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The discrete Green's function paradigm for two-way coupled Euler–Lagrange simulation

Published online by Cambridge University Press:  18 November 2021

J.A.K. Horwitz Open the ORCID record for J.A.K. Horwitz [Opens in a new window] ,
G. Iaccarino Open the ORCID record for G. Iaccarino [Opens in a new window] ,
J.K. Eaton Open the ORCID record for J.K. Eaton [Opens in a new window]  and
A. Mani Open the ORCID record for A. Mani [Opens in a new window]
J.A.K. Horwitz*
Affiliation:
Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550-0808, USA Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
G. Iaccarino
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
J.K. Eaton
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
A. Mani
Affiliation:
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: [email protected]
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Abstract

We outline a methodology for the simulation of two-way coupled particle-laden flows. The drag force that couples fluid and particle momentum depends on the undisturbed fluid velocity at the particle location, and this latter quantity requires modelling. We demonstrate that the undisturbed fluid velocity, in the low particle Reynolds number limit, can be related exactly to the discrete Green's function of the discrete Stokes equations. In addition to hydrodynamics, the method can be extended to other physics present in particle-laden flows such as heat transfer and electromagnetism. The discrete Green's functions for the Navier–Stokes equations are obtained at low particle Reynolds number in a two-plane channel geometry. We perform verification at different Reynolds numbers for a particle settling under gravity parallel to a plane wall, for different wall-normal separations. Compared with other point-particle schemes, the Stokesian discrete Green's function approach is the most robust at low particle Reynolds number, accurate at all wall-normal separations. To account for degradation in accuracy away from the wall at finite Reynolds number, we extend the present methodology to an Oseen-like discrete Green's function. The extended discrete Green's function method is found to be accurate within $6\,\%$ at all wall-normal separations for particle Reynolds numbers up to 24. The discrete Green's function approach is well suited to dilute systems with significant mass loading and this is highlighted by comparison against other Euler–Lagrange as well as particle-resolved simulations of gas–solid turbulent channel flow. Strong particle–turbulence coupling is observed in the form of turbulence modification and turbophoresis suppression, and these observations are placed in context of the different methods.

JFM classification

Mathematical Foundations: Computational methods Multiphase and Particle-laden Flows: Particle/fluid flow Turbulent Flows: Turbulence simulation
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 931 , 25 January 2022 , A3
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Akiki, G., Jackson, T.L. & Balachandar, S. 2017 Pairwise interaction extended point-particle model for a random array of monodisperse spheres. J. Fluid Mech. 813, 882–928.CrossRefGoogle Scholar
Bagchi, P. & Balachandar, S. 2004 Response of the wake of an isolated particle to an isotropic turbulent flow. J. Fluid Mech. 518, 95–123.CrossRefGoogle Scholar
Balachandar, S., Liu, K. & Lakhote, M. 2019 Self-induced velocity correction for improved drag estimation in Euler–Lagrange point-particle simulations. J. Comput. Phys. 376, 160–185.CrossRefGoogle Scholar
Batchelor, G.K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Battista, F., Mollicone, J.P., Gualtieri, P., Messina, R. & Casciola, C.M. 2019 Exact regularized point particle (ERPP) method for particle-laden wall-bounded flows in the two-way coupling regime. J. Fluid Mech. 878, 420–444.CrossRefGoogle ScholarPubMed
Benson, M., Tanaka, T. & Eaton, J.K. 2005 Effects of wall roughness on particle velocities in a turbulent channel flow. Trans. ASME J. Fluids Engng 127, 250–256.CrossRefGoogle Scholar
Blake, J.R. & Chwang, A.T. 1974 Fundamental singularities of viscous flow part I: the image systems in the vicinity of a stationary no-slip boundary. J. Engng Maths 8 (1), 23–29.CrossRefGoogle Scholar
Brenner, H. 1962 Effect of finite boundaries on the stokes resistance of an arbitrary particle. J. Fluid Mech. 12, 35–48.CrossRefGoogle Scholar
Cocco, R., Karri, S.B.R. & Knowlton, T. 2014 Introduction to fluidization. Chem. Engng Professionals Mag. 2014 (Nov.), 21–29.Google Scholar
Cortez, R. 2001 The method of regularized stokeslets. SIAM J. Sci. Comput. 23, 1204–1225.CrossRefGoogle Scholar
Costa, P., Brandt, L. & Picano, F. 2020 Interface-resolved simulations of small inertial particles in turbulent channel flow. J. Fluid Mech. 883, 1–19.CrossRefGoogle Scholar
Costa, P., Brandt, L. & Picano, F. 2021 Near-wall turbulence modulation by small inertial particles. arXiv:2102.11597 v1, pp. 1–10.Google Scholar
Cundall, P.A. & Strack, O.D.L. 1979 A discrete numerical model for granular assemblies. Geotechnique 29, 47–65.CrossRefGoogle Scholar
Ehrenpreis, L. 1954 Solution of some problems of division: part I. Division by a polynomial of derivation. Am. J. Maths 76, 883–903.CrossRefGoogle Scholar
Esmaily, M. & Horwitz, J.A.K. 2018 A correction scheme for two-way coupled point-particle simulations on anisotropic grids. J. Comput. Phys. 375, 960–982.CrossRefGoogle Scholar
Esmaily, M., Jofre, L., Mani, A. & Iaccarino, G. 2018 A scalable geometric multigrid solver for nonsymmetric elliptic systems with application to variable-density flows. J. Comput. Phys. 357, 142–158.CrossRefGoogle Scholar
Esmaily, M., Villafane, L., Banko, A.J., Iaccarino, G., Eaton, J.K. & Mani, A. 2020 A benchmark for particle-laden turbulent duct flow: a joint computational and experimental study. Intl J. Multiphase Flow 132, 103410.CrossRefGoogle Scholar
Frasca, M. & Khurshudyan, A.Z. 2019 A general representation for the Green's function of second-order nonlinear differential equations. Comput. Math. Meth. 1, e1038.CrossRefGoogle Scholar
Fukada, T., Takeuchi, S. & Kajishima, T. 2019 Estimation of fluid forces on a spherical particle for two-way coupling simulation based on the volume averaging. Intl J. Multiphase Flow 113, 165–178.CrossRefGoogle Scholar
Gualtieri, P., Picano, F., Sardina, G. & Casciola, C.M. 2015 Exact regularized point particle method for multiphase flows in the two-way coupling regime. J. Fluid Mech. 773, 520–561.CrossRefGoogle Scholar
Haberman, R. 2004 Applied Partial Differential Equations with Fourier Series and Boundary Value Problems. Pearson.Google Scholar
Hahn, D.W. & Ozisik, M.N. 2012 Heat Conduction. Wiley.CrossRefGoogle Scholar
Horne, W.J. & Mahesh, K. 2019 a A massively-parallel, unstructured overset method to simulate moving bodies in turbulent flows. J. Comput. Phys. 397, 108790.CrossRefGoogle Scholar
Horne, W.J. & Mahesh, K. 2019 b A particle-resolved direct-numerical simulation study of particle-laden turbulent channel flow using a massively parallel, unstructured, overset method. In 11th International Symposium on Turbulence and Shear Flow Phenomena (TSFP11) Southampton, UK, pp. 1–6.Google Scholar
Horwitz, J.A.K. 2018 Verifiable point-particle methods for two-way coupled particle-laden flows. PhD thesis, Stanford University.Google Scholar
Horwitz, J., Ganguli, S., Mani, A. & Lele, S. 2017 A correction procedure for thermally two-way coupled point-particles. Bull. Am. Phys. Soc. 62 (14), D37.00002.Google Scholar
Horwitz, J., Iaccarino, G., Eaton, J. & Mani, A. 2019 Comparison of Euler-Lagrange schemes in two-way coupled particle-laden channel flow. Bull. Am. Phys. Soc. 64 (13), H28.00003.Google Scholar
Horwitz, J.A.K. & Mani, A. 2016 Accurate calculation of Stokes drag for point-particle tracking in two-way coupled flows. J. Comput. Phys. 318, 85–109.CrossRefGoogle Scholar
Horwitz, J.A.K. & Mani, A. 2018 Correction scheme for point-particle models applied to a nonlinear drag law in simulations of particle-fluid interaction. Intl J. Multiphase Flow 101, 74–84.CrossRefGoogle Scholar
Horwitz, J.A.K. & Mani, A. 2020 Two-way coupled particle-turbulence interaction: effect of numerics and resolution on fluid and particle statistics. Phys. Rev. Fluids 5, 104302.CrossRefGoogle Scholar
Horwitz, J.A.K., Rahmani, M., Geraci, G., Banko, A.J. & Mani, A. 2016 Two-way coupling effects in particle-laden turbulence: how particle-tracking scheme affects particle and fluid statistics. In 9th International Conference on Multiphase Flow, Firenze, Italy.Google Scholar
Ireland, P.J. & Desjardins, O. 2017 Improving particle drag predictions in Euler–Lagrange simulations with two-way coupling. J. Comput. Phys. 338, 405–430.CrossRefGoogle Scholar
Johnson, P.L. 2020 Predicting the impact of particle-particle collisions on turbophoresis with a reduced number of computational particles. Intl J. Multiphase Flow 124, 103182.CrossRefGoogle Scholar
Johnson, P.L., Bassenne, M. & Moin, P. 2020 Turbophoresis of small inertial particles: theoretical considerations and application to wall-modelled large-eddy simulations. J. Fluid Mech. 883, A27.CrossRefGoogle Scholar
Kim, S. & Karrila, S.J. 2005 Microhydrodynamics: Principles and Selected Applications. Butterworth- Heinemann.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.CrossRefGoogle Scholar
Kok, J.F, Parteli, E.J.R., Michaels, T.I & Karam, D.B. 2012 The physics of wind-blown sand and dust. Rep. Prog. Phys. 75 (10), 1–72.CrossRefGoogle Scholar
Kulick, J.D., Fessler, J.R. & Eaton, J.K. 1994 Particle response and turbulence modification in fully developed channel flow. J. Fluid Mech. 277, 109–134.CrossRefGoogle Scholar
Lewy, H. 1957 An example of a smooth linear partial differential equation without solution. Ann. Maths 66, 155–158.CrossRefGoogle Scholar
Li, Y., Mclaughlin, J.B., Kontomaris, K. & Portela, L. 2001 Numerical simulation of particle-laden turbulent channel flow. Phys. Fluids 13, 2957–2967.CrossRefGoogle Scholar
Liron, N. & Mochon, S. 1976 Stokes flow for a stokeslet between two parallel flat plates. J. Engng Maths 10 (4), 287–303.CrossRefGoogle Scholar
Liu, K., Lakhote, M. & Balachandar, S. 2019 Self-induced temperature correction for inter-phase heat transfer in Euler-Lagrange point-particle simulation. J. Comput. Phys. 396, 596–615.CrossRefGoogle Scholar
Malgrange, B. 1956 Existence et approximation des solutions des equations aux deŕiveés partielles et des equations de convolution. Ann. Inst. Fourier 6, 271–355.CrossRefGoogle Scholar
Malmborg, V.B., Eriksson, A.C., Shen, M., Nilsson, P., Gallo, Y., Waldheim, B., Martinsson, J., Andersson, O. & Pagels, J. 2017 Evolution of in-cylinder diesel engine soot and emission characteristics investigated with online aerosol mass spectrometry. Environ. Sci. Technol. 1876–1885.CrossRefGoogle ScholarPubMed
Maxey, M. 2017 Simulation methods for particulate flows and concentrated suspensions. Annu. Rev. Fluid Mech. 49, 171–193.CrossRefGoogle Scholar
Maxey, M.R. & Riley, J.J. 1983 Equation of motion for a small rigid sphere in a uniform flow. Phys. Fluids 26 (4), 883–889.CrossRefGoogle Scholar
Mehrabadi, M., Horwitz, J.A.K., Subramaniam, S. & Mani, A. 2018 A direct comparison of particle-resolved and point-particle methods in decaying turbulence. J. Fluid Mech. 850, 336–369.CrossRefGoogle Scholar
O'Neill, M.E. 1964 A slow motion of viscous liquid caused by a slowly moving solid sphere. Mathematika 11, 67–74.CrossRefGoogle Scholar
Pakseresht, P., Esmaily, M. & Apte, S.V. 2019 A correction scheme for two-way coupled Euler-Lagrange wall-bounded flows. Bull. Am. Phys. Soc. 64 (13), H28.00008.Google Scholar
Pakseresht, P., Esmaily, M. & Apte, S.V. 2020 A correction scheme for wall-bounded two-way coupled point-particle simulations. J. Comput. Phys. 420, 109711.CrossRefGoogle Scholar
Paris, A.D. 2001 Turbulence attenuation in a particle-laden channel flow. PhD thesis, Stanford University.Google Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Poustis, J.-F., Senoner, J.-M., Zuzio, D. & Villedieu, P. 2019 Regularization of the lagrangian point force approximation for deterministic discrete particle simulations. Intl J. Multiphase Flow 117, 138–152.CrossRefGoogle Scholar
Sardina, G., Schlatter, P., Brandt, L., Picano, F. & Casciola, C.M. 2012 Wall accumulation and spatial localization in particle-laden wall flows. J. Fluid Mech. 699, 50–78.CrossRefGoogle Scholar
Segura, J.C. 2004 Predictive capabilities of particle-laden large eddy simulation. PhD thesis, Stanford University.Google Scholar
Shirian, Y., Horwitz, J. & Mani, A. 2021 On the convergence of statistics in simulations of stationary turbulent flows. arXiv:2110.10300.Google Scholar
Swan, J.W. & Brady, J.F. 2010 Particle motion between parallel walls: hydrodynamics and simulation. Phys. Fluids 22, 103301.CrossRefGoogle Scholar
Taneda, S. 1956 Experimental investigation of the wake behind a sphere at low Reynolds numbers. J. Phys. Soc. Japan 11 (10), 1104–1108.CrossRefGoogle Scholar
Vasseur, P. & Cox, R.G. 1977 The lateral migration of spherical particles sedimenting in a stagnant bounded fluid. J. Fluid Mech. 80, 561–591.CrossRefGoogle Scholar
Wang, G., Fong, K.O., Coletti, F., Capecelatro, J. & Richter, D.H. 2019 Inertial particle velocity and distribution in vertical turbulent channel flow: a numerical and experimental comparison. Intl J. Multiphase Flow 120, 1–16.CrossRefGoogle Scholar
Yamamoto, Y., Potthoff, M., Tanaka, T., Kajishima, T. & Tsuji, Y. 2001 Large-eddy simulation of turbulent gas–particle flow in a vertical channel: effect of considering inter-particle collisions. J. Fluid Mech. 442, 303–334.CrossRefGoogle Scholar
Yao, Y. & Capecelatro, J. 2018 Competition between drag and Coulomb interactions in turbulent particle-laden flows using a coupled-fluid-Ewald-summation based approach. Phys. Rev. Fluids 034301.CrossRefGoogle Scholar
Zeilberger, D. 2011 The discrete analog of the Malgrange-Ehrenpreis theorem. Preprint. arXiv:1107.4380 v1.CrossRefGoogle Scholar
Zhao, L.H., Andersson, H.I. & Gillissen, J.J.J. 2010 Turbulence modulation and drag reduction by spherical particles. Phys. Fluids 22, 081702.CrossRefGoogle Scholar