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ARCHIEF
OFFSHORE TECHNOLOGY CONFERENCE 6200 North Central Expressway
Dallas, Texas 75206
Technisché Hogeschool
Deift
PAPEROTO 2534
NUMBER
Wave
nteractiofl with a
Submerged Open-Bottom
Structure
By
Subrata K. Chakrabarti and Robert A. Naftzger,
Chicago Bridge & Iron Co.
THIS PAPER IS SUBJECT TO CORRECTION ©Copyright 1976
Offshore Technology Conference on behalf of.the American Instituteof Mining, Metallurgical,
and Petroleum Engineers, Inc. (Society ofMining Engineers, The Metallurgical Society and Societyof Petroleum Engineers). American Association of Petroleum Geologists, American lnstitute of Chemical Engineers, American
Society of Civil Engineers, American Society of Mechanical Engineers, Institute of Electrical and Electronics En-gineers, Marine Technology Society, Society of Exploration Geophysicists, and Society of Naval Architects and Marine Engineers.
This paper was prepared for presentation at the Eighth AnnualOffshore Technology Conference,
Houston, Tax., May 3-6, 1976. Permission to copy is restricted to an abstractof not more than 300 words.
Illustrations may not be copied. Such use of an abstract shouldcontain conspicuous acknowledgment of where and by
whom the paper is presented.
ABSTRACT
The interaction of sinusoidal waves with a large open bottom structure near the ocean floor is investigated. The.
wave diffraction problem is formulated within the framework of linearized potential. theory and solved numerically on a digital computer. The analysis is confirmed by the results of wave tank tests on a hemispherical shell and underwater storage tank models. It is found that both the horizontal and
ver-tical forces acting on an open struc-ture are lower than those on the corres-ponding sealed structure, the vertical force, in particular1 being an order of magnitude smaller over the practical range of wave periods. The inside pressure on the structure when slightly open correlates very well with the mean
bottom pressure on the structure when sealed, so that the results for the slightly open case can be obtained from the results for the sealed case.
References and illustrations at end of paper
INTRODUCTION
Increased offshore activities in recent years have led to the placement of a variety of structures on or near
the ocean bottom. These structures are subject to surface wave forces in both the horizontal and vertical directions and in many instances must be held by piling in addition to their own weight. Frequently, a piled structure is
scoured out at the bottom, leaving its underside exposed to the wave action.
Indeed, for a shell-like structure the entire 'inside' will be opened to the wave action. The large Khazzafl oil storage tanks (each one half million bbl in capacity), situated 59 miles offshore Dubai, fall into this category, for recent diver inspections have revealed
sizable openings under these structures. Within the framework of linear wave
theory this paper examines the effect of such openings upon the wave forces experienced by the Khazzan tanks and similar structures.
Different Green's function formuia tidns of the potential problem describ ing the wave interaction with a solid
within the fluid region and may be re-solved as follows:
body, defined by a closed surface, have been presented by John (5), Wehausen and Laitone (10), and Black (1), and numer-ical solutions for particular shapes have been obtained by a number of in-vestigators (2, 4, 6, 9). In this study,
however, we are primarily interested in the wave forces on a shell, defined by an open surface. The formulation of this problem is similar to that for a
solid, except that the solution is con-tained in a singular integral equation, rather than a regular Fredhoim equation. The numerical solution of such an equa-tion has been carried out earlier for a hemispherical shell (8). Here, this so-lution is extended to several open bo.t-torn structures of practical design, and the wave forces are compared to those for the corresponding sealed structures. Wave tank tests conducted on models of these structures show good agreement with the analysis.
In general, the analysis indicates that the pressure inside a slightly open shell matches closely the mean pressure at the bottom of the corresponding
sealed body, and this result is borne out by experimental tests on a slightly open hemisphere. Consequently, the
pressures and forces on a structure with gaps at the bottom can be estimated from the results for the body when sealed. ANALYSIS
Mathematical Formulation
The basic flow is assumed to be oscillatory and to have a large Reynolds number with respect to the submerged object, so that in the greater portion of the flow the viscous effects will be negligible compared to the inertial effects, and the flow can be regarded as irrotational. The incident wave is
chosen as the linear wave for which the flow is irrotational and incompressible, and the wave height, H, is small com-pared to the wave length, L. As a
re-sult, the fluid velocity can be repre-sented as the gradient of a scalar po-tential, , which satisfies Laplace's equation
V2(x,y,z,t) = 0 (1)
urn
R--
'vrtYiz))
= 0 (8)10 WAVE INTERACTION WITH A SUBMERGED OPEN-BOTTOM STRUCTURE OTC-2534
x,y,z,t) = IrnC4(x,y,z) e -let} (2)
(x,y,
z)
= ct'. (x,y,
z)
(x,y,z)
4,1(x,y,z)
-cosh(kd) cosh(ky) e (4)
where 4,j = the incident wave potential,
= the reflected or scattered velocity potential from the submerged object, a =
the circular wave frequency, k = the
wave number, d = the water depth, g =
the gravitational acceleration, t = time and x,y,z = the Cartesian coordinates of a point in the fluid region. Refer to Fig. 1 for the coordinate convention. Given the conditions satisfied by the
incident potential at the boundaries of the fluid, the reflected potential, 4,r' should then meet the following boundary conditions:
1. The normal velocity at the bottom
is zero.
= 0
aty=0
(5) The normal velocity of the fluid at the free surface is equal to the velocity of the surface itself. Inthe linear theory this condition re-duces to
- r(X,Y,z) = 0 at y = d
(6) where v = k tanh(kd).
The normal velocity at the surface of the object, S, cancels that due to the incident potential.
j(x1yz) -
(x,y,z) on 5 (7)Far from the object the reflected potential satisfies the radiation condition describing an outward
where R is the radial distance from the object.
The interaction between the wave and the submerged object is thus posed as a problem in potential theory which can be solved by the Green's function method. The Green's function, G,is
chosen as a singular potential which satisfies the same boundary conditions as the reflected potential at the free
surface (Eq. 6), the ocean bottom (Eq. 5) and infinity (Eq. 8). The integral and
series expressions for G are given in Ref. 10. In the case of a shell the
solution for the reflected potential, is given uniquely in terms of a weighted
integral of the normal derivative of G over the surface of the shell, S.
=
(9)
1
S.
in which S denotes the side of S chosen as positive, n is the outward normal to
S, and t is the difference in across
S. Applying Eq. 7 to the total velocity
potential, the following integral
equa-tion for is then obtained:
IS+
-4?r-(x,y,z) x,y,z on S (10)
in which the prime denotes that n is a function of x, y, and z (rather than F, r, and ?), and P indicates that the Cauchy principal value of the in-tegral is to be taken.
The dynamic pressure is given by Bernoulli's equation. For a large ob-ject the velocity squared term is g.en-erally small compared to the linear
term. However, in the case of a shell
it will not be small near the edges and may contribute significantly to the re-sultant forces when the wave length is large compared to the overall size of the shell. The difference in pressure
across the shell is given by
= pC)
-in which = -grad(), and p = mass
density of water.
Method of Solution
The integral representing the scattered wave (Eq. 9) should be dis-continuous across the shell surface.
Otherwise, would be identically
zero, and would equal j. Since the
singular parts of }s differ in sign
as x, y, and z approach S from opposite sides, such a discontinuity obtains.
The flow field is a simply connect-ed region and the flow itself is assumconnect-ed to be irrotational everywhere.
Conse-quently, will be single valued and
unique to within an arbitrary constant, and will be unique and vanish at the
edge of S.
The open-bottom structure, Fig. 1, ,:is considered to be symmetric with
re-spect to the vertical y-axis. The sur-face of the structure is defined in terms of the rectangular Cartesian coordinates
(x,y,z) shown in the figure, and its edge is marked by the angle, a, measured with respect to the ocean bottom.
The incident progressive wave can be resolved into two standing waves, one of which is symmetric with respect to the y-z plane (Fig. 1), while the other one is asymmetric. Due to the symmetry of S fore and aft, the solution
for E thereupon splits into symmetric and asymmetric parts. Because these two parts of 1 behave differently near the edge of S as a-'0, it is desirable to
obtain them separately. Thus, t is
constructed from separate solutions to Eq. 10 corresponding to the symmetric and asymmetric parts of the incident
potential, The solution is carried
out numerically, as described in the next section.
12 WAVE INTERACTION WITH A SUBMERGED OPEN-BOTTOM STRUCTURE
Since the flow is assumed symmetric with respect to the x-y plane, L4 and
42 are symmetric about this plane and the integrals over S can be written as integrals over one half of S. This portion of the surface is divided into N subsections whose linear dimensions are nearly equal (Fig.2). Each subsection is identified by the coordinates of its center and the normal direction and cur-vature at its center. The integrals over S are then replaced by summations of integrals over these subsections. The Mean Value Theorem for weighted means is applied to the unknown part in each integral, approximating the mean value of the unknown by its value at the center of the subsection. The re-sulting weighted integrals for the sub-sections are evaluated, yielding 2N real algebraic equations for the N mean
v'alues of jc and tc2 at the center of 3ach subsection. These equations may be rewritten in the complex matrix Eorm as follows:
= -4irV.
i,j
= l,2,..N
(14)Ln which Vj is the normal velocity due :0 the incident wave. The solutions
or L
and
L\q2 are obtained by the Lnversion of the complex matrix.=
-4irRe{[A..]
V.}
(15)
=
-41TIm{IA1F' v}
(16)1he dynamic pressure is then given by
=
Pa{_(1)sint)2).cos(at)}
+ velocity squared terms (17)
rom which the horizontal and vertical omponents of the force are obtained by uadrature.
TAVE TANK TESTS
escription of Models and Instrumenta-ion
Both
open and closed hemispherical models have been tested in a large wavetank. Several test runs were made with
different gaps at the bottom, and the differential pressures, as well as the horizontal and vertical forces, were measured. The details of these tests
are described in the previously publish paper (8). Two practical designs of underwater oil storage tanks have been
tested also. One of them was a hemis-pherical shell attached to a closed cir-cular box of square section, while the other one was a structure similar to the axi-syinmetric tanks placed offshore Dubaj. Both models.were made of 3/32 inch thick aluminum plate, and their geometry and dimensions are shown in Fig. 3.
In order to measure the net verti-cal loads on the models, the structures were supported on load cells. The load cells consisted of flat steel plates equipped with strain gages, the bending of the plates serving to measure the
load. The calibration of the load cells
was performed in the dry and checked in situ in the wave tank. The models were placed on three load cells equally spaced at 1200. A typical supporting system for the models is shown in Fig. 4a. The model is attached to an angle
supported by a bolt resting on the strain-gaged plate. The plate is sup-ported between two 1/8 inch rods and thus subject to bending from concentrated loads in the middle.
Test Procedure
The tests were performed in the Chicago Bridge and Iron Companyts wave tank, which is 250 ft. long, 33 ft. wide and 18 ft. deep. Its adjustable concrete floors can be set to any depth up to 18
feet. The hemispherical storage tank
model was tested in a water depth of 53 1/8 inches. The water depth was changed to 57 1/2 inches for the tests on the Khazzan-type storage tank model. The models were set up on a steel test slab near the middle of the wave tank. The vertical gages were preloaded by distributing 250 lbs. of dead weight
in-side the model. Thus, it was ensured that the gages would operate in the linear range and not slacken.
Sinusoidal waves were generated by a pneumatic type wave generator. The height of the free surface was recorded by a capacitance wave probe placed about
OTC-2534 Numerical Procedure
Thecomp1ex quantities L and G in Eq. 10 are written in terms of their real and imaginary parts as
= 1
+
(12)
'I
16 feet ahead of the model where the sur-face was essentially unaffected by the presence of the model. A sonic wave probe was positioned alongside the model to provide information on the phase relationship between the forces and waves. The layout of the model in
the wave, tank is shown in Fig. 4b. The.
models were tied horizontally in several directions to prevent movement.
The hemispherical storage tank of radius a was tested with a uniform gap size of 3/4 inch which is equivalent to an angle ct of 1.53 degrees. The depth to radius ratio in this case was d/a = 1.66, and the periods of the sinusoidal waves generated varied from 1.35 sec. to 3.00 sec. Thus, the range of the ka values was approximately 0.52 to 1.80.
The Khazzan type storage tank model had a gap of 1/2 inch, corresponding to a value of ct = 0.78 degrees, and was as-signed an equivalent value of d/a = 1.56. The wave period ranged from 1.25 sec.. to 3.5 sec., giving a range of ka values from 0.48 to 2.4. Usually more than one wave height was run at each period. For the purpose of presenting the test
re-sults, the forces at each period have been normalized with respect to the wave
ight and averaged. DISCUSSION OF RESULTS Analytical Results
The differential pressure at a point on the shell is obtained from Eq.
17. On the assumption that the wave
height is small àompared to the size of the structure the velocity-squared terms in Eq. 17 can be neglected for moderate and large values of ka, a being the mean radius of the shell. The irrotational solution requires that at the edge of
the shell E = 0 in order that remain
continuous; and thus, to a first approx-imation the differential pressure at the edge vanishes.
When the opening between. the edge of the shell and the bottom is small, the pressure inside the shell will be uniform. Consequently, for an axially
symmetric shape the horizontal force will be about the same as that on the corresponding sealed body. On the. other
hand, the vertical force on the shell will differ from that on the sealed
object by an amount equal to the pressure inside times the area of its base.
Garrison and Snider (3) demonstrated this result experimentally in measuring the fOrces on a hemispherical shell sus-pended by strain gages slightly above the wave tank floor. The pressure
in-side the shell was measured by a single pressure transducer. The vertical force on the sealed body was determined from
these measurements and good correlation between the theoretical and experimental. results was found.
In an earlier paper (8) it is shown that the pressure inside a hemispherical shell is approximated quite well by two mean pressures at the bottom of the
corresponding sealed hemisphere. One of these mean pressures is the arithmetic mean of the four stagnation point pres-sures at the bottom, fore and aft and to the sides. The other mean pressure is the first term of the Fourier series expansion of the pressure profile at the bottom:
= fPo) dO (18)
where 0 is the measure of the angle around the object and p(0) is the
pres-sure acting at the bottom. Although both mean pressures were found to
repre-sent accurately the inside pressure in the case of a deeply submerged_hemisphere it is believed that generally p repre-sents the best approximation to this pressure for a slightly open body.
Halkyard (4) holds to the same opinion in his discussion of the wave, forces on three dimensional open bOdies.
To demonstrate the validity of these notions the pressures and phase angles for a hemisphere are compared in Table 1 for both deep and shallow sub-mergence. In the former paper (8), the
inside pressure was computed from the
differential pressure at the top of the shell and the (outside) pressure at the top of the corresponding closed hemis-phere. In this case, the inside pres-sure is obtained from the difference in
the vertical forces on the two bodies divided by the base area and is believed to produce a more accurate result due to the approximations involved in the nu-merical analyses. It is seen in Table 1
that the correlation for both d/a>2 and d/a = l.25 is quite good in the range
.lkal.
Above ka = 1, the correlation is uncertain due to numerical diffi-culties, as the differential pressure on the shell and the (outside) pressure on the sealed body approach each other. Thus, any error in the numerical solutior is magnified many times in theirdiffer-ence. Table 1 suggests that the verti-cal force on a slightly raised shell can be obtained from the results of the corresponding sealed body, at least for ka up to 1. This hypothesis will be further verified by the experimental results in the following section. Experimental Results
Hemispheres of various sizes have been tested in both open and sealed po-sitions and the results have been re-ported already (8). Reasonably good correlation of the test data with the theory for the wave forces was obtained.
It is shown that a substantial reduction in the vertical force occurs when the structure is lifted even slightly off the bottom, particularly at the lower ka values (longer waves relative to the structure size).
In a test with a hemisphere raised about 1/2 degree off the bottom, local dynamic pressures also were measured at a few points on the shell. The results for d/a = 4.18 and ka = 0.48 are shown
in Fig. 5. The solid line represents the maximum theoretical pressure differ-ence across the shell while the dashed line is the maximum outside pressure on the corresponding sealed hemisphere. The open circles are the maximum values of the measured differential pressure. The overall correlation of the test data with the theory is seen to be good. Note
that the differential pressure is only 1/2 to 1/3 the outside pressure and vanishes at the bottom edge.
In the tests on a hemispherical model by Garrison and Snider (3) the inside pressure and net forces on the model were measured. The data obtained on the forces for d/a = 3 and a = 10 are presented in Fig. 6. The net vertical force on the shell shown in the figure was measured directly. The vertical
force on a closed hemisphere was then estimated by adding the inside pressure t.imes the hemisphere base area to the measured force. The maximum measured inside pressures are compared with the
theoretical pressure in Fig. 7. The theoretical pressures are found from the difference in the computed verti-cal forces on the sealed and open hem-isphere divided by the base area. The measured pressures are found to be generally higher than the theory, and hence the measured vertical forces on the seated hemisphere also show some-what higher values. Nevertheless, the overall correlation is reasonably good.
At CBI the vertical wave forces on two models of underwater storage tanks were measured. The results of these tests are compared with the theory in Figs. 8 and 9. A sketch of the model is: included in each figure. The values
of thedepth-toradius ratio and the
opening, a are shown in the respective figures. The solid lines are the theo-retical forces on the models and the open circles represent the measured forces. The theoretical forces on the corresponding sealed bodies are also shown in the figures as dashed lines, so that the reduction in the uplift due to the opening at the bottom can be seen. The mean radii chosen for calculating the nondimensional quantities are the radii at the bottom. For example, for the hemispherical model, the radius, a is taken as 32 inches, while for the Khazzan-type tank, a = 36.75 inches. The forces represent the maximum values and are normalized with respect to pga2H/2, where p is the mass density. of water, g is the gravitational accelera-tion and H is the incident wave height. The measured force data .is seen to compare well with the analytical re-sults in both the cases.
CONCLUSIONS
An analysis of the wave interaction with a bottom mounted shell-like struc-ture has been presented. Generally, the results of the anlaysis are in good agreement with the experimental data collected from wave tank tests on such structures. it is clearly demonstrated that at long wave periods a substantial reduction in the vertical wave force is achieved by having the structure open at the bottom. The analysis further shows that, when the opening at the bottom is
small, the pressure inside is uniform, fluctuating only with time, and can be
'OTC-'2 53 4 ----CHAKRABARTI,- .K. AND NAFTZGER, R. 115
approximated by the mean outside pres-sure at the bottom of the corresponding sea1ed'structure, at least for ka values up to 1. The vertical force on the open
structure will differ from that on the sealed tructure by this pressure times the base area, while the horizontal forces (to a first approximation) will be the same. Thus,. the forces on a slightly open body can be determined dirctly from those 'on the corresponding sealed body. Application of this result to two practical underwater storage tank concepts ha's been made, and the corre-lation with model test data indicates that the theory can be validly extended to other submerged shapes.
ACKNOWLEDGEMENTS
The authors would like to thank the CBI Papers Committee for tI-eir permission to present this paper (CBT 5338). The assistance of the members of the CBI Marine Research group in carrying out the wave tank tests is gratefully ac-knowledged. Thanks are also extended. to Robert Snider for providing a part of the hemispherical test data.
REFERENCES
Black, J.L., "Wave Forces on Verti-calAxisylilmetriC Bodies," Journal of Fluid Mechanics. (1975), Vol. 67, Part 2, pp. 369-376.
Garrison, C.J., and Rao, S.V., "Interaction of Waves with Sub-merged Objects," Journal of the Waterways, Harbor and Coastal Engineering Division, ASCE, VOl.
97, No. WW2, Proc. Paper 8111, May, 1971, pp. 259-277.
Garrison, C.J., and Snider, R.H., "Wa\e Forces on Large Submerged Tanks," Sea Grant Publication No.
210, Coastal and Ocean Engineering 5iision (Report No. 1l7-COE),
Texas A&M University, Jan., 1970. Halkyard, J.E., "Wave Force on a Submerged Object," Ph.D. Thesis, Department of Ocean Engineering, Massachusetts Institute of Tech-nology, 1971.
John, F.., "On 'the 'Motion of Float-ingBodis,.I," Comm. Pure and
Applied Math., Vol. 3, 1950, pp. 45-100.
Lebreton, J.C., and Cormault, P., "Wave Action on Slightly Immersed Structures, Some Theoretical and Experimental Conditions,"
Proceed-ings, Symposium Research on Wave Action, Deift Hydraulics Laboratory, Delft, The Netherlands, July, 1969,
34 pp.
Milgram, J.H., and Halkyard, J.E., "Wave Forces on Large. Objects in the Sea," Journal of Ship Research, June, '1971, pp. 115-124.
Naftzger, R.A., and' Chakrabarti, S.K., "Wave Forceson a Submerged Hemispherical Shell," Proceedings, Conference on Civil Engineëiñ in
the Oceans/Ill, University of Dela-ware, Newark, DelaDela-ware, June, 1975. Sommet, and Vignat, P.H.! "Com-plex Wave Action on Submerged
Bod-ies," Proceedings, Symposium Re-search On WaVe Action, Deift Hy-draulics Laboratory, Deift, The Netherlands, July, 1969, 31 pp. Wehausen, J.V., and Laitone, E.V.,
"Surface Waves," 'Encyclopedia of Physics, S. Flugge, ed., FlüidDy namics III, Vol. 9, Springer-Verlag Berlin, West Germany, 1960, pp.
TABLE I - COMPARISON OF THE PRESSURE INSIDE A SLIGHTLY2
OPEN HEMISPHERE WITH THE AVERAGE PRESSURE AROUND THE BOTTOM OF THE HEMISPHERE WHEN CLOSED
d/a = 1.25 d/a > 2.0
Fig. 1 - Definition sketch.
IpivI
Phase Anqleof p, deg.l'hase Angle of p1, deg.
IPivII
0.2 1.000 -0. 025 -0. 025 0.2 1.000 .0.4 1.001 -0.352 -0.356 0.4 0.999 0.6 1.003 -1.65 -1.68 0.6 0.996 0.8 1.005 -4.71 -4.86 0.8 0.991 1.0 1.003 -9.98 -10.58 1.0 0.980 1.2 0.981 -17.08 -19.47 1.2 0.952Fig. 2 - Segmentation of the sheH for the numerical solution.
- - .' . - - '- -'- "
-
--(a)
C?
--
- _.-___z-3 '2
(b)
BALL BEARINGS ;
/
- - _z,4c_ ., ANGLE STRAIN GAUGE ROD RUBBERV
(a)
\
-'TANK FLOOR WAVE 0-SlDE 0F MODELigi
c1z'1
IOD
BEACH(b)
tl/2 LEGEND 0 CAPACITANCE WAVE PRO BE 0 VERTICAL GAUGEFig. 4 - Wave tarlk test set-up.
BOTTOM OF MODEL
LEGEND THEORY EXR RAISED (a: .51°) 0
SEALED(a :0)
4.1$: 0.48
90'
-0.2 0.3 P max. pg(H/2)Fig.
5- Correlation
of the measured and theoretical
pressures at the center section of a hemispherical
she I I
.0.4
1800
0.05 0.02 0.01 01 02 0.5 10 .ka 0.05-0.02 0.01 p THgORY EXP RAISED (a:1) 0 SEALED (a:0) 2.0
Fig.
6
-Correlation of the measured and
theoretical pressures at the center section
of a hemispherical shell.
E a) 2.0- 1.0- 0.5- 0.2-0.1 0 0 2.0 0.1 02 05 1.0 ka 0.2 K a E a) 0.1 2.0 1.0 0.51.0 0.0 WAVE
--b-
EGE NDTHEORY
-EXR/ 0 RAISED a 10 d1:3 00 Q2 04 06 kaFig. 7 - Correlation of the maximum
measured
pressure inside an open hemisphere with the
theoretical mean pressure at the bottom of the hemisphere when closed.
08 10 2. 0. a 0.05 0.0 0.0 0.1 Q2 0.5 1.0 20 50 10.0 ka
Figs 8 - Correlation of the vertical
wave force amplitudes on a
hemis-pherical storage tank model (d/a1.66)_
measured vs theoretical
7-t
SEALED THEORY EXp 0 a :I. \._ I./
RAISE0(a,
-/
P
x E 2.0 1.0 0.5I
0.2 0 0.1 0.05 0.02 0.01SL
d,,..156 WAVE F a'-t
I
0'
0.5 1.0 2.0 5.0 10.0 kaFig. 9
-Correlation of the vertical
wave force
amplitudes on a Khazzan
type storage tank
model(d/a1.56)-measured vs theoretical.
THEO. EXP
RAISED (a:Q.7)-
0SEALED(a:0)
