Comptes Rendus
no. 2
Settling velocity of quasi-neutrally-buoyant inertial particles
Marco Martins Afonso1 ; Sílvio M.A. Gama1
1 Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
Comptes Rendus. Mécanique, Volume 346 (2018) no. 2, pp. 121-131.

We investigate the sedimentation properties of quasi-neutrally buoyant inertial particles carried by incompressible zero-mean fluid flows. We obtain generic formulae for the terminal velocity in generic space-and-time periodic (or steady) flows, along with further information for flows endowed with some degree of spatial symmetry such as odd parity in the vertical direction. These expressions consist in space-time integrals of auxiliary quantities that satisfy partial differential equations of the advection–diffusion–reaction type, which can be solved at least numerically, since our scheme implies a huge reduction of the problem dimensionality from the full phase space to the classical physical space.

Nous étudions les propriétés de sédimentation de particules inertielles dotées de flottabilité quasi neutre et transportées par un écoulement incompressible à moyenne nulle. Nous obtenons des formules génériques pour la vitesse terminale dans des écoulements en général périodiques en espace et en temps (ou statiques), avec d'ultérieures informations disponibles pour les écoulements dotés de symétries spatiales spécifiques, telles qu'une parité négative dans la direction verticale. Ces expressions consistent en des intégrales spatio-temporelles de quantités auxiliaires qui obéissent à des équations aux dérivées partielles du type advection–diffusion–réaction. Ces dernières peuvent être résolues au moins numériquement, car notre procédure implique une forte réduction de la dimensionnalité du problème, de l'espace des phases complet à l'espace physique classique.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2017.11.005
Keywords: Fluid dynamics, Inertial particles, Settling velocity, Quasi-neutral buoyancy, Steady/periodic/cellular flows, Brownian diffusivity
Mot clés : Dynamique des fluides, Particules inertielles, Vitesse de sédimentation, Flottabilité quasi neutre, Flux stationnaires/périodiques/cellulaires, Diffusivité brownienne

Marco Martins Afonso 1; Sílvio M.A. Gama 1

1 Centro de Matemática da Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal 
@article{CRMECA_2018__346_2_121_0,
     author = {Marco Martins Afonso and S{\'\i}lvio M.A. Gama},
     title = {Settling velocity of quasi-neutrally-buoyant inertial particles},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {121--131},
     publisher = {Elsevier},
     volume = {346},
     number = {2},
     year = {2018},
     doi = {10.1016/j.crme.2017.11.005},
     language = {en},
}
TY  - JOUR
AU  - Marco Martins Afonso
AU  - Sílvio M.A. Gama
TI  - Settling velocity of quasi-neutrally-buoyant inertial particles
JO  - Comptes Rendus. Mécanique
PY  - 2018
SP  - 121
EP  - 131
VL  - 346
IS  - 2
PB  - Elsevier
DO  - 10.1016/j.crme.2017.11.005
LA  - en
ID  - CRMECA_2018__346_2_121_0
ER  - 
%0 Journal Article
%A Marco Martins Afonso
%A Sílvio M.A. Gama
%T Settling velocity of quasi-neutrally-buoyant inertial particles
%J Comptes Rendus. Mécanique
%D 2018
%P 121-131
%V 346
%N 2
%I Elsevier
%R 10.1016/j.crme.2017.11.005
%G en
%F CRMECA_2018__346_2_121_0
Marco Martins Afonso; Sílvio M.A. Gama. Settling velocity of quasi-neutrally-buoyant inertial particles. Comptes Rendus. Mécanique, Volume 346 (2018) no. 2, pp. 121-131. doi : 10.1016/j.crme.2017.11.005. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.1016/j.crme.2017.11.005/

[1] M.W. Reeks On a kinetic equation for the transport of particles in turbulent flows, Phys. Fluids A, Volume 3 (1991) no. 3, pp. 446-456

[2] M.W. Reeks On the continuum equations for dispersed particles in nonuniform flows, Phys. Fluids A, Volume 4 (1992) no. 6, pp. 1290-1303

[3] E. Balkovsky; G. Falkovich; A. Fouxon Intermittent distribution of inertial particles in turbulent flows, Phys. Rev. Lett., Volume 86 (2001), pp. 2790-2793

[4] J. Bec Fractal clustering of inertial particles in random flows, Phys. Fluids, Volume 15 (2003), p. L81-L84

[5] M. Wilkinson; B. Mehlig Path coalescence transition and its applications, Phys. Rev. E, Volume 68 (2003)

[6] G. Falkovich; A. Pumir Intermittent distribution of heavy particles in a turbulent flow, Phys. Fluids, Volume 16 (2004), p. L47-L50

[7] I.M. Mazzitelli; D. Lohse Lagrangian statistics for fluid particles and bubbles in turbulence, New J. Phys., Volume 6 (2004), pp. 1-28

[8] M. Cencini; J. Bec; L. Biferale; G. Boffetta; A. Celani; A.S. Lanotte; S. Musacchio; F. Toschi Dynamics and statistics of heavy particles in turbulent flows, J. Turbul., Volume 7 (2006) no. 36, pp. 1-36

[9] R. Volk; E. Calzavarini; G. Verhille; D. Lohse; N. Mordant; J.-F. Pinton; F. Toschi Acceleration of heavy and light particles in turbulence: comparison between experiments and direct numerical simulations, Physica D, Volume 237 (2008), pp. 2084-2089

[10] G. Károlyi; Á. Péntek; I. Scheuring; T. Tél; Z. Toroczkai Chaotic flow: the physics of species coexistence, Proc. Natl. Acad. Sci., Volume 97 (2000), pp. 13661-13665

[11] C. Habchi; N. Dumont; O. Simonin Multidimensional simulation of cavitating flows in diesel injectors by a homogeneous mixture modeling approach, Atomiz. Spr., Volume 18 (2008) no. 2, pp. 129-162

[12] S. Matarrese; R. Mohayee The growth of structure in the intergalactic medium, Mon. Not. R. Astron. Soc., Volume 329 (2002), pp. 37-60

[13] G. Falkovich; A. Fouxon; M. Stepanov Acceleration of rain initiation by cloud turbulence, Nature, Volume 419 (2002), pp. 151-154

[14] A.S. Monin; A.M. Yaglom Statistical Fluid Mechanics, MIT Press, Cambridge, 1975

[15] M. Avellaneda; A. Majda An integral representation and bounds on the effective diffusivity in passive advection and turbulent flows, Commun. Math. Phys., Volume 138 (1991), pp. 339-391

[16] D.J. Horntrop; A. Majda Subtle statistical behavior in simple models for random advection–diffusion, J. Math. Sci. Univ. Tokyo, Volume 1 (1994), pp. 23-70

[17] A. Mazzino; M. Vergassola Interference between turbulent and molecular diffusion, Europhys. Lett., Volume 37 (1997) no. 8, pp. 535-540

[18] A. Mazzino; S. Musacchio; A. Vulpiani Multiple-scale analysis and renormalization for preasymptotic scalar transport, Phys. Rev. E, Volume 71 (2005)

[19] M. Cencini; A. Mazzino; S. Musacchio; A. Vulpiani Large-scale effects on meso-scale modeling for scalar transport, Physica D, Volume 220 (2006), pp. 146-156

[20] R. Ferrari; M. Nikurashin Suppression of eddy diffusivity across jets in the Southern ocean, J. Phys. Oceanogr., Volume 40 (2010), pp. 1501-1519

[21] M. Martins Afonso; A. Mazzino; S. Gama Combined role of molecular diffusion, mean streaming and helicity in the eddy diffusivity of short-correlated random flows, J. Stat. Mech., Volume 2016 (2016) no. 10

[22] M.R. Maxey; J.J. Riley Equation of motion for a small rigid sphere in a nonuniform flow, Phys. Fluids, Volume 26 (1983) no. 4, pp. 883-889

[23] R. Gatignol The Faxén formulae for a rigid particle in an unsteady non-uniform Stokes flow, J. Méc. Théor. Appl., Volume 1 (1983), pp. 143-160

[24] M.W. Reeks The relationship between Brownian motion and the random motion of small particles in a turbulent flow, Phys. Fluids, Volume 31 (1988), pp. 1314-1316

[25] S. Chandrasekhar Stochastic problems in physics and astronomy, Rev. Mod. Phys., Volume 15 (1943), pp. 1-89

[26] C.W. Gardiner Handbook of Stochastic Methods: For Physics, Chemistry and the Natural Sciences, Springer, Berlin, 1985

[27] H. Risken The Fokker–Planck Equation: Methods of Solutions and Applications, Springer, Berlin, 1989

[28] N.G. Van Kampen Stochastic Processes in Physics and Chemistry, Elsevier, Amsterdam, 2007

[29] M.R. Maxey; S. Corrsin Gravitational settling of aerosol particles in randomly oriented cellular flow fields, J. Atmos. Sci., Volume 43 (1986) no. 11, pp. 1112-1134

[30] M.R. Maxey The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields, J. Fluid Mech., Volume 174 (1987), pp. 441-465

[31] M.R. Maxey The motion of small spherical particles in a cellular flow field, Phys. Fluids, Volume 30 (1987), pp. 1915-1928

[32] L.P. Wang; M.R. Maxey Settling velocity and concentration distribution of heavy particles in homogeneous isotropic turbulence, J. Fluid Mech., Volume 256 (1993), pp. 27-68

[33] P.D. Friedman; J. Katz Mean rise rate of droplets in isotropic turbulence, Phys. Fluids, Volume 14 (2002), pp. 3059-3073

[34] J. Ruiz; D. Macías; P. Peters Turbulence increases the average settling velocity of phytoplankton cell, Proc. Natl. Acad. Sci., Volume 101 (2004), pp. 17720-17724

[35] M. Martins Afonso The terminal velocity of sedimenting particles in a flowing fluid, J. Phys. A, Volume 41 (2008) no. 38

[36] A. Babiano; J.H.E. Cartwright; O. Piro; A. Provenzale Dynamics of a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian systems, Phys. Rev. Lett., Volume 84 (2000), pp. 5764-5767

[37] C. Marchioli; M. Fantoni; A. Soldati Influence of added mass on anomalous high rise velocity of light particles in cellular flow field: a note on the paper by Maxey (1987), Phys. Fluids, Volume 19 (2007)

[38] T. Sapsis; G. Haller Instabilities in the dynamics of neutrally buoyant particles, Phys. Fluids, Volume 20 (2008) no. 017102, pp. 1-7

[39] P. Castiglione; A. Crisanti Dispersion of passive tracers in a velocity field with non-δ-correlated noise, Phys. Rev. E, Volume 59 (1999) no. 4, pp. 3926-3934

[40] S. Boi; A. Mazzino; P. Muratore-Ginanneschi Eddy diffusivities of inertial particles in random Gaussian flows, Phys. Rev. Fluids, Volume 2 (2017)

[41] M. Martins Afonso; A. Mazzino Point-source inertial particle dispersion, Geophys. Astrophys. Fluid Dyn., Volume 105 (2011) no. 6, pp. 553-565

[42] M. Martins Afonso; A. Mazzino; P. Muratore-Ginanneschi Eddy diffusivities for inertial particles under gravity, J. Fluid Mech., Volume 694 (2012), pp. 426-463

[43] T.H. Solomon; J.P. Gollub Chaotic particle transport in time-dependent Rayleigh–Bénard convection, Phys. Rev. A, Volume 38 (1988), pp. 6280-6286

[44] P. Castiglione; A. Mazzino; P. Muratore-Ginanneschi; A. Vulpiani On strong anomalous diffusion, Physica D, Volume 134 (1999), pp. 75-93

[45] P. Castiglione; A. Mazzino; P. Muratore-Ginanneschi Numerical study of strong anomalous diffusion, Physica A, Volume 280 (2000), pp. 60-68

[46] M. Linkès; M. Martins Afonso; P. Fede; J. Morchain; P. Schmitz Numerical study of substrate assimilation by a microorganism exposed to fluctuating concentration, Chem. Eng. Sci., Volume 81 (2012), pp. 8-19

[47] U. Frisch Turbulence, Cambridge University Press, Cambridge, 1995

[48] M. Avellaneda; M. Vergassola Stieltjes integral representation of effective diffusivities in time-dependent flows, Phys. Rev. E, Volume 52 (1995) no. 3, pp. 3249-3251

[49] L. Biferale; A. Crisanti; M. Vergassola; A. Vulpiani Eddy diffusivities in scalar transport, Phys. Fluids, Volume 7 (1995) no. 11, pp. 2725-2734

[50] K.H. Andersen; P. Castiglione; A. Mazzino; A. Vulpiani Simple stochastic models showing strong anomalous diffusion, Eur. Phys. J. B, Volume 18 (2000), pp. 447-452

[51] G.A. Pavliotis; A.M. Stuart Periodic homogenization for inertial particles, Physica D, Volume 204 (2005), pp. 161-187

[52] C.M. Bender; S.A. Orszag Advanced Mathematical Methods for Scientists and Engineers, McGraw–Hill, New York, 1978

[53] A. Bensoussan; J.-L. Lions; G. Papanicolaou Asymptotic Analysis of Periodic Structures, North-Holland, Amsterdam, 1978

[54] G.A. Pavliotis; A.M. Stuart Multiscale Methods: Averaging and Homogenization, Springer, Berlin, 2007

[55] M. Martins Afonso Anomalous diffusion for inertial particles under gravity in parallel flows, Phys. Rev. E, Volume 89 (2014) no. 6

[56] S. Boi; M. Martins Afonso; A. Mazzino Anomalous diffusion of inertial particles in random parallel flows: theory and numerics face to face, J. Stat. Mech., Volume 2015 (2015) no. 10

[57] G.G. Stokes On the theory of oscillatory waves, Trans. Cambridge Philos. Soc., Volume 8 (1847), pp. 441-473

[58] M.S. Longuet-Higgins Eulerian and Lagrangian aspects of surface waves, J. Fluid Mech., Volume 173 (1986), pp. 683-707

[59] F. Santamaria; G. Boffetta; M. Martins Afonso; A. Mazzino; M. Onorato; D. Pugliese Stokes drift for inertial particles transported by water waves, Europhys. Lett., Volume 102 (2013) no. 1

[60] A. Celani; M. Martins Afonso; A. Mazzino Point-source scalar turbulence, J. Fluid Mech., Volume 583 (2007), pp. 189-198

Cited by Sources:

Comments - Politique