Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-19T16:02:10.726Z Has data issue: false hasContentIssue false
  • English
  • Français

Inertial particle relative velocity statistics in homogeneous isotropic turbulence

Published online by Cambridge University Press:  05 March 2012

Juan P. L. C. Salazar  and
Lance R. Collins
Juan P. L. C. Salazar
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-7501, USA International Collaboration for Turbulence Research
Lance R. Collins*
Affiliation:
Sibley School of Mechanical and Aerospace Engineering, Cornell University, Ithaca, NY 14853-7501, USA International Collaboration for Turbulence Research
*
†Email address for correspondence: lc246@cornell.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In the present study, we investigate the scaling of relative velocity structure functions, of order two and higher, for inertial particles, both in the dissipation range and the inertial subrange using direct numerical simulations (DNS). Within the inertial subrange our findings show that contrary to the well-known attenuation in the tails of the one-point acceleration probability density function (p.d.f.) with increasing inertia (Bec et al., J. Fluid Mech., vol. 550, 2006, pp. 349–358), the opposite occurs with the velocity structure function at sufficiently large Stokes numbers. We observe reduced scaling exponents for the structure function when compared to those of the fluid, and correspondingly broader p.d.f.s, similar to what occurs with a passive scalar. DNS allows us to isolate the two effects of inertia, namely biased sampling of the velocity field, a result of preferential concentration, and filtering, i.e. the tendency for the inertial particle velocity to attenuate the velocity fluctuations in the fluid. By isolating these effects, we show that sampling is playing the dominant role for low-order moments of the structure function, whereas filtering accounts for most of the scaling behaviour observed with the higher-order structure functions in the inertial subrange. In the dissipation range, we see evidence of so-called ‘crossing trajectories’, the ‘sling effect’ or ‘caustics’, and find good agreement with the theory put forth by Wilkinson et al. (Phys. Rev. Lett., vol. 97, 2006, 048501) and Falkovich & Pumir (J. Atmos. Sci., vol. 64, 2007, 4497) for Stokes numbers greater than 0.5. We also look at the scaling exponents within the context of the model proposed by Bec et al. (J. Fluid Mech., vol. 646, 2010, pp. 527–536). Another interesting finding is that inertial particles at low Stokes numbers sample regions of higher kinetic energy than the fluid particle field, the converse occurring at high Stokes numbers. The trend at low Stokes numbers is predicted by the theory of Chun et al. (J. Fluid Mech., vol. 536, 2005, 219–251). This work is relevant to modelling the particle collision rate (Sundaram & Collins, J. Fluid Mech., vol. 335, 1997, pp. 75–109), and highlights the interesting array of phenomena induced by inertia.

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 696 , 10 April 2012 , pp. 45 - 66
Copyright
Copyright © Cambridge University Press 2012

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.)

Footnotes

‡

Present address: Centro de Engenharia da Mobilidade, Universidade Federal de Santa Catarina, Joinville, Brazil.

References

1. Abrahamson, J. 1975 Collision rates of small particles in a vigorously turbulent fluid. Chem. Engng Sci. 30, 1371–1379.CrossRefGoogle Scholar
2. Ayala, O., Rosa, B., Wang, L.-P. & Grabowski, W. W. 2008 Effects of turbulence on the geometric collision rate of sedimenting droplets. Part 1. Results from direct numerical simulation. New J. Phys. 10, 075015.CrossRefGoogle Scholar
3. Ayyalasomayajula, S., Warhaft, Z. & Collins, L. R. 2008 Modelling inertial particle acceleration statistics in isotropic turbulence. Phys. Fluids 20, 094104.Google Scholar
4. Balachandar, S. & Eaton, J. K. 2010 Turbulent dispersed multiphase flow. Annu. Rev. Fluid Mech. 42, 111–133.CrossRefGoogle Scholar
5. Bec, J., Biferale, L., Boffetta, G., Celani, A., Cencini, M., Lanotte, A. S., Musacchio, S. & Toschi, F. 2006a Acceleration statistics of heavy particles in turbulence. J. Fluid Mech. 550, 349–358.CrossRefGoogle Scholar
6. Bec, J., Biferale, L., Boffetta, G., Cencini, M., Musacchio, S. & Toschi, F. 2006b Lyapunov exponents of heavy particles in turbulence. Phys. Fluids 18, 091702.Google Scholar
7. Bec, J., Biferale, L., Cencini, M., Lanotte, A. S., Musacchio, S. & Toschi, F. 2007 Heavy particle concentration in turbulence at dissipative and inertial scales. Phys. Rev. Lett. 98, 084502.CrossRefGoogle ScholarPubMed
8. 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
9. Benzi, R., Ciliberto, S., Tripiccione, R., Baudet, C., Massaioli, F. & Succi, S. 1993 Extended self-similarity in turbulent flows. Phys. Rev. E 48, R29–R32.CrossRefGoogle ScholarPubMed
10. Berrut, J. P. & Trefethen, L. N. 2004 Barycentric Lagrange interpolation. Siam Rev. 46, 501–517.Google Scholar
11. Biferale, L. 2003 Shell models of energy cascade in turbulence. Annu. Rev. Fluid Mech. 35, 441–468.Google Scholar
12. 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.CrossRefGoogle Scholar
13. 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.CrossRefGoogle Scholar
14. Chaumat, L. & Brenguier, J.-L. 2001 Droplet spectra broadening in cumulus clouds. Part II. Microscale droplet concentration heterogeneities. J. Atmos. Sci. 58, 642–654.2.0.CO;2>CrossRefGoogle Scholar
15. Chen, L., Goto, S. & Vassilicos, J. C. 2006 Turbulent clustering of stagnation points and inertial particles. J. Fluid Mech. 553, 143–155.CrossRefGoogle Scholar
16. 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
17. Coleman, S. W. & Vassilicos, J. C. 2009 A unified sweep-stick mechanism to explain particle clustering in two- and three-dimensional homogeneous, isotropic turbulence. Phys. Fluids 21 (11), 113301.CrossRefGoogle Scholar
18. Collins, L. R. & Keswani, A. 2004 Reynolds number scaling of particle clustering in turbulent aerosols. New J. Phys. 6, 119.Google Scholar
19. 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
20. Ducasse, L. & Pumir, A. 2009 Inertial particle collisions in turbulent synthetic flows: quantifying the sling effect. Phys. Rev. Lett. 80, 066312.Google ScholarPubMed
21. Eaton, J. K. & Fessler, J. R. 1994 Preferential concentration of particles by turbulence. Intl J. Multiphase Flow 20, 169–209.Google Scholar
22. Faeth, G. M. 1996 Spray combustion phenomena. Int. Combust. Symp. 26 (1), 1593–1612.CrossRefGoogle Scholar
23. Falkovich, G., Fouxon, A. & Stepanov, M. G. 2002 Acceleration of rain initiation by cloud turbulence. Nature 419, 151–154.Google Scholar
24. Falkovich, G. & Pumir, A. 2007 Sling effect in collisions of water droplets in turbulent clouds. J. Atmos. Sci. 64, 4497.CrossRefGoogle Scholar
25. Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
26. Gibert, M., Xu, H. & Bodenschatz, E. 2010 Inertial effects on two-particle relative dispersion in turbulent flows. Euro. Phys. Lett. 90 (6), 64005.Google Scholar
27. Goto, S. & Vassilicos, J. C. 2008 Sweep-stick mechanism of heavy particle clustering in fluid turbulence. Phys. Rev. Lett. 100 (5), 035504.Google Scholar
28. Grabowski, W. W. & Vaillancourt, P. 1999 Comments on ‘Preferential concentration of cloud droplets by turbulence: effects on the early evolution of cumulus cloud droplet spectra’ by Shaw, Reade, Collins and Verlinde. J. Atmos. Sci. 56, 1433–1436.2.0.CO;2>CrossRefGoogle Scholar
29. Grassberger, P. & Procaccia, I. 1983 Characterizations of strange attractors. Phys. Rev. Lett. 50, 346–349.CrossRefGoogle Scholar
30. Grossmann, S., Lohse, D. & Reeh, A. 1997 Application of extended self-similarity in turbulence. Phys. Rev. E 56, 5473–5478.CrossRefGoogle Scholar
31. Gustavsson, K. & Mehlig, B. 2011 Distribution of relative velocities in turbulent aerosols. Phys. Rev. E 84, 045304.CrossRefGoogle ScholarPubMed
32. Hill, R. J. 2001 Equations relating structure functions of all order. J. Fluid Mech. 434, 379–388.CrossRefGoogle Scholar
33. Hill, R. J. 2006 Opportunities for use of exact statistical equations. J. Turbul. 7, N43.Google Scholar
34. Ijzermans, R. H. A., Reeks, M. W., Meneguz, E., Picciotto, M. & Soldati, A. 2009 Measuring segregation of inertial particles in turbulence by a full Lagrangian approach. Phys. Rev. E 80 (1), 015302.Google Scholar
35. Ishihara, T., Gotoh, T. & Kaneda, Y. 2009 Study of high-Reynolds-number isotropic turbulence by direct numerical simulation. Annu. Rev. Fluid Mech. 41, 165–180.CrossRefGoogle Scholar
36. Johansen, A., Oishi, J. S., Mac Low, M. M., Klahr, H. & Henning, T. 2007 Rapid planetesimal formation in turbulent circumstellar disks. Nature 448 (7157), 1022–1025.Google Scholar
37. Kolmogorov, A. N. 1941 The local structure of turbulence in an incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR 30, 299–303.Google Scholar
38. Li, W. I., Perzl, M., Heyder, J., Langer, R., Brain, J. D., Englemeier, K. H., Niven, R. W. & Edwards, D. A. 1996 Aerodynamics and aerosol particle deaggregation phenomena in model oral–pharyngeal cavities. J. Aerosol Sci. 27 (8), 1269–1286.Google Scholar
39. Malkiel, E., Abras, J. N., Widder, E. A. & Katz, J. 2006 On the spatial distribution and nearest neighbour distance between particles in the water column determined from in situ holographic measurements. J. Plankton Res. 28 (2), 149–170.Google Scholar
40. 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
41. 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
42. McQuarrie, D. A. 1976 Statistical Mechanics. Harper & Row.Google Scholar
43. Moody, E. G. & Collins, L. R. 2003 Effect of mixing on nucleation and growth of titania particles. Aerosol Sci. Technol. 37, 403–424.Google Scholar
44. Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
45. Reade, W. C. & Collins, L. R. 2000a Effect of preferential concentration on turbulent collision rates. Phys. Fluids 12, 2530–2540.CrossRefGoogle Scholar
46. Reade, W. C. & Collins, L. R. 2000b A numerical study of the particle size distribution of an aerosol undergoing turbulent coagulation. J. Fluid Mech. 415, 45–64.CrossRefGoogle Scholar
47. Saffman, P. G. & Turner, J. S. 1956 On the collision of drops in turbulent clouds. J. Fluid Mech. 1, 16–30.Google Scholar
48. Salazar, J. P. L. C. 2010 Investigation of inertial particle phenomena in homogeneous isotropic turbulence. PhD dissertation, Cornell University.Google Scholar
49. 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
50. Saw, E. W., Shaw, R. A., Ayyalasomayajula, S., Chuang, P. Y. & Gylfason, A. 2008 Inertial clustering of particles in high-Reynolds-number turbulence. Phys. Rev. Lett. 100, 214501.Google Scholar
51. Schmidt, M. & Lipson, H. 2009 Distilling free-form natural laws from experimental data. Science 324, 81–85.Google Scholar
52. Shaw, R. A. 2003 Particle-turbulence interactions in atmospheric clouds. Annu. Rev. Fluid Mech. 35, 183–227.Google Scholar
53. 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
54. She, Z.-S. & Leveque, E. 1994 Universal scaling laws in fully developed turbulence. Phys. Rev. Lett. 72, 336–339.Google Scholar
55. Shotorban, B. & Balachandar, S. 2006 Particle concentration in homogeneous shear turbulence simulated via Lagrangian and equilibrium Eulerian approaches. Phys. Fluids 18 (6), 065105.CrossRefGoogle Scholar
56. Shotorban, B. & Balachandar, S. 2009 Two-fluid approach for direct numerical simulation of particle-laden turbulent flows at small Stokes numbers. Phys. Rev. E 79 (5), 056703.CrossRefGoogle ScholarPubMed
57. Siebert, H., Gerashchenko, S., Gylfason, A., Lehmann, K., Collins, L. R., Shaw, R. A. & Warhaft, Z. 2010a Towards understanding the role of turbulence on droplets in clouds: in situ and laboratory measurements. Atmos. Res. 97 (4), 426–437.Google Scholar
58. Siebert, H., Shaw, R. A. & Warhaft, Z. 2010b Statistics of small scale velocity fluctuations in marine stratocumulus clouds. J. Atmos. Sci. 67, 262–273.Google Scholar
59. Squires, K. D. & Eaton, J. K. 1991 Preferential concentration of particles by turbulence. Phys. Fluids A 3, 1169–1178.Google Scholar
60. Sreenivasan, K. R. 1995 On the universality of the Kolmogorov constant. Phys. Fluids 7, 2778–2784.Google Scholar
61. Sundaram, S. & Collins, L. R. 1997 Collision statistics in an isotropic, particle-laden turbulent suspension. Part I. Direct numerical simulations. J. Fluid Mech. 335, 75–109.Google Scholar
62. Sundaram, S. & Collins, L. R. 1999 A numerical study of the modulation of isotropic turbulence by suspended particles. J. Fluid Mech. 379, 105–143.Google Scholar
63. Vaillancourt, P. A., Yau, M. K. & Grabowski, W. W. 2001 Microscopic approach to cloud droplet growth by condensation. Part I. Model description and results without turbulence. J. Atmos. Sci. 58, 1945–1965.Google Scholar
64. 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
65. Wang, L.-P., Wexler, A. S. & Zhou, Y. 2000 Statistical mechanical description and modelling of turbulent collision of inertial particles. J. Fluid Mech. 415, 117–153.Google Scholar
66. Warhaft, Z. 2000 Passive scalars in turbulent flows. Annu. Rev. Fluid Mech. 32, 203–240.Google Scholar
67. Warhaft, Z. 2009a Laboratory studies of droplets in turbulence: towards understanding the formation of clouds. Fluid Dyn. Res. 41, 011201.Google Scholar
68. Warhaft, Z. 2009b Why we need turbulence experiments at high Reynolds number. Fluid Dyn. Res. 41, 021401.Google Scholar
69. Wilkinson, M., Mehlig, B. & Bezuglyy, V. 2006 Caustic activation of rain showers. Phys. Rev. Lett. 97, 048501.CrossRefGoogle ScholarPubMed
70. Wood, A. M., Hwang, W. & Eaton, J. K. 2005 Preferential concentration of particles in homogeneous and isotropic turbulence. Intl J. Multiphase Flow 31, 1220–1230.Google Scholar
71. Yudine, M. I. 1959 Physical considerations on heavy-particle dispersion. Adv. Geophys. 6, 185–191.Google Scholar
72. Zaichik, L. I. & Alipchenkov, V. M. 2003 Pair dispersion and preferential concentration of particles in isotropic turbulence. Phys. Fluids 15, 1776–1787.Google Scholar
73. 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
74. Zhou, Y., Wexler, A. S. & Wang, L.-P. 2001 Modelling turbulent collision of bidisperse inertial particles. J. Fluid Mech. 433, 77–104.Google Scholar