E
Vefiical walls and low reflection
alternatives
Resultsof waveflume tests
M W McBride
N W H Allsop
P Besley
D Colombo
L Madurini
ReportlT 417
April 1995
E
L
HR Wallingford
Add.€€ end Regbt€.6dOltlcs: HR Wrlllnglord Ltd, Hollbery PerlqWelhglord, C)xonOXIO8BA
Tol: +4.4 (0)l4Sl 835381 Faxr+ 4,f (0)1491832t33
F.did*
h Eratna fld lsaroce, Hn wdhgtod t. . rtEty ffd
lbdd.ry
.l ltR wtahcbrd ourc Lld.
E
Contract
The work describedin this repon was funded by the Departmentof the
Environment
ConstructionDirectorate(DoE) underresearchcontractPECD
7ld298,'The hydraulicdesignof harbourentrances"and by the Department
of Transport. The DoE nominated officer for this contract was Mr P B
Woodheadand HR Wallingrford's
nominatedotficer was Dr W R Whhe,
ResearchDirector. The report is publishedon behalfof lhe DoE but any
opinionse)pressedare nol necessarilythoseof the fundingdepa ment. The
researchdescribedin lhis reporlwas caffiedout in the Portsand Estuaries
Group,managedby Dr J V Smallman,and the CoastalStructuresSection,
managedby ProfessorN W H Allsop,at HR Wallingford.The HRlob number
was DAS 0042.
Preparedby
(Jobtitle)
Approvedby
/.
tr.
I*-.*.-
0i*d'
o^,...,i,o...[he.\11}..
O
HR WallingfodLimited1996
tT4i7 20r'0e96
tr
Summary
Verticalwallsand low reflectionalternatives
Resullsof waveflumetests
M W McBride
N W H Allsop
P Besley
D Colombo
L Madurini
ReportlT 417
ADril1995
Wavesreflectedfrom vertical breakwalersor olher highlyreflectivestructures
formingportentrancescan causeseriousproblemslo vesselnavigation.This
oftenleadsto portclosuresas vesselsare unableor unwillinglo entsror leave
the port.
This reportdescribesall aspectsof a 2D physicalmodelstudy. This study
wasinitiatedto examinethe reflectionand oveftoppingpedormance
of various
'low reflection'
structuresunder iffegularwave attack. lt is these types of
structureswhich can be used in the modificationor constructionof port
enlrancesto reducewave reflections,and so improveconditionsfor vessel
navigation.
Four mainslructuresweretestedconsi$ingof a smoolhverticalwall,single
and double chamber wave screens and rock armour slopes. Various
aarangementsof these structures were tested to identify lhe optimum
configurations.
The studyfoundthat the use of singleand doublechamberwavescreensis
effectivein reducingwave reflectionsahd overtopping. Fromthe analysisof
these results,an empiricalformulawas derivedwhichoan be usedto descrbe
the reflection pedormance of such structures, within pmctical engineering
limits.
This report also describes a new technhue to examine the effect of 'low
reflection'structureson wave conditions. This was canied out through the
analysis of local wave $eepness. Prwiously a desoriptionof the reflec{ion
peformarrceand ove oppingof a struclurewas usedin the assessmentof th€
likelyeffectof wavereflec.tions,and hencethe r€lativechangesin waveheight,
on vessel navigation. However, vessel navigalion prcblems are not only
relatedto wav€ heightas the wav6 period is also significantin the creationof
hazatdousconditions. This new tecfinique enables both relalive changesin
wave heightand u/aveporiodto be considered,allowinga betterassessmed
of the effect on vessel navigationto be made,
tT 417 20/05'96
E
Notation
A"
a,A,b,B
B*
B/1"
c.(r)
cr
Dh
Dnso
f
h
h"
t\
H"
H"i
H*
FI
H.
H"(x)
Hu,o
lr-
k",K, t!
L
l*
l_
Mso
n!
o
o'
o.
o#
Re
R"
R/H,
f
sm
s"
s.(x)
sm
si
s,
T^
Armour crest freeboard,relativeto SWL
Empiricallyderivedcoefficients
WiJth of structureor elem6nt,usually width of voided
chamberor spacingbetweenpeforatedand solidscreens
Belativechamberdeplh
Reflectioncoefficient,definedH",/H",
Refleclioncoefficienlfunctionoverfrequency
Transmissioncoefficient,definedHo/H,,
Holediamelor,in wavescreen
Nominalparticlediameterdefined(MJpoJt"
Wavefrequency,inverseof waveperiod
Water depth
Water depth seawardof toe of struclure
Toe depth
Significantwave height,averageof highestone-thirdof all
wave heights
Incidenlsignificantwave height
Offshoresignificantwave height
Reflectedsignificantwave height
Transmittedsignificantwave heighl
Significantwave heightat measuremenlpoint 'x' metres
lrom slructure
Wave height, averageof highestone-tenthof all wave
heights
lribarrennumber= tan o/s-05
Coelficientsin reflectionpredictionequalion,equation(2)
Wave length
Wave length in the local water depth h
Wave length in the local wat€r depth l\
Medianunit mass,generallyof armourunits
Porosity(by area) of screen,area of holesas a percentage
of total screen area
Mean overtoppingdischarge per unit length of structur€
ms/s.m
Goda's dimensionleesdischargeparamoter= q(2gH!o105
Owen'sdimensionlessdischargeparameter= a / fi, g HJ
Franco'sdirnensionlessdischargeparameter= @(gH,t)o'
Reynolds nunber, usually defined in terms of nominal
armourunit dhm6ter,Dn,or scr€enthickness,t.
Struclurecr6st fro€boardrelatlveto SWL
Relativecrest height
Regressioncoeflicient
Mean wave steepness,defined2nl-lolgTn,'
for deep water
Peak wave steepness,defined2nF{,/gTo2
for deep water
Mean wave steepnessat probe, x metres sealvardof the
struclure
Mean irrcidentwave steepness
Incidentspectralenergy
Reflected spectral energy
Meanwaveperiod
lT 4t7 20/05/96
tr
Notation Contd
Tp
L
x
a (alpha)
B (beta)
A
p
p,,p",p,
Peakwave period(usuallyoffshore)
Screenthickness
poinl(waveprobe)fromstructure
Distanceof measurement
Structureslopeangleto the horizontal
Angleof waveattack,relaliveto lhe norrnalto struclure
Surfsimilarityparameter,or lribarrennumber,= tan o / s05
lribarrennumber= tan c / s,oE
Fractional density of (eg) armour with respect to (sea)
water
Massdensity
Massdensityof rock,concrete,or (sea)water
E
Contents
Titlepage
Contract
Summary
Notation
Contents
Introduction
1.1
Background
1.2
Scopeof work
1.3
Outlineof reoort . .
1
The physical model . .
2.1
Modelconstruclion
.....
2.1.1
Vefticalwall
2.1.2
Wavescreens/ peiorated caissons
2.1.3
Rockarmourslooes. . . . .
2.2
Wavegeneration
2.3
Datacollectionand analysis
2.3.1
Watersuiace elevations
2.3.2
Wavereflections
2.3.3
Waveovertopping
2.3.4
Wave pressures
2
I
1
2
J
.t
3
3
4
6
Descriptionof tests
3.1
3.2
6
6
6
Waveconditionsand calibration
Test methodology
Discussionof reflectionresults
4.'l
4,2
Verticalwalls
Wave screens/ perforatedcaissons
4.3
Rockarmourslopes. . . . .
9
Discussionof overtoppingrcsults .
5.1
Verticalwall
5.2
5.3
11
5.1.1
Analysisusing offshore wave heights
5.1.2
Analysis using inshorewaveheights
5.1.3
Sumnaty of resultsfor verticalwall
Wave screens/ perforatedcaissons
Rockarmourslopes
.....
Dfscusslon
of wat$ su acecondltlons....
S c a f €s f f e c t s
8
11
Concluslons
and Recommondations
8.1
Conclusions
8 . 1 . 1 R e f l e c t i ao n a l y s l.s. . . . .
8.1.2 Oveftopping
analysis
8 . 1 . 3 W a t esr u i a c ec o n d i t i o n s
R e c o m m e n d a l i.o. .n. s. .
11
11
12
12
14
,...14
........17
..,.17
..........17
.......17
....18
........19
.......19
rT41729re5/96
tr
. .t.s.
Acknowledgomen
10
References
......
19
.........2o
Tables
Table 1
Table 2
Table 3
Table 4
Table 5
Waveconditionsand wavescreenspacingsin termsof
B/L.
Test programmo
Wave reflectionsfromverlicalwall,Tesl Series100o
Coefficients in Equation (2) for tested screen
configuralions
Resultsof watersuface analysis
Figures
Figure1
Figure2
Figure3
Figure4
Figure5
Figure6
Figur€7
Figure8
Figure9
Figure10
Figure11
Figure12
Figure13
Figure14
Figure15
Figure16
Figure17
Figure18
Figure19
Figure20
Figure21
Figure22
Figure23
with bathymelry
Modelconfiguration
Model caisson with double chamberwave screen al
B. = 0.575m
Perforatedwave screen,n" = 2o"/o
Verticalwall with rock annourslope
Compadsonof refleclionanalysismethods
Overtoppingof simple vertical walls, comparisonof
Herben(1993)and Goda(1985)
Influenceof relativecrestheighton reflec'tions
Reflectionsfrom singlechamberwith hole diameters=
screenlhickness,n"=20%,curve1, eqn (6a)
Reflections
fromsinglechamberwithholediameters= 2
x screenthickness,n.=20%,curve2, eqn (6b)
Influenceof parl-deplhfloors,waterlevelat +1.61m
Reflectionsfrom doublechamberwith hole diametem=
2 x screenthickness,n!=2096,curue4
Influenceof part-depthfloors,waterlevelat +1.43m
Reflectionsfrom doublechamberwith hole diametes =
screen thickness,n"=20ol",curve 3
Beflectionperformanceof smoothand arnoured slopes,
afterAllsop(1990)
Reflectionsfrom armouredslopes in front of verlical wall
Piled quay ov6r armouredslope on partdepth caisson
Reflectionsfrom altemativecoal be h quays
ComparisonbetweenHebert's resultsand Series 1000
Simplevertft:alwalls,Series1000,Owen'sequation
Simpleverticalwalls,Series1000,Franco'sequation
Frarpo's €quation using offshore and inshore wave
heights
De Waal's data with Owen's equation,A=0.002 and
B=26.76
De Waal'sdata with Franco'sequation,a=0.03and b=
-2.O5
rr arT 20/05196
tr
Contents continued
Figure24
Figure25
Figure26
Figure27
Figure28
Figure29
Figure30
Figure31
Figure32
Figure3tt
Single and double chamber wave scroen structures,
Series2100,3100and 3200
Wave screen slructuEs,large holes,Series4200 and
4100
Wave screen structures,B*=0.40m,Series 2200 and
3Kr00
Caissonchamberswith raisedfloors. Series 500 and
6000
Ovenoppingof verlicalwall with armouredslopes
Watersudaceconditions- Verticalwall
Watersurfaceconditiom- Singlechamberwavescreen
Watersurfaceconditions- Doublechamberwavescreen
Watersurfaceconditions- Rockarmouredslooe
Analysis of scale effecls for wave screens tested wilh
smallwave heights
tr
1 Introduction
1.1 Background
Previousworkby McBrideet al (1993)discussedproblemsarisingfromwave
refleclionsin harbours.lt wasfoundthat theseproblemswerecausedby the
use of vertical or near verticallyfaced structureslo form harbourboundaries.
This work also reviewedvarious iypes of 'bw reflection'structureslor which
some perforrnancedata were known. That review was followed by studies
usingphysir:aland numericalmodelsof harbourenlranceswhichis reponed
by McBrideet al (1994 and 1995). Those studiessuggestedthat wave
disturbance prcblems in many harbour entrances would be substantially
reduced if wave reflectionsfrom lhe primary harbour struclures were kept
below 40"/.. The reflection performanceof most vedical brealoraters and
habour walls falls in the rangeof 90 to 100%,so substantialreductionin
reflectionsmay be required.
This report describes tests to quantify the reflectionand wertopping
performanceof vertical breakwatersand 'low reflectionafternatives',and
discussesthe analysis of the reflectionsand an initial analysis of the
ovedoppingmeasurements.
The workdescdbedin this reporthasbeenused
in lhe preparation
guidelines
ol
for the hydraulicdesignof harbourentrances,
McBrideet al (1996).
1.2 Scopeof work
The simpleststructureconsideredin theseflumelests was a smoothvertical
wall. The waveheigtrt,wavesteepness,and localwaterlevelwereallvaried.
Low reflectionstructureswere formedby mountingperloratedscreens in front
of the wall,or by addingan armouredrubbleslopein trontof the verticalwall.
The formerconfigurationsimulateda rangeof perloratedface caissons,some
of which had raised floors within the chambers. The latler configuration
simulatedthe additionol armourin fronl of an exislingwall,orthe substitlrtion
of an armouredslopg perhaps beneath a piled quay. The sections which
were tested are summarisedbelow:
Test Series
Struc{uretesled
1000
2000
3000
z!000
5000
6000
Vertlcal structureonly
Vertical wall and single wave screen
Ve ical wall and doublewave screen
Verticalwall and single/ doublewavescreen
Doublewavescreen,chamberfloors at 30%depths
Doublewave screen,chamberflooe at 30 and 50%
depths
Rock armour slope 'l:1.5. 2 ston6 berm width at
+1.71m
Rock armour slope 1:1.5,2 stoneberm widthat
+1.61m
7000
8000
tT417 20/0vS6
tr
1.3 Outlineof report
This report details all aspects of th6se llume lests. Chapter 2 provides a
descriptionol the physicalmodelconstruclionand layout,withthe detailsof
givenin Chapter3. Thediscussion
the waveconditionsandtest methodology
of the resultsis dividedinlothreechapters(4, 5 and 6), in orderlhat the thr€e
and waveconditions
overtopping
mainareasfor discussion,wavereflections,
with regardto navigation,are consideredindependently.Crossreferencing
betweenthese chapters has been used where it has been considered
appropriate.Chapter7 discussesthe potentialscale effectsand ChapterI
summarisesthe conclusionsof this work.
2
The physical model
2.1 Modelconstruction
TheZ-dimensional
te$s wereconductedin the DeepRandomWaveFlumeat
Wallingford.This flume is 52m long,and operateswith waterdepthsat the
paddlebetween0.8m and 1.75m. The flume is configuredto reducerereflectionof un-wantedwave energyfrom the test sectionby the use of
absobing side channels,either side of, and separatedfrom the central
channelby pedorateddividingwalls.
The bathymetryin front of lhe slructurewas formedin cementmorlarto a
uniformslopewitha gradientof 1:50,as shownin FigureL The bed levelat
the positionof the structurewas +1.00mrelativeto the waveflumefloor. In
frontof the test stnrcture,a numberof channelswere mouldedinto the sea
bed,parallelto, and seawardof lhe verticalwall. Thesechannelswereused
to securethe base of each pedoratedscreenat given spacingsfrom lhe
verticalwall. Examplespacingsbelweentwo perforatedscreensplacedin
fronl of the caissonare shown in Figure2, and the full range of screen
spacingsare summarisedin Table 1.
2.1.1
Verticalwall
The verticalwall was fomed using a caisson. The nuin caissonsectionwas
formedas a hollowbox in marineplywood,witha crestlev6lat +1.8025m,ie.
the crest was 0.8025m above the to€. The use of a hollow box enabled
equipmentto be installedinsidethe box,
overtopping
collectiory'measurement
reducingthe need for any extemal collectionor measurementdevices. The
seaward side of the box, which formed the vertical wall, was faced with a
stainlesssteel plateto aid the installationof pressuremeasurementequipment,
which was used in subsequenttesting to examinewave impact and uplift
pressures. The caisson section was only installed in the flume after wave
calibrations,so those wave measurementswere not contaminaledby arry
reflectedwaves. The caisson section was us6d alone for those iests on the
verticalwall.
2.1.2
Wavescreens/ pefioratedcaissons
Two pairs of wave screens were construded fiom marine plywood with a
thicknessof 25mm. Thesewerepedoratedto an areaporosityof n^= ZV/o.
The first pair of screenswere perforatedby circularholeswith a dhmeterof
25mm,equallo the screenthickness.The holesin the secondpairof screens
were lormed with a diameterof 50mm, twice lhe screenlhickness,as shown
in Figuro3. ldenticalpairsof screenswere constructed
lo enableteslingof
singleand doublechamberwavescreens. lA/heninstaltedin theirrespective
lT 417 20,05i€6
tr
channelsin the sea bed all screenshad a crest levelof 0.8025mabovethe
toe, +1.8025mrelativeto flume floor. The screenswere usedlo form voided
chambersof overallwidth,B*, of 0.575mor 0.40Om.Duing testson double
wave screens, the chamber was split into lwo chambers by a second
perforatedscreenal halfthe spacing,as shownin Figure2.
2.1.3
Rockarmourstopes
A comnronmodification
to sinple vedicalwallsin harboursis the construction
of a rubbleor rock armourslope againstthe wall. The roughand porous
andalso,
armourlayersdissipalewaveenergy,thus reducingwavereflections,
in marrycases,overtopping. However,for some combinationsof water level,
armourcrestlwel, and waveconditions,the slopecan leadlo an increasein
overtopping.Hence,rock armourslopeswere includedto explorethe potential
of addingsuchslopesin frontof an existingverticalwall. Previousstudiesby
Allsop& Hettiamchichi
(1988)andAllsop& Channel(1989),reponedby Allsop
(1990),providedetailsol reflec.tion
performancefor simplearmouredslopes
without wave walls or other retlectiveelements. Some indicationof lhe
potentialinfluenceof a vefiical wall behindthe slope are given by ad hoc
studiesof armouredslopesbeneatha piledquay,reportedby Allsop(1990).
Therefore, lwo rock armour slopes were lesled to examine the effects of
changinglhe armourcrest level,and hencethe relativearmourfreeboard.
Botharmourslopeswereplacedat 1:1.5 with armourwith a nominalparticle
diameter,D"so,of 0.074mm,and a creslbermwidthof 2D"ro.The lwo armour
bermlevels,whichweretesled,were+1.61m,see Figure4, and +1.71m,with
the caissoncrest at +1.8025m. Tests on this structurein sedes 7000 and
8000 usedwaterlevelsof +1.61mand +1.,ltlm.
2.2 Wavegeneration
Waveswere generatedby a slidingwedgepaddle,drivenby doubleacting
hydraulicrams. The paddleis computercontrolledusing softwaredeveloped
at HR Wallingford. This softwareenables either regularor randomwaves lo
be produced. The randomwave signals are generatedto malch any wave
spectrumthat can be specifiedat 16 equal frequenoyofdinates. JONSWAP
spec'trawere generatedfor all of the tests. The nominalwave heightsin Table
1 were generatedand measuredin the deep water sedion of the flume.
2.3 Datacollectionand analysls
2.3.1 Watersurfaceelevations
Measurementsof water suface elevationswere made using I twin wire wave
probes,locatedalongthe approachto lhe verticalwall. Speciraland statislical
anafysesallowedthe signifirxnt, Oj%, artd oth6r elitremevalues of the wave
height distril:utionto be determinedat each monitoredposition,together with
the m€anlvave periods. The measuremontsfrom three of these probeswere
used to delermine the reflection coeff'nient. descfibed in S€ction 2.3.2.
Analysis of total water eurfaceexcursionsand periodsfrom these and oth6r
probes enabledthe determinationof the likely effecl on vessel navigation,as
descrbedin Chapter6.
2.3.2
Wave reflections
Incidentand reflectedwave conditionswere measuredusing three of the twin
wire wave probes, hcated approximately2 wave lengths seaward of the
structure. Each test was run for 1000 waves and output from the 3 probe
array was analysed to give the reflection coefficientfunction, q(f), over a
rangeof wave periods,equivalentto 0.5 to 2.0T,, usingth6 methodof Gilbert
tt 417 2qO5/96
E
& Thompson(1978)basedon the approachof lGiima (1969). The overall
reflectioncoefficient,C,, was determinedby summingenergiesfor eachtest
condition.
'HR method',
Gilbert& Thompson'smelhodlor the reflectioncoefficient,the
has recentlybeencheckedagainstthe rnethodof Davidson(1992)at Plymodh
University,referredto as the 'Plymouthmelhod'. Davidsonarguesthat the
maindifferencebetweenthe methodslieswiththe mannerin whichtheyreiect
data which is conlaminatedwith singularities.Bolh methodswere usedto
analysewaves reflecledfrom armouredslopes in lhe same wave tlume at
Wallingrfod.The resultsof lhis comparisonare summarisedin the simple
comparisonshownin Figure5. Theseshowvery closeagreementoverthe
range tested. Two atternaliveregressionlines have been fitled, namely
y = 0.96xand y = |.04x - 0.o5. Bothtinesgavea regressioncoefficient,12,of
of
0.97. This simplecomparisonsuggeststhat the errorin lhe determinalion
about
5%.
less
than
from
either
method
is
likely
to
be
G,
2.3.3
Waveovertopping
the slructure,the
Duringmostol thesetests,the humberof wavesovertopping
mean
ov€dopping
discharges,
wave by waveovenoppingvolumes,and the
were each determined. The numberof waves overtoppingthe strLlcturewere
countedusingfour ovenoppingprobesspacedacrossthe widthof the flume.
The overtoppingdischarge was determinedfrom ovettoppingvolumes
measuredusing a weighingmechanismlocatedinsidethe caissonwhich
walerinto
formedthe verticalwall.A 100mmwidechutedirectedovertopping
of the weighingmechanismallowedthe measurement
thetank The sensitivity
of wave by wave ovenoppingdischarges. The mean overloppingdischarge
was calculaledat the end of each lest. When a high mean overtopping
dischargewas expectedduringa test,the weighingcell was rernovedand a
large reseruoirwas used to collect the water. The water was then pumped
inlo calibratedvolumetriccylinders.
Overtoppinqo{ vertical walls
The main methodto predict wave overtoppingof vertical walls is based on a
graphical method developed by Goda (1985). For a given approach bed
slope, m, and oflshore sea steepness, s". = ?rHJgT-z, a dimensionless
discharge,Q' = cy(2gH*t)05,is plottedagains hr/H-, whereH.. is the offshoE
significantwave height, tL is the water depth at the toe of the structureand Q
is the meanovenoppingdischargein nf/s per meirerun of sea wall.
Recenttests by Hedert (1993) have confirmedand extendedthis aPPtoach
for steeperwavesand shallowerbed slopes. These graphsare ploftedas bolines of Rd/H-, for examplea comparisonof Herbert's resuhs with Goda is
shownia Figure6. Dirnensionlessovertoppingdischargos,Q', ris€ to ma(ima
to the relative
aroundh,/H- = 1.2 to 1.7, which cone,spondapproximately
depthforthe maximuminshoresignmcar waveheight.Abore h'/H.o= 1.2to
1.7,lhe dimensionless
dischargeagainreduces,conespondingto the region
of depth-limited,and ther€fore, broken waves. lt is importantto note that
Goda's method relies on the offshore wave height, H-, the offshore wave
steepness,s",o,andthe estimationof a simplesea bed slope(m = 1:10,30or
100). The inshorewaveheighlis not usedin this meihod,but is impliedfrom
the offshorewaveheight,period,seabedslope,andthe wat€rdepthfiv€wave
lengthsotfshore.
tr4i7 2U05i/96
E
The rnain weaknessof these m€thodsis that the influenceof the bathymetry
can only be accountedfor by clroosingthe nearestsimple bed slope, m, and
intoryolating
betweengmphsforgivenwavesleepness,q* = 0.017,0.036and
0.045 withorjt dala for steeper waves. lt must also be noted thal these are
manualgraphicalrnethodsand thereforethey can not be readifyincorporated
intoothercalculationmethod,eg. risk or joinl probabilityanalysis.
Further wo* by Franco (1993) suggestslhat there may be a simple
relationship
betweenan alternative
dimensionless
discharge,Q#,and Ro/H.for
vefiicalwallsor perforatedverticalwalls,with and withoutcrestdetail,whore
a# = O1np"sys. The work was based on physir:alrbdel test results with
wavesteepnessin the rang60.018< sm< 0.038. Franm (1993)showedthat
overtopping
of a vedicalwallcan be describedby a simpleempiricalformula:
Q#=aexp(bRd/H*)
wherea=0.2andb=-4.29
(1)
Overtoppinqof slopinq walls
Studiescarriedout in the UK on a wide range of slopingwalls inlhe lale
1970s have shown that it is usaful lo express the discharge, Q , a s a
dimensionless
parameterQ*, and the cre$ freeboardas R', where:
o.=d/(T.sH")
(2a)
R'= R"/ fi- (sHJo')
(2b)
R" is the crest freeboardabove still water level, H. and T, are the significant
wave height and mean wave period respectively. These paranFters were
usedby Owen(1980)to suggesta simpleempirbalrelationshipbotweenQ*
and R" for simpleand bermedslop€s:
Q*=Aexp(-BR-/r)
(3)
Values of the enpirical coefficientsA and B dependon the slope angle.
Valuas of the relalive Eughness coetficient,r, which are less than 1.0 inply
o/ertopping levels below those of the equivalent smooth slope. Owen
suggestsvaluesof the relativeroughnesscoefficientfor differentannourtypos
and these are summarised below, based on run-up exp€rimer s in the
Netherlands:
Annour Type
Roughnessvalue,r
Smooth,inpermeable
Roughconcrete
Pitchedstone in fironar
Two hyers of rubble
L00
0.85
0.75 - 0.80
0.50 - 0.60
The methodsdevelopedfor simple sloping seawallshave been extendedby
Owen & Steele(1991)to includethe intluenceof wave retum walls,and by
Besleyet al (1993)to includethe effec{sof armouredslopes,and cresl benns.
Dischaqe reductionfactors take account of the influenceon lhe overtopping
dischargeof a bermor wavewall.
For annoured slopes, values of the run-up teduction factor r have been
derived direc.tlyby analysing the overtopping of various armoured slopes,
tTa17 20rc5D6
E
describedby Allsopet al (1994b). Altematively,new valuesof A and B can
be derivedfor specific structures,using r = 1.
2.3.4
Wavepressures
Measurementsof wave pressureson lhe front face of the caissons,lhe
underskle,and withinthe seawardtoe of the rubblemound,were madebut
are not discussedin this report. These resultsare reporledseparatelyby
McKennaet al (1994).
3
Descriptionof tests
3.1 wavg conditionsand calibration
The waveconditionsusedfor testingare summarisedin Table 1. Thistable
also showsthe spacingsforthe wavescreenelementsin termsof lhe relative
wave lengthBr/L. for each wave condition,where B* is the total chamber
depth,and \ is the meanperiodwave lengthcalculatedfor the waterdepth
at the structure,h,. Theprimaryspacingsof B" = 0.575mand0.400mallowed
lhe lests to cover the range 0.1 < B,/L" < 0.25. The wide rangeof water
depthsand waveconditionsensuredthat lhe wavesat the struclurecovered
the full rangelrom non-breaking
waves,breakingwaves,and depth'limited
coveredthe rangeof 0.02to 0.06.
waves.Theseasteepness,
sm,investigated
were measured
Waveconditionsat a positionequivalentto the breahrvaler
structure.
of the modelbreal$dater
duringcalibrationlests,beforeconstruction
Short seguencesof waves were generatedduring calibrationand were
determinedusing spectralanalysis. Once the nominalwave conditionhad
w€re then made using
been achieved,morecomprehensive
measurements
statistixl
methods.
This ensured
longersequencelengrths,
analysedusing
that extreme waves were reproduced corectly, and that the statistical
distributionof wave heigtrtswas recorded. Statisticalanalysis allowed the
to
significant,0.1%,and otherextremevaluesof lhe waveheightdistribution
poeition,
togelher
with
the
mean
wave
probe
be delerminedat each wave
pedods.Theselongwavesequenceswereusedduringtestingto ensurethat
extremewaves were corectly represented.
3.2 Test methodology
The test programmeis summarisedin Table 2 which outlines the 137 tests
conducted.All waveconditionspresentedin Table 1 were usedfor the vertical
wall alone, and for the single charnber wave screen with a spacing of
B* = 0.575m. The conbinationsof wave conditionsand water levelsusedfor
the sr$sequent lests were reduced based on analysis of the reflectionand
overtoppingresuftsfrom the inithl tests. This enabledthe test programmeto
be kept to a manageablesize and to conce rate the analysison comparable
sels of test conditions.
Test series 1000 and 2000 addressedthe vertical wall and a single chamber
wave screen with two spacings. Overtoppingat the highest water level.
of the
+1.71mwas eltremelyhigh,and was judgedto be unrepresentative
degreeof overtopping
usuallyencounteredin Europeanharbours.Therefore
no furthertestswerecarriedout at this waterlevel.
An initialanalysisof the resultsof Tes Series2000suggestedthatthe screen
spacingof B* = 0.575mwas th6 optimumfor the particularmnge of wave
rT417 20/05t96
E
conditionsused in lhese lests. This spacingwasthereforeusedin Test Series
3000 lo 6000 which involvedmodificationsto the chambersand the screens,
suchas variationsof the hole diameterin the porousscreens,and the depth
withinthe chambers. The hst two test series,7000 and 8000 exploredthe
effect of rock arnoured slopes placed against the vertical wall. Two cresl
levelswere lested al lwo water levels,and the resultsprovidedinfonnationon
lhe influenceof the relativ6crestfreeboard.
4
Discussionof reflectionresults
4.1 Verticalwalls
The resultsof the reflectionmeasurements
in front of the plainverticalwall,
Tesl Series 1000,are summarisedin Table 3. In most inslancesC, falls
between0.85and 0.90,with relativelylittleinfluenceof irrcidentwave height
or period. The water level, and cresl freeboardwere howeverof more
influence,as summarisedin Figure7.
In general lhe larger and steeper waves lead to a slight reductionof
reflections, due to higher levels of energy dissipation caused by wave
breaking.However,an increasein the waterlevelalso leadto a reductionof
reflections,but this was primarilydue to the increasedovertoppingresulting
fromthe lowerstructurefreeboard,R.. This is illustratedin Figure7 whereit
can be seenlhat valuesof C, are reducedat lowervaluesof R"/H".A simple
predictionmethodwas createdthroughthe use of regressionanalysison the
upperand lower regionsof this dala. This enabledan upperbound and a
simpleequationto be appliedto these resuhsas follows:
C, = 0.79+ 0.11RrrH"
C. = 0.90
for R./H"< 1.0
forR/H">1.0
(4a)
(4b)
4.2 Wavescreens/ perforatedcaissons
Analysisof prwbus lest resultsand of numericalmodelpredicnionsby Allsop
& McBride (1S93) has shown that the fiiost useful way of prosentingthe
reflec*ionperfonnanceof perforatedwavescreens is by plottingthe refl€ction
coefficientC, agaimt the relative screen spacingto wave length ratio, B*(.
Resultstor Test S€rie62000 are presentedin this way in Figure 8. Resuhs
from pra/ious studiesby Allsop& Steele(1990)and Allsop & Beresford(1993)
using a singlo slatted wave sseen in fronl of an inpermeable ser6€n,for a
habour at Cardiff,havealso been present€d.Only comparablet€sts from this
study were us€d, ie. those wherethe screen porositywas 2006,as the s*udy
covered a wide range of screen porosities. The results of lests with other
screenporositieswere discussedby Allsop& McBrile (1993). Inclusionof the
Cardiffresultsenabledthe rangeof data to be extendedbeyondthe nuximum
relativespacingof the rnost recent studies,ie. B*/l= < 0.27.
The resultsfollowthe generallrends irJentifiedprwiousg by Allsop & McBride
(1993),ie. the minimumvalueof C. is achiwed whenthe porousscreenis a
nnb bss than a quaner of a wave lengthfrom the solid wall, B*A = 0.20 to
0.25. The previous work dernonstratedfhat initial estimat€s of reflection
pedormancecanbe achievedusingnumericalmodelssuchas 'BARRIER2'by
Bennettet al (1992). However,it is clearlhat such modelscannotdescribe
the influence of many of the detailed rnoditicationssludied here, and that
simple enpirical methodsmay be more appropriatein some circumstances.
rT 417 2ql09p6
tr
Thesedata havethereforebeen describedby fininga new empirioalcurve,
givenby:
(5)
c , = s i n { l { " t ( 8 " / L J - k ,l ' } + q
wherelq, lq and ( are coefficientsdeterminedfrom the data and:
lq
lq
K
determinesthe shape of the curve; low values (600) result in a
shallow'u' shapedcurve,and highvalues(900)resultin moresleep
'v' shapedcurves;
specifiesthe value of B*/l- whichcoresponds with the lowestpoint
of lhe lowestvalue of C.:
giveslhe lowestvalueof C,.
lo be established
The useof the testresultsallowsvaluesof thesecoeflicients
in
Figure
8,
describes
the resultsfor
for eachconfiguralion.Curye1, shown
=
porosity
n" 20%and a holediameter
a singlechamberwavescreen,witha
equallo the screenthickness:
C, = sin { 910 t @,/LJ - 0.22512| + O.28
(6a)
This equationis valid 0.05 . B/L" . 0.32. The use ol equation(6a)outside
this rangeis notsupporledby the resultsconsideredhere,althoughits usefor
0 < 8,./l-.< 0.05 will probablygive reasonableresults. There is, however,
seldomneed for detailsof reflectionperformanceoutsideof these limitsas
suchslructureswill be uneconomic,
or ineffective.
The modellingof the other structuralvariations enables values of the
coefficients in equation (5) to be established for different slructural
configumtions. The resuhs,shown in Figure 9, for screens with larger
diameterholes,ie. twicethe screenthickness,showlhat the lowestvalueof
G. is increased,and the curue is slightly w'lrJened.Curue2 can be described
by the followingversionof equation(5):
C,= sin { 780 I (8,/l-")-o.znf
J +0.315
(6b)
Curv62, equaiion(6b),is shownas the solidline in Figure9, rvhereit maybe
comparedwith Curve 1, equation(6a), shown with crosses. The differerrces
betweenthese curuesare relativelysrmll. The other modificetionst€stedhave
also been analysedto producesimihr curves,and coefficientsfor equation(5)
havebeendedvedand are summarisedin Table4.
The effecl of using doublescreensis shown in Figure 11 by comparingCurue
1 for a single chamber (shown as crosses) with Gurve 3 for the double
chambers(shownas the solid line). Thesashow slightlylower r€flectionswith
a doublechanrberlor B,/L > 0.2, brJtrather higher r€flectionsfor 8"4* < 0.2.
The use of the double screen increasesthe struoturecost without signlflant
benefit in rcducing value of C,, or increasingthe range of B*/l- over which
reflectionsare low. The low€st coefficientfor the single screen, q = 0.28,
oceurs at BJI- = 0.225, compared to q = 0.265 for the double screen at
B*Q = 9.25. The improvedreflectionsgiven by lhe double screen for
B*/l- > 0.20are of relativelylittlebenefitas the designe.srnainrequirement
is to minimisethe slructurewidth B*, and hencecost. However,this effect is
lessmarkedin comparingsingleand doublescreenswithholesof a diameter
equalto twicethe screenlhickness,Figures9 and 11.
t7 117 WO5lg6
tr
The reflectionperformanceof voided caissom with pedorationsover the full
depthof the structureare relativelyinsensitiveto changesin waterlevel. The
last modificatiohsto the caissonstructures,wherethe voided chamberswere
filled to depths of 30% or 50% of the structure height, were ralher more
sensilivelo changesin water level,as shownin figures10 and 12.
In TestSeries5000and 6000,reflections
werem€asuredfor doublechambers
withthe baseof the chambersfilledto depthsof 30% or 507"of the chamber
height. The baseot th€ chamberswerefilledwithblocksand rubble,and the
hofesin the frontscreenwereblocked. For 3oo/o
restriction,
lhe chamberwas
filledto + 1.24m,and for 50%to +1.40m. In Sedes5000,bothchamberswere
filledto 30%,but in Series6000,the floor in the rearchamberwas raisedby
fillingto 50o/o,
givinga steppedarrangement.Eachslruclurewas te$ed with
waler levelsof +1.43mor +1.61m. In each inslance,the restrictft]nin the
volumeof activevoids of the chambersincreasedthe retleclioncoefficiont,
with the rnostob/ious offect occuring at the lowerwater level. The resultsof
tests in Series5000are shownin Figure12, with a comparisonof resuftsfor
the tull depthchambers.The influenceof the reslrictionsare quitesignificant,
feadingto an increasein reflectionsfrom C. = 30o/oto 40o/o
to C. = 50% to
607".
At lhe higherwaterlevel,+1.61m,lhe effectof this change,shownin Figure
13, is less marked,brJtstill of significance.For the same wave heightsand
sleepness,valuesof B/l-. changedueto lhe influ€nceof waterdepthon wave
lenglh. These reductionsof B*/\ themselvesmight be expectedto give
slightlyhighervaluesof C.. The largerarea of open screenat the greater
waterdepthgiveslowerreflectionsfor lhe partdepthchambers,bd C, is still
greaterthan lor the fulldepth chambers.
It must be notedthat the work describedin this sec.tionrelatesto simple,
vedical structures. Herrce,it nray be the case that other struc.tures,such as
caissonsconstrucledon rock rnounds,rnay give differentresults due, in part,
lo the influenceof the moundon the waves.
4.3 Rockarmout slopes
Roflectiohsfrom sloping structuresare less severe than trcm verti€l walls.
The reflectioncharacteristicsof annouredand srnoothslopesdependon wave
breaking on the slope, and are related to the surf paEmeter or lribarren
nurber, (,' or lr", defined €n'= lr,' = tanor's.os; where s,' = ZnH,/gT.2.
R€fl€ctionsfrom smooth or arlnourd sbpes nray be describedby a simple
formuladerivedby Seelig(1983):
c,=a6.,/(b+€*1
(7)
This equationwas later adaptedby Allsop (1990)for randomwaves. Tests by
Allsop& Channel(1989)derivedcoefficierfsa and b for smoothand anmured
slopes, with wave conditions in the ranges of 0.004 < s,n< 0.052, and
0.6 < Hd/AD"so
< 1.9. Resuhs,shownin Figur€14 for smoothand armoured
slopes, are describedby equatiom (8a) and (8b) respec{ively:
C, = o.9os,'2l (4.8+ \^2)
(8a)
c. = 0.64(_, / (e.es+ fi,)
(8b)
tT 417 20/!5,18
tr
The use of equation(8b) for rock armouredslopes is supportedby recent
analysis by Davidsonet al (1994), who checked predictionsusing the
froma rock
coefficientsderivedby Allsop(1990),againslfield measurements
armouredbreakwaterwith goodagreement.
Resultslrom Test Series7000and 8000are summarisedin Figure15 for the
diff6rentarmourcrestlevels. In each instance,the armourfreeboardabove
by lhe armoursize, D".0. Values
the waterlevel,Ab,is non-dimensionalised
of Ao/D"*variedfromzero,wherenearlyall waveshitlhe verticalwall,to 3.8,
where the verticalwall had relativelylittle influenceon the retlections.At
intermediate
are closeto
armourlevels,A,/D"* = 2.4 and 1.35,the reflections
lhose of the armourslope,but some influenceof the armourslopeand crest
bermis evidentin reflectionsbelowthosepredictedby equation(8b).
Armouredslopes are often used within harboursbeneathpiled decks or
platforms.Reflections
fromthis type of structureare generallycloseto those
predictedfor the armouredslope alone. However,some details of the
illustratedby the
slructuremay significantly
rnodifyits hydraulicperformance,
followingexamplefromAllsop(1990).
An unusualstructurewas consideredfor a majorcoal handlingquay. The
habour designrequiredlow levelsof waverefleclionsfromthe quays,but was
also in a region subject to earthquakesunder which conventionalpiled
relievingplatformsmightbe unstable. The two altemativestruclureswhich
werelestedin lhe designsludywerea simplearmouredslopeof 1:1.75under
a pileddeck,and a composilestructurewith armouredslopeon a part-depth
caisson.
The compositeslruclure,shown in Figur€ 16, incorporatestwo important
diflerenceswhencomparedwiththe simplearmouredslope. The lowerface
of the structureis vertical,whichmigfrtb6 expectedto increasereflections,and
the crest level of the armouredslope is relativelylow, hence,largerwavesmay
reachthe vertical step at the rear of the slope formedby the rear deck beam,
againtendingto increasereflec'tions.The reflectionsfrom this structureare
comparedwith the simplepredictioncurye for armouredslopes,equation(8b),
in Figure17.
SectionA, representingthe armouredslope of 1:2.5 over the part depth
caisson,as shownin Figure16, had an armourcrest bem on which larger
waves couH be absobed. These gave reflectionsclose lo, but below those
predicted by equation (8b). The reduced refleclions arose due to the
inlerterencebetweenthe wave compon€ntrefleotedby the vertical pa , and
ihat reflec*edby the slope. The struc'turalvarialion A2 had a crest detail in
which more wave ac-tionreachedthe crest beam, wttich lead to an increase
in refleotions. The altemative section, B, which used a full-depth slope of
greaterthan
1:1.75,bnt includedthe crestdetailof A2, tesultedin refleotions
the pred'Ftions.
This exanple demonslralesthat, whilst lhe reflectionperformahceof simple
slopes is relativelyeasy to predict,reflectionsfrom compositestrudures may
be significantlyinfluencedby relative water levels, panicularv where these
water levelsapproachthe level of the arrnourcrest, or of other structural
elements.
rT 4i7 20/o5i€6
E
5
Discussionof overtopping results
The mothodsdescribedin Section2.3.3haveeachbeen usedto analysethe
overloppingmeasurementsmadeduringthis test programme. Forthe vertical
wall only lesl case, ovenoppingdischargeswere comparedwilh predictions
madeusingGoda'scurves. The methodsof Owen(1980)and Franco(1993)
were usedto analyseall of the data sets describedbelow. The overtopping
measuredduringtestswiththe rubbleslopesin frontof the verticalwall have
been comparedwith the predictionmethodof Owen& Steele(1991).
5.1 Verticalwall
5.1.1 Analysisusingoffshorewaveheights
A seriesof generalpredictionlines,whichwas derivedby Herbert(1993)for
randomwaves,is plottedin Figure6, with thosederivedby Goda (1985)for
a verticalwallon a 1:30slope,withan incidentwavesteepnessof s- = 0.036.
Thedatafromthe presentset of tests,Series1000,for an approachslopewith
m = 1:50havebeencombinedwith He6ert's datafor s* = 0.06,and a slope
of m = 1:30,as shown in Figure18. Of(shorewaveconditiorrswere measured
and H* and T,* were used in the followinganalysis. No comparisonwith
Goda is possibleas the wave steepnessexceedss,* = L945.
The resuftsshowsgoodagreementbetweenthe two sludies,despitethe very
different relativedepths due to the differencein modelscales. Water depths
and waveheigtrtsusedin lhe presentstudyare approximately
3 times larger
thanthoseusedby Hel|]ert(1993).Again,the useol this rnethodsharesthe
disadvantages
identifiedin Section2.3.3,and requirescarefulinterpolation
for
the differentvaluesof R /H*.
5.1.2 Analysisusinginshorewaveheights
The overtoppingpedormanceof a wall in relativelyshallowwaterdependson
the inshore wave height, period and shape. Using the offshore wave
conditionsonly, as in the Goda melhod, confuseslhe analysis as it looses
informationon the form of the waves inshore. The form of the inshorewaves
deperds on the water depth, toreshoreslope, as well as lhe wave heighl and
pedod. In this sludy, the inshorewave conditionswere measurcdat th6 toe
of the struc'tureduring wave calibration. These wave condilions have been
used in lhe following analysisof the data from these tests.
The simplest analysis method is offered by plotting Owen's dimensionless
discharge and freeboad pammeters Qt and R' using exponential or
logadthmicaxes. Theseare shownin Figure19 usingthe inshoreuraveheight
Hn frcm the calibrations. The data shows a good rehtionship between lnQ.
and R*, so a simple regressionline has been fined, giving A = 0.002 and
B = 26.76 in eqmtion (3). This regression line is used in hter ligures to
compare the overtoppingperformanoeof the vertical walls with that of the
ahernativostructures.
The tests described by Franco (1993) were run in relatively deep water.
Details of inshorewave condilionswere not giv6n, but it may be appropriate
to assumethat inshorewaveconditionsweresimilarto the offshoreoonditions.
UsingFranco'srelationshipbetweenQ# and R"/Hnfor a ve irxl wall, equation
(1), the overto,ppingfor a simple vertical wall nny be predirIed with a = 02
and b = -4.295. Resultsfor Series1000are plottedagainstFranco'sequation
11
tT alT 20/05/16
tr
in Figure 20. Franco'sformuh underestimatesthe overtoppingdischarge,
panicularlyat largervalues of Rc/H"i. The diffarencemay bo due to the
relativelysmall range of relativefreeboards,0.9 < Bd/H"i< 2.2, for which
Francoderivedthe equation. In the presentstudya wider rangeof R"/H",is
used in test series1000,0.o3< Rd/H"i< 3.2. Aftemativevaluesof a = 0.03
and b = -2.05 in the Francoequationmay be derivedby fittingthe general
form of equation(1) to thesetest resufts.
It is possiblethat lhe inshoreand offshorewaveconditionsin Franco'stesls
this directly,but HR data
may havediffered.lt was not possibleto investigate
both
Hand H", as shownin
using
from Serieslooo have been re-plotted
Figure21. lt is clearthat Franco'sregressionline,witha = 0.2and b = '4.295,
lies belowdata plottedusingHd,but aboveand closerlo data plottedusing
the inshorewave
H*. These resultsconfirmthe importanceof establishing
heighlwithconfidence,
butfurtherworkwillbe requiredto extendthisanalysis.
by De Wad (1994)canalsobe comparedwiththe
Overtopping
measurements
oredictionlinesderivedfromSeries1000. However,De Waal'sinshorewave
condilionswerenot measureddirectly,but werecalculaled.Thesedata are
plottedagainstOwen'sequationusingA = 0.002and B = 26.76 derivedfor
Series1000,in Frgwe22, with good agreement.
betweenQ#
A similarexercisehasbeenrepeatedusingFranco'srelalionship
and B,iH",,again for De Waal's tesl results. The scatterof the data aPpears
a little wider, but equation (1) with a = 0.03 and b = -2.05 gives good
agreementwith De Waal'sdata. This is shownin Figure23.
5.1.3
Summaryof resultsfor verticalwall
Measurements
of meanovenoppingdischargefor a simplevedicalwall on a
1:50 bed slope have been plotted against the generalform of Owen's
equation,and this gavegoodagreementwith:
Q*=AexP(-BR.)
withA = 0.002andB = 26.76
(4)
The same measurementshave been plotted against the geneml form of
Frarrco'sequation,with good agr€ementfor:
Q#=a exp (b R/HJ
witha = 0.03and b = '2.05
(5)
by De Waal,
Resullsfrom conpletelyindependenttests in the Netherlands,
(5),
(a)
and
again
with
relatively
plotted
havealso been
againsl both equations
goodagreement.
5.2 Wavescreens/ perforatedcaissons
A number ol wave screen / perforated caisson configurationshave been
in Table1. All
inves{igated,
and the rnainstrucluralvariablesare summarised
perforatedscreenshad a porosityof n.= 29"/". Two hole sizes of 25mmand
50mm were used, equivalentto the screen thickness and lwice the screen
thickness,t".
In consideringlhe overtoppingperfornnnceof each $ructuralconfiguration
tested, both Owen's and Franco'smethodswere explored,but the best
discharge
agreementwasfoundwithOwen'smethod.The meanoveftopping
for each test has therefore been used to calculate values of Q- and R', in
12
lT 417 2qo5,l6
E
each instance,usingthe inshorewave height,Ho. Thesehavebeen plotted
as fnQ*againstR" for eachconfiguration
in Figures24lo 27.
The resultsfor singleanddoublescreens,Series2100and 3100/ 3200,have
been combinedin Figure24 as they show very close agreemont. ll is
inleroslingto notethat Allsopet al (1994)alsofoundrelativelylittleditference
in the reflectionperformance
of these configuErtions.
The overtoppingof the
perforatedcaissonsectionsis, however,significantlylowerthan that for the
simple vertical wall, sumrnarisedin Figure 24 by equation (4). This
improvement
is illustratedby fittingan equationof thistormto the resultsfrom
test Series2100,310o,and 3200:
Q*=Aexp(-BR")
with A = 0.005and B = 59.95
(6)
The effectof changingthe hole size, Dh,from 1.0t"to 2.0L,withoutcfranging
the porosity,rL, is shownin Figure25. The resultsfor the largerholesare
plottedagainstthe regressionline for the smallerholes,equation(6). The
changein the screenholesizedoesnot appearto giveanysignificantchange
in overloppingperformance.
A tnoreconfusingpictureariseswhen lhe performanceof perforatedcaissons
of different widlhs are compared. The perforated caisson structures
consideredso far were all of B* = 0.575m. Th6 width of lhe intederence
chamberslestedin Series2200and 3300werereducedto B" = 0.400m.The
overtoppingresults,shownin Figure26, when comparedwith lhose for the
wider chambersin Figure24, appearlo indicatea fudher reductionin the
ovenoppingdischargeover lhe whole of the paramelerrangetested. This
somewhatsuprising resuft suggests that the spacing B* may have a
signiftcar atfecl on the overtoppingdischarge,which is not accountedlor in
other parameters. lt was noted that the analysis ol wave reflectionsfrom
thesestrucluresby Allsopet al (1994)indicatedthat the reflectioncoefficient
C, is well correlatedwith the relativescreenspacingto localwave lengthBJL..
Careful examinationof the resuhsshown in Figures24 and 26 did not,
however,identifyany suchcl€artrend.
In Series 5000 and 6ff)0, the tests exploredthe influenceof restric.tingthe
(vertical)depthof the chambers,in doublescreencaissons,by raisingthe floor
level of the chambers. This rnaybe necessaryin the prolol)?e to ensurethat
there is sufficientvolume of fill materialto geneft e the weight force required
to resistslidingor ovenurning.
In the rnodel,impermeablernaterialwas placed in the base ol each chamber
of the caissonto act as ballast. The overtoppingresultrsare shown in Figure
27. At high water levels,the raised floor had relativelylittle influenoeon the
size of the chamber, and hence on the overtoppingperfonnanoefor deep
water coMitions. h was expec.tedthat a reduction in the water level from
+1.61mto 1.4{lmwouldrcducedischargessignifkxntly,simplyby virtueof the
increasein R. and hence R*, however,th6 differencssare Jelativelysmall. lt
may lheretore be concludedthat the raised floors do not have a significant
influenceon overtopping.
At the commencementof the wave screen lests it was notedthal the form of
overtopping,and probablyalso the meandischarge,were influenoedby the
manner and degree of compressionof the air lrapped betweenthe vertical
wall, the perforatedscreen,ard th6 caissonchamberroof. In preliminary
13
|Ta17 mrc8D6
E
tests, this effectwas reducedbecauseair was able lo escapefrom lhe caisson
through snrall gaps betw€enthe roof and side walls. These gaps were
thereforesealedand lhe movementof air was thereforectntrolled by the torm
of lhe slrucluretestedalone.
The testswereconductedundernorrnalwave attack,F = 0', whichensured
thal any escapeof air from the interferencechambercould be significantly
restrictedby the incidentwavefronl. In the prolotype,wavesare unlikelyto
nor will they attackpedectlyat F = 0'' Any small
be entirelylong-crested,
devialionsfmm this idealisedcasewill allowair lo escapesideways,and thus
limited
increaselhe dissipationwithinthe caisson,unlessthis is inlentionally
lt
therelore
cells.
is
likely
by dividingthe cais-son
intemallyintosmallseparate
perfomance,
and
that these tests give an upper limit lo the overtopping
waveattack.
overtopping
willbe furtherreducedunderobliqueor shorl-crested
5.3 Fockarmoulslopes
Previouswork has indicatedthat the additionof an rock armouredslope in
the likelihood
frontof a verticalwall mightincreaseovertopping,by increasing
armour
is relatively
if
the
of waverunup overthe slope. This nraybe the case
shown in Figure28,
impermeablelo wave action,but the measurements,
providereassurance
thal the armouredsections,whichweretesled,produced
overtopping
whichwas not higher,and was oftenwellbelow,that Predictedby
Owen'sequation,withvaluesof A and B for the slopeangleof lhe armour,ie.
for
1:1.5,and a roughnessfactor r = 0.50. Comparingthe measurements
thesecompositestructuresin Figure28, usingr = 0'50 for a simplearmoured
slope in €quation(3), gives close agr€ementat low values of R*, when
to lhe structure.At valuesof R* < 0.11, the
ovenoppingis relativelyinsensitive
overtoppingperlormanceof the compositestruclure is somewhatbelter than
wouldbe predictedfor a simplearmouredslope.
A comparisoncan also be rnadewiththe predidionlinederivedearlierfor the
simpleverticalwall,test Series1000. Al low R. values,overtoppingof the
compositearmouredsectionis againcomparable1othat of the vefiicalwall
predictedby equation(4). At highervaluesof R*, R* < 0.11,and pa icularly
at relativelylow water levels, the overtoppingperformanceof the armoured
slopeand wall are significantly
betterthan that of the verticalwall alone. The
rock armour is thereforemore efficieni becauseit dissipat€swave enerry as
it runs up lhe slope.
A weakness of this simple approach is shown by the sudden drop in
dimensionless
overtoppingpararneterQ* f'orthosetestswilh R* < 1.1. There
is probabtyan influenceof the relativeanrour crest freeboard,AE,rvhichhas
not been llentifted separately.
6 Discussionof watersurfaceconditions
This chapterdisanssesa new analysisapproachto describethe watersurface
'lowcondition. lt has been dweloped to identify more clearly the effect of
reflection' structureson lhe sea surface in front of breahi,aters, and other
polentiallyreflectiveharbourstructures.lt is the sea suface conditionwhich
hasa significanteffecton vesselnavigation,especiallyfor smallcrafl.
14
tI417 20/Os/96
E
Prwiously used melhods of analysis, such as those .elaled to the reflection
coefficient, do not clearly identify the influence of the structure on lhe sea
surface.Howaner,someguidanceon creatingan analysismethodfor lhe sea
surfacewas prcrridedby Klopmanand van der Meer(1994).Theydeveloped
a procedurewhichdescribesthe spatialvariationsof the localwave heights,
as tunclions of the relative distance seawad from a struclure. While the
variationof waveheightmaybe relevanlto somell/p6sof vessels,whichmay
have to navigale close to such a structure, a more imponant factor is lhe
combinationof wave height and period. Hence,the analysismethodof
Klopmanandvan der Meer(1984)wasadaptedto enablethe spatialvarialion
of wavesteepnessto be described,as a functionof the relativedistancelrom
a struclure.
of the wavesleepnessal each
This newtecfiniqueinvolvesthe determination
of the wave probes which were positionedseawardol the structure,as
describedin Section2.3.1. The steepnesswas calculatedas follows:
s.(x)=H"(x)/Ln
Where: s.(x)
H"(x)
Lh
is the meansteepnessat probeposition,'x' metresfrom
the structure
is the significantwaveheighlat probeposition,'x' melres
lrom the structure
is the wavelength,calculatedfor th€ meanwaveperiod,
measuredat the probe posilion ('x' metres from the
structure)and the still waterdepthat lhe probeposition,
using the Hunt (1979) approxirnationto the wave
dispersionequation.
The resultsof this analysisare presentedin Figures29 to 32, tor the following
structutalanangements:
-
a simpleveiical wall
a singlechamberuravescreen,with B*/l of 0.575manda holedhmeter
equal to lhe screen thickness
a double chamber wave soreen, with BJL" of 0.575m and a hole
diam€ter equal to the screenthickness
a rock slope,with a 1:1.5slopeand a crestlevelof +1.71m.
In these figur€s lhe mean wave steepnessat each probe is presentedas a
dimensionless variable which describes the change in wave steepness
compared with the incident steepness, s, , k|own as the rehtive sea
ste€pness and defined as s,(x/s.. These values are ploited againsl the
relativedistanceseawad of the sfuc{ure, definedas x(. Differentsymbols
have been usedfor each of lhe three incilent wave steepness,0.02, 0.04 and
0.06.
In general the results show that at distances greater than fwe wave lengths
seawardof the structure,the ratio of relativewave steepnesscloselymatches
lhat evPectedfor reflectedwave heights. The resultsof the tests showedthat
al lhese distancesfrom the structurethe local wave periodswere comparable
withthe incidentwaveperiod.
Howev€r,some of the struclureswhich were tesled demonstrat€dsignifi€nt
differencesin the spatialvariationof sea surface sleepness,even where lhe
15
tr417 2WOv96
tr
overall reflections,given by C,, were relativelysimilar' As discussedin
Chapter4, an examinationof the reflectionperformanceof the structures
of usinga singlechamberwave
wlrichweretested,showedthe effectiveness
screen.However,theseresuhsalsoshowedthatthe useof a 1:1.5 rockslope
provideda betteroverallreflectionperfonnance,
andtherewasonlya marginal
ditferencebetweenthe more expensiveoptionof a doublechamberwave
screenc.ompared
with the singlechamberversion.
A comparisonof the relativesleepness,resultingfrom the use of each
structure,for distancesgreaterthan three wave lengthsfrom the structure,
prodrcedthe followingresults:
-
-
the double chamber wave screen produced the rnost favourable
conditionswith a meanrelativesteepnessof 1.15.
the 1:1.5 rock slope provided only marginallyworse sea surface
conditions,with a greateroverallrelativesteepnessof 1.19.
the singlechamberwavescreenprovideda relativesteepnessof 1'22,
which illustratesthe potentialbenefitsof the double chamberwave
screen. Within the range of these results, the double chamber
anangemenlshoweda 10% inprovementin relativewave steepness
overlhe singlechamberversion.
the verticalwallproduceda meanrelativesleepnessof 1.41,whichforms
the upperboundaryof theseresuhs.
These resultsdemonslratethe benefitsof using a double chamberwave
screenover the singlechamberanangemenl,if such a low value of relative
sea steepnessis required.lt also reinforcesthe reflectionanalysisresultsfor
the useof rock armourslopes,afthough,as Previouslydiscussedby McBride
et al (1993), it is often necessaryto maintainthe vertical nature of the
structure,whichfor a rockarmourslopeis not possible.
Thisanalysisalso indicatedthe spatialvariabilityof the seasudacesteepness
at distancesof lessthanthreewavelengthsfromthe structure.In this region,
the resultsfor the verticalwall, Figure29, show that the wave $eepnessis
rapidlychanging.Forthiscasethe relativesea steepnessoftenexceeds1.5,
which,for an incidentsea sleepnessof o.04 would lead to very steep waves
and frequent wave breaking. These conditionswill be hazardousfor vessel
navigation,especiallysnrallcratt such as small fishingvesselsor leisurecraft.
The resuhsfor the singlechamberwave screen (Figure30) show a similar
variationin this region,thoughthe overall meansteepnessis lowerthan for the
verticalwall arangement. The structureswhichcreatethe bestconditionsin
this regionare the doublechamberwave screen (Figure31) and the rock
slope (Figure 32). The rock slope reducesthe local saa sudace steepness,
closeto the structure,to lessthanthe incidentsteepnessdue to the distance
that the slope exends trom the vertical wall of the structur6and the energy
dissipationcausedby the roughand porouslayers. Obviousv,it wouldbe
hazardousfor vesselsto navigatevery clos€to this type of slructuredue to the
risk of grounding.
The results of this analysis are summarisedin Table 5, with corresponding
valuesof C, for comparison.
l6
lT alT 20,05196
E
7
Scale effects
The use of hydraulicmodel test resuhsshould always be subiectto analysis
of tho potential influence of scale effects. Such effects are principally of
conoernwhere flows in conduits or porous layers may be unrealistically
influencedby viscous flow effects. In the lesting of coastal structures,the
most frequentc$ncernis for lhe effectsthat any distortedflows would haveon
the stability ol arnour units. Polentialsoale effects have been discussedby
many aulhors. Owen& Allsop(1983)and Owen& Briggs(1985)reviewed
previousstudiesat laboratories
in the USA,Denmark,and UK,and concluded
that scaleeffectsin the flow in th6 primaryarmouron rubblebreakwaters
will
be very low providedthat the armourunit Reynoldsnumber,definedin terms
of lhe nominalunit diameter,is kepl above approximatelyRe = 3x1CF,or
perhapsaboveRe = 3116. For the currentset of tests,in Series7OO0and
8000,the significantwaveheightsweregteaterthanor equalto 0.1m,and the
size of the rock armour was approximate! 0.07m. This implies that
Re > 6x1d, whichis well aboveeitherof the limits.
A similarargumentmay be pursuedfor flow in the holes in the perforated
wave screens,wherethe Reynoldsnumbermay be definedin terms of the
screenlhickness,t. In thesestudiesthe lowestvalueof Re is givenby Re =
2x1t. This is well above a lower limit of Re > 3xld which might be
postulatedfrom the studieson flow in porouslayers,and very dose to the
moreseverelimitof Re > 3x1d. Suchan analogyis howevera littleweakon
its own, so data from previousstudies at HR Wallingfod have been analysed
for potentialscale effects.
Site specificsludies conceminglhe performanceof a peforated wave screen
in a large wave disturbarre nrodel, explored potential scale effects by
determiningthe lowestwave height in the modelbelowwhich le\relsof energy
dissipationstart to changesignificantly.This was identifiedby plottingthe sum
of relative reflectedand transmittedwave energies(C,"* C.t) against model
wave height, see Figure 3il. For model wave heights abwe H. = o.02m,
equivalentto Be = 4x103,the energy dissipationresponseis flat, but begins
to rise for H. < 0.02m,when Re < 4x108.This b causedprirrcipallyas the flow
resistance of lhe screen increases prEducing greatsr reflectionsand less
relativedissipationwithinthe screen. This limit is very closeto the lower limit
postulatedearlier,and substiantiallybelowlhe low€rvalue of Re calculatedfor
these studies. Viscous scale effects are ther6{or6 unlikely to influenceany
conclusionsdrawn from these dudies.
8
Conclusionsand Recommendations
8.1 Concluslons
8.1.1 Reflectionanalysis
-
Rellectionsfrom sitrple ve ical walls generallyfall closeto q = 0.90, btx
may be reduoed by heavy o\rertopping. A simple reduction factor is
suggestedin equation(,ta)
-
The use of perforatedwave caissons or screens has been shown lo be
eff€ctive in reducing reflections from vonical walls. The reflectftrn
performanceof a range of single or double chamber caissons can be
F 417 mngE6
E
limitsby equation(5). Coefficients
describedwithinpracticalengineering
for this n6w equationhave been dedvedfrom the lest results,and are
givenin Table4, for a selectionof structuralanangements.
A comparisonof the performanceo{ singleand doublechamberwave
screensshowsthat the reflectionperformanceoI the single chamber
wavescreenis marginallybetterfor B*/\ < 0.2andworsefor B*/t- > 0.2.
Any improvements
in the refleclions,by includinga secondscreen,seem
unlikelyto outweighthe increasedcomplexityandcostof its construction.
The optimumdiameterof circularholes in the porousfront scteen is
equalto the thicknessof the screen. The use of largerholes,with a
diameterequal to twice the screen thickness,may lead to a slight
degradation
in reflectionperfonnance.
Small changesin water level do not significantlyeffect the reflective
peformanceof singleor doublechamberwavescreens,providedthatthe
chambersdo nol altersignificanllyoverthe waterlevelrange. The use
particularlyat
of partdepthchambersincreasesreflectionssignificantly,
lowerrelativewaterlevels.
Reflections
fromsimplearmouredslopesare generallywell'predictedby
(8b),
equation
except where the armour crest is relativelylow. For
configurationswith the armoured crest below abotX A"/D"= 1.3,
abovethosepredictedby eguation
reflectionsmayincreasesignificantly
(8b). However,this effectmay be mitigatedby providinga widerarmour
by supponinglhe armouredslopeon a
crest,or in somecircumstances
pandepthcaisson.
The use of measuredinshorewave heightsin the analysisot these
resultssubstantiallyimprovesthe correlationwith predictionmethods.
A simple analysisof potenthl scale effects indicatesthat the results of
these tests will not be significanllyinfluencedby viscousscale effects.
8.1.2 Overtopping
analysis
-
Good conelation has been found between sets of irdependent
measurementsof rNranovertoppingdischarge,and each set appearsto
be well fitted by a sirple equatbn for th6 o/ertopping of simple vertical
walls,basedon the InQ' v R* rehtionshipderivedby Owenfor simple
slop6s.
-
Revisionsto the Franco lormuh based on Q# and ffi, have been
dedvedto describeovettoppingof vertical walls, and again,these seem
to show good agreementwith the test dala.
-
Wave screem placed in front of a veltical wall signifioantlyreduce the
overtoppingdischarge,as well as reflections. The effec'tof the wave
screensinqeases as Ro/H.or R" increases.
-
The overtoppingof a vertical structureprotectedby wave screens,or a
peforaled caisson, does not appear to be significantlyaffec{ed by the
chamber.or wavescreenadditionof a secondinterference
18
tT itl7
20/0649€
E
-
Placomentof an armouredslope in fronl of the verticalwall generally
reducesovenopping,paflicularlyal values of R' < 1.1. When lhe
struclurefreeboardis low, wertopping is close to that predictedfor the
simple verticalwall and an armouredslope. At lower water levels,
overtoppingof the compositeslruc{ureis lower than predictedfor an
armouredslopealone,and subslanthllylowerthanthat predictedfor the
simpleverlicalwall.
8.1.3
Watersurfaceconditions
-
Al distancesof greaterthan five wavelen$hs seawardof the struclure,
the ratioof relativewave steepnesscloselymatchesthat expectedfor
reflectedwaveheights.
-
An analysisof watersurfacecondilionsal distancesof morethan lhree
wavelenghsfromthe structuredernonstraled
tho potehtialbenefitsof the
doublechamberwavescreenover the singlechamberversion.
-
The relativeseasteepnessresultsforthe verlicalwallal distancesof less
lhan lhree wave lenglhsseawardof the structureshowedthat rapidly
changingwavesteepnessleadsto very steepwavesand frequentwave
breaking.Theseconditionswouldbe hazardouslor srmll craft navigating
in this region.
8.2 Fecommendations
It is recommendedthat funher, more detailedanalysis is canied out on both
the overtoppingand sea steepnessanalysesdiscussedin this report. This is
necessary in order to provide a greater understandingof wave by wave
overtoppingand lhe spatial variation of sea sleepness. The further analysis
of the sea steepnessresultsrequiresparticularattentiondue to the potenlial
impac{on the navigationof smalland leisurecraft.
It is also recommended
that funhertestingand analysisis caniedout intoall
aspectsof lhis work. In particular,it is importantthat a comprehensive
sedes
of 3D physioal model tests are canied out to €nable the eflects of differing
anglesof waveinciderrceto be assessedon wave refleclions,o/ertoppingand
sea sleepness.
9 Acknowledgements
The work describedin this report is basedon work completedby mernbersof
the Ports and Estuaries, and Coastal Grcups of HR Wallingford for the
ConstruciionDirectomteof the Departmentof Erwironment,under research
contrac{s PECD7/6/263, 7ld29a and 7161312,and the Department of
Transporl. Furthertesting and analysishas been supporledby the European
OommunityMAST ll programme,under Sub-task4.3 ot lhe MCS proiect,and
lhe Universityof Shefiield.
19
tT417 20rev96
E
10 References
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E
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tr
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Table 3
T$t numbsr
Water level r€lativ€lo
flumeioor (m)
Wav€hebht,
H. (m)
Wal/e abopn€ss,
grn
R€ffocdon
coetticis q
t 001
1.43
0.10
0.02
0.902
tm2
t.43
0.10
o.o4
0.900
1003
1.43
o.20
o.02
0.8so
t@4
|,43
0.20
o.04
0.890
I OO5
1.43
0.20
o.06
o.an
1006
1.43
0.25
0.04
0.863
1006b
t.43
0,04
0.856
1007
t.43
0.06
0.857
1007b
1.43
0.06
0.861
1008
't.61
o .t o
0.02
0.886
1009
1.61
0.t0
0.04
o.895
tot0
1.6t
0.20
o.02
0.883
1011
t.6t
0.20
o.04
0.884
1.6t
0.20
o.06
0.868
t.6t
o25
o.04
0.875
1 0 t4
1.61
0.25
0.06
o.871
1t24
1,6t
o.30
0.o4
0,849
1@3
1,61
0.30
0.06
0.858
1022
1.61
0.t6
DH bi-mod
0.886
1025
1.61
0.28
HR bi-modI
o.844
1026
1.61
o.25
HF bi*nod 2
o,85.I
1015
1.70
o.t0
o,o2
o.a75
10t6
1.70
o.t0
0.04
0,879
10t7
't.70
0.20
o.a!2
0,849
1 0 tI
't.70
0.m
0.0.1
0.8s4
10t9
1,70
0.20
0.06
0.833
to20
|.70
0.25
0.04
0.83e
1@1
1.70
0.2s
o.06
0.817
'1o12
'tot3
lT 417 A/0G/95
tr
Table4
Coefficientsin Equation(2) for tested screenconfigurations
Hole dameter
Screen
Curve
't
'I
910
o.225
0.280
780
0.223
0.315
K.
Single
sct€onlhickness
Sing16
2 ' (soroenhickness)
Double
screenlhickn€ss
3
750
0.250
0.265
Doublo
2 . (scroonhickness)
4
7so
0.250
o.275
Tahle 5
Flesulfsof watersurtaceanalysis
LowgstrgcordedC.
Structurs
Mean,relativ€s€a staepn€ss
at > 3L, s€awardat structure
Verticalwall
1.41
Singlechamb€rwave scr€sn
1.8
0.2E
Doubleohamb€rua\re scr€€n
't.t6
o.27
Rockamour dop€
t.t9
o.23
tT417 27rcGl95
E
Figures
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Figure1 Modelconfigurationwith bathymetry
E
Figure2 Modelcaissonwith doublescreensat B*=Q.$/$p
E
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Figure 3
Perforatedwave screen,a" = 20o/o
tr
9r
c.!
I
o
E
-g
=
o
E
e
E
@.
Figure4 Verticalwall with rock armourslope
E
-o
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#
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=
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ce
c;dci
(porltzu
UH)-rJ
Figure5
Comparisonof reflectionanalysismethods
tr
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of
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at
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t"-\
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ct
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Figure6 Overtoppingof simple vertical walls, comparisonof
Herbert(1993)and Goda(1985)
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Figure7
Influenceof relativecrest heighton reflections
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Figure8
Reflectionsfrom single chamber with hole diameters =
screen thickness, n"=2oo/o,
curve 1, eqn (6a)
tr
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=
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+x
t;
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loH
Figure9
Reflectionsfrom singlechamberwith holediameter= 2 x
screenthickness,n"=20o/o,
curve 2, eqn (6b)
tr
a
O
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e
a
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Figure10 Influenceof partdepthfloors,waterlevelat+1.61m
tr
=
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Figure11 Reflectionsfrom doubtechamberwith hole diameter= 2
x screenthickness, fl"=2oo/o,
curve 4
E
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a
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Figure 12 lnfluence of part-depthfloors, water level at +1.43m
E
T
X
X
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x
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x
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x=
=
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Figure13 Reflectionsfrom double chamberwith hole diameter=
screenthickness,n.=20o/o,
curve 3
tr
tr
E
=
t-
---
!-=
a
k
\.
-€
tr
a
=.
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%
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a
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Figure14 Reflectionperformance
of smoothand armouredslopes,
afterAllsop(1990)
E
c)
tl
-
X
X
!-H
=
t-
<.
c\t
c
-
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co
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-
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Figure15 Reflectionsfrom armouredslopesin front of verticalwall
E
Figure16 Pitedquayover armouredslope on part-depthcaisson
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l-
c:E
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=
C
C\J
e
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Figure17 Reflectionsfrom alternativecoal berthquays
<
a
E
tn
OJ
a-
trl_)
=
6
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c
-45
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t lo
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tr)
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Figure 18 ComparisonbetweenHerbert'sresults and Series1000
tr
(r)
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Figure19 Simpleverticalwalls,Series1000,Owen'sequation
o
tr
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Figure20 Simpleverticalwalls,Series1000,Franco'sequation
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n
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Figure21 Franco's equation using offshore and inshore wave
heights
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(tst
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Figure 22 De Waal's data with Owen's equation, A=0.002 and
B=26.76
E
n
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Figure23 De Waal's data with Franco's equation, a=0.03 and
b=-2.05
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Figure24 Single and double chamber wave screen structures,
Series2100,3100and 3200
E
(n
o
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o
N
tt
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d
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Figure25 Wavescreenstructures,largeholes,series4200and4100
E
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Figure26 Wavescreenstructures,B*=0.40m,
Series2200and 3300
tr
ar)
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Figure27 Calssonchamberswith raisedfloors, Series 5000 and
6000
E
t/r
n|
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N-
;O
I
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+{
dv,
=
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l--
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Figure28 Overtoppingof verticalwall with armouredslopes
E
--J
x
xx
i )=J
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f*.
iT*'
#..,
rkd.
'1'
"€Pxs'a E
Sq k"T
"
X*e
s u x* *;
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(g) ttnsl(xlrs
uesul e0uuqr
a^tlBl0U
Figure29 Watersurfaceconditions- Verticalwall
X
<I)
-c
E
t-
-
E
=
<D
.-J
€-,
a:r:
C\
-
t9l
E
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X
XX
i )-
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X
xx
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Xx
=
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(n
:=
<LJ
>
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&E
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,
E
'sseudaals
(g)rlrsy(xJnrs
a^tlul0H
uasuto0uuql
Figure30 Watersurfaceconditions- Singlechamberwavescreen
E
-J
iu
-
-
Fx
x
A
X
_=
t_
(r_
*
x
><<
-
P
(g
LN
g<
X-
<D
>
E
C\J
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rll
X
E4
x4
X
E
'ssaudeels
pasttt oOuprll
a^tlslaH
ft) rusl(x)nrs
Figure31 Watersurface
conditions- Doublechamberwave
screen
tr
_-
<D
X
X
a-)4)x
X
&
<l)
-.c
x
=s-
x
!r{_
VX
X
-
a
X<
fi-a
*=
<D
>
<EFx
E i.
eE
X
a<
X
s'
E
'sseudeels
uos
urofupql
o^rlpla6
ftJ rursl(xls
Figure32 Watersurfaceconditions- Rockarmourslope
C\j
-
E
a
o
o
.- ' _ o o t o o
N
!
^
F-
r\
ci
-
dr +O
o ooo
ct ct ctct
0
o
o
L
P
I
o
!
ct;
cic
I
n
c
'(.,
d-
q
o
zJ'zJ
Figure33 Analysis of scale effects for wave screens tested with
small wavehelghts