l•q naval postoraduate school · experimental values of the hori- ... ponding experimental...

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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

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FROLE WT ROLE WT ROLE WT

Gravity Wave Forces

lWave Interaction with Underwater CylindricalBodies

Capacitance Wave Height Probe

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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

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

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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

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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

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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

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.~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' -

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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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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

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