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Multiphase Flows

On the Volume Fraction Effects of Inertial Colliding Particles in Homogeneous Isotropic Turbulence

[+] Author and Article Information
Martin Ernst1

Mechanische Verfahrenstechnik, Zentrum für Ingenieurwissenschaften,  Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germanymartin.ernst@iw.uni-halle.de

Martin Sommerfeld

Mechanische Verfahrenstechnik, Zentrum für Ingenieurwissenschaften,  Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germanymartin.sommerfeld@iw.uni-halle.de

1

Corresponding author.

J. Fluids Eng 134(3), 031302 (Mar 23, 2012) (16 pages) doi:10.1115/1.4005681 History: Received April 19, 2011; Revised December 20, 2011; Published March 20, 2012; Online March 23, 2012

The main objective of the present study is the investigation of volume fraction effects on the collision statistics of nonsettling inertial particles in a granular medium as well as suspended in an unsteady homogeneous isotropic turbulent flow. For this purpose, different studies with mono-disperse Lagrangian point-particles having different Stokes numbers are considered in which the volume fraction of the dispersed phase is varied between 0.001 and 0.01. The fluid behavior is computed using a three-dimensional Lattice-Boltzmann method. The carrier-fluid turbulence is maintained at Taylor microscale Reynolds number 65.26 by applying a spectral forcing scheme. The Lagrangian particle tracking is based on considering the drag force only and a deterministic model is applied for collision detection. The influence of the particle phase on the fluid flow is neglected at this stage. The particle size is maintained at a constant value for all Stokes numbers so that the ratio of particle diameter to Kolmogorov length scale is fixed at 0.58. The variation of the particle Stokes number was realized by modifying the solids density. The observed particle Reynolds and Stokes numbers are in between [1.07, 2.61] and [0.34, 9.79], respectively. In the present simulations, the fluid flow and the particle motion including particle-particle collisions are based on different temporal discretization. Hence, an adaptive time stepping scheme is introduced. The particle motion as well as the occurrence of inter-particle collisions is characterized among others by Lagrangian correlation functions, the velocity angles between colliding particles and the collision frequencies. Initially, a fluid-free particle system is simulated and compared with the principles of the kinetic theory to validate the implemented deterministic collision model. Moreover, a selection of results obtained for homogeneous isotropic turbulence is compared with in literature available DNS and LES results as well. According to the performed simulations, the collision rate of particles with large Stokes numbers strongly depends on the adopted volume fraction, whereas for particles with small Stokes numbers the influence of particle volume fraction is less pronounced.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 15

Averaged Lagrangian correlation functions of the particle velocities RP,u (τ) and their corresponding Lagrangian integral timescales τL,P as a function of the particle Stokes number StP  = 0.01).

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Figure 16

Influence of the volume fraction αP on the Lagrangian correlation functions of the particles RP,u (τ) in presence as well as in absence of inter-particle collisions (St = 9.78). In case of particle-particle collision, the ratio of the particle Lagrangian integral timescale to the mean time between two successive inter-particle collisions τL,PC is given as well.

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Figure 18

Comparison of the ratio of kinetic energy of the particle fluctuation motion kP to the turbulent kinetic energy of the flow field kF obtained by the present DNS (triangle: ReT  = 65.26) with results from other DNS, published by Sundaram and Collins [12] (circle: ReT  = 54.20) and Fede and Simonin [15] (square: ReT  = 34.10), depending on the Stokes number St (Note: (1) symbols of one kind indicate the results for the different volume fractions, and (2) the results of all three DNS are based on a one-way momentum coupling of the dispersed phase with the fluid flow).

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Figure 19

Probability density function of the relative velocity modulus of colliding particles |uPij | which is normalized by Kolmogorov velocity uK with the Stokes number St as a parameter (αP  = 0.01). In addition, the mean values of relative velocity modulus are printed for easy comparison.

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Figure 1

Velocity direction vectors of the D3Q19 model

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Figure 10

Probability density function of the free path between particle collisions depending on the particle Stokes number StP  = 0.01). The indicated particle mean free path λFP¯ is also normalized by the Kolmogorov length scale λK .

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Figure 11

Effect of the particle response behavior (i.e., St) on the mean time between two particle-particle collisions τC : Here, the particle response time τP is normalized by the constant Kolmogorov timescale τK and plotted against the Stokes number St with the solid volume fraction αP as a parameter.

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Figure 12

Averaged collision frequencies fC , which are normalized by their corresponding particle response times τP , as a function of the Stokes number St and solid volume fraction αP .

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Figure 13

Computed collision frequencies fC as a function of the Stokes number StP  = 0.01): Comparison of results obtained by direct numerical simulations (present study) with the analytical Saffman and Turner limit [2] as well as the kinetic theory limit [3].

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Figure 14

Ratio of the computed collision frequency to the collision frequency obtained from kinetic theory plotted against the Stokes number StInt which is based on the fluid Lagrangian integral timescale: Comparison of results obtained by direct numerical simulations (open and partly filled symbols: present study), large eddy simulations for different volume fractions (closed symbols: Laviéville [21]) and analytical approximations (solid line: Kruis and Kusters [5]). Note: Symbols of one shape represent a comparable Stokes number, e.g., circle, square or triangle. Moreover, symbols of one filling level (present DNS) indicate the results for different volume fractions, i.e.,

P  = 0.001, P  = 0.005, P  = 0.01.

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Figure 17

Particle Lagrangian integral timescales τL,P as function of the Stokes number St with the solid volume fraction αP as a parameter: The calculated timescales are normalized by the Kolmogorov timescale τK as well as the mean time between successive inter-particle collisions τC  .

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Figure 2

Program flow chart for the deterministic collision model (in analogy to Ref. [36]).

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Figure 3

Pictorial representation of two colliding particles [38]

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Figure 4

Fluctuation of collision rates N observed in a fluid-free particle system as a function of the nondimensional tracking time (closed symbols). The solid line indicates the theoretical reference value based on the kinetic theory, cf. Eq. 2.

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Figure 5

Comparison of the probability density functions of the velocity modulus between colliding particles with theoretical results from the kinetic theory. The kinetic theory corresponds to the velocity distribution of the injected particles.

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Figure 6

Fluid velocity field (vector plot) and particle field distribution (spheres, St = 2.57, αP  = 0.01) for a single plane in the computational domain.

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Figure 7

Three-dimensional energy spectrum of the turbulent flow field (solid line with symbols: ReT  = 65.26) and Kolmogorov spectrum (dashed line: universal Kolmogorov constant C = 1.5) as a function of the nondimensional wave number.

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Figure 8

Viscous dissipation spectrum computed from the present DNS (ReT  = 65.26) and plotted against the nondimensional wave number.

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Figure 9

Probability density functions of the fluid velocity fluctuations for the three velocity components averaged over a single eddy turnover time.

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Figure 20

Influence of the volume fraction αP on (a) the mean relative velocity modulus |uPij|¯ and (b) the mean particle velocity angles ϕ¯ between colliding particles as a function of the Stokes number St.

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