Laboratory Measurements on Turbulent PressureFluctuations in and
above Gravel BedsMartin Detert1; Volker Weitbrecht2; and Gerhard H.
Jirka,
F.ASCE3Abstract:Thestatisticsofpressureuctuationsaboveandwithinthreetypesofporousgranularbedssuchasingravelbedstreams,rivers,
and man-made canals are investigated by data gained via laboratory
ume experiments. The ow conditions examined include adiversity of
hydrodynamic loads that increase up to the point where single
grains are moving from time to time, without causing
severemodicationtothebedtextureandtherelatedpositionsofthepressuresensors.Analysisisperformedbymeansofhistogramsandspectral
techniques and vertical intensity proles. Two simplied equations
are found that describe the vertical decrease for the
standarddeviation of the measured uctuations indicating drag and
lift, respectively, nondimensionalized by the mean bed shear
stress. The formeructuation is described by a crude linear t,
whereas the latter clearly shows that the lift intensity decreases
exponentially in the porousbed with a decay distance of one to two
times the equivalent grain roughness. Within the subsurface layer
the standard deviation reachesa nonzero constant, mainly dominated
by long-wave pressure elds that are convected in the outer ow.
These ndings can be used infuture sediment transport models that
use force balance approaches to determine incipient motion
conditions.DOI: 10.1061/ASCEHY.1943-7900.0000251CE Database subject
headings: Open channel ow; Gravel; Turbulence; Power spectral
density; Laboratory tests; River beds.Author keywords: Open-channel
ow; Gravel bed; Turbulence; Pressure uctuation; Spectral density;
Interstitial.IntroductionThedynamicsofowovergravelbedsisdeterminedbythein-tricateinteractionbetweenturbulentvelocityeldsandpressureeldse.g.,
Hoand et al. 2005; Detert et al., Synoptic velocityand pressure
elds at the water-sediment interface of stream-beds, J. Fluid
Mech., unpublished, 2010. In order to understandthe detailed
mechanisms of bed instability leading to the
erosionofindividualparticles,
itisnecessarytodeterminethetemporalsequence of synoptic
velocityandpressure signals since bothquantities are jointly
responsible for the hydrodynamic forces onthe particles.However,
most of the past studies concerning gravel beds
un-derlyingopen-channel owfocusedonvelocitymeasurementsalone, such
as velocities at a point e.g., Aberle and Nikora
2006orvelocityeldse.g., Royetal. 2004;Hurtheretal. 2007.
Incontrast, veryfewstudieswithdetailedturbulent
pressuremea-surements have been reported to datee.g., Hoand 2005;
Smartand Habersack 2007. This limitation is due to the technical
dif-cultiesthathaveexistedsofarthatpreventedthedevelopmentofsufcientlysmall-scalehighresolutionpressuremeasurementdevices
in the laboratory.Understanding the genesis of the instantaneous
local pressureneeds a knowledge of the entire turbulent uid domain.
By takingthe divergence of the Navier-Stokes
momentumconservationequations, thePoissonequationfor
theuctuatingpressure pwithin an incompressible ow is obtained in
Reynolds decompo-sition, e.g., Chang et al. 1999 12p =
2uixjujxi+2xi xjuiuj uiuj 1where=fluid density; i , j =1; 2; 3
following the Einstein nota-tion; the spatial coordinates are xi;
and the velocity vectors are
ui.Thersttermontheright-handsideiscalledtherapidrespec-tively,
linear or mean-shear source term because it responds im-mediately
to a change in the mean velocity gradient. The secondtermis
calledtheslowor nonlinear or
turbulence-turbulencesourceterm.Itfollowsthatboththevelocitygradientsanduc-tuation
gradients in the entire ow domain inuencepat a cer-tainmoment, but
the impact of single sources decreases withdistance. Undisturbed,
direct pressuremeasurementswithintur-bulent ows are impossible, as
an ideal probe had to be innitesi-mally small in order not to
inuence the ow. Pointmeasurements usingaverysmall-scalemeasurement
setupre-cently conducted by Tsuji et al. 2007 are promising in
minimiz-ing this disturbance. Nonintrusive ultrasonic-based
pressure eldmeasurementshavebeenreportedby, e.g.,
HansandWindorfer2003 and Yu et al. 2005. However, truly complete
informationabout pressureuctuationsor
entirepressureeldsseemsonlyrealizable by numerical
simulations.Kim1989analyzedthepressureuctuationsinaturbulentchannel
owwithsmoothwallsobtainedfromdirect numerical1Postdoctoral
Hydraulic Engineer and Researcher, Laboratory of Hy-draulics,
HydrologyandGlaciology, ETHZurich,
Gloriastrasse37-39,CH-8092Zurich, Switzerland; formerly,
Institutefor Hydromechanics,Univ. of Karlsruhe, D-76128 Karlsruhe,
Germany corresponding author.E-mail: [email protected]
Engineer, Laboratory of Hydraulics, Hydrology and Glaci-ology,
ETHZurich, Gloriastrasse37-39, CH-8092Zurich, Switzerland;formerly,
Institute for Hydromechanics, Univ. of Karlsruhe, D-76128Karlsruhe,
Germany. E-mail: [email protected] Professor, Institute for
Hydromechanics, Univ. ofKarlsruhe, D-76128 Karlsruhe, Germany.Note.
This manuscript was submitted on April 16, 2009; approved onApril
2, 2010; publishedonlineonApril 15, 2010. Discussionperiodopenuntil
March1, 2011; separatediscussionsmust
besubmittedforindividualpapers. Thispaperispartofthe Journal of
Hydraulic Engi-neering,
Vol.136,No.10,October1,2010.ASCE,ISSN0733-9429/2010/10-779789/$25.00.JOURNAL
OF HYDRAULIC ENGINEERING ASCE / OCTOBER 2010 / 779J. Hydraul. Eng.
2010.136:779-789.Downloaded from ascelibrary.org by INDIAN INST OF
TECHNOLOGY - ROORKEE on 08/14/14. Copyright ASCE. For personal use
only; all rights reserved.simulationDNS. AsshowninFig. 1,
pincreasestowardthewall and reaches its maximum slightly above it.
A detailed analy-sis revealed that the slow source term is
substantially larger thantherapidterm, except veryclosetothewall,
yt/ 0.15notshown in Fig. 1, where=channel half-width, here equal to
theboundary-layer thickness. Analysis of pressure elds of ow
overand within a permeable bed was given by Breugem et al.
2006.They studied the inuence of wall permeability on turbulent
owson top and through a porous medium of cubes also by DNS.
Thevertical proleofpinthefreestreamroughlyagreeswiththatgiven by
Kim1989 but is supplemented by an exponential de-crease inside the
permeable wall see Fig. 1. The peak value justabove the permeable
wall is revealed to increase with Re=ud/ , where u=shear velocity;
d=characteristic
obstaclelengthscale;and=kinematicviscosity,andwiththeroughnessgeometry
function, =Vf / Vo, where Vf=volume of uid withinthetotal volumeVo.
Thisdependencyisattributedtoturbulenttransport across the wall
interface and the reduction in mean shearduetoaweakeningof
thewall-blockingandthewall-inducedviscouseffects,respectively.
Todate,anexperimentalvalidationoftheresultsfortheturbulenceintensityprolesfromBreugemet
al.2006 has not been available.Whereas the measurement of pressure
within a turbulent
owisextremelydifcultasanappropriatenoncontactmeasurementdevicedoesnot
exist, turbulencewall pressuresTWP canbemeasured as the measuring
technique can be incorporated into thewall.TWP
havemainlybeenstudiedineldsofacoustic, aero-nautic, or naval
applications, with a large number of works in the1970s and 1980s. A
review is given by Eckelmann 1988. Blake1970 measured
boundary-layer TWP with pinhole microphoneson both smooth and rough
boundaries. He showed that the
shapeofthepressurespectraisthesameforbothwalltypes,buttheydiffer in
their scaling. The former scales by a viscous length /
uandthelatterscalesbytheaveragegeometricroughnessheight.At the wall
the magnitude of p was found to be almost equal forsmooth and rough
walls. Emmerling 1973 used an opticalmethod to investigate the
instantaneous structure of wall-pressureelds. Zones of
high-amplitudepappeared in irregular time
in-tervalsandweretheoreticallyassociatedwithburstingphenom-ena.Maximalpressurepeaksofuptopmax=6pwereobserved.Thesevaluesareof
thesamemagnitudeastheonesfoundbySchewe1983, as he observed maximal
pressure peaks of up topmax=7p. Thesendingsimplythatthe TWP
canbecomeverylarge and therefore should dominate mass and momentum
transferin the case of a permeable wall and could possibly inuence
bedstability.
Characteristicwall-pressurestructureswerefoundwithhigh amplitudes,
whose sources are located in the buffer layer ofthe boundary layer.
The mean characteristic wavelength was iden-tied to be 145/ u,
indicating correlation with bursting phenom-ena. Fromthe
measuredprobabilitydensityhe
calculatedthatthesestructuresplayanimportantroleinthewallregionoftheboundarylayer.
A scalinglawforpwasgivenbyFarabeeandCasarella 1991.
Basedondatafromeight studies whichap-peared within 19701990 and
data from their own
measurements,theydevelopedarelationforpdependingonlnReandtheboundaryshear
o=u2. Klewicki et al. 2008 conrmedthisrelationshipbymeasurement
dataobtainedinthesalt playaofUtahs west desert, however, with a
slight change in the multipli-cative constants.The
aforementionedexplanations refer tothe pressure inauid or at a
wall. However, the hydrodynamic forces acting on
asinglegrainalsoexpressibleasforceperarea, i.e., pressureare hardly
comparable to the pointwise local pressure as the
owisindirectinteractionwiththeobstacle. Thetermsdragandliftare used
for the streamwise and vertical components, respectively,of the
hydrodynamic force. Many experimental
investigationswereundertakentoreveal thecharacteristicsandtopredict
thedragandliftforces.TherecentcontributionsbyHoand2005and coworkers
using single piezoresistive pressure sensors as wellas by
Schmeeckle et al. 2007 using force transducer techniquescomprise
up-to-date knowledge andadvancedmeasuringtech-niques.Vollmer et al.
2002 presented quasi-low-pass ltered
labora-torydatathatindicateanexponentialdecayofpressureuctua-tionswithinporousgranularbeds.
Smart
andHabersack2007gainedsophisticatedelddataofnear-bedandsubbedpressureuctuations
that showa near Gaussianfrequencydistribution,wherethenear-bedp3o.
However, uptonownoin-depthmeasurement
campaignwithunambiguousboundaryconditions,highly resolved in time
and space, has been performed to describetheverticalstatistical
propertiesofuctuatingpressureinandabove different porous beds. To
bridge this lack of knowledge isthemaincontributionofthepresent
paper,
wherethemeasuredverticalintensityofhydrodynamicloadsatandinporousgravelbeds
will be described statistically. This information will help
tounderstand the intensity and occurrence of physical
processeswithin gravel beds, especially to improve the analytical
predictionof the initial point of sediment entrainment.Experimental
SetupLaboratory FacilitiesThe experiments were carried out in a
rectangular laboratoryumeat theInstitutefor Hydromechanics IfH,
UniversityofKarlsruhe, withaneffectivelengthof17.0mandawidthofB=0.9
m. The water depth ranged from h=0.13 to 0.22 m. A right-handed
coordinate system is implied, where x is orientated in
thestreamwiseowdirection,yintheupwardvertical,andzinthetransverse
direction. x=0 holds at the middle of the measurementarea, y=0
denotes a nominal wall level, where an extrapolated logt of u y
would reach zero, and z=0 is located in the centerlineFig. 1.
Intensities of pressureuctuations. DNSresults for p/ otaken from
Kim 1989 smooth walls, Re=u/ =179 and Breu-gem et al. 2006 rough
permeable wall, Re=176, =0.60. Here,y=0refers to the smooth wall
and the roughness tops, respectively.780 / JOURNAL OF HYDRAULIC
ENGINEERING ASCE / OCTOBER 2010J. Hydraul. Eng.
2010.136:779-789.Downloaded from ascelibrary.org by INDIAN INST OF
TECHNOLOGY - ROORKEE on 08/14/14. Copyright ASCE. For personal use
only; all rights reserved.of the ume. The inlet was located at
x=10.5 m relative to themeasurement area to guarantee a fully
developed boundary layer.Theoutlet at x=+6.5
mwascontrolledbyavertical thinplateweir. Hence,
inuencesofbothinletandoutletwerenegligible.The bottom of the ume
had a slight slope of 0.05%. The waterdepth h was measured at three
points at x=9.33; 0.00;+4.42
mbyultrasonicprobes.Theyweremountedonexternalcylindrical water
tanks that were in hydraulic interconnectionwith the porous bed via
exible tubes 25 mm in diameter. Duetothisarrangement,
uctuationsandsmall-scaleoscillationsofthe actual water level were
low pass ltered.Bed MaterialThree different bed materials were
inserted: uniformgravel,gravel from the river Rhine, and spheres.
The river Rhine gravelwastakenfromagravel bankabout
10kmdownstreamoftheIffezheim barrage low water conditions,
02.08.2006. At this po-sitionanarmoringlayer withathickness of
onetotwostonediameters was found. The underlying material revealed
to be bi-modal, withcentersat d10=0.4 mmfor15%oftheweight
andd60=14.7 mmfor 85%of theweight. Table1summarizestheproperties of
the bed parameters. The grain sizes di were
obtainedbysieveanalysis. Theroughnessgeometryfunctionandthestone
densityswere determinedbyanexternal experimentalsetup. The
permeability coefcient kfwas obtained from Hazens1892 equation for
the uniform gravel and the Rhine gravel andfrom Kozemy-Carmans
equation Carman 1956 for the spheres.Measurement SetupFig. 2
illustrates the measurement setup. It consisted of an arrayof up to
16 miniaturized piezoresistive pressure sensors
MPPSslocatedwithinandslightlyabovethegravellayer.
Typically,thedataacquisitionwascarriedoutfor205s.
Additionalresultsre-garding the velocity regime using a two
dimensional 2D particleimage velocimetry PIV system and an acoustic
Doppler
currentprolerADCPcanbefoundinDetert2008andDetertetal.Synopticvelocityandpressureeldsatthewater-sedimentin-terface
of streambeds, J. Fluid Mech., unpublished, 2010.MPPSsFig. 3 shows
the MPPS geometry and components. The
principleoftheMPPSisbasedonthepiezoresistiveeffect.Incontrasttothe
piezoelectric effect, this effect only causes a change in
resis-tance, but it does not produce electrical charges. The core
of theMPPSis a micromechanical silicon wafer with implanted
pi-ezoresistors on its bending panel. For the MPPS used in this
studythe differential pressure is measured with reference to
atmo-sphericpressure. Thecomponentsfor theMPPSwereobtainedfromAktiv
Sensor GmbH, Berlin. The sensor elements ATD0.040-G00-BG-K1408 and
AUblank PGA-V0-D18Awere
as-sembledattheIfHtoadaptthemtotheirapplicationwithintheexperimental
ume. Depending on the conguration, the sensorsmeasure both the
surrounding static pressure and the surroundingdynamicpressure.
Thus, theoutput signal reectstheeffectiveforce per pinhole area in
the direction of the pressure tube.To miniaturize the pressure
transducer the amplifying blanketshad to be arranged in an external
box. Unfortunately, the length ofthe exible cables to the external
amplifying board could not beshorter than2.5mdue tothe boundaries
of the experimentalsetup. Thus, slight antenna-noise effects had to
be accepted. Flex-ible PVC tubes were used to provide atmospheric
pressure in thepickup, alsowithalengthof2.5m.
ThepickupsoftheMPPSwere encapsulated with slowly hardening epoxy
resin and sealedupwithclear varnishtomakethemwater resistant.
Thenalmeandiameterofonesensorheadwas15mm.Theready-builtsensorswerepoint
calibratedby AktivSensorGmbHto19VTable 1. Parameter of the Bed
Materials Uniform Gravel, Gravel from the River Rhine Armoring
Layer, and Spheres; the Weighted Mean of the WholeGrain-Size
Distribution Is Represented by dMeyer-Peter and Mller 1949; for
#uni dd50 and for #rhi dd70Run number Bedd15, d, d85mmkfm/ss103kg/
m3 Packing#uni Uniform gravel 7.7, 10.2, 13.2 0.390.02 0.7 2.46
Loose#rhi Rhine, armored 13.8, 26.1, 38.8 0.330.02 1.5 2.51
Loose#sph Spheres 25.4 0.26 1.6 1.36 DensestFig. 2. Sketch of the
experimental setup, dimensions in meters, not to scale: a view in
the streamwise direction, with 16 MPPS and both 2D PIVarrangements
of setups Aand B;b side view, where the positions of the one
dimensional ADCP probe can also be seenJOURNAL OF HYDRAULIC
ENGINEERING ASCE / OCTOBER 2010 / 781J. Hydraul. Eng.
2010.136:779-789.Downloaded from ascelibrary.org by INDIAN INST OF
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only; all rights
reserved.accordingto04kPawithatoleranceinaccuracyoflessthan1.0%full
scale. Temperatureerrors arecompensatedelectroni-cally, leading to
a remaining nonlinearity of typically 0.1% fullscale. The response
time guaranteedbythe manufacturer was10 ms, limitedbythesignal
conditioningbytheamplifyingblankets.
Toavoidaliasingeffectsduetohighfrequencynoise2 kHz, the recording
was made at f 2, 125.7 Hz, additionallysupportedbyafourth-order
Butterworthlow-pass lter
withacutofffrequencyof500HzDataTranslationSAK52-150-501-10. A
16-bit AD card Data Translation 321 allowed a theoreti-cal
resolutioncorrespondingto0.15Pa. Withintheume, thesensors were
locally xed on a grid to keep them in an accuratelydenedposition.
Testsunder umeconditionsshowedthat theMPPS were even able to react
within 2 ms. In one measurementarrangement somesensors
werealignedwithinthelaser sheet.Thelaser
doublepulseswereclearlyidentiableinthesignal.This unforeseen effect
was used to validate the synchronization ofthe PIV and the MPPS
signal.The MPPS are used in horizontal Fig. 4a and vertical
Fig.4barrangementstodistinguishbetweendragandlift uctua-tions.
Apositivepressuredeviationinthestreamwisedirectionover pressure
refers to a drag-force-like event, D=p Fig.4a. A negative pressure
deviation in the vertical direction
lowpressurereferstoalift-force-likeevent, L=pFig. 4b.
Inthefollowing,
thepressureuctuationsmeasuredintheverticaldirectionaredenotedbythestresstermLandinthehorizontaldirection
by the stress term D, respectively. However, it has to benoticed
that the sensor in the drag conguration Fig. 4a inu-ences the
pressure signal because it is protruding with half of itsdiameter
intothe oweld. It is assumedthat the differencebetween the
conguration in Figs. 4a and b is only of quantita-tive matter and
that the overall characteristics are very much com-parable.Flow
ConditionsTable2givestheowconditionsthatwereprovidedduringthemeasurements.
Qis the ow rate, and the bulk velocity is deter-minedbyU=Q/
Bh.Thelocationofthemaximumvelocityindistancetothebed, h,
isthroughout smallerthanh, indicatingthepresenceof slight
secondarycurrents. This phenomenonisunavoidableinumeexperiments
withanaspect ratioof B/ h=4.2Song and Graf 1994. However, a closer
inspection of thevelocity elds showed that the ow behaves like a 2D
ow in thenear-bedcenterlineregionfor details, seeDetert 2008,
Chap.4.2.1. The bulk Reynolds number is dened by Reh=Uh/ , withthe
kinematic viscosity of water =106m2/ s at 20C. The shearvelocity u,
the equivalent grain roughness ks, and the
zero-planedisplacementyytweredeterminedfromlogtsof u
yfromindependent PIVandADCPmeasurements, withtolerances of3% for
u.ResultsThe following three subsections concentrate on the uniform
owconditions of experiment number #uni6as these measurementsgave
representative results for all experimental runs of #sphi,#unii,
and#rhi9whilethefourthsubsectiongives turbulenceintensity proles
for the different experimental runs.Time SeriesFig. 5shows
synchronous time series of pressure
uctuationsptmeasuredwiththeexperimentalconditionsof#uni6overt=3.0
s. The pinholes of the four pressure pickups were posi-tionedat
y=+10; +5; 7; 22 mm. The MPPSat y=+5 mmFig. 4. Arrangements of the
MPPS: a MPPS as a drag indicator; bMPPS as a lift indicatorFig. 3.
Sketch of the head of an MPPS, units in millimetersTable 2.
Experimental Flow Conditions; the Integers1,3,6,9 at the End of the
Run Number Approximately Refer to the Ratios of Qi, Ui, or Reh,i;
at#uni9 the Provided Flow Led to Very Slight Sediment Transport,
Where Isolated Single Grains Were Observed to Be Moving
OccasionallyRunnumberQm3/ shmmUm/shmmum/sRehRehksmmyytmm#uni3 56.6
200 314 165 30 62.8 4.95 26.52 2.62#uni6 120.0 200 667 175 63 133.4
11.03 26.52 2.62#uni9 180.0 211 948 155 95 200.0 14.73 26.52
2.62#rhi9 180.0 215 930 155 86 200.0 13.33 15.53 6.62#sph1 18.6 129
160 125 13 20.6 1.88 20.51.5 5.50.5#sph3 56.6 199 316 170 30 62.9
5.10 20.51.5 5.50.5782 / JOURNAL OF HYDRAULIC ENGINEERING ASCE /
OCTOBER 2010J. Hydraul. Eng. 2010.136:779-789.Downloaded from
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Copyright ASCE. For personal use only; all rights
reserved.wasfacingupward, asshowninFig. 4b; thusit
indicatestheuctuatinglift. Theother
threesensorswerearrangedwiththepinhole facingupstream, as
depictedinFig. 4a. Thus, thesesensors indicate the uctuating drag.
For the measurements givenin Fig. 5, the sensors indicating D were
vertically aligned in thecenterline of the ume, whereas the L
sensor was positioned at alateral distance of 80 mm relative to the
others.ThesignalsgiveninFig. 5clearlyshowthedampingoftheamplitude
with increasing depth into gravel. As expected thedeepest sensor at
y=22 mm gives the smallest pressure uctua-tions. However, the
uctuations recorded by the second sensor aty=7 mmi.e.,
onegraindiameter belowtheroughnesstopscan hardly be distinguished
from the deepest sensor. More intensesmall-scale uctuations can be
seen only synchronous to positivepeaks of the most exposed sensor
at t =45.5; 45.8; 46.6; 47.4 s.Due to its lateral distance from the
other sensors, the signal of theL sensor slightly above the
roughness crest shows no correlationwiththeother sensors. However,
therecordeductuations areclearly more turbulent in comparison with
the sensors within thegravel bed. The signal of the most exposed
sensor gives the mostextremepressureuctuations. Maximal
positivepressurepeaksreach up to p/ o+40, whereas the negative
pressure peaks areless extreme, around p/ o20. This indicates that
positivepeaksofDaremoreextremethanthenegativepeaks, i.e., thesignal
is skewed. Acloser examinationof this phenomenonisgiven in the next
section.HistogramsTocharacterizetheprobabilitydistributionofextremevaluesofthe
pressure uctuations, histograms of the measured signals
arepresented in the following. They are compared with a
probabilitydensityfunctionPDF
derivedbyHoandandBattjes2006.ThisPDFiscapableofdescribinginstantaneousdragforcesonbed
roughness elements. It is distantly related to the2distribu-tion,
whichwasproposedbyPapanicolaouet al. 2002forthePDF of drag forces.
However, the 2distribution includes a sum-mation from 0 to innity,
whereas Hoands formulation is easierto use in practice.Hoands
formulation is derived by assuming a
characteristic,normallydistributednear-bedvelocityub,thatisproportionaltothe
drag force by FD=ubub. This single characteristic ub is seento be
the only source for drag forces. The PDF is derived asPFD =122FDexp
12FD/ sgnFD22where the noncentrality parameter =
ub/ub3givesthereciprocaloftherelativenear-bedturbulenceintensityruled
by ub. Fits to the mean and standard deviation are given byfit= 2+
1 exp 1.63 4andfit= 42+ 2 + exp 0.552 5ThePDFisnegativelyskewed,
i.e., eventsof FD0aremoreextreme. HoandandBattjes2006testedEq.
2againsttheirown measurements. As a proxy to the streamwise drag
force FD,theyusedthestreamwisepressuredifferential Dmeasuredat
acubicmodelstonewithanedgelengthof30mm. Theshapeofthe PDF was
predicted almost perfectly for lower drag intensitiesintherangeof
2D. However, slight differencesfor extremedragvalues were observed,
where the trendof D was betterdescribed by an adaptation of . In
his Ph.D. report, Hoand alsotested pressure measurements indicating
lift uctuations, L,against anadoptedversionof Eq. 2. However,
themeasureddistributionsofLdidnot followthetheoretical curve.
Instead,theyrevealedtobealmostGaussianshapedbetween2L,andbeyond
that the atness of the distribution is higher than given bya normal
distribution.InFigs. 68, Eq. 2hasbeenusedtodeterminethepresentPDFs
from the pressure measurements at the experimental
condi-tionsof#uni6forbotharrangementstorecordDandL. Thegures are
given in two different ways: on the left they are plottedwith
linear scales in order to evaluate the shape of the
distributionaround the mean; on the right they are plotted with
semilogarith-mic scales to better represent the shape of the tails.
Three differ-ent bed exposures are analyzed, y=7 mm in Fig. 6, y=0
in Fig. 7,and y20 mmin Fig. 8. Figs. 6 and 7 include plots ofHoands
PDFEq.2, adopted for both D and L.A denite near-bed cannot be found
for D and L sinceub and ub are subjected to a larger scatter. Thus
the results withEq. 2wereobtainedwith=2.7; 5.4
tocoverareasonablerangegainedbythevelocitymeasurements Detert 2008.
TheguresalsoincludeplotsofthestandardGaussianorstandard-normal
densitydistribution. Fig. 6revealsthat HoandsPDFEq. 2 provides a
good approach to both histograms of D andL at y=7 mm. In tendency,
=2.7 gives an acceptable approxi-mation to the measured shape.
However, negative deviations canbe seen around the mode value and
the negative tails for1.8D; L. In the semilogarithmic plot it can
be seen that thepositivetail of D isdescribedwell byvaluesof
=2.75.4,whereas the maxima of its negative tails are better
approximatedfor5.4. ForLasimilartendencyisrevealed.Thepositivetail
of L isdescribedwell byvaluesof 2.7,
whereasthemaximaofitsnegativetailareapproximatedfor5.4.
NotethatEq.2equalstheGaussiandistributionifitisappliedwith=. These
ndings lead to the following conclusions concern-ingthepropertiesof
pressureuctuationsmeasuredslightlyontopof thegravel bed.
1AsbothPDFsof D andL canbeapproximated by Eq. 2, here both the
local drag and the local liftmust be mainly inuenced by the
near-bed velocities. In the senseof Hoand 2005, the underlying
process is called a quasi-steadymechanism, as it is
mainlyduetolarge-scalevelocityuctua-Fig. 5.
Simultaneoustimeseriesof pressureuctuations pt for#uni6. Linewidths
fromthicktothincorrespondtoy=+10; +5;7; 22 mm, indicating the
uctuating part of the drag and lift asD; L; D; D. Here, thesensors
indicatingdraguctuations arevertically aligned; the sensor
indicating lift uctuations is at z=+80 mm relative to the other
sensorslight gray line.JOURNAL OF HYDRAULIC ENGINEERING ASCE /
OCTOBER 2010 / 783J. Hydraul. Eng. 2010.136:779-789.Downloaded from
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Copyright ASCE. For personal use only; all rights
reserved.tions.However,hismeasurementsdidnotshowthatthisdepen-dencyalsoholds
for L, althoughheexpectedthis for higherprotrusions Hoand 2005, p.
105. Note that Hoand had aslightlydifferent
setupwithcrushedstonesandarelativelargecubical housing of 30 mm for
his pressure sensors. 2 The posi-tivetailsof 3D; L arebetter
describedbyhigher relativenear-bedturbulenceintensities, 1/ ,
thanthenegativetails. Inother words, at higher
near-bedturbulenceintensities, thedragand the lift uctuation is
more likely to be positive than negative.Similar ndings hold for
the measured PDFs at the gravel tops,as shown in Fig. 7. Again,
=2.7 givesat least
intendencyanacceptableapproximationtothemeasuredshape,wherethepositivetailsofDandLarepredictedalmost
per-fectlybythisvalue.Thenegativetailsobtainabettermatchingforlowerturbulenceintensities,
whereDcanbeapproximatedby=5.4 and Lby=. Thus, the trend goes to
lower near-bed turbulence intensities. Nevertheless, the
quasi-steady mecha-nism is still dominant. As shown in Fig. 8, all
PDFs of D and Lloose their skewness deeper inthe gravel at y20 mm.
ThePDFs are well predicted by a simple Gaussian distribution, or
inthe sense of Hoands PDF Eq. 2 =holds, i.e., ubub. Thus,
turbulence due to near-bed velocity uctuationsplaysnofurther role.
FollowingHoand2005, e.g., Fig. 6.1,TWP uctuations can be seen as
the responsible mechanism now.TWP
areduetoconvectingpressureeldsandnotduetonear-bed velocity elds.
Consequently, a symmetrical i.e., a non-skewed PDF can be
expected.The PDFs measured at y=+7; 0; 20 mm revealed that
theinterfacial layer say, 0y/
ks1representsasharpboundary.Abovethisregion,
dragandliftuctuationsaremorerelatedtothenear-bedvelocity,
andbelowtheyaremorerelatedtoTWP.Thedeterminedhistograms showthat
thehydrodynamicloadsactingonthebedmaterial
canbedescribedbyHoandsPDFEq.2. Based on this information the
probability of grain par-ticles tobe unstable couldbe
estimatednumericallyinfuturestochasticsediment transport modelsthat
useforcebalanceap-proaches to determine conditions of incipient
motion.Spectral AnalysisInthefollowingthespectral distributionof
thesignalsisana-lyzed. Fig. 9 shows typical representative power
spectra Spp of themeasured signals at different vertical positions.
They refer to
thesamepressuresignalsthatwerealreadypresentedastimeseriesinFig. 5.
Additionally, Fig. 9includes asupplementarysignalrecorded at y=38
mm, deeper within the gravel layer. The
one-sidedspectraareestimatedbyWelchsaveragedmodiedperi-odogrammethodof
spectral estimation. Segment lengths of
n=21761.7swith50%overlapwereused,withHammingwin-dows of the same
length. The results were smoothed by a movingaverage window of 50
elements, with the shape preserved.Withanoverall viewof Fig.
9thedampingof thepressureuctuations within the gravel layer becomes
obvious as the spec-traof thesignalsof
themoreshelteredsensorsthroughout arebelow the more exposed ones.
Furthermore, the lower frequencies2 Hz also contain the larger part
of the turbulent energy. Onlythe spectrumof the signal
measuredbythe uppermost sensorclearly follows the expected
Kolmogorov-scaling tendency withinthe inertial subrange. At
frequencies off 10 Hz, its curve dec-lination conforms with the
classical 7/3 power law for pressuree.g.,Moninand Yaglom1975.
Thisscalingalsoroughlyholdsfor thespectraat y=22; 38 mm
withinthegravel. How-Fig. 6.
PDFsofmeasuredinstantaneouspressureuctuationsslightlyabovethegravel
crest at y=+7 mm#uni6: aDtenindependentsignals;b L14 independent
signals compared with Eq.2 and a Gaussian distribution normalized
by its respective standard deviationi784 / JOURNAL OF HYDRAULIC
ENGINEERING ASCE / OCTOBER 2010J. Hydraul. Eng.
2010.136:779-789.Downloaded from ascelibrary.org by INDIAN INST OF
TECHNOLOGY - ROORKEE on 08/14/14. Copyright ASCE. For personal use
only; all rights reserved.ever, sincetheturbulent
uctuationsaredampedwithinthepo-rous layer, the resulting curves are
shifted toward lower values ofSppand f, respectively. As all
spectradeeper inthebedwerefound nearly to resemble these two
spectra not shown here, it
isconcludedthatlittleadditionaldampingtakesplacedeeperthan12ks. The
variance p2, i.e., the area under the curves, staysalmost
constant.Incontrast tothisKolmogorovconformity, thesignalsmea-sured
in the roughness layer at y=+5; 7 mm reveal a
differ-entspectralbehaviorfortheturbulentenergycascade.Here,thecurves
show a decrease slower than f7/3. At f 30 Hz, the curveof the MPPS
aty=+5 mm even crosses the spectrum of the up-permost sensor. This
behavior can possibly be explained with anenergy transfer from the
horizontal to the vertical uctuations. Asthe freestream ow in the
streamwise direction is hindered due tosingle grains in the
roughness layer, it transforms to a more
threedimensional3Dowwithintheintersticesbetweenthegrains.Therefore,
kinetic energy from the streamwise ow is shifted to-wardthe
vertical andtransversal directions. Withinthe gravellayer,
theclassical 7/3power cascadethat indicatesisotropicbehavior is
enhanced by a transformation of small-scale turbulentkinetic
energy. However, it has to be noted that the validity of a7/3lawfor
pressureisnot generallyaccepted. AlthoughLeeand Sung 2002 and Hoand
2005, p. 106 solely found a
7/3powerlawfortheirTWPspectraasmeasuredunderfreeshearowbehindabackward-facingstep,
GotohandRogallo1999proposed a second range asf5/3. Lately, Tsuji et
al.2007 evenFig.
7.PDFsofmeasuredinstantaneouspressureuctuationsatthegraveltopsaty=0#uni6:aDtwoindependentsignals;bL14independent
signalsFig. 8. PDFs of measured instantaneous pressure uctuations
within the gravel bed at y20 mm #uni6, nine independent signals
left: linear;right: semilogarithmic scaleJOURNAL OF HYDRAULIC
ENGINEERING ASCE / OCTOBER 2010 / 785J. Hydraul. Eng.
2010.136:779-789.Downloaded from ascelibrary.org by INDIAN INST OF
TECHNOLOGY - ROORKEE on 08/14/14. Copyright ASCE. For personal use
only; all rights reserved.foundcharacteristicisotropyat exponents
of 5/ 3, but theynever observed a7/3 power law in their pressure
spectra at all.Inthe present measurements large-scale oscillations
with f1 Hz are observed that are not damped within the porousgravel
layer. It is hypothesized that these pressure uctuations
aredominated by a long-wave oscillating water level. By
neglectingsurfacetension, therst-order wavetheorygives
theresultingstandard deviation of the bed pressure due to surface
wavesp =ga2 coshkh6whereadenotesthewaveamplitude, andthewavenumber
isgiven by k=2/ withbeing the wavelength. Within the
tran-sitionfromdeeptoshallowwaterbetween0.05h/ 0.5,thecorresponding
wave frequency becomesf = 1g/k tanhkh 7Fig. 9 includes a plot of
Eqs.6 and7, where=0.254.0 mandaconstant small amplitudeof a=0.6
mmisassumed. Theplot matches the spectra at f 1 Hz reasonablywell.
Conse-quently, the long-wave oscillations of the outer ow are
hypoth-esized to dominate p within the gravel layer. However, it
cannotbeanswereddenitelywhethertheyareduetolong-waveoscil-lations
of the water level or if they are due to macropressure
eldsresultingfromcoherent owstructuresthat aresensedaslong-wave
oscillations by the Fourier-transformed long-term signal ofthe
MPPS.Twokinds of noise levels canbe identiedinthe spectra.Although
the recording was obtained by applying a low-pass l-ter,
highfrequencies still producealiasingnoise.
Theresultingpeakscanbeseeninabandof1030Hzinthespectraofthesensors
at y=22; 38 mm, where the signal was too small toabsorb these
interferences. The second source of noise is the
un-avoidablewhitenoise. Inthespectrait canbeidentiedat
ap-proximately102Pa2/ Hz. Consequently, aspectral separationof the
uctuating pressure signal can be made as follows:p,tot2= p,t2+
p,w2+ N28wherep,tot2denotes thetotal variance; p,t2=part
duetoturbu-lence; p,w2=part due to long wave oscillations; and
N2=part dueFig. 9. One-sided power spectra for single pressure
signals at differ-ent vertical positions#uni6.
Theareaunderthecurvesequalsthevariances 2. Line widths from thick
to thin correspond toy=+10; +5; 7; 22; 38 mm,
indicatingtheuctuatingpart ofthedragandlift as D; L; D; D; D recall
Fig. 5. Here, thesensors indicating drag uctuations are vertically
aligned; the sensorindicating lift uctuations is atz=80 mm relative
to the other sen-sorslightgrayline,seealsoFig.5.
Thedashedcurvedenotesthepossible inuence of long waves in the outer
ow Eqs. 6 and 7.The vertical line highlights the response time of
10 ms guaranteed bythemanufacturer.
Thedottedhorizontallinerefersapproximatelytothewhitenoiselevel.
Theinuenceofthelow-passlterwiththecutoff frequency at 500 Hz
becomes prominent forf 200 Hz.Fig. 10. Vertical proles of the
standard deviation of the drag and lift,D andL, for runs #unii
plotted with ks. a Scaled with open-channelow variableo. b Scaled
with seepage ow variableghu/ kf. The position of the roughness
crest yt=0 is indicated by the horizontal line.Filled symbols refer
toD; unlled symbols refer toL. The measured signals are ltered from
white noisesee level in Fig. 9.786 / JOURNAL OF HYDRAULIC
ENGINEERING ASCE / OCTOBER 2010J. Hydraul. Eng.
2010.136:779-789.Downloaded from ascelibrary.org by INDIAN INST OF
TECHNOLOGY - ROORKEE on 08/14/14. Copyright ASCE. For personal use
only; all rights reserved.to noise. Since N2is independent of the
ow conditions, the mea-sured signals at low turbulence intensities
are subjected to a lowsignal-to-noise ratio. Especially for
measurements within the bedfor #sphiand #uni3, the signicance
ofp,t2andp,w2is low.Thepower spectrainFig. 9givedetailedinsights
intothegenesis, damping, anddissipationof pressureuctuationsfor
aporous bedunderlyingopenchannel ow. This informationisimportant
for the verication of time resolved numerical simula-tionse.g.,
large eddy simulation.Turbulence Intensity ProlesIn Fig. 10,
vertical proles of the standard deviation of the pres-sure
uctuations indicating drag and lift, DandL, for all runsof #unii
are given. Fig. 10a shows that D and L can be scaledwitho.
Theturbulenceintensitiesobtainedatthedifferentowconditions match
appropriately in the roughness layer and in theouter ow. However,
within the subsurface layer deeper than1ksaconstant
nonzerovalueisreachedfor bothDandL.Thus, deeper in the gravel layer
the variances are independent ofFig.
11.VerticalprolesofturbulentdragintensitiesD/
oleftcolumnandturbulentliftintensitiesL/
orightcolumnfora#unii;b#rhi9; and c #sphi. The given data points
have had removed the contributions from both long-wave oscillations
in the seepage ow and whitenoise. The ts of Eqs. 10 and 9 are given
by dashed and continuous lines, respectively. Roughness crest and
grain diameter d are shown in thebackground. Filledunlled symbols
refer toD L. The thinner parallel lines give a vertical range
of0.25y/ ks.JOURNAL OF HYDRAULIC ENGINEERING ASCE / OCTOBER 2010 /
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Copyright ASCE. For personal use only; all rights reserved.the
orientation of the pressure pickup. Here, an appropriate scal-ing
is performed with respect to seepage ow, as can be seen inFig. 10b.
The normalizationwas done usinga seepage owvariableghu/ kf, where1
thehydrostaticpressuretermghincorporates the inuence of large-scale
structures that scale withwater depth h e.g., Bakker et al. 1994.
However, as h was hardlyvaried in these experiments, the inuence
cannot be seen directly.2 The constant kf considers seepage ow. 3 u
accounts for theouter ow. uis proportional tothe
bulkvelocityandconse-quently also to a convection velocity,
providing a further indica-tionthat
thelong-waveoscillationsinthesubsurfacelayer aredominated by
pressure elds that are convected in the outer
ow.TodoaseparateexaminationofDandLforthethreedif-ferent
bedmaterials, thevariancesshowninFigs. 11achavehad removed the
contribution from the long-wave oscillations inthe seepage ow
assumed to be constant, see Fig. 10 and whitenoiseseelevelinFig.
9byapplyingEq. 8.Thus, solelytheturbulent uctuations are
considered. At y5 mm marginal dif-ferences are recognized between D
and L, which is most likelydue to the effect that the
drag-indicating sensors are not protrud-inganymoreout of thegravel
crest. Consequently, plotsof Dalso containL below this horizon and
vice versa.For all three bed types, an exponential decay within the
rough-nesslayerbecomesobvious.Thisndingsubstantiatesthend-ings of
Vollmer et al. 2002 and Breugem et al. 2006, where
thepressureuctuationswerealsofoundtodecreaseexponentiallyinside the
bed. At the interface of the free ow to the porous bed,Breugemet
al. 2006 foundvalues p/ o=1.6Re=176, =0.60 andp/ o=3.0Re=500,
=0.95, indicatingthat p= fRe. A
ReynoldsdependencywasalsoproposedbyFarabeeand Casarella 1991, and a
comparable tendency is found in theactual data. Whereas the
reference values for #unii scatter aroundD/ o=9andL/
o=3,therespectiveratiosarelargerfor#rhi9and smaller for #sphi.
However, the statistical scatter is too largeto allow a detailed
analysis of a Reynolds dependency. Therefore,acurvettingonlywas
conductedfor thedataof #unii. Thevertical decay ofL could be
matched well by an exponential tL/o = 2.88 expyks/2.0
9Incontrast,theverticaldecayofDisnotwelldescribedbyanexponential t.
A simple linear description was appliedD/o = 6.89 + 11.84
y/ks10BothrelationsareplottedinFigs. 11ac. Withrespect
totheshortcomingsindeningaswellasindetectingtheorigininy,thelinesparallel
toEqs. 9 and10 giveavertical rangeof0.25y/ ks. Inprinciple, the
shapes of Dand Lappear ad-equatelyapproximatedbyEqs. 9and10,
independentofthetype of the analyzed granular bed. After showing
the PDF of theextremevaluesandtherelatedfrequencydistributions,
Eqs. 9and 10 allowestimating the size of the destabilizing
loads.Based on Eqs.9 and10 prediction formulas for sediment
en-trainment or washout effectsof nesediment
canbeimprovednow.ConclusionsInthisstudy, theturbulent
pressureuctuationsleadingtodragandlift forcesat thewater-sediment
interfacehavebeendeter-minedexperimentally. Thesmall
dimensionaswell asthehighsensitivity and accuracy of MPPSs give a
powerful tool to high-light turbulent forces onsinglegrains at
river beds. Themainoutcomes are as follows: Vertical
prolesofthepressureuctuationsat different
owconditionsscaleappropriatelywiththeactual shearstressoand
equivalent grain roughness ks. Open-channel ow turbulence strongly
inuences the standarddeviationofthepressuresignal, p,
aboveandintherough-ness layer, where the lift uctuations decay
exponentially withincreasing depth of cover.
Withinthesubsurfacelayer, preachesanonzeroconstant,however, mainly
dominated by long-wave pressure elds thatare convected in the outer
ow.These ndings help to understand the physical
processeswithinaporous gravel bed.
Theycanbeusedtoimprovetheprediction of sediment entrainment by
analytical approaches.
Es-peciallytheformulationdescribingthedecayofliftuctuationswithincreasingdepthof
cover will behelpful toestimatethewashout effects of ne sediments
from the pores of a stable
gravellayer.AcknowledgmentsThesupportbytheBaden-WrttembergResearchProgramSe-curing
a Sustainable Living Environment BWPLUS Grant
No.BWR25003withfundsoftheStateofBaden-Wrttembergisgratefully
acknowledged.NotationThe following symbols are used in this paper:a
wave amplitude;Bwidth of the ume;Ddrag indicator in units of
pressure;d characteristic grain diameter;digrain size quantiles of
i%sieve screening;FDdrag force;FLlift force;f frequency;g
gravitational acceleration;h water depth;i , j integer numbers or
dummy variable;k wave number;kfpermeability coefcient;ksequivalent
grain roughness after Nikuradse;Llift indicator in units of
pressure;n integer number;p pressure;Qow rate;Rehbulk Reynolds
number =Uh/ ;Reboundary layers Reynolds number =u/ ;Sppautospectra
ofp;t time;Ubulk velocity =Q/ Bh;u streamwise velocity;ubnear-bed
velocity;ushear velocity =o/ 0.5;Vototal volume;Vfvolume of uid;x
coordinate in the streamwise ow direction;788 / JOURNAL OF
HYDRAULIC ENGINEERING ASCE / OCTOBER 2010J. Hydraul. Eng.
2010.136:779-789.Downloaded from ascelibrary.org by INDIAN INST OF
TECHNOLOGY - ROORKEE on 08/14/14. Copyright ASCE. For personal use
only; all rights reserved.y vertical coordinate, zero crossing
gained byextrapolating the log law;ytvertical coordinate, zero
crossing at theroughness tops;z transverse coordinate;empirical
constant; channel half-width, here=boundary
layerthickness;hlocation of the maximum velocity in distanceto the
bed;noncentrality parameter;wavelength;fitt parameter; kinematic
viscosity; density of water;sdensity of stone;pstandard deviation
of pressure;Dstandard deviation of the drag-indicatingsignal
D;Lstandard deviation of the lift indicating signalL;ubstandard
deviation of ub;fitt parameter;p,tot2total variance;p,t2part
ofp,tot2due to turbulence;p,w2part ofp,tot2due tolong wave
oscillations;N2part ofp,tot2due to noise;oboundary shear
stress;roughness geometry function=Vf / Vo, with10;. . .temporal
average of;. . . spatial average of ;. . . temporal uctuating part
of ;. . .maxmaximum of ; andVnabla operator =/ x, / y, / z in
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