Direct numerical simulation of particulate flows with a fictitious domain method

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<ul><li><p>eol, Zh</p><p>s dooureyntionis dre abo</p><p>n nature and industrial appli-otioning buentlye owticulatparticlow-Redynam</p><p>2001; Aidun et al., 1998; Qi, 1999; Feng and Michaelides, 2005),the ctitious domain method (FD) (e.g. Glowinski et al., 1999,2001; Patankar et al., 2000; Yu et al., 2004; Hwang et al., 2004;Sharma and Patankar, 2005; Veeramani et al., 2007; Yu and Shao,2007), the immersed boundary (IB) method (e.g. Her and Sch-warzer, 2000; Uhlmann, 2005; Kim and Choi, 2006; Wang et al.,</p><p>2008), and the immersed interface method (e.g. Le et al., 2006;Xu and Wang, 2006). We note that the physalis method (Zhangand Prosperetti, 2005) is also a non-boundary-tted method, but</p><p>pointed by Glowinski et al. (1999). Glowinski et al. (1994a,b,1995, 1997) described the FD methods for the Dirichlet problemin which the boundary condition is enforced with the Lagrangemultiplier method, and they employed the methods to solve somedifferential equations and the incompressible viscous unsteadyows in complex or moving geometries. The Lagrange multiplierbased FD method was also used by Bertrand et al. (1997) and Tan-guy et al. (1996) to calculate the three-dimensional Stokes ows ofNewtonian and viscoplastic uids in a mixer. Glowinski et al.* Corresponding author. Tel.: +86 571 87951768; fax: +86 571 87951464.</p><p>International Journal of Multiphase Flow 36 (2010) 127134</p><p>Contents lists availab</p><p>l</p><p>l seE-mail address: yuzhaosheng@zju.edu.cn (Z. Yu).1988), the boundary element method (e.g. Phan-Thien et al.,1991) and the slender-body theory (e.g. Lin et al., 2003), theforce-coupling method (Lomholt et al., 2002; Lomholt and Maxey,2003), the physalis method (Zhang and Prosperetti, 2005) and sev-eral other boundary-tted and non-boundary-tted methods. Thetypical boundary-tted methods are the Arbitrary LagrangianEulerian (ALE) nite element method (Hu et al., 1992, 2001; Hu,1996) and the spacetime nite element method (Johnson andTezduyar, 1996). The non-boundary-tted methods include thelattice Boltzmann method (e.g. Ladd, 1994; Ladd and Verberg,</p><p>the uid ow is computed on a boundary-tted mesh and repeatedre-meshing and solution projection is normally required as theinterfaces move, whereas for the non-boundary-tted methods,the uid ow is computed on a stationary grid, thus eliminatingthe need for repeated re-meshing and projection (Joseph, 2002).As a result, the non-boundary-tted methods are, generally speak-ing, simpler and more efcient than the boundary-tted methods,in particular for the simulation of concentrated suspensions.</p><p>The ctitious domain (FD) method was initially developed tosolve partial differential equations in a complex geometry, as1. Introduction</p><p>Particulate ows are ubiquitous ications. Dynamic simulation of the mlate ows is capable of providmacroscopic information and conseqtant investigative tool for multiphasdynamic simulation methods for parposed in the literature: the point-methods (Crowe et al., 1998), the lbased methods such as Stokesian0301-9322/$ - see front matter 2009 Elsevier Ltd. Adoi:10.1016/j.ijmultiphaseow.2009.10.001of particles in particu-oth microscopic andhas become an impor-problems. A variety ofe ows have been pro-e-approximation-basedynolds-number-theory-ics (Brady and Bossis,</p><p>differs from others in that the general analytic solution of theStokes equations is used in the particle boundary region to transferthe no-slip condition from the particle surface to the adjacent gridnodes and compute the hydrodynamic force on the particle. Thistechnique has been shown to allow for accurate solution of partic-ulate ows up to a particle Reynolds number of one hundred oncoarse grids, and later adopted by Perrin and Hu (2008) in their ex-plicit nite difference scheme. For the boundary-tted methods,Review</p><p>Direct numerical simulation of particulat</p><p>Zhaosheng Yu *, Xueming ShaoDepartment of Mechanics, State Key Laboratory of Fluid Power Transmission and Contr</p><p>a r t i c l e i n f o</p><p>Article history:Received 10 October 2008Received in revised form 6 September 2009Accepted 19 October 2009Available online 27 October 2009</p><p>Keywords:Fictitious domainParticulate owsDirect numerical simulationNon-NewtonianNon-isothermal</p><p>a b s t r a c t</p><p>Our works on the ctitioureviewed, and particularlyow at moderately high Rbility to simulate the moand non-isothermal uidsreported in the literature amethod and the immersed</p><p>International Journa</p><p>journal homepage: www.ell rights reserved.ows with a ctitious domain method</p><p>ejiang University, 310027 Hangzhou, China</p><p>main method for the direct numerical simulation of particulate ows arerecent progresses in the simulations of the motion of particles in Poiseuilleolds numbers are reported. The method is briey described, and its capa-of spherical and non-spherical particles in Newtonian, non-Newtonian</p><p>emonstrated. In addition, the applications of the ctitious domain methodlso reviewed, and some comments on the features of the ctitious domainundary method are given.</p><p> 2009 Elsevier Ltd. All rights reserved.</p><p>le at ScienceDirect</p><p>of Multiphase Flow</p><p>vier .com/ locate / i jmulflow</p></li><li><p>of ctitious domain and immersed boundary refer to thenon-boundary-tted feature of the methods and any solid body</p><p>of M(1999) developed the distributed Lagrange multiplier based FDmethod (DLM/FD) to simulate particulate ows where the rigidparticles move freely. The key idea in this method is that the inte-rior domains of the particles are lled with the same uids as thesurroundings and the Lagrange multiplier (i.e. a pseudo body force)is introduced to enforce the interior (ctitious) uids to satisfy theconstraint of rigid-body motion (Glowinski et al., 1999). In theDLM/FD code of Glowinski et al. (2001), the nite element methodwas used to solve the uid equations, and a twice-coarser pressureelement with respect to the velocity element was adopted. Patan-kar et al. (2000) proposed a version of the DLM/FD technique thatimposed the deformation rate tensor in the particle domain equalto zero. Patankar (2001) developed a fast computation technique inwhich the ow elds were solved with a nite difference methodand the velocities of particle were obtained explicitly without iter-ation. This approach was later published by Sharma and Patankar(2005). Yu et al. (2002) substituted the Q1-P0 nite element meth-od for the twice-coarser pressure element based one, and Yu et al.(2004) further replaced the Q1-P0 nite element method with ahalf-staggered nite difference method for the solution of the owelds. Hwang et al. (2004) proposed a DLM/FD method for the sim-ulation of particle motion in a sliding bi-periodic box at low Rey-nolds numbers. Unlike other versions, where the DLM isdistributed in the solid domain and the operator-splitting schemeis adopted to simplify the computations, in the code of Hwanget al., the DLM is distributed only on the particle boundary, andthe whole system is solved all together which eliminates the split-ting error but increases the computational cost substantially. Shar-ma and Patankar (2004) proposed an iterative DLM scheme whichis suitable for low Re or quasi-Stokes simulations, and they appliedthis approach to the simulation of Brownian motion of particles.Veeramani et al. (2007) and Yu and Shao (2007) developed thenon-iterative DLM/FD schemes in which the particle velocitiesand the pseudo body force (Lagrange multiplier) were obtainedexplicitly, as in Sharma and Patankar (2005), but the derivedexpressions for the particle velocities and the computationalschemes for the ow problem were different for these three meth-ods. The ctitious domain method has been applied to various par-ticulate ows such as the sedimentation of 6400 circular particlesin a two-dimensional cavity in the RayleighTaylor instability (Panet al., 2001), the uidization of a bed of 1024 spherical particles(Pan et al., 2002), the sedimentation of a spherical particle in a ver-tical tube (Yu et al., 2004), the sedimentation of two spheroids (Panet al., 2005), the pattern formation of a rotating suspension of non-Brownian settling particles in a fully lled cylinder (Pan et al.,2007), the sedimentation of circular particles in viscoelastic uids(Singh et al., 2000; Yu et al., 2002), the sedimentation of sphericalparticles in viscoelastic uids (Singh and Joseph, 2000), shear-thinning uids (Yu et al., 2006a) and Bingham uids (Yu andWachs, 2007), the inertia-induced migration in a plane Poiseuilleow (Pan and Glowinski, 2002) and a circular Poiseuille ow(Yu et al., 2004; Pan and Glowinski, 2005; Shao et al., 2008), theparticle motion in a sliding bi-periodic frame (Hwang et al.,2004) and a planar elongational ow (Hwang and Hulsen, 2006),ow-induced crystallization of particle-lled polymers (Hwang etal., 2006), the rotation of a spheroid in a Couette ow (Yu et al.,2007a) and a pipe ow (Pan et al., 2008), the shear-induced migra-tion in a 2D circular Couette ow (Yu et al., 2007b), the heat trans-fer between particles and uids (Yu et al., 2006b), the motion ofoating particles (Singh and Joseph, 2005), and the electrophoresisof particles (Kadaksham et al., 2004). We note that the DLM/FDmethod has been extended by Baaijens (2001) and Yu (2005) tohandle the uid/elasticstructure interactions.</p><p>128 Z. Yu, X. Shao / International JournalCompared to the lattice Boltzmann method, the DLM/FD meth-od has an advantage of exibility in the sense that it can be easilyextended to any phenomenological equations such as the Poissonhas both volume (ctitious domain) and boundary (immersed).The non-boundary-tted methods can be classied into two fami-lies: the body-force-based methods and non-body-force-basedmethods, depending on whether or not a pseudo body force (ormomentum forcing) is introduced into the uid momentumequations to enforce the rigid-body motion constraint in the parti-cle domain or the no-slip condition on the particle boundary. All FDmethods mentioned above are the body-force-based methods. Inthe original DLM/FD methods, the Lagrange multiplier is deter-mined implicitly from the rigid-body motion constraint and solvedtogether with the particle and uid velocities. However, theLagrange multiplier can also be calculated explicitly from the frac-tional-step time scheme and some reasonable approximations, andit appears to lose its original sense of the Lagrange multiplier (Pat-ankar, 2001; Sharma and Patankar, 2005; Veeramani et al., 2007;Yu and Shao, 2007). The IB methods reported in the literature havebody-force-based and non-body-force-based versions. In the body-force-based version, the body force is calculated explicitly from thespring force scheme (Peskin, 2002; Her and Schwarzer, 2000) orthe direct-forcing scheme (Uhlmann, 2005; Kim and Choi, 2006).Many non-body-force-based IB methods have been proposed forthe solution of the ow in a complex geometry (Mittal and Iaccari-no, 2005), and in principle, these methods can be directly appliedto the uidparticle interactions since the hydrodynamic forceson the particles can be computed with the resolved ow elds.However, the body-force-based FD or IB methods have been pre-dominately used for the particulate ows. A probable reason is:the hydrodynamic forces on the particles are of most importanceto the simulation of the particulate suspensions, and it is difcultto obtain the accurate hydrodynamic forces via the computationof the uid stress on the particle boundary for the non-body-force-based IB methods unless ne mesh is used, whereasfor the body-force-based FD or IB method the hydrodynamic forcescan be determined from the integration of the pseudo bodyforce with reasonable accuracy even for coarse mesh, and further-more the explicit computation of the hydrodynamic forces isnormally not required to determine the particle velocities, whichmight be helpful to the robustness of the methods (Glowinski etal., 1999).</p><p>The primary aim of the present paper is to review our works onthe FD method. The rest of the paper is organized as follows: InSection 2, we describe briey our direct-forcing ctitious domain(DF/FD) method. In Section 3, we present a review of the applica-tions of our FD codes and some validations. The conclusions aregiven in the nal section.</p><p>2. Fictitious domain method</p><p>2.1. FD formulation</p><p>The DF/FD method is an improved version of our previous DLM/FD code. The DF/FD code is more efcient than the DLM/FD codesince the Lagrange multiplier and the particle velocities are ob-equation and the constitutive equations for a complex or non-Newtonian uid whereas the extension of the lattice Boltzmannmethod to a complex uid or the Poisson equation is much moreinvolved.</p><p>There is no essential difference between the so-called ctitiousdomain and immersed boundary methods particularly for theuidparticle (or structure) interaction problem since both terms</p><p>ultiphase Flow 36 (2010) 127134tained without iteration, and it has been demonstrated that theDF/FD method is equally accurate and robust as the DLM/FD meth-od (Yu and Shao, 2007). We briey describe the DF/FD method in</p></li><li><p>c c p</p><p>(area in case of two-dimensions) dened by Vp Vp=Ldc (d being</p><p>and the uid velocities un+1 at the Eulerian nodes are obtainedfrom:</p><p>un1 u Dtkn1 kn: 11In the above manipulations, the bi-linear (or tri-linear) function</p><p>is used to interpolate the uid velocity from the Eulerian nodes tothe Lagrangian nodes, and to distribute the pseudo body force fromthe Lagrangian nodes to the Eulerian nodes. The arrangements ofLagrangian points for circular, spherical and spheroidal particlesare depicted in Fig. 1. The Lagrangian (collocation) points are alsoused to compute the integral terms in (8) and (9) and the momentof inertia tensor (and possibly the mass center for particles ofunsymmetrical shape).</p><p>The articial repulsive force collision model (Glowinski et al.,1999), the lubrication model (Sierou and Brady, 2001) and thehard-sphere collision model (Crowe et al., 1998) have been imple-mented in our code (Yu et al., 2002, 2006b; Shao et al., 2008) toprevent the mutual penetration of particles. We notice that</p><p>tures, the settling velocities, the trajectories and the angular</p><p>of Multiphase Flow 36 (2010) 127134 129the dimensionality of the problem), and J the dimensionless mo-ment of inertia tensor dened by J J=qsLd2c The weak formula-tion of (1)(5) was derived by Glowinski et al. (1999).</p><p>2.2. Numerical schemes</p><p>A fractional-step time scheme is used to decouple the system(1)(5) into the following two sub-problems.</p><p>Fluid sub-problem for u*, p:</p><p>u unDt</p><p>r2u</p><p>2Rerp1</p><p>23u runu run1h i</p><p>r2un</p><p>2Rekn; 6</p><p>ru 0: 7A nite-difference-based projection method on a homogeneous</p><p>half-staggered grid is used for the solution of the above uid equa-tions (6) and (7). The diffusion problem (6) can be decomposedinto tri-diagonal systems with the ADI technique and the Poissonequation for the pressure arising from the projection scheme issolved with the FFT-based fast solver. All spatial derivatives arediscretized with the second-order central difference scheme.</p><p>Particle sub-problem for Un+1, xn+1:</p><p>qrVpUn1</p><p>Dt qr1 Vp</p><p>Un</p><p>DtFrg</p><p>g</p><p> ZP</p><p>u</p><p>Dtkn</p><p> dx;...</p></li></ul>

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