Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-16T17:52:34.178Z Has data issue: false hasContentIssue false
  • English
  • Français

Improved modelling of interfacial terms in the second-moment closure for particle-laden flows based on interface-resolved simulation data

Published online by Cambridge University Press:  24 November 2022

Yan Xia ,
Zhaosheng Yu Open the ORCID record for Zhaosheng Yu [Opens in a new window] ,
Zhaowu Lin  and
Yu Guo Open the ORCID record for Yu Guo [Opens in a new window]
Yan Xia
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
Zhaosheng Yu*
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
Zhaowu Lin
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
Yu Guo
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou 310027, PR China
*
†Email address for correspondence: yuzhaosheng@zju.edu.cn
Get access Rights & Permissions [Opens in a new window]

Abstract

Correlations for the interfacial terms in the fluid dissipation rate equation and Reynolds stress equations are established for particle-laden flows, based on data from the interfaced-resolved direct numerical simulations of particle sedimentation in a periodic domain at a density ratio ranging from 0.01 to 1000, a particle concentration ranging from 2.3 % to 30.2 % and a particle Reynolds number below 250. The correlations for the mean drag and the pseudo-turbulent kinetic energy are also reported, which are used for the modelling of the interfacial term in the fluid dissipation rate equation. The interfacial term correlations obtained are then incorporated in the Reynolds stress model (RSM) (i.e. second-moment closure) for the simulation of vertical turbulent channel flows laden with the finite-size particles at relatively low particle volume fractions. The results show that the RSM with new interfacial term correlations can quantitatively predict particle-induced turbulence enhancement or suppression in vertical channel flows.

JFM classification

Multiphase and Particle-laden Flows: Multiphase flow Turbulent Flows: Turbulence modelling
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 952 , 10 December 2022 , A25
Check for updates
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, T.B. & Jackson, R. 1967 Fluid mechanical description of fluidized beds. Equations of motion. Ind. Engng Chem. Fundam. 6 (4), 527–539.CrossRefGoogle Scholar
Ansys, Inc. 2013 ANSYS FLUENT theory guide. Release 15.0. Canonsburg.Google Scholar
Antal, S.P., Lahey, R.T. Jr & Flaherty, J.E. 1991 Analysis of phase distribution in fully developed laminar bubbly two-phase flow. Intl J. Multiphase Flow 17 (5), 635–652.CrossRefGoogle Scholar
Baker, M.C., Fox, R.O., Kong, B., Capecelatro, J. & Desjardins, O. 2020 Reynolds-stress modeling of cluster-induced turbulence in particle-laden vertical channel flow. Phys. Rev. Fluids 5 (7), 074304.CrossRefGoogle Scholar
Balachandar, S. & Eaton, J.K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42, 111–133.CrossRefGoogle Scholar
Barnocky, G. & Davis, R.H. 1988 Elastohydrodynamic collision and rebound of spheres: experimental verification. Phys. Fluids 31 (6), 1324–1329.CrossRefGoogle Scholar
Burns, A.D., Frank, T., Hamill, I. & Shi, J.M. 2004 The favre averaged drag model for turbulent dispersion in Eulerian multi-phase flows. In 5th International Conference on Multiphase Flow, ICMF, vol. 4, pp. 1–17. ICMF.Google Scholar
Capecelatro, J., Desjardins, O. & Fox, R.O. 2016 a Strongly coupled fluid-particle flows in vertical channels. I. Reynolds-averaged two-phase turbulence statistics. Phys. Fluids 28 (3), 033306.CrossRefGoogle Scholar
Capecelatro, J., Desjardins, O. & Fox, R.O. 2016 b Strongly coupled fluid-particle flows in vertical channels. II. Turbulence modeling. Phys. Fluids 28 (3), 033307.CrossRefGoogle Scholar
Cokljat, D., Slack, M., Vasquez, S.A., Bakker, A. & Montante, G. 2006 Reynolds-stress model for Eulerian multiphase. Prog. Comput. Fluid Dyn. Intl J. 6 (1–3), 168–178.CrossRefGoogle Scholar
Colombo, M. & Fairweather, M. 2015 Multiphase turbulence in bubbly flows: RANS simulations. Intl J. Multiphase Flow 77, 222–243.CrossRefGoogle Scholar
Crowe, C.T., Schwarzkopf, J.D., Sommerfeld, M. & Tsuji, Y. 2011 Multiphase Flows with Droplets and Particles. CRC Press.CrossRefGoogle Scholar
Crowe, C.T., Troutt, T.R. & Chung, J.N. 1996 Numerical models for two-phase turbulent flows. Annu. Rev. Fluid Mech. 28 (1), 11–43.CrossRefGoogle Scholar
Deen, N.G., Van Sint Annaland, M., Van der Hoef, M.A. & Kuipers, J.A.M. 2007 Review of discrete particle modeling of fluidized beds. Chem. Engng Sci. 62 (1–2), 28–44.CrossRefGoogle Scholar
Ding, J. & Gidaspow, D. 1990 A bubbling fluidization model using kinetic theory of granular flow. AIChE J. 36 (4), 523–538.CrossRefGoogle Scholar
Du Cluzeau, A., Bois, G. & Toutant, A. 2019 Analysis and modelling of Reynolds stresses in turbulent bubbly up-flows from direct numerical simulations. J. Fluid Mech. 866, 132–168.CrossRefGoogle Scholar
Fox, R.O. 2014 On multiphase turbulence models for collisional fluid–particle flows. J. Fluid Mech. 742, 368–424.CrossRefGoogle Scholar
Gidaspow, D. 1994 Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic Press.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
Hanjalić, K. & Launder, B.E. 1976 Contribution towards a Reynolds-stress closure for low-Reynolds- number turbulence. J. Fluid Mech. 74 (4), 593–610.CrossRefGoogle Scholar
Hanjalić, K. & Launder, B. 2011 Modelling Turbulence in Engineering and the Environment: Second-Moment Routes to Closure. Cambridge University Press.CrossRefGoogle Scholar
Hoyas, S. & Jiménez, J. 2008 Reynolds number effects on the Reynolds-stress budgets in turbulent channels. Phys. Fluids 20 (10), 101511.CrossRefGoogle Scholar
Ishii, M. & Hibiki, T. 2010 Thermo-Fluid Dynamics of Two-Phase Flow. Springer Science & Business Media.Google Scholar
Jeffrey, D.J. 1982 Low-Reynolds-number flow between converging spheres. Mathematika 29 (1), 58–66.CrossRefGoogle Scholar
Joshi, J.B. & Nandakumar, K. 2015 Computational modeling of multiphase reactors. Annu. Rev. Chem. Biomol. Engng 6, 347–378.CrossRefGoogle ScholarPubMed
Kataoka, I., Besnard, D.C. & Serizawa, A. 1992 Basic equation of turbulence and modeling of interfacial transfer terms in gas-liquid two-phase flow. Chem. Engng Commun. 118 (1), 221–236.CrossRefGoogle 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
Lu, H., Gidaspow, D., Bouillard, J. & Liu, W. 2003 Hydrodynamic simulation of gas–solid flow in a riser using kinetic theory of granular flow. Chem. Engng J. 95 (1–3), 1–13.Google Scholar
Lumley, J.L. 1979 Computational modeling of turbulent flows. Adv. Appl. Mech. 18, 123–176.CrossRefGoogle Scholar
Ma, T., Lucas, D. & Bragg, A.D. 2020 a Explicit algebraic relation for calculating Reynolds normal stresses in flows dominated by bubble-induced turbulence. Phys. Rev. Fluids 5 (8), 084305.CrossRefGoogle Scholar
Ma, T., Lucas, D., Jakirlić, S. & Fröhlich, J. 2020 b Progress in the second-moment closure for bubbly flow based on direct numerical simulation data. J. Fluid Mech. 883, A9.CrossRefGoogle Scholar
Ma, T., Santarelli, C., Ziegenhein, T., Lucas, D. & Fröhlich, J. 2017 Direct numerical simulation–based Reynolds-averaged closure for bubble-induced turbulence. Phys. Rev. Fluids 2 (3), 034301.CrossRefGoogle Scholar
Mansour, N.N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15–44.CrossRefGoogle Scholar
Masood, R.M.A., Rauh, C. & Delgado, A. 2014 CFD simulation of bubble column flows: an explicit algebraic Reynolds stress model approach. Intl J. Multiphase Flow 66, 11–25.CrossRefGoogle Scholar
Maxey, M. 2017 Simulation methods for particulate flows and concentrated suspensions. Annu. Rev. Fluid Mech. 49, 171–193.CrossRefGoogle Scholar
Mehrabadi, M., Tenneti, S., Garg, R. & Subramaniam, S. 2015 Pseudo-turbulent gas-phase velocity fluctuations in homogeneous gas-solid flow: fixed particle assemblies and freely evolving suspensions. J. Fluid Mech. 770, 210–246.CrossRefGoogle Scholar
Naot, D. 1970 Interactions between components of the turbulent velocity correlation tensor. Isr. J. Technol. 8, 259–269.Google Scholar
Peng, C., Ayala, O.M. & Wang, L.P. 2019 A direct numerical investigation of two-way interactions in a particle-laden turbulent channel flow. J. Fluid Mech. 875, 1096–1144.CrossRefGoogle Scholar
Poelma, C., Westerweel, J. & Ooms, G. 2007 Particle–fluid interactions in grid-generated turbulence. J. Fluid Mech. 589, 315–351.CrossRefGoogle Scholar
Politano, M.S., Carrica, P.M. & Converti, J. 2003 A model for turbulent polydisperse two-phase flow in vertical channels. Intl J. Multiphase Flow 29 (7), 1153–1182.CrossRefGoogle Scholar
Riella, M., Kahraman, R. & Tabor, G.R. 2019 Inhomogeneity and anisotropy in Eulerian–Eulerian near-wall modelling. Intl J. Multiphase Flow 114, 9–18.CrossRefGoogle Scholar
Risso, F., Roig, V., Amoura, Z., Riboux, G. & Billet, A.M. 2008 Wake attenuation in large Reynolds number dispersed two-phase flows. Phil. Trans. R. Soc. A: Math. Phys. Engng Sci. 366 (1873), 2177–2190.CrossRefGoogle ScholarPubMed
Rotta, J.C. 1951 Statistische theorie nichthomogener turbulenz. Z. Phys. 129 (6), 547–572.CrossRefGoogle Scholar
Rzehak, R. & Krepper, E. 2013 CFD modeling of bubble-induced turbulence. Intl J. Multiphase Flow 55, 138–155.CrossRefGoogle Scholar
Saffman, P.G. 1956 On the motion of small spheroidal particles in a viscous liquid. J. Fluid Mech. 1 (5), 540–553.CrossRefGoogle Scholar
Santarelli, C. & Fröhlich, J. 2015 Direct numerical simulations of spherical bubbles in vertical turbulent channel flow. Intl J. Multiphase Flow 75, 174–193.CrossRefGoogle Scholar
Santarelli, C., Roussel, J. & Fröhlich, J. 2016 Budget analysis of the turbulent kinetic energy for bubbly flow in a vertical channel. Chem. Engng Sci. 141, 46–62.CrossRefGoogle Scholar
Sato, Y. & Hishida, K. 1996 Transport process of turbulence energy in particle-laden turbulent flow. Intl J. Heat Fluid Flow 17 (3), 202–210.CrossRefGoogle Scholar
Sato, Y., Sadatomi, M. & Sekoguchi, K. 1981 Momentum and heat transfer in two-phase bubble flow–I. Theory. Intl J. Multiphase Flow 7 (2), 167–177.CrossRefGoogle Scholar
Schwarzkopf, J.D., Crowe, C.T. & Dutta, P. 2009 A turbulence dissipation model for particle laden flow. AIChE J. 55 (6), 1416–1425.CrossRefGoogle Scholar
Shao, X., Wu, T. & Yu, Z. 2012 Fully resolved numerical simulation of particle-laden turbulent flow in a horizontal channel at a low Reynolds number. J. Fluid Mech. 693, 319–344.CrossRefGoogle Scholar
Shi, P. & Rzehak, R. 2020 Solid-liquid flow in stirred tanks: Euler-Euler/RANS modeling. Chem. Engng Sci. 227, 115875.CrossRefGoogle Scholar
Shir, C.C. 1973 A preliminary numerical study of atmospheric turbulent flows in the idealized planetary boundary layer. J. Atmos. Sci. 30 (7), 1327–1339.2.0.CO;2>CrossRefGoogle Scholar
So, R.M.C., Aksoy, H., Yuan, S.P. & Sommer, T.P. 1996 Modeling Reynolds-number effects in wall-bounded turbulent flows. Trans. ASME J. Fluids Engng 118 (2), 260–267.CrossRefGoogle Scholar
Speziale, C.G., Sarkar, S. & Gatski, T.B. 1991 Modelling the pressure–strain correlation of turbulence: an invariant dynamical systems approach. J. Fluid Mech. 227, 245–272.CrossRefGoogle Scholar
Tang, Y., Peters, E.A.J.F., Kuipers, J.A.M., Kriebitzsch, S.H.L. & 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.Google Scholar
Tenneti, S. & Subramaniam, S. 2014 Particle-resolved direct numerical simulation for gas-solid flow model development. Annu. Rev. Fluid Mech. 46, 199–230.CrossRefGoogle Scholar
Troshko, A.A. & Hassan, Y.A. 2001 A two-equation turbulence model of turbulent bubbly flows. Intl J. Multiphase Flow 27 (11), 1965–2000.CrossRefGoogle Scholar
Tsuji, Y., Kawaguchi, T. & Tanaka, T. 1993 Discrete particle simulation of two-dimensional fluidized bed. Powder Technol. 77 (1), 79–87.CrossRefGoogle Scholar
Tsuji, Y., Tanaka, T. & Ishida, T. 1992 Lagrangian numerical simulation of plug flow of cohesionless particles in a horizontal pipe. Powder Technol. 71 (3), 239–250.CrossRefGoogle Scholar
Uhlmann, M. 2008 Interface-resolved direct numerical simulation of vertical particulate channel flow in the turbulent regime. Phys. Fluids 20 (5), 053305.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
Vreman, A.W. 2015 Turbulence attenuation in particle-laden flow in smooth and rough channels. J. Fluid Mech. 773, 103–136.CrossRefGoogle Scholar
Wang, L.P., Peng, C., Guo, Z. & Yu, Z. 2016 Flow modulation by finite-size neutrally buoyant particles in a turbulent channel flow. J. Fluids Engng 138 (4), 041306.CrossRefGoogle Scholar
Wen, C.Y. & Yu, Y.H. 1966 Mechanics of fluidization. In Chemical Engineering Progress Symposium Series, vol. 62, pp. 100–111.Google Scholar
Xia, Y., Lin, Z., Pan, D. & Yu, Z. 2021 Turbulence modulation by finite-size heavy particles in a downward turbulent channel flow. Phys. Fluids 33 (6), 063321.CrossRefGoogle Scholar
Xia, Y., Yu, Z., Lin, Z. & Guo, Y. 2022 a Model of interfacial term in turbulent kinetic energy equation and computation of dissipation rate for particle-laden flows. Phys. Fluids 34 (8), 083311.CrossRefGoogle Scholar
Xia, Y., Yu, Z., Pan, D., Lin, Z. & Guo, Y. 2022 b Drag model from interface-resolved simulations of particle sedimentation in a periodic domain and vertical turbulent channel flows. J. Fluid Mech. 944, A25.CrossRefGoogle Scholar
Yao, W. & Morel, C. 2004 Volumetric interfacial area prediction in upward bubbly two-phase flow. Intl J. Heat Mass Transfer 47 (2), 307–328.CrossRefGoogle Scholar
Yu, Z., Lin, Z., Shao, X. & Wang, L.P. 2016 A parallel fictitious domain method for the interface-resolved simulation of particle-laden flows and its application to the turbulent channel flow. Engng Appl. Comput. Fluid Mech. 10 (1), 160–170.Google Scholar
Yu, Z. & Shao, X. 2007 A direct-forcing fictitious domain method for particulate flows. J. Comput. Phys. 227 (1), 292–314.CrossRefGoogle Scholar
Yu, Z., Xia, Y., Guo, Y. & Lin, J. 2021 Modulation of turbulence intensity by heavy finite-size particles in upward channel flow. J. Fluid Mech. 913, A3.CrossRefGoogle Scholar
Yu, Z., Zhu, C., Wang, Y. & Shao, X. 2019 Effects of finite-size neutrally buoyant particles on the turbulent channel flow at a Reynolds number of 395. Appl. Math. Mech. 40 (2), 293–304.CrossRefGoogle Scholar
Zhu, H.P., Zhou, Z.Y., Yang, R.Y. & Yu, A.B. 2007 Discrete particle simulation of particulate systems: theoretical developments. Chem. Engng Sci. 62 (13), 3378–3396.CrossRefGoogle Scholar