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Wave forces on inclined tubes
Coastal Engineering, 1 (1977) 149--165 © Elsevier Scientific Publishing Company, Amsterdam -- Printed in The Netherlands
149
WAVE FORCES ON INCLINED TUBES
SU B R ATA K. C H A K R A B A R T I 1, A L L A N L. WOLBERT 2 and WILLIAM A. TAM 3
Chicago Bridge & Iron Company, Plainfield, Ill. (U.S.A.) (Received September 30, 1976; accepted January 31, 1977)
ABSTRACT Chakrabarti, S.K., Wolbert, A.L. and Tam, W.A., 1977. Wave forces on inclined tubes. Coastal Eng., 1: 149--165. Total forces due to sinusoidal progressive waves on a small circular tube arbitrarily oriented with respect to the wave direction are investigated. Two-component normal forces are measured in a wave-tank experiment. The in-line and transverse forces are resolved from these measurements. The lift coefficients are obtained from the transverse forces. The frequencies of the transverse forces are found to depend on the period parameter. The ratio of the m ax i m u m resultant force to the maximum in-line force on small sections is presented as a function of the period parameter for each tube position. The results show that the resultant force could be twice as much as the in-line force depending on the period parameter.
INTRODUCTION
Wave forces on vertical circular cylinders have received considerable attention in the past years. Many researchers have studied the effects of waves on cylinders of various sizes, mostly under experimental conditions. One of the most n o t e w o r t h y theoretical investigations has been the one by MacCamy and Fuchs (1954), who obtained a simple closed-form expression for wave forces including diffraction effects on a large vertical cylinder which is b o t t o m - m o u n t e d and surface-piercing. However, the formula is n o t valid when drag and lift effects are important, namely for smaller diameter cylinders. Morison et al. (1950) first introduced an empirical equation for the in-line force which is composed of two terms -- an inertia and a drag -- linearly added together. The inertia term includes an inertia coefficient while the drag term involves a drag co.efficient.Since then this formula has been extensively used in determining the inertia and drag coefficients from experimental data as well as in designing tubular members. It has also found its application in other submerged shapes. 1Head, Analytical Group, Marine Research and Development, Chicago Bridge & Iron Company, Plainfield, Ill. 2Manager of Marine Computer Facilities, Chicago Bridge & Iron Company, Plainfield, Ill. SDirector of Marine Research, Chicago Bridge & Iron Company, Plainfield, Ill.
150 While its use has been generally limited to small objects where diffraction and added mass effects are synonymous, its application to larger objects is not uncommon. In the latter cases, an effective inertia coefficient is employed in order to include the diffraction effect. Wiegel (1964) and Dean and Harleman (1966) have provided extensive reviews relating both to the application of the Morison equation for wave force evaluation and to the determination of the appropriate force coefficients. In general, however, investigation of inclined tubular members has been scarce. Borgman (1958) analytically extended the Morison equation in the absence of lift force to include the general orientation of a tube with respect to the wave direction. According to this formula, it is assumed that the forces on a tube act normal to the tube and can be expressed in terms of a normal velocity and a normal acceleration component. The normal velocity c o m p o n e n t results in both viscous and pressure forces while the tangential component causes only a shear force (skin friction) parallel to the axis (McCormick, 1973). The tangential components of the force are assumed small and hence negligible. Thus, the force vector in the generalized Morison equation having an inertia and a drag term is a function of the normal velocity and acceleration vectors. There have been other proposed formulas, e.g. by Dean and Harleman (1966); however, little is known about the inertia, drag and lift coefficients for an inclined cylinder. The experiments have generally been limited to the vertical and horizontal positions. One of the pioneering works in determining the values of the h y d r o d y n a m i c coefficients and their dependence on certain nondimensional numbers was successfully carried out by Keulegan and Carpenter (1958). They tested a submerged horizontal cylinder at the node of a standing wave so that the cylinder essentially experienced a one-dimensional flow. The measured horizontal forces were correlated with the Fourier series representation of the Morison equation and the mean values of the h y d r o d y n a m i c coefficients were evaluated. The flow effect around the cylinder was examined. The coefficients were shown to depend on the period parameter. Bidde (1971) studied the eddy-shedding process in the " l e e " of a vertical cylindrical pile due to oscillatory waves. The number of eddies was observed to depend on the period parameter or the Keulegan-Carpenter number. The lift force was found to be as much as 60% of the in-line force. Sarpkaya (1975) in a recent test verified the results of Keulegan and Carpenter through tests in a controlled environment. He tested cylindrical models across a one-dimensional sinusoidally oscillating flow in a U-tube. The velocity and acceleration fields in this controlled environment were nearly sinusoids. Similar analysis showed t h a t the dependence of the h y d r o d y n a m i c coefficients including the lift coefficients on the period parameter followed closely that found by Keulegan and Carpenter. Sarpkaya (1976) extended his previous test work to high Reynolds number and showed that the values of in-line mass and drag and transverse or lift coefficients are, indeed, dependent on the period parameter (i.e. Keulegan-Carpenter modulus) as well as Reynolds number or
151
equivalently a frequency parameter (defined as the ratio of the Reynolds number and Keulegan-Carpenter modulus). He tested both smooth and rough horizontal cylindrical models. Isaacson (1974) examined the transverse forces on a vertical cylinder. He observed the eddy formation " b e h i n d " the cylinder and found t h a t the number of eddies being formed is a function of the Keulegan-Carpenter modulus. He presented the first five components of the lift coefficients by representing the transverse force in a Fourier series form. The components were shown to be functions of the Reynolds number. AI-Kazily (1974) investigated wave forces on submerged pipelines. He computed the values of the empirical force coefficients in the Morison equation from the m a x i m u m measured forces on a horizontal cylinder normal to the two-dimensional wave flow. He observed large scatter in these coefficient values and concluded that the coefficients must be a function of several variables, e.g. wave height, wave length, as well as cylinder diameter and depth below SWL. In addition, they were found to vary over a given wave cycle. AIKazily also tested a tube inclined to the horizontal in the direction of the waves but did n o t present the values of the inertia and drag coefficients for the inclined cylinder. In the present study, a cylindrical model is tested at various orientations in order to determine the effects of the inertia, drag and lift forces on the tube. Measurements are obtained at two points on the cylinder conveniently located near the still water level so that some overlapping of data can be obtained while a m a x i m u m range of the test possible in the available wave tank is achieved. The measured in-line forces on the two points of the cylinder are utilized to derive the values of the inertia and drag coefficients which have been presented in an earlier paper (Chakrabarti et al., 1976}. In this paper, the effect of the transverse forces due to eddy-shedding and the total resultant forces on the tube are examined. A wave tank test was conducted with a 7.6 cm (3 inch) O.D., 3.05 m (10 ft.) long smooth tube placed in water at various orientations. Two 0.305 m (1 ft.) sections of the tube were instrumented to measure the two c o m p o n e n t normal forces on these sections. In addition, three c o m p o n e n t total orthogonal forces on the submerged portion of the t u b e due to waves were measured. Eight sets of runs were made, one for each orientation. At each angle, a series of wave periods and wave heights from the smallest to the largest possible sinusoidal waves were generated. The results for the vertical tube were presented in an earlier paper (Chakrabarti et al., 1976a), while the in-line forces on the inclined tubes were discussed by Chakrabarti et al., 1976b. The presentl paper gives the results for the transverse and total forces on the inclined tubes and is thus an extension of the earlier works (Chakrabarti et al., 1976b). The predominant frequency of the transverse forces on the tube at various orientations is compared with the wave frequency. The ratio of these two frequencies is indicative of the number of eddies formed which depends on the
152
p e r i o d p a r a m e t e r . T h e l i f t c o e f f i c i e n t s are c a l c u l a t e d f r o m t h e t r a n s v e r s e f o r c e s f o r t h e t u b e p l a c e d a l o n g t h e t a n k . T h e r e s u l t a n t f o r c e p r o f i l e s are p r e s e n t e d and the ratios of the maximum resultant force to the maximum in-line force are s h o w n t o i n c r e a s e s t e a d i l y w i t h t h e i n c r e a s i n g p e r i o d p a r a m e t e r f o r v a r i o u s tube orientations. '~ WAVE TANK TEST T h e w a v e t a n k i n w h i c h t h e t e s t s w e r e c o n d u c t e d is 7 6 . 2 m ( 2 5 0 ft.) l o n g , 1 0 . 1 m ( 3 3 ft.) w i d e a n d 5 . 5 m ( 1 8 ft.) d e e p . I t has a n a d j u s t a b l e c o n c r e t e f l o o r w h i c h was m a i n t a i n e d a t a 3 . 0 5 m ( 1 0 ft.) w a t e r d e p t h . T h e t u b e w a s placed a b o u t 0.16 m (0.52 ft.) off the b o t t o m . T h e details of the test set-up a n d i n s t r u m e n t a t i o n w e r e d e s c r i b e d b y C h a k r a b a r t i e t al. ( 1 9 7 6 a , b). The results f r o m the following seven t u b e o r i e n t a t i o n s are p r e s e n t e d here: (a) t w o t u b e e l e v a t i o n s , 4 5 ° a n d 6 0 ° ( w i t h r e s p e c t t o t h e h o r i z o n t a l ) i n a p l a n e across t h e t a n k ; (b) t h r e e e l e v a t i o n s , 4 5 °, 6 0 ° a n d 7 5 ° i n a p l a n e a l o n g t h e t a n k ; a n d (c) t w o e l e v a t i o n s , 4 5 ° a n d 6 0 ° i n a p l a n e m a k i n g 4 5 ° a n g l e t o t h e w a v e direction. NOTATION List of symbols CD CL
CM D
F~ fz
f0 N n
Ri Rmean T uX
Ux W w m
y Yl Y2 P
= drag coefficient = lift coefficient = inertia coefficient = tube diameter = total horizontal force on the tube = lift force frequency = resultant force on the tube sections per unit length = in-line force on the tube sections per unit length = vertical force on the tube sections per unit length = transverse force on the tube sections per unit length = incident wave frequency ffi number of runs = number of eddies formed = ratio of the total theoretical in-line force to the corresponding measured force on the tube per run = mean value for a particular tube position = wave period in-line component of the normal water particle velocity = time derivative of u x = normal water particle velocity vector = maximum normal water particle velocity = elevation measured from the tank floor = lower end elevation of the tube = upper end elevation of the tube = angle of tube inclination to the horizontal = mass density of water =
153
The t w o - c o m p o n e n t forces on the two 0.305 m (1 ft.) sections of the tube were measured normal to the tube. Therefore, at any time in a wave cycle the resultant normal force on these sections was known. This resultant force comprised of an inertia force component, a drag force c o m p o n e n t and a lift force component. The range of the test runs for each inclined tube position in the tank is given in terms of certain nondimensional quantities and is included in Table I. The period parameter is chosen as the independent parameter for the presentation of the results and is f o u n d to present the dependence of CM, CD, and CLaS well as the total forces reasonably well (see Notation). The Reynolds number (and hence the frequency number) is low and its range is also quite small. Hence the further dependence of the coefficients on the Reynolds number or the frequency number (as shown by Sarpkaya, 1976) cannot be determined. The wake parameter determines the importance of eddy-shedding while the diffraction parameter is small enough to exclude the effect of any wave scattering. DISCUSSION OF RESULTS A pair of CM, C D values per test run was obtained from each 0.305 m (1 ft.) section. These values of the coefficients are correlated with the corresponding period parameters. The period parameter at a p o i n t on a randomly oriented tube is defined in terms of the m a x i m u m normal velocity at that point, win, the tube diameter, D, and the wave period, T, as w m T/D. The values of C M and C D for each tube position obtained by the least square fit of the measured in-line force data were presented in the earlier paper (Chakrabarti et al., 1976b). The values of C M and C D in this m e t h o d are assumed constant over a cycle and any variation of these values within a cycle is ignored by choosing the best possible values. The residual force is about 10% of the measured and generally has twice the wave frequency. A mean curve is drawn through each of these plots representing a mean value of the coefficients at each wmT/D. These mean curves for CM and C D are shown in Fig.1. From the plots of C M and CD, it appears that the coefficients are functions of not only the period parameter but also the orientation of the tube. Since the effect of the lift forces in the direction of the waves is generally small the total in-line forces on the tube are computed using the mean curve as follows:
Fx" f Yl
p ~D2CM (wmT/D) u x + l2p D C D (wmT/D) Iw lu x dy/sin~b ¢ ¢ 0 (1) 4
where p = mass density of water, D = tube diameter, ~ = angle of elevation of " the tube to the horizontal, w m = m a x i m u m normal water particle velocity, w = normal water particle velocity vector, the bars denote its magnitude, Ux = its in-line component, and Ux = its time derivative. Yl and Y2 are the lower and upper end elevations of the inclined tube in water and C M and C D are con-
I
.38--12.9 1280--19800 1450--3390 .0114--.0673 .0259--.142 .015--.0682
WmT/D WmD/v D2/vT H/L H/d nD/L
Period p a r a m e t e r Reynolds number Frequency parameter Wave steepness Wave p a r a m e t e r Diffraction p a r a m e t e r
.071--13.5 290--22800 1440--4050 .0083--.0687 .0327--.143 .015--.0982
60 ° elev. along t a n k
.244--16.0 990--25000 1450--4060 .0093--095 .0311--.137 .015--.0982
75 ° elev. along tank
test
26 28 27 30 25 28
45 ° elevation 75 ° elevation 45 ° elevation 60 ° elevation 45 ° elevation 60 ° elevation
along tank along tank across t a n k across t a n k at 45 ° to d o w n - t a n k at 45 ° to d o w n - t a n k
No, of runs, N
Tube orientation rain. 0.656 0.917 0.709 0.750 0,913 0.985
max. 1.075 1.424 1.136 1.196 1.657 1.606
0.942 1.078 0.927 0.972 1.179 1.284
mean
Ratio = c o m p u t e d f o r c e / m e a s u r e d f o r c e
Ratio of the m a x i m u m total in-line c o m p u t e d force with the m a x i m u m total in-line m e a s u r e d force
T A B L E II
45 ° elev. along t a n k
formula
description
for the wave-tank
Range of test
parameters
Dimensionless p a r a m e t e r
Range of various dimensionless
TABLE
.228--14.6 930--23600 1450--4060 .011--.0604 .0313--.120 .015--.0982
45 ° elev. across t a n k
.248--15.7 1010--26200 1450--4070 .0115--.0579 .0313--.119 .015--.0982
60 ° elev. across t a n k
.4--13.3 1340--18600 1450--3380 .0126--.0698 .0202--.137 .015--.0682
45 ° elev. at 45 ° to down-tank
.0716--14.8 290--23100 1450--4060 .0058--.0648 .021--.145 .015---.0982
60 ° elev. at 45 ° to down-tank
t=a
155 LEGEND 6-
405: T? HORIZ?NTAL ALONG TANK
75 ° . . . . . . . .
"',
"~
~x
45 °
. . . .
45"
"
ACROSS
"
~ 45"
"
TO
DOWN
TANK
CD
2
CM ~x-
xx - -
Wrn____~T D
Fig. 1. Mean C M and C D values versus period parameter for a s m o o t h circular t u b e at various elevations.
sidered functions of the local period parameters. Note that the dependence of eq. 1 on ~ was omitted in the earlier paper (Chakrabarti et al., 1976b). The total in-line force obtained from eq. 1 is compared with the total measured in-line force and the ratio of the c o m p u t e d and the measured is calculated. The m a x i m u m and m i n i m u m values of the ratio, as well as the mean value, are shown in Table II. The correlation for the 60 ° to the horizontal along the tank runs cannot be shown due to some instrumental problems with an XYZ load cell in this case. The mean values shown in Table II are obtained from Rmean = V ~
(Ri) 2/g
(2)
i=l where N is the number of runs for a tube position and R i is the ratio in each case. Note that the measured force is predicted generally within 10%. The inline force was chosen for the calculation of C M and C D as well as for the total
156
force correlation since it is least affected by the lift force. However, for the two randomly oriented tube positions (45 ° to down tank) even these forces are affected by the eddy-shedding which may explain the larger scatter in the total in-line force correlation (Table II). For an arbitrarily oriented tube in the two-dimensional waves it is difficult to separate out the eddy-shedding forces from the measured transverse force. One means of deriving the lift forces due to eddy-shedding is to compute first the transverse (inertia plus drag) force using the CM and C D values calculated from the in-line forces and then subtracting it from the measured transverse force. The residual force may be attributed to the eddy-shedding process and wake formation " b e h i n d " the cylinder. However, this m e t h o d will carry any error in the C M and C D values to the lift forces. The lift coefficient, CL, obtained from the residual force then may have a large a m o u n t of error in it. For the tube inclined along the tank, however, the transverse force on the tube is strictly due to eddy-shedding. In these cases, a lift coefficient, C L, may be calculated from the transverse force fz on 0.305 m (1 ft.) sections by the following formula:
CL = (f z )max /( l pD 'wl 2 max )
(3)
where Iw] max is the m a x i m u m normal water particle velocity amplitude on the tube sections. The values of C L obtained this way for the three down-tank tube positions are presented as functions of the period parameter in Fig.2. Note t h a t the lift forces are generally quite small for low period parameter (w m T/D<4) and hence the scatter in C L is large. Values of CLwhich exceed the plot limits in this range have been omitted. The resultant force on a unit length of the tube sections in this case may be obtained from:
x y are the in-line and vertical force components obtained from the C M and C D values shown in Fig.1 while fz is the transverse force which in this tube position is totally due to wake formation in the " l e e " of the tube and is computed from a relationship similar to eq. 3. The calculations for the lift coefficient, C L, for the other tube positions (other than the down-tank positions) have n o t been carried out because of the difficulty in obtaining the pure lift (eddy-shedding) force in these cases. However, the lift force frequencies could be analyzed in these cases. For example, for the across-the-tank tubes the transverse (inertia plus drag) forces on the tube sections are computed from the mean C M and C D values in Fig.1 and these values are subtracted from the measured transverse forces on the tube sections. The residual forces approximately represent the vortex shedding effects. The measured transverse force profiles are converted to their energy spectra by a standard FFT routine. The number of data points chosen is 1024
157
0 D
DATA MEAN FITTED LINE 45" ELEV ALONG TANK
co o
oo
o
o
GO" ELEV ALONG TANK
o .
00
J
o
o o
u o o
o o o °~oo o
.J
o
o
.o o
o
o
o
o
/
°°
o ' - t o ~ - - ~ o ' ' ' ° ° oo o
Oo
oo
7'5' ELEV ALONG TANK o
o
o
O
WmT/O T%g,2. Lift force coefficients for the tube along the tank.
with a time increment of 0.04 seconds. A few sample plots are presented for various tube positions in Fig.3. The plots are shown in decreasing order of the period parameter for each tube position: (a) along the tank; (b) across the tank (measured normal transverse force is used here); and (c) at 45 ° to downtank. The values of the peak amplitude of the spectra are given in the figure. The profiles are chosen from both top and bottom section measurements at random. Note that the most prominent (or predominant) peak appears at or near a multiple of the wave frequency -- depending on the wave parameter. To examine the dependence of the predominant lift frequency on the period parameter, calculations are carried out for the ratio of the most predominant (the one having the highest spectral peak among several peaks present) lift
158
Wm T/O: 12.9 : 3 5 SEC
.,o%.O.~
~0
Wm T/O:6 4 T ; 2 7 5 SEC
T:3.5 SEC
T
~0 "x
~
WmT/D:51 T :3 0 SEC
45" ELEVATION G TANK
o.~
0
g
'0 >-
Wm~D:135 T:3.0 SEC
O WmT/D 31 T : 2 2 5 SEC
WrnT/D : 118 T : 2.5 SEC
~0 " X
z 60 ° ELEVATION ALONG TANK
A
(z z uJ
L,
0
'•O"
Wm T/D: 1G,O T ; 3 2 5 SEC.
~'0
A A, w m T/D : 8.6
W m ~D :12 O T:3.25 SEC,
~J
'0
t
Wm T/O: 42 T : 25 SEC
75 ° ELEVATION ALONG TANK
©
OO
1
1
2
1
,
2
FREOUENCY,Hz. (o)
T : 3.25
T
SEC
SEC.
~
T : 225 SEC.
L~
WmT/D- 12 T : 1 5 SEC
45 ° ELEVATION ACROSS TANK
..--...~L..~%.
_
,
A,
O
J. LJ
N
°~
~,nT~o:,~.,
%
~,,,5o:,,.5
I
Wr~T'D: G.O
O
WmT/o : O7 T:
1.5 SEC
60" ELEVATION ACROSS TANK
O
2
O
1
2
1
FREQUENCY . (b)
HZ
2
0
159
'0
WmT/D: 1 2 8
Wrn T/D : 9.3
T : 3.5
T : 325
SEC
IO
Wm
T/D: 61
T : 225
SEC
wm T~D: 24 T:20
SEC.
cj ~J L~
SEC
45 a
ELEVATrON
AT
45"
TO
DOWN
E
TANK
>-
(, c~ wmT/D : 10.9
e: ' © T : 3.5
T:325
SEC.
WmT/D : 2 9 T : 2.0
SEC.
SEC
O
T:20
SEC
LU
6 0 a ELEVATION AT
45"
DOWN
0
1
2
0
2
2
O
FREQUENCY-
TO TANK
O
Hz
(c)
Fig.3. Transverse force spectra for the tube sections of the 7.62 cm (3 inch) O.D. tube: (a) along the tank; (b) across the tank; (c) 45 ° to the down-tank.
frequency, fl to the wave frequency, f0. These ratios for each tube position are plotted against the period parameter in Fig.4. Note that the ratios are nearly all integer numbers. While the actual transition zone is somewhat uncertain the ratio seems to increase in steps with increasing period parameter. Isaacson (1974) observed that the number of eddies formed "behind" a cylinder is one less than this frequency ratio. Thus, no eddy is formed when the ratio n is 1; one eddy is formed for n = 2 and so on. Note also from Fig.3 that the total energy (area under the curve) for the lift force is small for low w m T/D values indicating that the lift force is small. When the lift force is significant, the resultant force differs significantly from the force obtained by the extended Morison equation. This is evident from examining the t w o measured components of the normal force on the tube sections. Sample plots of the resultant force profiles on the tube sections are presented in Fig.5 for each tube position in decreasing order of the period parameter. The same runs as in Fig,3 are chosen for this purpose for the tubes: (a) along the tank; (b) across the tank; and (c) at 45 ° to down-tank. The same trend as before is found. The profiles are quite irregular for large w m T/D values, but become more and more regular as w m T/D value decreases. The irregularity in the profiles is due to the presence of significant lift force. In fact, in some cases the resultant force is over 90% higher than the in-line force. For tubes along the tank the resultant force becomes the same as the in-line normal force at low Wmt/D (<4). For tubes across the tank the profile becomes regular with the in-line and the normal component of the vertical force remaining at low
160
X2 0o
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am
ee~
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el
e~
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i
e owo e
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ONe
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GO* E L E V . A L O N G TANK
•
coo
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45" ELEV. A L O N G TANK
•
•
,
J •
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?
. . . . . . . . .
? ' 1 "3"
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7 5 " ELEV. ALONG TANK
4 5 * ELEV. ACROSS TANK ooo
•
•
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h- z LL W
-~ ~ O / _z
,
,
,
,
,
6 0 * ELEV. A C R O S S TANK
t
. . . . .
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222
..,L.,J,(,,~
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2 • 2
3
2
./
,
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eeeo~
i
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45" ELEV. AT 4 5 t TO DOWN T A N K
? 75
~
o
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.
"~/~ /
g
2
s
w~T/D
lb
GO" ELEV. AT 4 5 " TO DOWN TANK
17
Fig.4. Ratio of the most predominant lift frequency to the wave frequency versus period parameter.
Thus, a definite trend is evident in these plots of the resultant force in relation to the in-line force with the period parameter. The maximum normal resultant force was obtained from the profiles of the type of Fig. 5. The maximum in-line force was computed as the average of all the peak values in a run. The ratio of these two forces is calculated for all cases. This ratio is plotted against the period parameter in Fig.6 for each tube w m T/D.
161 positiorL The ratio increases steadily with increasing w m T/ D values. The reason for this increase is t h a t the lift force due to eddy-shedding increases with the period parameter. At low w m T/D the ratio levels off to a certain limiting value depending on the position of the tube. For example, for a tube at an elevation of 45 ° down the tank (Fig.6a) in the absence of the lift force, the normal force is comprised of the in-line and vertical force so that the ratio of resultant normal to in-line becomes 1/sin 45 ° or 1.414. For the 60 ° alongthe-tank tube, the ratio is i / s i n 60 ° or 1.155 and for the 75 ° along-the-tank tube, it is I/sin 75 ° or 1.035. For across-the-tank tubes, it is 1.0. For the tubes at 45 ° to the down-tank, it is 1.206 for 45 ° elevation and 1.079 for 60 ° elevation. In these last two cases the ratio is n o t constant, but rather decreases slowly with the period parameter as shown in Fig.6 (f and g). Mean curves are drawn through the points representing the ratio. These mean curves give the multiplicative factor to be used at a particular w m T/D value in order to obtain the m a x i m u m resultant force on a tube section from the m a x i m u m in-line force on a submerged tube section in waves. Thus, for a section of a tube at 45 ° elevation along the tank, the factor is 1.57 for a w m T/D of 8.0, i.e. the m a x i m u m in-line force should be multiplied by 1.57 to obtain the m a x i m u m normal resultant force. Since the design of a tube should be based on the m a x i m u m force anticipated on the tube, the theoretical in-line force based on the mean C M and C D values should be increased by the applicable ratio given in Fig.6. CONCLUSIONS The resultant wave forces on a circular cylindrical member arbitrarily oriented in waves have been examined. Mean values of C M and C D for each tube orientation have been presented. For tubes inclined along the tank, the values of C L have been shown as functions of the period parameter. These values of the coefficients may be used to construct the resultant force profiles on a tube within the given period parameter. The transverse force spectra and the lift force frequency are presented for all tube positions. This information helps in identifying the eddy-shedding process. Since the two c o m p o n e n t normal forces on the tube sections were measured, it is possible to obtain the resultant forces on them. A few of the resultant force profiles have been plotted to show the dependence of their magnitudes on the period parameter. The ratios of the m a x i m u m resultant force to the m a x i m u m in-line force are shown to be functions of the period parameter. Mean lines are drawn through these points for each tube elevation. Thus, the total in-line (in the direction of the waves) force on a tube section may be c o m p u t e d from the given C M and C D values which may then be increased by these correction factors to obtain the m a x i m u m load on the tube section. Since the resultant forces and their h y d r o d y n a m i c coefficients are shown to be functions of the period parameter and since the period parameter follows
~
Fig. 5(a). Legend see p. 163.
z~
WmT/D:12.9
(a)
WmT/D:10.5 WmT/D:G.4
~
wm'r/D:5.1
~~lllt~_
CIRCLE REPRESENTSZEROFORCE SCALE:1": 0.73 kg.(1.(;Ib.) RADIAL
L?;2;2 N
45° ELEVATION ALONG TANK
163
CIRCLE REPRESENTS ZERO FORCE SCALE: 1": 0.73 k9 (1g Ib ) RADIAL
45 ° ELEVATION ACROSS TANK
Wm T/D: 14.~
'
Wm T/D: 11J;
Wm T/D: G,O
Wm T/O:1 2
GO" ELEVATION ACROSS TANK
(b) CIRCLE REPRESENTS ZERO FORCE SCALE 1 :O.73 kg (1G lb.) RADIAL
G
~
2
5
"
Wm "~D: 12.8
Wm T/D: 9,3
Wm T/D: G.1
Wm T/D: 2 4
Wm T/D : 14.B
Wm'i/D : 10 9 (c)
Wm T/D: e. 2
Wm r/o : 29
ELEVATION AT 45" TO OWN TANK
Fig. 5. Profile o f the resultant measured force normal t o the tube s e c t i o n s for the 7 . 6 2 c m (3 i n c h ) O.D. tube. The circle represents zero f o r c e line. T h e wave direction is s h o w n b y the arrows for tube: (a) along the tank; (b) across the tank; and (c) 4 5 ° t o the d o w n - t a n k .
164
o ~,
(cl) ~
o
~
o
o
o
.........
] " •
i
2
i
,
,
(b~ ]
° ° o
~
~
6 0 " ELEV.
~
ALONG
,;~
. . . . . . . . . . . .
]
a
,.o,,/ .
z
MEAN FITTED LINE 4 5 ° ELEV. A L O N G TANK
_
1.41
~
DATA
~
.
I
;
I
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.
.....
. .
.:
..o~0 ......
::
~.
._...o
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DO
- ......
,
~: )~
o
75 ° ELEV.
oO
_,2 . . . .
o
o
TANK
.
._____.
.....
Q
,
(¢1 2
GO° E L E V ACROSS
2
TANK
(f) o e
1
.
2
o
o
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I
°o
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o
4 5 ° ELEV. AT 4 5 ° TO
o
....
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,
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.
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(g)
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~
o
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^ ....
o
o
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o o
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o
o
o
o
o
o
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--
o
4
~, PERIOD
8 PARAMETER
ib ,
Iz
,4
g
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Fig. 6. Ratio of the m a x i m u m resultant normal force to the m a x i m u m in-line force on the 0.305 m (1 ft. ) tube sections against period parameter.
the Froude scale, scaling to other (e.g. prototype) sizes is straightforward However, the use of the results is limited by the relatively small range of the period parameter. Moreover, the Reynolds number is low and the dependence of the coefficients on the Reynolds number or the frequency number cannot
165 be s h o w n because o f t h e limited range o f the R e y n o l d s n u m b e r . Thus, t h e f u r t h e r e f f e c t o f large R e y n o l d s n u m b e r s o n t h e values o f t h e c o e f f i c i e n t s pres e n t e d is u n k n o w n . REFERENCES AI-Kazily, M.F., 1974. Wave forces on submerged pipelines. Proc. Int. Conf. Coastal Eng., 14th, Copenhagen, pp. 1863--1886. Bidde, D.D., 1971. Laboratory study of lift forces on circular piles. J. Waterw., Harbors Coastal Eng. Div., ASCE, 97 (WW4), Proc. Pap. 8495. Borgman, L.E., 1958. Computation of the ocean-wave forces on inclined cylinders. J. Geophys. Res. Trans., AGU, 39 (5): 885--888. Chakrabarti, S.K., Wolbert, A.L. and Tam, W.A., 1976a. Wave forces on vertical circular cylinder. J. Waterw., Harbors Coastal Eng. Div., ASCE, 102 (WW2): 203--221, Proc. Pap. 12140. Chakrabarti, S.K., Tam, W.A~ and Wolbert, A.L., 1976b. Total force on a randomly oriented tube due to waves. Proc. Ann. Offshore Technol. Conf., 8th, Houston, Texas, OTC 2495, pp. 433--447. Dean, tL G. and Aagaard, P.M., 1970. Wave forces: data analysis and engineering calculation method. J. Pet. Technol., 1970, pp. 368--375. Dean, R.G. and Harleman, D.R.F., 1966. Interaction of structures and waves. In: A. Ippen (Editor), Estuary and Coastline Hydrodynamics. McGraw-Hill, New York, N.Y. ]saacson, M. St. Q., 1974. The Forces on Circular Cylinders in Waves. Ph.D. Thesis, Dept, of Engineering, University of Cambridge, Cambridge, 227 pp. Keulegan, G.I-L and Carpenter, L.H., 1958. Forces on cylinders and plates in an oscillating fluid. J. Res., Natl. Bur. Standards, 60 (5): 423--440. MacCamy, R.C. and Fuchs, R.A., 1954. Wave forces on piles: a diffraction theory. Beach Erosion Board, Tech. Memo., 6 9 : 1 7 pp. McCormick, M.E., 1973. Ocean Engineering Wave Mechanics. Wiley Interscience, New York, N.Y., pp. 12--13. Morison, J.R., O'Brien, M.P., Johnson, J.W. and Schaaf, S.A., 1950. The forces exerted by surface waves on piles. J. Pet. Tech., Amer. Inst. Mining Eng., 189: 148--154. Sarpkaya, T., 1975. Forces on cylinders and spheres in a sinusoidally oscillating fluid. J. Appl. Mech., 1975, pp. 32--37. Sarpkaya, T., 1976. In-line and transverse forces on cylinders in oscillating flow at high Reynolds numbers. Proc. on the Eighth Ann. Offshore Technol. Conf., 8th, Houston, Texas, OTC 2533, pp. 95--108. Wiegel, R.L, 1964. Oceanographical Engineering. Prentice Hall, Engiewood Cliffs, N.J., 1964.
149
WAVE FORCES ON INCLINED TUBES
SU B R ATA K. C H A K R A B A R T I 1, A L L A N L. WOLBERT 2 and WILLIAM A. TAM 3
Chicago Bridge & Iron Company, Plainfield, Ill. (U.S.A.) (Received September 30, 1976; accepted January 31, 1977)
ABSTRACT Chakrabarti, S.K., Wolbert, A.L. and Tam, W.A., 1977. Wave forces on inclined tubes. Coastal Eng., 1: 149--165. Total forces due to sinusoidal progressive waves on a small circular tube arbitrarily oriented with respect to the wave direction are investigated. Two-component normal forces are measured in a wave-tank experiment. The in-line and transverse forces are resolved from these measurements. The lift coefficients are obtained from the transverse forces. The frequencies of the transverse forces are found to depend on the period parameter. The ratio of the m ax i m u m resultant force to the maximum in-line force on small sections is presented as a function of the period parameter for each tube position. The results show that the resultant force could be twice as much as the in-line force depending on the period parameter.
INTRODUCTION
Wave forces on vertical circular cylinders have received considerable attention in the past years. Many researchers have studied the effects of waves on cylinders of various sizes, mostly under experimental conditions. One of the most n o t e w o r t h y theoretical investigations has been the one by MacCamy and Fuchs (1954), who obtained a simple closed-form expression for wave forces including diffraction effects on a large vertical cylinder which is b o t t o m - m o u n t e d and surface-piercing. However, the formula is n o t valid when drag and lift effects are important, namely for smaller diameter cylinders. Morison et al. (1950) first introduced an empirical equation for the in-line force which is composed of two terms -- an inertia and a drag -- linearly added together. The inertia term includes an inertia coefficient while the drag term involves a drag co.efficient.Since then this formula has been extensively used in determining the inertia and drag coefficients from experimental data as well as in designing tubular members. It has also found its application in other submerged shapes. 1Head, Analytical Group, Marine Research and Development, Chicago Bridge & Iron Company, Plainfield, Ill. 2Manager of Marine Computer Facilities, Chicago Bridge & Iron Company, Plainfield, Ill. SDirector of Marine Research, Chicago Bridge & Iron Company, Plainfield, Ill.
150 While its use has been generally limited to small objects where diffraction and added mass effects are synonymous, its application to larger objects is not uncommon. In the latter cases, an effective inertia coefficient is employed in order to include the diffraction effect. Wiegel (1964) and Dean and Harleman (1966) have provided extensive reviews relating both to the application of the Morison equation for wave force evaluation and to the determination of the appropriate force coefficients. In general, however, investigation of inclined tubular members has been scarce. Borgman (1958) analytically extended the Morison equation in the absence of lift force to include the general orientation of a tube with respect to the wave direction. According to this formula, it is assumed that the forces on a tube act normal to the tube and can be expressed in terms of a normal velocity and a normal acceleration component. The normal velocity c o m p o n e n t results in both viscous and pressure forces while the tangential component causes only a shear force (skin friction) parallel to the axis (McCormick, 1973). The tangential components of the force are assumed small and hence negligible. Thus, the force vector in the generalized Morison equation having an inertia and a drag term is a function of the normal velocity and acceleration vectors. There have been other proposed formulas, e.g. by Dean and Harleman (1966); however, little is known about the inertia, drag and lift coefficients for an inclined cylinder. The experiments have generally been limited to the vertical and horizontal positions. One of the pioneering works in determining the values of the h y d r o d y n a m i c coefficients and their dependence on certain nondimensional numbers was successfully carried out by Keulegan and Carpenter (1958). They tested a submerged horizontal cylinder at the node of a standing wave so that the cylinder essentially experienced a one-dimensional flow. The measured horizontal forces were correlated with the Fourier series representation of the Morison equation and the mean values of the h y d r o d y n a m i c coefficients were evaluated. The flow effect around the cylinder was examined. The coefficients were shown to depend on the period parameter. Bidde (1971) studied the eddy-shedding process in the " l e e " of a vertical cylindrical pile due to oscillatory waves. The number of eddies was observed to depend on the period parameter or the Keulegan-Carpenter number. The lift force was found to be as much as 60% of the in-line force. Sarpkaya (1975) in a recent test verified the results of Keulegan and Carpenter through tests in a controlled environment. He tested cylindrical models across a one-dimensional sinusoidally oscillating flow in a U-tube. The velocity and acceleration fields in this controlled environment were nearly sinusoids. Similar analysis showed t h a t the dependence of the h y d r o d y n a m i c coefficients including the lift coefficients on the period parameter followed closely that found by Keulegan and Carpenter. Sarpkaya (1976) extended his previous test work to high Reynolds number and showed that the values of in-line mass and drag and transverse or lift coefficients are, indeed, dependent on the period parameter (i.e. Keulegan-Carpenter modulus) as well as Reynolds number or
151
equivalently a frequency parameter (defined as the ratio of the Reynolds number and Keulegan-Carpenter modulus). He tested both smooth and rough horizontal cylindrical models. Isaacson (1974) examined the transverse forces on a vertical cylinder. He observed the eddy formation " b e h i n d " the cylinder and found t h a t the number of eddies being formed is a function of the Keulegan-Carpenter modulus. He presented the first five components of the lift coefficients by representing the transverse force in a Fourier series form. The components were shown to be functions of the Reynolds number. AI-Kazily (1974) investigated wave forces on submerged pipelines. He computed the values of the empirical force coefficients in the Morison equation from the m a x i m u m measured forces on a horizontal cylinder normal to the two-dimensional wave flow. He observed large scatter in these coefficient values and concluded that the coefficients must be a function of several variables, e.g. wave height, wave length, as well as cylinder diameter and depth below SWL. In addition, they were found to vary over a given wave cycle. AIKazily also tested a tube inclined to the horizontal in the direction of the waves but did n o t present the values of the inertia and drag coefficients for the inclined cylinder. In the present study, a cylindrical model is tested at various orientations in order to determine the effects of the inertia, drag and lift forces on the tube. Measurements are obtained at two points on the cylinder conveniently located near the still water level so that some overlapping of data can be obtained while a m a x i m u m range of the test possible in the available wave tank is achieved. The measured in-line forces on the two points of the cylinder are utilized to derive the values of the inertia and drag coefficients which have been presented in an earlier paper (Chakrabarti et al., 1976}. In this paper, the effect of the transverse forces due to eddy-shedding and the total resultant forces on the tube are examined. A wave tank test was conducted with a 7.6 cm (3 inch) O.D., 3.05 m (10 ft.) long smooth tube placed in water at various orientations. Two 0.305 m (1 ft.) sections of the tube were instrumented to measure the two c o m p o n e n t normal forces on these sections. In addition, three c o m p o n e n t total orthogonal forces on the submerged portion of the t u b e due to waves were measured. Eight sets of runs were made, one for each orientation. At each angle, a series of wave periods and wave heights from the smallest to the largest possible sinusoidal waves were generated. The results for the vertical tube were presented in an earlier paper (Chakrabarti et al., 1976a), while the in-line forces on the inclined tubes were discussed by Chakrabarti et al., 1976b. The presentl paper gives the results for the transverse and total forces on the inclined tubes and is thus an extension of the earlier works (Chakrabarti et al., 1976b). The predominant frequency of the transverse forces on the tube at various orientations is compared with the wave frequency. The ratio of these two frequencies is indicative of the number of eddies formed which depends on the
152
p e r i o d p a r a m e t e r . T h e l i f t c o e f f i c i e n t s are c a l c u l a t e d f r o m t h e t r a n s v e r s e f o r c e s f o r t h e t u b e p l a c e d a l o n g t h e t a n k . T h e r e s u l t a n t f o r c e p r o f i l e s are p r e s e n t e d and the ratios of the maximum resultant force to the maximum in-line force are s h o w n t o i n c r e a s e s t e a d i l y w i t h t h e i n c r e a s i n g p e r i o d p a r a m e t e r f o r v a r i o u s tube orientations. '~ WAVE TANK TEST T h e w a v e t a n k i n w h i c h t h e t e s t s w e r e c o n d u c t e d is 7 6 . 2 m ( 2 5 0 ft.) l o n g , 1 0 . 1 m ( 3 3 ft.) w i d e a n d 5 . 5 m ( 1 8 ft.) d e e p . I t has a n a d j u s t a b l e c o n c r e t e f l o o r w h i c h was m a i n t a i n e d a t a 3 . 0 5 m ( 1 0 ft.) w a t e r d e p t h . T h e t u b e w a s placed a b o u t 0.16 m (0.52 ft.) off the b o t t o m . T h e details of the test set-up a n d i n s t r u m e n t a t i o n w e r e d e s c r i b e d b y C h a k r a b a r t i e t al. ( 1 9 7 6 a , b). The results f r o m the following seven t u b e o r i e n t a t i o n s are p r e s e n t e d here: (a) t w o t u b e e l e v a t i o n s , 4 5 ° a n d 6 0 ° ( w i t h r e s p e c t t o t h e h o r i z o n t a l ) i n a p l a n e across t h e t a n k ; (b) t h r e e e l e v a t i o n s , 4 5 °, 6 0 ° a n d 7 5 ° i n a p l a n e a l o n g t h e t a n k ; a n d (c) t w o e l e v a t i o n s , 4 5 ° a n d 6 0 ° i n a p l a n e m a k i n g 4 5 ° a n g l e t o t h e w a v e direction. NOTATION List of symbols CD CL
CM D
F~ fz
f0 N n
Ri Rmean T uX
Ux W w m
y Yl Y2 P
= drag coefficient = lift coefficient = inertia coefficient = tube diameter = total horizontal force on the tube = lift force frequency = resultant force on the tube sections per unit length = in-line force on the tube sections per unit length = vertical force on the tube sections per unit length = transverse force on the tube sections per unit length = incident wave frequency ffi number of runs = number of eddies formed = ratio of the total theoretical in-line force to the corresponding measured force on the tube per run = mean value for a particular tube position = wave period in-line component of the normal water particle velocity = time derivative of u x = normal water particle velocity vector = maximum normal water particle velocity = elevation measured from the tank floor = lower end elevation of the tube = upper end elevation of the tube = angle of tube inclination to the horizontal = mass density of water =
153
The t w o - c o m p o n e n t forces on the two 0.305 m (1 ft.) sections of the tube were measured normal to the tube. Therefore, at any time in a wave cycle the resultant normal force on these sections was known. This resultant force comprised of an inertia force component, a drag force c o m p o n e n t and a lift force component. The range of the test runs for each inclined tube position in the tank is given in terms of certain nondimensional quantities and is included in Table I. The period parameter is chosen as the independent parameter for the presentation of the results and is f o u n d to present the dependence of CM, CD, and CLaS well as the total forces reasonably well (see Notation). The Reynolds number (and hence the frequency number) is low and its range is also quite small. Hence the further dependence of the coefficients on the Reynolds number or the frequency number (as shown by Sarpkaya, 1976) cannot be determined. The wake parameter determines the importance of eddy-shedding while the diffraction parameter is small enough to exclude the effect of any wave scattering. DISCUSSION OF RESULTS A pair of CM, C D values per test run was obtained from each 0.305 m (1 ft.) section. These values of the coefficients are correlated with the corresponding period parameters. The period parameter at a p o i n t on a randomly oriented tube is defined in terms of the m a x i m u m normal velocity at that point, win, the tube diameter, D, and the wave period, T, as w m T/D. The values of C M and C D for each tube position obtained by the least square fit of the measured in-line force data were presented in the earlier paper (Chakrabarti et al., 1976b). The values of C M and C D in this m e t h o d are assumed constant over a cycle and any variation of these values within a cycle is ignored by choosing the best possible values. The residual force is about 10% of the measured and generally has twice the wave frequency. A mean curve is drawn through each of these plots representing a mean value of the coefficients at each wmT/D. These mean curves for CM and C D are shown in Fig.1. From the plots of C M and CD, it appears that the coefficients are functions of not only the period parameter but also the orientation of the tube. Since the effect of the lift forces in the direction of the waves is generally small the total in-line forces on the tube are computed using the mean curve as follows:
Fx" f Yl
p ~D2CM (wmT/D) u x + l2p D C D (wmT/D) Iw lu x dy/sin~b ¢ ¢ 0 (1) 4
where p = mass density of water, D = tube diameter, ~ = angle of elevation of " the tube to the horizontal, w m = m a x i m u m normal water particle velocity, w = normal water particle velocity vector, the bars denote its magnitude, Ux = its in-line component, and Ux = its time derivative. Yl and Y2 are the lower and upper end elevations of the inclined tube in water and C M and C D are con-
I
.38--12.9 1280--19800 1450--3390 .0114--.0673 .0259--.142 .015--.0682
WmT/D WmD/v D2/vT H/L H/d nD/L
Period p a r a m e t e r Reynolds number Frequency parameter Wave steepness Wave p a r a m e t e r Diffraction p a r a m e t e r
.071--13.5 290--22800 1440--4050 .0083--.0687 .0327--.143 .015--.0982
60 ° elev. along t a n k
.244--16.0 990--25000 1450--4060 .0093--095 .0311--.137 .015--.0982
75 ° elev. along tank
test
26 28 27 30 25 28
45 ° elevation 75 ° elevation 45 ° elevation 60 ° elevation 45 ° elevation 60 ° elevation
along tank along tank across t a n k across t a n k at 45 ° to d o w n - t a n k at 45 ° to d o w n - t a n k
No, of runs, N
Tube orientation rain. 0.656 0.917 0.709 0.750 0,913 0.985
max. 1.075 1.424 1.136 1.196 1.657 1.606
0.942 1.078 0.927 0.972 1.179 1.284
mean
Ratio = c o m p u t e d f o r c e / m e a s u r e d f o r c e
Ratio of the m a x i m u m total in-line c o m p u t e d force with the m a x i m u m total in-line m e a s u r e d force
T A B L E II
45 ° elev. along t a n k
formula
description
for the wave-tank
Range of test
parameters
Dimensionless p a r a m e t e r
Range of various dimensionless
TABLE
.228--14.6 930--23600 1450--4060 .011--.0604 .0313--.120 .015--.0982
45 ° elev. across t a n k
.248--15.7 1010--26200 1450--4070 .0115--.0579 .0313--.119 .015--.0982
60 ° elev. across t a n k
.4--13.3 1340--18600 1450--3380 .0126--.0698 .0202--.137 .015--.0682
45 ° elev. at 45 ° to down-tank
.0716--14.8 290--23100 1450--4060 .0058--.0648 .021--.145 .015---.0982
60 ° elev. at 45 ° to down-tank
t=a
155 LEGEND 6-
405: T? HORIZ?NTAL ALONG TANK
75 ° . . . . . . . .
"',
"~
~x
45 °
. . . .
45"
"
ACROSS
"
~ 45"
"
TO
DOWN
TANK
CD
2
CM ~x-
xx - -
Wrn____~T D
Fig. 1. Mean C M and C D values versus period parameter for a s m o o t h circular t u b e at various elevations.
sidered functions of the local period parameters. Note that the dependence of eq. 1 on ~ was omitted in the earlier paper (Chakrabarti et al., 1976b). The total in-line force obtained from eq. 1 is compared with the total measured in-line force and the ratio of the c o m p u t e d and the measured is calculated. The m a x i m u m and m i n i m u m values of the ratio, as well as the mean value, are shown in Table II. The correlation for the 60 ° to the horizontal along the tank runs cannot be shown due to some instrumental problems with an XYZ load cell in this case. The mean values shown in Table II are obtained from Rmean = V ~
(Ri) 2/g
(2)
i=l where N is the number of runs for a tube position and R i is the ratio in each case. Note that the measured force is predicted generally within 10%. The inline force was chosen for the calculation of C M and C D as well as for the total
156
force correlation since it is least affected by the lift force. However, for the two randomly oriented tube positions (45 ° to down tank) even these forces are affected by the eddy-shedding which may explain the larger scatter in the total in-line force correlation (Table II). For an arbitrarily oriented tube in the two-dimensional waves it is difficult to separate out the eddy-shedding forces from the measured transverse force. One means of deriving the lift forces due to eddy-shedding is to compute first the transverse (inertia plus drag) force using the CM and C D values calculated from the in-line forces and then subtracting it from the measured transverse force. The residual force may be attributed to the eddy-shedding process and wake formation " b e h i n d " the cylinder. However, this m e t h o d will carry any error in the C M and C D values to the lift forces. The lift coefficient, CL, obtained from the residual force then may have a large a m o u n t of error in it. For the tube inclined along the tank, however, the transverse force on the tube is strictly due to eddy-shedding. In these cases, a lift coefficient, C L, may be calculated from the transverse force fz on 0.305 m (1 ft.) sections by the following formula:
CL = (f z )max /( l pD 'wl 2 max )
(3)
where Iw] max is the m a x i m u m normal water particle velocity amplitude on the tube sections. The values of C L obtained this way for the three down-tank tube positions are presented as functions of the period parameter in Fig.2. Note t h a t the lift forces are generally quite small for low period parameter (w m T/D<4) and hence the scatter in C L is large. Values of CLwhich exceed the plot limits in this range have been omitted. The resultant force on a unit length of the tube sections in this case may be obtained from:
x y are the in-line and vertical force components obtained from the C M and C D values shown in Fig.1 while fz is the transverse force which in this tube position is totally due to wake formation in the " l e e " of the tube and is computed from a relationship similar to eq. 3. The calculations for the lift coefficient, C L, for the other tube positions (other than the down-tank positions) have n o t been carried out because of the difficulty in obtaining the pure lift (eddy-shedding) force in these cases. However, the lift force frequencies could be analyzed in these cases. For example, for the across-the-tank tubes the transverse (inertia plus drag) forces on the tube sections are computed from the mean C M and C D values in Fig.1 and these values are subtracted from the measured transverse forces on the tube sections. The residual forces approximately represent the vortex shedding effects. The measured transverse force profiles are converted to their energy spectra by a standard FFT routine. The number of data points chosen is 1024
157
0 D
DATA MEAN FITTED LINE 45" ELEV ALONG TANK
co o
oo
o
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WmT/O T%g,2. Lift force coefficients for the tube along the tank.
with a time increment of 0.04 seconds. A few sample plots are presented for various tube positions in Fig.3. The plots are shown in decreasing order of the period parameter for each tube position: (a) along the tank; (b) across the tank (measured normal transverse force is used here); and (c) at 45 ° to downtank. The values of the peak amplitude of the spectra are given in the figure. The profiles are chosen from both top and bottom section measurements at random. Note that the most prominent (or predominant) peak appears at or near a multiple of the wave frequency -- depending on the wave parameter. To examine the dependence of the predominant lift frequency on the period parameter, calculations are carried out for the ratio of the most predominant (the one having the highest spectral peak among several peaks present) lift
158
Wm T/O: 12.9 : 3 5 SEC
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Wm T/O:6 4 T ; 2 7 5 SEC
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Wm T/O: 42 T : 25 SEC
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6 0 a ELEVATION AT
45"
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0
1
2
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2
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FREQUENCY-
TO TANK
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Hz
(c)
Fig.3. Transverse force spectra for the tube sections of the 7.62 cm (3 inch) O.D. tube: (a) along the tank; (b) across the tank; (c) 45 ° to the down-tank.
frequency, fl to the wave frequency, f0. These ratios for each tube position are plotted against the period parameter in Fig.4. Note that the ratios are nearly all integer numbers. While the actual transition zone is somewhat uncertain the ratio seems to increase in steps with increasing period parameter. Isaacson (1974) observed that the number of eddies formed "behind" a cylinder is one less than this frequency ratio. Thus, no eddy is formed when the ratio n is 1; one eddy is formed for n = 2 and so on. Note also from Fig.3 that the total energy (area under the curve) for the lift force is small for low w m T/D values indicating that the lift force is small. When the lift force is significant, the resultant force differs significantly from the force obtained by the extended Morison equation. This is evident from examining the t w o measured components of the normal force on the tube sections. Sample plots of the resultant force profiles on the tube sections are presented in Fig.5 for each tube position in decreasing order of the period parameter. The same runs as in Fig,3 are chosen for this purpose for the tubes: (a) along the tank; (b) across the tank; and (c) at 45 ° to down-tank. The same trend as before is found. The profiles are quite irregular for large w m T/D values, but become more and more regular as w m T/D value decreases. The irregularity in the profiles is due to the presence of significant lift force. In fact, in some cases the resultant force is over 90% higher than the in-line force. For tubes along the tank the resultant force becomes the same as the in-line normal force at low Wmt/D (<4). For tubes across the tank the profile becomes regular with the in-line and the normal component of the vertical force remaining at low
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17
Fig.4. Ratio of the most predominant lift frequency to the wave frequency versus period parameter.
Thus, a definite trend is evident in these plots of the resultant force in relation to the in-line force with the period parameter. The maximum normal resultant force was obtained from the profiles of the type of Fig. 5. The maximum in-line force was computed as the average of all the peak values in a run. The ratio of these two forces is calculated for all cases. This ratio is plotted against the period parameter in Fig.6 for each tube w m T/D.
161 positiorL The ratio increases steadily with increasing w m T/ D values. The reason for this increase is t h a t the lift force due to eddy-shedding increases with the period parameter. At low w m T/D the ratio levels off to a certain limiting value depending on the position of the tube. For example, for a tube at an elevation of 45 ° down the tank (Fig.6a) in the absence of the lift force, the normal force is comprised of the in-line and vertical force so that the ratio of resultant normal to in-line becomes 1/sin 45 ° or 1.414. For the 60 ° alongthe-tank tube, the ratio is i / s i n 60 ° or 1.155 and for the 75 ° along-the-tank tube, it is I/sin 75 ° or 1.035. For across-the-tank tubes, it is 1.0. For the tubes at 45 ° to the down-tank, it is 1.206 for 45 ° elevation and 1.079 for 60 ° elevation. In these last two cases the ratio is n o t constant, but rather decreases slowly with the period parameter as shown in Fig.6 (f and g). Mean curves are drawn through the points representing the ratio. These mean curves give the multiplicative factor to be used at a particular w m T/D value in order to obtain the m a x i m u m resultant force on a tube section from the m a x i m u m in-line force on a submerged tube section in waves. Thus, for a section of a tube at 45 ° elevation along the tank, the factor is 1.57 for a w m T/D of 8.0, i.e. the m a x i m u m in-line force should be multiplied by 1.57 to obtain the m a x i m u m normal resultant force. Since the design of a tube should be based on the m a x i m u m force anticipated on the tube, the theoretical in-line force based on the mean C M and C D values should be increased by the applicable ratio given in Fig.6. CONCLUSIONS The resultant wave forces on a circular cylindrical member arbitrarily oriented in waves have been examined. Mean values of C M and C D for each tube orientation have been presented. For tubes inclined along the tank, the values of C L have been shown as functions of the period parameter. These values of the coefficients may be used to construct the resultant force profiles on a tube within the given period parameter. The transverse force spectra and the lift force frequency are presented for all tube positions. This information helps in identifying the eddy-shedding process. Since the two c o m p o n e n t normal forces on the tube sections were measured, it is possible to obtain the resultant forces on them. A few of the resultant force profiles have been plotted to show the dependence of their magnitudes on the period parameter. The ratios of the m a x i m u m resultant force to the m a x i m u m in-line force are shown to be functions of the period parameter. Mean lines are drawn through these points for each tube elevation. Thus, the total in-line (in the direction of the waves) force on a tube section may be c o m p u t e d from the given C M and C D values which may then be increased by these correction factors to obtain the m a x i m u m load on the tube section. Since the resultant forces and their h y d r o d y n a m i c coefficients are shown to be functions of the period parameter and since the period parameter follows
~
Fig. 5(a). Legend see p. 163.
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(a)
WmT/D:10.5 WmT/D:G.4
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L?;2;2 N
45° ELEVATION ALONG TANK
163
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45 ° ELEVATION ACROSS TANK
Wm T/D: 14.~
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Wm T/D: 11J;
Wm T/D: G,O
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GO" ELEVATION ACROSS TANK
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G
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Wm "~D: 12.8
Wm T/D: 9,3
Wm T/D: G.1
Wm T/D: 2 4
Wm T/D : 14.B
Wm'i/D : 10 9 (c)
Wm T/D: e. 2
Wm r/o : 29
ELEVATION AT 45" TO OWN TANK
Fig. 5. Profile o f the resultant measured force normal t o the tube s e c t i o n s for the 7 . 6 2 c m (3 i n c h ) O.D. tube. The circle represents zero f o r c e line. T h e wave direction is s h o w n b y the arrows for tube: (a) along the tank; (b) across the tank; and (c) 4 5 ° t o the d o w n - t a n k .
164
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8 PARAMETER
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Fig. 6. Ratio of the m a x i m u m resultant normal force to the m a x i m u m in-line force on the 0.305 m (1 ft. ) tube sections against period parameter.
the Froude scale, scaling to other (e.g. prototype) sizes is straightforward However, the use of the results is limited by the relatively small range of the period parameter. Moreover, the Reynolds number is low and the dependence of the coefficients on the Reynolds number or the frequency number cannot
165 be s h o w n because o f t h e limited range o f the R e y n o l d s n u m b e r . Thus, t h e f u r t h e r e f f e c t o f large R e y n o l d s n u m b e r s o n t h e values o f t h e c o e f f i c i e n t s pres e n t e d is u n k n o w n . REFERENCES AI-Kazily, M.F., 1974. Wave forces on submerged pipelines. Proc. Int. Conf. Coastal Eng., 14th, Copenhagen, pp. 1863--1886. Bidde, D.D., 1971. Laboratory study of lift forces on circular piles. J. Waterw., Harbors Coastal Eng. Div., ASCE, 97 (WW4), Proc. Pap. 8495. Borgman, L.E., 1958. Computation of the ocean-wave forces on inclined cylinders. J. Geophys. Res. Trans., AGU, 39 (5): 885--888. Chakrabarti, S.K., Wolbert, A.L. and Tam, W.A., 1976a. Wave forces on vertical circular cylinder. J. Waterw., Harbors Coastal Eng. Div., ASCE, 102 (WW2): 203--221, Proc. Pap. 12140. Chakrabarti, S.K., Tam, W.A~ and Wolbert, A.L., 1976b. Total force on a randomly oriented tube due to waves. Proc. Ann. Offshore Technol. Conf., 8th, Houston, Texas, OTC 2495, pp. 433--447. Dean, tL G. and Aagaard, P.M., 1970. Wave forces: data analysis and engineering calculation method. J. Pet. Technol., 1970, pp. 368--375. Dean, R.G. and Harleman, D.R.F., 1966. Interaction of structures and waves. In: A. Ippen (Editor), Estuary and Coastline Hydrodynamics. McGraw-Hill, New York, N.Y. ]saacson, M. St. Q., 1974. The Forces on Circular Cylinders in Waves. Ph.D. Thesis, Dept, of Engineering, University of Cambridge, Cambridge, 227 pp. Keulegan, G.I-L and Carpenter, L.H., 1958. Forces on cylinders and plates in an oscillating fluid. J. Res., Natl. Bur. Standards, 60 (5): 423--440. MacCamy, R.C. and Fuchs, R.A., 1954. Wave forces on piles: a diffraction theory. Beach Erosion Board, Tech. Memo., 6 9 : 1 7 pp. McCormick, M.E., 1973. Ocean Engineering Wave Mechanics. Wiley Interscience, New York, N.Y., pp. 12--13. Morison, J.R., O'Brien, M.P., Johnson, J.W. and Schaaf, S.A., 1950. The forces exerted by surface waves on piles. J. Pet. Tech., Amer. Inst. Mining Eng., 189: 148--154. Sarpkaya, T., 1975. Forces on cylinders and spheres in a sinusoidally oscillating fluid. J. Appl. Mech., 1975, pp. 32--37. Sarpkaya, T., 1976. In-line and transverse forces on cylinders in oscillating flow at high Reynolds numbers. Proc. on the Eighth Ann. Offshore Technol. Conf., 8th, Houston, Texas, OTC 2533, pp. 95--108. Wiegel, R.L, 1964. Oceanographical Engineering. Prentice Hall, Engiewood Cliffs, N.J., 1964.