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L•Q
NAVAL POSTORADUATE SCHOOLMonterey, California
••,•_#•D D C;
SEP 5 197
THESIS
WAVE FORMES ON A HORIZORIAL CIRCUlAR CYLINDER
by
Brian Thoas Perkis1on
Thesis Advisor: C. J. Garrison
June 1972
Approved for public release; distribution unlimited
NATIONIAt TECHNICALINFOrMArION SERVICE
-UNCLASSIFIED- Security Classification
14 K LINK A LINK B LINK C
FROLE WT ROLE WT ROLE WT
Gravity Wave Forces
lWave Interaction with Underwater CylindricalBodies
Capacitance Wave Height Probe
, •ov ,• NCLASSIFID
k; N oI01•• . . •,rt .as ~ c to
UTV1ASSTFX1M S~~Sectinwv C'la.ssification ..DOCIJNENT CONTROL DATA- R & D
(Stcurity classificeaion of title, body of ah.zttcf and ndexing annont ion must be entered when the overall repor, Is classlfied)
1. OrIGINATING ACTIVITY (Corporate author) Zs. GREOUP EUIT -S$ IAT
Naval Postgraduate SchoolMonterey, California 93940
3. REPORT TITLE
Wave Forces on a Horizontal Circular Cylinder
4. DESCRIPTIVE NOTES (7)ype of report and.inclusive dates)
Master's Thesis; June 19725. AUTHORISI (First name, middle Initial, last name)
Brian Thanas Perkdnson
6. REPORT OAYF 7&. TOTAL NO. OF PAGES 7b. NO. OF REFS
June 1972 88 9$a. CONTRACT OR GRANT NO. 94. ORIGINATOR'S REPORT NUMBER(S)
b. PROJECT NO.
c eb. OTHER REPORT NO(S) (Any other numbers that may be assignedthis report)
d.
10 DISTRIBUTION STATEMENT
Approved for public release; distribution unlimited.
11. SUPPLEMEN. ARY NOTES 12. SPONSORING MILITARY ACTIVITY
13. AOSTRACT
A horizontal circular cylinder, located near the floor of a t~n dimensionalwave channel, was subjected to a train of gravity waves. The wave height andhorizontal and vertical forces on the cylinder due to the incident waves weremeasured and presented in dimensionless form. Experimental values of the hori-zontal and vertical diirensionless force coefficients are presented as functionsof relative wave height and relative wave length. The dimensionless forcecoefficients predicted by a modified 1ýbrrison's equation are canpared with theexperimental data. Experimental wave lengths of frca tv~o to seventeen feet wereinvestigated.
The horizontal force coefficients vere found to vary linearly with waveheight over the range of wave lengths tested. The vertical force coefficientsdisplayed regions in which the force was both inertia dcndnated by a lift force.
FORM (PAGE 1)DD .. t547:3 UNULSI, NOV 65 •1 UNSIIASSIFI.EDS/N 0101 -807-681 1 S-curt CIssIICob A-31405
- .
Wave Forces om x Horizontal Circular Cylinder
* by
Brian Mhcas Perkinson
Lieate-nt C er,, United States NavySB.S,- United States Naval Acaderay, 1963
Su A ,,ittod in partial fulfillme-nt of therequirements for the degree of
;.ter of Science in Mechanical Engineering
fran theNAVAL POSTRADUATE SCHOOL
June 1972
Author %'# 7'
Approved by _______"_--,-"___ _6? Thesis---vior
Chairman, Departbent of Mechanical Engineering
/ .Ad.' Dean'" ~Academnic Dean
UABZE OF COCTSIM
I. I rI CrION *..................... ............ ...... ........ S
- II. THIEORETIC7 ANALYSIS ...... ].......
III. DES JRffIfN OF APPARATUS AND EXPER-MVJAL PROCEDUE ....... 17
A. APPARATUS ............ ........................... 17
B. TEST PROCEUIXRE . ........ ...... ...... .............. 30
IV. PRESENTATION OF RESULTS AND CONCLUSION .................... 35
A. EXPERIMENTAL ?RESULTS .................................. 35
B. OONCLUSIONS .................................... ... 45
APPENDIX A: Experinqntal Data ............ .................... ... 47
APPENDIX B: Wave Force-Height Plots ............................ 71
APPENDIX C: Ccoputer Progra .................. ......... ........ 83
LIST OF EFERENCES . ...................... s .......... ....... .. . 85
INITIAL DISTRIBUTION LIST ........................................ 86
FORM DD 1473 ........................ ......... .... ......... 87
3
LIST OF FIGURES
1. Definition Sketc& 12
2. wIave Cbhnnel ....................................... 18
3. Wave Ganerator Paddle ....................... ....... ....... 20
4c Wave Generator .... *............................. ..... ... . 21
5" Test mj4dule ....... 0. ........................................ 24
6. Horizontal Force Cantilever .................................. 25
7. Vertical Force Cantilever ............. ...... . ......... 26
8, Force Cantilevers .... . ... 27
9. wave iieight Probe.. .............. 0.....I..............0........... 31
10 i Wave Height Circuit Schenatic ........ ....................... 32
11. Cantilever Calibration Arrangement .................. 34
12. Wave Height-Force Traces h/a = 9.0 ........................... 36
13. Wave Hcight-Force Traces h/a = 7.0 ........................... 37
14. Wave Height-Force Traces h/a = 5A5 ........................... 38
15ý Wave Height-Force Traces h/a = 4.0 ........ o......39
16. Hoý;izontal Force Ccefficient . ... .... ........... ............ 42
17. Vertical Force Coefficient ................................... 44
18. Tbrizontal Force Coefficient ........................... 46
4
LIST OF MIBOIS
Sybol Quantity Units Dimension
a Padius of Cylinder ft L
b Cantilever Base Dimension ft L
Cd Drag Coefficient
CL Lift Coefficient - -
Cm Added Mas:; Coefficient - -
E Dibdulus of Elasticity lb/ft 2 F/L 2
F Horizontal Force lb F
F Maximum Horizontal Force lb FPxMax
Horizontal Force Coefficient - -x
F (+) W -) Vertical Force (Upward) (Downward) lb F
m Yaximu Vertical Force lb Fy max
f (+) (-) Vertical Force CoefficientY (Ulard) (Downward) - -
g Acceleration of Gravity ft/sec2 L/T2
h Water Dept-h ft L
H Wave Height ft L
"" Cantilever Beam Height Dimension ft L
k Wave Number (2ir/L) 1/ft I/L
K Stiffness lb/ft F/L
L Wave length ft L
£ Length of Cylinder ft L
b Cantilever Beam length DihensiL.. ft L
5
P Cantilever Load lb F
t ime or DuXration sec T
T Wave Period sec T
u Velocity in X-direction ft/sec L/T
dLocal Acceleration ft/sec2 I/T 2
E Strain ft/ft
p Mass Densi( l- hb-sec 2/ft4 F-T2/L4
1 Fluid Viscosity lb-sec/ft 2 F-TA 2
SWave Angular Frequency 1/sec 1/T
Velocity Potential ft 2/sec L2/T
6
The author wishes to thank the many peole who contributed of themselves
to this work. While there were runy, several standout, bissrs. George
Baxter, Joseph Beck and Thomas Christian of the Mechanical Engineering
Departm•nt who constructed the apparatus. Dr. Edward B. Thornton acted as
a counselor and friend throughout the project. Mrs. Deedie Sutherland
converted the handwritten scrawl into a finished manuscript.
Dr. Clarence J. Garrison, the advisor, is gratefully acknowledged for
his supervision and encouragemnt.
Finally, for Alice, my wife, Michael and Paul Perkinson, there is a
very special thank you for enduring.
7
I. INDUL-TION
Exploration of the ocean floor has received increased i.nterest in the
past several decades, particularly the area of the continental shelf. A
significant amount of attention has been devoted to offsnore oil explo-
ration, to the recovery of petroleum and to deployment of undersea
habitats. Often petroleum is located in remote areas of the world and
oil production involves use of very large submerged oil storage Lmks and
associated pipelines. In other locations petroleum has; been piped ashore,
through the surf zone, to shore facilities. Recently, interest has been
demonstrated [Il in the deployment of large subnerged nuclear power
plants. In the reference cited it is proposed that a nuclear plant be
placed on the ocean floor in two hundred and fifty feet of water.
As one might expect, this activity has generated considerable interest
in the interaction of gravity waves with suhmerged objects. For example,
the wave force exerted on vertical piling has been a topic of many investi-
gations during the past twenty years [21, [3], [4]. In these studies the
well known Morrison Equation [4] appears to be the most practical approach
to the calculation cf wave forces. This equation combines both a Crag and
inertia term, the valbes of which must be determined experimentally. Ar-
best, these two expe-tiiental - .ýfficients are dependent upon the geQ:M_ cry
of the pile a anplitude of the fluid motion, and it would appear that
twenty yea.s of research has only partially delineated this interdepender.:e.
In the case c'f large, deeply subnerged objects the morrison Equation
is also app1.72Žale and generally yields valid results for the horizontal
wave forces piov'ided diffrL..2.ion effects do not become appre.iable. Inthis ira tn :-:: Ine dj.--arded since the hydrodynamic force
i• ~8
. '. . ....._. __._
is pri-..arily inertial in nature. However, it is still necessary to know
the added mass coefficient as it depends on the body configuration and
the effect of the rigid bottan surface. Moreover, if the object is
bottan mounted the vertical fluid acceleration is zero, and accordingly,
the Morrison Equation yields zero vertical force, a result that is
obviously invalid.
A diffraction theory, valid for the calculation of forces exerted on
large suh1rged or semi-submerged bodies of arbitrary shape, which accounts
for both t:he effect of the free surface and bottcmn, has been developed by
Garrison and Chow [5]. Theoretical results were campared with corres-
ponding experimental results for to different practical su1nmerged tank
ooiifigarations and good agreement was found. Ibwever, the method is valid
for linear incident waves only; the nonlinear effects of large amplitude
waves can not be determined.
In the case of large objects, which are not so deeply submerged that
the free surface has a sizable effect, no theory is available which is
valid for large values of wave .'oight and few experimental results have
been published. However, one study has been. carried out by Johnson [6]
for horizontal wave forces acting on a bottaw-mounted, horizontal
- circular cylinder in long waves. Inasmuch as only long waves were con-
sidered, the results were dependent on the wave height and water depth
only and no consideration was given to intermsdiate wave lengths. More-
over, rather than correlate the forces with appropriate dimensionless
paramaters, Johnson used a regression analysis to correlate the forces.
onsequently, little light was shed on t/U. basic mechanism involved.
t9
Schiuler [7] measured wave folces acting on a sube~rged horizontalcircular cylinder due to rather small anplitude waves. Over the range of
wave heights considered in his tests, he found the amplitude of the hori-
zontal force to vary linearly with the wave height. At small wave heightsthe vertical force was found to vary harmonically with the wave motion,
the maxin •, amplitude of which increased linearly with the wave height.
At larger values of wave he:' jht, the vertical force amplitude tended toincrease in proportion to the wave height squared. This latter variation
of the wave force is caused by the velocity squared term occuring in
Bernoulli' s Equation.
The present investigation was an extension of Schiller's work to
include large wave heights. However, the present investigation was limi-
ted to the single configuration where the horizontal circular cylinder
was placed on the bottom. A transition section was placed in the wave
channel used by Schiller, at a location twenty feet fram the wavegenerator for purposes of providing a natural transition from intermediate
or deep water waves to shallow water waves of finite amplitude. The beach
slope was adjusted for negligible reflection as discussed i. the section
on experimental apparatus.
10
II. THEORETICAL ANALYSIS
The problem under consideration is depicted in Figure 1. A train of
regular waves of height H is considered to progress in the positive x
direction in water of deirth h. It is the present interest to determine
the horizontal and vertical components of wave force acting on the
horizontal circular cylinder in contact with the rigid bottom.
The exact analytical solution to this problem is extremely difficult
and, therefore, only an approximate analysis is attempted. However, it
is first instructive to carry out a dimimnsional analysis of all of the
pertinent parameters. It is known a ptioti that the maximum horizontal
or vertical force per unit length of a cylinder is dependent upon the
following variab)es:
F For__ya = f(h,a,L,H,pg,ji) (1)
£ 9
in which
H = wave heightk = cylinder lengthh = water deptha cylinder radiusL = wave lengthp = fluid densityg = gravitational acceleration
= fluid viscosity.
It may be noted at this point that a relationship exists between the
parameters associated with the incident wave (i.e., H, T, L and h, where
T denotes the period). Consequently, only t.-hree of the incident wave
parameters are needed in the dimensional analysis, i.e., either H, h and
L or H, h and T. Movreover, either of these th-.ee sets of parameters
1Ii
21
completely describes the incident wave regardless of ehiether or not it
is of small amplitude. For small amplitude waves the well-known
relationship
2 7T 1 Tanh! 2.h (2)
gT2 L L
exists between the parameters which describe the incident wave. For waves
of finite amplitude the relationship is more cmplicated but nevertheless
exists.
A dimensional analysis 6f the physical paramaters indicated in
Equation (1) gives the. following dimensionless parameters.
ý max Y =f (22a h H i (3)pga 2 k pga 2z L a 2a pvgih )
As pointed out by Garrison and Chav [5) the l&t term on the right hand
side of Equation (3) represents the ratio of the Froude number to
Reynolds number and, accordingly, is an indicator of the ratio of the
viscous to gravity forces. If this ntmiber is small, corresponding to
large scale flows with small viscosity, it may be assumed that this
parameter may not be important since the flow would be controlled primarily
by gravitational forces. In this case Equation (3) may be written as
FF 12 h
x max or -y max - f 2 --, h H4)
pga 2 z pga 2 z L'-- a 2a
13
SW-ever, as noted previously, the wave period may be used in place of the
wave length, in wAich case Equation (4) would appear in alternate form as
F Fo°r f Tor =~P-~--'-1(5)
pga22 pga29 h a 2a1
Equation (5) may also be written in a third form as
F F [92L 2 T hx a o ma x = f --- -If 1 (6)
pga2 h pga2 9 h H a)
In the experimental program an attempt is made at determining the corre-
lation of the dimensionless parameters, on the right hand side of
Equations (4), (5) aWd (6), with the force coefficients.
The complete analytical solution to the wave interaction problem is
quite complex. However, by use of Mzrrison's Equation [4], an approxi-
mate representation may be carried out. For this purpose, the velocity
potential associated with the incident wave is written as
H g Cosh k (y+h) Cos (k) at) (7)
2 a Cosh kh
where k = 24/L denotes the wave number and a = 2A•/r denotes the frequency.
FrAcm Equation (7) the horizontal ccnponent of velocity and acceleration
is obtained, respectively, by differentiation as,
u j- Hk Cosh k(y+h) Sin(kx -at) (8)2 a Coshkh
H gk Cosh k (y+h) Cos (kx - at) (9)
2 Cosh kh
14
The horizontal force acting on the cylinder is written, accotding to
rorrison's Equation, as
FX =Cd p 2a£ u2 + (1 + Cm) a2£6 (10)
2 m
where Cd denotes the drag coefficient and m denotes the added mass
coefficient. Hawever, for cases where the particle motion is small in
comparison to the cylinder diameter separation effects are small and it
is possible to disregard the drag contribution to the total force and
write Equation (10) as
Fx - ( + Cm) _k H Cos ( at) (11)
pga 2 £ 2 Cosh kh
Using the definition of k and a, Equation (11) may be written in terms
of the dimensionless parameters indicated in Equation (4) as
F_ _
Fx Max - = (i + Cm) 2wa7r(2-- .2•a h (2
pga 2 k(H/ 2a) L Cosh (-3-- (2
The vertical component of the force could be expressed in a form
similar to Equation (10), but the vertical ccaponent of acceleration on
the bottom is zero. It is supposed, therefore, that the vertical
cponent of force is associated with the lift force only. Accordingly
the vertical force may be written as
F Pu2 C 2aZ (13)y 2 C
15
in which CL denotes the lift coefficient and u the wave in, iced particle
velocity on the bottom. Substituting for the velocity fram Equation (8),
Equation (13) may be written as
Fy =2C ,ua !L2 Sin2 ort (14)pga2 Lsinh 2kh
Equations (12) and (14) contain two coefficients, namely the added mass
coefficient, C%, and the lift coefficient, CL. Values for those coeffi-
cients which account for both the bottcm and free surface are not
available in the literature. However, if the water is relatively deep
the free surface effect should be small. In this case the exact potential
flow value for d circular cylinder in contact with the rigid bottan has
been calculated by Garrison [8] as
C =2.29.m
For the same geometry, Daltnan and Helfinstein [9] have calculated the
lift coefficient by an approximate method. Their value is
CL = 4.48.
These values of the two coefficients may be used in Equations (12) and
(14) to obtain approximate expressions for the horizontal and vertical
force coefficients. It should be noted, however, that the assumptions
underlying these approximate relationships are met only when the scale
of the waves, i.e., the wave length and water depth are large in ccnpari-
son to the cylinder diameter. Moreover, since linear wave theory has been
used to determine the velocity and acceleration, the validity is also
restricted to small amplitude waves.
16
III. DESCRIPTION OF AoPPAAUS AND EXPER1T7E L PI_2EI
A. APPARATUS
In order to measure the wave forces acting on the horizontal cylinder,
experinents were carried out in a small wave channel. The basic wave
channel, as shown in Figure 2, consisted of three sections; a deep wrater
section, a shallow water section and a transition section, each fifteen
inches wide with a rectangular cross section. The overall chlamel length
was sixty-four feet.
Three quarter inch exterior plywood sheets were used for the channel
sides. Vertical rigidity was provided by bracing the plywood sheets with
2 x 4's at various intervals, every four feet in the shall, w water section,
every foot in the deep water section and every two feet in transition
section. IWo 2 x 4's were attached to the top of the vertical studs and
extended the length of the channel.
Since the bottll of the deep water section was resting on a concrete
foundation, only a single section of plywood was used for the channel
floor. A double floor was used in the shallow and transition sections.
This was done to provide extra strength. The double floor was constructed
fran two plywood sheets and two and a half inch separators. The sides of
these sections were bolted together, through the spacing of the double
bottcm, with three-eighths threaded rod. Steel angle spacers were placed
across the channel top to maintain the width dimension.
Prior to assenbly all wooden sections exposed to water were water-
proofed. The channel interior was then given two coats of Morewear
Vitri-Glaz 1320A. Dow Corning Sealant 780 was applied to all seams and
joints.
17
54Itif
It
It
If
If
If
Iif
IfIf
Ii
If
'I
If
It
F-A-
18
1. D"eep ater S.. .. tio. n
Sevzeral n-eths of increasing -qwave height and steeprss are
available. One of the most canmn, iethods is the generation of vrav in
a deep wzater regie and allo. then to transition, by shoaling, to a
sh.ac Tu+eri regiin. ay the proper choice of depths and shoal slope, the
transition can be such that there is no reflection from the shoal, and the
resultant shallow wvater •aves are in a near breaking condition. It was
this mathod of inc-reasing irie height that was chosen. As shown in
"Figure 2, the deep water section was sixteen feet long and four feet deep.
A wave generator was located tzxn and three-omurters feet from the closed
erd of the channel. In order to generate regular waves in the channel a
paddle type w-ave generator was 'used. *An aluminum plate, adequately
reLnforced: w-as hinged at the wave chixxnel flcor and attached by a pin
cnnec-Lion at the ton to a driving rod, Figure 3. One-eighth inch teflon
sheeting, four inches by fifty-one inches was located along the vertical
Sides cf the plate to prevent leakage from one side to the other. In the
space be-hind the wave paddle the vave motion was danped by use of baffle
plates. A tqo horsepaqer variable speed drive, with an output speed
range of twen4y tr. one hundred and eighty revolutions per minute, was
mounted on the top of the wave channel and fitted with a six and three-
varter inch radius face plate. The face plate was provided with ways
-o that the wave generator driving rod could be adjusted to any
eccentricity between zero and eight inches, Figure 4.
2. Transition Section
As previously noted, the wave height was increased by shoaling
deep waber waves. To accomp? ish this a 14.14 foot long ranp was
constructed and installed between the deep and shallow sections. This
19
4'v
I ~ ~ ~ ~ ~ * /.I~ 3.M,.AO PDL
'b-a
3 ~~'D- -
i qW
A;
provided for the transition -frcn deep water to shallkw water. In order
to reduce the amount of reflection from the transition raq., The slope
w-as set at 1:5. The change in depth due to the shoal was tao feet.
3. Shallai Water Sectau.
75ýe shall.ci water section had a depth of two feet and a length
of foyty-two feet. The test module was located twnty four feet from the
transition; thus, allowing enough length for the waves to reach- a fully
developed state follwing the transition, before reaching the test section.
At the end of the channel was located a variable slope beach for purposes
cf dissipating the wave motion. The beach cons;isted of metal shavings
held between two sheets of perforated nretal. These were separated by
two inch u.oden spacers. A solid sheet of one-eight inch aluminum was
attached to the bottom of the beach by three and five-eighths inch
separators. On the exposed surface of the beach was attached, parallel
to the wave fronts, one by one inch aluminum angles. The maximum slope
of the beach was 1:7.
a. Test b4dule
In order to provide a means of locating the test cylinder and
measuring the wave forces acting on it, a separate test module was
designed. The test module, consisting of the circular cylinder along with
its omi walls and floor, was constructed as an integral system in order
that it may be removed from the channel. This concept allowed for the
maintenance of close tolerances during manufacture and provided for easy
access. To accept the test module, the wave channel sides and floor were
recessed and made of plexiglass.
22
r. -- -- :••c - Tr•••:•• 'o• - --- -- -- -*•-''- -- -• - • • "'•"•÷ /• • • : "' '• • • • •° JY
SThe four inch diameter plexiglass cylinder was suspended
betyeen the walls of t1 test module by use of cantilever beams and
adjusted to approximately one-sixteenth inch above the floor, Figure 5.
It was knorm a pkiot that this sixteenth of an inch clearance would
cause a high velocit-y jet under the cylinder due to wave action- To
prevent this, a flexible plastic barrier was installed in the gap. The
barrier was held in place by 1'' ing material pressed into 0.007 inch
slot in the cylinder and the txzst module floor.
In order to susrxend the cylinder a small distance off the
wave channel floor and at the same time measure forces exerted on it by
surface waves, cantilever beam mounts were used as supzorts as shown inFigures 6 and 7. Buikheads were fixed in the cylinder at approximately
two inches from each end, and the fixed ends of the cantilever beams
bolted on these bulkheads. The free ends of the two beams were allowed
to protruze slightly beyond the end of the cylinder and into the plexi-
glass tb.st nodule walls. The free ends were si'pported by small self-
aligning ball bearings pressed into the plexiglass. Both beams were
fictua with strain gages and waterproofed using BIi Barrier 'C',
Figure 8. One of the beams with its largest cross sectional dimension
harizontal was used to measure the vertical force and the other with its
largest cross ectional dimension vertical was used to measure thej fhorcizontal force. Calibration tests showed the cross coupling to be
negligible.
This design presented sane unique ptcblems. The cantilever
beams had to be flexible enough to allow measurement of the forces and
stiff enough so that the natural frequency was large in canparison to
23
Io 6
4,A
I 1 24
6 , r
Vl)JURE 6. llORlU:(X!AL ~ ~.x:ir'
Os 41Cr
- &C6/
ao.
St I1C
rrJ
e CO
the excitation frequency. The characteristics of the strain gage bridge
a )d amplifier determined the minimum strain necessary to produce a
measurable output. The strain resulting on a cantilever from a given
load is
6PQ= (15)
where
= strain
P = load
S= cantilever beam length
E = modulus of elasticityb = cantilever beam base dimension
hb = cantilever beam height dimension
The relationship for the spring constant of a cantilever beam is
4h•3K = -(16)
By subsfituting Equa'.si (15) into (16) a relationship for the spring
xoonstant, in terms of the strain and cantilever dimension is determined
K p3 (17)E3E2b2h3
It is apparent that the maximum stiffness and, consequently, the maximum
natural frequency is obtained by making b, h and c as small as possible.
The minimum value of b was set by the width necessary to mount the strain
28
gages. The minimum value of e was determined by the maximun sensitivity
of the amplifier/recorder. The maximnm stiffness was then achieved by
making h as small as possible. However, Equation (15) shows that the
beam length decreases with b and h as expressed by
6P
Thus, the lower limit on h was finally determined by setting a reasonable
minumn value for the beam length.
In order to prevent the cylinder from rotating, an offset arm
was connected from the horizontal force cantilever to the test nodule.
A Farber bearing, AMS IM7, was used as a wheel at the test module end of
the arm to reduce friction.
4. Wave Height Probe
There are two types of probes in ccnmon usage for measuring wave
height; these are ccmmonly known as a capacitance and a resistance type
probe. The resistance type utilizes two vertical wires placed a small
distance apart; the amount of wetting of the wires varies the resistance
between them. This active element or "variable resistance" is wired
into one leg of a Wheatstone bridge and the signal read out on an
amplifier/recorder. Experience has shown, however, that the resistance
type probe is extremely sensitive and well suited to measuring small
amplitude waves but is rather nonlinear at larger amplitudes.
The capacitance type wave height probe, although less sensitive,
tends to be linear over large wave amplitudes and was, therefore, used
as the primary method of sensing wave height in the present study. The
29
probe consists essentially of a single insulated wire which acts as a
variable capacitor; the wire acts as one plate and the water the second
plate. As the water surface rises and falls, the capacitance varies
accordingly.
The probe used in the present study is shown in Figure 9. Located
at the botton of the foil-like support member was a three-eighths inch
diameter acrylic rod. The sensing wire was terminated at the bottan in
the acrylic rod and at the top at a nylon block. The tip of the acrylic
rod was made removable in order to facilitate changing the sensing wire.
A number of wire types were considered and tested prior to
selecting number 30 A.W.G. wire with polythermaleze insulation. This
wire provided the capability of measuring wave heights to less than one
thirty-second of an inch with good accuracy.
The electronic circuit utilized with this probe was obtained
in sddimatic form from Hewlett Packard and is shan in Figure 10. A
Hewlett Packard carrier amplifier 350 was used with the probe circuit.
B. TEST PROEDUME
With the channel filled to the desired level, the test section and
wave height probe were set in place. In each instance the probe was
immersed to a depth of nine inches. The arplqifiers were then adjusted
to balance the bridge.
With the use of a transverse mechanism the calibration of the wave
probe was accomplished. The probe inmersion was varied over a seven
inch range in half-inch increments. The gage output was measured both
on the Hewlett Packard recorder and one channel of a Brush recorder.
This provided the initial calibration information. Prior to and after
each set of runs the calibration was checked at several points.
30
IT
I &
A .R
.~44
N" 'y
A~ %
31
m1 PRJBE .l
o1. pf
_:Olpf
500AO. i 5 00 T
1oo00 - 0
' I - -- -ii I'V I
ALL CABLES ISHIELDED -.. .
H.P. Model 350-1100CCarrier Preamplifier
Tr: LINE-PLATE TRANSFORMER (UTC 0-7)
T2 - LINE-PLATE TRANSFORMER (UTC 0-12)
FIGUPE 10. WAVE HEIGHT CIRWUIT Si-2AmTIC
32
- - - - - - - - - - - - - -
The force cantilevers were calibrated by exerting a load on the
cylinder by means of a series of veights and a wire pully arrangment as
shon~ in Figure U1. Ile force was transmitted to the cylinder through a
series of tapped holes located around the circumference at three, nine
and twelve o'clock. Calibration information was obtained from the Bash
S- - recorder output by loading the cylinder in half-pound increments. As
with the wave height probe, calibration was checked prior to and after
each series of rmns.
With the calibration completed, the wave generator speed was adjusted
for the desired wave length. The wave height probe was located an integer
number of wave lengths from the cylinder. Its output was connected to
the horizontal grid of an oscilloscope. A second probe was then located
directly over the cylinder and its output connected to the vertical
oscilloscope grid. By observing the scope pattern the first probe could
be adjusted to be in phase with the waves at the cylinder. The second
probe was then removed and a data run ccamced. During each run the
wave generator drive eccentricity was varied from a half-inch to eight
inches, or to a point where the waves broke in the channel.
33
FIGUM 11.CANTILEVR CALIBP7ATI~~ AIRPANGMMT
34
IV. PrazIkT.TON OF I-ULTS .AND O2CI0SICM S
A. YE"ORDI7L IMUDTS
ZAs discussed in Theoretical Analysis, non-dimensional wave forces can
be represented as functions cf relative water depth (h/a), relative wave
length (2-a/L) or period parameter (h/gT2) and relative height parameter
(H/2a). In order to ascertain the effect of varying these parameters on
the horizontal and vertical ccaponent3 of wave force a test program was
established which, for a given run, held water depth and period constant
while varying the wave height. Four series of runs were conducted att different water depths corresponding to the relative water depths (h/a)
of 9.0, 7.0, 5.5 and 4.0. For each of these water depths, the relative
wave length (27ra/L) was varied over a range frcim 0.06 to 0.48.
Duie to the difficulty in presenting all the wave force traces, a
series of representative traces is shown in Figures 12, 13, 14 and 15.
These traces represent three wave lengths from the three deeper water
depths (h/a = 9.0, 7.0 and 5.5) and two wave lengths for the shallavest
water depth (h/a = 4.0). For each of the eleven runs, four wave heights
arranged in increasing order are shavn. The uppernmost trace represents
wave height, the middle vertical force and the bottom trace is horizontal
force.
The wave height traces show that the waves were generally sinusoidal
in nature. This was especially true at the shorter wave lengths. At the
greater wave lengths the wave trace departed frcn the sinusoidal form as
the wave height was increased. As can b3. seen in Figure 12, for the
greater lengths, as the height was increased the wave was characterized by
sharp peaks and long flat troughs.
35
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39
Before discussing the wave force traces a brief discussion of the
mechanism involved in the wave forces is in order. In Bernoulli's
equation two contributions to the pressure acting on the surface of the
cylinder are recogni. ed, an unsteady contribution associated with the time
derivative of the potential to the first power (vertical forces) and a
term associated witk the fLuid velocity squared. The velocity squared
term tends to be symmetric about the y axis and, consequently, contributes
little to the horizontal force. The inertia term is therefore dominant,
in the horizontal forces, and increases linearly with wave height H/2a.
Examination of the representative traces shows that the horizontal force
varied nearly sinusoidally with wave height for all cases except when the
wave length became very large. In the case of very long waves, the hori-
zontal force peaks are separated by a null region which corresponded to
the wave ;rough. The maximum value of the horizontal force coincided with
the maximu water particle acceleration associated with the incident wave,
Bquaation (9) .
For the vertical forces, there are two regimes, one in which the
inertial forces dominate and a second in which the velocity squared term
drannates. The inertial contribution to the pressure produces both posi-
tive and negative values of vertical for-e. This term tends to be linear
in wave height H/2a and have a maximum value of uplift when the cylinder
is under the wave trough. The -velocity squared term produces only upward
forces due to the non-synntrical flow about the cýiinder. This component
of vertical force is a funiction of the wave heicht squared (H/2a) 2 and
peaks occur at the poia.ts of maximum wave induced velocity occurring at
both crest and trouc~h of the wave. Examination of the vertical force
traces shows that for the longer waves the force was regular and had a
40
frequency twice that of the incident wave. Tne double frequency was
caused by the velocity maximums at the crests and troughs. The maxr'ixn
occurring under the wave crest in the longest waves was apparently
attributable to the fact that the velocity under the crest of a shallow
water wave is greater than at the trough. Here the upward force was lift
dominated. At intermediate wave lengths the reverse occurred. As the
wave length i shortened the inertial effects gradually took effect and
the mkaximum occurred at the trough where the tuo components were additive.
Finally, at very short wave lengths the inertial force became docinant
and the traces showed a sinusoidal behavior; the lift force associated with
the velocity squared term in Bernoulli's equation became negligible.
1. Horizontal Forces
As noted previously, the horizontal force was nearly sinusoidal for
the entire series of data runs. It is therefore possible to characterize
the force by the maximum value. In order to examine the variation in the
horizontal force with varying wave height, plots were constructed of the
non-dimensional force coefficient fx (fx = Fx X/pga2 k) versus the
relative wave height H/2a. These wave force versus wave height plots are
presented in Appendix B. It is apparent from the plots that horizontal
force coefficient is characterized by a linear variation with wave height.
Inasmuch as the variation of the force coefficient was linear with H/2a,
for a given wave length, it is possible to characterize the horizontal
force by the dimensionless parameter fx/ (H/2a), the slope of the wave
force-height plots.
The paraueter, Fx raJ Pga 2 (H/2a)] is presented as a function of the
relative wave length (2Tra/L) in Figure 16. This figure also presents the
result predicted by Forrison's analysis, Equation (12). For the larger
41
4.0
h/a = 4.0
3.0 A h/a = 5.5• 3.0
0 h/a = 7.0S0 h/a = 9.0
2.0
• 1.0
0.9
"0.8
0.3
0.2-_
0 .1 -....
0.06 0.08 0.1 0.2 0.3 0.4 0.5
27ra/LFIGLUE 16. HOPIZaWTAL FORCE uEFFICIENT
42
values of relative wave length (2Tra/L) the experimental results and
theory are in good agreement. This is generally as expected because for
the larger values of 2na/L the waves are well represented by linear theory
and the effect of the free surface is reduced due to the greater water
depth. It may be recalled that these were the assumptions upon which the
Morrison equation is based. However, for smaller values of 2na/L, the
experimental results are consistently high. This is not unexpected as the
waves have beccme shallow water waves and are not well represented by the
linear wave theory. Also, in the case of the smaller depths the effect of
the free surface becomes important.
2. Vertical Forces
As previously discussed, the vertical force may display two regimes,
one inertia dcminated and one dominated by the force associated with the
velocity squared term in Bernoulli's equation. The assumption having been
made in Equation (13) that the inertia forces would be negligible, the
vertical forces would be a function of the velocity squared only and hence
a function of (H/2a) 2 . It is difficult to characterize the force vari-
ations with wave height by a single term as two i-ypes of flow exist.
However, having made the assL-nption that the force was a function of
velocity squared, it was decided to use fy/(fl/2a)2 as the characteristic
quantity.Values of Fy m/Pga2 £ (H/2a) 2 were plotted against the relative wave
length (2lTa/L) and presented in Figure 17. The values of this coefficient
predicted by a lift force only are also shown in this figure. The agree-
ment between the theory and experimental results appears to be good for
the greater water depth (h/a = 9.0). However, as the water depth was
decreased the agreement became poorer and the experimental data displayed
43
2.0 h/a = 5.5
0I- h/a = 7. 0
0 h/a = 9. 0
-'-- . !TEOtff
1.00.9
0.7
0.5 1 i " _", _
0.4
0.3
"• 0.2
0.1
0.09
0.07
0.05
0.04
0.03
0.02
0.01 ___ _
0.02 0.04 0.06 0.1 0.2 0.3 0.5 0.7
2na/L
FIGURE 17. VERTICAL FORCE COEFFICIENT
44
higher force values than predicted by Equation (14); apparently owing to
the effect of the free surface and the non-linear effects of the incident
wave. Also, as the wave length decreased, or the parameter 2na/L increased,
it may be noted that the agreement is poor. This is apparently due to the
fact that the inertia canponent becomes daminant and this contribution to
the force was neglected in Equation (13).
As waves are often characterized by period rather than wave length,
Figure 18 presents the force coefficient as functions of a relative period
(h/gT2 ).
B. CONCLUSIONS
Fran the experimental results the follaiing conclusions are considered
warranted:
1. The horizontal forces on the cylinder, resulting fran the incident
gravity waves, increased linearly with wave height.
2. The vertical force showed two regimes, one inertial daninated and
one velocity squared dominated. The inertial force regime occurred at
shorter wave lengths, while the velocity squared term predaninated at the
longer wave lengths.
3. The maximum vertical, force was always smaller than the maximum
horizontal force and the da-inward vertical force was nearly zero.
4. For greater water depths the theoretical predictions appeared to
give valid results for both horizontal and vertical forces in the short
wave length range.
5. Horizontal forces became independent of the parameter, 2na/L, for
greater wave lengths.
45It
' -'
10.0 h/e = 4.09.0 A h/a = 5.5
7.0 -0 h/a = 7.0
0 h/a = 9.0
5.0
4.0 - -
3.0
2.0 _
1.0 . _(6 0.9C14 _•-0.7-L' -
C 0.5 _ _ _ _CL 0.4 _
0.3 _ ______
•x\0.2 -- -
0.1 .004.007:01 .020.08
00.10i0.004 0.007 0.01 0.02 0.04 0.06 0.08 0.1
2na/L
FIGURE 18. HORIZONTAL FORCE COEFFICIENT VERSUS PERIOD PARAMEIER
46
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21 APPENDIX B
-j1.50 Run 5 f
27ra/L = 0.i18 f M44 •, 1.25
U 1.00
0.75 -
0.50 A
0.25 AA
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
4 Run 61 .5 0 - R a6
h/a = 9.0"4 1.5 2na/L =0.1530
4 1.00
0)
8 0.75
�0.50
• -• o~~.25- • • _. /
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
71
1.50 RLn 7h/a= 9.02lra/L = 0.250
-- 1.2543
o 1.00
0.75 0
r 4 0.50 c
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
1.50 Run 8h/a = 9.02na/L = 0.300
• 1.25
1.00444-I
1- 0.75 -
o0.50 -
0.25
•-•---• • F- - 7--. I- I I I
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
772
1.50 Ran 9
h/a= 9.02a/L = 0.402
"4 1.25
4.' 1.00
O• 0.75
0.50
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
1.50 Rhn 101 1 h/a= 9.0
2wa/L = 0.4774-4 1.25
.'• 1.00
0.75
So 0.50
0.25 -
k( • •---z• I I ! I I
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative ',ave Height (H/2a)
73
1.50 Run ih/a= 9.021ra/L = 0.084
44 1.25
4"T 1.00
8 0.75
04 0.50
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave height (H/2a)
1.50 Ran 17h/a = 7.027ra/L = 0.086
tH 1.25
"8 1.00tJ-
0.75
S0.50
0.25
SI ! I W I 1 _ . i... I
0.2 0.4 0.6 0.8 1.0 1.2 1.4 L.6 1.8
Relative Wave Height (H/2a)
74
i.50 Ram 18h/a= 7.0
1.25 2ra/L = 0.105. 1.25
D " 1.00
*0
0.75
0.50
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
1.50 - Run 19h/a= 7.0 0 027ra/L 0.145
!!.• 1.25 -
1.00
0.75 -
0.50
0.25 A
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
75
1.50 Rn 20I h/a= 7.0
"2r•a/L = 0.1961.25-
0 1.00 0
0.75o 0.50
0. 250, I II II
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
1.50 Ran 21h/a = 7.027a/L = 0.295
- 1.25
1.00
0,1
to0
0.50 -
0.25 -
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
76
1.50 Ram 22h/a= 7.02a-L = 0.34244 1.25
0 1.00
* 0.75
o 0.50
0.25
Zrrr--•-T• I I I,-0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
1.50 AMn 23h/a= 7.02va/L = 0.447
Q 1.25
1.00
8 0.75
S 0.50D14
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (.4/2a)
77
1.50 R- nh/a= 5.52a/L= 0.078
C 1.25
S1.001-
0.75-S0.50
-A0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
I0IQ1.50 - Run 27
h/a= 5.5 o2wa/L= 0.148
ij ~ 1.25 -
1.00
* ~0.750
i)
o 0.50 0
0.25 / A A
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
78
1.50 RM 28
h/a= 5.52-za/L = 0.195
44 1.25
~43
100
0.75 -
10.50 o0.25 -
6II0.2 0.4 0.6 0.8 11.0 1.2 1.4 1.6 1.8
Relative Wave Height (H!2a)
1.50 Run 29h/a = 5.52na/L = 0.302
t 1.25
1.00
0.75 -
o 0.50
0.25 i
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
79
1.50 - RM 30h/a= 5.52na/L 0.397
4 1.25
4J
4 1.00
U4- 0.758
0.50
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
1.50 Rn 31h/a = .*.52¶a/, = 0.512
4. 1.25
4J
1.00--t
0.75
0.50
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
80
1.50 - 35h/a= 4.02.a/L = 0.143
4 1.25
;• 1.00 0
0m 0.75
o 0.50
0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)
1.50 Ibn 36 0Ii/a = 4.0
S 1.25 - 2na/L = 0.19/7
4J
@ 1.00i0
0.75
0.50
A0.25
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8
Relative Wave Height (H/2a)I 81
1 . 50 Ran 37h/a = 4.0
. 27ra/L = 0.29444• 1.25 ,
m 1.001
0.75 -
,• 0.50-
0.25 - /9/ ~~AA
SI ! • I I __ _ _ _ _ _ _
0.2 0.4 0,6 0.8 1.0 1.2 1.4 1.6 1.3
Relative Wax.e Height (H/2a)
1.50 Iort 38h/a 4.02ra!L = 0.419
44 1.25
o -
0.75 0
7
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1-6 1.8
Relative Wave Height (H/2a)
82
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84
LST OF REFERENCES
1. Marble, R. W., and Robideau, R. F., Thermal Field Resulting Fran anOffshore-Suhnerged Nuclear Electric Dower Generating Station, paperpresented at ASmE winter Annual Mseting, 28 November 1971.
2. Harlewan, D. R. F. and Shapiro, W. C., Experimental and AnalyticalStudies of Wave Forces on Offshore Structures, Part I, ITHydrodynamics Lab. T. R. No. 19, May 1955.
3. Carpenter, L. H. and Keulegan, G. H., "Forces on Cylinders and Platesin an Oscillating Fluid," Journal of Research of the National Bureauof Standards, v. 60, p. 423-440, May 1958.
4. Morrison, J. R., Johnson, J. W. and O'Brien, M. P., "ExperimentalStudies of Forces on Piles," Proc. Fourth Conf. Coastal Council onWave Research, Berkeley, California, p. 340-370, 1954.
5. Garrison, C. J. and Chow, P. Y., Forces Exerted on a Submerged OilStorage Tank by Surface Waves, paper presented at OTC, Dallas,Texas, 1 May 1972.
6. Johnson, R. E., "Regression W4del of Wave Forces on Ocean Outfalls,"Journal of the Waterways and Harbors Division Proceedings of theASCE, v. 96, No. VM-2, p. 289-305, May 1970.
7. Schiller, F. C., Wave Forces on a Suhnerqed Horizontal Cylinder,M.S. Thesis, Naval Postgraduate School, June 1971.
8. Ukrrison, C. J., "Added Mass of a Circular Cylinder in C6htact witha Rigid Boundary," Journal of Hydronautics, v. 6, No. 1, p. 59, JanJanuary 1972.
9. Dalton, C. an. Helfinstine, R. A., Potential Flow Past a Group ofCircular Cylinders," paper presented at Fluids EngineeringConference (ASML), Pittsburgh, 9 vay 1971.
8
85