Liquid–Solid Flows Using Smoothed Particle Hydrodynamics and the Discrete Element Method
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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 addi- tion 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 simu- late 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.