RIVERMECHANICS
SECONDEDITION
PIERREY.JULIEN
ColoradoStateUniversity
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DedicatedtomydeceasedmotherYolandeandmybrotherMichel
Prefacepage xi Notation xv
1PhysicalProperties1
1.1DimensionsandUnits1
1.2WaterProperties4
1.3SedimentProperties6 2MechanicsofRivers14
2.1EquationsGoverningRiverFlows14
2.2EquationsGoverningSedimentMotion33
2.3WhyDoRiversForm?41
3RiverBasins47
3.1RiverBasinCharacteristics47
3.2ExcessRainfallPrecipitation49
3.3SurfaceRunoff60
3.4SedimentSourcesandSedimentYield69
4RiverBasinDynamics83
4.1RainfallPrecipitation83
4.2RiverFlowDurationCurves96
4.3FloodFrequencyAnalysis101
4.4ExtremeFloods108
5SteadyFlowinRivers116
5.1Steady-UniformRiverFlow116
5.2SteadyNonuniformRiverFlow138
6UnsteadyFlowinRivers153
6.1SolitaryWavePropagation153
6.2KinematicandDynamicWaves154
6.3Flood-WaveCelerity155
6.4Flood-WavePropagation159
6.5Short-WavePropagation164
6.6FlowPulsesinRivers165
6.7Loop-RatingCurves167
7MathematicalRiverModels177
7.1RiverFlood-WavePropagation177
7.2Advection–DispersionofRiverContaminants186
7.3Aggradation–DegradationinRivers195
7.4NumericalRiverModels198
8HillslopeandRevetmentStability205
8.1HillslopeStability205
8.2RevetmentStability214
9RiverbankProtection230
9.1RiverbankStability230
9.2RiverFlow-ControlStructures247 10RiverEquilibrium260
10.1IrrigationCanalGeometry260
10.2DownstreamHydraulicGeometry262
10.3RiverMeandering276 11RiverDynamics292 11.1RiverResponse293
11.2RiverDegradation299
11.3RiverAggradation311 12PhysicalRiverModels330
12.1HydraulicSimilitude330
12.2Rigid-BedRiverModels333
12.3Sediment-TransportSimilitude338
12.4Mobile-BedRiverModels340 13StreamRestoration348
13.1WatershedEnvironment348
13.2ChannelRehabilitation351
13.3AquaticEnvironment358
13.4StreamRestorationGuidelines375 14RiverEngineering379
14.1RiverFloodControl379
14.2RiverClosureandLocalScour388
14.3WeirsandCanalHeadworks397
14.4BridgeScour402
14.5NavigableWaterways408
Preface
Waterisessentialtosustainlifeandriversaretrulyfascinating.Mostprosperouscitiesarelocatednearriverconfluencesandriverengineersmust designstructurestodrawbenefitsfromthe fl uvialsystemfordeveloping societies.Ideally,scientistsshoulddevelopnewmethodstoimproveengineeringdesign,whilepractitionersmustunderstandwhycertainstructures workandothersfail.Fundamentally,rivermechanicsrequiresunderstanding ofhydrodynamicforcesgoverningthemotionofwaterandsedimentincomplexriversystems.Additionally,the fluvialnetworkmustseekequilibriumin itsabilitytocarrywaterandtransportsedimentthroughdynamicriver systems.Nowadays,riverengineersareconcernednotonlyabouturban drainage, floodcontrol,andwatersupply,butalsoaboutwaterquality,contamination,andaquatichabitat.Thistextbookbroadensthisperspectiveby integratingknowledgeofclimatology,hydrology,andgeomorphology.
Thistextbookhasbeenpreparedforengineersandscientistsdevelopinga broad-basedtechnicalexpertiseinrivermechanics.Ithasbeenspecifically designedforgraduatestudents,forscholarsactivelypursuingscientific research,andforpractitionerskeepingupwithrecentdevelopmentsinriver engineering.Theprerequisitesforreadingitandmakinguseofitaresimply abasicknowledgeofundergraduate fluidmechanicsandofpartialdifferentialequations.Thetextbook ErosionandSedimentation fromCambridge UniversityPressservesasprerequisitematerialforthegraduatecourse, RiverMechanics,thatIhavetaughtatColoradoStateUniversityoverthe pastthreedecades.
MyteachingphilosophyhasbeendetailedinmyrecentHunterRouse lecture(Julien,2017).SketchI.1illustratesthekeypointsthatIseekto developamongmygraduatestudentsandpostdoctoraladvisees.
Observation field and lab.
Physics–chemistry
SketchI.1.Professionaldevelopmentinriverengineering
First,theessentialcomplementarityoftheoryandpracticecannotbeoveremphasized.Theorycanbestenhanceengineeringapplicationswhenthefundamentalunderstandinghasbeengroundedinpracticalobservations. Second,thereisaneedtodevelopthreemainpoles,whereobservations from fieldandlaboratoriesleadtophysicalunderstanding,priortomathematicalcalculations.Expertiseisdevelopedbyexpandingtheoverlapping areasofthesethreepoles.Finally,whiletheprocessesoflisteningandreadingareessentialtotheabilitytolearnandretainnewknowledge,myteachingemphasizesalsotheneedtodevelopverbalandwrittencommunication skills.Theabilitytoexpressdynamicthinkingisatremendousassetforany successfulprofessionalcareer.
Ratherthanbeingavoluminousencyclopedia,thistextbookscrutinizes selectedmethodswhichmeetpedagogicalobjectives.Thereissufficientmaterialfora45-hgraduate-levelcourse.Besidebasictheoryandlecture material,thechaptersofthisbookcontainvariousexercisesandproblems, datasetsandexamples,computerproblems,andcasestudies.Theyillustrate specificaspectsoftheprofessionfromtheoreticalderivations,throughexercisesandproblems,topracticalsolutionswiththeanalysisofcasestudies. Mostproblemscanbesolvedwithafewalgebraicequations;othersrequire theuseofcomputers.Nospecificcomputercodeorlanguageisrequired. Insteadofpromotingtheuseofcommercialsoftwarepackages,Istimulate students’ creativityandoriginalityindevelopingtheirowncomputerprograms.Throughoutthebook,asoliddiamond(♦)denotesequationsand problemsofparticularsignificance;doublediamond(♦♦)denotesthemost important.
Thebookcoverstopicsessentiallyfromthemountainstotheoceans:
Chapter1outlinesthephysicalpropertiesofwaterandsediment; Chapter2reviewsthegoverningequationsofmotionandsedimenttransport; Chapter3describesriverbasinsintermsofthesourceofwaterandsediment; Chapter4looksatriverbasindynamics; Chapter5treatsthesteady- flowconditionsincanalsandrivers;
Chapter6delvesinto flood-wavepropagationinrivers; Chapter7introducessomenumericalmethodsusedtosolveriverengineering problems;
Chapter8copeswithhillslopeandriverbankstability; Chapter9dealswithriverbankprotectionmeasures; Chapter10delineatesthehydraulicgeometryandequilibriuminalluvialrivers; Chapter11explainstheconceptsofriverdynamicsandresponse; Chapter12focusesonphysicalmodelingtechniques; Chapter13providesessentialknowledgeonstreamrestoration; Chapter14presentsseveralriverengineeringtechniques;and Chapter15coverswavesandtidesinriverestuaries.
MyteachinghasbeengreatlyinspiredbyDrs.MarcelFrenette,DarylB. Simons,HunterRouse,YvonOuellet,E.V.Richardson,JeanLouis Verrette,StevenR.Abt,JoseD.Salas,RichardEykholt,HsiehWenShen, JimRuff,CarlF.Nordin,JeanRousselle,andStanSchumm,aswellas manyothers.Theyhavegreatlyinfluencedmyprofessionaldevelopmentand universityteachingsince1979.IamalsothankfultoDrs.PhilCombs,Drew Baird,andPatrickO’Brienforsharingtheirpracticalexpertiseinriverengineering.Thisbookwouldnothavebeenthesamewithoutcontributionsand suggestionsfromacoupleofgenerationsofgraduatestudentsatColorado StateUniversity.Theyhelpedmetailorthistextbooktomeettheirneeds undertheconstraintsofquality,concision,andaffordability.JeanParent patientlydraftedallthe figures.Finally,ithasbeenarenewedpleasure tocollaboratewithMattLloyd,EstherMigueliz,andtheCambridge UniversityPressproductionstaff.
Notation
Symbols
ax, ay, az Cartesianacceleration
ar, aθ, az cylindricalaccelerations
a referenceelevation
a pierwidth
acent centrifugalacceleration
acor Coriolisacceleration
ai incrementalcross-sectionarea
aj 1, aj+1 upstream/downstreamboundarycoef ficientsofthe Leonardscheme at partialwatershedarea
aΘ projectionofthesubmergedweightintothe embankmentplane awaveamplitude
a, b coefficientsoftheresistanceequation
a, b, â, ^ b transformcoef ficientsfordurationcurves
A,B coefficientandexponentofthesedimentratingcurve
A surfacearea
Aa erroramplitudefactor
Asb surfaceareaofasettlingbasin At watersheddrainagearea
Ã, B wavecoef ficients
br river-bendcoefficient
B basechannelwidth
BCFbioconcentrationfactor
c wavecelerity
c * dimensionlesscelerity xv
cG groupcelerity
cu undrainedcohesion
C Chézycoef ficient
C sedimentconcentration
Ca referenceconcentration
ˆ
C croppingmanagementfactor
Cfl Courant–Friedrichs–Lewycondition
Ck griddispersionnumber
C0i upstreamsedimentconcentration
Cr runoffcoef ficient
Cu = uΔt/Δx Courantnumber
Cv, Cw, Cppm, Cmg/l sedimentconcentrationbyvolume,weight,partsper million,andmilligramsperliter
d10, d50 particlesizedistribution,% finerbyweight
dm effectiveriprapsize
ds particlesize
d* dimensionlessparticlediameter
D pipe/culvertdiameter
D headcutheight
Dd degree-days
Dp dropheightofagrade-controlstructure
Dx oxygendeficit
DO dissolvedoxygencontent
e voidratio
E specificenergy
E grosserosion
Etons expectedsoillossintons
Ê soillossperunitarea
E totalenergyofawave
E()exceedanceprobability
ΔE specificenergylostinahydraulicjump
f Darcy–Weisbachfrictionfactor
fl Laceysiltfactor
f (t)infiltrationrate
F force
F fetchlengthofwindwaves
FB buoyancyforce
Fc centrifugalforce
FD dragforce
Fg gravitationalforce
Fh hydrodynamicforce
Fi inertialforce
FL liftforce
FM momentumforce
Fp pressureforce
Fs shearforceinabend
FS submergedweightofaparticle
FVf ¼ V = gLfp fishFroudenumber
Fw weightofwater
FW weightofaparticle
F()nonexceedanceprobability
Fn(z)standardnormaldistribution
F(t)cumulativeinfiltration
Fa(t)actualcumulativeinfi ltration
Fp(t)potentialcumulativeinfiltration
FrFroudenumber
g gravitationalacceleration
G specificgravityofsediment
Grgradationcoef ficient
Gu universalgravitationconstant
h flowdepth
hc critical flowdepth
hd downstream flowdepth
hn normal flowdepth
hp pressureheadatthewettingfront
hr rainfalldepth
hs cumulativesnowmelt
ht tailwaterdepth
hu upstream flowdepth
hw partialelevationdroponawatershed
Δh localchangein flowdepth
H Bernoullisum
ΔH energylossoverameanderwavelength
Hc criticalhillslopesoilthickness
Ho(θm)Struvefunction
H s ¼ 2a waveheight
Hw elevationdroponawatershed i rainfallintensity
ib riverbedinfiltrationrate ie excessrainfallintensity if snowmeltrate i30 maximum30-minrainfallintensity j spaceindex
J0(θm)zeroth-orderBesselfunctionofthe firstkind
k decaycoefficient
k0 resistanceparameterforlaminaroverland flow
ks surfaceroughness
k ′ s grainroughnessheight
kt totalresistancetolaminaroverland flow
k wavenumber
K saturatedhydraulicconductivity
K conveyancecoefficient
^ K soilerodibilityfactor
K1, K2 coefficientsofthepierscourequation
Kb ratioofmaximumshearstressinabendtoastraight channel
Kc riprapcoef ficient
Kd dispersioncoefficient
Kd flood-wavediffusivity
Kd soil–waterpartitioncoef ficientorratioofsorbedto dissolvedmetals
KEaveragekineticenergyperunitarea
KG(T)frequencyfactoroftheGumbeldistribution
Knum numericaldispersioncoef ficient
Koc soil–waterpartitioncoef ficientnormalizedtoorganic carbons
Kow octanol–waterpartitioncoefficient
Kp plungingjetcoefficient
Kp(γ)frequencyfactorofthelog-PearsonIIIdistribution
KS ratioofthesedimentvolume
Ksj submergedjetcoefficient
Δl ¼ a=R meanannualmigrationrate
l1 to l4 momentarms
lc, ld momentarmsinradialstabilityofriverbends
L sinuousriverlength
L fieldrunofflength
La abutmentlength
LC50 lethalconcentrationresultingin50%mortality
Lf depthofthewettingfront
L0 normalizedchannellength
Lp pierlength
Lr riverlength
Lr lengthratio
Lsb settling-basinlength
^ L slope-lengthfactor
Lf fishlength
LM runoff-modelgrid-cellsize
LR gridsizeofrainfallprecipitation
LS correlationlengthofastorm
LW lengthscaleofawatershed
LΔ lengthofarrestedsalinewedge
m exponentoftheresistanceequation
mE massoftheEarth
mM massoftheMoon
ms sedimentmasserodedfromasinglestorm
M mass
M specificmomentum
M snowmeltrate
Mf meltfactor
M1, M2 firstandsecondmomentsofadistribution
M, N particle-stabilitycoefficients
M/N ratiooflifttodragmomentsofforce
n Manningcoefficient n
~
n normalvectorpointingoutsideofthecontrol volume
˜
n wavenumberindex
N numberofpointsperwavelength
N numberofstorms
N0(θm)Neumannfunction,orthezeroth-orderBesselY function
O()orderofanapproximation
p pressure
p()probabilitydensityfunction
pcl meanannualpercentagelateralmigrationrate
p0 porosity
p0e effectiveporosity
p0i initialwatercontent
p0r residualwatercontent
Δpc fractionofmaterialcoarserthan dsc
Δpi sedimentsizefraction
Δp0 changeinwatercontentatthewettingfront
P wettedperimeter
P()probability
ΔP powerlossinahydraulicjump
^ P conservationpracticefactor
P totalpowerofawave
PCBpolychlorinatedbiphenyls
PEaveragepotentialenergyperunitsurfacearea
P0 powerloss
PΔ gridPecletnumber
q unitdischarge
qbv unitsedimentdischargebyvolume
qbv ¼ qbv =ω0 ds dimensionlessunitsedimentdischarge
ql lateralunitdischarge
qm maximumunitdischarge
qs unitsedimentdischarge
qsj þ1 ; qsj upstreamanddownstreamunitsedimentdischarge
qt totalunitsedimentdischarge
Q riverdischarge
Qbv bedsedimentdischargebyvolume
Qp peakdischarge
Qs sedimentdischarge
r radialcoordinate
r * dimensionlessradiusofcurvature
r, θ, z cylindricalcoordinatesystem r lateral, θ downstream,and z upward
rQ dischargeratio
R risk
R radiusofcurvatureofariver
^ R rainfall-erosivityfactor
ΔRe excessrainfall
RE radiusoftheEarth
ReReynoldsnumber
Re* = u*ds/v grainshearReynoldsnumber
Rh hydraulicradius
Rm
Ro ¼ ω=κ u
minimumradiusofcurvatureofachannel
Rousenumber
S slope
^ S slope-steepnessfactor
SD specificdegradation
SDR sedimentdeliveryratio
Se effectivesaturation
S0, Sf, Sw bed,friction,andwater-surfaceslopes
S0x,S0y bed-slopecomponentsin x and y
Sr, Swr radialwater-surfaceslope
Sr dimensionlessradialslope
SFsafetyfactor
t time
t trapezoidalsectionparameter
Δt timeincrement
Δts timeincrementforsediment
ta cumulativetimewithpositiveairtemperature
te timetoequilibrium
tf cumulativedurationofsnowmelt
tf fishswimmingduration
tr rainfallduration
tr ¼ tr =tr normalizedstormduration
tt transversalmixingtime
tv verticalmixingtime
T periodofreturnofextremeevents
T waveperiod
T° temperature
T50 timeforhalfthechannel-widthchange
TE trapefficiency
Ts windstormduration
u, v velocityalongaverticalprofile
u average flowvelocity
u* shearvelocity
u*c criticalshearvelocity
Uf fishswimmingvelocity
Uw windspeed
vh migrationrateofheadcuts
vs localvelocityagainstthestone
vx, vy, vz localvelocitycomponents
V mean flowvelocity
Vc criticalvelocity
Vx, Vy, Vz Cartesianmean flowvelocitiesin x, y,and z
VΔ densimetricvelocity
Vθ downstreamvelocityincylindricalcoordinates
∀ volume
∀v ; ∀t volumeofvoidsandtotalvolume
W channelwidth
W weightofsoilperunitwidth
W, W0, We active,initialandequilibriumchannelwidth
Wm meanderwidth
W0 overlandplanewidth
x, y, z coordinatesusually x downstream, y lateral,and z upward
xr, yr, zr lengthratiosforhydraulicmodels
Xmax downstreamdistancewiththemaximumoxygen deficit
Δx gridspacing
X runofflength
Xc reachlength
Xe equilibriumrunofflength
Xmax maximumendurance fishswimmingdistance
yd, yu downstreamandupstreamwaveamplitude
Y sedimentyield
zb bedelevation
zw water-surfaceelevation
z* dimensionlessdepth
Δz scourdepth
GreekSymbols
α coef ficientofthestage–dischargerelationship
α, β parametersofthegammadistribution
αb deflectionangleofbarges
αe Coriolisenergycorrectionfactor
~ α ¼ 2π =N phaseangle
β exponentofthestage–dischargerelationship
β bedparticle-motionangle
βm momentumcorrectionfactor
γ speci ficweightofwater
γ skewnesscoef ficient
γm speci ficweightofawater–sedimentmixture
γmd dryspeci ficweightofawater–sedimentmixture
γs speci ficweightofsediment
Γ ¼ 1 þ 4kK =U 2 p dimensionlesssettlingparameter
Γ(x)gammafunction
δ anglebetweenstreamlineandparticledirection
δL ¼ lnðyd =yu Þ waveamplificationoverlength L0
ξ ratioofexceedanceprobabilities
ξr ¼ Wr =hr channelwidth–depthratio
~
ξ wavedisplacementinthe x direction
η sideslopestabilitynumber
~
η wavesurfaceelevation
ζ k n Fouriercoef ficients
κ vonKármánconstant
λ streamlinedeviationangle
λ wavelength
λf snowmeltintensity
λr = tr/te hydrographequilibriumnumber
λs significantwavelength
Λ meanderwavelength
μ dynamicviscosityofwater
ν kinematicviscosityofwater
φ angleofreposeofbedmaterial
ϕ latitude
Φ potentialfunctionforwaves
ρ massdensityofwater
ρm massdensityofawater–sedimentmixture
ρmd drymassdensityofawater–sedimentmixture
ρs massdensityofsediment
ρsea massdensityofseawater
Δρ massdensitydifference
Π = ln[ ln E(x)]doublelogarithmofexceedanceprobability
ω settlingvelocity
ωE angularvelocityoftheEarth
Ω sinuosity
ΩR ratioofcentrifugalforcetoshearforceinbends
θ downstreamorientationofchannel flow
θ angularcoordinate
θc criticalangleofthefailureplane
θj jetanglemeasuredfromthehorizontal
θm maximumorientationofchannel flow
θp floworientationangleagainstapier
θr raindropangle
θ0, Θ0 downstreambedangle
Θ1 sideslopeangle
Θ = (t tr)/te dimensionlesstime
σ stresscomponents
σ standarddeviation
σ ¼ 2π L0 =λ dimensionlesswavenumber
σ ′ effectivestress
σg gradationcoeffi cient
σx, σy, σz normalstresses(negativepressure)
σΔt standarddeviationofdispersedmaterial
σθ normalstressonaplaneatanangle θ fromthe principalstresses
~ σ angularfrequencyofsurfacewaves
τ shearstress
τ0, τb bedshearstress
τ0x, τ0y downstreamandlateralbedshearstresses
τbn bedshearstressatanormaldepth
τc criticalshearstress
τf failureshearstrengthofthesoil
τr radialshearstress
τ r dimensionlessradialshearstress
τs sideshearstress
τsc criticalshearstressonasideslope
τw windshearstress
τzx shearstressinthe x directioninaplane perpendicularto z
τ* Shieldsparameter
τ*c criticalvalueoftheShieldsparameter
τθ tangentialstressonaplaneatanangle θ fromthe principalstresses
ψ = q/ieL dimensionlessdischarge
Ψ reducedvariable
PhysicalProperties
Asanaturalscience,thevariabilityofriverprocessesmustbeexamined throughthemeasurementofphysicalparameters.Thischapterdescribes dimensionsandunits(Section1.1),physicalpropertiesofwater(Section1.2), andsediment(Section1.3).
1.1DimensionsandUnits
Physicalpropertiesareusuallyexpressedintermsofthefollowingfundamentaldimensions:mass(M),length(L),andtime(T).Temperature(T °)isalso sometimesconsidered.Thefundamentaldimensionofmassispreferredto thecorrespondingforce.
Thefundamentaldimensionsaremeasurableinquantifiableunits.Inthe SIsystemofunits,theunitsformass,length,time,andtemperatureare thekilogram(kg),themeter(m),thesecond(s),anddegreesKelvin(K).The Celsiusscale(°C)iscommonlypreferredinriverengineeringbecauseitrefers tothefreezingpointofwateras0°C.Theabbreviationsforcubicmetersper second(1cms = 1m3/s)andcubicfeetpersecond(1cfs = 1ft3/s)arecommonlyusedtodescribethe flowdischargeofariver.
ANewton(N)istheforcerequiredtoaccelerate1kgat1m/s2,or1N = 1kgm/s2.ThegravitationalaccelerationattheEarth’ssurfaceis g = 9.81m/s2. Theweightofonekilogramis F = mass × g = 1kg × 9.81m/s2 = 9.81N. Thepressureisgiveninpascalsfrom1Pa = 1N/m2.Theunitofwork(or energy)isthejoule(J),whichequalstheproductof1N × 1m.Theunit ofpowerisawatt(W),whichis1J/s.Prefixesindicatemultiplesorfractionsofunitsbypowersof10:
μðmicroÞ = 10 6 ; kðkiloÞ = 103 ; mðmilliÞ = 10 3 ; MðmegaÞ = 106 ; cðcentiÞ = 10 2 ; GðgigaÞ = 109 1
Forexample,sandparticlesarecoarserthan62.5 μmormicrons;gravels arecoarserthan2mm;and1megawatt(MW)equals1millionwatts (1,000,000or106 W).
IntheEnglishsystemofunits,thetimeunitisasecond,thefundamental unitsoflengthandmassare,respectively,thefoot(ft),equalto30.48cm, andtheslug,equalto14.59kg.Theforcetoaccelerateamassofoneslugat 1ft/s2 isapoundforce(lb).Inthistext,apoundalwaysreferstoaforce,not amass.TemperatureindegreesCelsius, T °C,isconvertedtothetemperatureindegreesFahrenheit, T °F,by T °F = 32.2°F + 1.8 T °C.
Variablesareclassifiedasgeometric,kinematic,dynamic,anddimensionlessvariables.AsshowninTable1.1,geometricvariablesdescribethegeometryintermsoflength,area,andvolume.Kinematicvariablesdescribethe
Table1.1. Geometric,kinematic,dynamic,anddimensionlessvariables
VariableSymbol Fundamental dimensionsSIunits
Geometric(L)
, L, T)
Speci ficgravity
ReynoldsnumberRe
Grainshear
ReynoldsnumberRe*
Shieldsparameter
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“The French Dressing-Room.” Stipple-Engraving by P. W. Tomkins, after Chas. Ansell.
(Published 1789. Size 7½″ × 9½″.)
From the collection of Frederick Behrens, Esq.
Plate XXXIV. “January” (“The Months”).
Stipple-Engraving by F. Bartolozzi, R.A., after Wm. Hamilton, R.A. (Published 1788. Size 10″ × 12″.)
From the collection of J. H. Edwards, Esq.
Plate XXXV. “Virtuous Love” (from Thomson’s “Seasons”).
Stipple-Engraving by F. Bartolozzi, R.A., after Wm. Hamilton, R.A. (Published 1793. Size 6¼″ × 5″.)
From the collection of Frederick Behrens, Esq.
Plate XXXVI. “The Chanters.” Stipple-Engraving by J. R. Smith, after Rev. Matthew W. Peters, R.A. (Published 1787. Size 7⅝″.)
From the collection of Frederick Behrens, Esq.
