Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-02T05:41:55.167Z Has data issue: false hasContentIssue false

Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 2. Comparison of theory and DNS

Published online by Cambridge University Press:  22 May 2019

Sarma L. Rani*
Affiliation:
Department of Mechanical and Aerospace Engineering, University of Alabama in Huntsville, Huntsville, AL 35899, USA
Rohit Dhariwal
Affiliation:
Department of Civil and Environmental Engineering, Duke University, Durham, NC 27708, USA
Donald L. Koch
Affiliation:
Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, NY 14853, USA
*
Email address for correspondence: sarma.rani@uah.edu

Abstract

Part 1 (Rani et al. J. Fluid Mech., vol. 871, 2019, pp. 450–476) of this study presented a stochastic theory for the clustering of monodisperse, rapidly settling, low-Stokes-number particle pairs in homogeneous isotropic turbulence. The theory involved the development of closure approximations for the drift and diffusion fluxes in the probability density function (p.d.f.) equation for the pair relative positions $\boldsymbol{r}$. In this part 2 paper, the theory is quantitatively analysed by comparing its predictions of particle clustering with data from direct numerical simulations (DNS) of isotropic turbulence containing particles settling under gravity. The simulations were performed at a Taylor micro-scale Reynolds number $Re_{\unicode[STIX]{x1D706}}=77.76$ for three Froude numbers $Fr=\infty ,0.052,0.006$, where $Fr$ is the ratio of the Kolmogorov scale of acceleration and the magnitude of gravitational acceleration. Thus, $Fr=\infty$ corresponds to zero gravity, and $Fr=0.006$ to the highest magnitude of gravity among the three DNS cases. For each $Fr$, particles of Stokes numbers in the range $0.01\leqslant St_{\unicode[STIX]{x1D702}}\leqslant 0.2$ were tracked in the DNS, and particle clustering quantified both as a function of separation and the spherical polar angle. We compared the DNS and theory values for the exponent $\unicode[STIX]{x1D6FD}$ characterizing the power-law dependence of clustering on separation. The $\unicode[STIX]{x1D6FD}$ from the $Fr=0.006$ DNS case are in reasonable agreement with the theoretical predictions obtained using the second drift closure (referred to as DF2). To quantify the anisotropy in clustering, we calculated the leading–order coefficient in the spherical harmonics expansion of the p.d.f. of pair relative positions. The coefficients predicted by the theory (DF2) again show reasonable agreement with those calculated from the DNS clustering data for $Fr=0.006$. However, we note that in spite of the high magnitude of gravity, the clustering is only marginally anisotropic both in DNS and theory. The theory predicts that the spherical harmonic coefficient scales with $\unicode[STIX]{x1D6FD}(=\unicode[STIX]{x1D6FD}_{2}St_{\unicode[STIX]{x1D702}}^{2})$, where $\unicode[STIX]{x1D6FD}_{2}$ is the ratio of the drift and diffusion flux coefficients. Since the drift flux, and thereby $\unicode[STIX]{x1D6FD}_{2}$, is seen to decrease with gravity for $St_{\unicode[STIX]{x1D702}}<1$, the anisotropy is also correspondingly diminished.

Type
JFM Papers
Copyright
© 2019 Cambridge University Press 

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

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 (7), 075015.Google Scholar
Bec, J., Homann, H. & Ray, S. S. 2014 Gravity-driven enhancement of heavy particle clustering in turbulent flow. Phys. Rev. Lett. 112 (18), 184501.10.1103/PhysRevLett.112.184501Google 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, 2032.10.1016/j.jcp.2006.10.018Google Scholar
Chun, J., Koch, D. L., Rani, S. L., Ahluwalia, A. & Collins, L. R. 2005 Clustering of aerosol particles in isotropic turbulence. J. Fluid Mech. 536, 219251.10.1017/S0022112005004568Google Scholar
Dhariwal, R. & Bragg, A. D. 2018 Small-scale dynamics of settling, bidisperse particles in turbulence. J. Fluid Mech. 839, 594620.10.1017/jfm.2018.24Google Scholar
Ireland, P. J., Bragg, A. D. & Collins, L. R. 2016 The effect of Reynolds number on inertial particle dynamics in isotropic turbulence. Part 2. Simulations with gravitational effects. J. Fluid Mech. 796, 659711.10.1017/jfm.2016.227Google Scholar
Ireland, P. J., Vaithianathan, T., Sukheswalla, P. S., Ray, B. & Collins, L. R. 2013 Highly parallel particle-laden flow solver for turbulence research. Comput. Fluids 76, 170177.10.1016/j.compfluid.2013.01.020Google Scholar
Onishi, R., Takahashi, K. & Komori, S. 2009 Influence of gravity on collisions of monodispersed droplets in homogeneous isotropic turbulence. Phys. Fluids 21 (12), 125108.10.1063/1.3276906Google Scholar
Parishani, H., Ayala, O., Rosa, B., Wang, L.-P. & Grabowski, W. W. 2015 Effects of gravity on the acceleration and pair statistics of inertial particles in homogeneous isotropic turbulence. Phys. Fluids 27 (3), 033304.10.1063/1.4915121Google Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.10.1017/CBO9780511840531Google Scholar
Rani, S. L., Gupta, V. K. & Koch, D. L. 2019 Clustering of rapidly settling, low-inertia particle pairs in isotropic turbulence. Part 1. Drift and diffusion flux closures. J. Fluid Mech. 871, 450476.10.1017/jfm.2019.204Google Scholar
Reade, W. C. & Collins, L. R. 2000 Effect of preferential concentration on turbulent collision rates. Phys. Fluids 12 (10), 25302540.10.1063/1.1288515Google Scholar
Witkowska, A., Juvé, D. & Brasseur, J. G. 1997 Numerical study of noise from isotropic turbulence. J. Comput. Acoust. 5 (03), 317336.10.1142/S0218396X97000186Google Scholar
Woittiez, E. J. P., Jonker, H. J. J. & Portela, L. M. 2009 On the combined effects of turbulence and gravity on droplet collisions in clouds: a numerical study. J. Atmos. Sci. 66 (7), 19261943.10.1175/2005JAS2669.1Google Scholar