Fluid flows containing solid particles are frequently encountered in industrial processes. Some examples are oil and gas pipelines and wells, pneumatic conveying, reactors, sedimentation and fluidization. Traditionally two modeling techniques for particle-fluid flows, known as Eulerian-Eulerian (E-E) and Eulerian-Lagrangian (E-L) have been used (see e.g. [1]). The E-E approach (see e.g. [2, 3]) treats the particulate phase as a continuous phase which penetrates and interacts with the fluid phase. Two sets of conservation equations are solved for fluid and particle phases; they are coupled by heat and momentum transfer. In the E-L technique (see e.g. [4, 5]) the fluid phase motion is calculated by solving continuity and momentum equations, whereas particles are treated as points and their motion calculated by tracking them integrating Newton's second law. By this method, particle collisions, which are crucial in dense flows, can be taken into account. The subject of this paper is a third approach, which is becoming feasible due to the increase of computational power. We call it "direct numerical simulation". Direct numerical simulation has recently become more popular because it offers the prospect of studying phenomena and mechanisms on the particle scale directly (see e.g. Kurose and Komori [6-11]). In this approach, fluid flow is calculated by solving the Navier-Stokes equations directly on a computational grid, which is fine relative to the particles. Hydrodynamic forces can be determined from the pressure distribution on the particle surfaces. Based on this simulation method which may require remeshing during calculations see e.g. [12-18]) or not (see e.g. [19-21]), a discussion on grid influence will be carried out. The finite element formulation with moving boundaries and interfaces has been used for calculations with remeshing, whereas Lagrange multiplier-based fictitious domain method has been used in [20-22]. Moreover, in [12-15, 17, 18] the gap between particles and particles with walls is modeled as film of liquid consisting of three [12-15] or one [17, 18] layers of finite elements. Additionally, a description on the techniques for treating surface boundaries and interstitial fluid between particles when they come to a collision will be discussed.

References:

[1] Crowe, C., Sommerfeld, M., Tsuji, Y., 1998. Multiphase Flows with Droplets and Particles. CRC Press

[2] Slater, S.A., Young, J.B., 2001. The calculation of inertial particle transport in dilute gas-particle flows. Int. J Multiphase Flow 27, 61-87

[3] Slater, S.A., Leeming, A.D., Young, J.B., 2003. Particle deposition from two-dimensional turbulent gas flows. Int. J. Multiphase Flow 29, 721-750

[4] Sommerfeld, M., Huber, N., 1999. Experimental analysis and modelling of particle-wall collisions. Int. J Multiphase Flow 25, 1457-1489

[5] Patankar, N.A., Joseph, D.D., 2001. Lagrangian numerical simulation of particulate flows. Int. J. Multiphase Flow 27, 1685-1706

[6] Kurose, R., Komori, S., 1999. Drag and lift on a rotating sphere in a linear shear flow. J. Fluid Mechanics 384, 183-206

[7] Johnson, T.A., Patel, V.C., 1999. Flow past a sphere up to a Reynolds number of 300. J. Fluid Mechanics 378, 19-70 [8] Kim, D.J., Choi, H., 2002. Laminar flow past a sphere rotating in the streamwise direction. J. Fluid Mechanics 461, 365-386

[9] Tsuji, T., Narutomi, R., Yokomine, T., Ebara, S., Shimizu, A., 2003. Unsteady three dimensional simulation of interactions between flow and two particles. Int. J. Multiphase Flow 29, 1431-1450

[10] Cate, A.T. , Derksen, J.J., Portela, L.M., van den Akker, H.E.A , 2004. Fully resolved simulations of colliding monodisperse spheres in forced isotropic turbulence. J. Fluid Mechanics 519, 33-271

[11] Cho, S.H., Choi, H.G., Yoo, J.Y., 2005. Direct numerical simulation of fluid flow laden with many particles. Int. J. Multiphase Flow 31, 435-451

[12] Johnson, A.A., Tezduyar T.E., 1996. Simulation of multiple falling in a liquid-filled tube. Comput. Methods Appl. Mech. Engrg. 134, 351-373

[13] Johnson, A.A., Tezduyar,T.E.,1997. 3D simulation of fluid-particle interactions with the number of particles reaching 100. Comput. Methods Appl. Mech. Engrg. 145, 301-321

[14] Johnson, A., Tezduyar, T., 2001. Methods for 3D computation of fluid-object interactions in spatially periodic flows. Comput. Methods Appl. Mech. Engrg. 190, 3201-3221

[15] Behr, M., Tezduyar, T., 2001. Shear-slip mesh update in 3D computation of complex flow problems with rotating mechanical components. Comput. Methods Appl. Mech. Engrg. 190, 3189-3200

[16] Patankar, N.A., Huang, P.Y., Ko, T., Joseph, D.D., 2001. Lift-off of a single in Newtonian and visco-elastic fluids by direct numerical simulation. J. Fluid Mechanics 438, 67-100

[17] Hu, H. H., 1996. Direct simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 22, 335-352

[18] Hu, H.H, Patankar, N.A., Zhu, M.Y., 2001. Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian-Eulerian technique. J. Computational Physics 169, 427-462

[19] Glowinski, R., Pan, T.W., Hesla, T.I., Joseph, D.D., 1999. A distributed Lagrange multiplier/fictitious domain method for particulate flows. Int. J. Multiphase Flow 25, 755-794

[20] Pan, T.W., Joseph, D.D., Bai, R., Glowinski, R., Sarin, V., 2002. Fluidization of 1024 spheres simulation and experiment. J. Fluid Mechanics 451, 169-191

[21] Pan, T.W., Glowinski, R., 2002. Direct simulation of the motion of neutrally buoyant circular cylinders in plane Poiseuille flow. J. Computational Physics 181, 260-279