Drag, turbulence, and diffusion in flow through emergent
vegetation
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Nepf, H. M. “Drag, turbulence, and diffusion in flow through
emergent vegetation.” Water Resources Research 35.2 (1999):
479. © 1999 American Geophysical Union
As Published
http://dx.doi.org/10.1029/1998WR900069
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American Geophysical Union
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Wed May 25 21:49:37 EDT 2016
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http://hdl.handle.net/1721.1/68641
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WATER RESOURCES RESEARCH,
VOL. 35, NO. 2, PAGES 479-489, FEBRUARY
1999
Drag, turbulence, and diffusion in flow through emergent
vegetation
H. M. Nepf
Ralph M. ParsonsLaboratory,Departmentof Civil and EnvironmentalEngineering,Massachusetts
Institute of
Technology,Cambridge
Abstract. Aquatic plantsconvertmean kinetic energyinto turbulentkinetic energyat the
scaleof the plant stemsand branches.This energytransfer,linked to wake generation,
affectsvegetativedrag and turbulenceintensity.Drawing on this physicallink, a model is
developedto describethe drag,turbulenceand diffusionfor flow throughemergent
vegetationwhich for the first time capturesthe relevantunderlyingphysics,and coversthe
natural rangeof vegetationdensityand stemReynolds'numbers.The model is supported
by laboratoryand field observations.
In addition,this work extendsthe cylinder-based
model for vegetativeresistanceby includingthe dependenceof the drag coefficient,
on the stempopulationdensity,and introducesthe importanceof mechanicaldiffusionin
vegetatedflows.
1.
Introduction
Freshwaterand saltwaterwetlandsprovide important transitionzonesbetweenterrestrialand aquaticsystems,
mediating
exchanges
of sediment[Phillips,1989],nutrients[Nixon,1980;
Barkoet al., 1991],metals[Orsonet al., 1992;Lee et al., 1991],
and other contaminants[Dixon and Florian, 1993]. Wetland
plants control these exchangesboth directly through uptake
and biologicaltransformationand indirectly by altering the
hydrodynamic
conditions[Kadlec,1995].The latter is the focus
of this study, specificallythe impact of vegetationon drag,
turbulenceintensity,and diffusivity.By providinga physically
basedmodelfor estimatingtheseparameters,the resultsof this
paper will improve the engineeringof wetland systemsfor
wastewaterand stormwatertreatment,an increasingly
popular
and more environmentallysustainablealternative to traditional facilities.
The additionaldragexertedby plantsreducesthe meanflow
withinvegetatedregionsrelativeto unvegetatedones[Kadlec,
1990;Shi et al., 1995]. This bafflingpromotessedimentaccumulationby reducingnear-bedstresses
and subsequent
erosion
[Ward et al., 1984;Leonardand Luther, 1995]. An increasein
drag can also lead to an increasein water depth and thus
residencetime, further influencingspecies'fate and biological
activity [Jadhavand Buchberger,1995;Kadlec, 1990]. Several
effortshave been made to model the additionaldrag usinga
modified Manning's equation [e.g., Guardo and Tomasello,
1995]. Although useful for its simplicity,this adaptation of
Manning'sequation revealslittle information about the flow
structurewithin and abovethe canopyand cannotrepresent
regionsof emergentvegetationor regionsof creepingor transitional flow [Kadlec, 1990; Jadhav and Buchberger,1995].
Other researchershavemodeledvegetativedragbasedon cylinder drag,either usingthe dragcoefficientas a fitting parameter or choosingit on the basisof valuesfor isolatedcylinders
asa functionof Reynoldsnumber[e.g.,Burkeand Stolzenbach,
1983;Hosokawaand Horie, 1992].This paper extendsthe cylCopyright1999 by the American GeophysicalUnion.
Paper number 1998WR900069.
0043-1397/99/1998WR900069509.00
479
inder-basedmodel by definingthe effect of stem population
densityon dragcoefficient.It canbe appliedacrossa rangeof
flow conditionsand for verticallyvariedvegetativedensity.
In addition to affectingthe mean velocity,vegetationalso
affectsthe turbulenceintensityand the diffusion.The conversionof mean kinetic energyto turbulentkinetic energywithin
stem wakes augmentsthe turbulenceintensity,and because
wake turbulenceis generatedat the stem scale,the dominant
turbulentlengthscaleis shifteddownward,relativeto unvegetated,open-channelconditions[Nepfet al., 1997].The combination of reducedvelocity and reduced eddy-scaleshould
reducethe in-canopymacroscalediffusionrelative to unvegetated regions.This reductionhas been observedfor aquatic
grasses[Worcester,
1995;Ackermanand Okubo,1993].
This paper developsand testsa physicallybasedmodel that
predictsthe turbulenceintensityand diffusionwithin emergent
vegetation.The model linksvegetativedrag,turbulenceintensity,and turbulent diffusion.Observationsof turbulenceintensity match predictionsbased on the drag model. Observed
diffusionrates,however,are greaterthan thosepredictedfor
turbulentdiffusionalone.To explainthe difference,thispaper
introducesthe processof mechanicaldiffusionfor vegetated
flows.Together the contributionsof turbulent and mechanical
diffusionfully describethe observations.
The newlydescribed
link between drag and turbulenceintensity,as well as the
introductionof mechanicaldiffusionas an importantdiffusive
processin vegetatedflows,can improveour understandingof
seedand larvae dispersalin coastalmarsh systemsas well as
our understandingof sedimentand contaminanttrapping.The
resultsof this study are also relevant to the broader classof
flow through distributedobjects,for example,pesticidedispersalwithin cropsand pollutant dispersalwithin cities.
Section2 introducesmodelsfor drag, turbulence,and diffusivityfor flow through emergentvegetation.Laboratory and
field experimentsdescribedin section3 provideobservations
whichsupportthesemodels.The comparisonof modelprediction and experimentalobservationis givenin section4. Finally,
the models are used to compare the mean flow, turbulence
intensity,and diffusivityin vegetatedand unvegetatedregions
(section5).
480
2.
NEPF: DRAG, TURBULENCE,
AND DIFFUSION
Model Development
2.1.
Drag Model for Emergent Vegetation
The drag force per fluid mass due to vegetationmay be
described
as
U2
Fr=[force]
massj
=51Cz>a
(1)
where p is the fluid density,U is the equivalentuniformvelocity, and Co is a bulk drag coefficientrepresentingthe array.
The vegetationdensity,a (per meter), is the projectedplant
area per unit volume. If the plants are modeledas cylinders,
dh
d
a = nd= AS2h
= AS2,
(2)
06 0.8•
0
2
4
6
8
10
•
12
14
16
,
18
2O
L/d
Figure 1. Contoursof dragcoefficient,Co on trailingcylinder, B, showthe suppressionof Codue to wake interaction.
Contours are based on data measured with two cylinders
(blackdots [Bokaianand Geoola,1984]) and on the observed
decayof wake interferencewith separationdistance[Blevins,
1994, pp. 177-181], which was used to set the longitudinal
trend for the abovecontoursbetweenL/d = 6 and 20. Red =
2600, and Co- 1.17 at T/d, L/d -• •. A region of Co<
where n is the number of cylindersper unit area, that is, stems 0 existsfor L/d < 2, T/d < 0.25.
per squaremeter;AS is the meanspacingbetweencylinders;d
is the cylinderdiameter;andh is the flow depth.From (2) we
candefinea dimensionless
population
density,
ad - d2/AS2.
For the cylindermodel ad representsthe fractionalvolume of
the flow domain occupiedby plants.Although it excludesthe
effectsof stemmorphologyand flexibility,thismodelprovides
a reasonablefirst step for exploringthe effect of vegetation
densityon drag.
The drag coefficientfor an isolatedelement,Co, is affected
by the wake structure, and thus dependson the Reynolds
number,Red = Ud/v, where v is the kinematicviscosity.In
coastaland freshwatersystems
Red - O(1-1000) [Leonard
and Luther, 1995;Hammer and Kadlec, 1986], a range which
covers both laminar
and turbulent
wake structure.
Williamson
[1992] describesthe details of this laminar to turbulent transition for an isolatedcylinderin a uniform flow. The cylinder
wake remains laminar up to Red • 200, although vortex
sheddingis initiated at Re d • 50. For Red >• 200, vortex
instabilitycausesthe wake to becometurbulent. Lateral shear
in the flow upstreamof an isolatedcylindercandelaythe onset
of vortex sheddingto Red = 200 [Tamura et al., 1980], and
becausevortexsheddingis a precursorto wake instability[Williamson,1992], the transitionto a turbulentwake will alsobe
delayed. Within a cylinder array, Nepf et al. [1997; ad =
0.008-0.07] observedthat vortex sheddingwas initiated at
Re d - 150-200, and attributedthe delayto shearassociated
with upstreamwakes.However, as the array densitydeclines
(ad --> 0), the criticalRed valuesshouldreturn to thoseof an
isolatedcylinder.From the above observations,the transition
from laminar to turbulent wake structurewithin the array is
expectedto occurat Red • 200. As is discussed
shortly,wake
structure(laminar versusturbulent) is importantin defining
the contributionof the wake to turbulent kinetic energyand
diffusionwithin the array.
The bulk drag coefficient,Co, is a function of population
density.To describethis relationship,we first considerthe
interactionof cylinderpairs. As summarizedin Figure 1, experimentson pairsof cylindershaveshownthat the wake of an
upstreamcylinder can suppressthe drag coefficient,Co, on
the trailing cylinder for both semi-infinite [Bokaian and
Geoola,1984;Blevins,1994]and finite cylinderlengths[Luo et
al., 1996], where Co is basedon the upstreamvelocity.This
effectincreasesasboth the lateral, T/d, andlongitudinal,L/d,
spacingbetween the cylindersdecreases,and it resultsfrom
two properties of the wake. First, the downstreamcylinder
experiencesa lower impactvelocitydue to the velocityreduc-
tion in the wake. Second,the turbulencecontributedby the
wake delaysthe point of separationon the downstreamcylinder, resultingin a lower pressuredifferential aroundthe cylinder andthusa lowerdrag [Zukauskas,1987;Luo et al., 1996].
Both of these wake characteristics
contribute
to the "shelter-
ing" effect describedby Raupach [1992] that diminishesthe
drag on downstreamelements.
Basedon the observations
for pairs of cylinders,we anticipate that the bulk drag coefficientfor an array, Cz>,will decreaseas the element spacingdecreasesor ad increases.To
explorethis trend, a numericalmodel was usedto extrapolate
the observations
made for cylinderpairs [Bokaianand Geoola,
1984;Re d = 2600] to estimatethe cumulativeshelteringand
bulk drag within a randomlyarrangedarray. Cylinderplacementwithin eachnumericalarraywasassignedusinga random
number generatorwith a placementresolutionof d/10. For
each cylinderi the local drag coefficient,Cz>i,was then assignedbasedon the proximityof the nearestupstreamneighbor. This assumesthat changesin Cz>iare setby the strongest
wake interaction(nearestcylinder)and neglectscumulative
effectsof multiple wake interaction.This is a reasonableassumptionfor sparsedistributions,that is, ad < 0.1 [Raupach,
1992]. The total drag,Fr, was estimatedas the sum of individual cylinderdrags,and usedwith (1) to estimateCz>.Fifty
realizationswere performedfor eachvalue of ad. The results,
presentedlater, indicate a strongdependenceCz>(ad) for
ad > 0.01. For comparison,severalstaggeredarray configurationswere alsoconsidered.Although different in detail, the
curvesCz>(ad) derivedfor the staggeredarraysalso showa
declinein Cz>within increasingad.
2.2. Turbulence Intensity Within Emergent Vegetation
Even for sparsepopulationsof emergent vegetation, the
productionof turbulencewithin stemwakes,Pw, exceedsthe
productionthroughbed shear,Ps, overmostof the flow depth
[Nepfetal., 1997;Burkeand Stolzenbach,
1983].On the basisof
this and the further assumptionof homogeneity,the turbulent
kinetic energy budget is reduced to a balance between the
wake productionand the viscousdissipation,e, that is,Pw • e.
This simplificationis confirmed experimentally,as discussed
shortly.The wake productionis estimatedas the work input,
FrU, whereFr is describedin (1), that is, per unit mass:
NEPF: DRAG, TURBULENCE, AND DIFFUSION
481
vegetatedflowsasthe obstruction
of the flowby stemscreates
similarvariationsin flow path. Considerparticlesreleasedat
(x, y) - (0, 0) that advectthroughthe array at a mean
velocity,U. For simplicityassumethat advectiondominates
diffusionin longitudinaltransportsuchthat in eachtime interval,At, the particlesmovedownstreama distance,Ax UAt. In the sametime, eachparticlehasprobability[aAx] of
• •
0 '•
• '"'-"' /AYB
encountering
a stem,andif a stemis encountered
the particle
is forcedto moverightor left alongthey axisa distanceAy Bt=t
•3d, where •3 is an O(1)-scale factor. On averageand after
Figure 2. Mechanicaldiffusionarisesfrom the physicalob- manysuchcollisionsthe particlesmove right and left with
structionof the flowby stems.When a stemis encountered
the equalprobability.
After N steps,whereN is large,the lateral
particlemustmovelaterallyAy -- d. For a largenumberof (y) positionof an individualparticleis givenby a Gaussian
particlesreleasedat t - 0, and after manysuchencounters,
with variance;
the distribution
of particlesat timet is Gaussian
withvariance, probabilitydistribution
oa • [ad]Udt, describing
a Fickiandiffusion
process.
o-2= [aAx](13d)2t/At= 132[ad]Sdt
(7a)
OoøoO
^,_-o
",--0ø
m
t
1
Pw= 5 Cz)aU3
(3)
[Hoelet al., 1972].For a largenumberof particlesthismodel
describes
a Fickiandiffusionprocessfor whichthe varianceof
the particledistribution
increases
linearlywith time,suchthat
the effectivediffusioncoefficientmay be taken to be
This assumesthat all of the energyextractedfrom the mean
flowthroughstem(cylinder)dragappearsasturbulentkinetic
Dm= -•-[ad]Sd.
(7b)
energy.
Thisassumption
islimitedbelowRea < 200 wherethe
viscousdrag,whichdissipates
meanflow energywithoutgen- Becausethe processes
of mechanical
and turbulentdiffusion
eratingturbulence,becomesincreasingly
important.As the areindependent,
theirvariances,
andthustheircontribution
to
fractionof viscous
dragincreases,
Pw decreases
fromthevalue the totaldiffusivity,
are additive.The totalhorizontaldiffusivgivenin (3).
Assuming
thecharacteristic
lengthscaleof theturbulence
is
setby the stemgeometry,
that is, d, the dissipation
ratewill
ity is then givenby
D=a[Cz•ad]U3+
[-•]ad, (8)
scale as
Ud
S • k3/2d
-1,
(4)
in whichthe net diffusivityhasbeennondimensionalized
using
the mean velocity,U, and cylinderdiameter,d. The scale
wherek isthe turbulentkineticenergyper unit mass[Tennekes
factora = a•a 2 combines
thoseappearingin (5) and (6).
andLumley,1990,p. 68].Equatingthiswith(3), theturbulence
intensityis givenby
x•_a•[C•ad]U3
U
'
3.
(5) 3.1.
Methods
Laboratory Experiments
Laboratoryexperiments
wereconducted
in a 24-m-longby
where a• is an O(1)-scale coefficient.Thus the turbulence 38-cm-wide,
glass-walled
flumewith a flow depthh = 15 cm,
intensity
increases
withbulkdragcoefficient
andwithpopula- shownin Figure 3. Model arrayswere constructed
from 1/4
tiondensity.
Note thatbecause
CD is a decreasing
functionof inch(0.64cm)woodencylinders
at densities
betweena - 1.2
population
density,
ad, theturbulence
intensity,
x/k/U, asso- and 10.5m-• (at/ = [0.008-0.07], andequivalent
to 200ciatedwithwakeproduction
maybe considered
a functionof
ad alone.
2.3. Diffusion Within Emergent Vegetation
The net diffusionwithin a cylinderarray is determinedby
J•
3m
•,
turbulentPrandtlnumberof 1, the turbulentdiffusivitymaybe
Flow
as
Straightener
Dt-- ol2x/•d,
(6)
5m
"-I
Dye Injection Velocity ImagingCamera
twoprocesses,
turbulentandmechanical
diffusion.
Assuming
a
written
Model Array
•11•
Probe
, _:t
IllIll
IIIIIII
IIIIII
IIIIIIIIIIIii
Ill III IIIlllllllJl
-•r-rTI
.7
IIIIIIIIIlllll
I I"
I
whereagainthecharacteristic
lengthscaleof theturbulence
is
setbycylinder(stem)diameter[Nepfetal., 1997].Thevelocity
scale,X/•, andthusthe turbulentdiffusivity
is linkedto the
arraydragthrough(5). Becausethe arraydiffusivityis not
isotropic[Nepfet al., 1997],the scalefactor,a2, will differ
Horizontal
Light
Sheet:
Position
ofImaging
Plane--•
Figure 3. Laboratoryexperiments
were conductedin a 5-mlong model arrayfitted into a 24 m recirculating
flume.A
continuousdye releasewas usedto observethe horizontal
diffusivityat mid-depth.The diffusivity
wasestimateby fitting
between horizontal and vertical diffusion.
the lateralprofilesof meanconcentration
at multiplelongituMechanicaldiffusion,D m, is commonin porousmediaflow dinal stationsto the theoreticalGaussiandistribution.Velocity
andreflectsthe dispersal
of fluidparticlesdueto variabilityin wasmeasuredusingboth acousticDoppler (ADV) and laser
individualflow pathsimposedby the tortuousityof the pore Doppler(LDV) techniques.
Surfaceslopethroughthe array
usingtwo0.2mmresolution
wavegauges
(WG).
spaces.
As shownin Figure2, thisconcept
canbe extended
to wasmeasured
482
NEPF: DRAG, TURBULENCE,
2000stemsm-2). The densities
wereselected
basedon actual
stempopulationsobservedin saltwater[Gambiet al., 1990]and
freshwaterenvironments[Kadlec,1990]. The cylinderswere
mountedinto half-inch(1.3 cm) Plexiglasboardsand cut long
enough to protrude through the water surface. The dowels
were arrangedrandomlyusinga lm templategeneratedby the
numericalprogramdescribedabove.The templatewasapplied
five times, end to end, to createthe total test sectionarray of
5 m length. The array extendedfrom wall to wall to prevent
accelerated
currents
close to the wall.
The
base boards
ex-
tended 3 m upstreamof the array and were tapered to the
bottom to eliminate flow disruption.Smoothinlet conditions
were achievedusingmats of rubberizedfiber to dampeninlet
turbulence, and a 0.5 m sectionof honey comb to eliminate
swirl.
The velocity components(u, v, w), correspondingto
streamwise,lateral, and vertical directions,respectively,were
measuredwithin the array usinga 3-D acousticDoppler velo-
AND DIFFUSION
Suu[cm
2 s]
101
loo
10-1
10-2
Frequency
(s-1)
Figure 4. Velocity spectrum,S,
for U = 5.5 cm/sand a =
0.028cm-•. Levelingof spectrum
between1 and4 Hz correspondsto stem-scalegeneration.An inertial sub-rangefit, solid
line, is usedto estimatedissipationrate.
cimeter(ADV) anda 2-D laserDopplervelocimeter(LDV) at
a crosssectionpositioned at the mid-length of the array. A
longitudinaltransectthroughthe array demonstratedthat this
positionwasrepresentativeof uniformflow conditions,that is,
not affected by the leading edge. Six-minute records were
collectedat 25 and 100 Hz for the ADV and LDV, respectively.While the ADV couldbe deployedwithout disturbance
to the array configuration,the LDV typically required the
displacementof two or three dowelsto clear its optical path.
Becausethe flow field within an arrayis inhomogeneous
at the
scaleof the arrayelements,a horizontalaverageis necessary
to
representthe mean,homogeneous
(in x andy) velocitystatistics [Kaimal and Finnigan, 1994, pp. 84-85]. Profiles were
measuredat five lateral positionsfor eachflow condition.The
mean and turbulentvelocitystatisticswere then averagedlaterally to find the representative,homogeneous
values[Nepfet
al., 1997]. The integraltime scale,T u, wasevaluatedfrom the
autocorrelationfunction and multiplied by U to form the Eulerian integral length scale:L u = UT u. The rate of turbulent
dissipation,s, was evaluatedfrom the velocity spectra,Suu,
usingthe inertial subrange
A pair of resistance-type
surfacedisplacementgauges(0.2
mm resolution),positioned5 cm upstreamand downstreamof
the array was used to measuredthe surface slope, Oh/Ox,
acrossthe entire array.The bulk dragcoefficient,Co, wasthen
evaluatedusingthe followingforce balance:
1 (•_) Oh
(1- ad)CBU
2+ 5 CDad U2=ghOx'
(9)
where the first term on the left is the drag contributedby the
base of the array, with a bed drag coefficient,CB = 0.001
[e.g.,Munsonet al., 1990,p. 673], the secondterm on the left
is the drag contributedby the stems,and h is the flow depth.
The drag contributed by the glasswalls is negligible and
droppedfrom (9) for simplicity.
The horizontal diffusivitywas estimatedusinga continuous
plumeof dye releasedthrougha 0.16 cm (1/16 inch) stainless
steel tube at mid-depth,midwidth, and 2 m from the leading
edge.The injectionvelocitywas carefullymatchedto the ambient flow to avoidjet-inducedmixing.The vertical centerline
of the evolvingdyeplumewasilluminatedby a laserlight sheet
Suu=A
,
which causedthe dye to fluorescewith an intensityproporwhere A • 1.5 is an experimentallydetermined,universal tional to the dye concentration[Monismithet al., 1990].Video
constantfor turbulent flows [e.g.,Kundu, 1990,p. 441]. As an imagesof the plumewere recordedand then transferredframe
example,Figure 4 showsa spectrummeasuredat mid-depth by frame to an image analysissoftwarepackageusinga frame
withinthe dowelarraywith U = 5.5 cm/sanda - 2.8 m-•. grabber.Using a 100-frame average,the mean lateral concenNote that the spectrum"levels"between1 and 4 Hz, indicating tration profile was recordedat multiple longitudinalpositions
a local-scaleinput of energy,here vortex sheddingand wake from 20 to 100 cm downstreamof the injectionpoint. For the
highestdensityarray only, it wasnecessaryto removethree to
production.For a singlecylinderwith d - 0.6 cm and U =
5.5 cm/sthe sheddingfrequencyfalls atfs - 1.8 Hz. However, five dowels at each measurement section to allow sufficient
the sheddingfrequencyis affected by the proximity of other penetrationof the light sheet.Finally, in regionsfor whichthe
cylinderswithin the array,potentiallydoublingas the cylinder lateral variance of the plume increasedlinearly in x, that is,
= constant
(Fickiandiffusion),
thediffusivity
wasesspacingdecreases[Weaver,1993]. Thus a range of shedding OO'2/OX
frequencies(here, 1.8-3.6 Hz) is observedwithin the random timated by fitting the averagedprofiles, containing150-200
array, reflectingthe range of local array densitiesand its im- points,to the theoreticalGaussianplume profile [e.g.,Fischer,
in this study,the vertical
pact on sheddingfrequency. For higher velocity casesthis 1979,chap.2]. Althoughnot discussed
wake-scalesignatureis pushedto higherfrequencies.For these diffusivitywas similarlyestimatedusinga verticallyorientated
by Nepf et al. [1997].
casesthe samplingrate was increaseto 100 Hz (using the light sheet.Theseresultsare discussed
LDV) to resolvethis signature.Finally, the vertical turbulent
transport,T = O(w'k)/Oz, and bed-shearproduction,Ps = 3.2. Field Experiments
A similar dye plume techniquewas used to measurediffu-u'w'(OU/Oz), terms were also evaluatedfrom individual
velocityrecordsand then averagedlaterally.The wake produc- sivityin the field within an emergentstandof Spartinaalternition was evaluatedfrom (3).
flora (smoothcordgrass)
in the Great Sippewisset
Marsh,Cape
18/32/3I
//]2/3œ-5/3
NEPF: DRAG, TURBULENCE,
AND DIFFUSION
483
positionsspacedat 10 cm acrossthe mid-depthof the plume.
At the startof eachposition30 swereneededto fully purgethe
lines,this data was later removedbefore further analysis.The
averageconcentrationat eachpositionwasdeterminedfrom 1
min of sampleddata, the shortesttime for which mean statistics were stable. The samplingtime was constrainedby the
need to completethe profilewithin a periodof steadycurrent.
The currentspeedwasestimatedbefore and after eachprofile
wascompleted,by measuringthe transittime of a smallpatch
of dyeinjectedat mid-depthacrossa short,fixeddistance.The
measurementwas repeated 10 times for each estimate.Over
the severalfield trials a range of velocityconditionswere observed,with U = 3.0-7.4 cm/sandrea = 200-600. Finally,
the diffusivitywas determinedby fitting the measuredprofile
points to a Gaussiandistributionusing at least squarestechnique.An exampleof this fitting procedureis givenin Figure
(b)
Fthodamino[ppb]
5b.
i
200
-
1
4.
Results
_
4.1.
Laboratory Observations of Bulk Drag Coefficient
Figure 6 presentsthe model results,Cz>(ad), for both random (solidline) and staggeredarrays(dashedline). The grey
shadingrepresentsone standarddeviationaround the mean
for the random array.Three configurationsof staggeredarray
1 O0
110
120
130
140
150
160
were considered,definedby n, the ratio of the longitudinalto
HorizontalCoordinate [cm]
lateral spacingbetween successrows. For both random and
Figure 5. (a) Field experimentat Sippewisset
Marsh, Cape staggeredarraysthe model predictsrelativelyconstantvalues
Cod. Dye injected at a continuousrate is measured 1.5 m of Cz> up to ad • 0.01 and a steady decline beyond this
downcurrentusinga continuouslyrecording,flow-throughflu- density.In general,the bulk drag coefficientdropsoff more
orometer.The samplingport is traversedhorizontallyto mea- rapidlyfor the staggeredarrays.This reflectsthe fact that wake
sure the plume concentrationprofile. (b) Determinationof shelteringis strongestandpersistsfor the greatestlongitudinal
diffusivityfrom field data usingexponentialfit. Concentration spacingwhen the cylindersare aligned,that is, T/d = 0 in
at eachpoint is averagedover 1 min. U - 5.5 cm/s,a = 2.1 Figure 1. This alignmentoccursintrinsicallyin staggeredarm -• and D = 0.73 cm2 s-•
rays,betweeneverym th row, but only stochastically
in random
100 -
,
ß
arrays.
Cod, Massachusetts.
This grasshas a simple cylindrical-like
morphology,and under normal tidal conditionsexhibitsonly
limited bending.Thus the stemsand leavesresemblethe cylinder array on which our model and laboratoryexperiments
were based.The stem densitywas varied through pruning to
observe conditions at three values, n -
96, 196, and 370
1.6 ........
,
........
'
"
'
'
o
o
o
1.2 ,--•,-'-•,•--o__•.-,...
•..
ø .,
o
stemsm-2. The stemdensitywasdetermined
by randomly
casting
a 0.1 m2 frameoverthevegetation
andcounting
the
CD
-
numberof stemswithinthe frame.Thiswasrepeatedfivetimes
for each condition. In addition, 100 stems were collected from
the site to determinean averageleaf count (3.2 +_0.3 leaves
per stem) and width (d = 0.69 _+ .03 cm). Thesecharacteristicswere then usedto estimatethe vegetationdensity,a 2.1, 4.3, and 8.2 m-•.
The tracerwaspreparedby mixing2 mL of a 20% Rhodamine dye solutionwith 10 L of marshwater. A portion of the
solution
was saved for later calibration.
The solution
0.8
0.4
10-3
was re-
leasedthrougha 1/4 inchL tube at mid-depth(h • 1 m) and
at the ambientflow speedusinga variable-speedpump. As
shownin Figure 5a, the dye injectiontablewasorientedwith its
long axis(2 m) perpendicularto the currentto eliminatethe
interferencefrom the legs.The plumeconcentrationwasmeasured1.5 m downstreamusinga recordingfluorometerin continuousflow mode.While the concentrationwas continuously
sampledat 10 Hz, internallyaveraged,and recordedat 1 Hz,
the sample port was progressivelymoved to six horizontal
n= 112 1 2
\
10-2
'
[]
1•0-1
ad
Figure 6. Wake interferencemodel predicts that the bulk
drag coefficient,Cz>, decreaseswith increasingarray density,
ad, for both random(solidline) and staggered(dashedlines)
arrays.The data alsorepresentboth staggered(opensymbols)
and random (solid symbols) arrays. A summary of data
sources,flow conditions,and symbolsin givenin Table 1. For
the staggeredarrayspitch is definedby n - ratio of longitudinalto lateralrow spacing,that is,n - 1 is a squarestaggered
array.
484
NEPF: DRAG, TURBULENCE,
AND DIFFUSION
Table 1. Summaryof ReynoldsNumber and Array Configuration
Source
R ea
Dunn et al. [1996]
Segineret al. [1976]
Kaysand London [1956]
Petryk[1969]
1,000-4,000
1,000
1,000
10,000
staggered,n
staggered,n
staggered,n
random
staggered,n
Zdravkovich
[1993]
1,000
staggered,
n - 1'
Presentstudy
4,000-10,000
random
Model
>•200
random
Symbolin
Figure 6
Configuration
= 1/2
= 1
= 1-3
open circle
open diamond
= 1, 2
triangle
squarewith slash
square
solid diamond
solid circle
solid line
dashed lines
staggered,n = 1/2-2
Here, n is ratio of longitudinalto lateral row spacingwithin staggeredarray.
*One third, 2/3, and fully staggered.
The model resultsare comparedto laboratoryobservations
made in this study,aswell as to a collectionof other studiesin
both random and staggeredarrays.A summaryof the Reynolds'numberand arrayconfigurationfor eachstudyis givenin
Table 1. Note that for Dunn et al. [1996], the rods are submerged, but that the drag coefficient,Cz•B in that study,is
definedon the basisof a verticalaverageoverthe heightof the
array only and so is roughly equivalentto Cry. For all other
studiesthe cylindersextend through the entire flow domain,
equivalentto an emergentcondition.A reasonableagreement
between the model and the experimentaldata is observed,
evenfor ad > 0.1 for whichthe assumptionthat the nearest
upstreamcylinderalone setsdegreeof wake shelteringmay be
violated [Raupach,1992]. The agreementbetweenmodel and
data suggests
that the suppression
of Cz• is correctlydescribed
by the wake shelter effectson which the model is based.Finally, althoughvalues of large ad were included for model
comparisons,
the relevantrangefor aquaticvegetationconditions is ad = [0.001-0.1].
4.2.
Laboratory Observations of Turbulence Structure
The assumptions
made earlier regardingthe productionand
the scale of turbulencewithin the array were confirmedby
observation.First,within the arraythe Eulerian integrallength
scale,L•,, wasfairly uniform over depthwith L u/d • 1.5, L•,
• 1 cm (Figure 7a). This value is diminishedfrom that observedfor unobstructedflow in the same channel,L•, = 5-6
cm, and supportsthe assumptionthat within the array turbulence is rescaledto the obstructiongeometry,that is, d. Consistentwith this, there is an excellentagreementbetween the
spectrum-based
estimatesof dissipationrate and thosebased
on the scaleassumptiont• - d (e.g., Figure 7b). Here, the
dissipationratesare normalizedby an equivalentshearveloc-
Finally,the dependence,suggested
by (5), of the turbulence
intensityon the dimensionless
parameter,Cz• ad, is confirmed
in Figure 9. The valuesshownare depth-averagedturbulence
intensity, excludingregions near the bed for which P• was
non-negligible.
From the data,a• = 0.9 _+0.1, in (5), with 95%
certainty.
4.3.
Diffusion: Laboratory and Field Observations
Figure 10 compareslaboratoryobservations
of diffusivityfor
conditions
with
and without
turbulent
wakes.
As described
above,the existenceof laminar wakeswas confirmedby the
absenceof vortex shedding.The dimensionlessdiffusivities
observedfor conditionsproducingturbulentwakes(opencircles)are more than an order of magnitudegreaterthan those
observedfor laminarwakes(solidcircles),and this difference
is attributedto enhancedmixing after the onsetof wake turbulence.
z/h
1
I
(a)
(b)
I
E, k3/2/ d
0.5 -
ity, Um= [ #h(dh/dx)] •/2, andtheflowdepth.
Figure 8 showsthe calculatedterms from the turbulent ki0
I
I
I
netic energy budget. The dissipationrate, s, and the wake
0
2
-4
-3
-2
-1
0
production,Pw, are nearly constantover depth and balance
one another,while the bed-shearproduction,Ps, andturbulent
L/d
u
[F_.,
k3/2/d] [h/ U1313]
transport,T, termsare negligibleexceptvery closeto the bed.
Theseobservations
further supportthe assumption
that turbulence levelsare controlledby the balanceof wake production Figure 7. Observationsverify that the turbulence length
scale,e, is set by the stemgeometry,that is, e --- d. (a) The
and dissipation.For comparison,the boundary-layer
scaling,s
Eulerian integral length scalenormalizedby the cylinderdi• Um3/(gZ),isalsoincluded
(solidline).It underpredicts
the ameter.(b) Estimatesof dissipationrate basedon the scaling
dissipationrate except close to the bottom where bed-shear e --• d (circles)matchthe observeddissipation
rate (squares)
production,Ps, increasesto competewith Pw. In both Figures basedon spectrumfit. Dissipationestimatesare scaledon an
7 and 8 constraintsof the facilitydid not permit measurements equivalent
shearvelocity,Um -- [#h(dh/dx)]•/2, and the
above z/h > 0.8.
averageflow depth,h, in the test section.
NEPF: DRAG, TURBULENCE,
AND DIFFUSION
Z/h
485
T'otal
DiffL•sion
=
1
i
I
Turbulent+ Mechanical
i
0.8
œ
D/Ud
Pw -
•:(z/h)'
|
0.6
0.1
0.4
,Ps
0.2
0
i
•
/
I
-3
-2
-1
/
[Ps'Pw'T,•][h/Urn
•]
/
Eqs..
5 &6
/•
o:2=0.9
•
• •,/"•1
Eq.7, [• =1
J
10-2
Figure 8. Selectedterms from the turbulent kinetic energy
7 for normalization.
Boundary-layer
scaling,
e --- U3m
(•Z)- 2,
is includedfor comparison(solidline).
•
,
J • , , il
ad
budgetfor a - 2.8 m-•. Dissipation
rate, e, (triangles)and
wake production,Pw (squares),balanceover mostof the flow
depth.Bed-shearproduction,Ps (diamonds),becomesimportant near the bed. Vertical transportof turbulent kinetic energy,T (circles),is negligibleoverthe entire depth.SeeFigure
•i=0.9
MechanicalDiffusion
/
0.01
/ Turbulent
Diffusion
/
10-•
Figure 10. Dimensionlessdiffusivityfrom laboratory observationswithin two wake regimes,Res - 400-2000 (open
circles)andRes = 60-90 (solidcircles),fittedto modelcurves
for mechanical(dashedline) and turbulent (dashed-dotted
line) diffusivity,aswell as the sumof both diffusionprocesses
(solidline). Field data,Res = 300-600 (triangles),alsofit the
model well.
The dashedlines in Figure 10 indicatethe two components
of diffusion, turbulent (dashed-dotted) and mechanical
(dashed),givenin (8); the solidline indicatesthe sumof these
two processes.The curve representingturbulent diffusionis
based on (5) and (6) and incorporates the dependence
Cr•(ad). The scalefactor a• = 0.9 (in (5)), is basedon direct
experimentalobservation(Figure 9). The secondscalefactor,
ot2 in (6), wasthenselectedto bestfit the observeddiffusionfor
the lowestvaluesof ad, whenmechanicaldiffusionis negligible
(or2 -- 0.9). The curverepresenting
mechanicaldiffusion,(7b),
mechanical
and turbulent
diffusion
curves fits the turbulent
wake observations
well (open circles),confirmingthe theoretical description.
The diffusivity measuredin SippewissetMarsh (Red =
300-600; Figure 10, triangles) is consistentwith both the
laboratorydata and the model for turbulent wake conditions,
althoughthe Re d in the field were on the low end of those
observed
in the lab. The normalized
diffusion
increases with
0.3
vegetationdensitywith a slopewhich matchesthe local slope
of the diffusionmodel.The averagefield diffusivitiesexceeded
the laboratory values by •30%. This may be attributed to
additionalsourcesof turbulencepresentin the field, for example, wind, or to differencesin morphology.However,the generally goodagreementbetweenthe laboratory(cylindermorphology)and the field data supportsthe use of this model for
the marsh grassSpartinaalternifiora.The model should also
work well for other simple morphology,such as reeds. For
more complicatedstem morphologythe relation Cr•(ad) requires more detailed evaluation,after which the model presentedhere providesthe basisfor characterizingturbulence
and diffusivity.
0.2
5.
uses the assumedscale factor, /3 = 1. The data for laminar
wakes(Res - 60-90) generally.fall abovethis line, and this
maybe attributedto the presenceof bed-generatedturbulence
evenin the absenceof vortexshedding.Alternatively,note that
for/3 = 2, alsoa reasonablescaleassumption,the mechanical
diffusionline falls directlythroughthe middle of the laminar
wake observations
(solid circles).Finally, the additionof the
•k/U
0.4
5.1.
Discussion
Comparison of Vegetated and Unvegetated Flow
Conditions
0.1
0.01
0.1
adC
Figure 9. Turbulence intensity within the stem array increasesto the 1/3 power of the dimensionless
parameter,Cr•
ad, as predictedby scalingarguments,Pw "' •. Res = 400900.
The modeldevelopedabovecharacterizes
drag,turbulence,
and diffusivityfor flows through emergent vegetation.This
modelwill now be usedto comparehydrodynamicconditions
in a regionof emergentvegetationwith conditionsin an equivalent regionof unvegetated,open-channelflow, denotedusing
subscripto. There are two key resultsof this comparison:(1)
Turbulenceintensityincreaseswith the introductionof sparse
486
NEPF: DRAG, TURBULENCE,
vegetation,but then decreasesas stem populationincreases
further. (2) Diffusivitywithin the vegetatedsystemis consistently lessthan in the equivalentunvegetatedsystem,owing
principallyto the decreasein eddyscalecausedby the presence
of the vegetation.
First, we considerthe mean flow. The steadyvelocity,U,
drivenby a free-surfacegradientis determinedfrom the force
balancegivenin (9). For simplicity,it is assumed
that Ca is not
affectedby the presenceof vegetation;that is, Ca is the same
in both the vegetatedand unvegetatedsystems.This assumption is not strictlycorrect,asthe ratio of bed-frictionvelocityto
mean flow speed,u ,/U, increasesslightlywith ad [Nepf et al.,
AND DIFFUSION
lOO
/:J
k/k • f(h/d)t
lO
k/k
o
• ........:..::.----.•.
.......y.jz...•...
................
:..
D/D
o
0.1
1997],suchthatCa = (u,/U) •/2 alsoincreases
slightly
with
vegetationdensity.Two valuesof bed-dragcoefficient,Ca =
0.001 and 0.005, equivalentto friction factors,f = 0.0080.04, are selectedto representconditionsof smoothand rough
earthenbeds.The open-channel
condition,Uo, is givenby (9)
with
ad -
0.
The depth-averaged
turbulentkineticenergy,k, reflectstwo
productionterms, the bed shear and the wake production.
When wake productiondominatesthe turbulentkineticenergy
is given by (5). When bed-shearproductiondominates,the
0.01
i iii•
10-5
i , , ••,,, , , , , ,,,,/
10-3
10-1
ad
Figure 11. Comparisonof turbulent kinetic energy,k, and
diffusivity,D, for vegetatedandunvegetated(subscript
o) flow
conditionswith the samevelocity.Ca - 0.005 (solidlines)
and 0.001 (dashed lines). The curvesrepresentqualitative
trendsonly.The turbulenceratio, k/ko, increasessteadilywith
turbulent
kineticenergyis givenby,k •/2 "• CaU2 [e.g.,Nezu stempopulation,at/, reflectingthe contributionof wake proand Rodi, 1986]. Combiningtheseproducesthe scalingequa- duction.The diffusionratio, D/Do, is affectedby reductionin
eddy scale,and so is a function of the ratio of flow depth to
vegetationscale,h/d.
tion
k-
(1 - ad)CBU2 q- [Coad]2/3U
2
(10)
(the terms to the left of the plus sign representbed-shear
= h. For densepopulations,AS < eo, the stemsbreak apart
production;those to the right representwake production)
channel-scale
eddies,reducingthe mixing-lengthscale,until at
which is correctin the limits of ad(-• 0 or 1) but may be
at/> 0.01, I• • d [Nepfetal., 1997].The modelassumes
that
oversimplified
when the two termsare comparablebecauseit
• varieslinearly from h to d betweentheselimits.
does not account for nonlinear interactions between the two
By using(9)-(11) the flowconditions
in a vegetatedchannel
processes,
for example,changesin bed sheararisingowingto
and an equivalent,unvegetatedchannelcannowbe compared.
the presenceof stems.Assumingthat Cz• = 1, wake producWe considerthree ratiosof depthto stemscale,h/d = 10, 20,
tion dominatesfor ad > 0.001 and 0.01, with Ca = 0.001
50, to be representativeof field conditions.Keep in mind that
and 0.005,respectively.
The equivalentopen-channelcondition
(10) and (11) are scalerelations,so that the parametersthey
is givenby (10)withad - 0, thatis,ko • CaUo
2.
describe,D/Do andk/ko, respectively,
provideonlyqualitative
Becauseturbulent and mechanicaldiffusionare indepenpredictionsof flow behavior.
dentFickianprocesses,
their contributionto total diffusion,D,
For simplicity,we first considerthe casefor whichthe flow
is additive, that is,
speedis the samein both the vegetatedandunvegetatedchanO '• kl/2eq-[ad]Sd.
(11) nels,that is, U = Uo (Figure 11). For equivalentflow speed
the turbulentkineticenergyratio, k/ko, is greaterthan one for
The first term representsthe turbulent diffusionscale,Dr, all valuesof at/> 0, andincreases
to O (10) at highvegetation
definedby the turbulentvelocityscale,k •/2, and a mixing density.This trend reflectsthe additionalsourceof turbulence
lengthscale,•?.Note that becausek is givenby (10), a scale providedby the stemwakes.Even thoughk/ko > 1, D/Do -<
relation, (11) is strictlya scalerelation as well, so that for 1 for most valuesof at/ (Figure 11). For sparsevegetation
simplicitywe have excludedpreviouslydefinedscalefactors. (smallat/), D/Do • 1 despitethe additionof wake-generated
The secondterm in (11) representsthe mechanicaldiffusion, turbulence. This occursbecauseeddies that are channel scale,
D,•, basedon (7b) excludingthe scalefactor, /3. The mixing h, can persistin sparsevegetation(as describedabove),and
lengthscale,•, dependsstronglyon the presenceof vegetation. becausethe additionalturbulentkineticenergyis introducedat
In unvegetatedflow, •?and thusD t increasewith the scaleof the stemscale,d << h. The larger, channel-scaleeddiesdomthe diffusingpatchuntil the largestlengthscaleis reached,that inatethe macroscalediffusivetransport.The additionof smallis, the lengthscaledefinedby the flow domain[Okubo,1971]. er-scale mixing does, however, smooth out local gradients
In contrast,in vegetatedflow the mixing-lengthscaleson the withinthe turbulentpatch[Nepfet al., 1997].As at/increases,
vegetationgeometry,• • d, for all patchsizesgreaterthan d the stemsimpingeon the channel-scaleeddies,and theseed[Nepfet al., 1997].To evaluate(11), we setthe followinglimits diesare brokenapart.All of the turbulenceis then rescaledto
on the mixinglength.First, for simplicitywe assumea mixing the stem geometry.The reduction in eddy-scalecausesthe
length,•o • h, for unvegetatedopen-channel
flow (ad = 0). relativediffusion,D/Do, to decrease,eventhoughthe relative
If only sparsevegetationis introducedto this flow, that is, turbulence,k/ko, continuesto increase.A similar trend was
vegetationwith spacingAS > •o, the channel-scale
eddies,•o, observed in a channel flow downstream of a mesh. Turbulence
persistand continueto dominatethe diffusivetransport.Thus intensitywas enhancedby the presenceof the mesh,yet the
for sparsevegetationthe dominantmixinglengthis still• = •o diffusivitywas diminishedbecausethe eddyscalewasreduced
NEPF: DRAG, TURBULENCE,
AND DIFFUSION
487
1
o.1
: 10 i
•:
O.Ol
0.1
i
o.1
10-6
(a)
10'4
ad
10'2
10'6
(b)
10'4
ad
10-6
10'2
(c)
10-4
10-2
ad
Figure 12. Comparisonof flow conditionswithinvegetatedandunvegetated(subscripto) regionsunderthe
sameforcing,that is, surfaceslope:(a) velocity,(b) turbulentkinetic energy,and (c) diffusivity.Bed drag
coefficients
of 0.01(solidlines)and0.005(dashedlines)areconsidered
aswell asthreeratiosof depth-to-stem
diameter.
[Hinze, 1987, pp. 448-449]. In Figure 11 the declinein eddy
The trendsin turbulence
intensity(kX/2/U)suggested
byFigscaleis initiated at ad •- [0.002, 0.001, 0.0002] for h/d = ures12a and 12bhaveimportantimplicationsfor canopyecol[ 10, 20, 50], respectively.
The transitionoccursearlier(small- ogy. For example, an increasein turbulenceintensitycould
er ad) in largerflow depths,becausethe largerchannel-scale benefit the vegetationby augmentingnutrient uptake and/or
eddies,eo '" h, are impactedat a largerstemspacing(AS). gas exchange[Andersonand Charters,1982]. Within wasteOnce the vegetationis sufficientlydense (ad •- 0.01) the treatment wetlands the turbulence level can control the rate at
stem-scaleturbulencedominatesand e --- d. With the length- whichcontaminantsare deliveredto vegetationsurfaceswhere
scalefixed,the continuedincreasein turbulenceintensitywith degradingmicrobeslive and thus may determine the rate of
increasingad producesan increasingD/Do. In addition,me- degradation[e.g.,Gantzeret al., 1991].Finally,increasedlevels
chanicaldiffusionbecomesimportantat thesehigh vegetative of turbulenceexperiencedat sparseplant densitymay explain
densities.
why patchyplantings,unlike denseones,do not promotethe
In the abovediscussion
(Figure 11), vegetatedand unveg- desiredsedimentaccumulationfor bank stabilization(J. Hart,
etated flow conditionswere comparedusing identical mean personal communication,1998), and may actually create
flow, U = Uo. For a more relevant but more complexcom- higherratesof erosion[Coppinand Richards,1990,p. 58].
The diffusivityratio for identical surface-slopeconditions
parison,we now considerthe two systems
underidenticalforcing, that is, identicalsurfaceslope,Oh/Ox. The velocity,tur- (Figure 12c) again reflectsthe importanceof reducededdy
bulence, and diffusion ratios for this condition are given in scalewithin the vegetatedregion.As describedabove,a rapid
Figure 12. Becausethe vegetationoffersadditionalresistance, declinein D/Do occursoncethe stemsimpingeon the channelthe velocitywithin the vegetatedchannelis alwayslessthan scaleeddies,and convertthem to progressively
smallerscales
that in the unvegetatedchannel,and the velocityratio, U/Uo, asat/increases.Once • --- d, for example,ad > 0.01, a more
decreasesas the vegetationdensity(ad) increases(Figure gradualdeclinein D/Do is observed,reflectingthe near bal12a). The drop in velocityratio is more rapid for greaterflow ance between decliningvelocity ratio, U/Uo, and increasing
depthsand smallervaluesof bed friction, CB, both of which contributionby mechanicaldiffusion,D m --- ad.
enhancethe fractionalcontributionof vegetativeresistance.
Changesin turbulentkineticenergy(Figure12b)reflectthe 5.2. Reynolds Number Dependence
A majority of the abovediscussiondescribesconditionsof
competingeffectsof reducedvelocityandincreasedturbulence
production,both of whichaccompany
the introductionof veg- Rea >• 200, for which Cr• is only a weak function of Rea
etation. These opposingtendenciesproduce a nonlinear re- [e.g.,Munsonet al., 1990,p. 614]. These conditionsare comsponsein which the turbulencelevelsinitially increasewith mon in coastalsystemssuchas the SippewissetMarsh usedin
increasingstem density,but eventually decreaseas ad in- this study [also, e.g., Leonard and Luther, 1995]. However,
creasesfurther.A similarresponsewaspredictednumerically conditionsof Re a < •- 200 are also common in the field, in
for flow throughemergentvegetation[Burkeand Stolzenbach, particular in freshwatersystemswhich are not exposedto
1983] and within bottom roughnesselements[Eckman,1990] strongtidal flows.For Rea < •- 200 wake productionis negand was observedin a flume studyof flow throughreal stems ligibleand the turbulentcomponentof diffusionwill be greatly
of ZosteraMarina [Gambi et al., 1990]. Note, however,that reduced,determinedsolely by the bed shear (and possible
(10) andFigure (12b) excludethe contributions
of turbulence wind shear).For theseconditions,the total diffusionis likely
generatedby wind and wave action,which may be presentin dominatedby mechanicaldiffusion.This study suggeststhat
the field. Becauseemergentvegetationdampswaveenergyand mechanical diffusion alone is still sufficient to maintain macsheltersthe water surfacefrom wind stress,both of thesepro- roscalediffusionat levelsabovethe molecularvalue. Finally,
cesses
would affectthe ratio k/ko by more readilyaugmenting the removal of wake turbulencebelow Re a •. 200 also alters
turbulencein open-channelregionsthan in vegetatedregions. the dynamicsof wake interferenceand thusthe relation Cr• =
488
NEPF: DRAG, TURBULENCE,
f(ad). Further studyis neededto understandbulk dragunder
these conditions.
AND DIFFUSION
Eckman,J., A model of passivesettlementby planktoniclarvae onto
bottoms of differing roughness,Limnol. Oceanogr.,35, 887-901,
1990.
6.
Conclusions
Fischer,H. B., E. J. List, R. Koh, J. Imberger,and N. Brooks,Mixing
in Inland and CoastalWaters,Academic,San Diego, Calif., 1979.
Gambi, M., A. Nowell, and P. Jumars, Flume observationson flow
dynamicsin Zosteramarina(eelgrass)beds,Mar. Ecol.Prog.Ser.,61,
This paper presentsa model for drag,turbulenceand diffu159-169, 1990.
sion within emergentvegetationwhich capturesthe relevant
Gantzer,C., B. Rittmann, and E. Herricks,Effect of long-termwater
underlyingphysics,and coversa natural range of vegetation
velocitychangeson streambedbiofilmactivity,WaterRes.,25, 15-20,
densityand stem Reynolds'numbers.This work also extends
1991.
the cylinder-based
modelfor vegetativeresistanceby including Guardo, M., and R. Tomasello, Hydrodynamicsimulationsof a constructedwetlands in south Florida, WaterResour.Bull., 31,687-701,
the dependenceof the bulk drag coefficient,Cz>,on the veg1995.
etativedensity,ad, for Red > • 200. The model,confirmedby
experimental
results,
shows
thatturbulence
intensity,
kl/2/U,
is principallydependenton the vegetativedrag, and that for
vegetativedensitiesassmallas 1% (ad = 0.01), the bed-drag
and bed-shear production are negligible compared to their
vegetationcounterparts.The fraction of mean energypartitioned to turbulencedependson the ratio of form drag to
viscousdrag on the stems,which in turn dependson the morphologyand flexibility of the stems,and the stem Reynolds
number.In addition,the modelpredictsthat turbulenceintensityincreaseswith the introductionof sparsevegetationowing
to the addition of wake productionbut then decreaseswith
increasingpopulationdensityas mean flow speedis reduced.
Becauseof the reductionin eddyscale,diffusionis reduced
within an array. Specifically,for vegetativedensitiesgreater
than 1%, turbulencescalesare controlledby vegetationgeometry, that is, I?--- d. The mixing-lengthscale,t?,is thusreduced
from open-channelconditions,I?o --- h or larger. This shift in
eddyscalecontrolsthe turbulentdiffusion,suchthat turbulent
diffusivityis reducedwithin the vegetatedregion,despitethe
fact that turbulenceintensitymay be increased.For higher
vegetationdensitymechanicaldiffusionbecomesimportant,
that is, diffusion arising from the obstructionof direct flow
pathsby the vegetation(arrayelements),andwill dominatethe
overalldiffusionof scalarspeciesfor ad > O(0.1) or for Red
<•
200.
Acknowledgments. This work was completed under NSF CAREER Award, EAR 9629259.The author would like to thank Becky
Zavistoski,Jenn Sullivan, Al Tarrell, and Enrique Vivoni for their
assistance
with laboratoryand field experiments;andboth Ole Madsen
and Dennis Lyn for providingthoughtfuleditorial comments.
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489
(ReceivedMay 6, 1998;revisedOctober21, 1998;
acceptedOctober21, 1998.)