Liquid–Solid Flows Using Smoothed Particle Hydrodynamics and the Discrete Element Method

Powder Technology 116Ž. 2001 204–213 www.elsevier.comrlocaterpowtec

Liquid–solid flows using smoothed particle hydrodynamics and the discrete element method

Alexander V. Potapov a,b,1, Melany L. Hunt a,), Charles S. Campbell b a DiÕision of Engineering and Applied Science, California Institute of Technology, Mail Code 104-44, Pasadena, CA 91125, USA b Department of Mechanical Engineering, UniÕersity of Southern California, Los Angeles, CA 90089-1453, USA

Received 31 March 2000; received in revised form 21 September 2000; accepted 21 September 2000

Abstract

This study presents a computational method combining smoothed particle hydrodynamicsŽ. SPH and the discrete element method Ž.DEM to model flows containing a viscous fluid and macroscopic solid particles. The two-dimensional numerical simulations are validated by comparing the wake size, drag coefficient and local heat transfer for flow past a circular cylinder at Reynolds numbers near 100. The central focus of the work, however, is in computing flows of liquid–solid mixtures, such as the classic shear-cell experiments of Bagnold. Hence, the simulations were performed for neutrally buoyant particles contained between two plates for different solid fractions, fluid viscosities and shear rates. The tangential force resulting from the presence of particles shows an increasing dependence on the shear rate as observed in the Bagnold experiments. The normal force shows large variations with time, whose source is presently unclear but independent of the direct collisions between particles and the walls. q 2001 Elsevier Science B.V. All rights reserved.

Keywords: Liquid–solid flows; Smoothed particle hydrodynamics; Discrete element method

1. Introduction interactions. By contrast, numerical simulations for dry granular flows have been extremely useful in developing Flows of a solid–liquid mixture—such as slurries and and evaluating constitutive relations; the technique de- debris flows—are common in industrial and geophysical scribed herein may provide similar insights into multiphase settingswx 1,2 . These flows, however, present a challenge mixtures. for both experimental and numerical research. In an exper- The flows investigated in this study are generally de- iment involving particulates, standard single-phase instru- scribed by Reynolds numbers, Re, greater than 1, where s r mentation such as hot-wire anemometry or laser-Doppler Re rffUD m using r as the fluid density, U as the velocimetry can only be used to measure the velocity characteristic velocity, d as the particle diameter, and m as distributions if the flows are dilute and the particles are the dynamic fluid viscosity. The ratio of the densities of r small. Measurements of the variations in solids concentra- the solid to fluid phases Ž.rsfr typically ranges from 1 to tions are equally problematic, and only recently have 10, and the solid fraction, f, may approach that of a non-intrusive techniques, such as NMR, been introduced to packed bed. Hence, these flows are quite distinct from study these flows. Similarly, numerical simulations of dilute or low-Reynolds number suspensions, and simula- solid–liquid flows are formidable because of the complex tions of the flows must include the inertial effects of both geometries and the uncertainties regarding the fluid–solid the fluid and particle phases, as well as the effects of fluid viscosity. This work introduces a Lagrangian computational scheme to simulate this type of fluid–solid flow. The scheme combines two methods that can be found in the ) q q Corresponding author. Tel.: 1-626-395-4231; fax: 1-626-568- Ž. 2719. literature: the discrete element method DEM and E-mail address: [email protected]Ž. M.L. Hunt . smoothed particle hydrodynamicsŽ. SPH . DEM has been 1 Currently at Conveyor Dynamics, Bellingham, WA. used extensively to simulate dry granular flows that ne-

0032-5910r01r$ - see front matter q 2001 Elsevier Science B.V. All rights reserved. PII: S0032-5910Ž. 00 00395-8 A.V. PotapoÕ et al.rPowder Technology 116() 2001 204–213 205 glect any effects of the interstitial fluidwx 3–5 . The tech- material with varying thermal conductivity and compared nique is similar to molecular dynamicsŽ. MD studies, as the results with exact solutions. each particle within the DEM simulation is treated sepa- A coupling of the SPH and the DEM methods provides rately and followed as it moves. Newton’s second law a technique that differs from other approaches found in the describes the motion of each particle subject to all the literature for liquid–solid flows. Several research groups forces acting on it, including external fields such as gravity have used the DEM method to compute flows with a or through contacts with other particles. Each equation of viscous fluid by including drag effectswx 16,17 . The drag motion is integrated over a fixed time step to find the forces are computed by using an averaged equation for the particle’s velocity and position. The normal contact forces fluid motion that includes a fluid–solid coupling term. For are often modeled as a damped spring, although more dense beds, the coupling term resembles the Ergun equa- complex models have also been introducedwx 5,6 . In addition for packed beds; for dilute systems, the term is based tion, the angular motion of the particles is also determined on the drag for a single particle in an infinite fluidwx 16 . A by relating the rate of change of the angular velocity to the finite element technique was used by Feng et al.wx 18,19 , moment acting on the particle. These DEM simulations and Huwx 20 to calculate the motion of the fluid and solid have allowed researchers to study the details of the flow, phases. This approach requires an unstructured grid to be including variations of velocity, velocity fluctuations and updated with each time step. A grid was also used in the solid fractions, and have been valuable in developing finite-difference solver by Kalthof and Herrmannwx 21 ; the constitutive relations and transport properties for dry gran- interaction between the solid and the fluid phases was ular flowswx 7,8 . introduced by integrating the stress tensor at the particle Smoothed particle hydrodynamicsŽ. SPH is a meshless surface using a series expansion. The model compared Lagrangian computational technique that was introduced in well for flow around a single settling particle and for flow the astrophysics community to simulate the movement of around a fixed lattice of particles. Fluid–solid suspensions masses of material in an unbounded three-dimensional have also been simulated at finite Reynolds numbers using spacewx 9–11 . Each computational point or particle has an a Lattice–Boltzmann methodwx 22,23 . This technique has associated mass, momentum and energy. The movement of been extended to higher Reynolds numberŽ approximately each particle results from the integration of Newton’s 100. for flow through a fixed array of cylinderswx 24 . At second law, subject to the forces that are imposed on the low Reynolds numbers, the Stokesian dynamics technique particle. For astrophysics applications, the forces result has become a standard tool for investigating suspensions from pressure gradients or body forces acting on the wx25 . particles. However, viscous shear forces have also been This paper presents the initial results of an ongoing includedwx 11–13 . A property within the flow, such as the study to simulate liquid–solid flows including heat transfer density, is obtained by using an averaging or smoothing by combining DEM and SPH; this approach does not algorithm that integrates over neighboring particles. Super- require a computational mesh, nor does it rely on empirical ficially, the SPH algorithm looks similar to MD or DEM models to couple the liquid and solid phases. The method- because the momentum equations for every SPH particle ology of the technique is briefly described below, although are integrated numerically subject to forces applied by the the details can be found separately in either the literature neighboring particles. The major difference between SPH for DEM or SPH. The first half of the paper involves the and MD or DEM is that the inter-particle forces in SPH verification of the technique by simulating flow around a are derived from the Navier–Stokes equations instead of two-dimensional cylinder. The second half involves the particle interaction laws. simulation of a two-dimensional liquid–solid flow without The SPH technique has been used extensively for com- gravity involving tens of thousands of SPH particles in a pressible flow problems, but has recently been applied to shear flow and up to 18 solid particles that each contains nearly incompressible flows for several different applica- hundreds of SPH particles. tionswx 12–14 . Monaghan wx 12 used the technique to simulate free surfaces flows of water by ensuring that the speed of sound within the fluid is much larger than the speed of 2. The numerical model the bulk flow of material. Morris et al.wx 13 and Zhu et al. wx14 modeled low Reynolds number flows, and computed the two-dimensional flow field for a Couette flow, laminar 2.1. Smoothed particle hydrodynamics flow in a channel, flow past a cylinder in a periodic lattice, and flow through a square or hexagonal lattice of cylin- As outlined below, the basic ideas describing SPH ders. By introducing SPH boundary particles, the simula- follow from the review by Monaghanwx 11 . In SPH the tion satisfied the no-slip condition along the surface solid fluid is defined by a finite set of Lagrangian points situated boundaries. These flows were compared with known solu- in the computational domain. Strictly, these are interpola- tions. In addition, the recent computations by Cleary and tion points used in integrating the Navier–Stokes equa- Monaghanwx 15 solved the heat conduction equation for a tions, but it is useful from the point of view of physical 206 A.V. PotapoÕ et al.rPowder Technology 116() 2001 204–213 intuition to consider these as small pieces of fluid that Hence, while the mass of each SPH particle is fixed, the interact with their neighbors through pressure and viscous density associated with that particle changesŽ essentially forces. Each fluid particle at position ra interacts to vary- because the local volume associated with that mass X ing degrees with its neighbors at position r based on the changes. . Hence, SPH simulations are naturally compress- < y X < separation distance between particles ra r and the value ible. Since the focus of this work involves incompressible y X of the kernel function WŽra r ,h.. This kernel function fluid flow, an artificial equation of state can be introduced depends on the smoothing length, h, which is of the order that approximates incompressibility by ensuring that the of the initial separation distance of the SPH particles. A Mach number Ž MsUrc using c as the fluid sound typical form for the kernel is as follows: speed. of a fluid flow is low. The equation of state used in this simulation follows that of Morris et al.wx 13 , 1 <

mb where v , v , P , and P are the velocity and pressure AŽ.r f AW Žr yr ,h . Ž.3 ab a b ababÝ r b b corresponding to positions raband r . The viscous term P is approximated using the results of the work by r ab with mass mbb, density , and position r band the value of Morris et al.wx 13 and Cleary and Monaghan wx 15 , r A at position rbbis A . The ratio m bbr can be viewed as m Ž.Ž.m qm r yr P= W the area or volume associated with the fluid particle in two s ba ba b aab y P abŽ.Ž.v av b 8 or three-dimensional calculations. In this way, the local rrŽ.r yr 2 fluid properties are defined from a set of neighboring ab a b points situated in the computational domain. The gradients, where maband m are the dynamic viscosity of the fluid = A, are computed in terms of the gradients of the kernel corresponding to positions raband r . This formulation of function by differentiation of Eq.Ž. 3 , the viscous term is not based on the second derivative of the kernel, but estimates the viscous diffusion termŽ and mb =AŽ.r s A = W Ž.4 later the thermal diffusion term in the energy equation. abaabÝ r b b through an integral approximation. As noted above, SPH can be used to solve for the where = W is the gradient of WŽ.r yr ,h evaluated aab a b thermodynamic temperature Ž.T within the solid and the with respect to position r . The mass of each particle is a liquid phases. The energy equationŽ dTrdtsa= 2 T where calculated from the total mass within the computational a is thermal diffusivity of the solid or the fluid. is volume and the number of SPH particles. approximated in a manner similar to that used for the The continuity equationŽ. d rrdtsyr=Pv is written viscous term in the momentum equation. The resulting using the identity =Pvsw=PŽ.r v yvP=rxwrr 12,13 x . SPH formulation is as followswx 15 , The density of a fluid particle at a evolves as found in the following equation dTmŽ.Ž.r qr r yr P=W ababababsa y Ý Ž.TabT d r dt rrŽ.r yr 2 a s y P= b ab a b Ý mbaŽ.v v b aabW Ž.5 dt b Ž.9 A.V. PotapoÕ et al.rPowder Technology 116() 2001 204–213 207

Eq.Ž. 9 is used for both fluid and solid SPH particles with friction coefficient. Hence, as the particle surfaces move the only difference being the value of the thermal diffusiv- relative to one another in the direction tangential to the ity a. contact point, the tangential spring will load until the In this study, the quintic kernel function is used as tangential force reaches the friction coefficient times the suggested by Morris et al.wx 13 , normal forceŽ. frictional force ; then the surfaces slip relative to one another against this frictional force. °Ž.Ž.Ž.3y x 55y62y x q15 1y x 5,0F x-1; The computations depend on values for the spring 7 ~ Ž.Ž.3y x 55y62y x ,1F x-2; constants and the damping constant, which determine the WxŽ.s 478p h2 ¢ Ž.3y x 5 ,2F x-3; coefficient of restitution and the largest stable integration 0, xG3. time step. Results from the discrete element simulations Ž.10 have been compared with experiments involving vibrating beds and shear cells, which guided the choices for the X wx with xs

tion of the normal vector n has to be the same from both fluid and solid side. Hence,

T yTTyT s aos o b qn kfsk Ž.14 ddab

where kfsand k are the values of the thermal conductivity for fluid and solid. Eq.Ž. 14 is used to calculate To. Once this temperature is known, the integration of the energy equation across the boundary proceeds in the same manner as for the calculation of the solid–fluid interaction forces. Hence, an artificial extrapolated value of the temperature is Fig. 1. Illustration of the implementation of the solid–fluid no-slip prescribed to the point for the integration of the energy boundary condition. equation at the point. Since the temperature distribution within the solid can be solved, the same procedure is used to calculate the actual temperatures in the interior nodes. The computational algorithm is similar to that used in wx where voais the velocity of the point O, v is the velocity the DEM simulations 4,8 . The computation is initiated by of the fluid SPH particle a, daband d are the distances randomly placing solid particles into the domain of the from the points a and b to the line tangent to the fluid–solid flow. The SPH particles are initially configured in a regu- interface passing through the point O. To calculate the lar arrangement. If any of the solid particles are touching, solid–fluid interaction forces in Eq.Ž. 7 , this velocity v the DEM interaction force is calculated between the two bart is used instead of the actual velocity with which the solid particles. Then the pressure and viscous interaction particle is moving. The fluid–solid interaction force using forces acting on each SPH particle are calculated. Using an v is also added with the opposite sign to the right-hand explicit time-stepping method, the equations of motions bart side of the equation of motion for the solid particle, Eq. are integrated first to find the velocities and then to find Ž.7. the positions for both the solid particles and for the SPH At the fluid–solid boundary where the thermal parame- particles. The time step is chosen as the minimum of either ters change abruptly, the boundary conditions are formu- the time-step associated with the DEM or the SPH calcula- lated based on the fact that heat flux, qn , from the fluid tions. For the DEM calculations, the time step depends on side has to be equal to the heat flux from the solid side. the speed of sound in the solid that can be calculated from

Using Fig. 1, and taking Tab, T and T obe temperatures of the coefficient of restitution, the mass of the solid particle the SPH particles a Ž.Ž.fluid , b solid and temperature of the and the spring constant. For the fluid, there are two point O Ž.fluid–solid interface , the heat flux in the direc- characteristic time scales relating to the viscous scale,

Fig. 2. Wake behind a cylinder for Res30. This figure is a close-up view of the cylinder; the actual flow domain is larger. A.V. PotapoÕ et al.rPowder Technology 116() 2001 204–213 209

Reynolds numbers greater than 1. The simulation was performed with a cylinder diameter equal to 15 mean spacings between the fluid SPH particles. Fig. 2 is a snapshot of a portion of the computed flow field showing the cylinder and its wake. The cylinder is dragged at a constant velocity through an initially stagnant fluid. Each line represents the velocity vector at that position, with the tail of the vector at the physical location of the SPH particle. Periodic boundary conditions are applied in the direction of the flow, and the distance from the periodic boundary to the cylinder center was never less than 10D. No-slip boundary conditions were applied to the upper and lower walls that bound the Fig. 3. Comparison of the computed wake length behind the cylinder for different Reynolds numbersŽ. markers against the experimental results simulation. The ratio of the cylinder diameter to the wall Ž.solid linewx 28 . spacing was set to 0.12 for which experimental data is availablewx 27 . Approximately 37,500 SPH particles were used to compute the flow, and the Mach number is 0.03. h2rŽ.mrr , or the scale associated with the speed of Fig. 3 is a plot of the wake length as a function of sound, hrc. In the current work, the time scale associated Reynolds for Res10, 20, 30, and 40 compared against with the sound speed is smaller than the viscous time scale experimentswx 28 . The error bars in Fig. 3 correspond to because of magnitude of the Reynolds number, the Mach one characteristic kernel size h, which is the minimum number and the size of a solid particle compared to the distance to which the wake size can be determined. The smoothing length. computational data corresponds to the experimental data with acceptable degree of accuracyŽ the error in the wake length is below the kernel size. . Computations were also 3. Flow around a circular cylinder made of the drag at Reynolds numbers from 10 to 50 by computing the total force exerted on the cylinder; in The study by Morris et al.wx 13 used the SPH technique accord with experimental results in that range, the drag for the flow around a cylinder arranged in a periodic coefficient was found to change little with Reynolds num- lattice. The results showed that the SPH method could be ber and the values compared quantitatively with experi- used to predict the velocity contours and pressure profiles mental values. Similar computations were also conducted for Reynolds numbersŽ. based on cylinder diameter D of at Reynolds numbers from 100 to 200, which show a 0.03 and 1. In this study, the SPH technique is used to recirculating vortex bubble behind a cylinder and the simulate two-dimensional flows around a cylinder for beginning of vortex shedding. However, a very large simu-

Fig. 4. Comparison of the computational results for local Nusselt number as a function of position around the cylinder with the experiments of Eckert and Soehngenwx 29 . 210 A.V. PotapoÕ et al.rPowder Technology 116() 2001 204–213 lation volume had to be used in these computations mak- based on the measured values minus the stresses measured ing them very slow, and they were abandoned soon after in the absence of particles. For small values of Ba, the r 22r r 2r the start of shedding was observed. dimensionless stresses P Žlm rp D .Žand T lm 2 The next step in the model verification is a check of the rp D . are a linear function of Ba; this ‘macroviscous’ heat transfer calculations for flow around a heated circular regime is characterized by Ba-40. For large values of cylinder with a constant wall temperature as shown in Fig. Ba, the dimensionless stresses depend quadratically on Ba; 4. The simulations were set up exactly as described above. this ‘grain inertia’ regime corresponds to Ba)450. A s rw y - - The local value of the Nusselt number, Nu qD ŽTw transitional regime is defined for 40 Ba 450. x Tk`. f , was measured as a function of the angle calculated Later experimental studies viewed the dependence of from the front stagnation point of the flow around the the dimensionless values of stresses on Bagnold number as cylinder. The local heat flux, q, was computed from the overly simplisticwx 31 , because of the additional effects relation given for the boundary condition in Eq.Ž. 14 . such as fluid turbulence, wall roughness, and the dilation The comparison is for Reynolds numbers of 23 and 120 of the material in a flexible experimental apparatus. How- corresponding with the experimental conditionswx 29 , and ever, the experiments by Savage and McKeownwx 31 did there is good agreement with the exception of the area verify the slope of this dependence being between 1.0 and surrounding the forward stagnation point of the cylinder in 2.0 for 25-Ba-490. The only other study that measured the data for Res123, where the error is about 12%. This the normal stresses is by Prasad and Kytomaawx 32 , but discrepancy is due the resolution that relates to the limited they limited their experimental range to the macro-viscous number of SPH fluid particles, in a manner similar to the regime. There have been no numerical simulations of this limits imposed on the resolution by the finite cell size for problem. mesh-based CFD techniques. Near the forward stagnation The combined SPHrDEM simulation was used to in- point the thermal boundary layer is thin compared to the vestigate the two-dimensional flow between two parallel SPH particle spacing. By using a larger ratio of the shearing plates to examine the dependence of the stresses cylinder diameter to the mean SPH particle spacing, the on the Bagnold number. The ratio of the diameter of the accuracy in that region could be improved. solid particles to the gap spacing is 0.16 and the total number of SPH particles is approximately 18,000. There is no gravity, and the density of the solid matches that of the 4. Shear flows of particles fluid. A typical picture of the SPH particle velocities after running for a period of time is presented in Fig. 5. Each As described in the Introduction, the focus of the work SPH point is shown by the dark dot; the line originating is in developing a numerical scheme to compute fluid–solid from this point is proportional to the instantaneous velocity flows such as the experiments performed in 1954 by Bagnoldwx 30 . In these experiments, Bagnold measured the forces generated by a shear flow of large neutrally buoyant wax particles in an alcohol–glycerin–water mixture for solid fractions between 0.13 and 0.62. The experiment involved an annular shear cell in which the outer wall was rotated. The inner cylinder was made of rubber, which enabled the normal force to be measured with a manometer. Although the inner wall was deformable, Bagnold assumed that the total volume was fixed. The shear stress was determined from measurements of the torque on the inner cylinder. From these experiments, both the dimensionless normal and tangential stresses were found to depend on a single non-dimensional parameter that is now referred to as the Bagnold number, Ba, defined as Bas 1r2 2 r l r PpD g m, where g is the shear rate and r is the density of solid particles. The ratio of the grain diameter to the mean free dispersion distance, l, is a function of the solid fraction, f, and found from the following relation, sw r 1r2 y xy1 l Ž.foof 1 , using f as the maximum possible static solid fraction.Ž Note in Bagnold’s formulation sw r 1r3 y xy1 this appears as l Ž.fo f 1 ; the change from an exponent of 1r3to1r2 reflects the two-dimensional Fig. 5. Velocity vectors for fluid and solid SPH particles in a shear flow Ž.HrLs1 . The top wall moves to the right, while the bottom wall nature of the current simulations relative to Bagnold’s moves to the left. The case presented on the picture corresponds to solid three-dimensional experiments.. The stresses P and T are fraction of 37%, and a shear rate of g Drcs5.05=10y3. A.V. PotapoÕ et al.rPowder Technology 116() 2001 204–213 211 of the point with direction of the line coinciding with the method and the SPH tensile instability, as they are associ- direction of the instantaneous velocity. Periodic boundary ated with positiveŽ. compressive, not tensile pressures. In conditions are used on the left and right boundaries. the experiments of Prasad and Kytomaawx 32 , large varia- To simulate the solid walls on the top and bottom of the tions in the normal stress as a function of time were noted. computation domain, several rows of particles move with Because Bagnoldwx 30 used a manometer to measure the the same prescribed speed but in the opposite directions. normal stress, any variation with time would be obscured. These particles are used to enforce the no-slip condition The dimensionless tangential force minus the viscous along the solid boundaries. The SPH particles are initially force that would be measured without the presence of introduced in the hexagonal packing. The solid particles particles as a function of Bagnold number is shown in Fig. are then superimposed on the pattern of the SPH particles. 7. The solid lines in the figure are for a slope of 1.0 and The simulation runs for at least 12 ‘revolutions’ of the cell, 2.0. The simulation results suggest that the tangential and during this time the normal and tangential force on the stress depends on the Bagnold number to a power between walls are recorded. The shear flow simulations were per- 1.0 to about 2.0, which is consistent with the experiments formed for the range of fluid viscosity with two values of of Bagnoldwx 30 , and is also consistent with behavior the non-dimensional shear rate Ž.g Drc of 5.05=10y3 observed in the earlier experimental studies of Savage and and 5.05=10y4. In addition, solid area fractions of 17% McKeownwx 31 . Those studies observed that, when scaled and 37% were used corresponding to 8 or 17 solid parti- this way, the slope changed from 1 at small Bagnold cles. number to 2 at larger Bagnold numbers. This data appears From the simulations it was observed that the tangential to fit to a straight line with a slope of 1.34; however, the force is established quickly, but that the normal force on first three points could also be fit to a line of slope 1, while the plate varies considerably with time. A typical plot of the last four fit to a line with slope 2, consistent with the the forces on the plates versus non-dimensional time Ž.g t previous findings by Bagnoldwx 30 . is presented in Fig. 6. The figure shows that the tangential The analysis by Bagnoldwx 30 suggested that the normal force varies little after a non-dimensional time of approxi- stresses for the high-Bagnold-number grain-inertia regime mately 5, while the variations in the normal force are Ž.a quadratic dependence of the stresses on the strain rate greater than the average value. There is no clear explana- result from collisional impulses of particles; hence, the tion for the variations in the normal stress, especially as tangential stress is linearly proportional to the normal these impulses are not reflected in the tangential force. If stress in a manner similar to that of a frictional solid. The these variations represent collisions between solid particles simulations, however, suggest that the mechanism for the and the wall, large and simultaneous tangential forces generation of the two stresses may differ since the varia- might also be expected. Also for the same reason, these tions with time differ significantly. The first culprit to be variations are unlikely to be an artifact of the simulation considered is that fluid motion, and not particle motion,

r 22r Fig. 6. Dimensionless normal and shear force ŽF lm rp D . on the walls of the shear cell as a function of time normalized by the shear rate for the conditions presented in Fig. 5 at Bas15. 212 A.V. PotapoÕ et al.rPowder Technology 116() 2001 204–213

Fig. 7. Dimensionless time-averaged tangential stress on the walls of shear cell as a function of the Bagnold number, Ba. Flow 1: g Drcs5.05=10y3 , fs0.37; flow 2: g Drcs5.05=10y4 , fs0.37; flow 3: g Drcs5.05=10y3 , fs0.17. generates the normal stresses. Note, since the simulations This paper presents initial results of the combined and the experiments by Bagnold use particles with the DEMrSPH algorithm to compute liquid–solid flows. The same density as that of the fluid, it is impossible from simulations capture the physical dependencies observed in scaling to distinguish between the contributions of the experiments. Additional work is needed to determine the fluid and solid inertia to the stresses. One possible source effect of the Mach number, the kernel function and size on of normal stresses is the acoustic impulse observed by the overall calculations. However, the initial calculations Zenit and Huntwx 33 . Hopefully, continued research with are promising, and indicate that the DEMrSPH method this model will lead to an understanding of this phe- may be an effective tool for modeling liquid–solid flows nomenon. for Reynolds numbers greater than 1. Ongoing work in- The SPH model breaks down when the solid particle cludes measurements of the variations in solid fraction, separation becomes smaller than a kernel width. The simu- velocity and velocity fluctuations across the gap of the lation technique should be patched in some manner to cell. In addition, future studies will also include calcula- handle this region. However, for the current results, no tions of the heat transfer across a gap in which the such correction has been made to the model. bounding surfaces are maintained at different temperatures.

5. Conclusions Acknowledgements

This paper presents a two-dimensional Lagrangian com- This work was partially supported by grants from the puter model to simulate fluid flow and heat transfer for a National Science FoundationŽ. CTS-9530357 and National solid–fluid medium. The motion of the fluid is simulated Aeronautics and Space AdministrationŽ. NAG3-2358 . using the smoothed particle hydrodynamics approach, while the solid particles are simulated by discrete element technique. The technique is coupled through the no-slip bound- References ary conditions applied to the boundaries and the solid surfaces of the particles. The technique was verified by wx1 R.M. Iverson, The physics of debris flows, Rev. Geophys. 35Ž. 1997 comparing with existing experimental results for the drag, 245–296. size of the wake and the heat transfer for flow around a wx2 C.A. Shook, M.C. Roco, Slurry Flow: Principles and Practices, circular cylinder. Butterworth–Heinemann, Reed Publishing, Boston, 1991. wx3 P.A. Cundall, O.D.L. Strack, A discrete numerical model for granu- The focus of the study, however, involves simulations lar assemblies, Geotechnique 29Ž. 1979 47–65. of a shear flow of neutrally buoyant particles. A compari- wx4 C.S. Campbell, Computer simulation of powder flows, in: K. Gotoh, son with the experiments of Bagnold show that the tangen- H. Masuda, K. HigashitaniŽ. Eds. , Powder Technology Handbook tial stress increases with the Bagnold number to a power vols. 777–794, Marcell Dekker, New York, 1997, 2nd edn. wx between 1 and 2. The normal stresses, however, vary 5 O.R. Walton, R.L. Braun, Viscosity, granular-temperature and stress calculations for shearing assemblies of inelastic, frictional disks, J. considerably with time. The reason for this is unclear, Rheol. 30Ž. 1986 949–980. although earlier measurements also notice large variations wx6 C. Thornton, K.K. Yin, Impact of elastic spheres with and without in the normal stress measurements. adhesion, Power Technol. 65Ž. 1991 153–165. A.V. PotapoÕ et al.rPowder Technology 116() 2001 204–213 213

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