Influence of shear stress fluctuation on bed particle mobilityNian-Sheng Cheng Citation: Phys. Fluids 18, 096602 (2006); doi: 10.1063/1.2354434 View online: http://dx.doi.org/10.1063/1.2354434 View Table of Contents: http://pof.aip.org/resource/1/PHFLE6/v18/i9 Published by the American Institute of Physics. Related ArticlesA dynamic hybrid Reynolds-averaged Navier StokesLarge eddy simulation modeling framework Phys. Fluids 24, 015103 (2012) Reynolds number effects on scale energy balance in wall turbulence Phys. Fluids 24, 015101 (2012) Direct numerical simulation of a high-Froude-number turbulent open-channel flow Phys. Fluids 23, 125108 (2011) Dispersion of ferrofluid aggregates in steady flows Phys. Fluids 23, 127102 (2011) Turbulence modification and heat transfer enhancement by inertial particles in turbulent channel flow Phys. Fluids 23, 123301 (2011) Additional information on Phys. FluidsJournal Homepage: http://pof.aip.org/ Journal Information: http://pof.aip.org/about/about_the_journal Top downloads: http://pof.aip.org/features/most_downloaded Information for Authors: http://pof.aip.org/authors Downloaded 14 Feb 2012 to 132.207.227.63. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissionsInuence of shear stress uctuation on bed particle mobilityNian-Sheng ChengaSchool of Civil and Environmental Engineering, Nanyang Technological University,Nanyang Avenue, 639798 SingaporeReceived 8 March 2006; accepted 15 August 2006; published online 27 September 2006Whether or not a sediment particle is entrained from a channel bed is associated with both averagebed shear stress and shear stress uctuation, the latter being ow-dependent and also related to bedirregularities. In the rst part of this study, a preliminary analysis of possible uctuations induced bybed roughness is presented for the case of an immobile plane bed comprised of unisized sediments.Theresultshowsthattheroughness-inducedvariationisgenerallycomparabletothatassociatedwithnear-bedturbulence, andbothvariations canbeapproximatedas log-normal interms ofprobability density distribution. The bed particle mobility is then analyzed by considering the effectsof shear stress uctuations. The relevant computations demonstrate that with increasing shear stressuctuations, the probability of the mobility of a bed particle may be enhanced or weakened. For thecase of lowsediment entrainment, the probability is increased by turbulence. However, theprobability may be reduced by the shear stress uctuation if the average bed shear stress becomesrelatively higher. This study shows that the critical condition for initial sediment motion could beoverestimatedif theShieldsdiagramisappliedfor theconditionof owswithhighturbulenceintensities. 2006 American Institute of Physics. DOI: 10.1063/1.2354434INTRODUCTIONFor a bed particle subject to turbulent ow, its mobilityis random, varying with time and also location. This is due tothe fact that drivingforces suchas shear stress actingatsediment beductuatetemporallyandspatiallyevenwithconstant time-mean magnitudes. Such random variations in-herent inthemotionof bedparticles implythat sedimenttransport rate is also closely associated with characteristics ofturbulent ows. However, relevant mechanisms are not wellunderstood. For example, widely cited bedload functionssuchasthosedevelopedbyEinstein,1Bagnold,2and Yalin3are all expressed by relating sediment transport rate to time-meanowparameters. Inthesepioneeringworks that ad-dress sediment transport problems, either stochasticallyordeterministically, the informationof turbulence was takeninto account only at the rst-order level, i.e., in the sense oftime-mean velocity or shear stress. This limitation was sub-stantiated by Grass and Ayoub4with their sediment transportdata collected for laminar and turbulent owconditions,which were designed at similar mean shear levels. Grass andAyoub also presented an excellent conceptual explanation oftheir observationwithaprobability-basedmodel that wasproposed for delineating turbulence-induced variations in theinstability of sediment bed.A recent extension of Grass and Ayoubs effort has beendone by Sumer et al.,5showing that a signicant increase inthebedloadtransportratecouldbeinducedbyextraturbu-lent excitation exerting on a sediment bed. Their experimentswereconductedforbothplanebedandripple-coveredbedconditions, with turbulence being characterized by shearstressandvelocityuctuations, respectively.Theplanebedwas prepared with ne sediment and thus served as a smoothowboundarypriortoinitiationofripples. Forthisplane-bed condition, it can be further shown that with reference tosediment transport inlaminar ow, theturbulence-inducedincrementinthetransportratecanbeevaluatedreasonablyusing a probabilistic model based on the log-normalfunction.6Stochasticconsiderationsgenerallyfacilitaterealizationof the fact that a turbulent ow quantity or sediment-relatedparameter with the same time-mean value may uctuate withdifferent amplitudes. Several such examples are available intheliteratureshowingapplicationsofstochastictheoriestosediment transport. The earliest contribution is due toEinstein1informulatingbedloadtransport rate. Recent ef-fortshavebeenprovidedindifferentaspects. Forexample,Cheng and Chiew7,8showed that both pickup probability andinitial suspensioncanbe reasonablydescribedusingnor-mallydistributednear-bedvelocities.Parkeretal.9reportedthat implementation of a probabilistic concept in reformulat-ing the Exner equation of sediment continuity would offer abetterunderstandingofhowstreamscreateaverticalstruc-ture of sediment bed. Kleinhans and van Rijns work10dem-onstrated that a probability-based approach can signicantlyimprovepredictionof bedloadtransport near theincipientmotion, whichotherwisemayencounter failurewhenem-ploying classical deterministic predictors such as the Meyer-Peter and Mueller formula. It is also noted that various prob-abilitydensityfunctionspdf havebeenassumedintheseresearchworks, andmost ofthemwereusedwithout theo-retical interpretation or experimental verication. The Gauss-ian or normal function appears popular compared to others; itmay apply to near-bed ows dominated largely by a combi-nation of turbulent events with equivalent sweeps andejections.7,8,10Ontheotherhand,someexperimentalobser-aElectronic mail: cnscheng@ntu.edu.sgPHYSICS OF FLUIDS 18, 096602 20061070-6631/2006/189/096602/7/$23.00 2006 American Institute of Physics 18, 096602-1Downloaded 14 Feb 2012 to 132.207.227.63. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissionsvations also uncover skewed probability density distributionsofsometurbulentquantities. Theskeweddistributionisbe-lieved to be closely related to turbulent coherent structures orgenerated by nonequivalent contributions of sweeps andejections. However, theoretical connections in this regard arelacking at present, which is why various approximationshavebeenmadeinliterature. Forexample, ininvestigatingincipient motion for three representative bed packing condi-tions, Papanicolaouet al.11reportedthat theinterrelation-ship of owand sediment can be accounted for by theChi-squaredistribution. Otherexamplesarethelog-normalfunction1214and more complicated functions includingthepolynomialapproximation15andthefourth-orderGram-Charlier function.16A uctuation in the bed shear stress is of course closelyrelated to the condition of turbulent ow, but also being sub-jecttothedetailofbedparticleconguration.Thelatterisparticularly important in the presence of a sediment bed thatishydrodynamicallyrough. Thisstudyaimstoperformananalysis of roughness-induced shear stress uctuation for thecaseofanimmobileplanebedcomprisedofunisizedsedi-ment particles. Then, a pdf is proposed to take into accountrandomvariations causedbybothturbulent owandbedroughness. With the derived pdf, the mobility of bed particlessubject to turbulence is nally discussed.ROUGHNESS-INDUCEDSHEARSTRESSFLUCTUATIONSGenerally, the near-bed ow condition is not only relatedtoturbulenceanditscoherent structurebut alsosubject tosediment-related irregularities. Both factors cause randomuctuationsinthebedshearstress. Thecomponent associ-atedwiththebedparticles maybedenedas roughness-induced uctuation1, while that associated with turbulencemay be referred to as turbulence-induced uctuation 2. As aresult, the bed shear stress uctuation is generally given by = m = 1 + 2, 1where m is the mean bed shear stress being averaged both intimeandspacedomains, or thedouble-averagebedshearstress.17,18In comparison with2, the uctuation1 is negli-gibleforasmoothbedcomprisedofneparticlesbut be-comesdominantlyimportantforlaminarowoveraroughbed.Here, itisassumedthatopenchannelowsoccuroveranimmobileat bedcomprisedof unisizedsediment par-ticles. Werst considerthat thevariationinthebedshearstressisonlyassociatedwiththebedparticleconguration.For this limited condition, turbulent effects can be excludedandthusthevariationcanbeindirectlyevaluatedundertheconditionoflaminarow.Incomparisonwiththetemporalturbulence uctuation normally considered at a certain point,theroughness-inducedvariationisconsideredhereonlyinthespatialdomain,i.e.,atwo-dimensionalplaneparalleltothe average bed surface.As sketched in Fig. 1, the presence of sediment particlescanhaveconsiderableeffectsontheow, but sucheffectsare limited to the area near the bed. The affected ow layercanbecalledtheroughnesssublayer,17,18withitsthicknessdenotedby. Itcangenerallybeexpectedthatforaplanebedof unisizedsediment, isproportional totheaverageroughness size, the latter being taken to be the particle diam-eter D. BasedontheresultsreportedbyRaupachet al.,17=25Dor on average3.5D.For the case of laminar ow, the velocity distribution isgiven byu = u*2h yh y22h2, 2where u* is shear velocity, is kinematic viscosity, h is owdepth, and yis measured from the average bed surface. Theaverage velocity in the roughness sublayer isU = 10udy = u*2h 2h 26h2. 3Ifis much smaller than h, Eq. 3 can be reduced toU = u*22=12, 4where1=u*2is the local bed shear stress and is the den-sity of uid. It is further assumed that the ow discharge perunit width within the roughness sublayer remains unchanged,q = U =122= const. 5This discharge can also be characterized in terms of averageparameters,q =1mm22, 6where 1m is the average bed shear stress and m is the aver-age thickness.With Eqs. 5 and 6, the relative bed shear stress can beassociatedwiththerelativethicknessoftheroughnesssub-layer,FIG. 1. Denition of roughness sublayer.096602-2 Nian-Sheng Cheng Phys. Fluids 18, 0966022006Downloaded 14 Feb 2012 to 132.207.227.63. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissions11m= m 2. 7For aat sediment bed, theuctuationinthebedsurfaceelevation and thus is random. With uniform sediment, it isreasonabletoemploytheGaussianfunctionfor describingthe resulting distribution. Therefore, the pdf of the variableis given byf =12exp m222, 8whereisthestandarddeviationthatcanbetakenastherms value of i.e.,rms. Using probability transformation,the pdf of can be derived from Eq. 8 based on the relationgiven by Eq. 7, yieldingf1 =18I1m13 exp 1 1m/122I2+ exp 1 + 1m/122I2, 9where I=rms/ m is dened as the intensity of the variationinthebedsurfaceelevation. A furtherprobabilitytransfor-mationfromEq. 9resultsinthefollowingpdf,whichap-plies for the normalized shear stress dened as1+=1/ 1m:f1+ =1I81+3exp 1 1+22I21++ exp 1 + 1+22I21+. 10Equation 10 shows that the pdf of 1+varies with the inten-sity of I if the randomness of the bed shear stress is inducedsolely by the bed irregularity.COMPARISON WITH FLOW-INDUCED SHEARSTRESS FLUCTUATIONSTheabove-derivedprobabilitydensitydistributionmaydiffer from that associated with near-bed turbulence. The lat-terhasbeenattemptedrecentlyforunidirectional turbulentows over a smooth boundary.13,14The relevant results dem-onstrated that the log-normal function can effectively repre-sent theprobabilitydensitydistributionof theturbulence-induced bed shear stress denoted by 2, which weremeasured for various ow conditions. The log-normal func-tion used is expressed asf2 =122exp + 2 ln2/2m2811orf2+ =122+ exp + 2 ln 2+28, 12where 2+=2/ 2m and =ln1+I22 with I2=2rms/ 2m the bedshearstressintensity, 2mtheaveragebedshearstress, and2rms the rms value of 2. It should be mentioned that 2includedinEqs. 11 and12 was printedbymistakeas21+I2in the previous works.6,13Equation 12showsthatforthecaseofasmoothbed,theprobabilityfunctionusedfordescribingtheturbulence-inducedshearstressvariationvariesonlywiththenear-bedturbulence intensitydenedbyI2. ChengandLaw14alsoindicatedthat thevalueofI2increaseswithreducingRey-noldsnumber,andathigherI2value,thestressdistributionappearstobemoreskewed. Intheextreme, apeakydistri-butionwouldoccur at verylowReynolds number, whichagrees with experimental observations.19The log-normalfunctionisalsocharacterizedbyitslongtailforpositivestresses, which serves to represent the intermittent nature ofReynolds stress contributions during bursting events.19To understand how Eq. 10 differs from Eq. 12, one ofthe simple ways is to nd rst the relation of I and I for thecase considered in this study. Using Eq. 7, the shear stressuctuation1 is1 = 1 1m =mm + 2 11m. 13Equation13 indicates that an increase in the thicknessoftheroughnesssublayerresultsinadecreaseinthelocalbedshearstress. Thisequationcanbesimpliedto1/ 1m2/ m by applying a power series expansion for the caseof much smaller thanm. Furthermore, by taking the rootmean square, one gets that I12I with I1=1rms/ 1m. Usingthis approximation, Eq. 10 can be compared to Eq. 12 byplotting them as pdfs against s for various I values, wheres=m / rmsis dened as the standardized bed shearstress. Note that the probability transformation used for plot-ting the graphs is based on the relation of pdfs =If+. AsshowninFig. 2, theresultssuggestthatbothpdfsareveryclose,particularlyforsmallshearstressintensities.Inotherwords, thelog-normal functioncanalsobeusedasagoodalternativetoEq. 9or10fordescribingtheprobabilitydensity distribution associated with roughness-induced shearstress uctuations. The advantage of the use of this similaritywill be demonstrated in the subsequent derivation for evalu-ating bed particle mobility.AsgiveninEq. 1,thebedshearstressuctuationforturbulent ows over a sediment bed can be generally consid-eredas asumof twocomponents, 1 and2. Sincebothcomponents canbeapproximatedas log-normal variables,thelog-normal pdf canbefurther usedfor describingthevariable . This is because a sum of log-normal random vari-ables is alsolog-normallydistributed, whichappears as areasonableassumptionwithmanysuccessfulapplications.20However, the extendeduse of the log-normal functionisbasedonthe approximationthat the twocomponents areindependent. The limitationof this approximationwill bediscussed later in this paper. From Eq. 1, it can be derivedthatrms2= 1rms2+ 2rms2. 14Dividing bym2, Eq. 14 is rewritten as096602-3 Inuence of shear stress uctuation Phys. Fluids 18, 0966022006Downloaded 14 Feb 2012 to 132.207.227.63. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissionsI2= I12+ I22, 15whereI =rms/ m, I1=1rms/ m, andI2=2rms/ m. Inthefol-lowing, the two kinds of uctuation intensities are comparedbased on limited information available in the literature.Cheng et al.13observed that for unidirectional ows overa smooth boundary, the bed shear stress uctuates dependingonlocal owconditions andalsoupstreameffects. Theirobservations also showed that I2 ranged from 0.212 to 1.015for open channel ows subject to extra turbulent eddies,whichwerearticiallygeneratedbysuperimposingexternalstructures including horizontally installed transverse pipeand grids above the channel bed. This range covers the regu-larcaseofuniformopenchannelowswithoutanydistur-bances imposed, of which I20.30.5.To estimate I1, the approximation of I1=2I is again in-vokedhere. Sincermsisthesameasthermsvalueoftheuctuation in the bed surface elevation, I1 can then be evalu-ated based on the bed conguration. To perform such calcu-lationsforillustrationpurposes,weconsiderherethreebedconditionswithregularparticlearrangement. Therst sce-nario, caseI, isthatthebedsurfacevariesinonedirectiononly, withbedparticles beingrepresentedbysemicircularcylinders closely connected. The second case concerns atwo-dimensional bedsurface, whichiscomprisedofhemi-spheres arranged in a square matrix. The last situation, caseIII, issimilartothesecond, but thehemispheresareposi-tioned in a rhombus fashion, which allows the least interpar-ticle space. A plan view of cases IIII is provided in Fig. 3.For the above three cases, a bed surface displacement isrst computed for the determination of the average bed level.This level is dened in such a way that the interstitial spaceof thesediment bedbelowit is equivalent totheparticlevolume above it. The rms value of the uctuation in the bedsurface elevation is then computed by the area-weighted av-erage with reference to the average bed level. The results aresummarizedinTableI, showingthat rms=0.1120.172Dfor the simpliedbedconditions. Obviously, the ratioofrms/ Dwould increase for a bed comprised of irregularsediment. For example, Nikora et al.18reported thatrms0.375Dfor a gravel-roughenedbed. However, suchinformation is generally not available for sediment beds ob-served in laboratory channels and natural rivers.If taking rms=0.10.4D and =25D, thenI1=2I=0.040.4. SuchI1valuescancauseanincreaseofup to 110% in the total uctuation given in Eq. 15, which isestimatedbasedontheobservedI2range.13This result isroughbut impliesthat theroughness-induceductuationinthe bed shear stress is generally considerable in comparisonFIG. 2. Comparisons of Eq. 12 withEq. 10 for different turbulence inten-sities. The solid lines are computed us-ing Eq. 12 and the dash-dot lines arecomputed using Eq. 10.FIG. 3. Plan view of particle congurations.096602-4 Nian-Sheng Cheng Phys. Fluids 18, 0966022006Downloaded 14 Feb 2012 to 132.207.227.63. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissionswith the turbulence-associated counterpart. Therefore, to takeintoaccount theeffects of bedshear stress uctuationonsediment transport, a double-average technique shouldbegenerally applied for the case of turbulent ows over a sedi-ment bed.PROBABILITY OF BED PARTICLE MOBILITYThemobilityof bedparticlescanbequantiedbyitsprobability dened as21p =minmaxfped, 16where the bed shear stress varies frommin tomax, and peistheprobabilityof mobilityassociatedonlywithrandombed particle conguration. For a plane bed comprised of uni-form sediment, the random variation of the bed mobility canbe approximatedas a hypothetical wave process,21whichyieldspe = 1 exp42em2 , 17whereemistheaveragevalueoftheminimumbedshearstress called effective shear stress that is required for a bedparticletomove. Itcanbeshownthat emissimilartothecritical shear stress dened by the Shields diagram but asso-ciated with sediment entrainment of higher intensity.22Asmentionedearlier, canbegenerallyapproximatedas a log-normal variable, and therefore its pdf is given byf =12exp + 2 ln/m28, 18where =ln1+I2 and I =rms/ m. Substituting Eqs. 17 and18 intoEq. 16 andthenintegratingfrom=0to=providesanexpressionforevaluatingtheprobabilityofthebed particle mobility subject to turbulent ow. It readsp = 1 012t exp + 2 lnt2842t2dt,19where=m/ em. Itisnotedthattwoimportantparametersare involved in this evaluation, namely the turbulence inten-sity Iand the relative bed shear stress .ThevariationsofpwiththetwoparametersfortypicalsituationsarepresentedinFig. 4. It canbeobservedfromFig. 4a that a continuous increase in the probability occursfor various turbulence intensities when the relative shearstress is gradually increased. However, an increase in turbu-lenceintensitymaynotalwaysresultinanenhancedprob-ability of the bed particle mobility. As shown in Fig. 4b, forrelativelylowshearstress e.g., =0.1, abedparticlebe-comes more unstable if the near-bed turbulence is enhanced.However, theoppositecaseoccurs for higher shear stresse.g., =2.0, where a bed particle would become morestable from increasing near-bed turbulence. Figure 4b alsoindicates that changes in p are not considerable for very highturbulence intensities, say I 2.The determination of depends on how to evaluate em.It can be shown thatem could be computed indirectly usingtheShieldscritical shearstressc.22Forexample, forbed-load transport in laminar ow, the dimensionless criticalshear stress canbe empiricallyrelatedtoshear Reynoldsnumber in the power form of*c=cD*mc,wherecis a co-efcient, mcisanexponent, *c=c/ gD isthedimen-sionlessshearstressusedforthecriticalconditionofinitialsediment motion, andD*=g/ 21/3Dis the dimensionlessparticle diameter, with Dthe sediment particle diameter,=s / , sthesediment density, theuiddensity,thekinematicviscosityof uid, andgthegravitationalacceleration. With a modied coefcient, the power relationmayalsobe appliedfor the effective shear stress, whichyields *e=eD*mewith *e=em/ gD and u*e=em/ . ToTABLE I. Estimated rms values of roughness sublayer thickness.CaseAverage bed level measureddownward from the top ofroughness elements rms/ DI 0.107D 0.112II 0.238D 0.172III 0.198D 0.148FIG. 4. Variations ofpwith two parameters,and I.096602-5 Inuence of shear stress uctuation Phys. Fluids 18, 0966022006Downloaded 14 Feb 2012 to 132.207.227.63. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissionsachievethesamepickupprobabilityforthecaseoftheini-tiation of sediment motion, by assuming that mc=me,Cheng22nallyobtainedthat e/ c=11.42. For simplicity,the same ratio of e/ c is also used here in this study. There-fore, =**e=*11.42*c, 20where *c can be computed using the following expression:23*c = 0.13D*0.392exp 0.015D*2+ 0.0451 exp 0.068D*. 21PlottedinFig. 5areaseries of contours of pvaluescomputedusingEq. 19fortwodifferentturbulenceinten-sities. As a reference curve, the Shields critical conditiongiven by Eq. 21 is also superimposed in this gure. It canbeseenthat theprobabilityofbedparticlemobilityvaries,which generally follows the trend of the Shields relation butwithmodications toa certainextent, dependingonbedshearstressuctuations. Asanexample,itisnotedthattheprobabilitythat ts tothe Shields curve is approximately0.6%forI =0.1, whileit increasesto1.1%forI =1.0. Thisdifferencesuggeststhat thecritical conditionfortheinitialsediment motionshouldbealteredfor different turbulenceintensities. Inother words, thecritical conditioncouldbeoverestimatedwhenapplyingtheconventional Shieldsdia-gram to ow conditions with strong turbulence.TofurtherunderstandEq. 19,anextremecaseisalsoconsidered here for which weak sediment transport occurs atsmallbedshearstresses. Thismakespossiblethefollowingapproximation for smallvalues:exp42t2 1 142t2. 22Substituting into Eq. 19 and integrating yieldsp =241 + I2. 23Equation 23 shows that turbulence-enhanced probability ofmobility is proportional to the squared turbulent intensity forthe condition of low bed shear stresses. In addition, it is alsonotedfromthederivationthatforthisextremecase,there-sult given by Eq. 23 also holds even if other pdfs than Eq.18 areusedfor this simplication. Figure6shows howwellEq. 19canbeapproximatedbyEq. 23forlowbedshear stresses. For example, if I =1.0, the simplicationgivenbyEq. 23 canbe consideredveryreasonable for0.1.DISCUSSIONSTheanalysesperformedinthisstudyarelimitedtothecondition of a plane bed comprised of uniform sediment. Thebedfailureisdescribedintermsofmobilityprobabilitybutforowconditionspriortoinitiationofbedformssuchasripples or dunes.This workcanbeimprovedbyincludingthepossibleinterrelationof thetwouctuationcomponents, onebeingrelated to bed roughness and the other due to ow variation.However, relevant information is not available at this stage.Consider the bedroughness beingrepresentedintwoex-treme ways, i.e., i by individual sediment particles and iiby large-scale bedforms such as dunes. In this study, we areconcerned with case I, which is relatively simple. Because ofFIG. 5. Probabilityof bedparticlemobilityplottedinD*-*plane. TheShieldscritical conditionfor initial sediment motionisdenotedbyblackdots.FIG. 6. Simplication of Eq. 19 for low bed shear stresses.096602-6 Nian-Sheng Cheng Phys. Fluids 18, 0966022006Downloaded 14 Feb 2012 to 132.207.227.63. Redistribution subject to AIP license or copyright; see http://pof.aip.org/about/rights_and_permissionsthe consideration being also limited to a at, immobile, uni-sized sediment bed, it is expected that there would be a mi-nor interaction between the two uctuation components.In contrast, for case II, the dimension of large-scale bed-formscouldbeintheorderofowdepth, whichismuchlargerthanthesizeofsediment particles. Inaddition, it isnoted that the large-scale bedform is closely associated withthegenerationofcoherent owstructures, whichmaythusdominantly affect the stress uctuation. For this extremecondition, anyignoringof theroughness-owinterrelationwould denitely compromise the applicability of analysis. Ifa similar analysis is performed, it would then require an ef-fectivecaptureofcoherentowstructures,forexample,byemployingthedecompositiontechniqueproposedbyRey-noldsandHussain24andsubsequentlyappliedbyotherre-searchers including Tamburrino and Gulliver.25Furthermore,the presence of large-scale roughness also modies the char-acteristics and thus probability density distributions of near-bed turbulence. For such conditions, information is needed toverify the reasonability of the assumption of log-normal vari-ables even in the sense of double-average.For the case of high shear stresses, this study shows thatan increase in the shear stress uctuation may yield a reduc-tionintheprobabilityof bedparticlemobility. Thisisanideal case considered here because high shear stress is oftenassociatedwithhighsediment transport ratewithbedformspresented. A physical interpretation of this result is not avail-able at present.Inthisstudy, theanalysisisperformedintermsofthebedshearstressratherthanowvelocities. Thisislargelybecause most of the existing sediment transport functions arepresented in terms of the time-mean bed shear stress. Manyfailures inapplyingtraditional bedloadformulas couldbedue to the ignoring of the bed shear stress uctuation. Someexperimental evidence was recently presented by Sumeret al.5If knowledge of the uctuation of the bed shear stressis available for various cases, then the computation of sedi-ment dischargecanbeimprovedifthetraditional formulasarestill engaged. Ontheother hand, it isnotedthat owvelocitiescouldbemoresuitableparametersfor exploringturbulence-relatedevents. However, itshouldbementionedthat a quantitative link of sediment discharge and turbulenceevents is unknown at this stage. This would limit any furtherapplication of the analysis if it is being performed based onow velocities.CONCLUSIONSForthecaseofplanesedimentbed, itisshowninthisstudy that the distribution of the roughness-induced bedshear stressvariationscanbeapproximatedaslog-normal.Such variations are generally comparable to those associatedwith near-bed turbulence, of which much information isavailable in the literature for the case of unidirectional owsover smoothboundaries. Toconsider the effects of shearstress uctuationonsediment transport, a double-averagetechnique should be generally applied for the case of turbu-lent ows over a sediment bed.Withthelog-normalpdf,theprobabilityofbedparticlemobility is then analyzed, showing that the mobility may beenhancedor weakenedbythebedshear stress uctuationdependingontheaverageshearstress. Iftheaverageshearstress is low, an increase in the shear stress uctuation mayresultinlargerprobabilities;whiletheprobabilitycouldbereduced with increasing uctuations if the average shearstress becomes higher. This result further implies that theShieldsdiagrammayoverestimatethecriticalconditionforinitial sediment motionfor ows subject tostrongturbu-lence.1H. A. Einstein, TheBed-loadFunctionfor Sediment TransportationinOpen Channel Flows U.S. Dept. of Agriculture, Washington, DC, 1950.2R. A. Bagnold, Thenatureof saltationandof bed-loadtransport inwater, Proc. R. Soc. London, Ser. A332, 473 1973.3M. S. Yalin, Mechanicsof Sediment Transport, 2nded. Pergamon, Ox-ford, 1977.4A. J. GrassandR. N. M. Ayoub, Bedloadtransport of nesandbylaminar and turbulent ow, in Proceedings of the 18th Coastal Engineer-ing Conference, Cape Town, South Africa, edited by B. L. Edge AmericanSociety of Civil Engineers, New York, 1983, pp. 15891599.5B. M. Sumer, L. H. C. Chua, N. S. Cheng, and J. 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