Note on the possible use of a perforated vertical wall breakwater

NOTE ON

_{THE POSSIBLE USE}

_{OF A}

PERFORATED, VERTICAL-. WALL BREAKA TER

By

G. E. Jarlan

Hydraujj.cs 'Laboratory _{National Research}

Council, Ottawa,

Canada

Introduction

with a solid, taking

Into account the viscàslty of the fluid than would be expeáted.

that In many cases sound penetrates

porous bodies more freely been the object _of

_exhistve

studies, particularly _{in acoustics.} Stokes and Kirchoff

Investigated the, motion of air in contact

arid the effect due _{to heat.}

They arrived at the conclusion

The problem of wave reflection at _{a porous wall has}

If a continuous _{perforated but flat,}

vertical wall Is placed _{perpendicular to the direction}

of propagation of

acoustic waves, part of _{the wave energy will}

be reflected wtiile

the remainder of the _{energy will be transmitted}

or dissipated through viscous friction and heat. _{The perforation}

may, for

Instance, consist _{of a series of uniform circular channels} of a diameter much smaller

than the wave length of the vibrations

imposed. _{VIScosity and heat}

losses may be _{neglected in a}

simplified stucy of the

characteristics of the reflected wave.

If the origin of _x

is taken at the face of the wall, the _one

dimensional equation _{of the wave}

propaga-.

L

tion may be written _[il

a =

eIoX

nt) ₊ BeI(koX - nt) (1) >x / where a =

ei(4x

+ nt) cos(k0x + nt)

Is the incident _{wave with k0}

e _{being the Sound velocity In air}

and

n _{the frequency.} The horizontal velocity component

_{Is given by}

u = c(_e 0X +

BsiOX

nt)

B

K(1)

- K0

K(1+g) + K0

-2-The ratio of u and a _{near the wall is found as a first}

approxImation by putting x O _{in (1) and (2) so that}

- _{, B being the reflection coefficient.}

For a thin wall, i.e. for _{a length of channel very} small compared to the _{wave length of the perturbation, the mean}

pressures are aproxiiately the sanie at both terminations _of the tube. If

(u

_{denotes the area of the unperforated part}

of the wall and cI'p the area of the perforated part, _{one has:}

n

6

c(B-1 (

6u

+ p)

which gives the relationship between _{the inside and tside} motion

Using the continuity equation,

(1

_(r

n a d 6 + K

J (

udr

= 0; 6=. cross section area

of the tube,

the expression for _{B, as derived by Rayleigh, may be obtained}

with m-

-where K

le a ñmction of

n and c, of the channel radius _r and of the speciflc heat of air. _{If one assumes K = K0 , then}

ni

2+m

for ni = 1, B = 1/3, and the reflection is small.

These results suggest that _{a similar phenomenon, as}

far as the reflection is _{concerned, mIght occur when water}

is

the fluid In oscIllatory motion,; _{Consider a chamber, one wall}

of which has a serles of perforations _{equally dIstant from each} other, placed In an infinite _{liquid.of depth equal to its width.} When the wave impinges s;aint theperfcrated wail, part of the

energy Is transmitted through

_the

_{circular holes, the rest} being reflected or dissipated _{through friction at the solid}

boundary. The chamber will then aJternately _{fill and empty,}

the oscillatory motion at the _{perforated wail being graduaJ.iy}

transformed Into a mass transport _{through the holes which}

forces the

lIq'Lid

to arrIve in the chamber in

_the

_form

of jets.

If

H

s the hydrostatic head rorrespondig

_{to the ma.lium of}

the crest elevation at the

ai.l,

the

velocity of the jet will

be

y =

_\J2gH

if H

Is coistant,

_The

_jet

pressure _Po to a pressure _{Pi through an orifice of} cross-section iill be accompanied by a change of _{momentum in the}

direction of the velocity0 _{It is difficult to analyze the jet}

problem 1f lt is to be studied as a manifold fibu with a

certain number of holes being submerged _{only part of the time} and where pressure fluctuations follow _{a harmonic law.}

Mass Oscillation in a Tube

The flow in a single tubé will now be examined, under the assumption that the motion Is _{laminar and the hole always} submerged, using a simple device which will _{help to retain the} physical principle Involved In such a casé. Two tanks filled

with water, open at the free surface, _{are connected by a tube,} sufficiently long with respect toits _{diameter. A plunger,}

subject to a harmonic motion, _{generates a forced wave in the}

tube. The system being initially at rest, the motion In the conduit results

from the superposition of an oscilla-tion proper to the system and of the

I forced oscillation. Experience shows

I that the first type of oscillation

I disappears after a while and that it is I the forcéd oscillation which Is of

oL_J

Interest. The liquid oscillates In every point of the cross-section of the

tube with .a period equal to the one

Imposed by the pistons. For a long and

narrow tube, one may observe that _{a phase difference tends to}

occur in the motion of the liquid.

With O X _{representing the axis of the tube, the}

Navier-Stokes equation is written as

Y:-!i'

J/Ò'V ,._/_V

òic

rer

t'SP'

where y is the velocity component along _{O X , is the} pressure gradient such that '- = , r Is an

arbitrary distance on the radius _{of the tube and t) is the} kinematic viscosity. V may be assumed independent of _{X if}

the tube is very long. _{The boundary conditions are simply}

y = O , r at the wall. exi(2) _{obtainéd for}

V the folowing express1on

which shows that the velocity _{in the tubé involves two}

terms of similar period and .amDlitude, out of phase by _{(R-r). A wave}

amplitude0 For R very large and neglecting _capillary zi

hysteresis, the relation _{gives V} =

. Hence, an

axial symmétric _{mass oscillation tak}

place in which the

velocity at the walls is _{greater than in the}

core (Richardson annular effect)

When the motion in the _{tube is turbulent, a jet forms} outside the tube. _{It is presumed that the}

theory of wakes may be applicab]e to the _{case of. the jet, completely}

self-preserving flow di _{pearing at a distance close}

to 50 diameters from _the orificeU'), The velocity distribution _{In the jet itself is} importánt if _{one wants to study its subsequent}

diffusion

within

the liquid0 _{Assuming the ratio of the}

Reynolds shear stress

to the mean velocity _{gradient to be constant,}

it Is possible to

use a velocity distribution _{similar to the}

one observed -In a wake. _{This velocity}

distribution may be represented by an error function with appropriate _{coefficients.}

The study of flow establishment (se _{Tolimlen) shows that the}

entrajnner1t o' the

surrounding fluid by the _{expanding turbulent region Is Inertially}

balanced because of a continuou3 reduction of _{the velocity}

with-in the jet Itself0 _{The shear exerted}

longitudinally as the jet

occurs may be defined theorétically

using the Prandtl _concept of mixing length. _{Practically, it Is}

impossible to measure the

eddy viscosity coefficient _{within the jet.}

If the circular

wall is not smooth, large _{eddy motion will take}

place originating

from the artIon _{f forces normal to the}

wall which In turn will

affect the edd- viscosity

and consequently the velocity gradient.

Across the surface of _{the hole, there exists an} average pressure from the incident _{gravity wave.}

This pressure acts as a driving force _{on the water in the hole,}

thereby causing variation in the level Inì _{the chamber.}

The flow In the tube being turbulent _{the resistance}

coefficient to flow must _be taken into acount.. _{Assuming the head loss}

term to be of the form

_r/2D(v/2

_{the differential}

equation of the motion of the free surface in the _{chamber would be}

tt

(i

2

L.b

where _{f = frictIon term, F}

driv±ng force. _{2 7 /T,}

-

= a distance,

y

vertical notion. _{This equatloc is not} easIly integrable because

_{of the nonlIner}

errn bu an

approprIate solution cn be obtamne _usir'

_{serles developmen}

and Rieman Int-egríition methods.

In order to fix the ideas, simplification may be made

whereby the above system _{may b,e represented by}

a niase-spring-resistance mechanical system _{subject to a harmoç4ç driving}

force F1 sin t. _{The equation of motion}

is

,<*wX zF,Sc-w

);

_,K=

'-t

A general expression _{for X is:}

x

,j f4#. (w t cp)

W.

,

,,-

2Kw

9

cú

t

4

The term q

= 1/

[:?

-1 #4Kc4/,3

w*)

-is the magnification _{factor. Plotting}

q _{against W , one}

obtains a.tuning _{curve which shows that the}

high damping _response

is fairly flat, _{After a calculation}

of the appropriate coeffi-cienta, being a purely resistance impedance, _{it is possible to} study the law of motion _{of the water level in}

the chamber.

Breaater

A large amplitude gravity wave reflecting _{on a plain} smooth vertical wall will _{create clapotis, the}

amplitude of

which may reach 190 percent of the incident _{wave height, the} accompanying wave set-up being _{of the order of 10}

to 20 percent of the same wave, Thus,

the pressures and subsequent dynamic loads exerted _{on the wall ere important,}

The stream lines of

the clapotis which

approximately represent the _{trajectories of} the water particles _{are such that the horizontal}

cponent of

the motion at the nose In the vicinity of

the structure are very important,

Consequently1 this horizontal force will tend to

exert in shallow water _{important shear stresses on the bottom,} Erosion may then occur which could endanger _{the stability of}

the vertical structure. _{Moreover, the breaking}

of the

clapotis is inevitable in

shallow water and the subsequent

elaStic forces developing _{In the structure}

end in the joints Is detrimental to _{good stability. A simple well-known}

rule

may thus be Inferred whereby

a vertical structure must _nevèr be placed In shallow _water.

¿p,

-6-One must then have _{recourse to rubble-mound} break-waters which dissipate _{the wave energy through}

mechanical

friction in the porous medium, _{the amount of energy passing}

through the breakwater being _{very small for short-period}

waves

However, in areas where _{no stone is available (and}

this occurs frequently In

Nova Scotia or Prince Edward Island) the construction of _{rubble mounds involves heavy}

transportation costs having a noticeable _{bearing on final cost and maintenance} budgets.

Consequently, it appears desirable to study the

feasibility of prefabricating a caisson which would offer _the

same advantages as rubble-mound _{breakwaters but would not} involve the disadvantages _{Inherent to the use of vertical} structures in shallow water.

In view of the above theoretical considerations, it

was felt that a caisson with

a perforated sea-side wall could

be used as a breakwater, _{provided that lt would}

not reflect

waves to a great extent and _{assuming that it could be}

adequately

designed to resist the dynamic loads exerted by waves. Two-Dimensional Model Studies

To study the conditions

of the reflection of the wave

against a perforated wall a parallelepipedic chamber was built Involving a plain vertical wall made of concrete to the scale 1/30 (FIg. i and 2) _{Tests were made In a}

wave flume, the equivalent prototype depth being 30 ft. The dimensions of te perforations _{were calculated In order to}

obtain a ratio

_{m = 0J9, the}

_length

and diameter of the holes representing about 1/loo of the _{wave length of a deep-water} wave of period 8 seconds. _{The impedance of the holes is thus}

sufficIent to ensure _{a noticeable phaeè shift}

between the wave

motion outside the chamber _{and the fluctuations}

of the water level Inside, _{The reflection coefficient was first}

investigated and it. was found that

this varied (for non-breaklng wave) as a

function of the wave _{camber, between 1G and. 20 percent.}

It was also observed that the phase shift between _{the maximum water}

elevation at the watt _{outside and inside the chamber varied} from 50 _{for small camber}

waves to 120° for high camber _waves,

the amplitude of the fluctuations In the _{chamberbeing always} smaller than the amplitude _{of the motiôn at the outside wall.} These results are _{indicative of' the}

effectiveness of the

Impedance for transforming

the oscillatory motion of the liquid into a mass transport. _{The horizontal}

component of the velocity

being largely reduced _{through jet diffusion,}

one maInly observes

in the chamber variation

of hy:rostatjc pressure somewhat similar to the one which _{would be observed in}

a surge tank

although the damping _{conditions of the level fluctuations are}

From this exploratry _{test enes lt can be inferred}

that the width of the _{chamber does not}

appear critical; in

-other words, the damping _{thaide the chamber was suffIcient}

80

that no resonance, similar _{to the case of the Helmholtz}

resonator, could develop. _{The selection of the width of the} chamber would thus _{appear to depend merely ori the wave heights} occurring at a given site.

From the fluid mechanics _{viewpoint, the}

_conditions

of dissipation of the _{et diffusing into a liquid niass subject}

to an oscillatory motion _{la a challenging problem and lt is not}

known how much the _{dissipated energy dampens}

the energy of the incoming wave. Howeve _{lt could be observed that with} suffi-ciently high wave camber,

the wave motion at the wall was

some-what reduced as compared _{to the incident wave height.}

The

diffusion of energy Int _{the liquid mass creates an extra} turbulence near the wall which is prone to exert a damping

effect on the wave front _{close to the perforated}

wall. It was also observed that as jets penetrated Into _{the chamber,}

bubbles of entrained _{aIr were carried down}

into the water by

the jet. _{The presence of this}

air should somewhat _{affect the} amplitude of the vertical motion of the free _{surface In the}

chamber (FIg, 3 to 9).

Bed Movement

Observations in a two-dimensional _{fltvne showed that,}

as the chamber empties,

a fairly Intense current perpendicular to the wall Is

_Induced

_{by the jets in the vicinity of the free}

surface. _{Although all holes involve}

a jet action, the latter is more important _{near the surface, which Is to}

be expected.

Hence, according to the _{principle of continuity,}

there exists

a circulation which takes place _{in a vertical plane.}

Some exploratory tests _{were run in connection with} the transport of material _{under the form of}

a bed load. _These tests showed that the _{material which penetrated}

Inside the

chamber was evacuated In

the vicinity of the surface through the upper holes, _{The turbulence induced}

In the chamber by _the Jets appeared to be sufficient to permit the solid particles

to be lifted by vertical upward currents taking _{place In the}

chamber, _{The material used in}

these tests _{was a fine sand of}

medIan diameter equivalent _{to 0.1 nni.}

Remarks

The presence of this surface current would be an asset in areas where ice

floes occur during storms since the drift induced by the jets _{iould tend to damp the}

momentüm acquired by the ice under _{the influence of combined wind and}

Run-Up

Figures 10 to lL. show _{a two-dimensional impact of}

a

wave against a plain vertical _{wall, the Incident wave height}

being the same as the _{one imposed for Figures}

3 to

5.

It may

be seen that the shape _{of the free surface is considerably}

different from the _{one observed in the case of}

a perforated

wall. _{The tests showed that}

the run-up over the structure

was not very large, even in the _{case of waves at the wall}

breaking partially0

Erosion at the Toe

Placing the structure _{on a rubble-mound mattress with}

natural slope, it was possible to observe that the stability

of the mound was not _{Impaired,. No stones were displaced}

for wave heights of 15 to 18 ft0, the total depth of water being

equivalent to

₃₆

ft. _{The average weight of the stone units}

was

equivalent to 200 lbs. _{No strong pumping action}

could be observed at the toe.

Combined Breakwater and Quay Wall

On the basis of the flume _{results It is feasible to} conceive a structure which could be used as a combined

break-water and quay unit, as indicated in Figure

_15.

_{Such a} struc-ture could be built 10 or

₁₅

_{ft0 above the highest}

tides. The design should involve bracing of the chamber made of _porous wall so as to allow equilibrium of the level to _{take place}

rapidly for a wave Impinging _{at the wall under a given incidence.} Such a structur3 should _{be competitive, in spite}

of the possible

difficulties involved in _{order to prevent Important}

shear

stresses from occurring around the holes and to take _{care of} accompanying phenomena such _{as transient cavitation developing}

at the walls of the holes,

BIbliography

1.

_Rayleig.

_{"Theory of Sound".}

(Dover) , New York.

20 Sexl "tTher der von E0G0 _{Richardson entdeckten Annular}

Effect". _{Zeitschrift der Physik,}

1930, p. 3L9.

30 Townsend _{"Structure of Turbulent Shear Flow".} Cambridge University Press.

¿ To1lrnien W0 _{"tTher die Entstchu.ng}

der Turbulenz, Nachr, Wigs., G8ttingen _(1929).

5

MacLachian, NW

_{"Theory of Vibrations".}

(Dover) New York.

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