Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-22T22:33:59.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Preferential concentration and relative velocity statistics of inertial particles in Navier–Stokes turbulence with and without filtering

Published online by Cambridge University Press:  06 June 2011

BAIDURJA RAY  and
LANCE R. COLLINS
BAIDURJA RAY
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14850, USA
LANCE R. COLLINS*
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14850, USA
*
†Email address for correspondence: lc246@cornell.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

The radial distribution function (RDF, a statistical measure of preferential concentration), and the relative velocity measured along the line-of-centres of two particles are the key statistical inputs to the collision kernel for finite-inertia particles suspended in a turbulent flow Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, p. 75). In this paper, we investigate the behaviour of these two-particle statistics using direct numerical simulation (DNS) of homogeneous isotropic turbulence. While it is known that the RDF for particles of any Stokes number (St) decreases with separation distance Sundaram & Collins (J. Fluid Mech., vol. 335, 1997, p. 75), Reade & Collins (Phys. Fluids, vol. 12, 2000, p. 2530), Salazar et al. (J. Fluid Mech., vol. 600, 2008, p. 245), we observe that the peak in the RDF versus St curve shifts to higher St as we increase the separation distance. Here, St is defined as the ratio of the particle's viscous relaxation time to the Kolmogorov time-scale of the flow. Furthermore, as found in a previous study Wang, Wexler, & Zhou (J. Fluid Mech., vol. 415, 2000, p. 117), the variance of the radial relative velocity (wr) is found to increase monotonically with increasing separation distance and increasing Stokes number; however, we show for the first time that the parameteric variation of the skewness of wr with St and r/η is qualitatively similar to that of the RDF, and points to a connection between the two. We then apply low-pass filters (using three different filter scales) on the DNS velocity field in wavenumber space in order to produce ‘perfect’ large-eddy simulation (LES) velocity fields without any errors associated with subgrid-scale modelling. We present visual evidence of the effect of sharp-spectral filtering on the flow structure and the particle field. We calculate the particle statistics in the filtered velocity field and find that the RDF decreases with filtering at low St and increases with filtering at high St, similar to Fede & Simonin (Phys. Fluids, vol. 18, 2006, p. 045103). We also find that the variation of the RDF with St shifts towards higher St with filtering at all separation distances. The variance of wr is found to decrease with filtering for all St and separation distances, but the skewness of wr shows a non-monotonic response to filtering that is qualitatively similar to the RDF. We consider the variation of the RDF and moments of wr with filter scale and find that they are approximately linear in the inertial range. We demonstrate that a simple model consisting of a redefinition of the St based on the time-scale of the filtered velocity field cannot recover the unfiltered statistics. Our findings provide insight on the effect of subgrid-scale eddies on the RDF and wr, and establish the requirements of a LES model for inertial particles that can correctly predict clustering and collisional behaviour.

JFM classification

Turbulent Flows: Isotropic turbulence Multiphase and Particle-laden Flows: Particle/fluid flow Turbulent Flows: Turbulence simulation
Type
Papers
Information
Journal of Fluid Mechanics , Volume 680 , 10 August 2011 , pp. 488 - 510
Copyright
Copyright © Cambridge University Press 2011

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

Aliseda, A., Cartellier, A., Hainaux, F. & Lasheras, J. C. 2002 Effect of preferential concentration on the settling velocity of heavy particles in homogeneous isotropic turbulence. J. Fluid Mech. 468, 77–105.CrossRefGoogle Scholar
Armenio, V., Piomelli, U. & Fiorotto, V. 1999 Effect of the subgrid scales on particle motion. Phys. Fluids 11, 3030–3042.CrossRefGoogle Scholar
Ayala, O., Rosa, B. & Wang, L. P. 2008 Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 2. Theory and parameterization. New J. Phys. 10, 075013.Google Scholar
Bec, J., Biferale, L., Cencini, M., Lanotte, A. S. & Toschi, F. 2010 Intermittency in the velocity distribution of heavy particles in turbulence. J. Fluid Mech. 646, 527–536.CrossRefGoogle Scholar
Bec, J., Celani, A., Cencini, M. & Musacchio, S. 2005 Clustering and collisions of heavy particles in random smooth flows. Phys. Fluids 17, 073301.CrossRefGoogle Scholar
Berrouk, A. S., Laurence, D., Riley, J. J. & Stock, D. E. 2007 Stochastic modelling of inertial particle dispersion by subgrid motion for LES of high Reynolds number pipe flow. J. Turbulence 8 (50).CrossRefGoogle Scholar
Berrut, J. P. & Trefethen, L. N. 2004 Barycentric Lagrange interpolation. Siam Rev. 46, 501–517.CrossRefGoogle Scholar
Bini, M. & Jones, W. P. 2007 Particle acceleration in turbulent flows: A class of nonlinear stochastic models for intermittency. Phys. Fluids 19 (3), 035104.Google Scholar
Boivin, M., Simonin, O. & Squires, K. D. 2000 On the prediction of gas–solid flows with two-way coupling using large eddy simulation. Phys. Fluids 12, 2080–2090.CrossRefGoogle Scholar
Brucker, K. A., Isaza, J. C., Vaithianathan, T. & Collins, L. R. 2007 Efficient algorithm for simulating homogeneous turbulent shear flow without remeshing. J. Comput. Phys. 225, 20–32.Google Scholar
Chun, J., Koch, D. L., Rani, S., Ahluwalia, A. & Collins, L. R. 2005 Clustering of aerosol particles in isotropic turbulence. J. Fluid Mech. 536, 219–251.CrossRefGoogle Scholar
Collins, L. R. & Keswani, A. 2004 Reynolds number scaling of particle clustering in turbulent aerosols. New J. Phys. 6, 119.CrossRefGoogle Scholar
Eaton, J. K. & Fessler, J. R. 1994 Preferential concentration of particles by turbulence. Intl J. Multiphase Flow 20, 169–209.CrossRefGoogle Scholar
Falkovich, G., Fouxon, A. & Stepanov, M. G. 2002 Acceleration of rain initiation by cloud turbulence. Nature 419, 151–154.Google Scholar
Falkovich, G. & Pumir, A. 2007 Sling effect in collisions of water droplets in turbulent clouds. J. Atmos. Sci. 64, 4497–4505.Google Scholar
Fede, P. & Simonin, O. 2006 Numerical study of the subgrid fluid turbulence effects on the statistics of heavy colliding particles. Phys. Fluids 18, 045103.Google Scholar
Fede, P., Simonin, O., Villedieu, P. & Squires, K. D. 2006 Stochastic modeling of turbulent subgrid fluid velocity along inertia particle trajectories. In Proceedings of the 2006 CTR Summer Program, Center for Turbulence Research, Stanford, CA, pp. 247–258.Google Scholar
Germano, M., Piomelli, U., Moin, P. & Cabot, W. H. 1991 A dynamic subgrid-scale eddy viscosity model. Phys. Fluids A 3, 1760–1765.CrossRefGoogle Scholar
Jin, G., He, G.-W. & Wang, L.-P. 2010 a Large-eddy simulation of turbulent collision of heavy particles in isotropic turbulence. Phys. Fluids 22, 055106.CrossRefGoogle Scholar
Jin, G., He, G.-W., Wang, L.-P. & Zhang, J. 2010 b Subgrid scale fluid velocity timescales seen by inertial particles in large-eddy simulation of particle-laden turbulence. Intl J. Multiphase Flow 36, 432–437, Brief Communication.Google Scholar
Johnson, R. W. (Ed.) 1998 Mathematics of fluid mechanics. In The Handbook of Fluid Dynamics. CRC Press, A1-A93.Google Scholar
de Jong, J., Salazar, J. P. L. C., Cao, L., Woodward, S. H., Collins, L. R. & Meng, H. 2010 Measurement of inertial particle clustering and relative velocity statistics in isotropic turbulence using holographic imaging. Intl J. Multiphase Flow 36, 324–332.CrossRefGoogle Scholar
Kuerten, J. G. M. 2008 Large-eddy simulation of particle-laden channel flow. In Quality and Reliability of Large-Eddy Simulations, pp. 367–378. Springer Netherlands.CrossRefGoogle Scholar
Marchioli, C., Salvetti, M. V. & Soldati, A. 2008 Some issues concerning large-eddy simulation of inertial particle dispersion in turbulent bounded flows. Phys. Fluids 20, 040603.CrossRefGoogle Scholar
Maxey, M. R. 1987 The gravitational settling of aerosol particles in homogeneous turbulence and random flow fields. J. Fluid Mech. 174, 441–465.CrossRefGoogle Scholar
Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a nonuniform flow. Phys. Fluids 26, 883–889.Google Scholar
McQuarrie, D. A. 1976 Statistical Mechanics. Harper & Row.Google Scholar
Metais, O. & Lesieur, M. 1992 Spectral large-eddy simulation of isotropic and stably stratified turbulence. J. Fluid Mech. 239, 157–194.Google Scholar
Patterson, G. S. & Orszag, S. A. 1971 Spectral calculation of isotropic turbulence: efficient removal of aliasing interactions. Phys. Fluids 14, 2538–2541.Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Pozorski, J. & Apte, S. V. 2009 Filtered particle tracking in isotropic turbulence and stochastic modeling of subgrid-scale dispersion. Intl J. Multiphase Flow 35, 118–128.CrossRefGoogle Scholar
Reade, W. C. & Collins, L. R. 2000 Effect of preferential concentration on turbulent collision rates. Phys. Fluids 12, 2530–2540.Google Scholar
Saffman, P. G. & Turner, J. S. 1956 On the collision of drops in turbulent clouds. J. Fluid Mech. 1, 16–30.Google Scholar
Salazar, J. P. L. C. & Collins, L. R. 2011 Inertial particle relative velocity statistics in homogeneous isotropic turbulence. J. Fluid Mech. (in review).Google Scholar
Salazar, J. P. L. C., de Jong, J., Cao, L., Woodward, S., Meng, H. & Collins, L. R. 2008 Experimental and numerical investigation of inertial particle clustering in isotropic turbulence. J. Fluid Mech. 600, 245–256.Google Scholar
Shaw, R. A., Reade, W. C., Collins, L. R. & Verlinde, J. 1998 Preferential concentration of cloud droplets by turbulence: effects on the early evolution of cumulus cloud droplet spectra. J. Atmos. Sci. 55, 1965–1976.Google Scholar
Shotorban, B. & Mashayek, F. 2005 Modeling subgrid-scale effects on particles by approximate deconvolution. Phys. Fluids 17, 081701.CrossRefGoogle Scholar
Shotorban, B. & Mashayek, F. 2006 a On stochastic modeling of heavy particle dispersion in large-eddy simulation of two-phase turbulent flow. In IUTAM Symposium on Computational Multiphase Flow, 4–7 October 2004, pp. 373–380. Argonne National Laboratory.CrossRefGoogle Scholar
Shotorban, B. & Mashayek, F. 2006 b A stochastic model for particle motion in large-eddy simulation. J. Turbulence 7 (18).CrossRefGoogle Scholar
Squires, K. D. & Eaton, J. K. 1991 Preferential concentration of particles by turbulence. Phys. Fluids A 3, 1169–1178.CrossRefGoogle 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.CrossRefGoogle Scholar
Sundaram, S. & Collins, L. R. 1999 A numerical study of the modulation of isotropic turbulence by suspended particles. J. Fluid Mech. 379, 105–143.CrossRefGoogle Scholar
Vaillancourt, P. A. & Yau, M. K. 2000 Review of particle–turbulence interactions and consequences for cloud physics. Bull. Am. Meteorol. Soc. 81, 285–298.Google Scholar
Wang, L. P., Chen, S., Brasseur, J. G. & Wyngaard, J. C. 1996 Examination of hypotheses in Kolmogorov refined turbulence theory through high-resolution simulations. I. Velocity field. J. Fluid Mech. 309, 113–156.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.Google Scholar
Wang, L.-P., Wexler, A. S. & Zhou, Y. 2000 Statistical mechanical description and modeling of turbulent collision of inertial particles. J. Fluid Mech. 415, 117–153.Google Scholar
Wang, Q. & Squires, K. D. 1996 Large eddy simulation of particle-laden turbulent channel flow. Phys. Fluids 8, 1207–1223.CrossRefGoogle Scholar
Wilkinson, M., Mehlig, B. & Bezuglyy, V. 2006 Caustic activation of rain showers. Phys. Rev. Lett. 97, 048501.Google Scholar
Witkowska, A., Brasseur, J. G. & Juvé, D. 1997 Numerical study of noise from isotropic turbulence. J. Comput. Acoust. 5, 317–336.CrossRefGoogle Scholar
Zaichik, L. I. & Alipchenkov, V. M. 2003 Pair dispersion and preferential concentration of particles in isotropic turbulence. Phys. Fluids 15, 1776–1787.Google Scholar
Zaichik, L. I. & Alipchenkov, V. M. 2009 Statistical models for predicting pair dispersion and particle clustering in isotropic turbulence and their applications. New J. Phys. 11, 103018.Google Scholar