Large Eddy Simulation of Sediment Deformation in a Turbulent Flow by Means of Level-Set Method

Large Eddy Simulation of Sediment Deformation in aTurbulent Flow by Means of Level-Set Method

Susanne Kraft1; Yongqi Wang2; and Martin Oberlack3

Abstract: Sediment transport in a turbulent channel flow over the sediment bed with a ripple structure is numerically simulated by means ofa large eddy simulation. The filtered Navier-Stokes equations for the channel flow and the filtered advection-diffusion equation with a settlingterm for the suspended sediment are numerically solved, in which the unresolved subgrid-scale processes are modeled by the dynamicsubgrid-scale model of Germano et al. The migration and deformation of the interface between the sediment bed and the fluid flow is capturedby the level-set method. The sediment erosion is approached by means of three different pickup relations postulated by van Rijn, Einstein, andYalin, respectively, partly modified by the authors. Generally, the sediment is entrained into the flow from locations where the shear stressexceeds a critical value—on the upstream slopes of ripple crests—and is advected downstream in suspension by the flow, until it settles againwhen the local flow condition cannot further transport it, e.g., on the lee sides of ripples. A global effect of these local processes is themigration of ripples. The numerical results on the fluid flow field and the sediment concentration distribution are discussed. The computedmigration speed of the ripples, which is only a fraction of the free stream velocity, is compared with known experimental data and a goodagreement is demonstrated. DOI: 10.1061/(ASCE)HY.1943-7900.0000439. © 2011 American Society of Civil Engineers.

CE Database subject headings: Channel flow; Velocity; Simulation; Eddies; Sediment.

Author keywords: Ripple; Channel flow; Migration velocity; Large eddy simulation; Level-set method.

Introduction

The bed of a flowing water body is usually composed of fine co-hesionless loose sediments and thus its form is not stable. Whenthe bed cannot resist the wall shear stress caused by the fluid flow,sediment is entrained and transported downstream (erosion). On thecontrary, in weaker flow areas in which the force of gravity of theparticles prevails, the suspended sediment settles at the bed surface(sedimentation). The erosion and sedimentation deform the riverbed, which affects the fluid flow and the associated sediment trans-port in a strongly coupled manner. Under certain flow conditions,e.g., when a wave passes over sand or silt in shallow water, or aninitial nonsmooth geometry structure of the sediment bed exists,ripple structures may be developed (Zanke 1999). Rows of ripplesform perpendicular to the flow direction of the water. Ripple struc-tures have a dynamic state of equilibrium, i.e., they keep the shapebut are not spatially fixed. They move downstream by erosion ofsediment from the upstream side of the ripple, and deposition on the

lee side. For given flow conditions, the ripples move with a certainmigration velocity, which is small compared with the mean fluidvelocity (Kühlborn 1993). Ripple formation occurs ubiquitouslyas natural processes both in interior and in marine flows. Thereexists a large scientific and economic interest to model this phe-nomenon exactly, hence to predict the effect of anthropogenicinterferences in water bodies.

The fundamental understanding of sedimentation and erosion aswell as the metastable equilibrium of these processes are not only oftheoretical interest, but also particularly of special importance forthe description of large-scale sedimentation. Small-scale local sed-imentation and erosion are responsible for global spacious changesboth in inland water bodies and in coastal regions. In many fields,erosion and sedimentation lead to problems. Erosion of coastalregions and also the sedimentation of river and channel beds areexamples.

The phenomenon of the ripple emergence and migration hasbeen experimentally analyzed in many works (e.g., Fürböter 1983;Kühlborn 1993). In recent years the numerical investigation hasalso gained significance because of the increase of computerachievement. Most of the works focus mainly on the sedimenttransport rather than on the ripple movement.

Chang and Scotti (2003) examined the influence of coherentstructures on particle transport in suspension over a fixed wavelikesurface with the help of the large eddy simulation (LES). They useda Lagrangian ansatz for the modeling of the particle movement andcomputed the movement of each individual particle. In a furtherwork, Chang and Scotti 2004 compared numerical results of theLES with those of the Reynolds-averaged Navier-Stokes (RANS)modeling, for which the k � ω closure model was employed. Itwas demonstrated that the RANS modeling could not reproducethe most important coherent structures near the flow bed andthe transport in suspension correctly. Similarly, Zedler and Street(2001) examined the influence of ripples on the sediment transportby means of the LES. They also used a sinusoidal wave as the

1Chair of Fluid Dynamics, Dept. of Mechanical Engineering,Technische Universität at Darmstadt, Petersenstrasse 30, 64287 Darmstadt,Germany.

2Chair of Fluid Dynamics, Dept. of Mechanical Engineering,Technische Universität at Darmstadt, Petersenstrasse 30, 64287 Darmstadt,Germany. (corresponding author). E-mail: wang@fdy.tu-darmstadt.de

3Chair of Fluid Dynamics, Dept. of Mechanical Engineering,Technische Universität at Darmstadt, Petersenstrasse 30, 64287 Darmstadt,Germany.; Center of Smart Interfaces, Technische Universität at Darmstadt,Petersenstrasse 32, 64287 Darmstadt, Germany; Graduate School of Com-putational Engineering, Technische Universität at Darmstadt, Dolivostrasse15, 64293 Darmstadt, Germany.

Note. This manuscript was submitted on September 14, 2010; approvedon April 11, 2011; published online on April 13, 2011. Discussion periodopen until April 1, 2012; separate discussions must be submitted for indi-vidual papers. This paper is part of the Journal of Hydraulic Engineering,Vol. 137, No. 11, November 1, 2011. ©ASCE, ISSN 0733-9429/2011/11-1394–1405/$25.00.

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space-fixed bed, but employed the Eulerian description and simu-lated the particle transport in suspension by the concentration dis-tribution with the help of the filtered convection-diffusion equation.At the sediment surface the erosion rate was computed with thehelp of the pickup function by van Rijn (1984a), whereby the criticalwall shear stress is based on a Shields-type criterion (Shields 1936).The results verify the observations of Soulsby et al. (1994) andBennett and Best (1995) that coherent structures are responsiblefor the sediment transport in suspension. In both the test cases ofZedler and Street (2001) and Chang and Scotti (2004), the employedripple surfaces deviate considerably from that emerging in nature.

In the case of the conceptual design of technical systems thestatistically averaged values of the turbulent flow parameters areusually sufficient, as the RANS modelings. An example is the workof Wu et al. (2000). They computed the flow in a curved channelwith a loose bed. Suspended-load transport is simulated through thegeneral convection-diffusion equation with an empirical settling-velocity term. This equation and the full RANS equations withthe k � ϵ turbulence model are solved numerically with a finite-volume method on an adaptive, nonstaggered grid. Bed-loadtransport is simulated with a nonequilibrium method (van Rijn1987) and the bed deformation is obtained from a depth-averagedmass-balance equation. The numerical results for the flow and sedi-ment transport in a 180°-channel bend with loose bed are in goodagreement with measurements.

The two-dimensional depth-averaged shallow-water equationrepresents a further modeling method. For most water-structuralrelevant simulations, the lateral extent is much larger than the depthof water. In such cases, a three-dimensional problem can be mod-eled by a two-dimensional depth-averaged shallow-water equation,and hence the numerical expenditure is reduced substantially.On the basis of this approximation, Duc et al. (2004) calculatedthe bed deformation in alluvial channels. The sediment transportmodule comprises semiempirical models of suspended load andnonequilibrium bed load. The bed-deformation module was basedon the mass balance for sediment. Comparison of the numericalresults with data from laboratory measurements demonstrated agood agreement. With this method, all information in the verticaldirection get lost, and thus secondary flows and separations cannotbe computed. A detailed understanding and description of the rip-ple emergence cannot be obtained on the basis of the shallow-waterequations.

In the present work, the sediment transport in suspension andthe ripple movement of a ripple structure in a channel flow arenumerically calculated using the LES and the level-set method,established by Osher and Sethian (1988) to track the motion ofthe interface between fluid and sediment bed. The interface speeddepends on the physics of the sediment transport. The sedimentparticles erode if the wall shear stress exceeds the critical wall shearstress. The erosion rate is calculated with three different pickupfunctions. The eroded particles are carried by the fluid in suspen-sion. The distribution of the suspended sediment concentration iscalculated with the convection-diffusion equation. The rate of sed-imentation depends on the concentration near the bed and thesettling velocity of the sediment. For the calculation of the criticalbed-shear stress, the Shields-type criterion (1936) and the approachof Zanke (2001) are employed, respectively, and tested. To reachmore reasonable ripple deformation, the classical pickup functionsare modified, with which the ripple development and movement atdifferent Reynolds numbers are analyzed. The numerical resultsshow a good agreement with experimental data.

Modeling of Sediment Transport

Sediment materials of many different shapes and particle sizeserode and contribute to sediment transport. Different ratios of thesettling velocity of the sediment to the critical shear stress deter-mine how they will be transported downstream. Basically two typesof transport are distinguished: suspended load and bed load. Sus-pended load is composed of fine-sediment particles suspended andtransported through the fluid flow. Bed load rolls slowly along theflow bed. These include the larger and heavier materials. Thereexists a small range in between: Saltation, which is a bouncelikemovement, occurs when large particles are suspended in the streamfor a short distance after which they fall to the bed, dislodging par-ticles from the bed. The dislodged particles move downstream fora short distance where they fall to the bed, again loosening bed-loadparticles upon impact. In this research we consider only the fine-sediment materials for which the suspended load is important. If thewall shear stress exceeds a critical value, the sediment is erodedand suspended. It will be transported in suspension, until it comesto a position, where the local flow conditions can not further trans-port it and then will settle again (Fig. 1). The exchange of sedimentparticles through the bed surface, consisting of erosion and sedi-mentation or pickup and deposition, is an essential part of thepresent modeling. Usually, the particle pickup and deposition arerepresented as separate subsystems, which can be combined byapplying the equation of continuity to determine the deformationof the sediment surface.

Erosion Rate

A substantial component for the description of the sediment trans-port is the knowledge of the beginning of movement of particles:erosion. The erosion of sediment begins when the shear stresson the bed surface, τw, exceeds the critical wall shear stress ofthe corresponding sediment material, τ c. A widely used procedurefor the determination of the beginning of entrainment of cohesion-less particles is represented by the Shields curve (1936; see also vanRijn 1984b), which is based on the results of numerous laboratorymeasurements with different grain sizes, densities and wall shearstresses. A critical Shields parameter (the dimensionless criticalshear stress) is defined by

θc ¼τ c

ρsgd; with s ¼ ρs � ρ

ρð1Þ

where d = mean particle diameter; and ρs and ρ = densities ofsediment and fluid, respectively. According to the Shields curvethe dimensionless critical shear stress can be approximated by dif-ferent functions for different ranges of the particle size (van Rijn1984b), e.g., θc ¼ 0:24�D�1 for the dimensionless particle diameter

Fig. 1. Schematic of the transport in suspension

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�D ¼ d=ðν2=sgÞ1=3 ≤ 4, in which ν = kinematic viscosity of thefluid.

Zanke (2001) developed an analytic ansatz, which makes itpossible to describe the beginning of sediment entrainment aloneby the angle of the internal friction and the turbulent fluctuations.According to his approach, for a laminar flow the dimensionlesscritical wall shear stress θc is determined only by the internal fric-tion angle of the sediment φ, given by

θc ¼ α tanφ ð2ÞFor natural sediments with 70% solid material the constant α takesthe value α≃ 0:7. The internal friction angle φ, or the angle ofinternal friction, is a measure of the ability of a unit of granularmaterial to withstand a shear stress. It is the angle, measuredbetween the normal force and resultant force, that is attained whenfailure just occurs in response to a shearing stress. For static cases,it is the natural angle of repose, the maximum critical angle thatthe heap of a granular material can make with the horizontal plane.For natural sand, φ≃ 30°.

For a turbulent flow, to erode the sediment, instead of only themean shear stress, the sum of the statistically averaged wall shearstress and its turbulent fluctuation needs to exceed the critical shearstress. This is equivalent to a reduced critical shear stress. There-fore, the effective critical shear stress is smaller than that for alaminar flow, given by

θeffc ¼ θc � θ0w ¼ α tanφ� θ0w ð3Þwhere θ0w = the root mean square (rms) turbulent fluctuation of thewall shear stress.

The aforementioned ansatzs of Shields and Zanke describe onlythe threshold value for the entrainment of sediments. The statementof the entrainment amount has not been dealt. In the literature manyformulations for determining the erosion rate can be found. In thepresent work, three approaches will be employed and compared.Because of the complexity of the phenomena involved, a completetheoretical description of particle entrainment (the pickup process)is not yet feasible. Many experiments were initiated by van Rijn(1984a) with various particle sizes and flow velocities to empiri-cally determine the sediment erosion rate (the pickup rate) E inmass per unit area and time. The achieved dimensionless erosionrate ϕp obeys a so-called pickup function

ϕp ≡ EρsðsgdÞ0:5

¼ 0:00033�D0:3T1:5; with T ¼ u2τw � u2τ cu2τ c

where uτ c ¼ffiffiffiffiffiffiffiffiffiffiτ c=ρ

pand uτw ¼ ffiffiffiffiffiffiffiffiffiffi

τw=ρp

denote the critical andactual wall shear velocities, respectively.

The approach of Einstein (1950) is stochastic. He did not as-sume a statistically averaged wall shear stress but instead supposedthat turbulent fluctuations will push the particles in motion. Thepickup rate can be expressed as

E ¼ αρsðsgdÞ0:5P ð5Þin which α = universal constant; and P = fraction of time duringwhich a sediment particle is suspended by the flow. Note that thisrelation contains no critical shear stress. While for small wall shearstress P is negligibly small, for sufficiently large wall shear stress Pwill rapidly reach its saturation value. In the present application weconsider P simply as a constant and the erosion will occur just asthe shear stress exceeds its critical value.

Yalin (1977), however, deduced his approach for the determi-nation of the erosion rate from statistically averaged flow param-eters. If the critical shear stress is exceeded, particles are entrained.

The number of eroded particles rises linearly with the wall shearvelocity. The erosion rate is computed by

E ¼ αρsuτw ð6ÞThe constant α should be determined by experiments.

Yalin (1977) and van Rijn (1984a) assumed in their formula-tions that the number of eroded particles increases with increasingwall shear velocity. Alan and Kennedy (see e.g., Yalin 1985) intheir experiments demonstrated that the flow near the sedimentbed is fully saturated when a certain wall shear velocity is reached,and the erosion rate converges to a certain value and does not risefurther. With this in mind, only the approach of Einstein doesjustice to these observations.

Transport in Suspension and Sedimentation

If cohesionless particles are eroded from the sediment bed andtransported in suspension, the settling velocity of the floating sedi-ment particles is a substantial component regarding the behavior ofthe particle transport. For sediments with a mean diameter d of100–1;000 μm, the settling velocity ws can be computed by meansof the approach of Zanke (1977):

ws ¼10νd

��1þ 0:01sgd3

�0:5

�ð7Þ

which is directed toward the gravity. This expression was derivedby solving the balance between the gravitational force and the dragresistance for a particle in a fluid at rest. In its derivation empiricaldata for the drag coefficient were employed.

It is assumed that concentrations of suspended sediment remainsmall. The sediment particles are considered as passive tracer insuspension. The particle-fluid and particle-particle interactionsare not considered. In the Euler description, the particle transportin suspension can be computed by the concentration distributionby means of the advection-diffusion equation (Zedler and Street2001), i.e.,

∂c∂t þ

∂∂xj

�ðuj � wsδj2Þc�

νσ∂c∂xj

�¼ 0 ð8Þ

in which c = local suspended sediment concentration; σ = Schmidtnumber and is usually assumed to be 1; and uj (j ¼ 1, 2, 3) = flowvelocity components. The coordinates x1, x2, x3 are, respectively,toward the main flow, the vertical (against the gravity), andthe homogeneous transverse directions. The three coordinates(x1, x2, x3) are also often represented by (x, y, z).

For the investigated free-surface channel flow, on the above freesurface of the flow domain no mass exchange happens, hence theboundary condition

νσ∂c∂x2 þ wsc ¼ 0 ð9Þ

is valid. Through the sediment surface, i.e., the interface betweenthe sediment and fluid, the sediment exchange depends on the ero-sion rate E and the sedimentation rate S by the condition�

νσgradcþ wsce2

�· n ¼ Sðn · e2Þ �

ð10Þ

where n = unit normal vector of the sediment surface directedtoward the fluid domain; and e2 = unit vertical vector againstthe gravity.

The sedimentation rate S, in volume per unit horizontal area andtime, is determined by

S ¼ wsc0 ð11Þ

in which c0 = suspended sediment concentration in the immediatevicinity to the sediment surface.

Large Eddy Simulation of Turbulent Flows

The fluid flow above the sediment bed obeys the general conser-vation laws. For incompressible flows, they are the Navier-Stokesequations

∂ui∂xi ¼ 0 ð12Þ

∂ui∂t þ

∂uiuj∂xj ¼ ν

∂∂xj

�∂ui∂xj þ

∂uj∂xi

�� ∂P∂xi ð13Þ

in which P ¼ p=ρþ gx2, with p = hydrodynamic pressure.LES is a common technique for simulating turbulent flows and

is employed in the present work. The large, geometry-dependenteddies are explicitly solved in a flow calculation, while the small,statistically universal eddies are implicitly accounted for by using asubgrid-scale (SGS) model. Mathematically, the velocity field isseparated into a resolved and subgrid part, representing large andsmall eddies, respectively. The resolved field satisfies the filteredNavier-Stokes equations

∂�ui∂xi ¼ 0 ð14Þ

∂�ui∂t þ

∂�ui�uj∂xj ¼ ν

∂∂xj

�∂�ui∂xj þ

∂�uj∂xi

�� ∂ �P

∂xi þ∂τ ij∂xj ð15Þ

The effect of the small eddies on the resolved filtered field isincluded through the SGS model in the emerging SGS stressτ ij ¼ uiuj � �ui�uj.

The simplest and most commonly used SGS model was formu-lated by Smagorinsky (1963), being of the form

τ ij ¼ 2νt�Sij ð16Þ

where �Sij = rate-of-strain tensor for the resolved scale defined by

�Sij ¼12

�∂�ui∂xj þ

∂�uj∂xi

�ð17Þ

and νt = SGS turbulent viscosity. The prescription for νt writes

νt ¼ ðCsΔÞ2j�Sj, where j�Sj ¼ffiffiffiffiffiffiffiffiffiffiffiffi2�Sij�Sij

qrepresents the magnitude

of the resolved rate strain; Cs = nondimensional coefficient calledthe Smagorinsky constant; and Δ = model length scale and was setproportional to the local grid spacing. The major merits of theSmagorinsky model are its manageability, its computational stabil-ity and the simplicity of its formulation (involving only one ad-justed parameter). However, although this model is found to giveacceptable results in LES of homogeneous and isotropic turbu-lence, with Cs ≃ 0:17 according to Lilly (1967), it is too dissipativewith respect to the resolved motions in the near-wall region becauseof an excessive eddy-viscosity arising from the mean shear (Moinand Kim 1982). The eddy viscosity predicted by Smagorinskyis nonzero in laminar flow regions; the model introduces spuriousdissipation which damps the growth of small perturbations and thusrestrains the transition to turbulence (Piomelli and Zang 1991).

The limitations of the Smagorinsky model have led to theformulation of more general SGS models. The best known of

these newer models may be the dynamic SGS (DSGS) modelof Germano et al. (1991). The dynamic model evaluates theSmagorinsky constant (from the resolved motions) as the localerror minimization progresses (Germano et al. 1991), and thusavoids the need to specify a priori, or tune, the value of Cs. Thismethodology yields a coefficient Csðx; tÞ that varies with positionand time and vanishes near the boundary with the correct behavior(Piomelli 1993). The dynamic procedure thus greatly improves thecapability of the original Smagorinsky model.

Similarly, applying the filtering to the concentration Eq. (8),one obtains the following equation for the resolved sedimentconcentration �c

∂ �c∂t þ ð�ui � wsδj2Þ

∂ �c∂xi ¼

νσ∂2 �c∂x2i �

∂χi

∂xi ð18Þ

where χi = SGS turbulent flux of the sediment concentration,modeled by

χi ¼ � νtσt

∂ �c∂xi ð19Þ

where σt = turbulent Schmidt number.

Level-Set Method and Numerical Procedure

To track the movable sediment surface, the level-set methods,proposed first by Osher and Sethian (1988), may be suitabletechniques. Level-set methods rely on two central propositions:first, the embedding of the interface as the zero level set of a higherdimensional function, and second, the extension of the interfacevelocity to this higher dimensional level-set function.

Consider the function ϕðx; tÞ : R3 × ½0;∞Þ, defined so that thezero level set ϕ ¼ 0 corresponds to the evolving interface. Theequation for the evolution of ϕ corresponding to the motion ofthe interface is given by

∂ϕ∂t þ Fj∇ϕj ¼ 0 ð20Þ

where F = propagating velocity of the interface along its normaldirection. The implicit representation tracks all the level setsthroughout the entire computational domain, even though interestis only confined to the zero level set itself. Adalsteinsson andSethian (1999) introduced the idea of the narrow-band approach,which limits work to a thin region around the zero level setand hence saves calculation costs substantially. The implicit em-bedding inherent in the level-set approach means that the velocityF that transports the interface must be extended throughout thecomputational domain, i.e., Fext, not just on the interface itself.Even with the use of the narrow-band approach, one must be ableto construct an extension velocity that advances the neighboringlevel sets.

To maintain a numerically stable solution of the ϕ-equation, thesigned distance function is used to build ϕ outside the interface(ϕ ¼ 0), known as initialization, in which the Eikonal equationj∇ϕj ¼ 1 is solved. Using the fast marching method, simultaneousconstructions of a signed distance function and an extensionvelocity Fext can be combined and performed very rapidly (seeAdalsteinsson and Sethian (1999) and Sethian (1996) for details).

In the present case, the velocity of the sediment surface F isdetermined by the mechanism of sedimentation and erosion,which particularly is dependent of the local wall shear stress. If theinterface-parallel wall shear stress τw is obtained, the erosion rate E

can be computed by means of the approaches of van Rijn, Einstein,or Yalin, i.e.,

E ¼�erosion rate according to ð4Þ; ð5Þ or ð6Þ for τω ≥ τ c0 for τω ≥ τ c

ð21ÞThe propagating velocity of the sediment surface can be speci-

fied by

F ¼ S⊥ � E=ρs ¼ c0wsðe2 · nÞ � E=ρs ð22Þwhere the quantities S⊥ and E=ρs = erosion and sedimentationvelocities in the direction normal to the sediment surface,respectively.

For our numerical simulations, the numerical code FASTEST-3D (flow analysis solving transport equations simulation turbu-lence 3 Dimensional) (FASTEST 2001) is employed to solve thefiltered Navier-Stokes Eqs. (14) and (15) for the velocity field. Thecode was originally developed by the company Invent Computingand later significantly extended from the collaborative researchcenter (SFB568) at the Technische Universität Darmstadt duringthe last 10 years. The flow solver is based on the finite-volumediscretization method. Generally, it can use a nonorthogonal, block,structured grid with hexahedral control volumes. The level-setalgorithms for capturing the sediment surface and the filteredadvection-diffusion equation with a settling term Eq. (18) forthe sediment transport in suspension have been implemented inFASTEST-3D. The distance function property of the ϕ-field ismaintained by the reinitialisation procedure, by means of the ex-tension velocity method and the fast marching method. The timediscretization is performed using a second-order implicit Crank-Nicolson scheme. The spatial discretization is of second-ordercentral scheme. The coupled system of equations is solved inFASTEST-3D using a strongly implicit procedure (SIP) (Leisterand Perić 1994).

Under the assumption that the influence of the sediment concen-tration on the fluid flow is insignificant, for each time step, the flowfield is first calculated by evolving Eqs. (14) and (15). Then theobtained flow field and the turbulent viscosity are employed toEqs. (18) and (19), and the concentration field of sediment inthe fluid is calculated. By solving the level-set Eq. (20), the positionof the sediment surface, indicated by the zero level set ϕ ¼ 0, isupdated for each time step.

Numerical Results and Discussion

The migration of ripples can be described qualitatively as follows.As the flow accelerates on the upstream ripple slopes and the wallshear stress exceeds a critical value, it begins to pick up sediment.The sediment is transported into the flow as suspended sediment.Due to the ripple geometry, a thin circulation zone forms on the leeside of the ripple, extending over the entire distance from the crestto the trough. The sediment is transported partially by the flowabove the crest and partially by the reverse flow near the bottomof the circulation zone, to the lee side of the ripple crests, where thenear-bed sediment settles out as the flow loses its ability to entrain it(see Fig. 2). A general effect of these local processes is the migra-tion of ripples in the fluid flow direction.

The goal of this work is the numerical description of thesediment transport and the ripple migration with the help of thelevel-set method. The ripple movement is determined by a dynamicequilibrium between the erosion and the deposition of sedimentparticles, which are controlled in turn by the local wall shear stress

and the concentration distribution of the sediment in suspension.The ripple geometry and the fluid flow above it are coupled. Thiscoupling is of decisive importance for the computation of the ripplemigration.

The formation of the ripple structure is a complicated, sponta-neous process and will be investigated only shortly at the end of thiswork. For the present numerical simulation, the ripple contour ob-tained by Haslinger (1993) in laboratory experiments is employed,approximated by

¼ 1� 2:107 · sin

�1:15 · exp

�� 2πx

�· sin

�πxL

��ð23Þ

for x ∈ ½0; L� within one wavelength, as the initial ripple form,where K = height of the ripple. The ripple height is taken asK ¼ 0:015 m, corresponding to sand with a grain diameter ofapprox. 0.1 mm.

A reference simulation has been performed using LES for a flowbetween a fixed upper plane and an unformed sinusoidal sedimentsurface in the form of yh ¼ y0 þ A sinð2πx=λÞ, with y0 ¼ 0:005 m,A ¼ 0:015 m, and λ ¼ 0:15 m. A bed-fitted nonorthogonal meshand a Cartesian (orthogonal) mesh, respectively, were employed.The computational domain had a length of L ¼ 0:3 m, a meandepth of H ¼ 0:06 m, and a width of B ¼ 0:15 m. The Reynoldsnumber R ¼ 6;000, referring to the half-mean depth, was chosen.Test simulations demonstrated that with a spatial resolution ofNL × NH × NB ¼ 120 × 120 × 30, the convergent numerical solu-tion was reached for the bed-fitted nonorthogonal mesh with alogarithmic refinement in the vertical direction near the bottomsurface. The Cartesian mesh numerical simulations for the sameconfiguration have also been performed with a spatial resolutionof NL × NH × NB ¼ 120 × 200 × 30, in which the mesh in theregion near the bottom with a depth of 4 cm, including the wholesediment surface, was refined. The numerical results obtained byemploying both the meshes are displayed and compared in Fig. 3for statistically averaged velocity �u1 and turbulent stress componentu01u

02. The comparison shows a good agreement.The numerical simulation for the same case, but using a standard

RANS modeling including the k � ϵ closure model, was also per-formed and the numerical results were compared with the referencesimulation. A large deviation of the RANS results from those ofLES was demonstrated, in particular, in the flow field in the sep-aration zone and the concentration distribution near the sedimentsurface. This demonstrates that the RANS framework cannot ac-count for the most important coherent structures, which are three-dimensional time-evolving objects. The RANS modeling filters outanything of that sort. It is for this reason that in this work a LESmodeling is employed.

In the following simulations using the LES modeling, the near-wall region till the viscous sublayer was resolved and no wall

Fig. 2. Flow over ripple

model was employed. We examined the LES simulation for a less-fine resolution in the wall-near region, but when employing theWerner-Wengle wall model, numerical results became worse.

Although the numerical code FASTEST can use a nonorthog-onal mesh, it is difficult to employ a nonorthogonal mesh whichfollows the deformation of the ripples. In the following study, aCartesian mesh is employed for the rectangular basin investigated(including the fluid and the sediment parts). As demonstratedpreviously, the Cartesian mesh can also provide sufficiently exactnumerical solutions if the near-wall region is sufficiently resolvedby a suitable mesh refinement. The position of the interface, whichcuts the cells, is determined by the zero level set function. Theboundary conditions at the interface are given by the immersedboundary technique (Tseng and Ferziger 2003). In the followingcomputations, the initial sediment configuration Eq. (23) is chosen.The modeling domain has a dimension of L ¼ 0:15 m in x (flow)direction and y (vertical) direction as well as 0.075 m in z (span-wise) direction, which is resolved by 100 × 120 × 30 control ele-ments, see Fig. 4. Simulations with the doubled resolutions in allspatial directions were also performed. Comparison demonstratedno obvious difference. For the value of the time step, we chose theCourant number to be so small that a convergent simulation isreached. Such Courant numbers are usually much smaller than thatrequired by the numerical stability. It is often the case that theCourant number ≪ 0:1.

The initial ripple peak located at x ¼ 0:03 m. In the literature,the coordinate x0 with the origin at the ripple crest, for the presentcase x0 ¼ x� 0:03, or its dimensionless quantity x0=K is usuallyemployed near the ripple contour where the grid is refined. Theboundary conditions for the fluid are no slip at the bed (the ripple

-0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

orthogonal meshbed-fitted mesh

-0.003 -0.002 -0.001 0

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35

-0.0025 -0.0015 -0.0005

(a) (b)

(c) (d)

Fig. 3. Vertical profiles of (a), (c) statistically averaged velocity �u1 and (b), (d) turbulent stress component u01u02 (right panels) at two different

positions x ¼ 3λ=4 (above panels) and x ¼ λ (below panels), respectively; results obtained by a bed-following nonorthogonal mesh are comparedwith those by a Cartesian (orthogonal) mesh

0.040.06

0.080.1

Fig. 4. Computational domain and grid distribution with refinementnear the ripple surface

surface), free slip at the top, and periodic on all lateral boundaries.A mean flow velocity (or volume flow rate) is specified.

The level-set method is a numerical technique for trackinginterfaces between two mediums. The level-set functions are de-fined and solved in the whole domain, i.e., on the both sides ofthe investigated interface, although only the zero level set (i.e.,interface) possesses physical meaning. For this purpose, the move-ment of the sediment phase is formally calculated (e.g., even as aNewtonian fluid), but the corresponding results are of no physicalmeaning (in fact, the sediment motion is negligible).

In the numerical simulations, the specified initial ripple formwill first be maintained unchanged and only the fluid flow fieldis computed until a statistically steady flow state is reached. Thenthe convection-diffusion equation for the suspended sediment con-centration is turned on together with the bottom boundary condi-tions, allowing sediment to be entrained from the bottom and settle,but the ripple geometry would still be kept unchanged artificially.Finally, with the attained statistically quasi-steady flow field andthe suspended-sediment concentration field as the initial condi-tions, the ripple migration and deformation will be coupled bymeans of the level-set method.

Flow Field

Because the movement of the ripple surface is much slower than thefluid flow, the time scales for the fluid flow is much smaller thanthese of the bed deformation. For each instantaneous ripple contour.it is considered that a statistically quasi-steady flow is attained.Here, the statistically steady flow for the given, spatially fixed rip-ple geometry is investigated, which is qualitative representative ofthe real flow over the movable ripple. The numerical simulationsfor three different mean velocities vm ¼ 0:3, 0.4, and 0:5 ms� 1,respectively, are performed. For the typical water depth, H ¼0:15 m, and a typical value of the water viscosity, ν ¼8:90 × 10�7 m2 s�2, the Reynolds number R ¼ VmH=ν takesthe values, respectively, R ¼ ð5:06; 6:74; 8:43Þ × 104. The un-steady flow fields are calculated until the statistically steady statesare attained. The statistically averaged wall shear stress distribu-tions along the ripple are displayed in Fig. 5 for three test cases.This wall-parallel shear stress depends on the fluid viscosity andthe wall-normal gradient of the flow velocity. The latter is com-puted from the flow velocity at the nearest-wall grid point dividedby the normal distance to the ripple surface. The correspondingtransversely and temporally (statistically) averaged flow fieldtogether with the streamlines, only for vm ¼ 0:4 ms�1, is shownin Fig. 6, drawn for the domain near the sediment. As expected,with the increase of the fluid flow rate, the wall shear stress onthe ripple surface increases rapidly. Generally, during the overflowof a ripple structure a pressure-induced reverse flow (the recircu-lation zone) is formed on the lee side of the ripple with separationand reattachment points which can be identified from the positionswith τw ¼ 0. The influence of the fluid flow rate on the separationand reattachment points is negligibly small. The separation point isnearly at the ripple crest in x ¼ 0:03 m (x0=K ¼ 0). The reattach-ment point is located at x ¼ 0:1 m (x0=K ¼ 4:7), which is in goodagreement with the experimental data of van Mierlo and Ruiter(1988), x0=K ¼ 5, and Raudkivi (1963), x0=K ¼ 4:7. The geometryof the ripple is consequently responsible for the position of the re-attachment point of the recirculation zone, as observed by Vanoniand Hwang as well as Karahan und Peterson (Wallisch 1996). Theshear stress is positive and highest on the upslopes of the ripplecrests. At the ripple troughs, the stress is negative and weaker,sometimes entraining sediment in the opposite direction to the flow,as will be shown subsequently. The minimum (the negative maxi-mum) of the wall shear stress located at x ¼ 0:076 m (x0=K ¼ 3:1)

is approximately 70% of the length of the separation zone, whichis somewhat below the value of 80% from experiments. Fig. 6shows that the recirculation zone is divided from the main flow bya zero streamline. In the lower half of the zone, i.e., the separationzone, a reverse flow occurs.

Even for an almost statistically steady turbulent flow, the instan-taneous flow field also demonstrates full instationarity. The trans-versely, but not temporally, averaged flow fields of the statisticallysteady turbulent flow for vm ¼ 0:4 ms�1 are displayed in Fig. 7 forthree different time instants. In contrast to the averaged velocityfield (Fig. 6), several circulation zones may occur in the separationzone. It can be recognized that behind the crest vortices are formed,which move downstream. A further circulation zone exists at thetrough. Also, this is a rather transient motion and is not locallystationary, but is limited by the rise of the upslope in the flowdirection.

Concentration Distribution in Suspension

In this next step, a stationary ripple structure is still employed, butthe advection-diffusion equation for the suspended sediment isturned on, allowing erosion and deposition. Three pickup equations

0.6vm=0,3 m/svm=0,4 m/svm=0,5 m/s

0.03 0.06 0.09 0.12

Fig. 5. Wall shear stress along the ripple for three different values ofthe mean flow velocity (top panel) and the fixed and steady ripplecontour (bottom panel)

Fig. 6. Contours of statistically averaged steady velocity field (ms�1)and streamlines for vm ¼ 0:4 ms�1

by Yalin, Einstein, and van Rijn are employed, respectively, and thecorresponding numerical results are compared. The constants in theequations of Einstein and Yalin must be determined by experi-ments. In the context of this work, the constants determined byvan Rijn are used. The α value in the model of Einstein is set to7:7 × 10�3 and in the model of Yalin to 8:8 × 10�3. The modelingis on the basis of the assumption that the bed material consists ofsand with uniform grain diameter, here chosen as 100 μm. For sedi-ments with this grain diameter, we have the inequality ws=u� < 0:7.According to Breusers and Raudkivi (1991), it can be assumedthat the substantial transport takes place in suspension and nocohesive force arises. For such a fine-sand sediment, the criticalwall shear stress according to the Shields criterion takes the valueτ c ¼ 0:149 Nm�1. If the local shear stress on the ripple surfaceexceeds this threshold value, the entrainment of the sedimentparticles begins.

In Fig. 8, the spanwise-averaged sediment concentration distri-butions along the ripple for three different times are depicted, forwhich vm ¼ 0:4 ms�1 and the pickup equation of Yalin has beenemployed for the erosion rate. In general, the flow entrains and car-ries sediment from the upslopes of the ripples, and then transports it

across the ripple crests to the recirculation zones at the troughs.This forms a fingerlike shape in the statistically quasi-steady con-centration field [Fig. 8(c)]. Similar results were also obtained byZedler and Street (2001). For this quasi-steady concentration dis-tribution, the high-concentration zone ends roughly halfway overthe trough where it appears to begin to diffuse with some of thesediment settling and some of it being transported upward intothe flow. Some sediment is also entrained at the bottom of therecirculation zones in the flow-adverse direction. This transportpattern relies on the mechanisms for releasing sediment fromthe bottom boundary and subsequently entraining it upward intothe flow domain. Between the two erosion-dominant zones, thesedimentation is more dominant. This feature coincides with thewall-parallel shear stress distribution along the ripple surface, asdisplayed in Fig. 5. Comparing the concentration distributionsfor different times after the sediment is released (Fig. 8) also showsthat the total sediment in suspension increases with increasing timeat the beginning time interval.

Fig. 8 shows that the concentration fields strongly vary tempo-rally at the initial phase. For the modeling of the steady ripplemigration, a quasi-stationary concentration distribution is neces-sary. The time series of the whole-domain averaged concentrations(a measure of the total sediment in suspension) for the three pickupmodels are displayed in Fig. 9. After a sufficiently long time, aquasi-equilibrium between erosion and sedimentation is reached,

Fig. 7. Instantaneous velocity fields and corresponding streamlines forthree different time instants

Fig. 8. Instantaneous snapshots of contours of the transverselyaveraged suspended-sediment concentration along the ripples for(a) t ¼ 1 s, (b) t ¼ 2 s, and (c) t ¼ 30 s obtained by means of themodel of Yalin; at t ¼ 30, statistically quasi-steady flow and concen-tration fields are reached

for which the averaged sediment amount in suspension does notsubstantially change with time but exhibits some small superim-posed fluctuations. It can also be seen that the van Rijn modelshows a large deviation from the results of the other two pickupequations by Yalin and Einstein. With the van Rijn model substan-tially less sediment is entrained, which corresponds to a slowermigration of the ripples.

Migration of Ripples

In the previous computations, the statistically quasi-steady flowand concentration fields are simulated for a fixed ripple form. Themigration of the ripples can now be investigated with these fieldsas initial conditions.

The pickup equations employed above are based on a Shields-type criterion, releasing sediment only where the statistically aver-aged shear stress exceeds a critical value. One may note that it maybe impossible with such a constant threshold value of the wall shearstress to generate a periodic translation migration of a ripple struc-ture. In the range near the reattachment point of the separation zone(τw ¼ 0), in which the shear stress falls always below the criticalstress, an entrainment will not happen. In fact, this phenomena isthe case in the region near the reattachment point (x ¼ 0:1 m) at thetail of the separation zone (Fig. 5). However, to generate a trans-lational movement of the ripple and simultaneously keep its shapeunchanged, the entrainment in this region is necessary. A firstnumerical result demonstrates a much slower migration speed inthis range than there is elsewhere. This means an obvious defor-mation of the ripple structure instead of a uniform periodic migra-tion. Because of space limitations, corresponding results are notpresented here. For details see Kraft (2009)

To overcome this limitation, we employ the approach by Zanke(2001), in which the critical shear stress is determined by the in-ternal friction angle of the sediment and the turbulent fluctuation ofthe shear stress [Eq. (3)]. Fig. 10 shows the rms fluctuation of thewall shear stress along the ripple. It can be seen that the stress fluc-tuation is notably high in the range of the reattachment point andvery low on the lee side. It may be expected that with the Zankecriterion instead of the constant critical shear stress from Shields, abetter, quasi-uniform translational migration may be reached.

The Zanke criterion cannot be applied directly to the originalpickup equation of van Rijn [see Eq. (4)]. According to the Zanke

approach in Eq. (3), the mean critical wall shear stress τ c may takea zero value, so that the denominator in (4) becomes zero. For thisreason, in the application of van Rijn’s pickup equation for the criti-cal wall shear stress emerging in the numerator, the Zanke param-eter is used, but in the denominator still the critical wall shear stresswithout the consideration of the turbulent fluctuation is employed.We call this the modified equation of van Rijn. The pickup equationof Yalin, Eq. (6), is not related to the critical wall shear stress. Toapply the Zanke approach, we postulate a modified pickup equationof Yalin type of form

E ¼8<:

αρshβðτw�τ c;ZankeÞ

for τw ≥ τ c;Zanke

0 for τw < τ c;Zankeð24Þ

where β ¼ 2 is chosen so that the erosion rate for a typical wallstress, e.g., τw ∈ ð1:5τ c; 3τ cÞ, is comparable to that from the origi-nal Yalin model (Kraft 2009). Similar to the pickup equation of vanRjin, the modified equation of Yalin is related to the difference ofthe shear stress, and the critical value is taken according to Zanke.In the following, the modified equations of van Rijn and Yalin andthe equation of Einstein together with the critical shear stress ofZanke will be employed for simulating the pickup rate.

In Fig. 11, the calculated ripple contours by means of the Zankeapproach instead of the Shields parameter for a given time,t ¼ 96 s, are compared with the initial ripple geometry. Generally,all three pickup models generate a translation migration of ripples.However, with the modified van Rijn model, the ripples are flat-tening out considerably, whereas the Einstein model can best keepthe ripple shape unchanged, i.e., cause a periodic migration. Themodel of Einstein, in fact, provides the most plausible results.The flattening of the upstream slope of the ripples decreases par-ticularly with the use of the model of Einstein. A substantial dif-ference of the pickup equation from Einstein to the other pickupequations is the assumption that the erosion rate no longer changeswhen a saturated concentration near the sediment bed is reached.For fine sand, this condition can be reached very promptly becauseof the small critical wall shear stress. This assumption is includedonly in the approach of Einstein and seems to be substantial forthe simulation of the ripple migration. The migration velocities ob-tained by the pickup models of Yalin and Einstein are very close to

0.0002

0.0004

0.0006

0.0008

0.0012

0 10 20 30 40 50 60

Fig. 9. Time series of the whole domain averaged concentrations bymeans of the models of Yalin (1), Einstein (2), and van Rijn (3)

0.05 0.1

0.5τw’

Fig. 10. Root mean square fluctuation of the wall shear stress along theripple for vm ¼ 0:4 ms�1

each other. On the contrary, the pickup equation of van Rijn yields amuch smaller migration velocity.

The ripple migration velocities for different values of the meanflow velocity have been calculated and are displayed and comparedwith the experimental results by Brooks and by Dillo and Zanke(Kühlborn 1993) in Fig. 12. The numerical results and the exper-imental data demonstrate an empirical power-law relation betweenthe ripple migration speed vr and the middle flow rate vm byvr ¼ Avnm, although the values of the power n obtained may befairly different. The modified models of Yalin and Einstein yieldn≃ 4:1, which is in very good agreement with the experimentalresults of Brooks. This can be observed from the parallel linesfor both in Fig. 12. In comparison with the experimental data of

Brooks, the vr � vm line obtained by Yalin and Einstein has a lightshifting to the right, which implies a somewhat lower ripple migra-tion speed than that observed by Brooks. This may be because thegiven initial ripple geometry has still not achieved its stationarycondition subject to the present flow conditions, which may leadto a reduction of the ripple migration speed. The results obtainedwith the modified model of van Rijn agree very well with theobserved values of Dillo, corresponding to n≃ 5:3, i.e., vr ¼0:00003 ms�1 with vm ¼ 0:4 ms�1, and vr ¼ 0:0001 ms�1 withvm ¼ 0:5 ms�1.

In summary, the modified Yalin and Einstein models (especiallythe Einstein model) may be most suitable for describing the ripplemigration because they generate a translational movement of theripple and simultaneously keep its shape unchanged, and the ob-tained migration speeds are close to experimental data betweenthose obtained by Brooks and by Dillo and Zanke.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

t=0st=96s

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

t=0st=96s

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

t=0st=96s

Fig. 11. Ripple migration by means of the models of (a) Yalin,(b) Einstein, and (c) Rijn

10 100

Dillo and ZankeBrooks

our simulation (Einstein and Yalin)our simulation (van Rijn)

Fig. 12. Velocity of ripple migration versus mean flow velocity

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

t=0st=40.2s

Fig. 13. Initial and final ripple contours by means of the modifiedpickup model of Yalin for vm ¼ 0:5 ms�1

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

SimulationExperiment

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

Fig. 14. Comparison of formed ripple contours derived by means of(a) the modified Yalin model, (b) the Einstein model, and (c) the mod-ified van Rijn model with the experimental data of Haslinger (1993)

Formation of Ripple Structures

In the following, an initial bed geometry is given in form of asinusoidal wave

h0ðxÞ ¼ 0:075 sin

�2πxλ

�½m� ð25Þ

The ripple structure developing from it will be analyzed. The rippleformation is investigated on the basis of the three already knownpickup relations: the modified equations of Yalin and van Rijn aswell as the equation of Einstein. Haslinger (1993) shows that forsand with a grain diameter of approximately 0.1 mm, a rippleheight of approximately 1.5 cm and a ripple length of 15 cmmay be formed, as indicated in Eq. (23). Simulations for the sandsediment of a particle size of 0.1 mm are performed with a meanflow velocity of vm ¼ 0:5 ms�1. For all simulations, it is observedthat erosion happens on the upstream side and the crest and theentrained sediment moves toward the flow direction. As an exam-ple, the initial and final ripple contours are displayed in Fig. 13only for the modified pickup model of Yalin. An aspect ratio ofthe upstream slope to the lee side from the initial one-half to thefinal approximately two-thirds is formed (see Fig. 13). After a suf-ficiently long time, a quasi-identical final ripple contour is reachedwith all three pickup models (see Fig. 14).

Comparing the developed ripple structures with the experimen-tal data (Haslinger 1993) shows a very good agreement (seeFig. 14). The modified approach of van Rijn results in an increasederosion both on the upstream slope and in the zone of the crest. Thisleads to a smaller ripple height.

Concluding Remarks

In the present work, by means of the large eddy simulation, theturbulent free-surface channel flow and the suspended concentra-tion field over a ripple-shape bed are calculated. By using the dif-ferent pickup relations for the sediment erosion rate and theapproaches for the critical shear stress, the migration of ripplebed structure are investigated employing the level-set method. Withthe modified pickup equations, the obtained ripple migration speeddemonstrates a good agreement with known experimental results.It is also shown that, starting from a given initial bed geometry,a metastable ripple contour, which is coincident with the experi-mental data, is formed. In general, this work has demonstratedthe ability of large eddy simulation and the level-set method tomodel the time evolution of fluid flow field, sediment transportabove ripples, and the migration of ripples. The gained results aresupported by findings in the literature.

To correctly model the sediment transport and the ripple migra-tion, a special pickup equation for the erosion rate together with anadaptational critical stress for initiation of sediment entrainmentis decisive. The common pickup equations are on the basis of thestatistically averaged bottom shear stress. For turbulent flows, apickup model that may relate to the instantaneous bottom shearstress and/or the fluctuation of the turbulent shear stress may benecessary. The modification of the pickup equations is only the firstattempt in this way and needs to be further verified.

Acknowledgments

The writers are indebted to the Deutsche Forschungsgemeinschaft(DFG) for the financial support through the grant OB 96/17-1.

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Large Eddy Simulation of Sediment Deformation in a Turbulent Flow by Means of Level-Set Method

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