Collision statistics of inertial particles in two-dimensional homogeneous isotropic turbulence with an inverse cascade

Media type: E-Article
Title: Collision statistics of inertial particles in two-dimensional homogeneous isotropic turbulence with an inverse cascade
Contributor: Onishi, Ryo; Vassilicos, J. C.
imprint: Cambridge University Press (CUP), 2014
Published in: Journal of Fluid Mechanics
Language: English
DOI: 10.1017/jfm.2014.97
ISSN: 0022-1120; 1469-7645
Keywords: Mechanical Engineering ; Mechanics of Materials ; Condensed Matter Physics
Origination:
Footnote:
Description: <jats:title>Abstract</jats:title><jats:p>This study investigates the collision statistics of inertial particles in inverse-cascading two-dimensional (2D) homogeneous isotropic turbulence by means of a direct numerical simulation (DNS). A collision kernel model for particles with small Stokes number (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline1" /><jats:tex-math>$\mathit{St}$</jats:tex-math></jats:alternatives></jats:inline-formula>) in 2D flows is proposed based on the model of Saffman & Turner (<jats:italic>J. Fluid Mech.</jats:italic>, vol. 1, 1956, pp. 16–30) (ST56 model). The DNS results agree with this 2D version of the ST56 model for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline2" /><jats:tex-math>$\mathit{St}\lesssim 0.1$</jats:tex-math></jats:alternatives></jats:inline-formula>. It is then confirmed that our DNS results satisfy the 2D version of the spherical formulation of the collision kernel. The fact that the flatness factor stays around 3 in our 2D flow confirms that the present 2D turbulent flow is nearly intermittency-free. Collision statistics for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline3" /><jats:tex-math>$\mathit{St}= 0.1$</jats:tex-math></jats:alternatives></jats:inline-formula>, 0.4 and 0.6, i.e. for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline4" /><jats:tex-math>$\mathit{St}<1$</jats:tex-math></jats:alternatives></jats:inline-formula>, are obtained from the present 2D DNS and compared with those obtained from the three-dimensional (3D) DNS of Onishi <jats:italic>et al.</jats:italic> (<jats:italic>J. Comput. Phys.</jats:italic>, vol. 242, 2013, pp. 809–827). We have observed that the 3D radial distribution function at contact (<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline5" /><jats:tex-math>$g(R)$</jats:tex-math></jats:alternatives></jats:inline-formula>, the so-called clustering effect) decreases for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline6" /><jats:tex-math>$\mathit{St}= 0.4$</jats:tex-math></jats:alternatives></jats:inline-formula> and 0.6 with increasing Reynolds number, while the 2D <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline7" /><jats:tex-math>$g(R)$</jats:tex-math></jats:alternatives></jats:inline-formula> does not show a significant dependence on Reynolds number. This observation supports the view that the Reynolds-number dependence of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline8" /><jats:tex-math>$g(R)$</jats:tex-math></jats:alternatives></jats:inline-formula> observed in three dimensions is due to internal intermittency of the 3D turbulence. We have further investigated the local <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline9" /><jats:tex-math>$\mathit{St}$</jats:tex-math></jats:alternatives></jats:inline-formula>, which is a function of the local flow strain rates, and proposed a plausible mechanism that can explain the Reynolds-number dependence of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline10" /><jats:tex-math>$g(R)$</jats:tex-math></jats:alternatives></jats:inline-formula>. Meanwhile, 2D stochastic simulations based on the Smoluchowski equations for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline11" /><jats:tex-math>$\mathit{St}\ll 1$</jats:tex-math></jats:alternatives></jats:inline-formula> show that the collision growth can be predicted by the 2D ST56 model and that rare but strong events do not play a significant role in such a small-<jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline12" /><jats:tex-math>$\mathit{St}$</jats:tex-math></jats:alternatives></jats:inline-formula> particle system. However, the probability density function of local <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline13" /><jats:tex-math>$\mathit{St}$</jats:tex-math></jats:alternatives></jats:inline-formula> at the sites of colliding particle pairs supports the view that powerful rare events can be important for particle growth even in the absence of internal intermittency when <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" xlink:type="simple" xlink:href="S0022112014000974_inline14" /><jats:tex-math>$\mathit{St}$</jats:tex-math></jats:alternatives></jats:inline-formula> is not much smaller than unity.</jats:p>

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