DEPARTMENT OF THE NAVY

BUREAU OF YARDS AND DOCKS CONTRACT NOy~l2561

John H. Carr P r oject Supervisor

Interim Report

July 1953

Hyd:t adynamic s Labora'tory Hydraulic Structures Division

California Institute of Technology Pasadena, California

Rep o r t prepared by: John G . Elli ott

LIST OF SYMBOLS

d = still water depth

H = wave height

L = wave length

t = time

T = wave period

v = horizont;J.l component of particle velocity

x = horizontal di s placement of a petrticle

z = mean orbit depth below still water

I. INTRODUCTION

The forces experienced by a body fixed in an unsteady fluid flow are

of two kinds; (1) drag force, which is due to the viscous nature of the fluid,

with magnitude p1·oportional to the square of the fluid velocity, and (2) in­

ertia force, which is a consequence of Newton's Second Law, with magni­

tude proportional to the fluid acceleration. Wave motion is a particular

case of unsteady flow in which the velocity and acceleration vary periodi­

cally with time, hence a theoretical expression for the force on a body,

such as a pile, in the presence of wave n>otion may be written:

where CD and CM are the drag and inertia {or mass) coefficients respec­

tively, p is the mass density of the fluid, A is the cross-sectional area,

and V the volun·1e of the body, and the velocity and acceleration are written

as functions of time to emphasize the time -dependency of the total force.

In order to determine the maximum force on the body, therefore, it

is necessary to maximize the entire expression with respect to time, and

it will be recognized that this requires a detailed knowledge of v(t) and

a(t).

In recent years, investigators at the University of California have

made much progress in the development of rational methods for computing

the wave forces on structures of engineering significance by experiment­

ally determining values of CD and CM. In this work, however, the as­

sumption was made that the time -dependent functions, a and v, could be

closely approximated by various the ore tical expressions.

It was the purpose of this investigation, therefore, to apply precise

methods of measurement to the correlation of experimentally-determined

velocity and acceleration time histories with accepted theoretical values.

It was not expected that major differences would be discovered, but in

view of the complicated nature of the expression for total force, it is ap­

parent that even small differences of amplitude or phase could result in

appreciably different maximum forces.

A further consideration in the undertaking of this interim project

was the availability of the major equipment required (channel, strobo­

scopic lighting, photographic) and personnel while waiting for the fabrica­

tion of the wave force balance.

-2-

II. EXPERI]v1ENTAL TECHNIQUE

General

The particle motion studies were performed in the 3-ft deep, 2-ft

wide and 100-ft long laboratory wave channel. The wave generator in this

channel is of the inverted ballistic -pendulum type which is capable of im­

parting particle motion to the water which is of the proper horizontal di­

mensions from surface to bottom. To absorb the wave energy and to pre­

vent unwanted reflections, a gravel beach was installed with a slope of

fifteen to one at the west end of the channel. A light-tight shed (Fig. 1)

was constructed over the channel to house the photographic equipment used

in taking the trajectory photographs . This shed was about 45 ft from the

wave machine where there is 1/2-in. plate glass on both sides of the channel.

For c onvenience, the shed had access from both sides of the channel. The

portions of the channel outside the shed were covered with 15-lb building

felt to exclude stray light.

Globules

Particle motion may be photographed by placing in the water an

opaque globule of ma~erial of the same density as the water. This glob­

ule must be large enough iind white enough to be readily discernible on a

photograph and yet STJlall enough compared with orbit dimensions to be­

have as a molecule rather than as a solid body. For the pu:-pose of this

study a particle diameter of 1/16 in. was d ete rmined to be the desirable

size .

The globule material fpund most satisfactory was a mixture of

Alkazene •n* (sp. gr. 1,2.94) and heptane (sp. gr, 0.684) with zinc oxide

added for whitene ss. The specific gravity of the mixture was regu~ated

by adding a few drops of heptane to lighten it or A.lkazene 47 to make it

h eavier. Droplets of this mixture were discharged into the center plane

of the channel by means oi a hypodermic syringe (Fig. 2) which had a long,

fine metal tube with a tip of 0.024-in. internal diameter. The vertical

position of the tip was adjustable , thus permitting a complete range of

particle depths.

The procedure use d to release a single particle into the water was

to push enough fluid through the tip to form the d e sired size of particle 1

*A Dow Che mical Company product .

-3-

and then to simply pull the tip away leaving the globule at its initial depth.

Figure 3 shows a globule on the tip just prior to being released.

Photographic

To record the orbital motion of the globule, multiple exposure photo­

graphs using stroboscopic lights were taken at rates of twenty and thirty

exposures per s ec ond. A 4 x 5 view camera was securely fastened to a

special fix.:!d mount and the photographs were made through the glass wall

of the channel. The focal distance fron> camera to the center plane of the

chann~l was 4 ft. The procedure in making an exposure was to open the

shutter for one complete wave cycle while the stroboscopic lights wer~

flashing at a prede t e nnine d fixed rate. The timing of one wave cycle was

made convenient by a sp~cial timing light conne cted to the wave machin~.

The stroboscopic lights w...:re mounted in 12-in. long tubes which beamed

the light and prevented stray light from reaching the film. The lights

w e re plac~d above the c hannel to give overhead illumination of the globules .

The stroboscopic lighting equipment was loaned by the Hydrodynamics

Laboratory of the Institute campus where it had been d~veloped for cavita­

tion research, projectile sinking rate studies, etc. The: equipment employs

a 4000-volt powe r supply and thyratron pulser circuit to furnish an input

of 2 watt-seconds of energy per flash per lamp. The lamps are General

Elec tric FT-125 zenon-filled tubes, and the flash duration is 3 to 4 micro­

s ;.:c onds.

The grid (Fig . 3) which was used on all the photographs was obtained

by double exposure. A special frame supported the grid in the central

plane of the channe l. Illumination was from the back and was obtaine d by

the use of two photofloods. To minimize the number of ins tallations of

the grid, several films were exposed with each operation. After these

exposures, the grid w:.ts r<.!lnoved, thus clearing the way for the orbit

globules which were plc:..ceJ in th~ same plaue as the grid had previously

occupied . The grid was made by etching a 20- x 24-in. exposed and de­

vclope d photographic plate on the emulsion sid,:. Clear lac quer protected

the e mulsion while: the grid was in the water.

Wave Height M eas uremc:nts

The wave h e ights corrc sponding to each trajectory photograph w e r e

obtaine d by the use of two subme rge nce elements placed upstrearn and

Fig

. I

-D

arkr

oom

fo

r tr

ajec

tory

pho

togr

aphs

Fig

. 3 -

Glo

bule

p

rio

r to

rel

ease

F

ig. 2

-G

lobu

le

inje

ctio

n as

sem

bly

-4-

downstream from the t e st section of the channe l. The wave h e ights were

recorded as usual on an oscillograph, a manual coding device correlating

the wave records with the traje ctory photographs.

III. TEST PROGRAM

Since the investigation was an interim project, time limitations pre­

vented a complete investigation of all wave parameters and orbit depths,

so only three typical wave conditions were tested . These were the wave

lengths of 8.4, 14.3, and 24.7 ft with corresponding heights of 3.56, 5.42,

and 5.80 in . , respectively, the water depth in all cases being 24.0 in.

S even orbit depths we re observed for each wave length, so that a total of

21 photographs was necessary for the entire series of runs. To obtain 21

sharply focused photog1·aphs, however, it was necessary to take over 50

photographs in all. This was due mainly to the fact that it was difficult to

make the globule stay within the very small depth of field of the camera

lens. Best results were obtained by releasing the globule in still water,

starting the wave machine, and then making the one-cycle exposure im­

mediately after the wave became stable.

The small depth of field of the lens was an advantage in determin­

ing whether the globule was in the plane of the reference grid. At 4-ft

focal distance, the depth of field of the 8-in. Ektar l e ns was one -half inch,

and since only globules within this range were analyzed, the e rror of hori­

zontal displacements was less than one percent. Becaus ~ this is the main

source of e rror in obtaining the trajectory photographs, it is believed that

the experimental technique as described in this report is a valuable con­

tribution to experimental wave r esearch.

-5-

IV. ANALYSES OF TRAJECTORY PHOTOGRAPHS

Figure 4 shows typical trajectory photographs. Since the time be­

tween the "dots" is equal, the change in the horizontal component of veloc­

ity is readily apparent due to the change in the horizontal spacing of the

dots . To measure the horizontal spacing accurately, the original negative

was placed on a comparator and viewed through a microscope; by reading

tangents to the dots, horizontal measurements could be obtained to

±0. 0003 in. on the film. For a typical measurement this error amounts

to one -half of one percent. With the dimensions of the grid known, the

absolute globule displacement in the channel was computed quite accurate­

ly. Similarly, the mean depth of the orbit could be obtained simply by

noting its position on the grid.

The displacement of the globules can be expressed as:

where x 1, etc., are displacements from a datum line, the velocity as:

dx v = = ..

dt

wht:!re At is the time between light flashe s; and the acceleration as:

= (xl - xz) - (xz - x3)

At

By t a king first and second differences as indicated above, velocities and

accelerations were computed and the results are discussed in the next

section.

• . . . .

Fig. 4 - Typical trajectory photographs

-6-

V. RESULTS AND DISCUSSION

Velocities and acc e lerations as functions of time were computed for

e ach of the 21 test runs. Figure 5 shows typical plots for each of the three

wave l e ngths teste d, the com;_:>arison curves being those given by the Stokes

irrotational theory. Approximate equations for the velocity and accele r­

ation as given by this the ory1 • 2 • 3

are:

dx =

dt

+

+

d2

x =

dt2

TTH h 2'T!' (d + z)

c o s L cos

T sinh 271' d

L

LT (sinh 2.TT d ) 2 L

[-(7TH) 2 h 4'Tt'(d+z)

cos L

2LT 2'T!'d 2

(sinh L )

1

2

2'T!' 2

H cosh 271' (d + z)

+ L

T2 sinh 271' d --L

471' 3H2 [- 1

LT 2(sinh Z~d ) 2 2

3 +

4

sin

3 +

4

h47T(d+z) c os L

271' d 2 (sinh L }

271' t

T

h 47T(d+z) c os L

(sinh 271' d/ L J

sin

4TTt

T

47Tt ---T

The point where t/T = 0 . 50 w a s us e d a s the common point in drawing the

e xpe rim e ntal and the or e tical c urves .

A plot of the maximum positive and negative acc e l e rations and v e ­

locities against orbit d epth ratio (Fig . 6) shows v e ry good agreement b e ­

tween the e xperin!ent and the the ory of Stokes. As was e xpected, the

agreement was b e st in the case of the shorte st wave l e ngth (Fig. 6a}. For

this wave l e ngth the Stoke s e quations r e duc e to nearly a sine and cosine

40

0.0

I N

'20

u ., .. ';;"

• v

.2 0

~ "' Oi u :I2o I li

40

0.0

0.0

4

0

N'2

0 u ., ~

r:::

+

c:

0 -~

0

~0

0 0

~-

~ .!! "' ~20

40

0:0

0.2

5

t/T

0

.5

y ;-:

--..0

........ l'h

. o•

n c•

"

I u

•

1\.

"' 0

~

0"

(a)

z =

I 2

.25

in.

d =

24

.00

tn

T =

1.

35 s

ec

H =

3.

56in

. L

=

8.4

It

I I

I 0

.5

t (s

ec)

I v

v v

0 0

_....

-: I

u 0

I>

~~ -~-

----

--- I

o E

xpu

imen

tol

Oot

o

0.75

--0 --

'1>

lv

"<>

't-."

'

"" /

..........

. .....-

1.0

0.25

..,/ ~

0 .,."'

"' 0

, 0

0 ~

0 .-

'o

1.0

o-~

v ...D

1.0 2

0

v 0

:I 0

I

u .. .. ._ c:: o: ·;:; ~

0

.. >

20

~

0

"(~

..

400.

0

~0

u ., ~+

c:: "" .2 0

~ ~-

, ..

/.

v

~-o

u ~2

0 I

40 0

0

t/T

0.

5

v ./

0

v 0 ~r:

.....

.

~~

~

0 c

0

--<

0

0 0

-I-

-0

""' ,"

' 0

0 0

[S,

0

(c)

~

z =

9.

95 in

"' d

= 2

4.0

0in.

T

=

3.2

0 se

c H

=

5.8

0in

. L

= 2

4 7

It

I I

I

--

Acc

eler

atio

n

!(

sec)

Velo

city

0.2

5

t/T

0.

5 0.

75

/c ...

..-r)

-.......

. ~ "\

0 -

, 0

() ...

.. .._

, ...

.

; c

o-

, , ~

' a'~

,•

r-c',

"~

0

,"'

0

" ll'

,,.

..... ,

u (b

) ~~

0

v z

=

16.3

7 ln

.

"' v

d =

24.

00in

. T

=

2.0

0se

c ' ~

..JV

v H

=

5.42

1n.

L=

14

.3

It

I I

__

_ ! _

_ ~ _

__

L_

_

0.5

t (s

ec)

1.0

0.7

5

i ....

.. 0

... 0 ...

_ ... 0

... ... 0

... ... ... ~

~

0 .....

.. /'

_Q_

"

0 v

?I~

c-o

./

;--~

0 ~ v

0

2.0

1.5

-

0 c

0

,.....

o•

n 0

-C

C

-0

3.0

1.0

0

f-"-'

<-

20

0

u ., .. ._ +

-~

o--

-~

0 20

u 0 Oi

>

Fig

. 5 -

Com

pari

son

of h

oriz

onta

l ve

loci

ty a

nd

acce

lera

tion

with

Sto

kes

theo

ry

./'

I.020

v

10 u ., ~ c::

0::

;.

"'-c

. ~-o

-

"' u 0 ., >

10

--

20

2.

0

)(

0 ~

40

20

+ 0

20

40

"' + ~ 0 "' ' .S

c: 0

~20 ... QJ u u <t

)(

0 ~40

40

20

+ 0

20

f--

-

-

f--

f--

f--

f--

-

-

40 00

I I I <....._ (a) u --T = 1.35 sec

d = 24.00 in. H = 3.56 in.

L = 8.4 f t

~-- - o - --<1--- - o - ---

-t--

·~

()

I I I - 0

(b) T = 2.00 sec d = 24.00 1n.

H = 5.42 in.

L = 14. 3 ft o-· -- (J'-

o- --· o--

0 ' () -I I I - 0

(c) 0

T = 3 .20 sec d = 24.00 in .

H = 5 .80 in . - -- - rr 0 L = 24. 7 ft

o-. -- - o

'-' 0

.10 .20 .30 o Experimental Data

- 0 ~

- o 1- - o - -- o--

--o 1- - o • --- o--

0 0

-- .,._ __ --- --~ -- :-o-- ~-- ""tl -

0 0

0 0

-- 0 - --- - 0--

1--- ~<r- --- - o-

z/d .40 .50 - - Acceleration

0

o •- ~--

o--1- --o

~

0 -"

~ w 0

- "'0 -- - o- 1- o-

0 0 0

- - ~ -- - 0 - - -· 0

- -· o-- - o- - --o

1,-, v

.60 .70 - ---Velocity

Fig. 6 - Comparison of maximum horizontal velocity and acceleration with Stokes theory

.80

-7-

function {Fig. 5a).

While the agreement of the maximum value was good in all cases,

the position of the maximum acceleration deviated by about 20 degrees

from the theoretical for the 24.7-ft wave (Fig. 5c). This difference oc­

curred at both the plus and minus maxima and was apparent at all the

orbit depths of this wave length. It should be noted that the second ac­

celeration maximum {Fig. 5c) for the 24.7 -ft wave was very apparent in

the experimental data.

Figure 4 shows that the orbits of the globules do not close in the

vertical or horizontal direction. The direction of horizontal nonclosure

is opposite to that expected by mass transport and, therefore, must be

attributed to "closed" channel effects. Since the horizontal nonclosure is

small compared with the total o1·bit diameter, the gap was neglected in

the results reported here. Any vertical nonclosure of the orbit must be

attributed to a globule which is rising or falling due to incorrect density.

This vertical nonclosure, however, will have little effect on the horizon­

tal displacements except as it may slightly affect the orbit depth. For

this reason, the vertical gap was also neglected in analyzing the photo­

graphs.

-8-

VI. CONCLUSIONS

An approximation of the Stokes wave theory was compared for

several wave conditions. Keeping in mind the limited nature of the in­

vestigation, the following conclusions seem justified:

1. The theoretical and experimental horizontal particle ve­

locitizs and accelerations show very good agreement for waves

of d/L > 0. 14 and H/L < 0. 035.

2. For waves approaching d/L = 0. 08, a shift can be expected

in the location of the maximum acceleration although the value

of this maximum may still agree with the theory.

3. Maximum values of horizontal velocity show agrecm\::nt down

to at least d/L = 0. 08 without any appreciable shift in location.

In summary, it may be state d that the Stokes equations should give

good values for horizontal particle velocity and acceleration down to at

l east d/L:: 0. 14 for waves of ordinary steepness. As expected, best

agreement between theory and measurement occurs with the shorter

waves. Since, however, the tests de scribed begin to show disagree­

m~nt between theory and experiment for the longer wave s, it appears that

future investigations should be undertaken to test the longer waves more

thoroughly. The technique described in this report might also be applied

to the investigation of waves of large steepness near the breaking point,

where accelerations are high and available theories are more in question.

-9-

REFERENCES

l. Stokes, G. G., "On the Theory of Oscillatory Waves 11,

Trans. Cambridge Philosophical Society, Vol. VIII, 1 84 7 • p. 4 41.

2. Wiegel, R. L. and Johnson, J. W., "Elements of Wave Theory", Proceedings of First ConferencE: on Coastal Engineering, 1951, Council on Wave Research, The Engin(;ering Foundation.

3. Beach Erosion Board, "The Mechanics of Deep Water, Shallow Water, and Breaking Waves", Tech. Memo. No. 40, March 1953.