Modelling granular sediment transport over water-worked gravels

Modelling granular sediment transport 1069

Copyright © 2005 John Wiley & Sons, Ltd. Earth Surf. Process. Landforms 30, 1069–1076 (2005)

Earth Surface Processes and LandformsEarth Surf. Process. Landforms 30, 1069–1076 (2005)Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/esp.1277

Letter to ESEX

Modelling granular sediment transport overwater-worked gravelsRichard J. Hardy*Department of Geography, Science Site, South Road, Durham, DH1 3LE, UK

AbstractThis paper reports a numerical model that calculates the transport paths of individualgranular particles over a natural water-worked gravel surface. Surface topography is meas-ured using digital photogrammetry and incorporated into a computational fluid dynamics(CFD) scheme by the application of a porosity algorithm. The predicted hydraulics determinethe forces acting on each particle enabling the particle trajectory to be solved by Langrangiantransport equations. The methodology demonstrates the importance of including complextopography in the numerical scheme as topographically induced hydraulic acceleration anddeceleration affect both the transport mechanism and subsequent trajectory of the particle.Hydraulically induced hops, where the particle is entrained into a localized shear layeraround a bed particle, are demonstrated to be important in transporting granular sedimentacross the bed and in re-starting sediment transport if the particle has fallen into a hydrau-lically sheltered region. Copyright © 2005 John Wiley & Sons, Ltd.

Keywords: CFD; Langrangian transport equations; granular sediment transport

Introduction

A numerical description of the hydraulics and sediment transport processes operating in a fluvial environment is farfrom complete due to the complex interactions between form, flow and sediment transport. The structure of near-bed turbulence determines the dynamics of sediment transport and the development of bed forms through complexfeedback relations (Clifford et al., 1992; Ashworth, 1996). However, an analytical description of turbulent flow andsediment entrainment, transport and deposition is subject to significant errors due to the poor parameterization ofphysical processes (McEwan and Heald, 2001). Furthermore, velocity profiles are frequently used to determine themean boundary shear stress (McLean et al., 1999) which may be inadequate to describe the effect of bed roughness onflow characteristics (Papanicolaou et al., 2001). Such averaging obscures the complex, non-linear interaction of wakedecay, boundary-layer development and topographically induced acceleration and deceleration; this leads to an inaccu-rate estimate of boundary stress, particularly skin friction, which is essential in sediment transport. Entrainment andmotion of sediment grains are driven by turbulent structures producing high instantaneous fluid forces on the lowersurfaces of grains (Nelson et al., 2001). These events depend on the flow structure near the particle, which isdetermined by the particle exposure to the flow and the pocket geometry of the particle, in as much as they correlatepoorly with measurements made upstream of, or over, the particle (Nelson et al., 2001).

Granular material can be transported by: (1) suspension, an advanced stage of transport (Graf, 1998) where the particlesare rarely in contact with the bed; (2) saltation, the unsuspended transport of particles in the form of consecutive hopswithin the near-bed region which is generally considered the dominant mode of bedload transport (Nino and Garcia,1994); and (3) rolling and sliding, which occur to a lesser extent mainly near the threshold of entrainment andin between individual saltation events (Bridge and Dominic, 1984). A frequently used approach to predicting thetrajectories of saltating particles numerically is the application of Langrangian equations governing particle motion(e.g. Tsuchiya, 1969; Reizes, 1978; Hayashi and Ozaki, 1980; Murphy and Hooshiari, 1982; Van Rijn, 1984; Wibergand Smith, 1985; Nino and Garcia, 1994; Lee and Hsu, 1994). This approach has previously been applied to gain anunderstanding of the kinematic and geometric characteristics of the particle saltation process, which in turn can be

*Correspondence to: R. J. Hardy,Department of Geography,Science Site, South Road,Durham, DH1 3LE, UK.E-mail: [email protected]

1070 R. J. Hardy


used to develop a model of bedload transport (Yalin, 1963). Several of the models have predicted mean values ofsaltation characteristics (saltation length, height and particle velocity) which agreed well with experimental results.

The recurring limitation with all these individual particle-based transport models is that the hydraulic representationis calculated by averaged velocity profiles over a homogenous bed. Recent research in computational fluid dynamics(CFD) has developed a mass flux scaling algorithm based upon a cell porosity approach (Hardy et al., in press) thatenables the prediction of both time-averaged hydraulics over a water-worked gravel surface (Lane et al., 2003, 2004)and time-dependent flow predictions from the application of large eddy simulation (Hardy and Lane, 2004). Thedevelopment of this method has allowed an understanding of the way in which a complex gravel-bed surface interactswith the associated three-dimensional flow field. Furthermore, it provides a high-resolution prediction of the hydraulicforces acting on each individual particle. Here a sediment transport model is described where individual particlemotion is mathematically modelled through trajectories based on Langrangian equations driven by hydraulic forcescalculated from a CFD scheme.

Methodology

The sediment transport schemeThe model determines the particle transport trajectory and velocity, in a Lagrangian framework, by:

∂∂x

tUp

p = (1)

where vector xp is the particle position and Up is the particle velocity. The particle velocity is calculated by a particlemomentum equation:

mU

tp

p∂∂

= Dp(U − Up) + mpg − Vp∇p (2)

where mp = particle mass, Up = particle velocity, t = time, Dp = drag function, U = continuous-phase instantaneousvelocity (U = Uc + U′c, Uc is the continuous-phase average velocity and U ′c is the turbulent fluctuation component, g =gravity, Vp = particle volume, ∇p = continuous-phase pressure gradient. At present, the Basset force and the Magnusforce are neglected. The drag function (Dp) assumes that the particles are spherical and calculated by:

Dp =1

2ρApCD|U − Up | (3)

where Ap is the particle projected area, (πdp2)/4, and CD is the drag coefficient calculated by the correlation of Clift

et al. (1988):

CD Re

( Re ) Re

= + ⋅ +⋅

+ ⋅ ×⋅

− ⋅

241 015

0 42

1 1 425 100 687

4 116(4)

where Re is the particle Reynolds number.The effect of turbulent fluctuations on a particle’s trajectory is calculated with the stochastic turbulence model of

Gosman and Ioannides (1981) which applies a Reynolds decomposition approach where the fluctuating component iscalculated assuming a normal distribution with a mean of zero and a standard deviation of √(2k/3), where k is theturbulence kinetic energy. The fluctuating component U′c acts over a time interval which is defined as the minimumlifetime of the local eddy which the particle is traversing (Ludwig et al., 2004).

Interaction with bed topography is based upon the application of a restitution coefficient which is the scaling factorbetween the outgoing normal velocity and the incoming normal velocity for the centre of mass of the particle(Schmeeckle et al., 2001). Previous work has demonstrated sensitivity to this value (see Schmeeckle et al. (2001) fora review). In this application a value of 0·5 is used as it the mid-value of the data of Wiberg and Smith (1985). Finally,a stagnation criterion is applied where the particle trajectory will stop being calculated if the particle has fallen into ahydraulically stagnant region which is based on a pre-defined criterion based on zero movement over a number ofiterations. The bed topography (through the porosity algorithm) is then modified as it is assumed that the particles



Figure 1. The location of the particles (marked by a plus sign) at the inlet.

Table I. The hydraulic condition used in the experiments

Flow velocity (m s−1) 0·155 0·300 0·435Flow depth (m) 0·200Flume width (m) 0·300Froude number (−) c. 0·11 c. 0·21 c. 0·31Reynolds number (−) c. 13 000 c. 25 000 c. 37 000

have been deposited. At present, for computational simplicity in calculating the particle trajectory, the largest particlethat can be modelled is less than or equal to the resolution of numerical discretization (i.e. the mesh resolution).

Experimental set-upThe topography and hydraulic predictions used here are based upon yet unpublished flume experiments. Theexperiments consisted of water-worked gravel in a 10 × 0·3 × 0·4 m flume. A 0·72 m long section was studied. For thisregion, a digital elevation model (DEM) was collected using digital photogrammetry. The hydraulic conditions usedin the experiments are given in Table I. Two-dimensional velocity measurements were collected on a spatial resolutionof 0·002 m × 0·002 m and a temporal resolution of 15 Hz using digital particle imaging velocimetry (DPIV) and usedto provide inlet boundary conditions (Figure 1) as well as to validate the hydraulic scheme.

The representation of the gravel DEM in a CFD framework is shown in Figure 2, which is incorporated into acomputational mesh on a resolution of 0·002 × 0·002 × 0·002 m (361 × 130 × 100 = 4 693 000 grid cells). The porosityalgorithm for the flow treatment used has previously been reported, demonstrated and validated against acoustic Dopplervelocimetry (ADV) (Lane et al., 2003, 2004) and DPIV data (Hardy et al., in press) over different complex topographies.In this study a steady-state solution for the hydraulics is obtained using a RNG two-equation k–ε turbulence model.

Twenty particles enter the domain at the inlet at pre-specified locations where the initial particle velocity is dependentupon the hydraulic inlet boundary conditions (Figure 1 and Table I). The physical characteristics of the particles are

1072 R. J. Hardy


Figure 2. The predicted sediment transport paths. The scale refers to the vertical height of the individual particles in the digitalelevation model. (a) The particles entering the domain close to the bed. (b) The particles entering the domain at the mid level.(c) The particles entering the domain at the highest level

given in Table II. Each particle trajectory is then tracked until it either leaves the domain through the outlet or isdeposited.

Results

In these results, I focus on the 0·435 m s−1 experiment with a particle size of 0·5 mm as they demonstrate the mostdynamic particle movement and the key features of granular sediment transport over a gravel bed. The particles wereintroduced at four different vertical levels (Figure 1) and their trajectories are shown in Figure 2. The particles close to



the bed (Figure 2a) are transported in skimming flow over the gravel particles. Two particles are transported to the endof the domain while two others are transported approximately 0·2 m before being deposited in hydraulically shelteredlocations. As the vertical heights at which the particles enter the domain increase (Figure 2b and c) the main mech-anism for transport is suspension, but there is a general drift towards the bed, and the particles travel until they collidewith the bed. The transport mechanism is discussed in the next section.

A more detailed analysis of a particle trajectory is demonstrated in Figure 3. The initial starting point is high in theflow (Figure 3a) and the particle travels at a rate that approximates the depth-averaged velocity (c. 0·5 m s−1) down-stream although the particle is moving towards the bed as the particle vertical velocity is always below zero (Fig-ure 3b). The particle descends in suspension (A in Figure 3c) although the transport mechanism in this region is aproduct of the inlet boundary conditions (the starting height of the particle is unnaturally high and the particle isunlikely to reach this height again under these hydraulic conditions). At a downstream distance of 0·37 m from theinlet, the particle collides with the bed and the particle vertical velocity becomes positive. The particle transportmechanism then appears to be saltation where there are two ‘hops’ (region B in Figure 3c). Then the transportmechanism changes for a third time (region C in Figure 3c) into hydraulically induced hops. In this region theparticles are entrained upwards in a localized shear layer over the bed particle (Figure 3a), where a distinctive particlevelocity signal can be observed. When the particle collides with the bed there is a change from a negative verticalvelocity to positive and the particle moves upwards until it is entrained into the shear flow. The downstream com-ponent of particle velocity then increases dramatically as the sediment particle is transported over the bed form. Aseries of the velocity signatures detecting the hydraulically induced hops are marked by the dashed line betweenFigure 3b and c. Finally, the particle moves into a stagnant region (region B in Figure 3c) where a classic saltationtrajectory is observed.

Figure 4 demonstrates more localized processes of granular sediment transport. The figure shows a view down theDEM from the inlet. The particle is transported in a localized area of low flow and a region of possible deposition infront of a large pebble. However, the localized hydraulics in front of the particle, a possible suppressed saddle pointvortex (Hunt et al., 1978), transport the particle within a confined area until a tertiary vortex (Hunt et al., 1978)entrains the particle into the shear flow over the particle and into a hydraulically induced hop. This demonstrates thespatial variability and the hydraulic processes operating in these environments. It is demonstrated that a spatialdifference of millimetres can affect the forces acting on the particle and the possible transport mechanism. Once theparticle transported over the large clast, the transport mechanism changes to saltation and the particle moves downslope.

Discussion and Potential Applications

The results demonstrate the importance of micro-topography for the generation of localized flow structures, which inturn determine the transport direction of the particles as well as the associated transport mechanisms. In none of theexperiments did a particle travel the full length of the experiment by saltation. This is not surprising due to thedifference in size between the transported particles and the particles on the bed. However, transport paths are observedwhich have similar trajectories to saltating particles (Figure 3c) although they are driven by different mechanisms. Inthis transport mechanism the particle collides with the bed and bounces, or local vortex structures lift the particle untilit is entrained into a shear layer over an individual particle. They are therefore hydraulically induced. This mechanismcannot be defined as saltation, which Bagnold (1973) described as the unsteady suspended transport of particles byfluid flow, in the form of consecutive hops, in which no upward impulses are imparted to the particle other than thoseattributed to successive contacts between them and the bed. This implies that hydrodynamic lift is not important in thesaltation process and rules out the effect of turbulence as the mechanism to sustain saltation (Nino et al., 1994).However, these results have demonstrated that the saltation process can be restarted by localized hydraulic forces inthat after a hydraulically induced hop, the particle can saltate if the bed topography is smooth enough.

The model presented herein provides a new methodology to determine granular sediment transport over a naturalriver bed. However, the present scheme is by no means complete and there are several future developments under way.

Table II. The physical characteristics of the sediment particles

Particle size (m) 0·002, 0·001, 0·0005Density (kg m−3) 2650·0 (quartz)Shape SphericalRestitution coefficient (–) 0·5

1074 R. J. Hardy


Recent research has enabled the calculation of time-dependent hydraulics over similar gravel topographies by theapplication of large eddy simulation (LES) (Hardy and Lane, 2004). In these numerical simulations it was demon-strated that local topographic forcing of flow determined both the time and length scales of the turbulent structures,which would have significant implications for both local and temporal derivations of bed shear stress. Therefore, if themodel was enclosed in a LES–CFD framework a temporal derivation of the flow forces acting on an individualparticle could be determined. In theory, this would allow the Basset term to be resolved. The other major area where

Figure 3. The predicted characteristics of a sediment transport path of an individual particle. (a) The trajectory of the particleacross the DEM. (b) The velocity components of the particle. The downstream (u-) component is the solid line, the lateral (v-)component is dotted and the vertical (w-) component is dashed–dotted line. (c) The distance downstream against the verticalheight about the flume bed. There is a vertical exaggeration of about six



Figure 4. The localized sediment dynamics of an individual particle

the scheme requires improvement is that, at present, the scheme considers spherical particles with no rotation duringtransport. This leads to two main limitations. Primarily, granular particles are rarely spherical and therefore the presentscheme requires natural particle shapes to be considered, through a modification in the drag term. Secondly as there isno particle rotation, the particle’s angular velocity is zero, preventing the calculation of the Magnus lift force. Thismay be significant in calculating the saltation distance. However, the main problem in estimating the Magnus term isthat the particle angular velocity is an external variable that must be estimated a priori as it is controlled mainly by theparticle collision with the bed, so making it a highly random variable.

These limitations aside, the potential applications of this methodology are considerable. First, it provides a frame-work for exploring exactly how particles of different sizes move over bed surfaces of different composition andarrangement as well as how this composition and arrangement feeds back onto hydraulic forcing. The use of numer-ical porosity is crucial here as sediment deposition (or entrainment) can be explicitly represented through a change inporosity, so allowing simultaneous bed evolution without remeshing. This has been the limitation of applying com-putational fluid dynamics to sediment transport problems, especially given the severe influence of poor mesh geometryupon numerical diffusion and stability. Whilst a study of this resolution cannot address larger-scale questions of riverdynamics, it can be used as a tool to parameterize process representation for such questions. Second, we know thatthere is a strong interaction between fine sediment dynamics and the ecological quality of river beds. This methodprovides a tool for understanding how fine sediment particles infiltrate into the gravel matrix and how this subse-quently affects the hyporheic zone of the river bed.

AcknowledgementsThe author was funded under NERC fellowship NER/J/S/2002/00663. The author wishes to thank Dr JH Chandler, University ofLoughborough, for the loan of the Kodak DCS 460 digital camera and Dr PE Carbonneau, INRS Quebec, Canada, for undertakingthe photogrammetry to generate the DEM. Dr Andrew Nicholas and an anonymous referee provided very helpful comments on anearlier version of this manuscript.

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Modelling granular sediment transport over water-worked gravels