooring tower, risers, tension le~. With these applications in mind, a test was designed with a cylinder hinged at the bottom so that it was free to oscillate in a vertical plane and the tube was oscillated mechanically through still water in a harmonic fashion. A short instrumented section was incorporated into the cylinder in order to measure the inline and transverse forces locally. In addition, the angular motion and overall reaction Received February, 1983. Written discussion closes Dec. 1983.
226
Applied Ocean Research, 1983, Vol. 5, No. 4
at the hinge of the cylinder were measured. An articulated tower can be vertical or inclined in its equilibrium position due to steady loads on it. Therefore, the cylinder was tested at vertical (0 degrees) and two inclined (30 and 45 degrees) positions to investigate the effect o f the tower angle on the hydrodynamic coefficients. The derivation of the motion of a tower in waves depends on the knowledge o f a few hydrodynamic coefficients, namely, inertia coefficient, drag coefficient, added mass coefficient and lift coefficients. The purpose of this paper is to present the values of added mass, drag and lift coefficients applicable to an articulated tower. The top of the cylinder was attached to a mechanical forcing device and oscillated in a vertical plane in a pure sinusoidal motion. During all tests, the instrumented section was submerged throughout the oscillation. The elevation of the bottom of the cylinder was adjusted so that the instrumented section of the cylinder was always at the same elevation and the top of the cylinder was always above the water. The angular motion was measured at the base of the cylinder. Since the measured motion is sinusoidal, the velocity and acceleration at any point on the cylinder can be simply obtained by employing the differential equations of the harmonic motion. The twocomponent forces on the instrumented section were measured. The measured inline forces were used to obtain the added mass and drag coefficients. The transverse force on the instrumented section was used to calculate the lift coefficient and the lift force frequencies. These hydrodynamic coefficients are plotted as functions of the corresponding Keulegan-Carpenter parameter and Reynolds number. They are then used to predict the reaction at the bottom joint. The predicted bottom joint load is compared with the measured load.
THEORETICAL BACKGROUND For a fixed tower in waves, theempirical Morison equation for wave forces is written in terms of the local water 0309-1708183/040226-08 $02.00 © 1983 CML Publications
Hydrodynamic coefficients of a harmonically oscillated tower: S. K. Chakrabarti, D. C. Cotter and A. R. Libby particle velocity and acceleration components as
f = CMf~ + Cold
O)
fi=pn--D2fi
(2)
where
4 and
fD = ½pDIulu
(3)
in which f = force per unit length, p = mass density of water, D = tower diameter, u and fi = local water particle velocity and acceleration and CM, CD = inertia and drag force coefficients respectively. This expression for wave forces on a fixed cylinder was modified to describe reaction forces on an oscillating tube in still water as follows: rr
1
f =rr£# + CAP 4D22 +-CDDI212 +
(4)
in which m is the mass of the tube, x, ~ and £ are the tangential displacement, velocity and acceleration of the tube and CA = a d d e d mass coefficient, Co = drag coefficient, k = spring constant due to buoyancy. A linear damping term (which is generally small for small tubes under consideration) may be added to the right hand side of equation (4).
oscillation of the cylinder was measured with a Rotary Variable Differential Transformer (RVDT). The loads in the three orthogonal directions were measured by an X Y Z load cell of 25 lbf. capacity. Slots were provided in the mounting brackets to allow for alignment of the cross-tank axes of the instrumented section in the tower and the X Y Z load cell with the cylinder's axis of rotation. All alignments were performed and confirmed in the dry on a calibration stand. To allow the top of the cylinder to remain above the water for the three different test angle orientations, the articulated joint was mounted on top of a different height pedestal for each of the three test conditions of the tower ((3°, 30 ° and 45 ° to vertical). This allowed the tank water depth to remain constant throughout the testing. The pedestal was bolted to a rectangular steel plate that was weighted with lead for added stability. Three large bolts were attached to the plate that could be used to level the plate.
C. Mechanical forcing device This test plan required that the cylinder be oscillated at a known frequency and amplitude in still water. This oscillation was achieved by a mechanical forcing device which was composed of hydraulic piston, a pendulum, and a Moog electro-hydraulic servo-control device (see Figs. 1 and 2). The hydraulic piston was attached to the pendulum and causes it to move back and forth in a prescribed
DESCRIPTION OF THE MODEL
A. Tower The test structure is a circular cylinder with an outside diameter of 3 in. and length of 9 ft. The test tower contains three sections. The bottom section is 6 ft long and made from 6061-T6 aluminium with a thickness of 0.065 in. The bottom element contains a buoyant section consisting of two PVC plates with O-rings separated by a 0.25 in. ID PVC tube. In this manner a buoyant section is maintained and the wiring from the instrumented section above can be passed through the section to the computer data acquisition system. The centre section of the test tower is the instrumented section for measuring locally applied loads. This load cell has a 3 in. outer diameter, is 12 in. in length and measures loads in two orthogonal directions. The upper section has a length of 2 ft and is composed of the same material and thickness as the lower section and a buoyancy section is also maintained in this section by circular PVC plates with O-rings. In this case, no wiring need be passed through the section and no central tubing is required. A Plexiglas shroud is attached to the top of the tower to insure that the tower never becomes completely submerged during oscillation. An 0.5 in diameter rod which is threaded on one end and smooth on the other end is threaded through the top cap of the test tower. This rod is used for connection to the mechanical drive device.
Hydraulic
C y l i n d e r -
V Hydr. Pump
Wovctck (Freq..Oscillator')
~1 ~
~
I~--
Electro Hydraulic Moog $¢rvo Valve
Electronic Scrvo Cong=roller
XYZ Load C¢)1
-- 1 Foot Test S¢ctlon
B. Articulated ]oint An articulated joint was designed to support the lower end of the cylinder. The support joint allowed the test cylinder unrestricted rotation about one axis only and was oriented so that the cylinder oscillated in a plane parallel to the tank walls. The articulated joint was instrumented such that the cylinder rotation and support loads in three orthogonal directions could be measured. The angular
RVDT ( Rol.ary Variable Dif fcrcntlol "l"r'Qnsf (
Figure 1.
Lood C¢11.
Mechanical drive schematic
Applied Ocean Research, 1983, Vol. 5, No. 4
227
Hydrodynamic coefficients of a harmonically oscillated tower: S. K. Chakrabarti, D. C Cotter and A. R. Libby 1. Tower loads An X Y Z load cell with a capacity of 25 lbf. was attached to the upper restraint. This load cell measured the load in three orthogonal directions. In this case, during measurement one axis was aligned with the cylinder axis. During all tests the load at the articulated joint was measured. This load was also measured by a 25 lbf. X Y Z load cell but in this case one of the axes was always aligned with the vertical. These load cells are calibrated by applying weights in given directions to the load cells and measuring the electrical output signal.
2. Tower motion During the test runs for the mechanically oscillated cylinder, the cylinder was allowed to rotate in one plane about an articulated joint. The oscillation of the cylinder about the axis orthogonal to the vertical was monitored by an RVDT. This instrument is calibrated together with the articulated joint in the dry by moving the joint through known angles and measuring the output from the RVDT.
3. Instrument section
Figure 2.
Photograph o f test set-up
The loads at discrete intervals of the cylinder were determined by the instrumented section. This load cell was designed and developed in-house by CBI and measures the loads on a test section that is 1 ft long. It consists of a 3 in. diameter tube floating on a central core. The 3 in. cylinder is attached to two cantilever beams which deflect as loads are applied to the cylinder. Strain gauges mounted to these cantilever beams output an electrical signal which varies with the bending of the beam and thus the load. This load cell is also calibrated in the dry in a calibration stand. TEST RUNS
motion. The pivot point of the pendulum was attached to a rigid support structure mounted on the wave tank bridge. The bottom of the pendulum was attached to a machined stainless steel rod at the top of the cylinder with an assembly that included a ball-bushing that allowed free longitudinal movement of the assembly along the rod and a pair of radial bearings that allowed the assembly to rotate with the cylinder. The attachment assembly was instrumented with an X Y Z load cell similar to the one supporting the bottom of the cylinder. This load cell measured the cross-tank and down-tank driving loads normal to the cylinder axis and any load induced in the direction of the cylinder axis due to friction in the ball-bushing. The frequency and amplitude of the cylinder oscillation was controlled by the Moog servo-control unit. The Moog received a feedback signal from the RVDT at the bottom of the cylinder and adjusted the hydraulic piston motion as necessary to cause the feedback signal to match a reference signal from a Wavetek generator. The Wavetek is simply an electronic device which outputs a pure sine wave of known frequency and amplitude.
D. Transducers The following measurements were made: the loads in three orthogonal directions at the restraining points of the cylinder, the loads at two orthogonal directions at the specified instrumented section along the axis of the cylinder and the angular displacement of the cylinder.
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Applied Ocean Research, 1983, Vol. 5, No. 4
The frequency of the mechanical system was varied from 0.1 Hz to 0.8 Hz. At each frequency the amplitude of oscillation varied from 5 to 30 degrees. A total of 132 runs were made during these tests. The average KeuleganCarpenter number varied from 0.40 to 44.65 and the average Reynolds number varied from 2000 to 40000. During all phases of the test low pass filters with a cutoff frequency of 5 Hz were used to separate the desired signal from the electrical noise inherent in the data acquisition system. In addition a numerical filter with a 3 Hz cut-off frequency was used to separate mechanical noise from the signal in the in-line direction. DATA ANALYSIS When the tower is oscillated mechanically, the equation of motion is given by equation (4). Rewriting
fM = -- m2 -- kx + CAfA + CDfD
(5)
where 7r
f A = - - P 4D22;
1
fo=--2PDlYclf¢
in which x = rO, r = distance from the pin to the center of instrumented section, 0 = cylinder oscillation angle (measured, sinusoidal), k = s p r i n g constant from net buoyancy. In equation (5) the left hand side fM, fA andfD are known from the measured values. The velocity ~ = ar0 and 5( = o2rO are obtained by shifting the phase angle by 90 ° of the measured tower oscillation in which o = oscilla-
Hydrodynamic coefficients of a harmonically oscillated tower." S. K. Chakrabarti D. C Cotter and A. R. Libby ting circular frequency. The quantities, Ca and CD are obtained from the least square assumptions. In matrix form, the equations for CA and CO are written as
LfDfA
f~ J Co
IfMfoJ
TOWER ANGLE TO
VERTICAL - DEGREES
(6) ~i
where the summation over i = 1 . . . . . N (N = no. of data points in a cycle) applies to all terms. The solution is obtained by matrix inversion. The lift coefficients in the transverse direction are calculated from the measured transverse force. The coefficients related to the harmonic components of the lift force are obtained by a Fourier series analysis using the equation
BOTTOM:PIN
o
LOA[~ X - D I R E C t I O N
i
Ni
-
POUNDS
i
i
TOTAL EOAD ON ONE FOOT SECTION
DIRECTI:ON - POUNDS
M
fL = E CLmfDm m=l
(7)
where M = no. of Fourier components of lift coefficients, Czm, and fDm are the Fourier components off/;. The most predominant lift frequencies are obtained by taking the Fast Fourier Transform (FFT) of the lift force profile and choosing the frequency of the spectral peak.
POUNDS
ooi TOTAL LOAD ON ONE FOOT SECTION t-DIRECTiON d ........................................ i..................... i................. i..................... i~ ................ !
~O,O0
I
2',00
q',OO
DISCUSSION OF RESULTS The 3 in. diameter cylinder with the instrumental section was oscillated in still water by an electro-mechanical servocontrol device. It was oscillated in still water about a pin joint such that it resembled an inverted pendulum. The angular displacement at the pin was measured in time along with the driving force causing the cylinder oscillation, the inline and transverse loads on the instrumented section and the loads in three directions at the pin, that is, the inline, transverse and vertical loads. Even after the 5 Hz electronic filter was applied, spikes were left in the data. Although the cylinder angle was a smooth sinusoidal function of time, the loads in the inline direction demonstrated high frequency noise. This noise occurred because of servo-control device's tendency to overcompensate for errors between the reference and feedback signals and therefore oscillate about the mean reference value. This 'hunting' had little effect on the cylinder's angular position (the feedback to the drive system) but did cause cylinder accelerations that appeared as spikes in the inline loads, especially where the reference signal was changing most rapidly. The solution was to apply a 3 Hz digital signal filter to the inline loads and motions only after the data had been collected. These spikes did not occur in the transverse direction and hence, these loads were not faltered and thus, allowed any high frequency transverse loads to remain. These high frequency transverse loads are caused not by the mechanical system but by the lift forces and therefore should remain. See Fig. 3 for the filtered data after the 3 Hz digital t'dter was applied. The desired results of the mechanical oscillation tests were estimations of the added mass coefficient (CA), the drag coefficient (CD) and the lift force coefficient (CL). These coefficients are formulated as functions of the Keulegan-Carpenter number (KC) and the Reynolds number (Re) and plotted versus these non-dimensional numbers in Figs. 4-9. Various trends for these coefficients can be found from these plots. Figures 4 - 6 demonstrate the dependence of CA on the Keulegan-Carpenter number through various angles from vertical. The added mass coefficient was found to be non-dependent on the Re
6',OO
TIME -
Figure 3.
8',00
tO,O0
12,00
SECONDS
Sample of filtered test data
=cov
d~
I
Ol o
, ~
F~,ure 4.
o ~o
, is
r~-~
2'o KC
15
~0
i
F~-a i z ~ , !
~o
4'o
15
CA for cylinder at 0 °
I
d~
~o
~
I : cov
[~
l'~i
20
A
o o
i
~o
KC
Figure 5.
CA for cylinder at 30 °
I
I : COV
~9
0
o KC
Figure 6.
1'5
A
is
CA for cylinder at 45 °
Applied Ocean Research, 1983, Vol. 5, No. 4
229
Hydrodynamic coefficients o f a harmonically oscillated tower: S. K. Chakrabarti, D. C Cotter and A. R. Libby
number. In general, the added mass coefficient seems to decrease with angle for low KC numbers, that is, less than eight. In the mid-range the orientation angle of the cylinder seemed to make very little difference in the result. At high KC numbers, that is, greater than 20, added mass coefficients seemed to decrease with angle once more. Curves presented in Figs. 4-6 are mean curves drawn through the data points developed from the test. In general, there was scatter to these points about the mean line shown by the coefficient of variation (COV) at various values of KC in Figs. 4-6. The COV is computed by letting the collection of estimates for Ca at a given KC number be Yi (i = 1,2, 3 , . . . , iV) and the chosen mean value of the estimate be p. Then the random error of the estimate, Yi, is defined by the standard deviation of the estimate about the mean value, random error = Oy =
ai
o
~ "2
0
/
° //.
O.
/ e .
--
l#I/
LEGEND
/ #
~
o
......... - -
{3 - -
- -
~__
lb
1~
Re t ,,
RANGE ,i
"
"
2ODO0
- 30,000
30,000
-
,i
40.000-
0
10,000
2'o
-10,000 - 20,000 40,000 50,000
a'5
:70
:i~
2o
KC
Figure 8. at 30 °
Co versus K C f o r various R e ranges - cylinder
(8)
(Yi - Y) 1
The coefficient of variation is defined as the normalized random error, _
Oy
COV -
o
(9)
--
P The values of COV are shown in Figs. 4-6 at equal intervals of KC = 5. Contrary to the added mass coefficient, the drag coefficient shows a strong dependence on the Re number for the Re number range tested. It was found that there is comparatively little scatter of the data points about the mean line for the drag coefficient versus the Re number. The drag coefficient showed similar scatter when displayed versus the Kuelegan-Carpenter number but less scatter when plotted versus the Reynold's number. Accordingly the data was broken up into Reynold's number regions and plotted versus the Kuelegan-Carpenter number in Figs. 7-9. This reduced the scatter in the data and demonstrates the dependence of the drag coefficient on both of the nondimensional parameters. A computer program was written to compare the theoretical wave loads on the cylinder with the measured loads at the top and bottom, that is, at the driving arm of the mechanical oscillation device and at the restraining point on the pin. In this program, the cylinder is segmented into 20 sections and for each section and for each wave cycle a particular KC number is derived depending on the distance from the pin, the maximum angular velocity of the cylinder, the period of the oscillation, and the diameter of
C o, L E G E N D
el'
Re
=
Re
= 10,OO0-20,OOO
0-10,0OO
R e = 20,OOO-30,OO0 R e = 30,OO0-40,OO0
0 0
Figure 9. at 45 °
%
,b
KC
?s
~o
17~
CD versus K C f o r various R e ranges - cylinder
g
Z rid tl_
~. ~g
I ~L!A L) er-
o
iE
ei"
2E tO.
bJ
oo. LEGEND Re "~r R A N G E ,i ...........
o - -
o
"
~b
0
¢5
O
-
bJ ~E
10,OO0
10.000 - 20,000 20,00030.000 30,000 - 40,000 > 40,000
2'o
~
8¢" 3'o
g~
go
KC
Figure 7. at 0 °
230
CD versus K C f o r various R e ranges
Applied Ocean Research, 1983, Vol. 5, No. 4
cylinder
~.oo
~'.oo e'.oo Ib.oo tb.oo 2b.oo CRLCULRTED MAXIMUH F O R C E -- P O U N D S
Figure lO. Correlation o f m a x i m u m measured pin loads - cylinder at 30 °
theoretical
2b,.Do
and
Hydrodynamic coefficients o f a harmonically oscillated tower: S. K. Chakrabarti, D. C Cotter and A. R. Libby the cylinder. Then for each KC number and for each segment an added mass coefficient and a drag coefficient are determined from the mean curves of these coefficients versus the KC number. The loads from each segment are determined using these two coefficients and the time history of the angular motion at the pin. The loads for each section are then summed up and compared to the driving load and the restraining load at the pin. An example of these correlations is presented in Fig. 10. In this figure, a perfect correlation would lie along the 45 ° line. The bordering lines on each side of this line are the plus or minus 10% lines. It is seen that the correlation results are quite good for the smaller loads but as the loads become larger the theoretical loads are larger than the measured loads. It is felt that this deviation arises because the loads on the upper portion of the cylinder predominate over the lower portion of the cylinder. In those sections above the instrumented section where the drag load is quite high, the KC number is higher than for those cases experienced by the instrumented section because the velocity is larger. In those cases where the KC number required for the segment was greater than that presented by the mean curve, a linear extrapolation was used to develop the value for the drag or added mass coefficients from the mean curves. This could result in considerable error. The dependence of the lift force coefficient (CL) on the KC number is demonstrated in Figs. 11-16. It can be seen in these figures that the KC number range decreased for every increasing angle tested. For these mechanical oscillation tests, the KC number was strictly dependent on the amplitude of the oscillations. The maximum possible oscillations amplitude decreased with increasing angle in order to keep the instrumented section from leaving the water. Therefore, for each increasing angle the range of KC LEGEND ..........
Fundamental 2 ~ Harmonic 3r~ 4t__~
<5
O
5~
"2 ~ t oo. t~c5
c5
o o
5
I0
15
20 KC
25
35
CL versus KC for cylinder at 0 °
Figure 11.
LEGEND
o o
. . . . . . . .
Fundamentals 2 n~-~° H o r m o n i ¢ 4~ 5 I~
O
0
j
30
•
d"
~
o O
Figure 12.
, 5
• 10
....
• 15 ~C
• 20
--
= 25
CL versus KC for cylinder at 30 °
• 30
40
o
LEGEND
o.
.........
Fundamant01 2~
Harmonic
3~ 4L"
6'
a
5~
O'
5
10
15
20
25
KC
Figure 13.
CL versus KC for cylinder at 45 °
number tested decreased. In these figures there are two types of charts which enable the designer to determine the coefficients for the lift forces. Figures 11-13 are the first type of presentation for the lift force coefficients. In order to get this figure, the time history of the transverse load was broken up into its Fourier components. These loads were then divided by the factor p x ~ D / 2 where p is the density of the water, x m is the maximum normal velocity. This term is much like the term for the drag contribution in Morison's equation. Coefficients up to the fifth harmonic are presented because the energy tends to decrease rapidly beyond these harmonics. The second type of chart is presented in Figs. 14-16 which is the frequency ratio plot. In order to make this plot, the FFT of the time history of the transverse load and the angular motion were taken. For each test the frequency that contained the maximum energy of the transverse force and the angular motion data was determined and the ratio of these two frequencies was calculated. This did not mean there was no energy at other frequencies for the transverse load. This is simply the frequency with the maximum energy for each test for the transverse loads on the instrumented section. In most cases other peaks were present; they simply were not as large as the one chosen. In practice this ratio is usually considered an integer value. If the ratio is equal to 1, then the fundamental frequency or first harmonic is dominant. If it is equal to 2, then the second harmonic is dominant, etc. The test data shows that the frequency ratio takes on an integer value for low KC numbers. As the KC number increases the ratios tend to cluster around the integer values but become far more confused. At higher KC numbers, the trend persists but the basic assumption of an integer value relationship seems to begin to break down. The choice of method of determining the lift force is then up to the user and is dependent upon the type of information that can be used. For example, the user would first determine the KC number expected for any particular segment of the structure. Having this KC number he can then go into the second type of chart as in Fig. 14, and determine which frequency is dominant. With this information he can proceed to the first type of chart, Fig. 11 and choose the lift force coefficient for the KC number and harmonic required. Alternately, the user can determine a KC number and go to the first chart and pick out the five coefficients for the first five harmonics of the lift force. The phase angles of the lift force components are arbitrary. Therefore, whether one frequency is used or all five, the phases for these individual lift force components can be chosen from a random number generator. It can be seen in
Applied Ocean Research, 1983, Vol. 5, No. 4
231
Hydrodynamic coefficients o f a harmonically oscillated tower." S. K. Chakrabarti, D. C. Cotter and A. R. Libby Figs. 11-13 that the lift force coefficient for the second harmonic that is, C/,(2) tends to take on the largest value. In addition, the peak value for a particular harmonic tends to occur at higher KC number for increasing number of the harmonic.
50-
40*
30-
1
20-
l
10-
/I
0 0
FREQUENCY RATIO (TRANSVERSE / WAVE)
Figure 14. Dominant lift force frequency versus KC for cylinder at 0 ° 30-
25"
20-
"~15-
1 m
10-
0
I
0
FREQUENCY
RATIO (TRANSVERSE / WAVE)
Figure 15. Dominant lift force frequency versus KC for cylinder at 30 ° 25'
20-
15 ¸
l
u
10'
B 0
-
FREQUENCY
RATIO(TRANSVERSE / W A V E )
Figure 16. Dominant lift force frequency versus KC for cylinder at 45 °
232
Applied Ocean Research, 1983, Vol. 5, No. 4
CONCLUSIONS AND RECOMMENDATIONS The following conclusions may be drawn from the results presented here. 1. It is shown that the added mass coefficient is a function of the non-dimensional Keulegan-Carpenter number (KC) and not of the Reynolds number (Re) for the region of Reynolds number tested. 2. The hydrodynamic drag coefficients of a moving cylinder in still water showed clear dependence on Reynolds number in addition to the Keulegan-Carpenter number. 3. According to Froude's law, the Keulegan-Carpenter number between a prototype and its model is the same. The range of KC numbers attained in these tests will cover a wide range of prototype situations. Thus, as far as KC numbers are concerned, the mean curves of hydrodynamic coefficients presented in this report are directly applicable in a design. On the other hand, the coefficients are found to be at least weak functions of Re number (within the small range of test data). The Reynolds number between a prototype and its model differ significantly. In laboratory tests of this type, the largest Re number that can be generated is of the order of l0 s while many design conditions require an Re of about 1 0 7 . The trend of the coefficients with Re number should be determined from other similar tests (e.g., ocean tests, forced water oscillation, etc.) for extrapolation purposes. 4. Coefficients generated by an analysis of a particular location on the tower are similarly applicable over the entire length of the tower when these coefficients are formulated as functions of the Kuelegan-Carpenter number. The values of the coefficients should be varied over the length of a tower. 5. The drag coefficient shows a marked dependence on the Reynolds number while the added mass coefficient does not. This dependence becomes important at large KC numbers. 6. The lift (or transverse force) coefficients are presented in terms of its first five harmonics (by a Fourier series analysis). These harmonic coefficients are found to be functions (through some scatter) of the KC number. Since the lift force profiles are somewhat random in nature, these time histories of lift force may be reconstructed from the mean curves presented in the report if each harmonic is assigned an arbitrary ,phase angle. Thus, the results can be applied to a numerical analysis program quite easily. 7. The plot of the predominant lift force frequencies vs. KC number can be used to study the eddy-shedding process. Note that the predominant lift force frequency is almost an even multiple of the wave frequency (except at the higher end of the KC values where the flow becomes more turbulent). When the ratio is one, no eddies are shed (but probably a weak eddy formed) and the lift force is found to be regular. For a ratio of two, one eddy is shed alternatively on either side of the tower. For the ratio of three, two eddies are shed, and so on. 8. The largest value of the lift coefficient occurs at the second harmonic of the motion. The value of the KC number at which a particular coefficient and harmonic of
Hydrodynamic coefficients o f a harmonically oscillated tower: S. K. Chakrabarti, D. C Cotter and A. R. Libby the lift force becomes dominant increases with the harmonic. In other words, the higher the KC number, the higher the dominating harmonic of the lift force. 9. The test tower was restrained from lateral (or transverse) motion and, hence, the coefficients presented here are applicable to cylindrical members in a planar motion. If a member is free to oscillate laterally as well as inline, then the application of these coefficients directly without further corroboration may not be justified.
N Re )3 0 p o Oy
= = = =
No. of date points in a cycle Reynolds number Mean value of variables Yi (i = 1 , 2 . . . . . N ) Angular displacement of cylinder from equilibrium position = Water density = Circular frequency = Standard deviation of a variable setyi (i = 1,2 . . . . . N)
REFERENCES LIST OF SYMBOLS Force per unit length of structures
f fo fA,h
= Drag force for drag coefficient equal to 1.0 = Inertia force for inertia force coefficient equal to
JcL fM
=
k
=
m r x
= cA co cL
= = =
D KC M
= = =
1.0 Lift force Total measured force on instrumented section Restoring forces on instrumented section due to buoyancy Mass of instrumented section Distance from pin to centre of instrumented section Linear tangential displacement of instrumented section from equilibrium position Linear velocity of instrumented section Linear acceleration of instrumented section Maximum linear velocity during the cycle Added mass coefficient Drag coefficient Lift force coefficient Diameter of instrumented section Keulegan-Carpenter number No. of Fourier components
1. Sarpkaya, T. In-line and transverse forces on cylinders in oscillating flow at high Reynolds numbers. Proceedings on the Eighth Offshore Technology Conference, Houston, Texas, OTC 2533, 1976, pp. 95-108 2. Sarpkaya, T. Hydrodynamic forces on various multiple tube riser configurations, Proceedings on the Eleventh Offshore Technology Conference, Houston, Texas, OTC 3539, 1979, pp. 1603-1606 3. Garrison, C. J., Field, J. B. and May, M. D. l)rag and inertia forces on a cylinder in periodic flow, Journal o f the Waterway, Port, Coastal and Ocean Div., ASCE, Vol. 103, No. WW2, May, 1977 4. Rains, C. P. and Chakrabarti, S. K. Mechanical excitation of offshore tower model, Journal o f the Waterways, Harbors and Coastal Engineering Div., ASCE, Vol. 98, No. WWl, February, 1972, pp. 35-47 5. Laird, A. D. K. and Warren, R. P. Groups of vertical cylinders oscillating in water, Journal o f the Engineering Mechanics Division, ASCE, Vol. 89, No. EM1, February, 1963, pp. 25-35 6. Loken, A. E., Torset, P. P., Mathiassen, S. and Arnesan, T. Aspects of hydrodynamic loading in design of production risers, Proceedings on Eleventh Offshore Technology Conference, Houston, Texas, OTC 3538, 1979, pp. 1591-1601 7. Bearman, P. W. and Graham, J. M. R. Hydrodynamic forces on cylindrical bodies in oscillatory flow, Proceedings of the Second International Conference on Behaviour of Offshore Structures, London, England, August, 1979, pp. 309-322
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