THEORETICAL & APPLIED MECHANICS LETTERS 3, 021007 (2013)

Numerical modeling of fluid-particle interaction in granular mediaJidong Zhao,a) and Tong ShanDepartment of Civil and Environmental Engineering, Hong Kong University of Science and Technology,Clearwater Bay, Kowloon, Hong Kong, China

(Received 6 December 2012; accepted 11 January 2013; published online 10 March 2013)

Abstract Fluid-particle interaction underpins important behavior of granular media. Particle-scalesimulation may help to provide key microscopic information governing the interaction and offerbetter understanding of granular media as a whole. This paper presents a coupled computationalfluid dynamics and discrete element method (CFD–DEM) approach for this purpose. The granularparticle system is modeled by DEM, while the fluid flow is simulated by solving the locally averagedNavier–Stokes equation with CFD. The coupling is considered by exchanging such interaction forcesas drag force and buoyancy force between the DEM and CFD. The approach is benchmarked by twoclassic geomechanics problems for which analytical solutions are available, and is further applied tothe prediction of sand heap formation in water through hopper flow. It is demonstrated that thekey characteristic of granular materials interacting with pore water can be successfully capturedby the proposed method. c⃝ 2013 The Chinese Society of Theoretical and Applied Mechanics.[doi:10.1063/2.1302107]

Keywords granular media, fluid-particle interaction, coupled CFD–DEM, single particle settling, 1Dconsolidation, sand pile, pressure dip

Fluid-particle interaction is accountable for a widerange of complex phenomena relevant to granular ma-terials. Pore fluid in a granular medium may fluctuateor flow under various forces, which could lead to parti-cle motion. This may sometimes be advantageous, suchas in the case of sand production in oil reservoir, or be-come adverse, like in the case of internal/surface erosionof embankment dams and soil slopes which is widely re-garded as the major trigger of instability and failureof these structures.1 While traditional continuum the-ories of porous media have been limited in many waystowards better understanding the interaction betweenfluid and particles, a particle-scale modeling approachcould provide quantitative characterization of the mi-croscopic behavior of the interaction and could facili-tate the establishment of general methods for reliablescale-up, design and control of different particulate sys-tems and processes.2 A number of collective attemptshave been made on the modeling of fluid-particle inter-action, among which discrete element method (DEM)plays a key role. In particular, numerical approachescombining the computational fluid dynamics (CFD) andDEM (CFD–DEM) prove to be advantageous over manyother options such as lattice-Boltzman and DEM cou-pling (LB–DEM) method and direct numerical simula-tion (DNS) coupled DEM (DNS–DEM) in terms of com-putational efficiency and numerical convenience.3 TheCFD–DEM method has gained some success in appli-cation to chemical and mining engineering and powderindustries.2 Meanwhile, there has been a growing inter-est in the community of geomechanics in exploring theparticle-level information on fluid-particle interaction,in sought for key mechanisms and mitigating measuresfor various geotechnical hazards such as landslide and

a)Corresponding author. Email: jzhao@ust.hk.

debris flow. With limited effort being made in this field,this paper aims to develop a coupled CFD–DEM numer-ical method to model problems relevant to geotechnicalengineering.

Following a typical solution procedure, we solve theNewton’s equations for the motion of the particle sys-tem by DEM, and the Navier–Stokes equation by CFDfor the fluid flow, by considering suitable coupling be-tween DEM and CFD.1,3,4 The key to the coupling be-tween CFD and DEM is a proper consideration of theparticle-fluid interaction forces. While the study is fo-cused on granular materials interacting with fluid withrelatively low Reynolds number, the interaction forcesbeing considered here include the drag force and buoy-ancy force only. The expression of drag force F d fora system of particles in fluid used by Tsuji et al.4 isemployed in this study

F d =1

8Cdρπd

2p

(U f −Up

) ∣∣U f −Up∣∣n1−χ, (1)

where dp is the diameter of the considered particle, Up

is the velocity of the particle, U f is the average veloc-ity of a fluid cell, ρ is an averaged fluid density de-fined by ρ = αρw + (1− α) ρa, α = vw/vc = 1 − va/vc,n = vvoid/vc = 1 − vp/vc. n defines the porosity (voidfraction), α is the volume fraction of water phase in acell, vw is water phase volume, va is air phase volume,vvoid is the total volume of air and water phase, vp isthe volume of particles occupied in a cell and vc is thetotal volume of a cell. Evidently, if α = 1, the cell willbe fully occupied by water. If α = 0, it indicates anall-air cell. If 0 < α < 1, it corresponds to either a celllocating at the air–water interface or there are particlesin it. Cd and χ depends on the Reynolds number of the

021007-2 J. D. Zhao, and T. Shan Theor. Appl. Mech. Lett. 3, 021007 (2013)

particle Rep

Cd =

(0.63 +

4.8√Rep

)2

,

χ = 3.7− 0.65 exp

[− (1.5− lgRep)

2

2

], (2)

where the particle Reynolds number is determined by1

Rep =nρdp

∣∣U f −Up∣∣

µ. (3)

The following expression is adopted for the buoy-ancy force

F b =1

6πρd3pg, (4)

where g is the gravity constant.In the coupling CFD–DEM scheme, the fluid phase

is discretized with a typical cell size five to ten times ofthe average particle diameter. At each time step, theDEM code employs a Hertzian contact law in conjunc-tion with Coulomb’s friction law and solves the New-ton’s motion equation of the system to provide suchinformation as the position and velocity of each indi-vidual particle. The positions of particles are matchedwith the fluid cell to calculate relevant information ofeach cell such as the porosity. By following the coarse-grid approximation method proposed by Tsuji et al.,4

the locally averaged Navier–Stockes equation is solvedby the CFD program for the averaged velocity and pres-sure for each cell by the InterDyMFoam solver. Theseobtained average values for velocity and pressure of acell are then used to determine the drag force and buoy-ancy force acting on the particles in that cell. Iterativeschemes have to be followed to ensure the convergenceof relevant quantities such as fluid velocity and pres-sure. When a converged result is obtained, informationof fluid-particle interaction forces will be taken into ac-count for the next step calculation of the DEM part.

Ideally, information on the interaction forces shouldbe exchanged once after each step of calculation forDEM or CFD. This, however, may be computationallyextremely expensive in practice. For the problems tobe treated in this paper, numerical experience indicatesfor each CFD computing step, exchange of informationafter 100 steps of DEM calculation will ensure sufficientaccuracy and efficiency. If the time steps for DEM andCFD are sufficiently small, more steps for DEM are nor-mally acceptable.

The CFD–DEM program has been first bench-marked by two classic geomechanics problems whereanalytical solutions are available: the single particlesettling problem and the one dimensional (1D) consol-idation problem. In the single particle settling prob-lem, a spherical particle of 1 mm diameter is droppedfrom 45 mm high from the center of a container withL×W×H = 20 mm × 10 mm × 50 mm. The container

0 0.1 0.2 0.3 0.4 0.5

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0

-0.1

t/s

Vel

oci

ty/(m

.s-

1)

Simulation

Analytical solution5

Fig. 1. Comparison of the CFD–DEM predicted particlesettling velocity with the analytical Stokes solution for aspherical single particle settling into water.

Fig. 2. Predicted dissipation of excess pore pressure byCFD–DEM for the 1D consolidation problem in comparisonwith Terzaghi’s analytical solution.

is half-filled with water. The settling velocity of the par-ticle will be of major interest. In Fig. 1, the predictedsettling velocity of the particle dropping from the airand settling into the water of the container is comparedwith the analytical solution by Stokes5. Evidently, thenumerical prediction captures well the sharp reductionof velocity when the particle hits the water surface andmakes entry into the water (t = 0.075 s) and bouncesback from the bottom of the container (t = 0.235 s). Italso agrees well with the analytical solution.

The second benchmark problem is the classical 1Dconsolidation problem in soil mechanics. Terzaghi6 hasderived a 1D consolidation theory governing the settle-ment and the dissipation of excess pore pressure in aone-way drained soil layer, which will be employed tobenchmark our numerical prediction. A soil column of100 equal size spheres saturated with water is consid-ered in the simulation of the 1D consolidation problem.All spheres are initially placed at the center line of thecolumn without any overlap. When they are emergedin water, the gravitational force and the buoyancy forceare switched on to allow the particles to settle to a hy-drostatic state. Once the initial consolidation is fin-

021007-3 Numerical modeling of fluid-particle interaction in granular media Theor. Appl. Mech. Lett. 3, 021007 (2013)

Fig. 3. Simulated process of formation of sand pile into awater container through hopper flow.

ished, a surface load is then applied at the top of thecolumn. The simulated dissipation process of normal-ized excess pore pressure (P/P0, where P0 is the surfacecharge pressure) across the height of the soil column zis presented in Fig. 2, in comparison with the analyticalsolution by Terzaghi’s theory. Tv in Fig. 2 denotes anormalized time used by Terzaghi.6 The overall predic-tions are in good agreement with the Terzaghi’s analyt-ical solution.

The CFD–DEM program has also been employed toinvestigate the formation of sand pile in water throughhopper flow. Figure 3 presents a snapshot of the simu-lated process of the hopper flow of particles into waterto form a sand pile. In simulating the sand pile problem,we have considered the rolling resistance between par-ticles as well. To ensure the formation of proper sandpile, in particular in the case without rolling friction, asmall round baffle is used to bound the receiving panelas can be seen from Fig. 3. The vertical pressure profilesat the base of the sand pile for the different cases arepresented in Fig. 4. In this figure, the vertical pressurehas been normalized by ρgH, where H is the pinnacleheight of the sand pile, g is times by the gravity accel-eration, and ρ is the average density of the sand pile.As can be seen, the overall profile of the pressure inthe wet cases is slightly higher than the dry cases. Thepresence of water, however, may moderately reduce thepressure dip as compared to the dry cases. Considera-tion of rolling resistance may lead to enhanced pressuredip, and the difference is more appreciable in the wetcases than in the dry cases.

Figure 5 further presents a comparison of the con-tact force network in the sand pile formed in the dryand wet cases (rolling resistance is considered in theboth cases). In the dry cases, appreciable inclined ori-entated strong force chains are observed to form an archaround the centre part of the sand pile which leads toa significant pressure dip. In the wet cases, the forcechains are more vertically oriented and this prevent theformation of a strong arch to shield the bottom centreparticles which evidently renders the pressure dip lessstrong as compared to the dry cases.

0 0.2 0.4 0.6 0.8 1.0

1.0

0.8

0.6

0.4

0.2

0

rtanφ/z

Norm

alized v

ert

ical st

ress

µr=0, dryµr=0, wetµr=0.1, dryµr=0.1, wet

µ=0.7

Fig. 4. Predicted vertical stress profile for dry and wetcases, with and without consideration of rolling resistance.

(a) The dry cases

(b) The wet cases

Contact force0.005 130.004 000.002 000.000 92

Fig. 5. Comparison of the contact force network in the sandpile.

In summary, a CFD–DEM method has been devel-oped to simulate the fluid-particle interaction in gran-ular materials. The method has been benchmarked bytwo classic geomechanics problems and has been fur-ther applied to the investigation of sand pile formationin water. The numerical results well demonstrate thecapability of the CFD–DEM program for particle-fluidsystem modeling.

This work was supported by the Research Grants Coun-

cil of Hong Kong (622910).

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