Abstract
We study the settling of finite-size rigid spheres in quiescent fluid and in sustained homogeneous isotropic turbulence (HIT) by direct numerical simulations using an immersed boundary method to account for the dispersed solid phase. We consider semi-dilute and dense suspensions of rigid spheres with solid volume fractions \(\phi =0.5{-}10\%\), solid-to-fluid density ratio \(R=1.02\), and Galileo number (i.e., the ratio between buoyancy and viscous forces) \(Ga=145\). In HIT, the nominal Reynolds number based on the Taylor microscale is \(Re_{\lambda } \simeq 90\), and the ratio between the particle diameter and the nominal Kolmogorov scale is \((2a)/\eta \simeq 12\) (being a the particle radius). We find that in HIT the mean settling speed is less than that in quiescent fluid for all \(\phi \). For \(\phi =0.5\%\), the mean settling speed in HIT is \(8\%\) less than in quiescent fluid. However, by increasing the volume fraction the difference in the mean settling speed between quiescent fluid and HIT cases reduces, being only \(1.7\%\) for \(\phi =10\%\). Indeed, while at low \(\phi \) the settling speed is strongly altered by the interaction with turbulence, at large \(\phi \) this is mainly determined by the (strong) hindering effect. This is similar in quiescent fluid and in HIT, leading to similar mean settling speeds. On the contrary, particle angular velocities are always found to increase with \(\phi \). These are enhanced by the interaction with turbulence, especially at low \(\phi \). In HIT, the correlations of particle lateral velocity fluctuations oscillate around zero before decorrelating completely. The time period of the oscillation seems proportional to the ratio between the integral lengthscale of turbulence and the particle characteristic terminal velocity. Regarding the mean square particle displacement, we find that it is strongly enhanced by turbulence in the direction perpendicular to gravity, even at the largest \(\phi \). Finally, we investigate the collision statistics for all cases and find the interesting result that the collision frequency is larger in quiescent fluid than in HIT for \(\phi =0.5{-}1\%\). This is due to frequent drafting–kissing–tumbling events in quiescent fluid. The collision frequency becomes instead larger in HIT than in still fluid for \(\phi =5{-}10\%\), due to the larger relative approaching velocities in HIT, and to the less intense drafting–kissing–tumbling events in quiescent fluid. The collision frequency also appears to be almost proportional to the estimate for small inertial particles uniformly distributed in space, though much smaller. Concerning the turbulence modulation, we find that the mean energy dissipation increases almost linearly with \(\phi \), leading to a large reduction of \(Re_{\lambda }\).
Similar content being viewed by others
References
Aliseda, A., Cartellier, A., Hainaux, F., Lasheras, J.C.: Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 468, 77–105 (2002)
Ardekani, M.N., Costa, P., Breugem, W.P., Brandt, L.: Numerical study of the sedimentation of spheroidal particles. Int. J. Multiph. Flow 87, 16–34 (2016)
Bagchi, P., Balachandar, S.: Effect of turbulence on the drag and lift of a particle. Phys. Fluids (1994-present) 15(11), 3496–3513 (2003)
Brenner, H.: The slow motion of a sphere through a viscous fluid towards a plane surface. Chem. Eng. Sci. 16(3), 242–251 (1961)
Breugem, W.P.: A second-order accurate immersed boundary method for fully resolved simulations of particle-laden flows. J. Comput. Phys. 231(13), 4469–4498 (2012)
Byron, M.L.: The rotation and translation of non-spherical particles in homogeneous isotropic turbulence. arXiv:1506.00478 (2015)
Chrust, M.: Etude numérique de la chute libre d’objets axisymétriques dans un fluide newtonien. Ph.D. thesis, Strasbourg (2012)
Csanady, G.: Turbulent diffusion of heavy particles in the atmosphere. J. Atmos. Sci. 20(3), 201–208 (1963)
Ern, P., Risso, F., Fabre, D., Magnaudet, J.: Wake-induced oscillatory paths of bodies freely rising or falling in fluids. Ann. Rev. Fluid Mech. 44, 97–121 (2012)
Feng, J., Hu, H.H., Joseph, D.D.: 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 (1994)
Fornari, W., Ardekani, M.N., Brandt, L.: Clustering and increased settling speed of oblate particles at finite Reynolds number. J. Fluid Mech. 848, 696–721 (2018)
Fornari, W., Formenti, A., Picano, F., Brandt, L.: The effect of particle density in turbulent channel flow laden with finite size particles in semi-dilute conditions. Phys. Fluids (1994-present) 28(3), 033,301 (2016)
Fornari, W., Picano, F., Brandt, L.: Sedimentation of finite-size spheres in quiescent and turbulent environments. J. Fluid Mech. 788, 640–669 (2016)
Fornari, W., Picano, F., Sardina, G., Brandt, L.: Reduced particle settling speed in turbulence. J. Fluid Mech. 808, 153–167 (2016)
Fortes, A.F., Joseph, D.D., Lundgren, T.S.: Nonlinear mechanics of fluidization of beds of spherical particles. J. Fluid Mech. 177, 467–483 (1987)
Garside, J., Al-Dibouni, M.R.: Velocity-voidage relationships for fluidization and sedimentation in solid-liquid systems. Ind. Eng. Chem. Process Des. Dev. 16(2), 206–214 (1977)
Good, G., Ireland, P., Bewley, G., Bodenschatz, E., Collins, L., Warhaft, Z.: Settling regimes of inertial particles in isotropic turbulence. J. Fluid Mech. 759, R3 (2014)
Hampton, R., Mammoli, A., Graham, A., Tetlow, N., Altobelli, S.: Migration of particles undergoing pressure-driven flow in a circular conduit. J. Rheol. 41(3), 621–640 (1997)
Homann, H., Bec, J., Grauer, R.: Effect of turbulent fluctuations on the drag and lift forces on a towed sphere and its boundary layer. J. Fluid Mech. 721, 155–179 (2013)
Huisman, S.G., Barois, T., Bourgoin, M., Chouippe, A., Doychev, T., Huck, P., Morales, C.E.B., Uhlmann, M., Volk, R.: Columnar structure formation of a dilute suspension of settling spherical particles in a quiescent fluid. Phys. Rev. Fluids 1(7), 074,204 (2016)
Jenny, M., Dušek, J., Bouchet, G.: Instabilities and transition of a sphere falling or ascending freely in a Newtonian fluid. J. Fluid Mech. 508, 201–239 (2004)
Kawanisi, K., Shiozaki, R.: Turbulent effects on the settling velocity of suspended sediment. J. Hydraul. Eng. 134(2), 261–266 (2008)
Lambert, R.A., Picano, F., Breugem, W.P., Brandt, L.: Active suspensions in thin films: nutrient uptake and swimmer motion. J. Fluid Mech. 733, 528–557 (2013)
Lashgari, I., Picano, F., Breugem, W.P., Brandt, L.: Channel flow of rigid sphere suspensions: particle dynamics in the inertial regime. Int. J. Multiph. Flow 78, 12–24 (2016)
Maxey, M.: The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441–465 (1987)
Murray, S.P.: Settling velocities and vertical diffusion of particles in turbulent water. J. Geophys. Res. 75(9), 1647–1654 (1970)
Nielsen, P.: Turbulence effects on the settling of suspended particles. J. Sediment. Res. 63(5), 835–838 (1993)
Picano, F., Breugem, W.P., Brandt, L.: Turbulent channel flow of dense suspensions of neutrally buoyant spheres. J. Fluid Mech. 764, 463–487 (2015)
Richardson, J., Zaki, W.: The sedimentation of a suspension of uniform spheres under conditions of viscous flow. Chem. Eng. Sci. 3(2), 65–73 (1954)
Squires, K.D., Eaton, J.K.: Preferential concentration of particles by turbulence. Phys. Fluids A Fluid Dyn. (1989–1993) 3(5), 1169–1178 (1991)
Sundaram, S., Collins, L.R.: Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations. J. Fluid Mech. 335, 75–109 (1997)
Uhlmann, M., Doychev, T.: 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 (2014)
Vincent, A., Meneguzzi, M.: The spatial structure and statistical properties of homogeneous turbulence. J. Fluid Mech. 225, 1–20 (1991)
Wang, L.P., Maxey, M.R.: Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 256, 27–68 (1993)
Yang, T., Shy, S.: The settling velocity of heavy particles in an aqueous near-isotropic turbulence. Phys. Fluids (1994-present) 15(4), 868–880 (2003)
Yang, T., Shy, S.: Two-way interaction between solid particles and homogeneous air turbulence: particle settling rate and turbulence modification measurements. J. Fluid Mech. 526, 171–216 (2005)
Yeo, K., Dong, S., Climent, E., Maxey, M.R.: Modulation of homogeneous turbulence seeded with finite size bubbles or particles. Int. J. Multiph. Flow 36(3), 221–233 (2010)
Yin, X., Koch, D.L.: Hindered settling velocity and microstructure in suspensions of solid spheres with moderate Reynolds numbers. Phys. Fluids 19(9), 093,302 (2007)
Zaidi, A.A., Tsuji, T., Tanaka, T.: Direct numerical simulation of finite sized particles settling for high Reynolds number and dilute suspension. Int. J. Heat Fluid Flow 50, 330–341 (2014)
Zhan, C., Sardina, G., Lushi, E., Brandt, L.: Accumulation of motile elongated micro-organisms in turbulence. J. Fluid Mech. 739, 22–36 (2014)
Acknowledgements
This work was supported by the European Research Council Grant No. ERC-2013-CoG-616186, TRITOS and by the Swedish Research Council (VR). Computer time was provided by SNIC (Swedish National Infrastructure for Computing). The support from the COST Action MP1305: Flowing matter is also acknowledged.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fornari, W., Zade, S., Brandt, L. et al. Settling of finite-size particles in turbulence at different volume fractions. Acta Mech 230, 413–430 (2019). https://doi.org/10.1007/s00707-018-2269-1
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-018-2269-1