Impulse and particle dislodgement under turbulent flow conditionsAhmet O Celik Panayiotis Diplas Clinton L Dancey and Manousos Valyrakis Citation Physics of Fluids (1994-present) 22 046601 (2010) doi 10106313385433 View online httpdxdoiorg10106313385433 View Table of Contents httpscitationaiporgcontentaipjournalpof2224ver=pdfcov Published by the AIP Publishing

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Impulse and particle dislodgement under turbulent flow conditionsAhmet O Celik1 Panayiotis Diplas1 Clinton L Dancey2 and Manousos Valyrakis1

1Department of Civil and Environmental Engineering Baker Environmental Hydraulics LaboratoryVirginia Tech Blacksburg Virginia 24061 USA2Department of Mechanical Engineering Baker Environmental Hydraulics LaboratoryVirginia Tech Blacksburg Virginia 24061 USA

Received 22 December 2009 accepted 11 March 2010 published online 20 April 2010

In this study we investigated the role of turbulence fluctuations on the entrainment of a fullyexposed grain near threshold flow conditions Experiments were carried out to measuresynchronously the near bed flow velocity and the particle movement for a range of flow conditionsand resulting particle entrainment frequencies We used a simplified bed geometry consisted ofspherical particles to reduce the complexities associated with the variations in the bed and flowdetails in an effort to identify the underlying dominant physical mechanism An analysis wasperformed based on common force approximations using near bed flow velocity Turbulencefluctuations were treated as impulses which are products of magnitude and duration of appliedforce It is demonstrated that besides the magnitude of the instantaneous forces applied on asediment grain their duration is important as well in determining whether a particle will beentrained by a turbulent flow event Frequency of particle entrainment varied remarkably withminute changes in gross flow parameters Impulse imparted on the sediment grain by turbulent flowwas found to be well represented by a log-normal distribution We obtained a log-normalprobability density function pdf dependent on only the coefficient of variation of the impulseimpulse intensity Relation of the impulse intensity to the particle Reynolds number Re wasestablished The sensitivity of the computed impulse to the critical force level as well as theinfluence of the critical impulse level on the dislodgement events was explored Particleentrainment probabilities were found using the derived pdf as well as experimental observations anda good agreement between the two is reported Implications of the presented impulse concept andour experimental findings for sediment mobility at low bed shear stress conditions are alsodiscussed copy 2010 American Institute of Physics doi10106313385433

I INTRODUCTION

The incipient motion condition for granular material de-fines the stability of erodible beds and constitutes the centralproblem for sediment transport in rivers coastal areas andatmospheric flows Shieldsrsquo deterministic framework1 em-ploying time-space average bed shear stress to describe thedriving hydrodynamic forces near the bed is the most widelyused practical tool and has been for over 70 years2 On theother hand as a result of decades of work the literatureacknowledges the significance of momentary high turbulentforces on mobilizing the sediment grains at incipientconditions3ndash8 These forces occur randomly in space and timedue to the turbulent flow near the bed9 It is advocated thatthe movement of a grain begins when the local instantaneousturbulent forces overcome the resisting forces which are alsostatistical in nature8 Occasional sediment movement is stillpossible under turbulent flow conditions where the mean hy-drodynamic force is not large enough to entrain theparticles1011 In addition Paintal10 using long observationperiods showed under flow conditions well below conven-tional critical conditions such as those proposed byShields12 that not only the random movements of bed ma-terial are observed but a small increase in the bed meanshear stress causes a significant increase in the movement ofbed material Paintal10 reported that at these low stresses the

bedload transport rate increases with the 16th power of theboundary shear stress Helland-Hansen et al12 and Hofland13

also made observations under low shear stress and low mo-bility conditions those are in qualitative agreement withPaintalrsquos10 findings

Incipient motion criteria utilizing gross flow characteris-tics such as those proposed by Shields do not account forthe force fluctuations and therefore are not sufficient to de-scribe the phenomenon at incipient conditions This view hasled many researchers to argue that a unique threshold level interms of bed shear stress does not exist at which the grainmovement suddenly begins1114ndash16

Many researchers in an effort to overcome the limita-tions of the time-averaged wall shear stress approach ex-plored the role of turbulent velocity and the resulting fluctu-ating hydrodynamic forces near the bed on the particledislodgement particularly for incipient conditions Severaldeterministic and stochastic approaches have been proposedas a result17ndash22 Common in these approaches is the impor-tance of the magnitude of peaks in the local flow velocity thestreamwise component in particular and resulting instanta-neous forces acting on individual grains in mobilizing theseparticles Despite these efforts and partially due to the ex-perimental difficulties in observing the threshold of sedimentmovement with synchronized flow measurements which

PHYSICS OF FLUIDS 22 046601 2010

1070-6631201022404660113$3000 copy 2010 American Institute of Physics22 046601-1

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also lead to subjectivity in defining the bed mobility21718 theprecise determination of the incipient condition remains elu-sive Moreover the nature of the processes that are causingthe phenomenon observed by Paintal10 has not been ex-plained by the recent methods so far

The sensitivity of the interactions between local turbu-lent flow and mobile sediment to even a minute interferenceobserved during incipient motion experiments suggests thatthe investigation of threshold of particle movement requiresthe use of nonintrusive particle tracking techniques1823

Balakrishnan24 using a video camera together with localflow velocity measurements provided evidence contrary tothe general consensus and demonstrated that not all localflow velocity fluctuations well above the mean value result inparticle dislodgement under incipient flow conditions A re-cent study examined this phenomenon further under wellcontrolled laboratory conditions and demonstrated that it isnot the magnitude of the applied force alone that serves asthe necessary and sufficient condition for particle entrain-ment but rather the combination of force and duration orimpulse25 The implication here is that the turbulence in-duced momentary peak forces acting on the bed materialmust last long enough to cause entrainment Although resultsreported by Diplas et al25 represent the behavior of a testparticle in a well controlled environment this conclusion im-plies that the impulse potential of a turbulent stream is rel-evant to the inception of bed material motion rather thansimply the magnitude of the instantaneous forces or time-space-averaged bed shear stress

The goal of this study is to carry out laboratory flumeexperiments to further investigate the influence of impulseon entrainment of a single grain for a range of flow strengthsand resulting entrainment frequencies This is pursued byexamining long duration data sets of synchronously mea-sured local streamwise flow velocity and associated impulsevalues and the entrainment of a mobile test particle Theformer was measured via the use of a laser Doppler veloci-meter LDV while the latter was monitored using a laser-based particle tracking technique

Tests were performed in fully developed uniform openchannel flow at near threshold conditions In this study in aneffort to simplify the phenomenon to its most elemental formand facilitate the development of cause and effect relationswhile retaining the physics which dominate the entrain-ment of an isolated fully exposed spherical particle wasexamined The preferred mode of particle movement underthese conditions is rolling consistent with observations madein the field for rounded or semirounded particles subjected tonear threshold conditions411

The impulse concept and the detection of impulses areexplained in the following section The experimental meth-ods are described in detail in Sec III Statistics and probabil-ity density function pdf of impulses detected in variousruns are presented in Sec IV A We discuss in Secs IV Band IV C the sensitivity of the total number of impulses ob-served and the number of impulses with a potential to yieldparticle entrainment to the critical force and critical impulselevels respectively Following these analyses the probabilityof particle entrainment is explored in Sec IV D Implications

of our findings to sediment movement at low bed shearstresses are discussed in Sec V Conclusions are given inSec VI

II IMPULSE CONCEPT

Despite the evidence available in the literature on thedynamical significance of the magnitude of fluctuating localturbulent forces in mobilizing the bed material the durationsover which these turbulent forces act received no attentionthus far with the exception of Diplas et al25 They providedevidence that the impulse imparted by near bed turbulentevents has to be larger than a critical impulse level to entraina sediment grain from rest25 Understanding the statisticalcharacteristics of impulse imparted by turbulent flow ratherthan just those of the local forces or bed shear stress is es-sential for describing the incipient bed material motion Inorder to detect impulses we used a simple approach in whichthe force time history is obtained from the local flow veloc-ity as described in the following section

A Impulse detection

Here the focus is on a simple setting where the grainsare spherical and a mobile particle rests on densely packedidentical size particles as shown in Fig 1 For the givenconfiguration it is assumed that the forces acting on the mo-bile particle are submerged particle weight WS and hydro-dynamic force F assumed to act through the center of grav-ity of the sphere2627 Drag force FD is the prevailinghydrodynamic force component in line with the flow direc-tion The lift force FL is neglected here since drag domi-nates grain dislodgement for highly exposed particles2829

Integrating the forcing function Ft describing the timehistory of hydrodynamic force acting upon a particle be-tween times t1 and t2 the impulse related to particle entrain-ment is given as

Impulse = t1

t2

Ftdt = FT with

1Ft Fcr between t1 and t2

where T= t2minus t1 is the total duration of the applied force Fis the time average force over duration T and Fcr is theminimum critical force required to initiate bed materialmotion obtained from the resting equilibrium conditions18

FIG 1 Definition sketch of the forces acting on a spherical particle restingon identical size densely packed spheres side view left and top viewright of the bed geometry

046601-2 Celik et al Phys Fluids 22 046601 2010

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as discussed later Since FD is the only hydrodynamic forcecomponent considered here Eq 1 is treated in the stream-wise direction and corresponding Fcr the critical drag forceFD cr is determined as follows

FD cr necessary to first move the spherical grain is ob-tained by a moment balance about the contact points of thetest particle with the base particles18 This is illustrated inFig 1 where O1 and O2 indicate the contact points betweenthe mobile grain and the two downstream base particlesFD cr can be derived from

FD cr = fVWScosX

Zminus sin 2

where the hydrodynamic mass coefficient18 fV

= 1+05 Sminus S is the density of the particle is thedensity of water X and Z are the lever arms aligned with andnormal to the bed respectively and is the angle betweenthe channel bed and the horizontal plane For steady flowsthe relationship between FD and local flow velocity in thestreamwise direction u upstream of the submerged body isgiven by

FD = 12u2CDA 3

where CD is the drag coefficient and A is the projected grainarea perpendicular to flow direction It is reasonable to esti-mate the relative level of the instantaneous FD and its tem-poral variation through the approximation FDtu2t2830

Consequently the u2 time history is utilized in the analysis toestimate the drag force time history The critical conditionfor the minimum drag force to initiate motion in terms offlow velocity then can be derived using Eqs 2 and 3 toyield

ucr2 =

2

CDAfVWScos

X

Zminus sin 4

Here fv is 143 for the Teflonreg ball that is used in this studyand CD is assumed to be 0928 The critical u2 value obtainedfrom Eq 4 is used in our analysis to detect those events in

the u2 FD time series which exceed the minimum re-quired threshold value ucr

2 as illustrated in Fig 2All events with u2ucr

2 are detectable within the u2 timeseries together with their times of occurrence and durationsTi duration over which u2ucr

2 In addition time-averageu2 values u2i can be computed for each such event withu2ucr

2 representative of the average drag force FDi of animpulsive event angle brackets denote averaging over im-pulse duration Consequently relative impulse event magni-tudes Ii= u2iTi hereafter referred to as ldquoimpulserdquo can beobtained Those events Ii associated with particle entrain-ments can also be identified experimentally by simulta-neously measuring the particle movement together with utAccordingly experiments described in the following sectionwere performed to obtain data sets of local flow velocity andparticle entrainment pairs

III EXPERIMENTS

Incipient motion experiments were conducted in a205 m long and 06 m wide flume located in the BakerEnvironmental Hydraulics Laboratory at Virginia TechSporadic entrainment events of a fully exposed mobileTeflonreg spherical particle of diameter d=127 mm with aspecific gravity of 23 resting on two layers of well packedidentical diameter glass spheres was monitored togetherwith the local flow velocity The test section was 14 m down-stream from the channel entrance to ensure fully developedturbulent flow conditions Streamwise component of the lo-cal flow velocity u was measured at one diameter upstreamof the test particle along its centerline with a LDV systemFigure 3 illustrates this arrangement Entrainment of the mo-bile test particle was recorded utilizing a separate laser-basedsystem that detects its displacement

Another component of this setup was a retaining pinlocated 15 mm downstream from the mobile particle seeFig 3 to prevent it from being transported downstream fromthe measurement location This retaining pin permitted thegrain to return to its original position under the action ofgravity This simple but important feature allowed for con-tinuous records of entrainment episodes to be obtained with-out manual intervention In this fashion sets of particle en-trainment and local flow velocity data pairs were obtainedfor various flow conditions

FIG 2 Representation of the impulse events in the u2 time series The ithevent is characterized by u2i and Ti values representing force magnitudeand duration the product of which is impulse= u2iTi corresponds to theshaded rectangular area below the u2 line t0 and tn were determined byinterpolating the adjacent data points in the u2 time series The vertical linebetween the t0 and tn indicates that the particle movement was observedduring the ith event

FIG 3 Side view right and top view upper left corner sketches ofthe mobile test particle and pocket geometry diameter of the grainsd=127 mm

046601-3 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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A Incipient particle motion detection

The particle tracking system used in this study employsa single low power 25ndash30 mW HendashNe laser source and aphotodetector positioned similar to an ldquoelectric-eyerdquo ar-rangement with the voltage output of the photodetector di-rectly related to the position of the mobile grain The cali-bration of the HendashNe system was performed in situ using amicrometer and resulted in a resolution of less than 10 mover the 15 mm full streamwise range of particle motionThe LDV and entrainment signals were recorded synchro-nously via a multichannel signal processor under variousflow conditions during the experiments reported here Thesampling frequency in these experiments varied between 250and 700 Hz A fraction of the photodetector voltage outputfrom run E1 see Table I for summary is shown in Fig 4together with simultaneous records of u2 and Ii u2iTiIn this example record both rocking and pivoting of theparticle about the contact points are shown We use the term

ldquopivotingrdquo to indicate those grain movement events whichcorrespond to full grain dislodgement

The photodetector voltage output was used to identifywithin the u2 and Ii time records the instants when a specificlevel of particle movement occurred This is illustrated inFig 4 After detecting the instants when the grain moved abinary 01 signal with ldquo1rdquo indicating any detectable particlemovement was constructed Fig 4 Note that the impulseplot in the middle of Fig 4 includes data for events withu2ucr

2 onlyThe typical behavior of the test particle during the ex-

periments was observed to be as follows When the test par-ticle dislodges it rolls downstream over the valley formed bythe pocket arrangement until it reaches the retaining pin seeFig 4 instants B and C respectively The particle tempo-rarily remains positioned against the pin until the flow in-duced forces are not strong enough to maintain the positionof the particle Fig 4 instants C and D respectively Thenthe particle falls back to its original pocket as shown in Fig4 between instants D and E We also observed occasionalldquorockingrdquo events of the test particle in several runs an ex-ample of which is shown in Fig 4 instant A At these in-stants the particle moves within the pocket but does notreach the retaining pin

B Experimental procedure

We conducted flume experiments under eight differentuniform flow conditions that resulted in various time aver-age particle entrainment frequencies fE The following ex-perimental procedure was repeated for each flow conditionFirst the experiment was run for sufficiently long time toobtain a stable fE value 120 min Subsequently under

TABLE I Summary of the test conditions for entrainment experiments

RunU

msH

cm Re

fE

Entminumean

ms TI

E1 045 75 424 0011 693 025 027

E2 043 82 413 0010 573 024 027

E3 043 90 399 0010 42 024 026

E4 041 79 398 0010 206 023 027

E5 042 83 385 0009 133 023 027

E6 040 86 377 0009 052 023 026

E7 041 91 372 0008 024 023 026

E8 039 87 364 0008 014 022 027

FIG 4 From top to bottom representative time series of u2 impulse u2iTi and photodetector output from run E1 Dashed vertical lines in the top twoplots indicate detected particle movements Secondary vertical axes in the top two plots binary 01 signal Explanation of the solid vertical lines in the bottomplot a beginning of a rocking event b beginning of a pivoting event c instant when the test particle reached the retaining pin d instant when the testparticle started rolling back to its original pocket and e instant when the particle reached its original pocket

046601-4 Celik et al Phys Fluids 22 046601 2010

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the same flow condition the local flow velocity and entrain-ment signals were recorded simultaneously for 15 min Atthe completion of the run velocity profile measurementswere obtained one diameter upstream of the test particle butwith the test particle removed from its pocket The bed slopewas kept constant at 025 throughout the experiments

IV RESULTS AND ANALYSIS

The analysis reveals that more than 90 of the particleentrainments in runs E1ndashE4 and all of the entrainments in theother four runs were associated with events having u2ucr

2 This high percentage indicates that our assumption about thedominance of FD on the entrainment of a fully exposed par-ticle is reasonable It also gives validity to the approach em-ployed here for calculating the ucr

2 value although some un-certainty remains because of the fact that the drag coefficientCD is not precisely known28

Table I summarizes the flow conditions It provides thedepth average flow velocity U flow depth H particleReynolds number Re=ud where u is the friction ve-locity and is the kinematic viscosity u was obtained us-ing the Clauser method31 shields stress =0 sminusgdwhere 0 is time-average bed shear stress g is gravitationalacceleration fE umean which is the time average value of u ata point one diameter upstream of the test particle along its

center and turbulence intensity given by urms umean whereurms is the root-mean-square of the turbulent velocity fluctua-tions u

Significant variation in the frequency of grain entrain-ment occurs with only minute changes in the gross flow pa-rameters Table I shows that a 14 increase in Re 35increase in bed shear stress 0u2

is accompanied by anearly 50-fold increase in fE This result is in qualitativeagreement with the findings of Paintal10 and Helland-Hansenet al12 and consistent with more limited flume observationsof Hofland13 It is also noted that the variation in the turbu-lence intensity measured in the immediate vicinity of theparticle for all eight experiments is almost negligible Thesefindings exemplify the inadequacies of incipient motionmodels that employ time-space average flow parameters andsuggest that deterministic and stochastic models that dependon local turbulence intensity to define the threshold of par-ticle movement must be used cautiously

Events for which u2ucr2 were obtained for all runs

E1ndashE8 by performing the analysis described in Sec II A viaan impulse detection code The statistics of detected impulseevents are presented in Table II for E1ndashE8 The histogramsof u2 u2i representing drag force averaged over the impulseduration for which u2ucr

2 Ti which is the impulse durationand Ii impulse from run E1 are given in Fig 5 A number ofpdfs has been proposed for u2 in studies concerning the sedi-ment entrainment181921 Based on the assumption that thedrag force is proportional to u2 in combination with aGaussian distribution for u Papanicolaou18 proposed a chi-squared distribution to describe instantaneous drag forcewhich was later modified by Hofland and Battjes30 to ac-count for the relative turbulence intensity But the distribu-tion of the impulse impulse duration or force magnitudeassociated with impulse events to the writersrsquo knowledgehas never been examined in sediment transport research Thehistogram of impulse is positively skewed with a long tail asshown in Fig 5 These distribution characteristics are due tovery high magnitude but rare impulse events The rare im-pulse events described by relatively long durations and arange of u2i values are in fact those responsible for particlemovement see Sec IV B It is important to note that theduration of impulse events Ti shows more than an order-of-

TABLE II Summary of the impulse parameters obtained from 15 min runs

Run

Impulse I= u2T

Imean

m2 s10minus3Istd

m2 s10minus3 Skewness Flatness

E1 239 242 101 33 147

E2 231 229 099 38 111

E3 2 19 095 35 114

E4 21 207 097 36 126

E5 211 196 093 38 11

E6 216 18 084 29 125

E7 19 166 087 28 109

E8 185 157 085 27 9

FIG 5 Histograms of u2 u2i Ti and Ii from left to right for the run E1 Nearly 280 000 data points counts for u2 and total of 1978 data points for u2iTi and Ii are represented in each histogram

046601-5 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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magnitude variation but a relatively narrow range ofu2i013ndash02 m2 s2 in all runs Fig 5 The variability ofevent durations is much more pronounced compared to forcemagnitudes and thus the durations to a greater extent influ-ence the impulse values

A Distribution of impulse

The impulse statistics and histograms indicate that theimpulse distribution is heavily skewed A suitable functionthat describes data with such properties is the log-normaldistribution function Cheng and Law32 proposed a log-normal distribution to describe the bed shear stress fluctua-tions Another example of the use of log-normal distributionfunction in sediment research is provided by Wu et al33

Recently Mouri et al34 reported log normality in energyfluctuations obtained from velocity measurements for a widerange of turbulence scales at various flow conditions It isgenerally argued that the origin of log normality lies in themultiplicative stochastic processes ie as a result of inde-pendent stochastic variables inherent to turbulent flowphenomena3435 Impulse the product of turbulent forcesand their durations near the bed may be described by alog-normal distribution If the impulse is nondimensionalized

with its ensemble average value I= I Imean then the log-normal probability density function for the dimension-

less impulse I as a function of the impulse intensity= Istd Imean is given by

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2 5

The derivation of Eq 5 is given in the Appendix Note that=I the latter being the intensity of the dimensionless im-

pulse I Table II gives the calculated values for alleight runs A linear relationship exists between and Re forthe limited range of uniform flow conditions tested hereFig 6 A linear regression approach provides the followingexpression

=Re

417 6

with a coefficient of determination of 085

Figure 7 shows the influence of on the distribution of IEq 5 Equation 5 approaches a normal distribution for02 For higher values on the other hand the distribu-tion becomes positively skewed For all values that are ofinterest here Table II and shown in Fig 7 the difference inthe right tails of pdfs is clear and significant These devia-tions in the shape of the tail with small variations in essen-tially dictate the probability of occurrence of extreme im-pulse events above a high critical level As it will bediscussed in the next section critical impulse level25 itselfwhich has different values in dimensionless form foreach run is also crucial in determining the probability ofoccurrence of impulse events with a potential to entrain theparticle

Measured pdfs of impulses obtained from normalizedhistograms are compared with Eq 5 in Fig 8 for all eightruns values were determined from the data for each par-ticular run and were used in Eq 5 It can be seen that thederived pdf describes the data well In the construction of thepdfs shown in Fig 8 the tail of the distribution for highimpulse is emphasized This tail characterizes the extremeimpulse events and hence it is most relevant to particle en-trainment for near threshold conditions as discussed later inSec IV D The selection of the optimum number of bins25 used to resolve the tail necessarily results in poorresolution at the other extreme of the distribution the risingleg and peak In order to show the tail sections better asemilogarithmic plot of Eq 5 together with all the experi-mental data is presented in Fig 9 covering the range of testconditions =07 and 11 Despite the scatter of data espe-

cially at higher I values I3 Eq 5 predicts the overalltrend well Since the rising leg and the peak in the pdfs arenow involved in rest of the analysis no uncertainty due tothe differenceslack of data points in this section of the pdfwas introduced Therefore no attempts such as employingarbitrary values for different sections of the pdf have beenmade

FIG 6 Relationship between impulse intensity and particle Reynoldsnumber Re

FIG 7 Plots of the function given by Eq 5 for a range of values

046601-6 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

Furthermore low amplitude impulse events where

I1 in the rising legs of the pdfs are characterized by com-binations of very short Ti and very high u2i values or viceversa Close inspection of the data reveals that the former

may not be detected accurately even at turbulent resolvingsampling frequencies ie peaks occurring within less than5 ms The latter are on the other hand extremely rareGiven that the analysis is sensitive to the features of detected

FIG 8 Comparison of Eq 5 with measured pdfs for E1ndashE8 Solid lines are used to show pdfs obtained from Eq 5

046601-7 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

impulses the following Secs IV B and IV C are presented toexamine the role of critical u2 level ucr

2 on the number andstatistics of the detected impulses

B Critical u2 and number of impulses

Values of u2i are plotted with respect to correspondingTi in Fig 10 for all impulse events and including the eventsassociated with full particle dislodgement the latter areshown in the figure with solid circles for runs E1 and E4Plots given in Fig 10 clearly demonstrate that in agreementwith the findings of Diplas et al25 although FDFD cr

is necessary for the initiation of motion this condition aloneis not sufficient to predict full grain dislodgement Contraryto common force-balance models and other stochasticapproaches only particular combinations of u2i andTi u2ucr

2 with moderate to large durations yielded graindislodgement In fact the most recent incipient motionmodels161920 would significantly overestimate the particleentrainment rate under similar flow and bed conditions Thisis true since all of the detected events with u2ucr

2 for in-stance all 1978 events in run E1 are routinely expected toentrain the particle in such mechanistic models because theseevents would cause very high instantaneous drag forces onthe particle On the contrary our experiment shows that lessthan 7 of these events caused particle entrainment in runE1 This finding is consistent with Balakrishnanrsquos24 observa-tions The total number of impulse events nT for whichu2ucr

2 and the number of impulse events u2ucr2 that

yield particle dislodgement nE both observed over a 15 minsampling period are given for all runs in Table III Note that

nE=nP+nR where nP and nR are the number of pivoting androcking events respectively As shown in Table III the dif-ference between the nT and nE values is dramatic for all runsOverall less than an-order-of magnitude increase in nT cor-responds to two-orders-of-magnitude increase in the nE Thisobservation supports the idea that the force magnitude aloneis not sufficient to describe the particle entrainment

In order to investigate the origins of this remarkablevariability in the number of impulses with minute changes inthe flow conditions the role of critical u2 ucr

2 level on thenumber and statistics of the detected impulses was examinedThis was done artificially not by running additional experi-ments with different grain density or size but rather by ad-justing the level of ucr

2 in the analysis of the current series ofexperiments This approach is comparable to that by varyingthe gross flow parameters slightly as was the procedure in theactual experiments In this analysis data from run E1 wereused Eight arbitrarily chosen critical u2 values slightlyabove and below the original critical level ucr

2 13u2 asillustrated in Fig 11a were selected to identify impulseevents now associated with drag force above the new criticallevel Table IV gives the critical u2 value that is used theresulting number of detected impulses mean and standarddeviations of u2i Ti and impulse respectively and the im-pulse intensity The row with bold font in Table IV indicatesthe results from the conditions where the original critical u2

obtained from Eq 4 was used to detect the impulsesIt is observed that the number of detected impulses is

very sensitive to the chosen critical level of u2 Slightly shift-

TABLE III Number of impulse events and particle movements observed for15 min Note that nE=nP+nR

Run No of impulse events nT

No of particle movements nE

Pivoting nP Rocking nR

E1 1978 104 29

E2 1554 86 17

E3 1262 63 16

E4 1101 31 15

E5 722 20 8

E6 416 8 3

E7 362 4 4

E8 249 1 1

FIG 9 Semilogarithmic plot of measured pdfs from all eight runs Equation5 is also presented with =07 and 11 for comparison

FIG 10 u2i vs Ti plots 1978 data points from run E1 left 1101 datapoints from run E4 right Black circles indicate u2i Ti combinations thatare associated with full particle dislodgement pivoting

FIG 11 a Illustration of the approach used for varying the critical u2 bNumber of detected impulses vs the ratio of critical u2 level used to theoriginal ucr

2

046601-8 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

ing the critical u2 level up or down results in a significantchange in the number of detected impulses The number ofimpulses increases by an order of magnitude when the criti-cal u2 level is shifted downward by only about 40 asshown in Fig 11b The latter also corresponds to a hypo-thetical 25 decrease in the particles density Eq 4 Thisis due to the fact that as the critical u2 is reduced thoseevents that were initially below ucr

2 now become importantie detectable The significant increase in the number ofimpulses in this analysis similar to what was observed in theoriginal experiments suggests that the temporal features re-sulting from the underlying structure of turbulence togetherwith the critical u2 level plays a crucial role in the significantvariation of number of impulse events observed here

The u2i mean as well as Ti std increase linearly with in-creasing critical u2 level while the u2i std and Ti mean

decrease The change in Ti mean with the critical u2 levelis however much more pronounced compared to the changein u2i mean The impulse parameters also do not vary asmuch as the number of impulses Impulse intensity= Ii std Ii mean increases in a linear fashion as the criticallevel moves down mostly due to the decrease in the standarddeviation of impulse As the critical u2 level approaches theaverage of u2 the impulse intensity increases Together withour earlier findings shown in Fig 7 this observation sug-gests that the distribution of impulse becomes more skewedfor relatively lower critical u2 levels In the case of highercritical u2 levels however the smaller peaks are not detectedand the distribution of impulse including only those extremeevents with high impulse magnitudes will approach a nor-mal distribution

It is apparent that the number of impulse events thosewhere u2ucr

2 is sensitive to ucr2 but it is also clear that not

all impulse events yield grain dislodgement In the next sec-tion we investigate the relationship between those events thatare above ucr

2 and critical impulse25

C Critical impulse and number of grain entrainments

Figures 9 and 10 imply that the turbulent channel flowcontains flow structures with a potential to apply a range ofimpulse values varying over an order of magnitude A thresh-old impulse value under the given bed and particle condi-

tions can be defined by identifying an impulse level abovewhich particle entrainment takes place25 This constant im-pulse is approximated by a value of u2iTi in the overlapregion where both grain movement and no movement areobserved on the plot of impulse u2iTi with respect to u2i

Fig 12 This region lies between the ldquoimpulserdquo values of00034 and 00095 m2 s

The computed critical impulse level is obtained from theaverage over this region as shown in Fig 12 data fromE1ndashE8 are included The obtained u2iTi value approxi-mates the threshold of impulse Icr considering that all ofthe runs were performed under a single bed configurationand thus under the same threshold of movement conditionThe uncertainty in detecting the critical impulse level in ouranalysis is attributed to the variations in the instantaneousdrag coefficient28

In order to investigate the role of the critical impulselevel the previous analysis using artificial critical u2 levelsSec IV B was extended That is a corresponding criticalimpulse level for each critical u2 level Table IV was esti-mated as follows The density of the particle correspondingto each of the critical u2 levels given in Table IV was deter-mined using Eq 4 Note that here the size of the particle

TABLE IV Summary of the results from conditions where various u2 critical values were used for run E1

Critical u2

m2 s2

Numberof detected impulses

nTu2i mean

m2 s2u2i std

m2 s2Ti mean

sTi std

sIi mean

m2 sIi std

m2 s

0160 427 0176 0010 0013 0008 00022 00016 072

0144 925 0161 0010 0014 0011 00023 00019 083

0133 1532 0150 0010 0015 0013 00023 00022 096

0130 1748 0147 0010 0016 0014 00024 00023 096

0126 1978 0143 0011 0016 0015 00024 00024 101

0123 2302 0140 0011 0017 0016 00024 00025 104

0119 2615 0136 0011 0017 0016 00024 00026 108

0116 2948 0133 0011 0017 0017 00024 00026 108

0096 5456 0114 0012 0021 0023 00025 00032 128

FIG 12 Iiu2iTi vs u2i plot The region where movement and no move-ment areas overlap is shown with a gray band between the impulse values of00034 and 00095 m2 s Horizontal arrow indicates the critical impulselevel

046601-9 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

as discussed later Since FD is the only hydrodynamic forcecomponent considered here Eq 1 is treated in the stream-wise direction and corresponding Fcr the critical drag forceFD cr is determined as follows

FD cr necessary to first move the spherical grain is ob-tained by a moment balance about the contact points of thetest particle with the base particles18 This is illustrated inFig 1 where O1 and O2 indicate the contact points betweenthe mobile grain and the two downstream base particlesFD cr can be derived from

FD cr = fVWScosX

Zminus sin 2

where the hydrodynamic mass coefficient18 fV

= 1+05 Sminus S is the density of the particle is thedensity of water X and Z are the lever arms aligned with andnormal to the bed respectively and is the angle betweenthe channel bed and the horizontal plane For steady flowsthe relationship between FD and local flow velocity in thestreamwise direction u upstream of the submerged body isgiven by

FD = 12u2CDA 3

where CD is the drag coefficient and A is the projected grainarea perpendicular to flow direction It is reasonable to esti-mate the relative level of the instantaneous FD and its tem-poral variation through the approximation FDtu2t2830

Consequently the u2 time history is utilized in the analysis toestimate the drag force time history The critical conditionfor the minimum drag force to initiate motion in terms offlow velocity then can be derived using Eqs 2 and 3 toyield

ucr2 =

2

CDAfVWScos

X

Zminus sin 4

Here fv is 143 for the Teflonreg ball that is used in this studyand CD is assumed to be 0928 The critical u2 value obtainedfrom Eq 4 is used in our analysis to detect those events in

the u2 FD time series which exceed the minimum re-quired threshold value ucr

2 as illustrated in Fig 2All events with u2ucr

2 are detectable within the u2 timeseries together with their times of occurrence and durationsTi duration over which u2ucr

2 In addition time-averageu2 values u2i can be computed for each such event withu2ucr

2 representative of the average drag force FDi of animpulsive event angle brackets denote averaging over im-pulse duration Consequently relative impulse event magni-tudes Ii= u2iTi hereafter referred to as ldquoimpulserdquo can beobtained Those events Ii associated with particle entrain-ments can also be identified experimentally by simulta-neously measuring the particle movement together with utAccordingly experiments described in the following sectionwere performed to obtain data sets of local flow velocity andparticle entrainment pairs

III EXPERIMENTS

Incipient motion experiments were conducted in a205 m long and 06 m wide flume located in the BakerEnvironmental Hydraulics Laboratory at Virginia TechSporadic entrainment events of a fully exposed mobileTeflonreg spherical particle of diameter d=127 mm with aspecific gravity of 23 resting on two layers of well packedidentical diameter glass spheres was monitored togetherwith the local flow velocity The test section was 14 m down-stream from the channel entrance to ensure fully developedturbulent flow conditions Streamwise component of the lo-cal flow velocity u was measured at one diameter upstreamof the test particle along its centerline with a LDV systemFigure 3 illustrates this arrangement Entrainment of the mo-bile test particle was recorded utilizing a separate laser-basedsystem that detects its displacement

Another component of this setup was a retaining pinlocated 15 mm downstream from the mobile particle seeFig 3 to prevent it from being transported downstream fromthe measurement location This retaining pin permitted thegrain to return to its original position under the action ofgravity This simple but important feature allowed for con-tinuous records of entrainment episodes to be obtained with-out manual intervention In this fashion sets of particle en-trainment and local flow velocity data pairs were obtainedfor various flow conditions

FIG 2 Representation of the impulse events in the u2 time series The ithevent is characterized by u2i and Ti values representing force magnitudeand duration the product of which is impulse= u2iTi corresponds to theshaded rectangular area below the u2 line t0 and tn were determined byinterpolating the adjacent data points in the u2 time series The vertical linebetween the t0 and tn indicates that the particle movement was observedduring the ith event

FIG 3 Side view right and top view upper left corner sketches ofthe mobile test particle and pocket geometry diameter of the grainsd=127 mm

046601-3 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

A Incipient particle motion detection

The particle tracking system used in this study employsa single low power 25ndash30 mW HendashNe laser source and aphotodetector positioned similar to an ldquoelectric-eyerdquo ar-rangement with the voltage output of the photodetector di-rectly related to the position of the mobile grain The cali-bration of the HendashNe system was performed in situ using amicrometer and resulted in a resolution of less than 10 mover the 15 mm full streamwise range of particle motionThe LDV and entrainment signals were recorded synchro-nously via a multichannel signal processor under variousflow conditions during the experiments reported here Thesampling frequency in these experiments varied between 250and 700 Hz A fraction of the photodetector voltage outputfrom run E1 see Table I for summary is shown in Fig 4together with simultaneous records of u2 and Ii u2iTiIn this example record both rocking and pivoting of theparticle about the contact points are shown We use the term

ldquopivotingrdquo to indicate those grain movement events whichcorrespond to full grain dislodgement

The photodetector voltage output was used to identifywithin the u2 and Ii time records the instants when a specificlevel of particle movement occurred This is illustrated inFig 4 After detecting the instants when the grain moved abinary 01 signal with ldquo1rdquo indicating any detectable particlemovement was constructed Fig 4 Note that the impulseplot in the middle of Fig 4 includes data for events withu2ucr

2 onlyThe typical behavior of the test particle during the ex-

periments was observed to be as follows When the test par-ticle dislodges it rolls downstream over the valley formed bythe pocket arrangement until it reaches the retaining pin seeFig 4 instants B and C respectively The particle tempo-rarily remains positioned against the pin until the flow in-duced forces are not strong enough to maintain the positionof the particle Fig 4 instants C and D respectively Thenthe particle falls back to its original pocket as shown in Fig4 between instants D and E We also observed occasionalldquorockingrdquo events of the test particle in several runs an ex-ample of which is shown in Fig 4 instant A At these in-stants the particle moves within the pocket but does notreach the retaining pin

B Experimental procedure

We conducted flume experiments under eight differentuniform flow conditions that resulted in various time aver-age particle entrainment frequencies fE The following ex-perimental procedure was repeated for each flow conditionFirst the experiment was run for sufficiently long time toobtain a stable fE value 120 min Subsequently under

TABLE I Summary of the test conditions for entrainment experiments

RunU

msH

cm Re

fE

Entminumean

ms TI

E1 045 75 424 0011 693 025 027

E2 043 82 413 0010 573 024 027

E3 043 90 399 0010 42 024 026

E4 041 79 398 0010 206 023 027

E5 042 83 385 0009 133 023 027

E6 040 86 377 0009 052 023 026

E7 041 91 372 0008 024 023 026

E8 039 87 364 0008 014 022 027

FIG 4 From top to bottom representative time series of u2 impulse u2iTi and photodetector output from run E1 Dashed vertical lines in the top twoplots indicate detected particle movements Secondary vertical axes in the top two plots binary 01 signal Explanation of the solid vertical lines in the bottomplot a beginning of a rocking event b beginning of a pivoting event c instant when the test particle reached the retaining pin d instant when the testparticle started rolling back to its original pocket and e instant when the particle reached its original pocket

046601-4 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

the same flow condition the local flow velocity and entrain-ment signals were recorded simultaneously for 15 min Atthe completion of the run velocity profile measurementswere obtained one diameter upstream of the test particle butwith the test particle removed from its pocket The bed slopewas kept constant at 025 throughout the experiments

IV RESULTS AND ANALYSIS

The analysis reveals that more than 90 of the particleentrainments in runs E1ndashE4 and all of the entrainments in theother four runs were associated with events having u2ucr

2 This high percentage indicates that our assumption about thedominance of FD on the entrainment of a fully exposed par-ticle is reasonable It also gives validity to the approach em-ployed here for calculating the ucr

2 value although some un-certainty remains because of the fact that the drag coefficientCD is not precisely known28

Table I summarizes the flow conditions It provides thedepth average flow velocity U flow depth H particleReynolds number Re=ud where u is the friction ve-locity and is the kinematic viscosity u was obtained us-ing the Clauser method31 shields stress =0 sminusgdwhere 0 is time-average bed shear stress g is gravitationalacceleration fE umean which is the time average value of u ata point one diameter upstream of the test particle along its

center and turbulence intensity given by urms umean whereurms is the root-mean-square of the turbulent velocity fluctua-tions u

Significant variation in the frequency of grain entrain-ment occurs with only minute changes in the gross flow pa-rameters Table I shows that a 14 increase in Re 35increase in bed shear stress 0u2

is accompanied by anearly 50-fold increase in fE This result is in qualitativeagreement with the findings of Paintal10 and Helland-Hansenet al12 and consistent with more limited flume observationsof Hofland13 It is also noted that the variation in the turbu-lence intensity measured in the immediate vicinity of theparticle for all eight experiments is almost negligible Thesefindings exemplify the inadequacies of incipient motionmodels that employ time-space average flow parameters andsuggest that deterministic and stochastic models that dependon local turbulence intensity to define the threshold of par-ticle movement must be used cautiously

Events for which u2ucr2 were obtained for all runs

E1ndashE8 by performing the analysis described in Sec II A viaan impulse detection code The statistics of detected impulseevents are presented in Table II for E1ndashE8 The histogramsof u2 u2i representing drag force averaged over the impulseduration for which u2ucr

2 Ti which is the impulse durationand Ii impulse from run E1 are given in Fig 5 A number ofpdfs has been proposed for u2 in studies concerning the sedi-ment entrainment181921 Based on the assumption that thedrag force is proportional to u2 in combination with aGaussian distribution for u Papanicolaou18 proposed a chi-squared distribution to describe instantaneous drag forcewhich was later modified by Hofland and Battjes30 to ac-count for the relative turbulence intensity But the distribu-tion of the impulse impulse duration or force magnitudeassociated with impulse events to the writersrsquo knowledgehas never been examined in sediment transport research Thehistogram of impulse is positively skewed with a long tail asshown in Fig 5 These distribution characteristics are due tovery high magnitude but rare impulse events The rare im-pulse events described by relatively long durations and arange of u2i values are in fact those responsible for particlemovement see Sec IV B It is important to note that theduration of impulse events Ti shows more than an order-of-

TABLE II Summary of the impulse parameters obtained from 15 min runs

Run

Impulse I= u2T

Imean

m2 s10minus3Istd

m2 s10minus3 Skewness Flatness

E1 239 242 101 33 147

E2 231 229 099 38 111

E3 2 19 095 35 114

E4 21 207 097 36 126

E5 211 196 093 38 11

E6 216 18 084 29 125

E7 19 166 087 28 109

E8 185 157 085 27 9

FIG 5 Histograms of u2 u2i Ti and Ii from left to right for the run E1 Nearly 280 000 data points counts for u2 and total of 1978 data points for u2iTi and Ii are represented in each histogram

046601-5 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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magnitude variation but a relatively narrow range ofu2i013ndash02 m2 s2 in all runs Fig 5 The variability ofevent durations is much more pronounced compared to forcemagnitudes and thus the durations to a greater extent influ-ence the impulse values

A Distribution of impulse

The impulse statistics and histograms indicate that theimpulse distribution is heavily skewed A suitable functionthat describes data with such properties is the log-normaldistribution function Cheng and Law32 proposed a log-normal distribution to describe the bed shear stress fluctua-tions Another example of the use of log-normal distributionfunction in sediment research is provided by Wu et al33

Recently Mouri et al34 reported log normality in energyfluctuations obtained from velocity measurements for a widerange of turbulence scales at various flow conditions It isgenerally argued that the origin of log normality lies in themultiplicative stochastic processes ie as a result of inde-pendent stochastic variables inherent to turbulent flowphenomena3435 Impulse the product of turbulent forcesand their durations near the bed may be described by alog-normal distribution If the impulse is nondimensionalized

with its ensemble average value I= I Imean then the log-normal probability density function for the dimension-

less impulse I as a function of the impulse intensity= Istd Imean is given by

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2 5

The derivation of Eq 5 is given in the Appendix Note that=I the latter being the intensity of the dimensionless im-

pulse I Table II gives the calculated values for alleight runs A linear relationship exists between and Re forthe limited range of uniform flow conditions tested hereFig 6 A linear regression approach provides the followingexpression

=Re

417 6

with a coefficient of determination of 085

Figure 7 shows the influence of on the distribution of IEq 5 Equation 5 approaches a normal distribution for02 For higher values on the other hand the distribu-tion becomes positively skewed For all values that are ofinterest here Table II and shown in Fig 7 the difference inthe right tails of pdfs is clear and significant These devia-tions in the shape of the tail with small variations in essen-tially dictate the probability of occurrence of extreme im-pulse events above a high critical level As it will bediscussed in the next section critical impulse level25 itselfwhich has different values in dimensionless form foreach run is also crucial in determining the probability ofoccurrence of impulse events with a potential to entrain theparticle

Measured pdfs of impulses obtained from normalizedhistograms are compared with Eq 5 in Fig 8 for all eightruns values were determined from the data for each par-ticular run and were used in Eq 5 It can be seen that thederived pdf describes the data well In the construction of thepdfs shown in Fig 8 the tail of the distribution for highimpulse is emphasized This tail characterizes the extremeimpulse events and hence it is most relevant to particle en-trainment for near threshold conditions as discussed later inSec IV D The selection of the optimum number of bins25 used to resolve the tail necessarily results in poorresolution at the other extreme of the distribution the risingleg and peak In order to show the tail sections better asemilogarithmic plot of Eq 5 together with all the experi-mental data is presented in Fig 9 covering the range of testconditions =07 and 11 Despite the scatter of data espe-

cially at higher I values I3 Eq 5 predicts the overalltrend well Since the rising leg and the peak in the pdfs arenow involved in rest of the analysis no uncertainty due tothe differenceslack of data points in this section of the pdfwas introduced Therefore no attempts such as employingarbitrary values for different sections of the pdf have beenmade

FIG 6 Relationship between impulse intensity and particle Reynoldsnumber Re

FIG 7 Plots of the function given by Eq 5 for a range of values

046601-6 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

Furthermore low amplitude impulse events where

I1 in the rising legs of the pdfs are characterized by com-binations of very short Ti and very high u2i values or viceversa Close inspection of the data reveals that the former

may not be detected accurately even at turbulent resolvingsampling frequencies ie peaks occurring within less than5 ms The latter are on the other hand extremely rareGiven that the analysis is sensitive to the features of detected

FIG 8 Comparison of Eq 5 with measured pdfs for E1ndashE8 Solid lines are used to show pdfs obtained from Eq 5

046601-7 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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impulses the following Secs IV B and IV C are presented toexamine the role of critical u2 level ucr

2 on the number andstatistics of the detected impulses

B Critical u2 and number of impulses

Values of u2i are plotted with respect to correspondingTi in Fig 10 for all impulse events and including the eventsassociated with full particle dislodgement the latter areshown in the figure with solid circles for runs E1 and E4Plots given in Fig 10 clearly demonstrate that in agreementwith the findings of Diplas et al25 although FDFD cr

is necessary for the initiation of motion this condition aloneis not sufficient to predict full grain dislodgement Contraryto common force-balance models and other stochasticapproaches only particular combinations of u2i andTi u2ucr

2 with moderate to large durations yielded graindislodgement In fact the most recent incipient motionmodels161920 would significantly overestimate the particleentrainment rate under similar flow and bed conditions Thisis true since all of the detected events with u2ucr

2 for in-stance all 1978 events in run E1 are routinely expected toentrain the particle in such mechanistic models because theseevents would cause very high instantaneous drag forces onthe particle On the contrary our experiment shows that lessthan 7 of these events caused particle entrainment in runE1 This finding is consistent with Balakrishnanrsquos24 observa-tions The total number of impulse events nT for whichu2ucr

2 and the number of impulse events u2ucr2 that

yield particle dislodgement nE both observed over a 15 minsampling period are given for all runs in Table III Note that

nE=nP+nR where nP and nR are the number of pivoting androcking events respectively As shown in Table III the dif-ference between the nT and nE values is dramatic for all runsOverall less than an-order-of magnitude increase in nT cor-responds to two-orders-of-magnitude increase in the nE Thisobservation supports the idea that the force magnitude aloneis not sufficient to describe the particle entrainment

In order to investigate the origins of this remarkablevariability in the number of impulses with minute changes inthe flow conditions the role of critical u2 ucr

2 level on thenumber and statistics of the detected impulses was examinedThis was done artificially not by running additional experi-ments with different grain density or size but rather by ad-justing the level of ucr

2 in the analysis of the current series ofexperiments This approach is comparable to that by varyingthe gross flow parameters slightly as was the procedure in theactual experiments In this analysis data from run E1 wereused Eight arbitrarily chosen critical u2 values slightlyabove and below the original critical level ucr

2 13u2 asillustrated in Fig 11a were selected to identify impulseevents now associated with drag force above the new criticallevel Table IV gives the critical u2 value that is used theresulting number of detected impulses mean and standarddeviations of u2i Ti and impulse respectively and the im-pulse intensity The row with bold font in Table IV indicatesthe results from the conditions where the original critical u2

obtained from Eq 4 was used to detect the impulsesIt is observed that the number of detected impulses is

very sensitive to the chosen critical level of u2 Slightly shift-

TABLE III Number of impulse events and particle movements observed for15 min Note that nE=nP+nR

Run No of impulse events nT

No of particle movements nE

Pivoting nP Rocking nR

E1 1978 104 29

E2 1554 86 17

E3 1262 63 16

E4 1101 31 15

E5 722 20 8

E6 416 8 3

E7 362 4 4

E8 249 1 1

FIG 9 Semilogarithmic plot of measured pdfs from all eight runs Equation5 is also presented with =07 and 11 for comparison

FIG 10 u2i vs Ti plots 1978 data points from run E1 left 1101 datapoints from run E4 right Black circles indicate u2i Ti combinations thatare associated with full particle dislodgement pivoting

FIG 11 a Illustration of the approach used for varying the critical u2 bNumber of detected impulses vs the ratio of critical u2 level used to theoriginal ucr

2

046601-8 Celik et al Phys Fluids 22 046601 2010

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ing the critical u2 level up or down results in a significantchange in the number of detected impulses The number ofimpulses increases by an order of magnitude when the criti-cal u2 level is shifted downward by only about 40 asshown in Fig 11b The latter also corresponds to a hypo-thetical 25 decrease in the particles density Eq 4 Thisis due to the fact that as the critical u2 is reduced thoseevents that were initially below ucr

2 now become importantie detectable The significant increase in the number ofimpulses in this analysis similar to what was observed in theoriginal experiments suggests that the temporal features re-sulting from the underlying structure of turbulence togetherwith the critical u2 level plays a crucial role in the significantvariation of number of impulse events observed here

The u2i mean as well as Ti std increase linearly with in-creasing critical u2 level while the u2i std and Ti mean

decrease The change in Ti mean with the critical u2 levelis however much more pronounced compared to the changein u2i mean The impulse parameters also do not vary asmuch as the number of impulses Impulse intensity= Ii std Ii mean increases in a linear fashion as the criticallevel moves down mostly due to the decrease in the standarddeviation of impulse As the critical u2 level approaches theaverage of u2 the impulse intensity increases Together withour earlier findings shown in Fig 7 this observation sug-gests that the distribution of impulse becomes more skewedfor relatively lower critical u2 levels In the case of highercritical u2 levels however the smaller peaks are not detectedand the distribution of impulse including only those extremeevents with high impulse magnitudes will approach a nor-mal distribution

It is apparent that the number of impulse events thosewhere u2ucr

2 is sensitive to ucr2 but it is also clear that not

all impulse events yield grain dislodgement In the next sec-tion we investigate the relationship between those events thatare above ucr

2 and critical impulse25

C Critical impulse and number of grain entrainments

Figures 9 and 10 imply that the turbulent channel flowcontains flow structures with a potential to apply a range ofimpulse values varying over an order of magnitude A thresh-old impulse value under the given bed and particle condi-

tions can be defined by identifying an impulse level abovewhich particle entrainment takes place25 This constant im-pulse is approximated by a value of u2iTi in the overlapregion where both grain movement and no movement areobserved on the plot of impulse u2iTi with respect to u2i

Fig 12 This region lies between the ldquoimpulserdquo values of00034 and 00095 m2 s

The computed critical impulse level is obtained from theaverage over this region as shown in Fig 12 data fromE1ndashE8 are included The obtained u2iTi value approxi-mates the threshold of impulse Icr considering that all ofthe runs were performed under a single bed configurationand thus under the same threshold of movement conditionThe uncertainty in detecting the critical impulse level in ouranalysis is attributed to the variations in the instantaneousdrag coefficient28

In order to investigate the role of the critical impulselevel the previous analysis using artificial critical u2 levelsSec IV B was extended That is a corresponding criticalimpulse level for each critical u2 level Table IV was esti-mated as follows The density of the particle correspondingto each of the critical u2 levels given in Table IV was deter-mined using Eq 4 Note that here the size of the particle

TABLE IV Summary of the results from conditions where various u2 critical values were used for run E1

Critical u2

m2 s2

Numberof detected impulses

nTu2i mean

m2 s2u2i std

m2 s2Ti mean

sTi std

sIi mean

m2 sIi std

m2 s

0160 427 0176 0010 0013 0008 00022 00016 072

0144 925 0161 0010 0014 0011 00023 00019 083

0133 1532 0150 0010 0015 0013 00023 00022 096

0130 1748 0147 0010 0016 0014 00024 00023 096

0126 1978 0143 0011 0016 0015 00024 00024 101

0123 2302 0140 0011 0017 0016 00024 00025 104

0119 2615 0136 0011 0017 0016 00024 00026 108

0116 2948 0133 0011 0017 0017 00024 00026 108

0096 5456 0114 0012 0021 0023 00025 00032 128

FIG 12 Iiu2iTi vs u2i plot The region where movement and no move-ment areas overlap is shown with a gray band between the impulse values of00034 and 00095 m2 s Horizontal arrow indicates the critical impulselevel

046601-9 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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A Incipient particle motion detection

The particle tracking system used in this study employsa single low power 25ndash30 mW HendashNe laser source and aphotodetector positioned similar to an ldquoelectric-eyerdquo ar-rangement with the voltage output of the photodetector di-rectly related to the position of the mobile grain The cali-bration of the HendashNe system was performed in situ using amicrometer and resulted in a resolution of less than 10 mover the 15 mm full streamwise range of particle motionThe LDV and entrainment signals were recorded synchro-nously via a multichannel signal processor under variousflow conditions during the experiments reported here Thesampling frequency in these experiments varied between 250and 700 Hz A fraction of the photodetector voltage outputfrom run E1 see Table I for summary is shown in Fig 4together with simultaneous records of u2 and Ii u2iTiIn this example record both rocking and pivoting of theparticle about the contact points are shown We use the term

ldquopivotingrdquo to indicate those grain movement events whichcorrespond to full grain dislodgement

The photodetector voltage output was used to identifywithin the u2 and Ii time records the instants when a specificlevel of particle movement occurred This is illustrated inFig 4 After detecting the instants when the grain moved abinary 01 signal with ldquo1rdquo indicating any detectable particlemovement was constructed Fig 4 Note that the impulseplot in the middle of Fig 4 includes data for events withu2ucr

2 onlyThe typical behavior of the test particle during the ex-

periments was observed to be as follows When the test par-ticle dislodges it rolls downstream over the valley formed bythe pocket arrangement until it reaches the retaining pin seeFig 4 instants B and C respectively The particle tempo-rarily remains positioned against the pin until the flow in-duced forces are not strong enough to maintain the positionof the particle Fig 4 instants C and D respectively Thenthe particle falls back to its original pocket as shown in Fig4 between instants D and E We also observed occasionalldquorockingrdquo events of the test particle in several runs an ex-ample of which is shown in Fig 4 instant A At these in-stants the particle moves within the pocket but does notreach the retaining pin

B Experimental procedure

We conducted flume experiments under eight differentuniform flow conditions that resulted in various time aver-age particle entrainment frequencies fE The following ex-perimental procedure was repeated for each flow conditionFirst the experiment was run for sufficiently long time toobtain a stable fE value 120 min Subsequently under

TABLE I Summary of the test conditions for entrainment experiments

RunU

msH

cm Re

fE

Entminumean

ms TI

E1 045 75 424 0011 693 025 027

E2 043 82 413 0010 573 024 027

E3 043 90 399 0010 42 024 026

E4 041 79 398 0010 206 023 027

E5 042 83 385 0009 133 023 027

E6 040 86 377 0009 052 023 026

E7 041 91 372 0008 024 023 026

E8 039 87 364 0008 014 022 027

FIG 4 From top to bottom representative time series of u2 impulse u2iTi and photodetector output from run E1 Dashed vertical lines in the top twoplots indicate detected particle movements Secondary vertical axes in the top two plots binary 01 signal Explanation of the solid vertical lines in the bottomplot a beginning of a rocking event b beginning of a pivoting event c instant when the test particle reached the retaining pin d instant when the testparticle started rolling back to its original pocket and e instant when the particle reached its original pocket

046601-4 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

the same flow condition the local flow velocity and entrain-ment signals were recorded simultaneously for 15 min Atthe completion of the run velocity profile measurementswere obtained one diameter upstream of the test particle butwith the test particle removed from its pocket The bed slopewas kept constant at 025 throughout the experiments

IV RESULTS AND ANALYSIS

The analysis reveals that more than 90 of the particleentrainments in runs E1ndashE4 and all of the entrainments in theother four runs were associated with events having u2ucr

2 This high percentage indicates that our assumption about thedominance of FD on the entrainment of a fully exposed par-ticle is reasonable It also gives validity to the approach em-ployed here for calculating the ucr

2 value although some un-certainty remains because of the fact that the drag coefficientCD is not precisely known28

Table I summarizes the flow conditions It provides thedepth average flow velocity U flow depth H particleReynolds number Re=ud where u is the friction ve-locity and is the kinematic viscosity u was obtained us-ing the Clauser method31 shields stress =0 sminusgdwhere 0 is time-average bed shear stress g is gravitationalacceleration fE umean which is the time average value of u ata point one diameter upstream of the test particle along its

center and turbulence intensity given by urms umean whereurms is the root-mean-square of the turbulent velocity fluctua-tions u

Significant variation in the frequency of grain entrain-ment occurs with only minute changes in the gross flow pa-rameters Table I shows that a 14 increase in Re 35increase in bed shear stress 0u2

is accompanied by anearly 50-fold increase in fE This result is in qualitativeagreement with the findings of Paintal10 and Helland-Hansenet al12 and consistent with more limited flume observationsof Hofland13 It is also noted that the variation in the turbu-lence intensity measured in the immediate vicinity of theparticle for all eight experiments is almost negligible Thesefindings exemplify the inadequacies of incipient motionmodels that employ time-space average flow parameters andsuggest that deterministic and stochastic models that dependon local turbulence intensity to define the threshold of par-ticle movement must be used cautiously

Events for which u2ucr2 were obtained for all runs

E1ndashE8 by performing the analysis described in Sec II A viaan impulse detection code The statistics of detected impulseevents are presented in Table II for E1ndashE8 The histogramsof u2 u2i representing drag force averaged over the impulseduration for which u2ucr

2 Ti which is the impulse durationand Ii impulse from run E1 are given in Fig 5 A number ofpdfs has been proposed for u2 in studies concerning the sedi-ment entrainment181921 Based on the assumption that thedrag force is proportional to u2 in combination with aGaussian distribution for u Papanicolaou18 proposed a chi-squared distribution to describe instantaneous drag forcewhich was later modified by Hofland and Battjes30 to ac-count for the relative turbulence intensity But the distribu-tion of the impulse impulse duration or force magnitudeassociated with impulse events to the writersrsquo knowledgehas never been examined in sediment transport research Thehistogram of impulse is positively skewed with a long tail asshown in Fig 5 These distribution characteristics are due tovery high magnitude but rare impulse events The rare im-pulse events described by relatively long durations and arange of u2i values are in fact those responsible for particlemovement see Sec IV B It is important to note that theduration of impulse events Ti shows more than an order-of-

TABLE II Summary of the impulse parameters obtained from 15 min runs

Run

Impulse I= u2T

Imean

m2 s10minus3Istd

m2 s10minus3 Skewness Flatness

E1 239 242 101 33 147

E2 231 229 099 38 111

E3 2 19 095 35 114

E4 21 207 097 36 126

E5 211 196 093 38 11

E6 216 18 084 29 125

E7 19 166 087 28 109

E8 185 157 085 27 9

FIG 5 Histograms of u2 u2i Ti and Ii from left to right for the run E1 Nearly 280 000 data points counts for u2 and total of 1978 data points for u2iTi and Ii are represented in each histogram

046601-5 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

magnitude variation but a relatively narrow range ofu2i013ndash02 m2 s2 in all runs Fig 5 The variability ofevent durations is much more pronounced compared to forcemagnitudes and thus the durations to a greater extent influ-ence the impulse values

A Distribution of impulse

The impulse statistics and histograms indicate that theimpulse distribution is heavily skewed A suitable functionthat describes data with such properties is the log-normaldistribution function Cheng and Law32 proposed a log-normal distribution to describe the bed shear stress fluctua-tions Another example of the use of log-normal distributionfunction in sediment research is provided by Wu et al33

Recently Mouri et al34 reported log normality in energyfluctuations obtained from velocity measurements for a widerange of turbulence scales at various flow conditions It isgenerally argued that the origin of log normality lies in themultiplicative stochastic processes ie as a result of inde-pendent stochastic variables inherent to turbulent flowphenomena3435 Impulse the product of turbulent forcesand their durations near the bed may be described by alog-normal distribution If the impulse is nondimensionalized

with its ensemble average value I= I Imean then the log-normal probability density function for the dimension-

less impulse I as a function of the impulse intensity= Istd Imean is given by

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2 5

The derivation of Eq 5 is given in the Appendix Note that=I the latter being the intensity of the dimensionless im-

pulse I Table II gives the calculated values for alleight runs A linear relationship exists between and Re forthe limited range of uniform flow conditions tested hereFig 6 A linear regression approach provides the followingexpression

=Re

417 6

with a coefficient of determination of 085

Figure 7 shows the influence of on the distribution of IEq 5 Equation 5 approaches a normal distribution for02 For higher values on the other hand the distribu-tion becomes positively skewed For all values that are ofinterest here Table II and shown in Fig 7 the difference inthe right tails of pdfs is clear and significant These devia-tions in the shape of the tail with small variations in essen-tially dictate the probability of occurrence of extreme im-pulse events above a high critical level As it will bediscussed in the next section critical impulse level25 itselfwhich has different values in dimensionless form foreach run is also crucial in determining the probability ofoccurrence of impulse events with a potential to entrain theparticle

Measured pdfs of impulses obtained from normalizedhistograms are compared with Eq 5 in Fig 8 for all eightruns values were determined from the data for each par-ticular run and were used in Eq 5 It can be seen that thederived pdf describes the data well In the construction of thepdfs shown in Fig 8 the tail of the distribution for highimpulse is emphasized This tail characterizes the extremeimpulse events and hence it is most relevant to particle en-trainment for near threshold conditions as discussed later inSec IV D The selection of the optimum number of bins25 used to resolve the tail necessarily results in poorresolution at the other extreme of the distribution the risingleg and peak In order to show the tail sections better asemilogarithmic plot of Eq 5 together with all the experi-mental data is presented in Fig 9 covering the range of testconditions =07 and 11 Despite the scatter of data espe-

cially at higher I values I3 Eq 5 predicts the overalltrend well Since the rising leg and the peak in the pdfs arenow involved in rest of the analysis no uncertainty due tothe differenceslack of data points in this section of the pdfwas introduced Therefore no attempts such as employingarbitrary values for different sections of the pdf have beenmade

FIG 6 Relationship between impulse intensity and particle Reynoldsnumber Re

FIG 7 Plots of the function given by Eq 5 for a range of values

046601-6 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

Furthermore low amplitude impulse events where

I1 in the rising legs of the pdfs are characterized by com-binations of very short Ti and very high u2i values or viceversa Close inspection of the data reveals that the former

may not be detected accurately even at turbulent resolvingsampling frequencies ie peaks occurring within less than5 ms The latter are on the other hand extremely rareGiven that the analysis is sensitive to the features of detected

FIG 8 Comparison of Eq 5 with measured pdfs for E1ndashE8 Solid lines are used to show pdfs obtained from Eq 5

046601-7 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

impulses the following Secs IV B and IV C are presented toexamine the role of critical u2 level ucr

2 on the number andstatistics of the detected impulses

B Critical u2 and number of impulses

Values of u2i are plotted with respect to correspondingTi in Fig 10 for all impulse events and including the eventsassociated with full particle dislodgement the latter areshown in the figure with solid circles for runs E1 and E4Plots given in Fig 10 clearly demonstrate that in agreementwith the findings of Diplas et al25 although FDFD cr

is necessary for the initiation of motion this condition aloneis not sufficient to predict full grain dislodgement Contraryto common force-balance models and other stochasticapproaches only particular combinations of u2i andTi u2ucr

2 with moderate to large durations yielded graindislodgement In fact the most recent incipient motionmodels161920 would significantly overestimate the particleentrainment rate under similar flow and bed conditions Thisis true since all of the detected events with u2ucr

2 for in-stance all 1978 events in run E1 are routinely expected toentrain the particle in such mechanistic models because theseevents would cause very high instantaneous drag forces onthe particle On the contrary our experiment shows that lessthan 7 of these events caused particle entrainment in runE1 This finding is consistent with Balakrishnanrsquos24 observa-tions The total number of impulse events nT for whichu2ucr

2 and the number of impulse events u2ucr2 that

yield particle dislodgement nE both observed over a 15 minsampling period are given for all runs in Table III Note that

nE=nP+nR where nP and nR are the number of pivoting androcking events respectively As shown in Table III the dif-ference between the nT and nE values is dramatic for all runsOverall less than an-order-of magnitude increase in nT cor-responds to two-orders-of-magnitude increase in the nE Thisobservation supports the idea that the force magnitude aloneis not sufficient to describe the particle entrainment

In order to investigate the origins of this remarkablevariability in the number of impulses with minute changes inthe flow conditions the role of critical u2 ucr

2 level on thenumber and statistics of the detected impulses was examinedThis was done artificially not by running additional experi-ments with different grain density or size but rather by ad-justing the level of ucr

2 in the analysis of the current series ofexperiments This approach is comparable to that by varyingthe gross flow parameters slightly as was the procedure in theactual experiments In this analysis data from run E1 wereused Eight arbitrarily chosen critical u2 values slightlyabove and below the original critical level ucr

2 13u2 asillustrated in Fig 11a were selected to identify impulseevents now associated with drag force above the new criticallevel Table IV gives the critical u2 value that is used theresulting number of detected impulses mean and standarddeviations of u2i Ti and impulse respectively and the im-pulse intensity The row with bold font in Table IV indicatesthe results from the conditions where the original critical u2

obtained from Eq 4 was used to detect the impulsesIt is observed that the number of detected impulses is

very sensitive to the chosen critical level of u2 Slightly shift-

TABLE III Number of impulse events and particle movements observed for15 min Note that nE=nP+nR

Run No of impulse events nT

No of particle movements nE

Pivoting nP Rocking nR

E1 1978 104 29

E2 1554 86 17

E3 1262 63 16

E4 1101 31 15

E5 722 20 8

E6 416 8 3

E7 362 4 4

E8 249 1 1

FIG 9 Semilogarithmic plot of measured pdfs from all eight runs Equation5 is also presented with =07 and 11 for comparison

FIG 10 u2i vs Ti plots 1978 data points from run E1 left 1101 datapoints from run E4 right Black circles indicate u2i Ti combinations thatare associated with full particle dislodgement pivoting

FIG 11 a Illustration of the approach used for varying the critical u2 bNumber of detected impulses vs the ratio of critical u2 level used to theoriginal ucr

2

046601-8 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

ing the critical u2 level up or down results in a significantchange in the number of detected impulses The number ofimpulses increases by an order of magnitude when the criti-cal u2 level is shifted downward by only about 40 asshown in Fig 11b The latter also corresponds to a hypo-thetical 25 decrease in the particles density Eq 4 Thisis due to the fact that as the critical u2 is reduced thoseevents that were initially below ucr

2 now become importantie detectable The significant increase in the number ofimpulses in this analysis similar to what was observed in theoriginal experiments suggests that the temporal features re-sulting from the underlying structure of turbulence togetherwith the critical u2 level plays a crucial role in the significantvariation of number of impulse events observed here

The u2i mean as well as Ti std increase linearly with in-creasing critical u2 level while the u2i std and Ti mean

decrease The change in Ti mean with the critical u2 levelis however much more pronounced compared to the changein u2i mean The impulse parameters also do not vary asmuch as the number of impulses Impulse intensity= Ii std Ii mean increases in a linear fashion as the criticallevel moves down mostly due to the decrease in the standarddeviation of impulse As the critical u2 level approaches theaverage of u2 the impulse intensity increases Together withour earlier findings shown in Fig 7 this observation sug-gests that the distribution of impulse becomes more skewedfor relatively lower critical u2 levels In the case of highercritical u2 levels however the smaller peaks are not detectedand the distribution of impulse including only those extremeevents with high impulse magnitudes will approach a nor-mal distribution

It is apparent that the number of impulse events thosewhere u2ucr

2 is sensitive to ucr2 but it is also clear that not

all impulse events yield grain dislodgement In the next sec-tion we investigate the relationship between those events thatare above ucr

2 and critical impulse25

C Critical impulse and number of grain entrainments

Figures 9 and 10 imply that the turbulent channel flowcontains flow structures with a potential to apply a range ofimpulse values varying over an order of magnitude A thresh-old impulse value under the given bed and particle condi-

tions can be defined by identifying an impulse level abovewhich particle entrainment takes place25 This constant im-pulse is approximated by a value of u2iTi in the overlapregion where both grain movement and no movement areobserved on the plot of impulse u2iTi with respect to u2i

Fig 12 This region lies between the ldquoimpulserdquo values of00034 and 00095 m2 s

The computed critical impulse level is obtained from theaverage over this region as shown in Fig 12 data fromE1ndashE8 are included The obtained u2iTi value approxi-mates the threshold of impulse Icr considering that all ofthe runs were performed under a single bed configurationand thus under the same threshold of movement conditionThe uncertainty in detecting the critical impulse level in ouranalysis is attributed to the variations in the instantaneousdrag coefficient28

In order to investigate the role of the critical impulselevel the previous analysis using artificial critical u2 levelsSec IV B was extended That is a corresponding criticalimpulse level for each critical u2 level Table IV was esti-mated as follows The density of the particle correspondingto each of the critical u2 levels given in Table IV was deter-mined using Eq 4 Note that here the size of the particle

TABLE IV Summary of the results from conditions where various u2 critical values were used for run E1

Critical u2

m2 s2

Numberof detected impulses

nTu2i mean

m2 s2u2i std

m2 s2Ti mean

sTi std

sIi mean

m2 sIi std

m2 s

0160 427 0176 0010 0013 0008 00022 00016 072

0144 925 0161 0010 0014 0011 00023 00019 083

0133 1532 0150 0010 0015 0013 00023 00022 096

0130 1748 0147 0010 0016 0014 00024 00023 096

0126 1978 0143 0011 0016 0015 00024 00024 101

0123 2302 0140 0011 0017 0016 00024 00025 104

0119 2615 0136 0011 0017 0016 00024 00026 108

0116 2948 0133 0011 0017 0017 00024 00026 108

0096 5456 0114 0012 0021 0023 00025 00032 128

FIG 12 Iiu2iTi vs u2i plot The region where movement and no move-ment areas overlap is shown with a gray band between the impulse values of00034 and 00095 m2 s Horizontal arrow indicates the critical impulselevel

046601-9 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

the same flow condition the local flow velocity and entrain-ment signals were recorded simultaneously for 15 min Atthe completion of the run velocity profile measurementswere obtained one diameter upstream of the test particle butwith the test particle removed from its pocket The bed slopewas kept constant at 025 throughout the experiments

IV RESULTS AND ANALYSIS

The analysis reveals that more than 90 of the particleentrainments in runs E1ndashE4 and all of the entrainments in theother four runs were associated with events having u2ucr

2 This high percentage indicates that our assumption about thedominance of FD on the entrainment of a fully exposed par-ticle is reasonable It also gives validity to the approach em-ployed here for calculating the ucr

2 value although some un-certainty remains because of the fact that the drag coefficientCD is not precisely known28

Table I summarizes the flow conditions It provides thedepth average flow velocity U flow depth H particleReynolds number Re=ud where u is the friction ve-locity and is the kinematic viscosity u was obtained us-ing the Clauser method31 shields stress =0 sminusgdwhere 0 is time-average bed shear stress g is gravitationalacceleration fE umean which is the time average value of u ata point one diameter upstream of the test particle along its

center and turbulence intensity given by urms umean whereurms is the root-mean-square of the turbulent velocity fluctua-tions u

Significant variation in the frequency of grain entrain-ment occurs with only minute changes in the gross flow pa-rameters Table I shows that a 14 increase in Re 35increase in bed shear stress 0u2

is accompanied by anearly 50-fold increase in fE This result is in qualitativeagreement with the findings of Paintal10 and Helland-Hansenet al12 and consistent with more limited flume observationsof Hofland13 It is also noted that the variation in the turbu-lence intensity measured in the immediate vicinity of theparticle for all eight experiments is almost negligible Thesefindings exemplify the inadequacies of incipient motionmodels that employ time-space average flow parameters andsuggest that deterministic and stochastic models that dependon local turbulence intensity to define the threshold of par-ticle movement must be used cautiously

Events for which u2ucr2 were obtained for all runs

E1ndashE8 by performing the analysis described in Sec II A viaan impulse detection code The statistics of detected impulseevents are presented in Table II for E1ndashE8 The histogramsof u2 u2i representing drag force averaged over the impulseduration for which u2ucr

2 Ti which is the impulse durationand Ii impulse from run E1 are given in Fig 5 A number ofpdfs has been proposed for u2 in studies concerning the sedi-ment entrainment181921 Based on the assumption that thedrag force is proportional to u2 in combination with aGaussian distribution for u Papanicolaou18 proposed a chi-squared distribution to describe instantaneous drag forcewhich was later modified by Hofland and Battjes30 to ac-count for the relative turbulence intensity But the distribu-tion of the impulse impulse duration or force magnitudeassociated with impulse events to the writersrsquo knowledgehas never been examined in sediment transport research Thehistogram of impulse is positively skewed with a long tail asshown in Fig 5 These distribution characteristics are due tovery high magnitude but rare impulse events The rare im-pulse events described by relatively long durations and arange of u2i values are in fact those responsible for particlemovement see Sec IV B It is important to note that theduration of impulse events Ti shows more than an order-of-

TABLE II Summary of the impulse parameters obtained from 15 min runs

Run

Impulse I= u2T

Imean

m2 s10minus3Istd

m2 s10minus3 Skewness Flatness

E1 239 242 101 33 147

E2 231 229 099 38 111

E3 2 19 095 35 114

E4 21 207 097 36 126

E5 211 196 093 38 11

E6 216 18 084 29 125

E7 19 166 087 28 109

E8 185 157 085 27 9

FIG 5 Histograms of u2 u2i Ti and Ii from left to right for the run E1 Nearly 280 000 data points counts for u2 and total of 1978 data points for u2iTi and Ii are represented in each histogram

046601-5 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

magnitude variation but a relatively narrow range ofu2i013ndash02 m2 s2 in all runs Fig 5 The variability ofevent durations is much more pronounced compared to forcemagnitudes and thus the durations to a greater extent influ-ence the impulse values

A Distribution of impulse

The impulse statistics and histograms indicate that theimpulse distribution is heavily skewed A suitable functionthat describes data with such properties is the log-normaldistribution function Cheng and Law32 proposed a log-normal distribution to describe the bed shear stress fluctua-tions Another example of the use of log-normal distributionfunction in sediment research is provided by Wu et al33

Recently Mouri et al34 reported log normality in energyfluctuations obtained from velocity measurements for a widerange of turbulence scales at various flow conditions It isgenerally argued that the origin of log normality lies in themultiplicative stochastic processes ie as a result of inde-pendent stochastic variables inherent to turbulent flowphenomena3435 Impulse the product of turbulent forcesand their durations near the bed may be described by alog-normal distribution If the impulse is nondimensionalized

with its ensemble average value I= I Imean then the log-normal probability density function for the dimension-

less impulse I as a function of the impulse intensity= Istd Imean is given by

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2 5

The derivation of Eq 5 is given in the Appendix Note that=I the latter being the intensity of the dimensionless im-

pulse I Table II gives the calculated values for alleight runs A linear relationship exists between and Re forthe limited range of uniform flow conditions tested hereFig 6 A linear regression approach provides the followingexpression

=Re

417 6

with a coefficient of determination of 085

Figure 7 shows the influence of on the distribution of IEq 5 Equation 5 approaches a normal distribution for02 For higher values on the other hand the distribu-tion becomes positively skewed For all values that are ofinterest here Table II and shown in Fig 7 the difference inthe right tails of pdfs is clear and significant These devia-tions in the shape of the tail with small variations in essen-tially dictate the probability of occurrence of extreme im-pulse events above a high critical level As it will bediscussed in the next section critical impulse level25 itselfwhich has different values in dimensionless form foreach run is also crucial in determining the probability ofoccurrence of impulse events with a potential to entrain theparticle

Measured pdfs of impulses obtained from normalizedhistograms are compared with Eq 5 in Fig 8 for all eightruns values were determined from the data for each par-ticular run and were used in Eq 5 It can be seen that thederived pdf describes the data well In the construction of thepdfs shown in Fig 8 the tail of the distribution for highimpulse is emphasized This tail characterizes the extremeimpulse events and hence it is most relevant to particle en-trainment for near threshold conditions as discussed later inSec IV D The selection of the optimum number of bins25 used to resolve the tail necessarily results in poorresolution at the other extreme of the distribution the risingleg and peak In order to show the tail sections better asemilogarithmic plot of Eq 5 together with all the experi-mental data is presented in Fig 9 covering the range of testconditions =07 and 11 Despite the scatter of data espe-

cially at higher I values I3 Eq 5 predicts the overalltrend well Since the rising leg and the peak in the pdfs arenow involved in rest of the analysis no uncertainty due tothe differenceslack of data points in this section of the pdfwas introduced Therefore no attempts such as employingarbitrary values for different sections of the pdf have beenmade

FIG 6 Relationship between impulse intensity and particle Reynoldsnumber Re

FIG 7 Plots of the function given by Eq 5 for a range of values

046601-6 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

Furthermore low amplitude impulse events where

I1 in the rising legs of the pdfs are characterized by com-binations of very short Ti and very high u2i values or viceversa Close inspection of the data reveals that the former

may not be detected accurately even at turbulent resolvingsampling frequencies ie peaks occurring within less than5 ms The latter are on the other hand extremely rareGiven that the analysis is sensitive to the features of detected

FIG 8 Comparison of Eq 5 with measured pdfs for E1ndashE8 Solid lines are used to show pdfs obtained from Eq 5

046601-7 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

impulses the following Secs IV B and IV C are presented toexamine the role of critical u2 level ucr

2 on the number andstatistics of the detected impulses

B Critical u2 and number of impulses

Values of u2i are plotted with respect to correspondingTi in Fig 10 for all impulse events and including the eventsassociated with full particle dislodgement the latter areshown in the figure with solid circles for runs E1 and E4Plots given in Fig 10 clearly demonstrate that in agreementwith the findings of Diplas et al25 although FDFD cr

is necessary for the initiation of motion this condition aloneis not sufficient to predict full grain dislodgement Contraryto common force-balance models and other stochasticapproaches only particular combinations of u2i andTi u2ucr

2 with moderate to large durations yielded graindislodgement In fact the most recent incipient motionmodels161920 would significantly overestimate the particleentrainment rate under similar flow and bed conditions Thisis true since all of the detected events with u2ucr

2 for in-stance all 1978 events in run E1 are routinely expected toentrain the particle in such mechanistic models because theseevents would cause very high instantaneous drag forces onthe particle On the contrary our experiment shows that lessthan 7 of these events caused particle entrainment in runE1 This finding is consistent with Balakrishnanrsquos24 observa-tions The total number of impulse events nT for whichu2ucr

2 and the number of impulse events u2ucr2 that

yield particle dislodgement nE both observed over a 15 minsampling period are given for all runs in Table III Note that

nE=nP+nR where nP and nR are the number of pivoting androcking events respectively As shown in Table III the dif-ference between the nT and nE values is dramatic for all runsOverall less than an-order-of magnitude increase in nT cor-responds to two-orders-of-magnitude increase in the nE Thisobservation supports the idea that the force magnitude aloneis not sufficient to describe the particle entrainment

In order to investigate the origins of this remarkablevariability in the number of impulses with minute changes inthe flow conditions the role of critical u2 ucr

2 level on thenumber and statistics of the detected impulses was examinedThis was done artificially not by running additional experi-ments with different grain density or size but rather by ad-justing the level of ucr

2 in the analysis of the current series ofexperiments This approach is comparable to that by varyingthe gross flow parameters slightly as was the procedure in theactual experiments In this analysis data from run E1 wereused Eight arbitrarily chosen critical u2 values slightlyabove and below the original critical level ucr

2 13u2 asillustrated in Fig 11a were selected to identify impulseevents now associated with drag force above the new criticallevel Table IV gives the critical u2 value that is used theresulting number of detected impulses mean and standarddeviations of u2i Ti and impulse respectively and the im-pulse intensity The row with bold font in Table IV indicatesthe results from the conditions where the original critical u2

obtained from Eq 4 was used to detect the impulsesIt is observed that the number of detected impulses is

very sensitive to the chosen critical level of u2 Slightly shift-

TABLE III Number of impulse events and particle movements observed for15 min Note that nE=nP+nR

Run No of impulse events nT

No of particle movements nE

Pivoting nP Rocking nR

E1 1978 104 29

E2 1554 86 17

E3 1262 63 16

E4 1101 31 15

E5 722 20 8

E6 416 8 3

E7 362 4 4

E8 249 1 1

FIG 9 Semilogarithmic plot of measured pdfs from all eight runs Equation5 is also presented with =07 and 11 for comparison

FIG 10 u2i vs Ti plots 1978 data points from run E1 left 1101 datapoints from run E4 right Black circles indicate u2i Ti combinations thatare associated with full particle dislodgement pivoting

FIG 11 a Illustration of the approach used for varying the critical u2 bNumber of detected impulses vs the ratio of critical u2 level used to theoriginal ucr

2

046601-8 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

ing the critical u2 level up or down results in a significantchange in the number of detected impulses The number ofimpulses increases by an order of magnitude when the criti-cal u2 level is shifted downward by only about 40 asshown in Fig 11b The latter also corresponds to a hypo-thetical 25 decrease in the particles density Eq 4 Thisis due to the fact that as the critical u2 is reduced thoseevents that were initially below ucr

2 now become importantie detectable The significant increase in the number ofimpulses in this analysis similar to what was observed in theoriginal experiments suggests that the temporal features re-sulting from the underlying structure of turbulence togetherwith the critical u2 level plays a crucial role in the significantvariation of number of impulse events observed here

The u2i mean as well as Ti std increase linearly with in-creasing critical u2 level while the u2i std and Ti mean

decrease The change in Ti mean with the critical u2 levelis however much more pronounced compared to the changein u2i mean The impulse parameters also do not vary asmuch as the number of impulses Impulse intensity= Ii std Ii mean increases in a linear fashion as the criticallevel moves down mostly due to the decrease in the standarddeviation of impulse As the critical u2 level approaches theaverage of u2 the impulse intensity increases Together withour earlier findings shown in Fig 7 this observation sug-gests that the distribution of impulse becomes more skewedfor relatively lower critical u2 levels In the case of highercritical u2 levels however the smaller peaks are not detectedand the distribution of impulse including only those extremeevents with high impulse magnitudes will approach a nor-mal distribution

It is apparent that the number of impulse events thosewhere u2ucr

2 is sensitive to ucr2 but it is also clear that not

all impulse events yield grain dislodgement In the next sec-tion we investigate the relationship between those events thatare above ucr

2 and critical impulse25

C Critical impulse and number of grain entrainments

Figures 9 and 10 imply that the turbulent channel flowcontains flow structures with a potential to apply a range ofimpulse values varying over an order of magnitude A thresh-old impulse value under the given bed and particle condi-

tions can be defined by identifying an impulse level abovewhich particle entrainment takes place25 This constant im-pulse is approximated by a value of u2iTi in the overlapregion where both grain movement and no movement areobserved on the plot of impulse u2iTi with respect to u2i

Fig 12 This region lies between the ldquoimpulserdquo values of00034 and 00095 m2 s

The computed critical impulse level is obtained from theaverage over this region as shown in Fig 12 data fromE1ndashE8 are included The obtained u2iTi value approxi-mates the threshold of impulse Icr considering that all ofthe runs were performed under a single bed configurationand thus under the same threshold of movement conditionThe uncertainty in detecting the critical impulse level in ouranalysis is attributed to the variations in the instantaneousdrag coefficient28

In order to investigate the role of the critical impulselevel the previous analysis using artificial critical u2 levelsSec IV B was extended That is a corresponding criticalimpulse level for each critical u2 level Table IV was esti-mated as follows The density of the particle correspondingto each of the critical u2 levels given in Table IV was deter-mined using Eq 4 Note that here the size of the particle

TABLE IV Summary of the results from conditions where various u2 critical values were used for run E1

Critical u2

m2 s2

Numberof detected impulses

nTu2i mean

m2 s2u2i std

m2 s2Ti mean

sTi std

sIi mean

m2 sIi std

m2 s

0160 427 0176 0010 0013 0008 00022 00016 072

0144 925 0161 0010 0014 0011 00023 00019 083

0133 1532 0150 0010 0015 0013 00023 00022 096

0130 1748 0147 0010 0016 0014 00024 00023 096

0126 1978 0143 0011 0016 0015 00024 00024 101

0123 2302 0140 0011 0017 0016 00024 00025 104

0119 2615 0136 0011 0017 0016 00024 00026 108

0116 2948 0133 0011 0017 0017 00024 00026 108

0096 5456 0114 0012 0021 0023 00025 00032 128

FIG 12 Iiu2iTi vs u2i plot The region where movement and no move-ment areas overlap is shown with a gray band between the impulse values of00034 and 00095 m2 s Horizontal arrow indicates the critical impulselevel

046601-9 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

magnitude variation but a relatively narrow range ofu2i013ndash02 m2 s2 in all runs Fig 5 The variability ofevent durations is much more pronounced compared to forcemagnitudes and thus the durations to a greater extent influ-ence the impulse values

A Distribution of impulse

The impulse statistics and histograms indicate that theimpulse distribution is heavily skewed A suitable functionthat describes data with such properties is the log-normaldistribution function Cheng and Law32 proposed a log-normal distribution to describe the bed shear stress fluctua-tions Another example of the use of log-normal distributionfunction in sediment research is provided by Wu et al33

Recently Mouri et al34 reported log normality in energyfluctuations obtained from velocity measurements for a widerange of turbulence scales at various flow conditions It isgenerally argued that the origin of log normality lies in themultiplicative stochastic processes ie as a result of inde-pendent stochastic variables inherent to turbulent flowphenomena3435 Impulse the product of turbulent forcesand their durations near the bed may be described by alog-normal distribution If the impulse is nondimensionalized

with its ensemble average value I= I Imean then the log-normal probability density function for the dimension-

less impulse I as a function of the impulse intensity= Istd Imean is given by

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2 5

The derivation of Eq 5 is given in the Appendix Note that=I the latter being the intensity of the dimensionless im-

pulse I Table II gives the calculated values for alleight runs A linear relationship exists between and Re forthe limited range of uniform flow conditions tested hereFig 6 A linear regression approach provides the followingexpression

=Re

417 6

with a coefficient of determination of 085

Figure 7 shows the influence of on the distribution of IEq 5 Equation 5 approaches a normal distribution for02 For higher values on the other hand the distribu-tion becomes positively skewed For all values that are ofinterest here Table II and shown in Fig 7 the difference inthe right tails of pdfs is clear and significant These devia-tions in the shape of the tail with small variations in essen-tially dictate the probability of occurrence of extreme im-pulse events above a high critical level As it will bediscussed in the next section critical impulse level25 itselfwhich has different values in dimensionless form foreach run is also crucial in determining the probability ofoccurrence of impulse events with a potential to entrain theparticle

Measured pdfs of impulses obtained from normalizedhistograms are compared with Eq 5 in Fig 8 for all eightruns values were determined from the data for each par-ticular run and were used in Eq 5 It can be seen that thederived pdf describes the data well In the construction of thepdfs shown in Fig 8 the tail of the distribution for highimpulse is emphasized This tail characterizes the extremeimpulse events and hence it is most relevant to particle en-trainment for near threshold conditions as discussed later inSec IV D The selection of the optimum number of bins25 used to resolve the tail necessarily results in poorresolution at the other extreme of the distribution the risingleg and peak In order to show the tail sections better asemilogarithmic plot of Eq 5 together with all the experi-mental data is presented in Fig 9 covering the range of testconditions =07 and 11 Despite the scatter of data espe-

cially at higher I values I3 Eq 5 predicts the overalltrend well Since the rising leg and the peak in the pdfs arenow involved in rest of the analysis no uncertainty due tothe differenceslack of data points in this section of the pdfwas introduced Therefore no attempts such as employingarbitrary values for different sections of the pdf have beenmade

FIG 6 Relationship between impulse intensity and particle Reynoldsnumber Re

FIG 7 Plots of the function given by Eq 5 for a range of values

046601-6 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

Furthermore low amplitude impulse events where

I1 in the rising legs of the pdfs are characterized by com-binations of very short Ti and very high u2i values or viceversa Close inspection of the data reveals that the former

may not be detected accurately even at turbulent resolvingsampling frequencies ie peaks occurring within less than5 ms The latter are on the other hand extremely rareGiven that the analysis is sensitive to the features of detected

FIG 8 Comparison of Eq 5 with measured pdfs for E1ndashE8 Solid lines are used to show pdfs obtained from Eq 5

046601-7 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

impulses the following Secs IV B and IV C are presented toexamine the role of critical u2 level ucr

2 on the number andstatistics of the detected impulses

B Critical u2 and number of impulses

Values of u2i are plotted with respect to correspondingTi in Fig 10 for all impulse events and including the eventsassociated with full particle dislodgement the latter areshown in the figure with solid circles for runs E1 and E4Plots given in Fig 10 clearly demonstrate that in agreementwith the findings of Diplas et al25 although FDFD cr

is necessary for the initiation of motion this condition aloneis not sufficient to predict full grain dislodgement Contraryto common force-balance models and other stochasticapproaches only particular combinations of u2i andTi u2ucr

2 with moderate to large durations yielded graindislodgement In fact the most recent incipient motionmodels161920 would significantly overestimate the particleentrainment rate under similar flow and bed conditions Thisis true since all of the detected events with u2ucr

2 for in-stance all 1978 events in run E1 are routinely expected toentrain the particle in such mechanistic models because theseevents would cause very high instantaneous drag forces onthe particle On the contrary our experiment shows that lessthan 7 of these events caused particle entrainment in runE1 This finding is consistent with Balakrishnanrsquos24 observa-tions The total number of impulse events nT for whichu2ucr

2 and the number of impulse events u2ucr2 that

yield particle dislodgement nE both observed over a 15 minsampling period are given for all runs in Table III Note that

nE=nP+nR where nP and nR are the number of pivoting androcking events respectively As shown in Table III the dif-ference between the nT and nE values is dramatic for all runsOverall less than an-order-of magnitude increase in nT cor-responds to two-orders-of-magnitude increase in the nE Thisobservation supports the idea that the force magnitude aloneis not sufficient to describe the particle entrainment

In order to investigate the origins of this remarkablevariability in the number of impulses with minute changes inthe flow conditions the role of critical u2 ucr

2 level on thenumber and statistics of the detected impulses was examinedThis was done artificially not by running additional experi-ments with different grain density or size but rather by ad-justing the level of ucr

2 in the analysis of the current series ofexperiments This approach is comparable to that by varyingthe gross flow parameters slightly as was the procedure in theactual experiments In this analysis data from run E1 wereused Eight arbitrarily chosen critical u2 values slightlyabove and below the original critical level ucr

2 13u2 asillustrated in Fig 11a were selected to identify impulseevents now associated with drag force above the new criticallevel Table IV gives the critical u2 value that is used theresulting number of detected impulses mean and standarddeviations of u2i Ti and impulse respectively and the im-pulse intensity The row with bold font in Table IV indicatesthe results from the conditions where the original critical u2

obtained from Eq 4 was used to detect the impulsesIt is observed that the number of detected impulses is

very sensitive to the chosen critical level of u2 Slightly shift-

TABLE III Number of impulse events and particle movements observed for15 min Note that nE=nP+nR

Run No of impulse events nT

No of particle movements nE

Pivoting nP Rocking nR

E1 1978 104 29

E2 1554 86 17

E3 1262 63 16

E4 1101 31 15

E5 722 20 8

E6 416 8 3

E7 362 4 4

E8 249 1 1

FIG 9 Semilogarithmic plot of measured pdfs from all eight runs Equation5 is also presented with =07 and 11 for comparison

FIG 10 u2i vs Ti plots 1978 data points from run E1 left 1101 datapoints from run E4 right Black circles indicate u2i Ti combinations thatare associated with full particle dislodgement pivoting

FIG 11 a Illustration of the approach used for varying the critical u2 bNumber of detected impulses vs the ratio of critical u2 level used to theoriginal ucr

2

046601-8 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

ing the critical u2 level up or down results in a significantchange in the number of detected impulses The number ofimpulses increases by an order of magnitude when the criti-cal u2 level is shifted downward by only about 40 asshown in Fig 11b The latter also corresponds to a hypo-thetical 25 decrease in the particles density Eq 4 Thisis due to the fact that as the critical u2 is reduced thoseevents that were initially below ucr

2 now become importantie detectable The significant increase in the number ofimpulses in this analysis similar to what was observed in theoriginal experiments suggests that the temporal features re-sulting from the underlying structure of turbulence togetherwith the critical u2 level plays a crucial role in the significantvariation of number of impulse events observed here

The u2i mean as well as Ti std increase linearly with in-creasing critical u2 level while the u2i std and Ti mean

decrease The change in Ti mean with the critical u2 levelis however much more pronounced compared to the changein u2i mean The impulse parameters also do not vary asmuch as the number of impulses Impulse intensity= Ii std Ii mean increases in a linear fashion as the criticallevel moves down mostly due to the decrease in the standarddeviation of impulse As the critical u2 level approaches theaverage of u2 the impulse intensity increases Together withour earlier findings shown in Fig 7 this observation sug-gests that the distribution of impulse becomes more skewedfor relatively lower critical u2 levels In the case of highercritical u2 levels however the smaller peaks are not detectedand the distribution of impulse including only those extremeevents with high impulse magnitudes will approach a nor-mal distribution

It is apparent that the number of impulse events thosewhere u2ucr

2 is sensitive to ucr2 but it is also clear that not

all impulse events yield grain dislodgement In the next sec-tion we investigate the relationship between those events thatare above ucr

2 and critical impulse25

C Critical impulse and number of grain entrainments

Figures 9 and 10 imply that the turbulent channel flowcontains flow structures with a potential to apply a range ofimpulse values varying over an order of magnitude A thresh-old impulse value under the given bed and particle condi-

tions can be defined by identifying an impulse level abovewhich particle entrainment takes place25 This constant im-pulse is approximated by a value of u2iTi in the overlapregion where both grain movement and no movement areobserved on the plot of impulse u2iTi with respect to u2i

Fig 12 This region lies between the ldquoimpulserdquo values of00034 and 00095 m2 s

The computed critical impulse level is obtained from theaverage over this region as shown in Fig 12 data fromE1ndashE8 are included The obtained u2iTi value approxi-mates the threshold of impulse Icr considering that all ofthe runs were performed under a single bed configurationand thus under the same threshold of movement conditionThe uncertainty in detecting the critical impulse level in ouranalysis is attributed to the variations in the instantaneousdrag coefficient28

In order to investigate the role of the critical impulselevel the previous analysis using artificial critical u2 levelsSec IV B was extended That is a corresponding criticalimpulse level for each critical u2 level Table IV was esti-mated as follows The density of the particle correspondingto each of the critical u2 levels given in Table IV was deter-mined using Eq 4 Note that here the size of the particle

TABLE IV Summary of the results from conditions where various u2 critical values were used for run E1

Critical u2

m2 s2

Numberof detected impulses

nTu2i mean

m2 s2u2i std

m2 s2Ti mean

sTi std

sIi mean

m2 sIi std

m2 s

0160 427 0176 0010 0013 0008 00022 00016 072

0144 925 0161 0010 0014 0011 00023 00019 083

0133 1532 0150 0010 0015 0013 00023 00022 096

0130 1748 0147 0010 0016 0014 00024 00023 096

0126 1978 0143 0011 0016 0015 00024 00024 101

0123 2302 0140 0011 0017 0016 00024 00025 104

0119 2615 0136 0011 0017 0016 00024 00026 108

0116 2948 0133 0011 0017 0017 00024 00026 108

0096 5456 0114 0012 0021 0023 00025 00032 128

FIG 12 Iiu2iTi vs u2i plot The region where movement and no move-ment areas overlap is shown with a gray band between the impulse values of00034 and 00095 m2 s Horizontal arrow indicates the critical impulselevel

046601-9 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

Furthermore low amplitude impulse events where

I1 in the rising legs of the pdfs are characterized by com-binations of very short Ti and very high u2i values or viceversa Close inspection of the data reveals that the former

may not be detected accurately even at turbulent resolvingsampling frequencies ie peaks occurring within less than5 ms The latter are on the other hand extremely rareGiven that the analysis is sensitive to the features of detected

FIG 8 Comparison of Eq 5 with measured pdfs for E1ndashE8 Solid lines are used to show pdfs obtained from Eq 5

046601-7 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

impulses the following Secs IV B and IV C are presented toexamine the role of critical u2 level ucr

2 on the number andstatistics of the detected impulses

B Critical u2 and number of impulses

Values of u2i are plotted with respect to correspondingTi in Fig 10 for all impulse events and including the eventsassociated with full particle dislodgement the latter areshown in the figure with solid circles for runs E1 and E4Plots given in Fig 10 clearly demonstrate that in agreementwith the findings of Diplas et al25 although FDFD cr

is necessary for the initiation of motion this condition aloneis not sufficient to predict full grain dislodgement Contraryto common force-balance models and other stochasticapproaches only particular combinations of u2i andTi u2ucr

2 with moderate to large durations yielded graindislodgement In fact the most recent incipient motionmodels161920 would significantly overestimate the particleentrainment rate under similar flow and bed conditions Thisis true since all of the detected events with u2ucr

2 for in-stance all 1978 events in run E1 are routinely expected toentrain the particle in such mechanistic models because theseevents would cause very high instantaneous drag forces onthe particle On the contrary our experiment shows that lessthan 7 of these events caused particle entrainment in runE1 This finding is consistent with Balakrishnanrsquos24 observa-tions The total number of impulse events nT for whichu2ucr

2 and the number of impulse events u2ucr2 that

yield particle dislodgement nE both observed over a 15 minsampling period are given for all runs in Table III Note that

nE=nP+nR where nP and nR are the number of pivoting androcking events respectively As shown in Table III the dif-ference between the nT and nE values is dramatic for all runsOverall less than an-order-of magnitude increase in nT cor-responds to two-orders-of-magnitude increase in the nE Thisobservation supports the idea that the force magnitude aloneis not sufficient to describe the particle entrainment

In order to investigate the origins of this remarkablevariability in the number of impulses with minute changes inthe flow conditions the role of critical u2 ucr

2 level on thenumber and statistics of the detected impulses was examinedThis was done artificially not by running additional experi-ments with different grain density or size but rather by ad-justing the level of ucr

2 in the analysis of the current series ofexperiments This approach is comparable to that by varyingthe gross flow parameters slightly as was the procedure in theactual experiments In this analysis data from run E1 wereused Eight arbitrarily chosen critical u2 values slightlyabove and below the original critical level ucr

2 13u2 asillustrated in Fig 11a were selected to identify impulseevents now associated with drag force above the new criticallevel Table IV gives the critical u2 value that is used theresulting number of detected impulses mean and standarddeviations of u2i Ti and impulse respectively and the im-pulse intensity The row with bold font in Table IV indicatesthe results from the conditions where the original critical u2

obtained from Eq 4 was used to detect the impulsesIt is observed that the number of detected impulses is

very sensitive to the chosen critical level of u2 Slightly shift-

TABLE III Number of impulse events and particle movements observed for15 min Note that nE=nP+nR

Run No of impulse events nT

No of particle movements nE

Pivoting nP Rocking nR

E1 1978 104 29

E2 1554 86 17

E3 1262 63 16

E4 1101 31 15

E5 722 20 8

E6 416 8 3

E7 362 4 4

E8 249 1 1

FIG 9 Semilogarithmic plot of measured pdfs from all eight runs Equation5 is also presented with =07 and 11 for comparison

FIG 10 u2i vs Ti plots 1978 data points from run E1 left 1101 datapoints from run E4 right Black circles indicate u2i Ti combinations thatare associated with full particle dislodgement pivoting

FIG 11 a Illustration of the approach used for varying the critical u2 bNumber of detected impulses vs the ratio of critical u2 level used to theoriginal ucr

2

046601-8 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

ing the critical u2 level up or down results in a significantchange in the number of detected impulses The number ofimpulses increases by an order of magnitude when the criti-cal u2 level is shifted downward by only about 40 asshown in Fig 11b The latter also corresponds to a hypo-thetical 25 decrease in the particles density Eq 4 Thisis due to the fact that as the critical u2 is reduced thoseevents that were initially below ucr

2 now become importantie detectable The significant increase in the number ofimpulses in this analysis similar to what was observed in theoriginal experiments suggests that the temporal features re-sulting from the underlying structure of turbulence togetherwith the critical u2 level plays a crucial role in the significantvariation of number of impulse events observed here

The u2i mean as well as Ti std increase linearly with in-creasing critical u2 level while the u2i std and Ti mean

decrease The change in Ti mean with the critical u2 levelis however much more pronounced compared to the changein u2i mean The impulse parameters also do not vary asmuch as the number of impulses Impulse intensity= Ii std Ii mean increases in a linear fashion as the criticallevel moves down mostly due to the decrease in the standarddeviation of impulse As the critical u2 level approaches theaverage of u2 the impulse intensity increases Together withour earlier findings shown in Fig 7 this observation sug-gests that the distribution of impulse becomes more skewedfor relatively lower critical u2 levels In the case of highercritical u2 levels however the smaller peaks are not detectedand the distribution of impulse including only those extremeevents with high impulse magnitudes will approach a nor-mal distribution

It is apparent that the number of impulse events thosewhere u2ucr

2 is sensitive to ucr2 but it is also clear that not

all impulse events yield grain dislodgement In the next sec-tion we investigate the relationship between those events thatare above ucr

2 and critical impulse25

C Critical impulse and number of grain entrainments

Figures 9 and 10 imply that the turbulent channel flowcontains flow structures with a potential to apply a range ofimpulse values varying over an order of magnitude A thresh-old impulse value under the given bed and particle condi-

tions can be defined by identifying an impulse level abovewhich particle entrainment takes place25 This constant im-pulse is approximated by a value of u2iTi in the overlapregion where both grain movement and no movement areobserved on the plot of impulse u2iTi with respect to u2i

Fig 12 This region lies between the ldquoimpulserdquo values of00034 and 00095 m2 s

The computed critical impulse level is obtained from theaverage over this region as shown in Fig 12 data fromE1ndashE8 are included The obtained u2iTi value approxi-mates the threshold of impulse Icr considering that all ofthe runs were performed under a single bed configurationand thus under the same threshold of movement conditionThe uncertainty in detecting the critical impulse level in ouranalysis is attributed to the variations in the instantaneousdrag coefficient28

In order to investigate the role of the critical impulselevel the previous analysis using artificial critical u2 levelsSec IV B was extended That is a corresponding criticalimpulse level for each critical u2 level Table IV was esti-mated as follows The density of the particle correspondingto each of the critical u2 levels given in Table IV was deter-mined using Eq 4 Note that here the size of the particle

TABLE IV Summary of the results from conditions where various u2 critical values were used for run E1

Critical u2

m2 s2

Numberof detected impulses

nTu2i mean

m2 s2u2i std

m2 s2Ti mean

sTi std

sIi mean

m2 sIi std

m2 s

0160 427 0176 0010 0013 0008 00022 00016 072

0144 925 0161 0010 0014 0011 00023 00019 083

0133 1532 0150 0010 0015 0013 00023 00022 096

0130 1748 0147 0010 0016 0014 00024 00023 096

0126 1978 0143 0011 0016 0015 00024 00024 101

0123 2302 0140 0011 0017 0016 00024 00025 104

0119 2615 0136 0011 0017 0016 00024 00026 108

0116 2948 0133 0011 0017 0017 00024 00026 108

0096 5456 0114 0012 0021 0023 00025 00032 128

FIG 12 Iiu2iTi vs u2i plot The region where movement and no move-ment areas overlap is shown with a gray band between the impulse values of00034 and 00095 m2 s Horizontal arrow indicates the critical impulselevel

046601-9 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

impulses the following Secs IV B and IV C are presented toexamine the role of critical u2 level ucr

2 on the number andstatistics of the detected impulses

B Critical u2 and number of impulses

Values of u2i are plotted with respect to correspondingTi in Fig 10 for all impulse events and including the eventsassociated with full particle dislodgement the latter areshown in the figure with solid circles for runs E1 and E4Plots given in Fig 10 clearly demonstrate that in agreementwith the findings of Diplas et al25 although FDFD cr

is necessary for the initiation of motion this condition aloneis not sufficient to predict full grain dislodgement Contraryto common force-balance models and other stochasticapproaches only particular combinations of u2i andTi u2ucr

2 with moderate to large durations yielded graindislodgement In fact the most recent incipient motionmodels161920 would significantly overestimate the particleentrainment rate under similar flow and bed conditions Thisis true since all of the detected events with u2ucr

2 for in-stance all 1978 events in run E1 are routinely expected toentrain the particle in such mechanistic models because theseevents would cause very high instantaneous drag forces onthe particle On the contrary our experiment shows that lessthan 7 of these events caused particle entrainment in runE1 This finding is consistent with Balakrishnanrsquos24 observa-tions The total number of impulse events nT for whichu2ucr

2 and the number of impulse events u2ucr2 that

yield particle dislodgement nE both observed over a 15 minsampling period are given for all runs in Table III Note that

nE=nP+nR where nP and nR are the number of pivoting androcking events respectively As shown in Table III the dif-ference between the nT and nE values is dramatic for all runsOverall less than an-order-of magnitude increase in nT cor-responds to two-orders-of-magnitude increase in the nE Thisobservation supports the idea that the force magnitude aloneis not sufficient to describe the particle entrainment

In order to investigate the origins of this remarkablevariability in the number of impulses with minute changes inthe flow conditions the role of critical u2 ucr

2 level on thenumber and statistics of the detected impulses was examinedThis was done artificially not by running additional experi-ments with different grain density or size but rather by ad-justing the level of ucr

2 in the analysis of the current series ofexperiments This approach is comparable to that by varyingthe gross flow parameters slightly as was the procedure in theactual experiments In this analysis data from run E1 wereused Eight arbitrarily chosen critical u2 values slightlyabove and below the original critical level ucr

2 13u2 asillustrated in Fig 11a were selected to identify impulseevents now associated with drag force above the new criticallevel Table IV gives the critical u2 value that is used theresulting number of detected impulses mean and standarddeviations of u2i Ti and impulse respectively and the im-pulse intensity The row with bold font in Table IV indicatesthe results from the conditions where the original critical u2

obtained from Eq 4 was used to detect the impulsesIt is observed that the number of detected impulses is

very sensitive to the chosen critical level of u2 Slightly shift-

TABLE III Number of impulse events and particle movements observed for15 min Note that nE=nP+nR

Run No of impulse events nT

No of particle movements nE

Pivoting nP Rocking nR

E1 1978 104 29

E2 1554 86 17

E3 1262 63 16

E4 1101 31 15

E5 722 20 8

E6 416 8 3

E7 362 4 4

E8 249 1 1

FIG 9 Semilogarithmic plot of measured pdfs from all eight runs Equation5 is also presented with =07 and 11 for comparison

FIG 10 u2i vs Ti plots 1978 data points from run E1 left 1101 datapoints from run E4 right Black circles indicate u2i Ti combinations thatare associated with full particle dislodgement pivoting

FIG 11 a Illustration of the approach used for varying the critical u2 bNumber of detected impulses vs the ratio of critical u2 level used to theoriginal ucr

2

046601-8 Celik et al Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

ing the critical u2 level up or down results in a significantchange in the number of detected impulses The number ofimpulses increases by an order of magnitude when the criti-cal u2 level is shifted downward by only about 40 asshown in Fig 11b The latter also corresponds to a hypo-thetical 25 decrease in the particles density Eq 4 Thisis due to the fact that as the critical u2 is reduced thoseevents that were initially below ucr

2 now become importantie detectable The significant increase in the number ofimpulses in this analysis similar to what was observed in theoriginal experiments suggests that the temporal features re-sulting from the underlying structure of turbulence togetherwith the critical u2 level plays a crucial role in the significantvariation of number of impulse events observed here

The u2i mean as well as Ti std increase linearly with in-creasing critical u2 level while the u2i std and Ti mean

decrease The change in Ti mean with the critical u2 levelis however much more pronounced compared to the changein u2i mean The impulse parameters also do not vary asmuch as the number of impulses Impulse intensity= Ii std Ii mean increases in a linear fashion as the criticallevel moves down mostly due to the decrease in the standarddeviation of impulse As the critical u2 level approaches theaverage of u2 the impulse intensity increases Together withour earlier findings shown in Fig 7 this observation sug-gests that the distribution of impulse becomes more skewedfor relatively lower critical u2 levels In the case of highercritical u2 levels however the smaller peaks are not detectedand the distribution of impulse including only those extremeevents with high impulse magnitudes will approach a nor-mal distribution

It is apparent that the number of impulse events thosewhere u2ucr

2 is sensitive to ucr2 but it is also clear that not

all impulse events yield grain dislodgement In the next sec-tion we investigate the relationship between those events thatare above ucr

2 and critical impulse25

C Critical impulse and number of grain entrainments

Figures 9 and 10 imply that the turbulent channel flowcontains flow structures with a potential to apply a range ofimpulse values varying over an order of magnitude A thresh-old impulse value under the given bed and particle condi-

tions can be defined by identifying an impulse level abovewhich particle entrainment takes place25 This constant im-pulse is approximated by a value of u2iTi in the overlapregion where both grain movement and no movement areobserved on the plot of impulse u2iTi with respect to u2i

Fig 12 This region lies between the ldquoimpulserdquo values of00034 and 00095 m2 s

The computed critical impulse level is obtained from theaverage over this region as shown in Fig 12 data fromE1ndashE8 are included The obtained u2iTi value approxi-mates the threshold of impulse Icr considering that all ofthe runs were performed under a single bed configurationand thus under the same threshold of movement conditionThe uncertainty in detecting the critical impulse level in ouranalysis is attributed to the variations in the instantaneousdrag coefficient28

In order to investigate the role of the critical impulselevel the previous analysis using artificial critical u2 levelsSec IV B was extended That is a corresponding criticalimpulse level for each critical u2 level Table IV was esti-mated as follows The density of the particle correspondingto each of the critical u2 levels given in Table IV was deter-mined using Eq 4 Note that here the size of the particle

TABLE IV Summary of the results from conditions where various u2 critical values were used for run E1

Critical u2

m2 s2

Numberof detected impulses

nTu2i mean

m2 s2u2i std

m2 s2Ti mean

sTi std

sIi mean

m2 sIi std

m2 s

0160 427 0176 0010 0013 0008 00022 00016 072

0144 925 0161 0010 0014 0011 00023 00019 083

0133 1532 0150 0010 0015 0013 00023 00022 096

0130 1748 0147 0010 0016 0014 00024 00023 096

0126 1978 0143 0011 0016 0015 00024 00024 101

0123 2302 0140 0011 0017 0016 00024 00025 104

0119 2615 0136 0011 0017 0016 00024 00026 108

0116 2948 0133 0011 0017 0017 00024 00026 108

0096 5456 0114 0012 0021 0023 00025 00032 128

FIG 12 Iiu2iTi vs u2i plot The region where movement and no move-ment areas overlap is shown with a gray band between the impulse values of00034 and 00095 m2 s Horizontal arrow indicates the critical impulselevel

046601-9 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

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12817312576 On Wed 20 Nov 2013 200812

remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

ing the critical u2 level up or down results in a significantchange in the number of detected impulses The number ofimpulses increases by an order of magnitude when the criti-cal u2 level is shifted downward by only about 40 asshown in Fig 11b The latter also corresponds to a hypo-thetical 25 decrease in the particles density Eq 4 Thisis due to the fact that as the critical u2 is reduced thoseevents that were initially below ucr

2 now become importantie detectable The significant increase in the number ofimpulses in this analysis similar to what was observed in theoriginal experiments suggests that the temporal features re-sulting from the underlying structure of turbulence togetherwith the critical u2 level plays a crucial role in the significantvariation of number of impulse events observed here

The u2i mean as well as Ti std increase linearly with in-creasing critical u2 level while the u2i std and Ti mean

decrease The change in Ti mean with the critical u2 levelis however much more pronounced compared to the changein u2i mean The impulse parameters also do not vary asmuch as the number of impulses Impulse intensity= Ii std Ii mean increases in a linear fashion as the criticallevel moves down mostly due to the decrease in the standarddeviation of impulse As the critical u2 level approaches theaverage of u2 the impulse intensity increases Together withour earlier findings shown in Fig 7 this observation sug-gests that the distribution of impulse becomes more skewedfor relatively lower critical u2 levels In the case of highercritical u2 levels however the smaller peaks are not detectedand the distribution of impulse including only those extremeevents with high impulse magnitudes will approach a nor-mal distribution

It is apparent that the number of impulse events thosewhere u2ucr

2 is sensitive to ucr2 but it is also clear that not

all impulse events yield grain dislodgement In the next sec-tion we investigate the relationship between those events thatare above ucr

2 and critical impulse25

C Critical impulse and number of grain entrainments

Figures 9 and 10 imply that the turbulent channel flowcontains flow structures with a potential to apply a range ofimpulse values varying over an order of magnitude A thresh-old impulse value under the given bed and particle condi-

tions can be defined by identifying an impulse level abovewhich particle entrainment takes place25 This constant im-pulse is approximated by a value of u2iTi in the overlapregion where both grain movement and no movement areobserved on the plot of impulse u2iTi with respect to u2i

Fig 12 This region lies between the ldquoimpulserdquo values of00034 and 00095 m2 s

The computed critical impulse level is obtained from theaverage over this region as shown in Fig 12 data fromE1ndashE8 are included The obtained u2iTi value approxi-mates the threshold of impulse Icr considering that all ofthe runs were performed under a single bed configurationand thus under the same threshold of movement conditionThe uncertainty in detecting the critical impulse level in ouranalysis is attributed to the variations in the instantaneousdrag coefficient28

In order to investigate the role of the critical impulselevel the previous analysis using artificial critical u2 levelsSec IV B was extended That is a corresponding criticalimpulse level for each critical u2 level Table IV was esti-mated as follows The density of the particle correspondingto each of the critical u2 levels given in Table IV was deter-mined using Eq 4 Note that here the size of the particle

TABLE IV Summary of the results from conditions where various u2 critical values were used for run E1

Critical u2

m2 s2

Numberof detected impulses

nTu2i mean

m2 s2u2i std

m2 s2Ti mean

sTi std

sIi mean

m2 sIi std

m2 s

0160 427 0176 0010 0013 0008 00022 00016 072

0144 925 0161 0010 0014 0011 00023 00019 083

0133 1532 0150 0010 0015 0013 00023 00022 096

0130 1748 0147 0010 0016 0014 00024 00023 096

0126 1978 0143 0011 0016 0015 00024 00024 101

0123 2302 0140 0011 0017 0016 00024 00025 104

0119 2615 0136 0011 0017 0016 00024 00026 108

0116 2948 0133 0011 0017 0017 00024 00026 108

0096 5456 0114 0012 0021 0023 00025 00032 128

FIG 12 Iiu2iTi vs u2i plot The region where movement and no move-ment areas overlap is shown with a gray band between the impulse values of00034 and 00095 m2 s Horizontal arrow indicates the critical impulselevel

046601-9 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

remains the same A theoretical initial velocity Vinit of theparticle with the calculated density as a result of impulse inthe streamwise direction was estimated from the potentialenergy that the particle gains when it is elevated by anamount of 13z 04 mm normal to the flow direction as itreaches the retaining pin that is Vinit=2g13z For simplicitythe energy losses due to friction were ignored for this esti-mate Using the change in the linear momentum of the par-ticle the impulse required the critical level was determinedfor each calculated particle density using Vinit and Eqs 13 and 4 The assumption here is that the impulse deter-mined for each particle density represents a minimum criti-cal impulse level that causes a detectible particle movementThe critical impulse level Icr calculated in this fashion forthe original particle density was 00033 m2 s We note thatthis value is near the lower impulse threshold detected usingthe actual observations of particle movement Fig 12

Using the estimated artificial critical impulse levels thedata for each case in Table IV were reanalyzed to identify foreach critical u2 value the corresponding number of impulseevents above the approximate critical impulse level Theanalysis was performed using data from runs E1 and E5Additionally the number of impulse events above the criticalimpulse level of 00033 m2 s was also determined for alleight runs The results are presented in Fig 13 where thenumber of impulse events above critical Ii Icr per minuteare plotted with respect to the total number of impulse eventswith u2ucr

2 per minute for all critical u2 values tested usingthe data from runs E1 and E5 In addition the number ofactual particle movements nE observed in each of the eightruns was also plotted in the figure against the original nT

Table IV There is clearly a strong relationship between thenumber of events above critical impulse and the total numberof impulse events nT regardless of the approach used todetect them provided that the theoretical Icr is used The

functional relationship between the two variables and thecoefficient of determination are given in Fig 13 as an insetThis conformity indicates that the significant increase in bothnumber of impulse events and the number of events abovecritical impulse with minute changes in either flow param-eters or grain density is not a coincidence but a phenomenoninherent to turbulent flow-particle interactions However theactual number of ball movements is found to be lower thanthe number of events above a critical impulse with a poten-tial to move the ball black circles and plus signs in Fig 13are compared One obvious reason for this difference is thefact that the calculated critical impulse used here is near thelower observed threshold for particle movement as men-tioned earlier Fig 12 The next section examines the prob-ability of particle movement taking into consideration thecritical impulse values obtained here

D Probability of particle entrainment

Probability of particle entrainment has received muchattention in the literature9171836 It is commonly argued thatthe probability of exceeding of a given threshold level isequal to the probability of particle entrainment18 Here wehypothesize that the probability p of occurrence of flowevents that exceed the impulse threshold Fig 12 is equal tothe probability of particle entrainment pE this concept isillustrated in Fig 14 The former can be evaluated directlyusing Eq 5 with the critical impulse value in the form of

Icr= Icr Ii mean To examine the validity of this hypothesis thedata for each of the series E1ndashE8 were analyzed using threedifferent critical impulse levels obtained from Fig 12 theupper lower and average values associated with the overlapregion in the figure

We approximate the probability pE by the relative fre-quency of impulse events that yield particle dislodgementie

pE nE

nT 7

p and pE values obtained in this fashion are compared in Fig15 A good agreement is achieved between the two probabil-ity values especially for higher entrainment rates when thecritical impulse is taken as 00063 m2 s Fig 12 For com-parison p values obtained using the upper and lower limits

FIG 13 The plot of number of impulse events above critical impulse permin vs total number of impulse events above critical u2 per min Data pointswith black circles are from all eight runs where a constant Icr was usedWhite and gray circles indicate results from runs E1 and E5 respectivelywhere various Icr values were used Data with the plus sign indicate theactual particle movements nE vs nT observed in each run

FIG 14 Illustration of the probability analysis The probability that a flowevent will generate a level of impulse that exceeds a specified critical levelIcr is indicated by the shaded area and is assumed to be equal to the prob-ability of particle entrainment pE

046601-10 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

of the impulse threshold Fig 12 are also presented withrespect to pE for all runs to reflect the effect on p of theuncertainty in the critical impulse As shown in Fig 15 thevariation of p with Icr is significant Nevertheless the goodagreement between p and pE using Icr=00063 m2 s con-firms that the pdf given by Eq 5 describes the data welland suggests that despite the significant variability inparticle entrainment frequency consistent probability valuesfor particle entrainment are obtained near threshold condi-tions when the frequency of impulse ie nT is taken intoaccount

V IMPLICATIONS OF THE IMPULSE CONCEPTFOR LOW MOBILITY CONDITIONS

Bedload measurements constitute an important compo-nent in the development of reliable formulae relating theamount of transported bed material to the flow parametersPredictions of the limiting case of nearly zero bedload trans-port the threshold of motion or critical condition are evenmore challenging The complexities at very low mobilityconditions are associated with flow turbulence and bed mi-crotopography parameters the former being most prominentIn addition the paucity of dependable data obtained underlow shear stress and low bed material movement conditionshinders understanding of sediment transport processes at in-cipient conditions Given these facts Paintalrsquos10 well knownobservation still remains to be explained

We have shown in this study that for near threshold con-ditions the particle entrainment rate increases significantlydue to impulse events with only minute increases in the flowstrength interpreted in terms of boundary shear stress orsimilar parameters The Shields stress values for instance inour tests varied between 0008 and 0011 a change accom-panied by nearly two orders of magnitude increase in thenumber of full particle dislodgements This behavior exhibitsstrong qualitative similarities with the observations ofPaintal10 and Helland-Hansen et al12 Figure 16 shows par-ticle movement at low Shields stresses using as a surrogatemeasure the number of impulses per minute above a criticalimpulse obtained from our experiments see Sec IV CDimensionless bed load parameter q reported by Paintal10

and Helland-Hansen et al12 are included in the same figurefor comparison purposes Despite the variability in flow andbed material parameters employed by Paintal10 and Helland-Hansen et al12 Fig 16 shows a nearly two-orders-of-magnitude increase in the number of impulses with a poten-tial to move the grains per minute and in the dimensionlessbed load values over a low Shields stress range The ob-served similarity in the trends see the exponents shown inthe inset of Fig 16 helps explain the well known yet poorlyunderstood phenomenon observed first by Paintal10 It alsocontributes to the validation of the impulse concept pre-sented here for sediment entrainment at low shear conditions

It is well understood now that the mean bed shear stressother than being a good descriptor of the mean forces cannotexplain the local processes and resulting sediment movementat low shear stress conditions The magnitude of the localhydrodynamic forces and their distributions form the basis ofmany stochastic incipient motion models to overcome thecomplexities of fluid-mobile boundary interactions ieu2ucr

2 Our results on the other hand clearly show thatimpulse has far greater dynamical significance in terms ofinitial sediment movement

This is clearly seen that the impulse events above criticalIi Icr rather than the peak forces with u2ucr

2 are re-sponsible for particle dislodgement Fig 12 Within thegiven shear stress range a variability that is in close quali-tative agreement with the findings of Paintal10 and Helland-Hansen et al12 is observed in the number of impulse events

FIG 15 Probability of particle entrainment vs probability of exceedance ofcritical impulse

FIG 16 Dimensionless bed load parameter vs shields stress left y-axisfrom Refs 10 and 12 Note that data only in the range between 0005 and0016 were used The number of impulse events above critical impulse permin right y-axis vs shields stress is also plotted

046601-11 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

above critical Ii Icr and not the number of peak forces forwhich u2ucr

2 see Figs 13 and 16 We believe thereforethat the impulse concept presented here represents a signifi-cant step toward establishing incipient motion models withbetter predictive abilities

Because the impulse is the germane criterion for describ-ing the dislodgement of fully exposed as well as fully hiddenparticle configurations25 it is expected to remain valid forthe more general cases of variable local bed topography Fur-thermore determination of a critical impulse level dependson local bed conditions In natural settings grain size shapeexposure and packing density vary greatly in a probabilisticmanner8 making also the resistance critical impulse ofgrains to vary spatially Therefore the threshold of real sedi-ment movement in both space and time requires consider-ation of both the impulse potential of the turbulent flow andthe distribution of critical impulse Nevertheless future workwill be necessary to demonstrate this result conclusively

VI CONCLUSIONS

We showed for a range of flow conditions and particleentrainment rates that the impulse is a better suited parameterto describe the role of turbulence fluctuations on the particlemovement for the given bed configuration and under incipi-ent conditions Using a particle tracking technique that offershigh temporal and spatial resolution in detecting the motionof a single test particle we found that the particle entrain-ment rate is extraordinarily sensitive to minute changes inthe gross flow parameters This finding is in agreement withwell known observations of others available in the literature

A log-normal distribution was found to describe the im-pulse data well Figs 8 and 9 Equation 5 for the log-normal distribution is a function of impulse intensity alone which can be determined experimentally or using Eq6 for the given Re range Equation 6 is therefore usefulfor further studies under similar bed and low shear stressconfigurations

We demonstrated that the impulse imparted by the tur-bulent stream on the mobile grain has to be sufficiently largein order to cause dislodgement Combinations of force dura-tion and magnitude that entrained the particle were observedto be in general above a critical impulse value Fig 12 Thedurations of applied force varied an order of magnitude forcorresponding relatively small range of the force u2 valuesThe resulting strong dependency of impulse values on theduration of applied forcing suggests that the magnitude ofdrag force is necessary but not sufficient to characterize thethreshold of motion

We also investigated the role of ucr2 representing the

minimum drag force necessary to entrain the particle on thenumber of detected impulses Additionally the role of criticalimpulse level on the number of impulse events with a poten-tial to move the grain was examined The results showed thatminute changes in the ucr

2 and the critical impulse leads tosignificant variations in the number of total impulse eventsand events with a potential to dislodge the grain respec-tively This analysis substantiates and helps explain the ob-

served nearly two-orders-of-magnitude increase in the par-ticle movement rate with 35 increase in the bed shearstress

The derived pdf Eq 5 of dimensionless impulse wasused together with an approximated critical impulse level tofind the probability of particle entrainment Probabilities ob-tained in this fashion were checked against experimentallydetermined particle entrainment probabilities and a goodagreement was achieved when an approximated critical im-pulse level was used

Finally the significant increase in the number of impulseevents above critical impulse with small increases in the bedshear stress was found to be qualitatively similar to the sig-nificant increase in the bed load transport observed by othersunder comparable low mobility conditions

Our results using the impulse concept link the character-istics of turbulent flow to particle entrainment and despitethe simplified bed geometry employed have far reachingconsequences in describing the inadequacies of existing in-cipient motion models as well as improving our ability tomodel a wide range of river mechanics and other geomor-phology related phenomena

ACKNOWLEDGMENTS

The support of the National Science Foundation GrantNos EAR-0439663 and EAR-0738759 and Army ResearchOffice for this study is gratefully acknowledged

APPENDIX DERIVATION OF THE PDF FOR IMPULSE

The log-normal distribution for a random variable x isdefined by

fx =1

2xexpminus

1

213 lnx minus

2 A1

where and are the mean and standard deviation of thevariablersquos natural logarithm lnx xmean and xstd representingthe mean and standard deviation of the variable x and

cv =xstd

xmean A2

where cv is the coefficient of variation of x and can bedefined as

= ln xmean

1 + 13 xstd

xmean2 = ln13 xmean

1 + cv2 A3

and

=ln1 + 13 xstd

xmean2 = ln1 + cv

2 A4

respectively37 Equation A1 using Eqs A2ndashA4 can berewritten as

046601-12 Celik et al Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812

fx =1

2 ln1 + cv2x

expminus13ln13 x

xmean + ln1 + cv

22

2 ln1 + cv2

A5

The pdf given by Eq A5 can be transformed37 for a newdimensionless random variable x which is defined as

x =x

xmean A6

The pdf transformation yields

fx =1

2 ln1 + cv2x

expminusln x + ln1 + cv

22

2 ln1 + cv2

A7

Equation A7 is dependent only on cv Here x is the impulseI= u2T and cv is the impulse intensity given by

cv = =Istd

Imean A8

If the impulse is nondimensionalized as follows

x = I =I

Imean A9

Eqs A7ndashA9 then yield

fI =1

2 ln1 + 2Iexpminus

ln I + ln1 + 22

2 ln1 + 2

A10

Note that =I the latter being the intensity of the dimen-

sionless impulse I Equation A10 shown above is the pdf of

dimensionless impulse I as a function of

1A Shields 1936 ldquoAnwendung der aehnlichkeitsmechanik und der turbu-lenzforschung auf die eschiebebewegungrdquo Mitteilungen der Preussiischenversuchsanstalt fur wasserbau und schiffhau Heft 26 Berlin in GermanEnglish translation by W P Ott and J C van Uchelen Publication No167 Hydrodynamics Lab California Inst of Technology Pasadena CA1950

2J M Buffington and D R Montgomery ldquoA systematic analysis of eightdecades of incipient motion studies with special reference to gravel-bedded riversrdquo Water Resour Res 33 1993 1997

3A A Kalinske ldquoMovement of sediment as bed load in riversrdquo TransAm Geophys Union 28 615 1947

4A J Sutherland ldquoProposed mechanism for sediment entrainment by tur-bulent flowsrdquo J Geophys Res 72 6183 doi101029JZ072i024p061831967

5A D Heathershaw and P D Thorne ldquoSea-bed noises reveal role of tur-bulent bursting phenomenon in sediment transport by tidal currentsrdquo Na-ture London 316 339 1985

6J Nelson R L Shreve S R McLean and T G Drake ldquoRole of near-bedturbulence structure in bed-load transport and bed-form mechanicsrdquo WaterResour Res 31 2071 doi10102995WR00976 1995

7B M Sumer L H C Chua N S Cheng and J Fredsoe ldquoInfluence ofturbulence on bed load sediment transportrdquo J Hydrol Eng 129 5852003

8J W Kirchner W E Dietrich F Iseya and H Ikeda ldquoThe variability ofcritical shear stress friction angle and grain protrusion in water-workedsedimentsrdquo Sedimentology 37 647 1990

9A J Grass in The Influence of Boundary Layer Turbulence on the Me-chanics of Sediment Transport edited by B M Sumer and A MullerBalkema Rotterdam 1983 pp 3ndash17

10A S Paintal ldquoConcept of critical shear stress in loose boundary openchannelsrdquo J Hydr Div 9 91 1971

11J D Fenton and J E Abbot ldquoInitial movement of grains on a stream bedThe effect of relative protrusionrdquo Proc R Soc London 352A 5231977

12E Helland-Hansen P C Klingeman and R T Milhous ldquoSediment trans-port at low Shields-paramter valuesrdquo J Hydr Div 100 261 1974

13B Hofland ldquoRock amp Roll Turbulence-induced damage to granular bedprotectionsrdquo PhD thesis Department of Civil Engineering and Geo-sciences Delft University of Technology The Netherlands 2005

14H A Einstein and E A El-Samni ldquoHydrodynamic forces on a roughwallrdquo Rev Mod Phys 21 520 1949

15A J Grass ldquoInitial instability of fine bed sandrdquo J Hydr Div 96 6191970

16I McEwan and J Heald ldquoDiscrete particle modeling of entrainment fromflat uniformly sized sediment bedsrdquo J Hydrol Eng 127 5882001

17A B Shvidchenko and G Pender ldquoFlume study of the effect of relativedepth on the incipient motion of coarse uniform sedimentsrdquo Water ResourRes 36 619 doi1010291999WR900312 2000

18A N Papanicolaou P Diplas N Evaggelopoulos and S FotopoulosldquoStochastic incipient motion criterion for spheres under various bed pack-ing conditionsrdquo J Hydrol Eng 128 369 2002

19N S Cheng and Y M Chiew ldquoPick-up probability for sediment entrain-mentrdquo J Hydrol Eng 124 232 1998

20M W Schmeeckle and J M Nelson ldquoDirect numerical simulation ofbedload transport using a local dynamic boundary conditionrdquo Sedimen-tology 50 279 2003

21F C Wu and Y J Chou ldquoRolling and lifting probabilities for sedimententrainmentrdquo J Hydrol Eng 129 110 2003

22S Vollmer and M G Kleinhans ldquoPredicting incipient motion includingthe effect of turbulent pressure fluctuations in the bedrdquo Water Resour Res43 W05410 doi1010292006WR004919 2007

23M S Yalin River Mechanics Pergamon Oxford 199224M Balakrishnan ldquoThe role of turbulence on the entrainment of a single

sphere and the effects of roughness on fluid-solid interactionrdquo PhD dis-sertation Virginia Polytechnic Institute and State University BlacksburgVA 1997

25P Diplas C L Dancey A O Celik M Valyrakis K Greer and T AkarldquoThe role of impulse on the initiation of particle movement under turbu-lent flow conditionsrdquo Science 322 717 2008

26J Gessler in River Mechanics edited by H W Shen Water ResourcesForth Collins 1971 Chap 7

27C R White ldquoThe equilibrium of grains on the bed of a streamrdquo Proc RSoc London Ser A 174 322 1940

28M W Schmeeckle J M Nelson and R L Shreve ldquoForces on stationaryparticles in near-bed turbulent flowsrdquo J Geophys Res 112 F02003doi1010292006JF000536 2007

29J S Bridge and S J Bennett ldquoA model for the entrainment and transportof sediment grains of mixed sizes shapes and densitiesrdquo Water ResourRes 28 337 doi10102991WR02570 1992

30B Hofland and J Battjes ldquoProbability density function of instantaneousdrag forces and shear stresses on a bedrdquo J Hydrol Eng 132 11692006

31T Song W H Graf and U Lemmin ldquoUniform flow in open channelswith movable gravel bedrdquo J Hydraul Res 32 861 1994

32N S Cheng and A W K Law ldquoFluctuations of turbulent bed shearstressrdquo J Eng Mech 129 126 2003

33B S Wu A Molinas and P Y Julien ldquoBed-material load computationsfor nonuniform sedimentsrdquo J Hydraul Eng 130 1002 2004

34H Mouri A Hori and M Takaoka ldquoLarge-scale lognormal fluctuationsin turbulence velocity fieldsrdquo Phys Fluids 21 065107 2009

35E Limpert W A Stahel and M Abbt ldquoLog-normal distributions acrossthe sciences Keys and cluesrdquo Bioscience 51 341 2001

36C L Dancey P Diplas A Papanicolaou and M Bala ldquoProbability ofindividual grain movement and threshold conditionrdquo J Hydraul Eng128 1069 2002

37E J Gumbel Statistics of Extremes Columbia University Press NewYork 1958

046601-13 Impulse and particle dislodgement under turbulent flow conditions Phys Fluids 22 046601 2010

This article is copyrighted as indicated in the abstract Reuse of AIP content is subject to the terms at httpscitationaiporgtermsconditions Downloaded to IP

12817312576 On Wed 20 Nov 2013 200812