Hostname: page-component-8448b6f56d-mp689 Total loading time: 0 Render date: 2024-04-23T22:03:12.287Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-way coupled stochastic model for dispersion of inertial particles in turbulence

Published online by Cambridge University Press:  18 April 2012

Madhusudan G. Pai  and
Shankar Subramaniam
Madhusudan G. Pai*
Affiliation:
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
Shankar Subramaniam*
Affiliation:
Department of Mechanical Engineering, Iowa State University, Ames, IA 50011, USA
*
†Email addresses for correspondence: paim@ge.com, shankar@iastate.edu
†Email addresses for correspondence: paim@ge.com, shankar@iastate.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Turbulent two-phase flows are characterized by the presence of multiple time and length scales. Of particular interest in flows with non-negligible interphase momentum coupling are the time scales associated with interphase turbulent kinetic energy transfer (TKE) and inertial particle dispersion. Point-particle direct numerical simulations (DNS) of homogeneous turbulent flows laden with sub-Kolmogorov size particles report that the time scale associated with the interphase TKE transfer behaves differently with Stokes number than the time scale associated with particle dispersion. Here, the Stokes number is defined as the ratio of the particle momentum response time scale to the Kolmogorov time scale of turbulence. In this study, we propose a two-way coupled stochastic model (CSM), which is a system of two coupled Langevin equations for the fluctuating velocities in each phase. The basis for the model is the Eulerian–Eulerian probability density function formalism for two-phase flows that was established in Pai & Subramaniam (J. Fluid Mech., vol. 628, 2009, pp. 181–228). This new model possesses the unique capability of simultaneously capturing the disparate dependence of the time scales associated with interphase TKE transfer and particle dispersion on Stokes number. This is ascertained by comparing predicted trends of statistics of turbulent kinetic energy and particle dispersion in both phases from CSM, for varying Stokes number and mass loading, with point-particle DNS datasets of homogeneous particle-laden flows.

JFM classification

Multiphase and Particle-laden Flows: Fluidized beds Multiphase and Particle-laden Flows: Gas/liquid flow Multiphase and Particle-laden Flows: Multiphase flow
Type
Papers
Information
Journal of Fluid Mechanics , Volume 700 , 10 June 2012 , pp. 29 - 62
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: GE Global Research, One Research Circle, Niskayuna, NY 12309, USA.

References

1. Ahmed, A. M. & Elghobashi, S. 2000 On the mechanisms of modifying the structure of turbulent homogeneous shear flows by dispersed particles. Phys. Fluids 12 (11), 2906–2930.CrossRefGoogle Scholar
2. Ahmed, A. M. & Elghobashi, S. 2001 Direct numerical simulation of particle dispersion in homogeneous turbulent shear flows. Phys. Fluids 13 (11), 3346–3364.CrossRefGoogle Scholar
3. Alder, B. J. & Wainwright, T. E. 1970 Decay of the velocity autocorrelation function. Phys. Rev. A 1 (1), 18–21.CrossRefGoogle Scholar
4. Amsden, A. A., O’Rourke, P. J. & Butler, T. D. 1989 KIVA-II: a computer program for chemically reactive flows with sprays. Tech. Rep. LA-11560-MS. Los Alamos National Laboratory.CrossRefGoogle Scholar
5. Boivin, M., Simonin, O. & Squires, K. 1998 Direct numerical simulation of turbulence modulation by particles in isotropic turbulence. J. Fluid Mech. 375, 235–263.CrossRefGoogle Scholar
6. Chagras, V., Oesterle, B. & Boulet, P. 2005 On the heat transfer in gas–solid pipe flows: effects of collision induced alterations of the flow dynamics. Intl J. Heat Mass Transfer 48, 1649–1661.CrossRefGoogle Scholar
7. Chen, X.-Q. & Pereira, J. C. F. 1997 Efficient computation of particle dispersion in turbulent flows with a stochastic–probabilistic model. Intl J. Heat Mass Transfer 40 (8), 1727–1741.CrossRefGoogle Scholar
8. Corrsin, S. 1959 Progress report on some turbulent diffusion research. Adv. Geophys. 6, 161–164.CrossRefGoogle Scholar
9. Csanady, G. T. 1963 Turbulent diffusion of heavy particles in the atmosphere. J. Atmos. Sci. 20, 201–208.2.0.CO;2>CrossRefGoogle Scholar
10. Fadlun, E. A., Verzicco, R., Orlandi, P. & Mohd-Yusof, J. 2000 Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. J. Comput. Phys. 161 (1), 35–60.CrossRefGoogle Scholar
11. Gao, Z. & Mashayek, F. 2004 Stochastic modeling of evaporating droplets polydispersed in turbulent flows. Intl J. Heat Mass Transfer 47, 4339–4348.CrossRefGoogle Scholar
12. Gardiner, C. W. 1983 Handbook of Stochastic Methods. Springer.CrossRefGoogle Scholar
13. Garg, R., Narayanan, C. & Subramaniam, S. 2009 A numerically convergent Lagrangian–Eulerian simulation method for dispersed two-phase flows. Intl J. Multiphase Flow 35 (4), 376–388.CrossRefGoogle Scholar
14. Gosman, A. D. & Ioannides, E. 1983 Aspects of computer simulation of liquid fueled combustors. J. Engine Res. 6 (7), 482–490.Google Scholar
15. Gouesbet, G., Berlemont, A. & Picart, A. 1984 Dispersion of discrete particles by continuous turbulent motions. Extensive discussion of the Tchen’s theory, using a two-parameter family of Lagrangian correlation functions. Phys. Fluids 27 (4), 827–837.CrossRefGoogle Scholar
16. Groszmann, D. E. & Rogers, C. B. 2004 Turbulent scales of dilute particle-laden flows in microgravity. Phys. Fluids 16 (12), 4671–4684.CrossRefGoogle Scholar
17. Haworth, D. C. & Pope, S. B. 1986 A generalized Langevin model for turbulent flows. Phys. Fluids 29, 387–405.CrossRefGoogle Scholar
18. Hu, H., Zhu, M. Y. & Patankar, N. A. 2001 Direct numerical simulations of fluid–solid systems using the arbitrary Lagrangian–Eulerian technique. J. Comput. Phys. 169 (2), 427–462.CrossRefGoogle Scholar
19. Kloeden, P. & Platen, E. 1992 Numerical Solution of Stochastic Differential Equations. Springer.CrossRefGoogle Scholar
20. Lu, Q. Q. 1995 An approach to modeling particle motion in turbulent flows – I. Homogeneous, isotropic turbulence. Atmos. Environ. 29 (3), 423–436.CrossRefGoogle Scholar
21. L’vov, V. S., Ooms, G. & Pomyalov, A. 2003 Effect of particle inertia on turbulence in a suspension. Phys. Rev. E 67, 046314.CrossRefGoogle Scholar
22. Mashayek, F. 1999 Stochastic simulations of particle-laden isotropic turbulent flow. Intl J. Multiphase Flow 25, 1575–1599.CrossRefGoogle Scholar
23. Mashayek, F., Jaberi, F. A., Miller, R. S. & Givi, P. 1997 Dispersion and polydispersity of droplets in stationary isotropic turbulence. Intl J. Multiphase Flow 23, 337–355.CrossRefGoogle Scholar
24. Maxey, M. R. & Riley, J. J. 1983 Equation of motion for a small rigid sphere in a non-uniform flow. Phys. Fluids 26, 883–889.CrossRefGoogle Scholar
25. Mei, R., Adrian, R. J. & Hanratty, T. J. 1991 Particle dispersion in isotropic turbulence under Stokes drag and Basset force with gravitational settling. J. Fluid Mech. 225, 481–495.CrossRefGoogle Scholar
26. Pai, M. G. & Subramaniam, S. 2006 Modeling interphase turbulent kinetic energy transfer in Lagrangian–Eulerian spray computations. Atomiz. Sprays 16 (7), 807–826.CrossRefGoogle Scholar
27. Pai, M. G. & Subramaniam, S. 2007 Modeling droplet dispersion and interphase turbulent kinetic energy transfer using a new dual-timescale Langevin model. Intl J. Multiphase Flow 33 (3), 252–281.CrossRefGoogle Scholar
28. Pai, M. G. & Subramaniam, S. 2009 A comprehensive probability density function formalism for multiphase flows. J. Fluid Mech. 628, 181–228.CrossRefGoogle Scholar
29. Pascal, P. & Oesterlé, B. 2000 On the dispersion of discrete particles moving in a turbulent shear flow. Intl J. Multiphase Flow 26, 293–325.CrossRefGoogle Scholar
30. Pope, S. B. 1985 PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11, 119–192.CrossRefGoogle Scholar
31. Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
32. Pope, S. B. 2002 Stochastic Lagrangian models of velocity in homogeneous turbulent shear flow. Phys. Fluids 14 (5), 1696–1702.CrossRefGoogle Scholar
33. Pozorski, J. & Minier, J.-P. 1999 Probability density function modeling of dispersed two-phase turbulent flows. Phys. Rev. E 59 (1), 855–863.CrossRefGoogle Scholar
34. Sawford, B. L. & Yeung, P. K. 2001 Lagrangian statistics in uniform shear flow: direct numerical simulation and Lagrangian stochastic models. Phys. Fluids 13, 2627.CrossRefGoogle Scholar
35. Simonin, O. 1996a Continuum modeling of dispersed turbulent two-phase flows. Part 1. General model description. Tech. Rep. Lecture Series. Von Kármán Institute of Fluid Dynamics.Google Scholar
36. Simonin, O. 1996b Continuum modeling of dispersed turbulent two-phase flows. Part 2. Model predictions and discussion. Tech. Rep. Lecture Series. Von Kármán Institute of Fluid Dynamics.Google Scholar
37. Snyder, W. H. & Lumley, J. L. 1971 Some measurements of particle velocity autocorrelation functions in a turbulent flow. J. Fluid Mech. 48, 41–71.CrossRefGoogle Scholar
38. Squires, K. D. & Eaton, J. K. 1991 Measurements of particle dispersion obtained from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 226, 1–35.CrossRefGoogle Scholar
39. 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
40. Ten Cate, A., Derksen, J. J., Portela, L. M. & Van Den Akker, H. E. A. 2004 Fully resolved simulations of colliding monodisperse spheres in forced isotropic turbulence. J. Fluid Mech. 519, 233–271.CrossRefGoogle Scholar
41. Truesdell, G. C. & Elghobashi, S. 1994 On the two-way interaction between homogeneous turbulence and dispersed solid particles. II. Particle dispersion. Phys. Fluids 6 (3), 1405–1407.CrossRefGoogle Scholar
42. Wells, M. R. & Stock, D. E. 1983 The effects of crossing trajectories on the dispersion of particles in a turbulent flow. J. Fluid Mech. 136, 31–62.CrossRefGoogle Scholar
43. Xu, Y. & Subramaniam, S. 2006 A multiscale model for dilute turbulent gas–particle flows based on the equilibration of energy concept. Phys. Fluids 18, 033301.CrossRefGoogle Scholar
44. Xu, Y. & Subramaniam, S. 2007 Conservative interphase turbulent kinetic energy transfer in particle-laden flows. Phys. Fluids 19, 085101.CrossRefGoogle Scholar
45. Xu, Y. & Subramaniam, S. 2010 Effect of particle clusters on carrier flow turbulence: A direct numerical simulation study. Flow Turbul. Combust. 85, 735–761.CrossRefGoogle Scholar
46. Yeung, P. K. & Pope, S. B. 1989 Lagrangian statistics from direct numerical simulations of isotropic turbulence. J. Fluid Mech. 207, 531–586.CrossRefGoogle Scholar
47. Yusof, M. J. 1996 Interaction of massive particles with turbulence. PhD thesis, Cornell University.Google Scholar