Advances in Water Resources 48 (2012) 47–54

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Advances in Water Resources

journal homepage: www.elsevier .com/ locate/advwatres

Application of a simple power law for transport ratio with bimodal distributionsof spherical grains under oscillatory forcing

Kevin Holway a,1, Christopher S. Thaxton a,⇑, Joseph Calantoni b,2

a Department of Physics and Astronomy, Appalachian State University, 525 Rivers Street, Boone, NC 28608, USAb Marine Geosciences Division, Naval Research Laboratory, Stennis Space Center, MS, USA

a r t i c l e i n f o

Article history:Available online 4 May 2012

Keywords:MorphodynamicsBimodalSediment transportCoarse grainsModeling

0309-1708/$ - see front matter � 2012 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.advwatres.2012.04.003

⇑ Corresponding author. Tel.: +1 (0) 828 262 6836;E-mail addresses: kh73432@appstate.edu (K. Holw

(C.S. Thaxton), joe.calantoni@nrlssc.navy.mil (J. Calan1 Tel.: +1 (0) 828 262 3090; fax: +1 828 262 2049.2 Tel.: +1 (0) 228 688 4435; fax: +1 228 688 4853.

a b s t r a c t

Morphodynamic models of coastal evolution require relatively simple parameterizations of sedimenttransport for application over larger scales. Calantoni and Thaxton (2008) [6] presented a transportparameterization for bimodal distributions of coarse quartz grains derived from detailed boundary layersimulations for sheet flow and near sheet flow conditions. The simulation results, valid over a range ofwave forcing conditions and large- to small-grain diameter ratios, were successfully parameterized witha simple power law that allows for the prediction of the transport rates of each size fraction. Here, wehave applied the simple power law to a two-dimensional cellular automaton to simulate sheet flowtransport. Model results are validated with experiments performed in the small oscillating flow tunnel(S-OFT) at the Naval Research Laboratory at Stennis Space Center, MS, in which sheet flow transportwas generated with a bed composed of a bimodal distribution of non-cohesive grains. The work pre-sented suggests that, under the conditions specified, algorithms that incorporate the power law may cor-rectly reproduce laboratory bed surface measurements of bimodal sheet flow transport while inherentlyincorporating vertical mixing by size.

� 2012 Elsevier Ltd. All rights reserved.

1. Introduction

The modeling of morphodynamic changes in the nearshore hashistorically focused on predicting changes in bed elevation drivenby local spatial gradients in sediment transport [1,2]. However,even in the absence of bulk transport gradients, sediment grainsorting by size is ubiquitous. Grain size-dependent transport andthe surficial distribution of sediments by size may play a criticalrole in determining bed friction for circulation models, in additionto influencing bedform and channel development, evolution, andmigration. Recent attempts to incorporate fractional transportschemes into coupled wave-current-sediment models for the bot-tom boundary layer divide the bed into a user-defined number ofsize classes where conventional bedload transport formulae arecomputed using a median grain size (D50) for each class [3]. Othersemploy stochastic models for rivers with dunes [4,5]. Althoughsuch attempts have demonstrated skill, they do not incorporatethe effects of dynamic granular sorting during bedload transportresulting in vertical gradation of sediment by size within in thebed.

ll rights reserved.

fax: +1 828 262 2049.ay), thaxtoncs@appstate.edu

toni).

A simple, predictive power-law,

qL

qS¼ K

ML

MS

� �a

; ð1Þ

was developed [6] using results obtained from a discrete particlemodel (DPM) [7] for the bed-parallel transport rates of grains bysize in a bimodal distribution of coarse sediments under sheet flow(or near sheet flow) conditions given forcing from idealized mono-chromatic waves characteristic of the surf zone. Here, ML and MS arethe local mass abundances (kg) of large and small grains and qL andqS are the horizontal transport rates (kg m�1 s�1) of the large andsmall grains, respectively. The suggested values from simulationsfor the coefficients are K = 5.34 and a = 1.47 [4]. The power law(1) results from the fact that, through direct simulation of the gran-ular dynamics, the DPM reproduces, at high-resolution, rapid verti-cal kinematic sorting of grains by size that has been observed in thelaboratory [8–10]. The power-law was used to accurately hindcastthe relative transport rates from previous laboratory studies[9,10] with bimodal size ratios as high as 4.6:1 and having fluidforcing conditions similar to but not identical to the simulations.

Here, we have incorporated the power law (1) into a two-dimensional cellular automaton, BEDCELL, to simulate sheet flowtransport. Unlike traditional solvers of advection-diffusion partialdifferential equations that transport residual mass based on differ-ential relationships, or fractional transport approaches that do the

Fig. 1. Shown is a side view, schematic of the test section in the S-OFT drawn toscale. The total length of the test section is 200 cm. The sediment well is 30 cmdeep. The lower 20 cm of the sediment well was filled with sand. The upper 10 cmwas filled with the nylon material where the initial locations of the SMALL (red) andLARGE (white) grains are indicated. The ramps denoted at the ends of the test

48 K. Holway et al. / Advances in Water Resources 48 (2012) 47–54

same for each fraction independently, BEDCELL transports all massdictated by the net and fractional transport rates within the activelayer while incorporating the dynamic vertical sorting implicit in(1). In addition, BEDCELL tracks vertical mass distributions per sizefraction, producing sedimentation layers and/or eroding existinglayers depending on the temporal characteristics of the fluid forc-ing. BEDCELL is designed to function for small temporal and spatialscales; however, it is numerically efficient and may readily beadapted to support larger-scale modeling than is presented here.

Herein, we validate BEDCELL for sheet flow conditions in whicha bed comprised of a bimodal size distribution of noncohesivegrains is driven by a monochromatic oscillatory forcing producinga uniform net transport rate. Initial and boundary conditions, aswell as model control parameters, are set to match a set of exper-iments performed in the small oscillatory flow tunnel (S-OFT) atthe Naval Research Laboratory at the Stennis Space Center, MS.The time-dependent surface sediment distribution measured inthe S-OFT experiments compared well with that computed by BED-CELL. Our results suggest that, under the conditions specified, algo-rithms that incorporate the power law may correctly reproducelaboratory bed surface measurements of bimodal sheet flow trans-port while inherently incorporating vertical mixing by size.

section were constructed of cobble filled with sand to minimize fluid turbulence.The dashed lines from the camera location above the test section denote the extentof the field of view on the bed. The light gray window in the right half of the testsection denotes the field of view of the camera when looking through the sidewall.Shown in the inset (a) is a photo of the camera array mounted above the S-OFT testsection with the camera depicted in the schematic labeled. Shown in inset (b) is aphoto of the initial bed surface taken through the acrylic lid of the S-OFT.

2. Methods

2.1. Experimental facility

Experiments were performed in the small oscillatory flow tun-nel (S-OFT) at the Naval Research Laboratory at the Stennis SpaceCenter, MS, in which sheet flow transport was generated with abed composed of a bimodal size distribution of non-cohesivegrains. The sediment grains were nylon (material density1170 kg m�3) cylinders having a height equal to their diameter,D = 1.0 mm (white) and D = 0.5 mm (red). Sediments will be re-ferred to as large (1.0 mm) and small (0.5 mm) for the remainderof the manuscript. The S-OFT was filled with fresh water and thetemperature was nominally 25.6 �C during the experiments.

The S-OFT has a 200 cm-long test section with a 25 cm-widechannel. The upper portion of the sediment well was filled to adepth of 10 cm with the nylon grains. The entire bed was filledwith small grains except for a 25 cm-wide rectangular ‘‘plug’’ oflarge grains located 100 cm from the ‘‘downstream’’ end of the S-OFT and 75 cm from the ‘‘upstream’’ end of the S-OFT (Fig. 1).The bed was smoothed to achieve a flat surface prior to theexperiments.

Flow in the S-OFT is driven by a piston connected to a flywheelwith a slider and connecting rod. The stroke length of the pistonwas chosen to produce an excursion amplitude in the test sectionof 8.74 cm. The variable frequency inverter was set to produce aperiod of 2.4 s. Under these forcing conditions, the maximum freestream velocity u0 reached 23.2 cm/s, and the inherent asymmetryin acceleration produced a net sediment transport toward the pis-ton in the ‘‘downstream’’ direction. The piston was turned on andrun for about 25 periods (60 s) before being shutdown. The pistonreaches the set frequency within two complete revolutions afterstart up.

A Canon™ EOS 7D digital SLR camera outfitted with a fisheyelens was mounted 100 cm directly above the center point of thebed surface, with the tangential plane of the lens parallel to thebed surface. High definition video of the sediment bed was ac-quired in 1080 p (1920 � 1080 pixels) at 30 fps. The camera cap-tured 72 frames/period for a total of 1800 frames over 25 periods(60 s). The camera field of view at the bed was roughly 120 cm cen-tered along the 200 cm test section (16 pixels/cm). The camera wascalibrated with a standard checkerboard calibration sheet laid atop

the bed with the S-OFT filled and sealed. Images from the fisheyelens were remapped to a rectilinear projection prior to processing.

In a second experiment, the same Canon™ EOS 7D digital SLRcamera with a standard zoom lens was used to image throughthe acrylic sidewall of the S-OFT such that the tangential plane ofthe camera lens was parallel to the sidewall. An identical sedimentbed was constructed and identical piston forcing was used whilehigh-definition video was acquired in 720p at 60 fps. The field ofview (indicated in Fig. 1) had spatial dimensions of 15.4 cm wideby 10.1 cm tall (83 pixels/cm).

2.2. Numerical methods

BEDCELL is a 2-D automaton that resolves longitudinal and ver-tical sorting of grains by size during bedload transport. Input toBEDCELL includes forcing and bed information. Forcing input cancurrently take two forms: (1) a time series of net transport rates(kg m�1 s�1) along the longitudinal domain and (2) a time seriesof wave statistics from which net transport rates along the longitu-dinal domain can be derived from commonly-used bedload trans-port formulae (e.g., Meyer-Peter, Bagnold, van Rijn, Yalin, Madsenand others).

The user initializes the bed by defining an overall bed mass, lon-gitudinal length, and depth. The bed is then segmented it into anynumber of columns of fixed longitudinal length Dx. All columnscan be segmented into multiple vertical layers of fixed or varyingdepth, resulting in a 2-D matrix of cells. Each layer within individ-ual columns can be initialized to any grain size distribution quan-tified through a mean concentration and concentration profile asdescribed below.

Instead of solving advection-diffusion partial differential equa-tions (PDEs), BEDCELL models transport through advection and dif-fusion cellular algorithms that require less computationalresources than full PDE solvers for very large spatial and temporalscales. As such, BEDCELL propagates mass with a computationalvelocity of one longitudinal column per iteration – the net

K. Holway et al. / Advances in Water Resources 48 (2012) 47–54 49

transport rates determine the mean sediment velocity whichmatches the computational velocity. As a result, the ratio Dx/Dtis fixed but the user can specify either the spatial or temporal res-olution as desired.

The vertical cell heights may change over time as the bedevolves. During bed evolution, cell mass M (kg) is tracked, as wellas the relative abundance of each grain size—a given cell contains amass MS [kg] of small grains and a mass ML (kg) of large grains. Therelative abundance is represented as a mean cell concentration Cdefined as

C ¼ MS

M; ð2Þ

In general, the concentration CðyÞ varies with depth in the cell. Foreach cell that is updated within a given time step, a linear concen-tration profile is imposed prior to bed iteration with

CðyÞ ¼ Ch=2ðh� yÞ for C < 0:5; ð3Þ

CðyÞ ¼ 1� ð1� CÞh=2

y for C > 0:5;

where h (m) is the cell height and y (m) is the vertical position mea-sured from the bottom of the cell. The use of (3) is an approximationbased on observations [6,9,11].

2.2.1. Advection algorithmThe depth of motion (DOM), which quantifies the maximum

depth to which the bed is energized during a given time span(e.g., one wave period), is governed by the applied shear stresswhich in turn can be determined from the time series of waveamplitude and skewness. BEDCELL can receive as input a time ser-ies of wave statistics from which, at each time step, the (DOM)k isdetermined from look-up tables of simulations from Calantoni andThaxton [6] for each of the k = 1, . . . , N columns, where N is thenumber of columns, each of width Dx. Regardless of the existingvertical stratification of cells in the bed, an ‘active layer’ is createdfrom the mass in the bed from the surface down to the DOM. If theactive layer splits a single cell (i.e., the active layer does not reachthe bottom of a cell), the large and small grain mass abundancesthat dynamically participate in that time step are determined bythe linear concentration profile within only the fraction of the cellthat is consumed by the active layer. If the active layer consumesmultiple cells, the abundances within the active layer are deter-mined from the large and small grain masses and the mean con-centrations within cells that are completely encompassed by theactive layer, plus the abundance governed by the concentrationprofile of the active fraction of the bottom-most cell.

From the total mass abundance by grain size within the newly-formed active layer cell of a given column, the ratio of ML/MS, isused in (1) to determine the transport ratio, qL/qS, in that activelayer cell. The actual values of qL and qS are uniquely determinedby the closure relation,

q ¼ qL þ qS; ð4Þ

where q (kg m�1 s�1) has been provided as input or is determinedfrom fluid forcing conditions. The transport rates qL and qS are con-verted to the ‘entrained’ masses (kg) of large and small grain masses(kg) to be advected from column k to column k + 1 (or k � 1,depending on net transport direction). However, not all grains aretransported from the active layer. The active layer comprises grainsexperiencing oscillatory sheet flow motion and is used to determinethe mass ratio ML/MS for use in (1), but generally contains moremass than that which experiences a net transport. The portion ofmass within the active layer that is not transported to column

k + 1 is the ‘remnant’ mass in column k. Entrained mass from col-umn k � 1 is fully mixed with the remnant mass in column k, cre-ating a new active layer cell with a newly determined meanconcentration (2) and concentration profile (3). Prior to time itera-tion and in the absence of diffusion, this new active layer is depos-ited onto the top of column k.

2.2.2. Diffusion algorithmAfter advection has occurred, diffusion may be applied to the

active layer cells. Three tunable parameters used for diffusioninclude:

(1) %D = the percentage of the active layer mass in column k thatis to be diffused.

(2) n = the number of cells in each direction over which a Gauss-ian-distributed mass deposition is applied, corresponding tothe ‘‘distance’’ of diffusion. For example, if n = 1, half of %Dfrom column k is diffused to column k � 1 and half to col-umn k + 1. If n = 3, 34% of %D is diffused to column k ± 1,14% to column k ± 2, and 2% to column k ± 3.

(3) b = a proportionality constant relating diffusion to transportratios for large and small grains.

Using discrete particle simulations [7], the ratio of diffusivity totransport for both large and small grains is quantified [11] with

ðD=qÞLðD=qÞS

¼ b orDL

DS¼ b

qL

qS; ð5Þ

where D is diffusivity (m2 s�1) and b = 1 when ML/MS < 0.75, b = 3when 0.75 < ML/MS < 0.84, and b = 4 when ML/MS > 0.84. Since theratios ML/MS and qL/qS within the active layer are known, theproportion of %D comprising large vs. small grains is determinedfrom (5).

3. Results

3.1. Experimental results

To quantify advection of the plug of large grains along the120 cm field of view of the camera, a photometric center of mass(pCOM) was computed for the large (white) grains over time. Foreach video frame, the image was decomposed into red, blue, andgreen (RGB) channels. For analysis, the B-channel was used sincethe B-channel intensity contrast was much greater than the R-and G-channels for all frames. The B-channel also produced highercontrast than a gray-scaled version of the original frames. The B-channel intensities were averaged across the center portion ofthe transverse width of the channel to generate a longitudinalintensity profile for each channel per frame. Due to edge effectsand lighting positioned near the walls of the S-OFT, only the center190 pixels of the 390 pixel-wide frames were used for the trans-verse averaging. A longitudinal array of intensity standard devia-tions was also generated per frame to quantify transverseintensity variation. The standard center-of-mass equation was ap-plied to the overall longitudinal intensity profile to yield a pCOMposition (cm) per frame. The standard deviation of the overall lon-gitudinal intensity profile per frame was also computed to quantifylongitudinal variations. For comparison of results to BEDCELL sim-ulations, a rolling average routine was applied to the time series ofpCOM – a rolling average width of 500 frames best damped theoscillatory behavior of the pCOM time signal without loss of infor-mation. Fig. 2 shows the pCOM and rolling average for the experi-ment lasting 25 periods (1800 frames, 60 s) relative to the initialpCOM position.

Fig. 2. The photometric center of mass (pCOM) and rolling average (over 500 frames) for the experiment, run for 25 periods (1800 frames, 60 s), relative to the initial pCOMposition.

Fig. 3. Sample frame of the active bed, taken through the acrylic sidewall of theS-OFT by the Canon™ EOS 7D digital SLR camera with a standard zoom lens. High-definition video was acquired in 720p at 60 fps. The spatial dimensions of the fieldof view were 15.4 cm wide by 10.1 cm tall (83 pixels/cm). Overlaid is thecomputational grid used by the PIV method to extract sediment velocities.

50 K. Holway et al. / Advances in Water Resources 48 (2012) 47–54

3.2. BEDCELL simulations

In BEDCELL, column width was defined to be 2.5 cm such that48 cells spanned the 120 cm-long camera field of view. The timestep size was 1 wave period (2.4 s). Grain size and material density,as well as domain and experimental geometries, were set in BED-CELL to match the experimental run. To directly compare BEDCELLresults with the rolling averaged pCOM experimental results, thepCOM time series was sampled at intervals of 72 frames to yield25 pCOM values, one per wave period.

No direct measurements of the net sediment transport ratewere performed. Consequently, the transport rate input to BED-CELL was tuned to optimize results. A suggested range of net sed-iment transport rates for use in the BEDCELL simulations wereestimated from video acquired through the sidewall during a sep-arate experiment under identical conditions (see Fig. 3). A sedi-ment velocity field time series ~uðx; y; tÞ was generated from thevideo images using a time-resolved particle image velocimetry(PIV) technique [12] (Fig. 3). The depth-dependent horizontalvelocity profile ~uðy; tÞ of a representative wave period was deter-mined by averaging along rows of the grid and ensemble-averagingover several wave periods. The velocity phase map shown in Fig. 4was computed using the depth-dependent horizontal velocityprofile.

The net sediment transport, qNET (kg m�1 s�1), was computed asa time-averaged, vertically-integrated flux,

qNET ¼q0

T

Xy;t

uy;t DyDt; ð6Þ

where T is the wave period (s), and uy,t is the depth- and time-dependent horizontal sediment velocity (m/s). The bulk density,q0 (kg m�3), is known to be depth- and time-dependent due tothe dynamic behavior of the packing fraction, fp (where q0 = fpqand q is the material density (kg m�3)). Additionally, since thebed dilates during motion, the vertical limits of summation shouldchange. In lieu of considering the full dynamic nature of the prob-lem, we employed a range of constant values for bed heights and

packing fractions to generate a suite of BEDCELL simulations.Choosing 0.60 as a maximum value of the packing fraction c andallowing the vertical summation limits to vary from (�0.50 cm,+0.50 cm) up to (�2.5 cm, +0.75 cm), a range of possible net trans-port rates (kg m�1 s�1) emerged as

0:0024 kg m�1 s�1 < qNET < 0:0072 kg m�1 s�1: ð7Þ

A suite of 30 unique BEDCELL simulations was performed, spanninga multi-parameter space over net transport rate qNET, diffusion rate%D, and diffusion distance n. Specifically, qNET was specified as0.0030, 0.0036, or 0.0040 kg m�1 s�1; %D ranged from 0% to 40%in increments of 10; and n was set to 2 or 3 cells. The transport rateqNET was tuned to match experimental results and kept within thebounds estimated in (7). The value of %D was constrained due toexpectations that the bed evolution would not be diffusion-domi-nated. A maximum diffusion distance of n = 3 was determined bythe maximum piston excursion amplitude of 8.74 cm which would

Fig. 4. Ensemble-averaged sediment velocity field time series uðy; tÞ generated from the side wall video images using a time-resolved particle image velocimetry (PIV)technique. The vertical position y = 0 corresponds to the at-rest bed height.

K. Holway et al. / Advances in Water Resources 48 (2012) 47–54 51

not physically allow transport more than 3 BEDCELL columns (oflength Dx ¼ 2:5 cm) in either direction. Center-of-mass (COM) andlongitudinal standard deviation time series were produced for thelarge grains for comparison to the pCOM from the experiment.

For each simulation in the suite, the BEDCELL COM displace-ment time series was fitted with linear least squares to the exper-iment pCOM time series. Also, the BEDCELL longitudinal standarddeviation time series was fitted to the experiment longitudinalstandard deviation time series. For the COM/pCOM fit, the R2 val-ues ranged from 0.9817 to 0.9853 and the Mean Standard Error(MSE) ranged from 0.0386 to 0.0480. For the standard deviationfit, the R2 values ranged from 0.9871 to 0.9942 and the MSE rangedfrom 0.0002 to 0.0005. Fig. 5 shows the rolling, sampled pCOM dis-placement time series from experiment and the COM displacementtime series from BEDCELL with qNET = 0.0036, %D = 20%, and n = 3 inwhich the R2 value was 0.9839 and MSE was 0.0420. Also shownare the longitudinal standard deviations from BEDCELL and exper-iment, each normalized to their maximum values, in which the R2

value was 0.9916 and MSE was 0.0003. Fig. 6 shows the scatterplots of BEDCELL COM vs. experimental pCOM as well as the BED-CELL vs. experiment normalized longitudinal standard deviationfor the qNET = 0.0036, %D = 20%, and n = 3 simulation. Fig. 7 showsthe top-down image of the bed from the experiment initially (pa-nel A, top) and after 25 periods when the bed had come to restand after bubbles had been cleared from the top inner surface ofthe tank (panel A, bottom). Bed images are shown in comparisonto a surface layer rendering from BEDCELL before (panel B, top)and after (panel B, bottom) the simulation for qNET = 0.0036,%D = 20%, and n = 3.

The rate of advection of the pCOM from BEDCELL was twice assensitive to changes in the diffusion parameter n as %D over therange of values used, with better overall statistical fits with theexperimental COM for lower values of both n and %D. The longitu-dinal standard deviation from BEDCELL was equally sensitive tochanges in n and %D over the range of values used, with betteroverall statistical fits for n = 3 and 20% < %D < 30%.

4. Discussion

Although the photometric center of mass (pCOM) is not a truecenter of mass but rather a weighted average of intensity valuesfrom the video frames, we expect the intensity of the white (large)grains in the background of red (small) grains to correspond wellwith the large and small grain mass distribution in space and timein the experiments. BEDCELL transported the center of mass (COM)of the plug of large grains with skill compared to the experimen-tally determined photometric center of mass (pCOM) forqNET = 0.0036, %D = 20%, and n = 3. Similar R2 and mean square er-ror values were produced for 11 other simulations within the range0.0032 < qNET < 0.0040, 20% < %D < 30%, and 2 < n < 3 (full suite re-sults not shown) after 25 periods. The range of qNET values fallswell within the estimated bounds from (7). Also, the best rangefor %D is consistent with the expectation that the experimental set-up would not be diffusion-dominated during bed evolution. Valuesfor n>3 were not accessible since the maximum piston excursionamplitude was fixed at 8.74 cm.

The normalized longitudinal standard deviation represents thedegree to which the distribution of surface grains by size variedalong the longitudinal direction. Although longitudinal standarddeviation does not quantify diffusion in the presented experimentsand simulations, longitudinal standard deviation and diffusion arerelated. For example, because the initial configurations had sharp,discontinuous boundaries between the red (small) bed and thewhite (large) plug, the standard deviation is expected to be maxi-mized. As the bed evolves both within BEDCELL and in the exper-iment, the standard deviation is expected to drop since thediffusion works to smooth-out the sharp discontinuities. The diver-gence in time between the BEDCELL and experimental standarddeviation for the displacement of the COM was expected due tothe contribution of transient effects at startup and shutdown andas well as boundary effects (see Fig. 5).

The experimental pCOM translate more slowly during earlyand latter portions of the experiment due to the start-up and

Fig. 5. (Top) Center of mass displacement (cm) time series from BEDCELL with qNET = 0.0036 kg m�1 s�1, %D = 20%, and n = 3, as well as the rolling, sampled photometriccenter of mass (pCOM) displacement (cm) time series from experiment. (Bottom) The longitudinal standard deviation time series from BEDCELL and experiment, eachnormalized to their maximum values.

Fig. 6. (Top) Scatter plot of the BEDCELL center of mass vs. experimental photometric center of mass. (Bottom) Scatter plot of the BEDCELL vs. experiment normalizedlongitudinal standard deviation.

52 K. Holway et al. / Advances in Water Resources 48 (2012) 47–54

shut-down of the piston. Experimental data was gathered and ana-lyzed out to 25 periods, soon after which the piston eventuallystopped, but the pCOM transport rates dropped continually duringthe shut-down cycles (beyond 25 wave periods).

Since no experimental data was obtained to validate the evolu-tion of the vertical distribution of grains by size within the bed,

statements concerning BEDCELL’s ability to correctly predict verti-cal stratification in time remain speculative. However, since BED-CELL correctly models longitudinal dynamics within theconditions stated, we believe that further studies on vertical grada-tion based on (1) are merited. Fig. 8 shows the vertical stratifica-tion of sediment concentration by size within the active layer for

Fig. 7. The top-down image of the bed from experiment (A) and a surface layer rendering from BEDCELL (B) initially (Top) and after 48 periods (Bottom). Longitudinal lengthis 120 cm and net sediment transport is to the right.

Fig. 8. Rendering of the vertical concentration profile from BEDCELL after 25 periods with qNET = 0.0036 kg m�1 s�1, %D = 20%, and n = 3. The domain width is 48 cells(120 cm), 2.5 cm per cell. The top layer is the active layer of height 1.40 cm – the bottom layer is the upper-most layer within the bed. White represents large grains, redrepresents small grains. Net transport is to the right.

K. Holway et al. / Advances in Water Resources 48 (2012) 47–54 53

qNET = 0.0036, %D = 20%, and n = 3 at the end of the simulation.Since qNET did not vary in space or time, the active layer is the onlyvertical layer that experienced changes in vertical concentration bysize. As expected, large grains are preferentially sorted to the top ofthe active layer as governed by (1), which is consistent with obser-vations during the experimental runs.

5. Conclusions

From the results presented, we have confirmed the ability ofBEDCELL to accurately predict longitudinal sediment transport ofa bimodal distribution of grains for the given experiment condi-tions. Goodness of fit was found, over a range of BEDCELL parame-ters, between the BEDCELL COM time series and the experimentpCOM time series. Expected relationships in time between theBEDCELL and experiment longitudinal variations were also mea-sured. Results suggest that the Calantoni–Thaxton power law (1),as implemented in BEDCELL, can be coupled to a simple advec-tion-diffusion solver with skill in modeling active layer net andfractional transport.

Moreover, unlike traditional solvers of advection-diffusion par-tial differential equations that transport residual mass based ondifferential relationships, or fractional transport approaches thatdo the same for each fraction independently, BEDCELL transportsall mass dictated by the net and fractional transport rates withinthe active layer, incorporating dynamic vertical sorting implicitin (1). Validating BEDCELL for surface advection and diffusion isthe first step toward larger-scale implementation of (1).

Although the S-OFT experiments are manually intensive andtime consuming, more experiments similar to those presentedare recommended to validate the Calantoni–Thaxton power lawfor a wider range of material properties and forcing conditions.However, future simulations are planned that introduce spatially-and time-varying net sediment transport functions for comparisonto historical near shore field sediment transport data.

Acknowledgements

KH was supported by the Naval Research Enterprise Intern Pro-gram (NREIP 2011) sponsored by the American Society for Engi-neering Education (ASEE) at the Naval Research Laboratory. KH

54 K. Holway et al. / Advances in Water Resources 48 (2012) 47–54

and CT were supported by the NASA North Carolina Space Grant(2010 & 2011), the National Science Foundation SSTEM program(NSF #0754923), and the Appalachian State University Office ofStudent Research, Boone, NC. JC was supported under base fundingto the Naval Research Laboratory from the Office of Naval Research(PE#61153N). The authors wish to thank In Mei Sou and Tim Koo-ney for their support in obtaining the laboratory measurements.

References

[1] Gallagher EL, Elgar S, Guza RT. Observations of sand bar evolution on a naturalbeach. J Geophys Res 1998;103(C3):3203–15.

[2] Elgar S, Gallagher EL, Guza RT. Nearshore sandbar migration. J Geophys Res2001;106(C6):11623–7.

[3] Warner JC, Sherwood CR, Signell RP, Harris CK, Arango HG. Development of athree-dimensional, regional, coupled wave, current, and sediment-transportmodel. Comput Geosci 2008;34:1284–306.

[4] Blom A, Ribberink JS, Parker G. Vertical sorting and the morphodynamics ofbed form-dominated rivers: A sorting evolution model, J Geophys Res – EarthSci 2008;113(F1):F01019. DOI: 10.1029/2006JF000618.

[5] Blom A, Parker G, Ribberink JS, de Vriend, HJ. Vertical sorting and themorphodynamics of bed-form-dominated rivers: An equilibrium sortingmodel Author(s). J Geophys Res-Earth Sci 2006;111(F1):F01006. DOI:10.1029/2004JF000175.

[6] Calantoni J, Thaxton CS. Simple power law for transport ratio with bimodaldistributions of coarse sediments under waves. J Geophys Res2008;113:C03003. http://dx.doi.org/10.1029/2007JC004237.

[7] Drake TG, Calantoni J. Discrete particle model for sheet flow sedimenttransport in the nearshore. J Geophys Res 2001;106:19859–68.

[8] Sakai T, Goto H. Vertical grading of mixed-size grains in sheetflow regimeunder oscillatory flows. Coast Eng 2000:2766–79.

[9] Dibajnia M, Watanabe. A Moving layer thickness and transport rate of gradedsand. Coast Eng 2000, Conference Proceedings, July 16–21, Sydney, Australia2000. p. 2752–65.

[10] Hassan WN, Ribberink JS. Transport processes of uniform and mixed sands inoscillatory sheet flow. Coast Eng 2005;52:745–70.

[11] Thaxton C. Investigations of grain size dependent sediment transportphenomena on multiple scales. PhD dissertation. North Carolina StateUniversity, Raleigh, NC, 2004.

[12] Dobson DW. Real-time riverine particle image velocimetry. PhD dissertation.University of Southern Mississippi, Hattiesbug, MS, 2011.