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Non-monotonic effect of Stokes number on turbulence modulation in particle-laden channel flow

Published online by Cambridge University Press:  24 July 2025

Zi-Mo Liao Open the ORCID record for Zi-Mo Liao [Opens in a new window] ,
Fenghui Lin Open the ORCID record for Fenghui Lin [Opens in a new window] ,
Luoqin Liu Open the ORCID record for Luoqin Liu [Opens in a new window] ,
Nan-Sheng Liu Open the ORCID record for Nan-Sheng Liu [Opens in a new window]  and
Xi-Yun Lu Open the ORCID record for Xi-Yun Lu [Opens in a new window]
Zi-Mo Liao
Affiliation:
State Key Laboratory of High Temperature Gas Dynamics, School of Engineering Science, University of Science and Technology of China, Hefei 230027, PR China
Fenghui Lin
Affiliation:
State Key Laboratory of High Temperature Gas Dynamics, School of Engineering Science, University of Science and Technology of China, Hefei 230027, PR China
Luoqin Liu
Affiliation:
State Key Laboratory of High Temperature Gas Dynamics, School of Engineering Science, University of Science and Technology of China, Hefei 230027, PR China
Nan-Sheng Liu*
Affiliation:
State Key Laboratory of High Temperature Gas Dynamics, School of Engineering Science, University of Science and Technology of China, Hefei 230027, PR China
Xi-Yun Lu*
Affiliation:
State Key Laboratory of High Temperature Gas Dynamics, School of Engineering Science, University of Science and Technology of China, Hefei 230027, PR China
*
Corresponding authors: Nan-Sheng Liu, lns@ustc.edu.cn; Xi-Yun Lu, xlu@ustc.edu.cn
Corresponding authors: Nan-Sheng Liu, lns@ustc.edu.cn; Xi-Yun Lu, xlu@ustc.edu.cn
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Abstract

The effect of Stokes number on turbulence modulation in particle-laden channel flow is investigated through four-way coupled point-particle direct numerical simulations, with the mass loading fixed at 0.6 and the friction Stokes number $St^+$ varying from 3 to 300. A full transition pathway is observed, from a drag-enhanced to a drag-reduced regime, eventually approaching the single-phase state as $St^+$ increases towards 300. A set of transport equations for the particle phase is derived analytically to characterise the interphase coupling, within the framework of the point-based statistical description of particle-laden turbulence. By virtue of this, two dominant mechanisms are identified and quantitatively characterised: a positive, particle-induced extra transport that decreases monotonically with increasing $St^+$, and a negative, particle-induced extra dissipation that varies non-monotonically with $St^+$. The coupling of these two mechanisms leads to a direct contribution of the particle phase to the shear stress balance, the turbulent kinetic energy budgets and the Reynolds stress budgets. Consequently, as $St^+$ increases, the self-sustaining cycle of near-wall turbulence transitions from being augmented to being suppressed and, eventually, returns to the single-phase state. This gives rise to an indirect effect, manifested as a non-monotonic modulation of Reynolds shear stress and turbulence production rate. Taken together, complex interplays between particle-modified turbulent transport, particle-induced extra transport and extra dissipation are analysed and summarised, providing a holistic physical picture composed of consistent interpretations of turbulence modulation induced by small heavy particles.

JFM classification

Multiphase and Particle-laden Flows: Multiphase and Particle-laden Flows Turbulent Flows: Turbulent Flows

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JFM Papers
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Journal of Fluid Mechanics , Volume 1015 , 25 July 2025 , A55
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© The Author(s), 2025. Published by Cambridge University Press

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References

Bagnold, R.A. 2012 The Physics of Blown Sand and Desert Dunes. Courier Corporation.Google Scholar
Balachandar, S. & Eaton, J.K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42 (1), 111–133.10.1146/annurev.fluid.010908.165243CrossRefGoogle 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, 160–185.10.1016/j.jcp.2018.09.033CrossRefGoogle Scholar
Battista, F., Gualtieri, P., Mollicone, J.P., Salvadore, F. & Casciola, C.M. 2023 Drag increase and turbulence augmentation in two-way coupled particle-laden wall-bounded flows. Phys. Fluids 35 (4), 045133.10.1063/5.0141964CrossRefGoogle Scholar
Bec, J., Gustavsson, K. & Mehlig, B. 2024 Statistical models for the dynamics of heavy particles in turbulence. Annu. Rev. Fluid Mech. 56 (1), 189–213.10.1146/annurev-fluid-032822-014140CrossRefGoogle Scholar
Berk, T. & Coletti, F. 2024 An analytical model for the slip velocity of particles in turbulence. J. Fluid Mech. 996, A1.10.1017/jfm.2024.620CrossRefGoogle Scholar
Bernardini, M. 2014 Reynolds number scaling of inertial particle statistics in turbulent channel flows. J. Fluid Mech. 758, R1.10.1017/jfm.2014.561CrossRefGoogle Scholar
Bernardini, M., Modesti, D., Salvadore, F. & Pirozzoli, S. 2021 STREAmS: a high-fidelity accelerated solver for direct numerical simulation of compressible turbulent flows. Comput. Phys. Commun. 263, 107906.10.1016/j.cpc.2021.107906CrossRefGoogle Scholar
Bijlard, M.J., Oliemans, R.V.A., Portela, L.M. & Ooms, G. 2010 Direct numerical simulation analysis of local flow topology in a particle-laden turbulent channel flow. J. Fluid Mech. 653, 35–56.10.1017/S0022112010000169CrossRefGoogle Scholar
Boivin, M., Simonin, O. & Squires, K.D. 1998 Direct numerical simulation of turbulence modulation by particles in isotropic turbulence. J. Fluid Mech. 375, 235–263.10.1017/S0022112098002821CrossRefGoogle Scholar
Bragg, A., Swailes, D.C. & Skartlien, R. 2012 Drift-free kinetic equations for turbulent dispersion. Phys. Rev. E 86 (5), 056306.10.1103/PhysRevE.86.056306CrossRefGoogle ScholarPubMed
Bragg, A.D., Richter, D.H. & Wang, G. 2021 a Mechanisms governing the settling velocities and spatial distributions of inertial particles in wall-bounded turbulence. Phys. Rev. Fluids 6 (6), 064302.10.1103/PhysRevFluids.6.064302CrossRefGoogle Scholar
Bragg, A.D., Richter, D.H. & Wang, G. 2021 b Settling strongly modifies particle concentrations in wall-bounded turbulent flows even when the settling parameter is asymptotically small. Phys. Rev. Fluids 6 (12), 124301.10.1103/PhysRevFluids.6.124301CrossRefGoogle Scholar
Brandt, L. & Coletti, F. 2022 Particle-laden turbulence: progress and perspectives. Annu. Rev. Fluid Mech. 54 (1), 159–189.10.1146/annurev-fluid-030121-021103CrossRefGoogle Scholar
Brooke, J.W., Hanratty, T.J. & McLaughlin, J.B. 1994 Freeflight mixing and deposition of aerosols. Phys. Fluids 6 (10), 3404–3415.10.1063/1.868398CrossRefGoogle Scholar
Capecelatro, J. & Desjardins, O. 2013 An Euler-Lagrange strategy for simulating particle-laden flows. J. Comput. Phys. 238, 1–31.10.1016/j.jcp.2012.12.015CrossRefGoogle Scholar
Capecelatro, J., Desjardins, O. & Fox, R.O. 2015 On fluid-particle dynamics in fully developed cluster-induced turbulence. J. Fluid Mech. 780, 578–635.10.1017/jfm.2015.459CrossRefGoogle Scholar
Capecelatro, J., Desjardins, O. & Fox, R.O. 2016 Strongly coupled fluid-particle flows in vertical channels. I. Reynolds-averaged two-phase turbulence statistics. Phys. Fluids 28 (3), 033306.10.1063/1.4943231CrossRefGoogle Scholar
Capecelatro, J., Desjardins, O. & Fox, R.O. 2018 On the transition between turbulence regimes in particle-laden channel flows. J. Fluid Mech. 845, 499–519.10.1017/jfm.2018.259CrossRefGoogle Scholar
Caporaloni, M., Tampieri, F., Trombetti, F. & Vittori, O. 1975 Transfer of particles in nonisotropic air turbulence. J. Atmos. Sci. 32 (3), 565–568.10.1175/1520-0469(1975)032<0565:TOPINA>2.0.CO;22.0.CO;2>CrossRefGoogle Scholar
Caraman, N., Borée, J. & Simonin, O. 2003 Effect of collisions on the dispersed phase fluctuation in a dilute tube flow: experimental and theoretical analysis. Phys. Fluids 15 (12), 3602–3612.10.1063/1.1619136CrossRefGoogle Scholar
Chou, P.Y. 1945 On velocity correlations and the solutions of the equations of turbulent fluctuation. Q. Appl. Maths 3 (1), 38–54.10.1090/qam/11999CrossRefGoogle Scholar
Costa, P., Brandt, L. & Picano, F. 2021 Near-wall turbulence modulation by small inertial particles. J. Fluid Mech. 922, A9.10.1017/jfm.2021.507CrossRefGoogle Scholar
Crowe, C.T., Schwarzkopf, J.D., Sommerfeld, M. & Tsuji, Y. 2011 Multiphase Flows with Droplets and Particles. CRC Press.10.1201/b11103CrossRefGoogle Scholar
Crowe, C.T., Sharma, M.P. & Stock, D.E. 1977 The particle-source-in cell (PSI-CELL) model for gas-droplet flows. J. Fluids Eng. 99 (2), 325–332.10.1115/1.3448756CrossRefGoogle Scholar
Cui, Y., Zhang, H. & Zheng, X. 2024 Turbulence modulation by charged inertial particles in channel flow. J. Fluid Mech. 990, A2.10.1017/jfm.2024.508CrossRefGoogle Scholar
Dave, H. & Kasbaoui, M.H. 2023 Mechanisms of drag reduction by semidilute inertial particles in turbulent channel flow. Phys. Rev. Fluids 8 (8), 084305.10.1103/PhysRevFluids.8.084305CrossRefGoogle Scholar
Dritselis, C.D. 2016 Direct numerical simulation of particle-laden turbulent channel flows with two- and four-way coupling effects: budgets of Reynolds stress and streamwise enstrophy. Fluid Dyn. Res. 48 (1), 015507.10.1088/0169-5983/48/1/015507CrossRefGoogle Scholar
Eaton, J.K. 2009 Two-way coupled turbulence simulations of gas-particle flows using point-particle tracking. Int. J. Multiph. Flow 35 (9), 792–800.10.1016/j.ijmultiphaseflow.2009.02.009CrossRefGoogle Scholar
Eaton, J.K. & Longmire, E.K. 2016 Turbulence interactions. In Multiphase Flow Handbook, pp. 729–752.CRC Press.10.1201/9781315371924-12CrossRefGoogle Scholar
Elghobashi, S. 1991 Particle-laden turbulent flows: direct simulation and closure models. Appl. Sci. Res. 48 (3), 301–314.10.1007/BF02008202CrossRefGoogle Scholar
Elghobashi, S. 1994 On predicting particle-laden turbulent flows. Appl. Sci. Res. 52 (4), 309–329.10.1007/BF00936835CrossRefGoogle Scholar
Epanechnikov, V.A. 1969 Non-parametric estimation of a multivariate probability density. Theory Probab. Appl. 14 (1), 153–158.10.1137/1114019CrossRefGoogle 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, 960–982.10.1016/j.jcp.2018.09.009CrossRefGoogle 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. Int. J. Multiphase Flow 132, 103410.10.1016/j.ijmultiphaseflow.2020.103410CrossRefGoogle Scholar
Evrard, F., Denner, F. & van Wachem, B. 2020 Euler-Lagrange modelling of dilute particle-laden flows with arbitrary particle-size to mesh-spacing ratio. J. Comput. Phys. X 8, 100078.Google Scholar
Fong, K.O., Amili, O. & Coletti, F. 2019 Velocity and spatial distribution of inertial particles in a turbulent channel flow. J. Fluid Mech. 872, 367–406.10.1017/jfm.2019.355CrossRefGoogle Scholar
Fox, R.O. 2007 Introduction and Fundamentals of Modeling Approaches for Polydisperse Multiphase Flows. Springer Vienna, 1–40.Google Scholar
Fox, R.O. 2012 Large-eddy-simulation tools for multiphase flows. Annu. Rev. Fluid Mech. 44 (1), 47–76.10.1146/annurev-fluid-120710-101118CrossRefGoogle Scholar
Fox, R.O. 2014 On multiphase turbulence models for collisional fluid-particle flows. J. Fluid Mech. 742, 368–424.10.1017/jfm.2014.21CrossRefGoogle Scholar
Fox, R.O., Posey, J.W., Houim, R.W. & Laurent, F. 2024 A kinetic-based model for polydisperse, high-speed, fluid-particle flows. Intl J. Multiphase Flow 171, 104698.10.1016/j.ijmultiphaseflow.2023.104698CrossRefGoogle Scholar
Frankel, A., Pouransari, H., Coletti, F. & Mani, A. 2016 Settling of heated particles in homogeneous turbulence. J. Fluid Mech. 792, 869–893.10.1017/jfm.2016.102CrossRefGoogle 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. Int. J. Multiphase Flow 113, 165–178.10.1016/j.ijmultiphaseflow.2019.01.009CrossRefGoogle 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–L76.10.1063/1.1516779CrossRefGoogle Scholar
Gao, W., Samtaney, R. & Richter, D.H. 2023 Direct numerical simulation of particle-laden flow in an open channel at Re τ = 5186. J. Fluid Mech. 957, A3.10.1017/jfm.2023.26CrossRefGoogle Scholar
Gerolymos, G.A. & Vallet, I. 2014 Pressure, density, temperature and entropy fluctuations in compressible turbulent plane channel flow. J. Fluid Mech. 757, 701–746.10.1017/jfm.2014.431CrossRefGoogle Scholar
Gore, R.A. & Crowe, C.T. 1989 Effect of particle size on modulating turbulent intensity. Int. J. Multiphase Flow 15 (2), 279–285.10.1016/0301-9322(89)90076-1CrossRefGoogle Scholar
Gottlieb, S. & Shu, C.-W. 1998 Total variation diminishing Runge-Kutta schemes. Math. Comput. 67 (221), 73–85.10.1090/S0025-5718-98-00913-2CrossRefGoogle Scholar
Gualtieri, P., Battista, F., Salvadore, F. & Casciola, C.M. 2023 Effect of stokes number and particle-to-fluid density ratio on turbulence modification in channel flows. J. Fluid Mech. 974, A26.10.1017/jfm.2023.851CrossRefGoogle 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, 520–561.10.1017/jfm.2015.258CrossRefGoogle Scholar
Guha, A. 1997 A unified Eulerian theory of turbulent deposition to smooth and rough surfaces. J. Aerosol Sci. 28 (8), 1517–1537.10.1016/S0021-8502(97)00028-1CrossRefGoogle Scholar
Guha, A. 2008 Transport and deposition of particles in turbulent and laminar flow. Annu. Rev. Fluid Mech. 40 (1), 311–341.10.1146/annurev.fluid.40.111406.102220CrossRefGoogle Scholar
Hetsroni, G. 1989 Particles-turbulence interaction. Int. J. Multiphase Flow 15 (5), 735–746.10.1016/0301-9322(89)90037-2CrossRefGoogle Scholar
Hoomans, B.P.B., Kuipers, J.A.M., Briels, W.J. & van Swaaij, W.P.M. 1996 Discrete particle simulation of bubble and slug formation in a two-dimensional gas-fluidised bed: a hard-sphere approach. Chem. Eng. Sci. 51 (1), 99–118.10.1016/0009-2509(95)00271-5CrossRefGoogle Scholar
Horwitz, J.A.K. 2023 Improved force models for Euler-lagrange computations. In Modeling Approaches and Computational Methods for Particle-Laden Turbulent Flows, (ed S. Subramaniam & S. Balachandar), pp. 265–298. Academic Press.10.1016/B978-0-32-390133-8.00015-3CrossRefGoogle 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, 85–109.10.1016/j.jcp.2016.04.034CrossRefGoogle 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. Int. J. Multiphase Flow 101, 74–84.10.1016/j.ijmultiphaseflow.2018.01.003CrossRefGoogle 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 (10), 104302.10.1103/PhysRevFluids.5.104302CrossRefGoogle Scholar
Huang, P.G., Coleman, G.N. & Bradshaw, P. 1995 Compressible turbulent channel flows: DNS results and modelling. J. Fluid Mech. 305, 185–218.10.1017/S0022112095004599CrossRefGoogle Scholar
Ireland, P.J. & Desjardins, O. 2017 Improving particle drag predictions in Euler-Lagrange simulations with two-way coupling. J. Comput. Phys. 338, 405–430.10.1016/j.jcp.2017.02.070CrossRefGoogle Scholar
Jenkins, J.T. & Savage, S.B. 1983 A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles. J. Fluid Mech. 130 (–1), 187–202.10.1017/S0022112083001044CrossRefGoogle Scholar
Jie, Y., Cui, Z., Xu, C. & Zhao, L. 2022 On the existence and formation of multi-scale particle streaks in turbulent channel flows. J. Fluid Mech. 935, A18.10.1017/jfm.2022.8CrossRefGoogle 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, 103182.10.1016/j.ijmultiphaseflow.2019.103182CrossRefGoogle 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, A27.10.1017/jfm.2019.865CrossRefGoogle Scholar
Kiger, K.T. & Pan, C. 2002 Suspension and turbulence modification effects of solid particulates on a horizontal turbulent channel flow. J. Turbul. 3 (1), 019.10.1088/1468-5248/3/1/019CrossRefGoogle Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds number. J. Fluid Mech. 177, 133–166.10.1017/S0022112087000892CrossRefGoogle Scholar
Kremer, G.M. 2010 An Introduction to the Boltzmann Equation and Transport Processes in Gases. Springer.10.1007/978-3-642-11696-4CrossRefGoogle Scholar
Kuerten, J.G.M. 2016 Point-particle DNS and LES of particle-laden turbulent flow – a state-of-the-art review. Flow Turbul. Combust. 97 (3), 689–713.10.1007/s10494-016-9765-yCrossRefGoogle Scholar
Kuerten, J.G.M., Van Der Geld, C.W.M. & Geurts, B.J. 2011 Turbulence modification and heat transfer enhancement by inertial particles in turbulent channel flow. Phys. Fluids 23 (12), 123301.10.1063/1.3663308CrossRefGoogle Scholar
Kuerten, J.G.M. & Vreman, A.W. 2015 Effect of droplet interaction on droplet-laden turbulent channel flow. Phys. Fluids 27 (5), 053304.10.1063/1.4921492CrossRefGoogle Scholar
Kulick, J.D. 1994 On the Interactions Between Particles and Turbulence in a Fully-Developed Channel Flow in Air. Stanford University.Google 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, 109–134.10.1017/S0022112094002703CrossRefGoogle 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), 143–159.10.1007/s00348-002-0485-9CrossRefGoogle Scholar
Lee, J. & Lee, C. 2015 Modification of particle-laden near-wall turbulence: effect of stokes number. Phys. Fluids 27 (2), 023303.10.1063/1.4908277CrossRefGoogle Scholar
Lee, J. & Lee, C. 2019 The effect of wall-normal gravity on particle-laden near-wall turbulence. J. Fluid Mech. 873, 475–507.10.1017/jfm.2019.400CrossRefGoogle Scholar
Li, Y., McLaughlin, J.B., Kontomaris, K. & Portela, L. 2001 Numerical simulation of particle-laden turbulent channel flow. Phys. Fluids 13 (10), 2957–2967.10.1063/1.1396846CrossRefGoogle Scholar
Liao, Z.-M., Chen, L.-B., Wan, Z.-H., Liu, N.-S. & Lu, X.-Y. 2024 GPU acceleration of four-way coupled PP-DNS for compressible particle-laden wall turbulence. Intl J. Multiphase Flow 176, 104840.10.1016/j.ijmultiphaseflow.2024.104840CrossRefGoogle Scholar
Lifschitz, E.M. & Pitajewski, L.P. 1981 Physical kinetics. In Course of Theoretical Physics, vol. 10, Butterworth-Heinemann.Google Scholar
Lin, F., Song, J., Liu, N., Wan, Z., Lu, X.-Y. & Khomami, B. 2024 Maximum drag enhancement asymptote in turbulent Taylor-couette flow of dilute polymeric solutions. J. Non-Newton. Fluid Mech. 323, 105172.10.1016/j.jnnfm.2023.105172CrossRefGoogle Scholar
Lube, G., Breard, E.C.P., Esposti-Ongaro, T., Dufek, J. & Brand, B. 2020 Multiphase flow behaviour and hazard prediction of pyroclastic density currents. Nat. Rev. Earth Environ. 1 (7), 348–365.10.1038/s43017-020-0064-8CrossRefGoogle Scholar
Mallouppas, G. & van Wachem, B. 2013 Large eddy simulations of turbulent particle-laden channel flow. Intl J. Multiphase Flow 54, 65–75.10.1016/j.ijmultiphaseflow.2013.02.007CrossRefGoogle Scholar
Mansour, N.N., Kim, J. & Moin, P. 1988 Reynolds-stress and dissipation-rate budgets in a turbulent channel flow. J. Fluid Mech. 194, 15–44.10.1017/S0022112088002885CrossRefGoogle Scholar
Marchioli, C., Picciotto, M. & Soldati, A. 2007 Influence of gravity and lift on particle velocity statistics and transfer rates in turbulent vertical channel flow. Intl J. Multiphase Flow 33 (3), 227–251.10.1016/j.ijmultiphaseflow.2006.09.005CrossRefGoogle Scholar
Marchioli, C. & Soldati, A. 2002 Mechanisms for particle transfer and segregation in a turbulent boundary layer. J. Fluid Mech. 468, 283–315.10.1017/S0022112002001738CrossRefGoogle Scholar
Marchioli, C., Soldati, A., Kuerten, J.G.M., Arcen, B., Tanière, A., Goldensoph, G., Squires, K.D., Cargnelutti, M.F. & Portela, L.M. 2008 Statistics of particle dispersion in direct numerical simulations of wall-bounded turbulence: results of an international collaborative benchmark test. Intl J. Multiphase Flow 34 (9), 879–893.10.1016/j.ijmultiphaseflow.2008.01.009CrossRefGoogle Scholar
Maxey, M.R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441–465.10.1017/S0022112087000193CrossRefGoogle Scholar
Maxey, M.R. & Riley, J.J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26 (4), 883–889.10.1063/1.864230CrossRefGoogle Scholar
Mena, S.E. & Curtis, J.S. 2020 Experimental data for solid-liquid flows at intermediate and high stokes numbers. J. Fluid Mech. 883, A24.10.1017/jfm.2019.836CrossRefGoogle Scholar
Modesti, D. & Pirozzoli, S. 2016 Reynolds and Mach number effects in compressible turbulent channel flow. Int. J. Heat Mass Transf. 59, 33–49.Google Scholar
Motoori, Y., Wong, C. & Goto, S. 2022 Role of the hierarchy of coherent structures in the transport of heavy small particles in turbulent channel flow. J. Fluid Mech. 942, A3.10.1017/jfm.2022.327CrossRefGoogle Scholar
Pan, Q., Xiang, H., Wang, Z., Andersson, H.I. & Zhao, L. 2020 Kinetic energy balance in turbulent particle-laden channel flow. Phys. Fluids 32 (7), 073307.10.1063/5.0012570CrossRefGoogle Scholar
Paris, A.D. 2001 Turbulence Attenuation in a Particle-Laden Channel Flow. Stanford University.Google Scholar
Pirozzoli, S. 2010 Generalized conservative approximations of split convective derivative operators. J. Comput. Phys. 229 (19), 7180–7190.10.1016/j.jcp.2010.06.006CrossRefGoogle Scholar
Pope, S.B. 2000 Turbulent Flows. Cambridge Univ. Press.Google 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, 138–152.10.1016/j.ijmultiphaseflow.2019.04.021CrossRefGoogle Scholar
Reeks, M.W. 1983 The transport of discrete particles in inhomogeneous turbulence. J. Aerosol Sci. 14 (6), 729–739.10.1016/0021-8502(83)90055-1CrossRefGoogle Scholar
Reeks, M.W. 1991 On a kinetic equation for the transport of particles in turbulent flows. Phys. Fluids A 3 (3), 446–456.10.1063/1.858101CrossRefGoogle Scholar
Reeks, M.W. 1992 On the continuum equations for dispersed particles in nonuniform flows. Phys. Fluids 4 (6), 1290–1303.10.1063/1.858247CrossRefGoogle Scholar
Richter, D.H. 2015 Turbulence modification by inertial particles and its influence on the spectral energy budget in planar Couette flow. Phys. Fluids 27 (6), 063304.10.1063/1.4923043CrossRefGoogle Scholar
Richter, D.H. & Sullivan, P.P. 2013 Momentum transfer in a turbulent, particle-laden Couette flow. Phys. Fluids 25 (5), 053304.10.1063/1.4804391CrossRefGoogle Scholar
Richter, D.H. & Sullivan, P.P. 2014 Modification of near-wall coherent structures by inertial particles. Phys. Fluids 26 (10), 103304.10.1063/1.4900583CrossRefGoogle Scholar
Rossetti, S.J. & Pfeffer, R. 1972 Drag reduction in dilute flowing gas-solid suspensions. AIChE J. 18 (1), 31–39.10.1002/aic.690180107CrossRefGoogle Scholar
Ruan, Y. & Xiao, Z. 2024 On the shear-induced lift in two-way coupled turbulent channel flow laden with heavy particles. J. Fluid Mech. 984, A24.10.1017/jfm.2024.214CrossRefGoogle Scholar
Rupp, D.A., Mortimer, L.F. & Fairweather, M. 2023 Stokes number and coupling effects on particle interaction behavior in turbulent channel flows. Phys. Fluids 35 (11), 113307.10.1063/5.0173863CrossRefGoogle Scholar
Saffman, P.G. 1962 On the stability of laminar flow of a dusty gas. J. Fluid Mech. 13 (1), 120–128.10.1017/S0022112062000555CrossRefGoogle Scholar
Salinas, J., Balachandar, S., Shringarpure, M., Fedele, J., Hoyal, D. & Cantero, M. 2020 Soft transition between subcritical and supercritical currents through intermittent cascading interfacial instabilities. Proc. Natl. Acad. Sci. U.S.A. 117 (31), 18278–18284.10.1073/pnas.2008959117CrossRefGoogle ScholarPubMed
Salinas, J.S., Balachandar, S., Shringarpure, M., Fedele, J., Hoyal, D., Zuniga, S. & Cantero, M.I. 2021 Anatomy of subcritical submarine flows with a lutocline and an intermediate destruction layer. Nat. Comm. 12 (1), 1649.10.1038/s41467-021-21966-yCrossRefGoogle Scholar
Sato, Y. & Hishida, K. 1996 Transport process of turbulence energy in particle-laden turbulent flow. Intl J. Heat Mass Transfer 17 (3), 202–210.Google Scholar
Schiller, V.L. 1933 Über die grundlegenden berechnungen bei der schwerkraftaufbereitung. Z. Vereines Deutscher Inge. 77, 318–321.Google 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, 136–149.10.1016/j.ijmultiphaseflow.2016.06.023CrossRefGoogle Scholar
Shook, C.A. & Roco, M.C. 2013 Slurry Flow: Principles and Practice. Elsevier.Google Scholar
Soldati, A. & Marchioli, C. 2009 Physics and modelling of turbulent particle deposition and entrainment: review of a systematic study. Intl J. Multiphase Flow 35 (9), 827–839.10.1016/j.ijmultiphaseflow.2009.02.016CrossRefGoogle Scholar
Solé, A., Martorell, I. & Cabeza, L.F. 2015 State of the art on gas-solid thermochemical energy storage systems and reactors for building applications. Renew. Sustain. Energy Rev. 47, 386–398.10.1016/j.rser.2015.03.077CrossRefGoogle Scholar
Song, J., Zhu, Y., Lin, F., Liu, N. & Khomami, B. 2023 Turbulent Taylor-couette flow of dilute polymeric solutions: a 10-year retrospective. Philos. Trans. R. Soc. A 381 (2243), 20220132.10.1098/rsta.2022.0132CrossRefGoogle ScholarPubMed
Soo, S.L. 1967 Fluid dynamics of multiphase systems . Blaisdell, Waltham, Mass.Google Scholar
Spalart, P.R., Moser, R.D. & Rogers, M.M. 1991 Spectral methods for the Navier–Stokes equations with one infinite and two periodic directions. J. Comput. Phys. 96 (2), 297–324.10.1016/0021-9991(91)90238-GCrossRefGoogle Scholar
Subramaniam, S. 2020 Multiphase flows: rich physics, challenging theory, and big simulations. Phys. Rev. Fluids 5 (11), 110520.10.1103/PhysRevFluids.5.110520CrossRefGoogle Scholar
Subramaniam, S. & Balachandar, S. 2022 Modeling Approaches and Computational Methods for Particle-Laden Turbulent Flows. Elsevier.Google Scholar
Sundaram, S. & Collins, L.R. 1997 Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations. J. Fluid Mech. 335, 75–109.10.1017/S0022112096004454CrossRefGoogle 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–S193.10.1007/s003480070020CrossRefGoogle Scholar
Swailes, D.C. & Darbyshire, K.F.F. 1997 A generalized Fokker–Planck equation for particle transport in random media. Physica A 242 (1), 38–48.10.1016/S0378-4371(97)00195-7CrossRefGoogle Scholar
Swailes, D.C., Sergeev, Y.A. & Parker, A. 1998 Chapman-Enskog closure approximation in the kinetic theory of dilute turbulent gas-particulate suspensions. Physica A 254 (3), 517–547.10.1016/S0378-4371(98)00063-6CrossRefGoogle Scholar
Tanaka, T. & Eaton, J.K. 2008 Classification of turbulence modification by dispersed spheres using a novel dimensionless number. Phys. Rev. Lett. 101 (11), 114502.10.1103/PhysRevLett.101.114502CrossRefGoogle ScholarPubMed
Tenneti, S. & Subramaniam, S. 2014 Particle-resolved direct numerical simulation for gas-solid flow model development. Annu. Rev. Fluid Mech. 46 (1), 199–230.10.1146/annurev-fluid-010313-141344CrossRefGoogle Scholar
Tsuji, Y. & Morikawa, Y. 1982 LDV measurements of an air-solid two-phase flow in a horizontal pipe. J. Fluid Mech. 120, 385–409.10.1017/S002211208200281XCrossRefGoogle 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, 417–434.10.1017/S0022112084000422CrossRefGoogle Scholar
Vincenti, W.G. & Kruger, C.H. 1965 Introduction to Physical Gas Dynamics. Wiley.Google Scholar
Vreman, A.W. 2007 Turbulence characteristics of particle-laden pipe flow. J. Fluid Mech. 584, 235–279.10.1017/S0022112007006556CrossRefGoogle Scholar
White, C.M. & Mungal, M.G. 2008 Mechanics and prediction of turbulent drag reduction with polymer additives. Annu. Rev. Fluid Mech. 40 (1), 235–256.10.1146/annurev.fluid.40.111406.102156CrossRefGoogle Scholar
Wilson, W.E. 1942 Mechanics of flow with noncolloidal, inert solids. Trans. Am. Soc. Civ. Eng. 107 (1), 1576–1586.10.1061/TACEAT.0005556CrossRefGoogle Scholar
Xu, A., Xu, B.-R. & Xi, H.-D. 2024 Particle transport and deposition in wall-sheared thermal turbulence. J. Fluid Mech. 999, A15.10.1017/jfm.2024.936CrossRefGoogle Scholar
Yang, F.L. & Hunt, M.L. 2006 Dynamics of particle-particle collisions in a viscous liquid. Phys. Fluids 18 (12), 121506.10.1063/1.2396925CrossRefGoogle Scholar
Zhang, W.-H., Zhang, H.-N., Wang, Z.-M., Li, Y.-K., Yu, B. & Li, F.-C. 2022 Repicturing viscoelastic drag-reducing turbulence by introducing dynamics of elasto-inertial turbulence. J. Fluid Mech. 940, A31.10.1017/jfm.2022.255CrossRefGoogle 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, 32–59.10.1017/jfm.2012.492CrossRefGoogle Scholar
Zhao, L.H., Andersson, H.I. & Gillissen, J.J.J. 2010 Turbulence modulation and drag reduction by spherical particles. Phys. Fluids 22 (8), 081702.10.1063/1.3478308CrossRefGoogle Scholar
Zhou, T., Zhao, L., Huang, W. & Xu, C. 2020 Non-monotonic effect of mass loading on turbulence modulations in particle-laden channel flow. Phys. Fluids 32 (4), 043304.10.1063/5.0002114CrossRefGoogle Scholar
Zhu, Y., Wan, Z., Lin, F., Liu, N., Lu, X. & Khomami, B. 2023 Maximum drag enhancement asymptote in spanwise-rotating viscoelastic plane Couette flow of dilute polymeric solutions. J. Fluid Mech. 958, A15.10.1017/jfm.2023.75CrossRefGoogle Scholar
Zisselmar, R. & Molerus, O. 1979 Investigation of solid-liquid pipe flow with regard to turbulence modification. Chem. Eng. J. 18 (2), 233–239.Google Scholar