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Microstructure-informed probability-driven point-particle model for hydrodynamic forces and torques in particle-laden flows

Published online by Cambridge University Press:  12 August 2020

Arman Seyed-Ahmadi Open the ORCID record for Arman Seyed-Ahmadi [Opens in a new window]  and
Anthony Wachs Open the ORCID record for Anthony Wachs [Opens in a new window]
Arman Seyed-Ahmadi
Affiliation:
Department of Chemical & Biological Engineering, The University of British Columbia, 2360 East Mall, Vancouver, BCV6T 1Z3, Canada
Anthony Wachs*
Affiliation:
Department of Chemical & Biological Engineering, The University of British Columbia, 2360 East Mall, Vancouver, BCV6T 1Z3, Canada Department of Mathematics, The University of British Columbia, 1984 Mathematics Road, Vancouver, BCV6T 1Z2, Canada
*
†Email address for correspondence: wachs@mail.ubc.ca
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Abstract

We present a novel deterministic model that is capable of predicting particle-to-particle force and torque fluctuations in a fixed bed of randomly distributed monodisperse spheres. First, we generate our dataset by performing particle-resolved direct numerical simulations (PR-DNS) of arrays of stationary spheres in moderately inertial regimes with a Reynolds number range of $2 \leq \textit {Re} \leq 150$ and a solid volume fraction range of $0.1 \leq \phi \leq 0.4$. The key idea exploited by our model is that, while the arrangement of neighbours around each particle is uniform and random, conditioning forces or torques exerted on a reference sphere to specific ranges of values results in the emergence of significantly non-uniform distributions of neighbouring particles. Based on probabilistic arguments, we take advantage of the statistical information extracted from PR-DNS to construct force/torque-conditioned probability distribution maps, which are ultimately used as basis functions for regression. Given the locations of surrounding particles as input to the model, our results demonstrate that the present probability-driven framework is capable of predicting up to 85 % of the actual observed force and torque variation in the best cases. Since the precise location of each particle is known in an Eulerian–Lagrangian (EL) simulation, our model would be able to estimate the unresolved subgrid force and torque fluctuations reasonably well, and thereby considerably enhance the fidelity of EL simulations via improved interphase coupling.

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow Mathematical Foundations: Computational methods
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 900 , 10 October 2020 , A21
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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References

REFERENCES

Aidun, C. K. & Clausen, J. R. 2010 Lattice-Boltzmann method for complex flows. Annu. Rev. Fluid Mech. 42 (1), 439–472.CrossRefGoogle Scholar
Akiki, G., Jackson, T. L. & Balachandar, S. 2016 Force variation within arrays of monodisperse spherical particles. Phys. Rev. Fluids 1 (4), 044202.CrossRefGoogle Scholar
Akiki, G., Jackson, T. L. & Balachandar, S. 2017 a Pairwise interaction extended point-particle model for a random array of monodisperse spheres. J. Fluid Mech. 813, 882–928.CrossRefGoogle Scholar
Akiki, G., Moore, W. C. & Balachandar, S. 2017 b Pairwise-interaction extended point-particle model for particle-laden flows. J. Comput. Phys. 351, 329–357.CrossRefGoogle Scholar
Bagchi, P. & Balachandar, S. 2002 Effect of free rotation on the motion of a solid sphere in linear shear flow at moderate $Re$. Phys. Fluids 14 (8), 2719–2737.CrossRefGoogle Scholar
Bagchi, P. & Balachandar, S. 2003 Effect of turbulence on the drag and lift of a particle. Phys. Fluids 15 (11), 3496–3513.CrossRefGoogle Scholar
Balachandar, S. 2009 A scaling analysis for point–particle approaches to turbulent multiphase flows. Intl J. Multiphase Flow 35 (9), 801–810.CrossRefGoogle Scholar
Batchelor, G. K. 1972 Sedimentation in a dilute dispersion of spheres. J. Fluid Mech. 52 (02), 245.CrossRefGoogle Scholar
Beck, A., Flad, D. & Munz, C.-D. 2019 Deep neural networks for data-driven LES closure models. J. Comput. Phys. 398, 108910.CrossRefGoogle Scholar
Beetstra, R., van der Hoef, M. A. & Kuipers, J. A. M. 2007 Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres. AIChE J. 53 (2), 489–501.CrossRefGoogle Scholar
Bogner, S., Mohanty, S. & Rüde, U. 2015 Drag correlation for dilute and moderately dense fluid-particle systems using the lattice Boltzmann method. Intl J. Multiphase Flow 68, 71–79.CrossRefGoogle Scholar
Capecelatro, J. & Desjardins, O. 2013 An Euler–Lagrange strategy for simulating particle-laden flows. J. Comput. Phys. 238, 1–31.CrossRefGoogle Scholar
Chacón, J. E. & Duong, T. 2018 Multivariate Kernel Smoothing and Its Applications. CRC.CrossRefGoogle Scholar
Collins, L. R. & Keswani, A. 2004 Reynolds number scaling of particle clustering in turbulent aerosols. New J. Phys. 6, 119–119.CrossRefGoogle Scholar
Coveney, P. V., Dougherty, E. R. & Highfield, R. R. 2016 Big data need big theory too. Phil. Trans. R. Soc. A 374 (2080), 20160153.CrossRefGoogle ScholarPubMed
Dash, S. M. & Lee, T. S. 2015 Two spheres sedimentation dynamics in a viscous liquid column. Comput. Fluids 123, 218–234.CrossRefGoogle Scholar
Duraisamy, K., Iaccarino, G. & Xiao, H. 2019 Turbulence modeling in the age of data. Annu. Rev. Fluid Mech. 51 (1), 357–377.CrossRefGoogle Scholar
Ergun, S. 1952 Fluid flow through packed columns. Chem. Engng Prog. 48, 89–94.Google Scholar
Esteghamatian, A., Bernard, M., Lance, M., Hammouti, A. & Wachs, A. 2017 Micro/meso simulation of a fluidized bed in a homogeneous bubbling regime. Intl J. Multiphase Flow 92, 93–111.CrossRefGoogle Scholar
Esteghamatian, A., Euzenat, F., Hammouti, A., Lance, M. & Wachs, A. 2018 A stochastic formulation for the drag force based on multiscale numerical simulation of fluidized beds. Intl J. Multiphase Flow 99, 363–382.CrossRefGoogle Scholar
Faxen, H. 1923 Die bewegung einer starren kugel langs der achse eines mit zaher flussigkeit gefullten rohres. Ark. Mat. Astr. Fys. 17, 1–28.Google Scholar
Feng, J., Hu, H. H. & Joseph, D. D. 1994 a Direct simulation of initial value problems for the motion of solid bodies in a Newtonian fluid. Part 2. Couette and Poiseuille flows. J. Fluid Mech. 277, 271–301.CrossRefGoogle Scholar
Feng, J., Hu, H. H. & Joseph, D. D. 1994 b Direct simulation of initial value problems for the motion of solid bodies in a newtonian fluid. Part 1. Sedimentation. J. Fluid Mech. 261, 95–134.CrossRefGoogle Scholar
Fornari, W., Ardekani, M. N. & Brandt, L. 2018 Clustering and increased settling speed of oblate particles at finite Reynolds number. J. Fluid Mech. 848, 696–721.CrossRefGoogle Scholar
Fortes, A. F., Joseph, D. D. & Lundgren, T. S. 1987 Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177, 467–483.CrossRefGoogle Scholar
Gatignol, R. 1983 The faxén formulas for a rigid particle in an unsteady non-uniform stokes-flow. J. de Méc. Théor. et Appl. 2 (2), 143–160.Google Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic.Google Scholar
Glowinski, R., Pan, T. W., Hesla, T. I. & Joseph, D. D. 1999 A distributed Lagrange multiplier/fictitious domain method for particulate flows. Intl J. Multiphase Flow 25 (5), 755–794.CrossRefGoogle Scholar
Götz, J., Iglberger, K., Stürmer, M. & Rüde, U. 2010 Direct numerical simulation of particulate flows on 294 912 processor cores. In Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis, pp. 1–11. IEEE Computer Society.CrossRefGoogle Scholar
Guo, X., Li, W. & Iorio, F. 2016 Convolutional neural networks for steady flow approximation. In Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 481–490. ACM.CrossRefGoogle Scholar
Hasimoto, H. 1959 On the periodic fundamental solutions of the stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5 (2), 317–328.CrossRefGoogle Scholar
Hastie, T., Tibshirani, R. & Friedman, J. 2009 The Elements of Statistical Learning. Springer.CrossRefGoogle Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001 a The first effects of fluid inertia on flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 213–241.CrossRefGoogle Scholar
Hill, R. J., Koch, D. L. & Ladd, A. J. C. 2001 b Moderate-Reynolds-number flows in ordered and random arrays of spheres. J. Fluid Mech. 448, 243–278.CrossRefGoogle Scholar
Hinton, G., Deng, L., Yu, D., Dahl, G., Mohamed, A., Jaitly, N., Senior, A., Vanhoucke, V., Nguyen, P., Sainath, T. & Kingsbury, B. 2012 Deep neural networks for acoustic modeling in speech recognition: the shared views of four research groups. IEEE Signal Process. Mag. 29 (6), 82–97.CrossRefGoogle Scholar
van der Hoef, M. A., Beetstra, R. & Kuipers, J. A. M. 2005 Lattice-Boltzmann simulations of low-Reynolds-number flow past mono- and bidisperse arrays of spheres: results for the permeability and drag force. J. Fluid Mech. 528, 233–254.CrossRefGoogle Scholar
van der Hoef, M. A., van Sint Annaland, M., Deen, N. G. & Kuipers, J. A. M. 2008 Numerical simulation of dense gas-solid fluidized beds: a multiscale modeling strategy. Annu. Rev. Fluid Mech. 40, 47–70.CrossRefGoogle Scholar
van der Hoef, M. A., van Sint Annaland, M., Kuipers, J. A. M. 2004 Computational fluid dynamics for dense gas–solid fluidized beds: a multi-scale modeling strategy. Chem. Engng Sci. 59 (22–23), 5157–5165.CrossRefGoogle Scholar
Horne, W. J. & Mahesh, K. 2019 A massively-parallel, unstructured overset method to simulate moving bodies in turbulent flows. J. Comput. Phys. 397, 108790.CrossRefGoogle Scholar
Hornik, K., Stinchcombe, M. & White, H. 1989 Multilayer feedforward networks are universal approximators. Neural Networks 2 (5), 359–366.CrossRefGoogle Scholar
Hu, H. H., Patankar, N. A. & Zhu, M. Y. 2001 Direct numerical simulations of fluid–solid systems using the arbitrary Lagrangian–Eulerian technique. J. Comput. Phys. 169 (2), 427–462.CrossRefGoogle Scholar
Jiang, Y., Kolehmainen, J., Gu, Y., Kevrekidis, Y. G., Ozel, A. & Sundaresan, S. 2019 Neural-network-based filtered drag model for gas-particle flows. Powder Technol. 346, 403–413.CrossRefGoogle Scholar
Kajishima, T. & Takiguchi, S. 2002 Interaction between particle clusters and particle-induced turbulence. Intl J. Heat Fluid Flow 23 (5), 639–646.CrossRefGoogle Scholar
Kidanemariam, A. G., Chan-Braun, C., Doychev, T. & Uhlmann, M. 2013 Direct numerical simulation of horizontal open channel flow with finite-size, heavy particles at low solid volume fraction. New J. Phys. 15 (2), 025031.CrossRefGoogle Scholar
Kriebitzsch, S. H. L., van der Hoef, M. A. & Kuipers, J. A. M. 2013 Fully resolved simulation of a gas-fluidized bed: a critical test of DEM models. Chem. Engng Sci. 91, 1–4.CrossRefGoogle Scholar
Krizhevsky, A., Sutskever, I. & Hinton, G. E. 2012 Imagenet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems 25 (ed. F. Pereira, C. J. C. Burges, L. Bottou & K. Q. Weinberger), pp. 1097–1105. Curran Associates.Google Scholar
Kutz, J. N. 2017 Deep learning in fluid dynamics. J. Fluid Mech. 814, 1–4.CrossRefGoogle Scholar
Langrené, N. & Warin, X. 2019 Fast and stable multivariate kernel density estimation by fast sum updating. J. Comput. Graph. Stat. 28 (3), 596–608.CrossRefGoogle Scholar
LeCun, Y., Bengio, Y. & Hinton, G. 2015 Deep learning. Nature 521 (7553), 436–444.CrossRefGoogle ScholarPubMed
Ling, J., Kurzawski, A. & Templeton, J. 2016 Reynolds averaged turbulence modelling using deep neural networks with embedded invariance. J. Fluid Mech. 807, 155–166.CrossRefGoogle Scholar
Ling, J. & Templeton, J. 2015 Evaluation of machine learning algorithms for prediction of regions of high Reynolds averaged Navier–Stokes uncertainty. Phys. Fluids 27 (8), 085103.CrossRefGoogle Scholar
Louge, M., Lischer, D. J. & Chang, H. 1990 Measurements of voidage near the wall of a circulating fluidized bed riser. Powder Technol. 62 (3), 269–276.CrossRefGoogle Scholar
Ma, M., Lu, J. & Tryggvason, G. 2015 Using statistical learning to close two-fluid multiphase flow equations for a simple bubbly system. Phys. Fluids 27 (9), 092101.CrossRefGoogle Scholar
Maulik, R., San, O., Rasheed, A. & Vedula, P. 2018 Data-driven deconvolution for large eddy simulations of Kraichnan turbulence. Phys. Fluids 30 (12), 125109.CrossRefGoogle Scholar
Maxey, M. 2017 Simulation methods for particulate flows and concentrated suspensions. Annu. Rev. Fluid Mech. 49 (1), 171–193.CrossRefGoogle Scholar
Maxey, M. R. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883.CrossRefGoogle Scholar
Mittal, R. & Iaccarino, G. 2005 Immersed boundary methods. Annu. Rev. Fluid Mech. 37 (1), 239–261.CrossRefGoogle Scholar
Moore, W. C., Balachandar, S. & Akiki, G. 2019 A hybrid point-particle force model that combines physical and data-driven approaches. J. Comput. Phys. 385, 187–208.CrossRefGoogle Scholar
Papoulis, A., Pillai, S. U. & Pillai, S. U. 2002 Probability, Random Variables, and Stochastic Processes. McGraw-Hill.Google Scholar
Percus, J. K. & Yevick, G. J. 1958 Analysis of classical statistical mechanics by means of collective coordinates. Phys. Rev. 110 (1), 1–13.CrossRefGoogle Scholar
Peskin, C. S. 1977 Numerical analysis of blood flow in the heart. J. Comput. Phys. 25 (3), 220–252.CrossRefGoogle Scholar
Prahl, L., Hölzer, A., Arlov, D., Revstedt, J., Sommerfeld, M. & Fuchs, L. 2007 On the interaction between two fixed spherical particles. Intl J. Multiphase Flow 33 (7), 707–725.CrossRefGoogle Scholar
Raissi, M. & Karniadakis, G. E. 2018 Hidden physics models: machine learning of nonlinear partial differential equations. J. Comput. Phys. 357, 125–141.CrossRefGoogle Scholar
Raissi, M., Perdikaris, P. & Karniadakis, G. E. 2019 Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707.CrossRefGoogle Scholar
Raissi, M., Wang, Z., Triantafyllou, M. S. & Karniadakis, G. E. 2018 Deep learning of vortex-induced vibrations. J. Fluid Mech. 861, 119–137.CrossRefGoogle Scholar
Rettinger, C., Godenschwager, C., Eibl, S., Preclik, T., Schruff, T., Frings, R. & Rüde, U. 2017 Fully resolved simulations of dune formation in riverbeds. In International Supercomputing Conference, pp. 3–21. Springer.CrossRefGoogle Scholar
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidisation. Part I. Trans. Inst. Chem. Engrs 32, 35–53.Google Scholar
Ross, S. M. 2010 A First Course in Probability. Pearson Prentice Hall.Google Scholar
Saffman, P. G. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22 (2), 385–400.CrossRefGoogle Scholar
Sangani, A. S. & Acrivos, A. 1982 Slow flow through a periodic array of spheres. Intl J. Multiphase Flow 8 (4), 343–360.CrossRefGoogle Scholar
Schiller, L. & Naumann, A. 1933 Über die grundlegenden berechnungen bei der schwerkraftaufbereitung. Z. Vereines Deutscher Inge. 77, 318–321.Google Scholar
Scott, D. W. 1992 Multivariate Density Estimation: Theory, Practice, and Visualization. Wiley.CrossRefGoogle Scholar
Sekar, V. & Khoo, B. C. 2019 Fast flow field prediction over airfoils using deep learning approach. Phys. Fluids 31 (5), 057103.CrossRefGoogle Scholar
Silverman, B. W. 1986 Density Estimation for Statistics and Data Analysis. Taylor & Francis.CrossRefGoogle Scholar
Squires, K. 2007 Point-Particle Methods for Disperse Flows, pp. 282–319. Cambridge University Press.Google Scholar
Subramaniam, S. 2013 Lagrangian–Eulerian methods for multiphase flows. Prog. Energy Combust. Sci. 39 (2-3), 215–245.CrossRefGoogle Scholar
Subramaniam, S. & Balachandar, S. 2018 Towards Combined Deterministic and Statistical Approaches to Modeling Dispersed Multiphase Flows, pp. 7–42. Springer.Google Scholar
Succi, S. & Coveney, P. V. 2019 Big data: the end of the scientific method? Phil. Trans. R. Soc. A 377 (2142), 20180145.CrossRefGoogle ScholarPubMed
Sun, L., Gao, H., Pan, S. & Wang, J.-X. 2019 Surrogate modeling for fluid flows based on physics-constrained deep learning without simulation data. Comput. Meth. Appl. Mech. Engng 361, 112732.CrossRefGoogle Scholar
Tang, Y., Lau, Y. M., Deen, N. G., Peters, E. A. J. F. & Kuipers, J. A. M. 2016 Direct numerical simulations and experiments of a pseudo-2D gas-fluidized bed. Chem. Engng Sci. 143, 166–180.Google Scholar
Tang, Y., Peters, E. A., Kuipers, J. A., Kriebitzsch, S. H., van der Hoef, M. A. 2015 A new drag correlation from fully resolved simulations of flow past monodisperse static arrays of spheres. AIChE J. 61 (2), 688–698.CrossRefGoogle Scholar
Tenneti, S., Garg, R. & Subramaniam, S. 2011 Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres. Intl J. Multiphase Flow 37 (9), 1072–1092.CrossRefGoogle Scholar
Uhlmann, M. & Doychev, T. 2014 Sedimentation of a dilute suspension of rigid spheres at intermediate Galileo numbers: the effect of clustering upon the particle motion. J. Fluid Mech. 752, 310–348.CrossRefGoogle Scholar
Wachs, A. 2010 PeliGRIFF, a parallel DEM-DLM/FD direct numerical simulation tool for 3D particulate flows. J. Engng Maths 71 (1), 131–155.CrossRefGoogle Scholar
Wachs, A., Hammouti, A., Vinay, G. & Rahmani, M. 2015 Accuracy of finite volume/staggered grid distributed lagrange multiplier/fictitious domain simulations of particulate flows. Comput. Fluids 115, 154–172.CrossRefGoogle Scholar
Wen, C. Y. & Yu, Y. H. 1966 Mechanics of fluidization. Chem. Engng Prog. Symp. Series 62, 100–111.Google Scholar
Wertheim, M. S. 1963 Exact solution of the Percus-Yevick integral equation for hard spheres. Phys. Rev. Lett. 10 (8), 321–323.CrossRefGoogle Scholar
Willen, D. P. & Prosperetti, A. 2019 Resolved simulations of sedimenting suspensions of spheres. Phys. Rev. Fluids 4 (1), 014304.CrossRefGoogle Scholar
Willen, D. P. & Sierakowski, A. J. 2019 Resolved particle simulations using the physalis method on many GPUs. Comput. Phys. Commun. 250, 107071.CrossRefGoogle Scholar
Witten, I. H., Frank, E., Hall, M. A. & Pal, C. J. 2016 Data Mining: Practical Machine Learning Tools and Techniques. Elsevier Science.Google Scholar
Wu, J.-L., Xiao, H. & Paterson, E. 2018 Physics-informed machine learning approach for augmenting turbulence models: a comprehensive framework. Phys. Rev. Fluids 3 (7), 074602.CrossRefGoogle Scholar
Wylie, J. J. & Koch, D. L. 2000 Particle clustering due to hydrodynamic interactions. Phys. Fluids 12 (5), 964–970.CrossRefGoogle Scholar
Xie, C., Wang, J., Li, H., Wan, M. & Chen, S. 2019 a Artificial neural network mixed model for large eddy simulation of compressible isotropic turbulence. Phys. Fluids 31 (8), 085112.Google Scholar
Xie, C., Wang, J., Li, K. & Ma, C. 2019 b Artificial neural network approach to large-eddy simulation of compressible isotropic turbulence. Phys. Rev. E 99 (5), 053113.Google ScholarPubMed
Yang, X. I. A., Zafar, S., Wang, J.-X. & Xiao, H. 2019 Predictive large-eddy-simulation wall modeling via physics-informed neural networks. Phys. Rev. Fluids 4 (3), 034602.CrossRefGoogle Scholar
Yin, X. & Koch, D. L. 2007 Hindered settling velocity and microstructure in suspensions of solid spheres with moderate Reynolds numbers. Phys. Fluids 19 (9), 093302.CrossRefGoogle Scholar
Yin, X. & Koch, D. L. 2008 Velocity fluctuations and hydrodynamic diffusion in finite-Reynolds-number sedimenting suspensions. Phys. Fluids 20 (4), 043305.CrossRefGoogle Scholar
Yoon, D.-H. & Yang, K.-S. 2007 Flow-induced forces on two nearby spheres. Phys. Fluids 19 (9), 098103.CrossRefGoogle Scholar
Zaidi, A. A. 2018 Particle velocity distributions and velocity fluctuations of non-Brownian settling particles by particle-resolved direct numerical simulation. Phys. Rev. E 98 (5), 053103.CrossRefGoogle Scholar
Zaidi, A. A., Tsuji, T. & Tanaka, T. 2014 Direct numerical simulation of finite sized particles settling for high Reynolds number and dilute suspension. Intl J. Heat Fluid Flow 50, 330–341.CrossRefGoogle Scholar
Zhong, W., Yu, A., Liu, X., Tong, Z. & Zhang, H. 2016 DEM/CFD-DEM modelling of non-spherical particulate systems: theoretical developments and applications. Powder Technol. 302, 108–152.CrossRefGoogle Scholar
Zhou, Y. & Alam, Md. M. 2016 Wake of two interacting circular cylinders: a review. Intl J. Heat Fluid Flow 62, 510–537.Google Scholar