Hostname: page-component-586b7cd67f-dsjbd Total loading time: 0 Render date: 2024-11-28T05:50:54.277Z Has data issue: false hasContentIssue false
  • English
  • Français

Modulation of turbulence intensity by heavy finite-size particles in upward channel flow

Published online by Cambridge University Press:  19 February 2021

Zhaosheng Yu Open the ORCID record for Zhaosheng Yu [Opens in a new window] ,
Yan Xia ,
Yu Guo Open the ORCID record for Yu Guo [Opens in a new window]  and
Jianzhong Lin
Zhaosheng Yu*
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou310027, PR China
Yan Xia
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou310027, PR China
Yu Guo
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou310027, PR China
Jianzhong Lin
Affiliation:
State Key Laboratory of Fluid Power and Mechatronic Systems, Department of Mechanics, Zhejiang University, Hangzhou310027, PR China
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

It has been recognized that, generally, large particles enhance the turbulence intensity, while small particles attenuate the turbulence intensity. However, there has been no consensus on the quantitative criterion for particle-induced turbulence enhancement or attenuation. In the present study, interface-resolved direct numerical simulations of particle-laden turbulent flows in an upward vertical channel are performed with a direct forcing/fictitious domain method to establish a criterion for turbulence enhancement or attenuation. The effects of the particle Reynolds number ($Re_p$), the bulk Reynolds number ($Re_b$), the particle size, the density ratio and the particle volume fraction on the turbulence intensity are examined at $Re_b=5746$ (i.e. $Re_\tau =180.8$) and 12 000 ($Re_\tau =345.9$), the ratio of the particle radius to the half channel width $a/H=0.05\text {--}0.15$, the density ratio 2–100, the particle volume fraction $0.3\,\%$–$2.36\,\%$ and $Re_p < 227$. Our results indicate that at low $Re_p$ the turbulent intensity across the channel is all diminished; at intermediate $Re_p$ the turbulent intensity is enhanced in the channel centre region and attenuated in the near-wall region; and at sufficiently large $Re_p$ the turbulent intensity is enhanced across the channel. The critical $Re_p$ increases with increasing bulk Reynolds number, particle size and particle–fluid density ratio, while increasing with decreasing particle volume fraction, particularly for the channel centre region. Criteria for enhancement or attenuation are provided for the total turbulence intensity in the channel and the turbulence intensity at the channel centre, respectively, and both are shown to agree well with the experimental data in the literature. The reason for the dependence of the critical particle Reynolds number on the other parameters is discussed.

JFM classification

Multiphase and Particle-laden Flows: Particle/fluid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 913 , 25 April 2021 , A3
Check for updates
Copyright
© The Author(s), 2021. 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

REFERENCES

Ansys, Inc. 2013 ANSYS FLUENT Theory Guide. Release 15.0. Canonsburg.Google Scholar
Ardekani, M.N., Costa, P., Breugem, W.-P., Picano, F. & Brandt, L. 2017 Drag reduction in turbulent channel flow laden with finite-size oblate spheroids. J. Fluid Mech. 816, 43–70.CrossRefGoogle Scholar
Auton, T.R. 1987 The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199–218.CrossRefGoogle Scholar
Auton, T.R., Hunt, J.C.R. & Prud'Homme, M. 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241–257.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
Bellani, G., Byron, M.L., Collignon, A.G., Meyer, C.R. & Variano, E.A. 2012 Shape effects on turbulent modulation by large nearly neutrally buoyant particles. J. Fluid Mech. 712, 41–60.CrossRefGoogle Scholar
Biegert, E., Vowinckel, B. & Meiburg, E. 2017 A collision model for grain-resolving simulations of flows over dense, mobile, polydisperse granular sediment beds. J. Comput. Phys. 340, 105–127.CrossRefGoogle Scholar
Brändle de Motta, J.C., Breugem, W.-P., Gazanion, B., Estivalezes, J.-L., Vincent, S. & Climent, E. 2013 Numerical modelling of finite-size particle collisions in a viscous fluid. Phys. Fluids 25 (8), 083302.CrossRefGoogle Scholar
Caporaloni, M., Tampieri, F., Trombetti, F. & Vittori, O. 1975 Transfer of particles in nonisotropic air turbulence. J. Atmos. Sci. 32 (3), 565–568.2.0.CO;2>CrossRefGoogle Scholar
Clift, R., Grace, J.R. & Weber, M.E. 2005 Bubbles, Drops, and Particles. Courier Corporation.Google Scholar
Costa, P., Boersma, B.J., Westerweel, J. & Breugem, W.-P. 2015 Collision model for fully resolved simulations of flows laden with finite-size particles. Phys. Rev. E 92 (5), 053012.CrossRefGoogle ScholarPubMed
Costa, P., Brandt, L. & Picano, F. 2020 Interface-resolved simulations of small inertial particles in turbulent channel flow. J. Fluid Mech. 883, A54.CrossRefGoogle Scholar
Costa, P., Picano, F., Brandt, L. & Breugem, W.-P. 2018 Effects of the finite particle size in turbulent wall-bounded flows of dense suspensions. J. Fluid Mech. 843, 450–478.CrossRefGoogle Scholar
Crowe, C.T. 2000 On models for turbulence modulation in fluid–particle flows. Intl J. Multiphase Flow 26 (5), 719–727.CrossRefGoogle Scholar
Crowe, C.T., Schwarzkopf, J.D., Sommerfeld, M. & Tsuji, Y. 2011 Multiphase Flows with Droplets and Particles. CRC Press.CrossRefGoogle Scholar
Eaton, J.K. 2009 Two-way coupled turbulence simulations of gas-particle flows using point-particle tracking. Intl J. Multiphase Flow 35 (9), 792–800.CrossRefGoogle Scholar
Elghobashi, S. & Truesdell, G.C. 1993 On the two-way interaction between homogeneous turbulence and dispersed solid particles. I: turbulence modification. Phys. Fluids A: Fluid Dyn. 5 (7), 1790–1801.CrossRefGoogle Scholar
Fornari, W., Formenti, A., Picano, F. & Brandt, L. 2016 The effect of particle density in turbulent channel flow laden with finite size particles in semi-dilute conditions. Phys. Fluids 28 (3), 033301.CrossRefGoogle Scholar
Fornari, W., Kazerooni, H.T., Hussong, J. & Brandt, L. 2018 Suspensions of finite-size neutrally buoyant spheres in turbulent duct flow. J. Fluid Mech. 851, 148–186.CrossRefGoogle Scholar
Fukagata, K., Iwamoto, K. & Kasagi, N. 2002 Contribution of Reynolds stress distribution to the skin friction in wall-bounded flows. Phys. Fluids 14 (11), L73–L76.CrossRefGoogle Scholar
Gao, H., Li, H. & Wang, L.-P. 2013 Lattice Boltzmann simulation of turbulent flow laden with finite-size particles. Comput. Maths Applics. 65 (2), 194–210.CrossRefGoogle Scholar
Garcia-Villalba, M., Kidanemariam, A.G. & Uhlmann, M. 2012 DNS of vertical plane channel flow with finite-size particles: Voronoi analysis, acceleration statistics and particle-conditioned averaging. Intl J. Multiphase Flow 46, 54–74.CrossRefGoogle 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
Goes Oliveira, J.L., van der Geld, C.W.M. & Kuerten, J.G.M. 2017 Concentration and velocity statistics of inertial particles in upward and downward pipe flow. J. Fluid Mech. 822, 640–663.CrossRefGoogle Scholar
Gondret, P., Lance, M. & Petit, L. 2002 Bouncing motion of spherical particles in fluids. Phys. Fluids 14 (2), 643–652.CrossRefGoogle Scholar
Gore, R.A. & Crowe, C.T. 1989 Effect of particle size on modulating turbulent intensity. Intl J. Multiphase Flow 15 (2), 279–285.CrossRefGoogle Scholar
Hertz, H.R. 1882 Uber die beruhrung fester elastischer korper und uber die harte. In Verhandlung des Vereins zur Beforderung des GewerbefleiBes, Berlin p. 449. Duncker.CrossRefGoogle Scholar
Hetsroni, G. 1989 Particles-turbulence interaction. Intl J. Multiphase Flow 15 (5), 735–746.CrossRefGoogle Scholar
Hosokawa, S. & Tomiyama, A. 2004 Turbulence modification in gas–liquid and solid–liquid dispersed two-phase pipe flows. Intl J. Heat Fluid Flow 25 (3), 489–498.CrossRefGoogle Scholar
Hosokawa, S. & Tomiyama, A. 2009 Multi-fluid simulation of turbulent bubbly pipe flows. Chem. Engng Sci. 64 (24), 5308–5318.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
Jeffrey, D.J. 1982 Low-Reynolds-number flow between converging spheres. Mathematika 29 (1), 58–66.CrossRefGoogle Scholar
Kajishima, T., Takiguchi, S., Hamasaki, H. & Miyake, Y. 2001 Turbulence structure of particle-laden flow in a vertical plane channel due to vortex shedding. JSME Intl J. Ser. B Fluids Therm. Engng 44 (4), 526–535.CrossRefGoogle Scholar
Kempe, T. & Fröhlich, J. 2012 Collision modelling for the interface-resolved simulation of spherical particles in viscous fluids. J. Fluid Mech. 709, 445–489.CrossRefGoogle Scholar
Kiger, K.T. & Pan, C. 2002 Suspension and turbulence modification effects of solid particulates on a horizontal turbulent channel flow. J. Turbul. 3 (19), 1–17.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
Kussin, J. & Sommerfeld, M. 2002 Experimental studies on particle behaviour and turbulence modification in horizontal channel flow with different wall roughness. Exp. Fluids 33 (1), 143–159.CrossRefGoogle Scholar
Lee, S.L. & Durst, F. 1982 On the motion of particles in turbulent duct flows. Intl J. Multiphase Flow 8 (2), 125–146.CrossRefGoogle Scholar
Li, Y., McLaughlin, J.B., Kontomaris, K. & Portela, L. 2001 Numerical simulation of particle-laden turbulent channel flow. Phys. Fluids 13 (10), 2957–2967.CrossRefGoogle Scholar
Lin, Z.-W., Shao, X.-M., Yu, Z.-S. & Wang, L.-P. 2017 b Effects of finite-size heavy particles on the turbulent flows in a square duct. J. Hydrodyn. 29 (2), 272–282.CrossRefGoogle Scholar
Lin, Z., Yu, Z., Shao, X. & Wang, L.-P. 2017 a Effects of finite-size neutrally buoyant particles on the turbulent flows in a square duct. Phys. Fluids 29 (10), 103304.CrossRefGoogle Scholar
Liu, C., Tang, S., Shen, L. & Dong, Y. 2017 Characteristics of turbulence transport for momentum and heat in particle-laden turbulent vertical channel flows. Acta Mechanica Sin. 33 (5), 833–845.CrossRefGoogle Scholar
Lucci, F., Ferrante, A. & Elghobashi, S. 2010 Modulation of isotropic turbulence by particles of Taylor length-scale size. J. Fluid Mech. 650, 5–55.CrossRefGoogle Scholar
Luo, K., Luo, M. & Fan, J. 2016 On turbulence modulation by finite-size particles in dilute gas-solid internal flows. Powder Technol. 301, 1259–1263.CrossRefGoogle Scholar
Ma, T., Lucas, D., Jakirlić, S. & Fröhlich, J. 2020 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
Maeda, M., Hishida, K. & Furutani, T. 1980 Velocity distributions of air-solids suspension in upward pipe flow: effect of particles on air velocity distribution. In Transactions of the Japan Society of Mechanical Engineers, Series B, vol. 46 (412), pp. 2313–2320. Japan Society of Mechanical Engineers.CrossRefGoogle Scholar
Maxey, M. 2017 Simulation methods for particulate flows and concentrated suspensions. Annu. Rev. Fluid Mech. 49, 171–193.CrossRefGoogle Scholar
Mena, S.E. & Curtis, J.S. 2020 Experimental data for solid–liquid flows at intermediate and high stokes numbers. J. Fluid Mech. 883, A24.CrossRefGoogle Scholar
Mindlin, R.D. 1953 Elastic spheres in contact under varying oblique forces. J. Appl. Mech. 20, 327–344.Google Scholar
Muramulla, P., Tyagi, A., Goswami, P.S. & Kumaran, V. 2020 Disruption of turbulence due to particle loading in a dilute gas–particle suspension. J. Fluid Mech. 889, A28.CrossRefGoogle Scholar
Noguchi, K. & Nezu, I. 2009 Particle–turbulence interaction and local particle concentration in sediment-laden open-channel flows. J. Hydro-Environ. Res. 3 (2), 54–68.CrossRefGoogle Scholar
Parthasarathy, R.N. & Faeth, G.M. 1990 Turbulence modulation in homogeneous dilute particle-laden flows. J. Fluid Mech. 220, 485–514.CrossRefGoogle 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
Peng, C. & Wang, L.-P. 2019 Direct numerical simulations of turbulent pipe flow laden with finite-size neutrally buoyant particles at low flow Reynolds number. Acta Mechanica 230 (2), 517–539.CrossRefGoogle Scholar
Petersen, A.J., Baker, L. & Coletti, F. 2019 Experimental study of inertial particles clustering and settling in homogeneous turbulence. J. Fluid Mech. 864, 925–970.CrossRefGoogle Scholar
Picano, F., Breugem, W.-P. & Brandt, L. 2015 Turbulent channel flow of dense suspensions of neutrally buoyant spheres. J. Fluid Mech. 764, 463–487.CrossRefGoogle Scholar
Reeks, M.W. 1983 The transport of discrete particles in inhomogeneous turbulence. J. Aerosol Sci. 14 (6), 729–739.CrossRefGoogle Scholar
Righetti, M. & Romano, G.P. 2004 Particle–fluid interactions in a plane near-wall turbulent flow. J. Fluid Mech. 505, 93–121.CrossRefGoogle Scholar
Saber, A., Lundström, T.S. & Hellström, J.G.I. 2015 Turbulent modulation in particulate flow: a review of critical variables. Engineering 7 (10), 597.CrossRefGoogle Scholar
Saffman, P.G.T. 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22 (2), 385–400.CrossRefGoogle Scholar
Saito, I., Watanabe, T. & Gotoh, T. 2019 A new time scale for turbulence modulation by particles. J. Fluid Mech. 880, R6.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
Schreck, S. & Kleis, S.J. 1993 Modification of grid-generated turbulence by solid particles. J. Fluid Mech. 249, 665–688.CrossRefGoogle Scholar
Serizawa, A., Kataoka, I. & Michiyoshi, I. 1975 Turbulence structure of air-water bubbly flow-II. Local properties. Intl J. Multiphase Flow 2 (3), 235–246.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
Shokri, R., Ghaemi, S., Nobes, D.S. & Sanders, R.S. 2017 Investigation of particle-laden turbulent pipe flow at high-Reynolds-number using particle image/tracking velocimetry (piv/ptv). Intl J. Multiphase Flow 89, 136–149.CrossRefGoogle Scholar
Squires, K.D. & Eaton, J.K. 1990 Particle response and turbulence modification in isotropic turbulence. Phys. Fluids A: Fluid Dyn. 2 (7), 1191–1203.CrossRefGoogle Scholar
Suzuki, Y., Ikenoya, M. & Kasagi, N. 2000 Simultaneous measurement of fluid and dispersed phases in a particle-laden turbulent channel flow with the aid of 3-D PTV. Exp. Fluids 29 (1), S185–S193.CrossRefGoogle Scholar
Tanaka, T. & Eaton, J.K. 2008 Classification of turbulence modification by dispersed spheres using a novel dimensionless number. Phys. Rev. Lett. 101 (11), 114502.CrossRefGoogle ScholarPubMed
Ten Cate, A., 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 Mech. 519, 233–271.CrossRefGoogle 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
Theofanous, T.G. & Sullivan, J. 1982 Turbulence in two-phase dispersed flows. J. Fluid Mech. 116, 343–362.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. & Morikawa, Y. 1982 LDV measurements of an air—solid two-phase flow in a horizontal pipe. J. Fluid Mech. 120, 385–409.CrossRefGoogle Scholar
Tsuji, Y., Morikawa, Y. & Shiomi, H. 1984 LDV measurements of an air-solid two-phase flow in a vertical pipe. J. Fluid Mech. 139, 417–434.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
Vreman, A.W. 2015 Turbulence attenuation in particle-laden flow in smooth and rough channels. J. Fluid Mech. 773, 103–136.CrossRefGoogle Scholar
Vreman, A.W. & Kuerten, J.G.M. 2018 Turbulent channel flow past a moving array of spheres. J. Fluid Mech. 856, 580–632.CrossRefGoogle Scholar
Wang, B. 2010 Inter-phase interaction in a turbulent, vertical channel flow laden with heavy particles. Part II: two-phase velocity statistical properties. Intl J. Heat Mass Transfer 53 (11-12), 2522–2529.CrossRefGoogle Scholar
Wang, G., Abbas, M. & Climent, É. 2017 Modulation of large-scale structures by neutrally buoyant and inertial finite-size particles in turbulent Couette flow. Phys. Rev. Fluids 2 (8), 084302.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, 103105.CrossRefGoogle Scholar
Wang, L.-P. & Maxey, M.R. 1993 Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256, 27–68.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
Wu, T.-H., Shao, X.-M. & Yu, Z.-S. 2011 Fully resolved numerical simulation of turbulent pipe flows laden with large neutrally-buoyant particles. J. Hydrodyn. 23 (1), 21–25.CrossRefGoogle Scholar
Xia, Y., Xiong, H., Yu, Z. & Zhu, C. 2020 a Effects of the collision model in interface-resolved simulations of particle-laden turbulent channel flows. Phys. Fluids 32 (10), 103303.CrossRefGoogle Scholar
Xia, Y., Yu, Z. & Deng, J. 2019 A fictitious domain method for particulate flows of arbitrary density ratio. Comput. Fluids 193, 104293.CrossRefGoogle Scholar
Xia, Y., Yu, Z. & Guo, Y. 2020 b Interface-resolved numerical simulations of particle-laden turbulent channel flows with spanwise rotation. Phys. Fluids 32 (1), 013303.CrossRefGoogle Scholar
Yu, Z., Lin, Z., Shao, X. & Wang, L.-P. 2016 b 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., Lin, Z., Shao, X. & Wang, L.-P. 2017 Effects of particle-fluid density ratio on the interactions between the turbulent channel flow and finite-size particles. Phys. Rev. E 96 (3), 033102.CrossRefGoogle ScholarPubMed
Yu, Z., Phan-Thien, N. & Tanner, R.I. 2004 Dynamic simulation of sphere motion in a vertical tube. J. Fluid Mech. 518, 61–93.CrossRefGoogle 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, W., Vinkovic, I. & Buffat, M. 2016 a Finite-size particles in turbulent channel flow: quadrant analysis and acceleration statistics. J. Turbul. 17 (11), 1048–1071.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. Maths Mech. 40 (2), 293–304.CrossRefGoogle Scholar
Zade, S., Lundell, F. & Brandt, L. 2019 Turbulence modulation by finite-size spherical particles in newtonian and viscoelastic fluids. Intl J. Multiphase Flow 112, 116–129.CrossRefGoogle Scholar
Zhao, L.H., Andersson, H.I. & Gillissen, J.J.J. 2010 Turbulence modulation and drag reduction by spherical particles. Phys. Fluids 22 (8), 081702.CrossRefGoogle Scholar
Zhao, L., Andersson, H.I. & Gillissen, J.J.J. 2013 Interphasial energy transfer and particle dissipation in particle-laden wall turbulence. J. Fluid Mech. 715, 32–59.CrossRefGoogle Scholar
Zhu, H.-Y., Pan, C., Wang, J.-J., Liang, Y.-R. & Ji, X.-C. 2019 Sand-turbulence interaction in a high-Reynolds-number turbulent boundary layer under net sedimentation conditions. Intl J. Multiphase Flow 119, 56–71.CrossRefGoogle Scholar
Zhu, C., Yu, Z., Pan, D. & Shao, X. 2020 a Interface-resolved direct numerical simulations of the interactions between spheroidal particles and upward vertical turbulent channel flows. J. Fluid Mech. 891, A6.CrossRefGoogle Scholar
Zhu, C., Yu, Z. & Shao, X. 2018 Interface-resolved direct numerical simulations of the interactions between neutrally buoyant spheroidal particles and turbulent channel flows. Phys. Fluids 30 (11), 115103.CrossRefGoogle Scholar
Zhu, C., Yu, Z., Shao, X. & Deng, J. 2020 b Interface-resolved numerical simulations of particle-laden turbulent flows in a vertical channel filled with Bingham fluids. J. Fluid Mech. 883, A43.CrossRefGoogle Scholar