We proposed a direct numerical simulation model of granular soils and seepage flow by combining the discrete element method and the lattice Boltzmann method. The MRT model was introduced in order to obtain stable solutions of fluid flow under high Reynolds number condition. The PS model, which retains a local operation at each fluid node and keep from intensive increasing the computational costs for the calculation of collision term, was also introduced as a solid-fluid coupled model. We show the effectiveness of the PS-MRT lattice Boltzmann model through several validation tests

Fukumoto, Y.

Murakami, A.

Ohtsuka, S.

English

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PS-MRT Lattice Boltzmann model for direct simulation of granular soils and seepage flow

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537PS-MRT Lattice Boltzmann model for direct simulation of granular soils and seepage flowIV International Conference on Particle-based Methods – Fundamentals and Applications PARTICLES 2015 E. Oñate, M. Bischoff, D.R.J. Owen, P. Wriggers & T. Zohdi (Eds)     PS-MRT LATTICE BOLTZMANN MODEL FOR DIRECT SIMULATION OF GRANULAR SOILS AND SEEPAGE FLOW Y. FUKUMOTO ¹, S. OHTSUKA ² AND A. MURAKAMI ³  ¹Nagaoka University of Technology 1603-1, Kami-tomioka, Nagaoka-shi, Niigata 940-2188, Japan yfukumoto@vos.nagaokaut.ac.jp  ² Nagaoka University of Technology 1603-1, Kami-tomioka, Nagaoka-shi, Niigata 940-2188, Japan ohtsuka@vos.nagaokaut.ac.jp  ³ Kyoto University Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan akiram@kais.kyoto-u.ac.jp  Key words: Multi-phase Flow, Seepage Failure, DEM, LBM. Abstract. We proposed a direct numerical simulation model of granular soils and seepage flow by combining the discrete element method and the lattice Boltzmann method. The MRT model was introduced in order to obtain stable solutions of fluid flow under high Reynolds number condition. The PS model, which retains a local operation at each fluid node and keep from intensive increasing the computational costs for the calculation of collision term, was also introduced as a solid-fluid coupled model. We show the effectiveness of the PS-MRT lattice Boltzmann model through several validation tests.  1 INTRODUCTION The solid particle-fluid multiphase flow and its multi-physics phenomena can be found in a lot of scientific fields: fluidization in chemical engineering, transport of blood cell in bio-engineering, sedimentation and erosion in environmental sciences, sand production in resource engineering and so on. These physical phenomena are not well understood because of the wide variety and the complexity of the particle-fluid or the particle-particle interactions. The particle-fluid systems are often experimentally observed by using an X-ray tomography [1] [2] and a high speed camera [3], but observation capacity and available information at particle-resolution by using such apparatuses are still limited. In addition to experimental approaches, the development of numerical simulation can help our understanding of such complex particle-fluid systems. Because of the importance of capturing properly the interactions between particle and fluid, micro-scale numerical method which can deal with the fluid flow at less particle scale is needed. In contrast, macro-scale methods have less computationally load and are suitable for an industrial application, but require a local averaging which loses the essential details of the fluid flow. The former type of the direct methods is intently improved and new findings are obtained mainly in research fields such as chemical engineering, bio-engineering and soft matter physics. 538Y. Fukumoto, S. Ohtsuka and A. Murakami  2 In geo-mechanics, a particle-fluid system also exists in the form of solid particles and pore liquids or gases, which are characteristic of non-Brownian and highly concentrated suspensions. A deep understanding of such a system is a key to predict and control the various phenomena, such as sand boiling, weathering of rocks, internal erosion and liquefaction of foundation. However, compared to other research fields, there are still a few 3-D applications of a direct particle-fluid solution in geotechnical engineering or civil engineering, despite the importance of consideration of the particle-fluid interaction. The 3-D condition is especially important for the soil structure because an additional special treatment to handle the zero permeability is required for a 2-D congested granular system. In this study, we proposed an effective direct simulation method in three dimensions for both soils and seepage flow by using the DEM (Discrete Element Method) and the LBM (Lattice Boltzmann Method).  Firstly, in order to stabilize the flow analysis and to improve the accuracy of the non-slip boundary condition, the MRT (Multiple Relaxation Time) model [4] was introduced into the LBGK equation, which is also called the SRT (Single Relaxation Time) model and is standard solution of the non-compressible fluids. Secondlly, PS (Partially Saturated) model [5] was chosen as a solution of the moving boundary, which can maintain the inherent parallel nature of the lattice Boltzmann equation. These two LB models and the DEM were combined, and the validation of the PS-MRT LB model was performed through several types of simulations.   2 MRT LB MODEL FOR SEEPAGE FLOW The lattice Boltzmann method is one of the CFD methods and an alternative to the N-S equation. In the LBM, the velocity moments of the virtual fluid particles, f, having finite directions are placed at each node, and the behaviors of them are governed by a propagation phase and a collision phase from node to node. When the velocity moments have Q directions, f is defined as {f | =0,1,…Q-1}, where  is the number of the discrete velocities depending on the choice of the model for the velocity moment. With the consideration of the precision and the numerical efficiency, the D3Q19 model for the three dimensions is used in this study. The solution of the LBM is governed by the following lattice Boltzmann equation.       tttttt  GxΩxfcxf  ,,, . (1) The value for x is the position of the node which is being calculated, t is the time, t is the discrete time and x is the grid space. c is the grid velocity, which is calculated by x/t. One of the right term for  indicates the collision operator and G indicates the forcing term. For the SRT (single relaxation time) LB model [6], which is a standard solution of the LBM,  is given as       teqtt ,,1, xfxfIxΩ   , (2) where  is the relaxation time coefficient, I is the identity matrix and feq is the equilibrium distribution function in the velocity space. The kinetic viscosity of the fluid  is calculated by c2t (-0.5)/3.  Here, it is well known that the SRT model often results in numerical instability when 539Y. Fukumoto, S. Ohtsuka and A. Murakami  3 fluids have relatively low viscosities. Especially, serious instability problemes occur in three dimensional flows in the case where physics may not be adequately resolved due to compurtational constraints. In order to obtain the numerical stability, we introduce the MRT (multiple relaxation time) model  [4] as an alternative to the SRT model. In the MRT model, by choosing different time scales to represent changes in the various physical processes due  to collisions, the stability of the LBM can be significantly improved. The collision operator for the MRT model is as follows.       teqtt ,,ˆ1, xmxmSMxΩ  , (3) where M is a Q×Q matrix which linearly transform the velocity space, f, to the moment space, m: f = M-1 m, m = M f.  Ŝ is the diagonal matrix where Ŝ = diag (0, s1, s2, , s4, 0, s4, 0, s4, s9, s2, s9, s2, s9, s9, s9, s9, s16, s16, s16). meq is the equilibrium distribution function in the moment space. It should be noted here that the value for s9 is equal to , and other diagonal components of Ŝ is arbitrarily-assigned.  3 PS-MRT LB MODEL FOR PARTICLE-FLUID COUPLING In order to perform the coupled particle-fluid simulations in the framework of the LBM, the PS (partially saturated) model allowing momentum transfer inside the solid phase, whose concept is originally suggested by Noble and Torczynski [5], is used. This model enables to deal with moving solid-liquid boundary and to calculate the hydrodynamic force acting on the solid obstacle. Compared to other coupled model [7] [8] [9], this method has advantages of being able to retain a local operation at each node and of keeping from intensive increasing the computational costs for the calculation of collision term. In this approach, the collision operator in the SRT model as shown in Equation (1) is reformulated to account for the solid fraction at a node, B, and the additional term, Λ .           tteqtt ,ˆ,,ˆ, xΛBxfxfBIxΩ  , (4) where B̂  is the diagonal matrix whose diagonal components are B and Λ is the additional term considering the velocities of the nodes inside the solid phase. Then, the value for Λ , which is the -th component of Λ , is given as follows.  ),,(),(),(),( tffftf peqeq xuux      (5) where up is the velocity of the solid including both translation and rotation motion of the corresponding particle. The notation - is the opposite direction of . According to above equations, when | up | = 0 and B = 1, the bounce-back rule, i.e., the non-slip condition at solid-fluid boundary is obtained. When the PS model is introduced into the MRT model, the collision operator is given as the following equation.           tteqtt ,ˆ,,ˆ1ˆ, xΛBxmxmSMBIxΩ  . (6) We call this equation PS-MRT LB model in this study. Equation (6) also retains a local 540Y. Fukumoto, S. Ohtsuka and A. Murakami  4 operation at each node in the same as Equation (4). For this reason, it is easy to parallelize the LB solution in the frame of PS-MRT LB model.  4 COLLISION LAW AND MOTION OF SOIL PARTICLE The handling of the collision law and the motion of the soil particle are presented in this subsection. The collision law for a contact force Fcon and a contact torque Tcon is governed by the DEM, where the contact logic is followed by the Voigt model. The normal repulsive force is assumed to be proportional to the overlap distance, and a dissipative component is set to be proportional to the relative normal velocity between particles. For the calculation of the tangential force, in addition to the same way as the normal force, the Coulomb law of friction is also considered. The value for Fcon and Tcon are given by summing up the change of momentum inside the solid phase All of the calculations in the following subsections are performed on the graphic processing unit (NVIDIA GeForce GTX TITAN). Parallelized algorithm for the DEM suggested by Nishiura and Sakaguchi [10] is incorporated into the coupled particle-fluid code. Detailed parallelization methods relative to the coupling scheme on a many core architecture is not discussed in this literature.  5 DRAG FORCE ON SOIL PARTCLE At the first step in the validation of the PS-MRT LB model, we simulate the obstacle flow, as shown in Figure 1, and obtain the correlation between drag force coefficient CD and Reynolds number Re. Fluid flows are generated in positive direction of y-coordinate by imposing a constant velocity boundary condition on one side of zx plane, while the other side of zx plane is free outflow boundary. Grid space x is 1.0×10-5 m and the system size is 150x ×300x ×150x . Diameter of the sphere d is 30x. The density of the fluid f is 1000 kg/m3 and the kinetic viscosity of the fluid  is 1.0×10-6 m2/s.    Figure 1: Model setup for the drag force acting on the sphere     As a result of the analysis, the relationships between CD and Re for the PS-SRT model and the PS-MRT model are plotted in Figure 2. From this figure, it is shown that drag force coefficient CD obtained from the numerical simulation is corresponding with the empirical equation: CD = 24/Re + 6/ (1+Re-0.5) + 0.4. Compared with two models, stability solution can be obtained in the range of Re < 150 in both cases, and remarkable differences between two models are not observed. 541Y. Fukumoto, S. Ohtsuka and A. Murakami  5    Figure 2: Drag force coefficient CD vs. Reynolds number Re  6 SEEPAGE FLOW IN GRANULAR SOILS At the next step in the validation of the PS-MRT LB model, we obtain the correlation between friction factor in porous media  and Reynolds number Re. Figure 3 shows the model setup for the seepage flow in granular soils, which are prepared by the packing analysis by using the DEM. The number of soil particles consisting of the porous media is about 1,000 and the diameter of the soil particle d is 16x. Fluid flows are generated in positive direction of y-coordinate by imposing a constant velocity boundary condition on one side of zx plane, while the other side of zx plane is free outflow boundary, in the same manner as the analysis of the previous subsection. Grid space x is 1.0×10-5 m and the system size is 128x ×196x ×128x . The area occupied with the soil particles is 128x ×128x ×128x . The density of the fluid f is 1000 kg/m3 and the kinetic viscosity of the fluid  is 1.0×10-6 m2/s.    Figure 3: Model setup for the seepage flow in granular soils  Figure 4 shows the relationships between  and Re with the empirical equations:  = 150/Re (Re<10, Blake-Kozeny eq.),  = 150/Re+1.75 (10<Re<103, Ergun eq.). From the figure, the PS-MRT LB model can simulate the flow in the porous media in the range of Re < 30. By contrast, the PS-SRT LB model cannot obtain stability solution more than Re = 2. 542Y. Fukumoto, S. Ohtsuka and A. Murakami  6 Thus, the PS-MRT LB model has an adavantages to stably simulate in the flow inside the complex boundary.    Figure 4: Model setup for the drag force acting on the sphere  7 PARTICLE SETTLING The last simulation, as shown in Figure 5, illustrates the ability of the PS-MRT LB model to handle large systems of (104). The number of the soil particles is 12,000 and the particle diameter d is 4x. Grid space x is 4.0×10-5 m and the system size is 160x ×160x ×250x . The density of the fluid f is 1000 kg/m3, the density ratio of the solid to the fluid is 2.5, and the kinetic viscosity of the fluid  is 1.0×10-5 m2/s. The Stokes settling velocity is 2.1×10-3 m/s and Reynolds number Re is about 0.3.    Figure 5: Snapshots of the simulation of particle settling 543Y. Fukumoto, S. Ohtsuka and A. Murakami  7  Figure 5 shows some snapshots during the settling simulation for different elapsed time. The core of the soil particles first swiftly settles followed by the rest of the particles. On the other hand, particles near the vertical walls fall very slowly because the strong interaction occurs between particles and walls. This type of particle settling can be observed in the previous research [11], in which a fictitious domain approach is employed for the coupling method. Therefore, the PS-MRT LB model also can obtain qualitative numerical results in such a large particle-fluid system.  8 CONCLUSIONS In geo-mechanics, a particle-fluid system exists in the form of solid particles and pore liquids or gases, which are characteristic of non-Brownian and highly concentrated suspensions. For such complex particle-fluid systems, the PS-MRT LB model can be one of the effective way to directly solve both the soil particles and the seepage flow at pore scale. If some parallelization schemes are installed into the simulation code, we can apply the direct numerical model to not only the two-dimensional but also the three-dimensional phenomena.  REFERENCES [1] Sukop, M.C., Huang, H., Lin, C.L., Deo, M.D., Oh, K. and Miller, J.D., 2008. Distribution of multiphase fluids in porous media: Comparison between lattice Boltzmann modeling and micro-x-ray tomography. Physical Review E, 77(2), 026710. [2] Moreno-Atanasio, R., Williams, R. A. and Jia, X., 2010. Combining X-ray microtomography with computer simulation for analysis of granular and porous materials. Particuology, 8(2), 81-99. [3] Li, X., Hunt, M. L. and Colonius, T., 2012. A contact model for normal immersed collisions between a particle and a wall. Journal of Fluid Mechanics, 691, 123-145. [4] D'Humieres, D., 2002. Multiple relaxation time lattice Boltzmann models in three dimensions. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 360(1792), 437-451. [5] Noble, D. R. and Torczynski, J. R., 1998. A lattice-Boltzmann method for partially saturated computational cells. International Journal of Modern Physics C, 9(08), 1189-1201. [6] Qian, Y.H., D'Humi'eres, D. and Lallemand, P., 1992. Lattice BGK Models for Navier-Stokes Equation. Europhysics Letters, 17, 479. [7] Ladd, A. J., 1994a. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 1. Theoretical foundation. Journal of Fluid Mechanics, 271, 285-309. [8] Ladd, A. J., 1994b. Numerical simulations of particulate suspensions via a discretized Boltzmann equation. Part 2. Numerical results. Journal of Fluid Mechanics, 271, 311-339. [9] Feng, Z.G. and  Michaelides, E. E., 2004. The immersed boundary-lattice Boltzmann method for solving fluid-particles interaction problems. Journal of Computational Physics, 195(2), 602-628. 544Y. Fukumoto, S. Ohtsuka and A. Murakami  8 [10] Nishiura, D. and Sakaguchi, H., 2011. Parallel-vector algorithms for particle simulations on shared-memory multiprocessors. Journal of Computational Physics, 230(5), 1923-1938. [11] Gallier, S., Lemaire, E., Lobry, L. and Peters, F., 2014. A fictitious domain approach for the simulation of dense suspensions. Journal of Computational Physics, 256, 367-387. 

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PS-MRT Lattice Boltzmann model for direct simulation of granular soils and seepage flow

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