# Pairwise interaction extended point-particle model for a random array of monodisperse spheres

## Abstract

This study introduces a new point-particle force model that attempts to account for the hydrodynamic influence of the neighbouring particles in an Eulerian–Lagrangian simulation. In previous point-particle models the force on a particle depends only on Reynolds number and mean volume fraction. Thus, as long as the mean local volume fraction is the same, the force on different particles will be estimated to be the same, even though the precise arrangement of neighbours can be vastly different. From direct numerical simulation (DNS) it has been observed that in a random arrangement of spheres that were distributed with uniform probability, the particle-to-particle variation in force can be as large as the mean drag. Since the Reynolds number and mean volume fraction of all the particles within the array are the same, the standard models fail to account for the significant particle-to-particle force variation within the random array. Here, we develop a model which can compute the drag and lateral forces on each particle by accounting for the precise location of a few surrounding neighbours. A pairwise interaction is assumed where the perturbation flow induced by each neighbour is considered separately, then the effects of all neighbours are linearly superposed to obtain the total perturbation. Faxén correction is used to quantify the force perturbation due to the presence of the neighbours. The single neighbour perturbations are mapped in the vicinity of a reference sphere and stored as libraries. We test the pairwise interaction extended point-particle (PIEP) model for random arrays at two different volume fractions of$$\unicode[STIX]{x1D719}=0.1$$and 0.21 and Reynolds numbers in the range$$16.5\leqslant Re\leqslant 170$$. The PIEP model predictions are compared against drag and lift forces obtained from the fully resolved DNS simulations performed using the immersed boundary method. Although not perfect, we observe the PIEP model prediction to correlate much better with the DNS results than the classical mean drag model prediction.

- Authors:

- Publication Date:

- Research Org.:
- Univ. of Florida, Gainesville, FL (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1534390

- DOE Contract Number:
- NA0002378

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Fluid Mechanics

- Additional Journal Information:
- Journal Volume: 813; Journal ID: ISSN 0022-1120

- Publisher:
- Cambridge University Press

- Country of Publication:
- United States

- Language:
- English

- Subject:
- Mechanics; Physics

### Citation Formats

```
Akiki, G., Jackson, T. L., and Balachandar, S.
```*Pairwise interaction extended point-particle model for a random array of monodisperse spheres*. United States: N. p., 2017.
Web. doi:10.1017/jfm.2016.877.

```
Akiki, G., Jackson, T. L., & Balachandar, S.
```*Pairwise interaction extended point-particle model for a random array of monodisperse spheres*. United States. doi:10.1017/jfm.2016.877.

```
Akiki, G., Jackson, T. L., and Balachandar, S. Thu .
"Pairwise interaction extended point-particle model for a random array of monodisperse spheres". United States. doi:10.1017/jfm.2016.877.
```

```
@article{osti_1534390,
```

title = {Pairwise interaction extended point-particle model for a random array of monodisperse spheres},

author = {Akiki, G. and Jackson, T. L. and Balachandar, S.},

abstractNote = {This study introduces a new point-particle force model that attempts to account for the hydrodynamic influence of the neighbouring particles in an Eulerian–Lagrangian simulation. In previous point-particle models the force on a particle depends only on Reynolds number and mean volume fraction. Thus, as long as the mean local volume fraction is the same, the force on different particles will be estimated to be the same, even though the precise arrangement of neighbours can be vastly different. From direct numerical simulation (DNS) it has been observed that in a random arrangement of spheres that were distributed with uniform probability, the particle-to-particle variation in force can be as large as the mean drag. Since the Reynolds number and mean volume fraction of all the particles within the array are the same, the standard models fail to account for the significant particle-to-particle force variation within the random array. Here, we develop a model which can compute the drag and lateral forces on each particle by accounting for the precise location of a few surrounding neighbours. A pairwise interaction is assumed where the perturbation flow induced by each neighbour is considered separately, then the effects of all neighbours are linearly superposed to obtain the total perturbation. Faxén correction is used to quantify the force perturbation due to the presence of the neighbours. The single neighbour perturbations are mapped in the vicinity of a reference sphere and stored as libraries. We test the pairwise interaction extended point-particle (PIEP) model for random arrays at two different volume fractions of$\unicode[STIX]{x1D719}=0.1$and 0.21 and Reynolds numbers in the range$16.5\leqslant Re\leqslant 170$. The PIEP model predictions are compared against drag and lift forces obtained from the fully resolved DNS simulations performed using the immersed boundary method. Although not perfect, we observe the PIEP model prediction to correlate much better with the DNS results than the classical mean drag model prediction.},

doi = {10.1017/jfm.2016.877},

journal = {Journal of Fluid Mechanics},

issn = {0022-1120},

number = ,

volume = 813,

place = {United States},

year = {2017},

month = {1}

}

Works referenced in this record:

##
Drag law for monodisperse gas–solid systems using particle-resolved direct numerical simulation of flow past fixed assemblies of spheres

journal, November 2011

- Tenneti, S.; Garg, R.; Subramaniam, S.
- International Journal of Multiphase Flow, Vol. 37, Issue 9

##
Magnetic resonance imaging of liquid flow and pore structure within packed beds

journal, July 1997

- Sederman, A. J.; Johns, M. L.; Bramley, A. S.
- Chemical Engineering Science, Vol. 52, Issue 14

##
Preferential concentration of heavy particles: A Voronoï analysis

journal, October 2010

- Monchaux, R.; Bourgoin, M.; Cartellier, A.
- Physics of Fluids, Vol. 22, Issue 10

##
Immersed boundary method with non-uniform distribution of Lagrangian markers for a non-uniform Eulerian mesh

journal, February 2016

- Akiki, G.; Balachandar, S.
- Journal of Computational Physics, Vol. 307

##
Accurate calculation of Stokes drag for point–particle tracking in two-way coupled flows

journal, August 2016

- Horwitz, J. A. K.; Mani, A.
- Journal of Computational Physics, Vol. 318

##
Analyzing preferential concentration and clustering of inertial particles in turbulence

journal, April 2012

- Monchaux, Romain; Bourgoin, Mickael; Cartellier, Alain
- International Journal of Multiphase Flow, Vol. 40

##
The transverse force on a spinning sphere moving in a viscous fluid

journal, November 1961

- Rubinow, S. I.; Keller, Joseph B.
- Journal of Fluid Mechanics, Vol. 11, Issue 03

##
Shake-The-Box: Lagrangian particle tracking at high particle image densities

journal, April 2016

- Schanz, Daniel; Gesemann, Sebastian; Schröder, Andreas
- Experiments in Fluids, Vol. 57, Issue 5

##
Effect of coal particle size distribution on packed bed pressure drop and gas flow distribution

journal, July 2006

- Keyser, M.; Conradie, M.; Coertzen, M.
- Fuel, Vol. 85, Issue 10-11

##
On the dispersed two-phase flow in the laminar flow regime

journal, November 1964

- Zuber, N.
- Chemical Engineering Science, Vol. 19, Issue 11

##
Direct Numerical Simulation of Flow Past Elliptic Cylinders

journal, March 1996

- Mittal, R.; Balachandar, S.
- Journal of Computational Physics, Vol. 124, Issue 2

##
Local volume fraction fluctuations in random media

journal, February 1997

- Quintanilla, J.; Torquato, S.
- The Journal of Chemical Physics, Vol. 106, Issue 7

##
The sedimentation of a suspension of uniform spheres under conditions of viscous flow

journal, April 1954

- Richardson, J. F.; Zaki, W. N.
- Chemical Engineering Science, Vol. 3, Issue 2

##
An immersed boundary method with direct forcing for the simulation of particulate flows

journal, November 2005

- Uhlmann, Markus
- Journal of Computational Physics, Vol. 209, Issue 2

##
A hybrid approach for simulating turbulent collisions of hydrodynamically-interacting particles

journal, July 2007

- Ayala, Orlando; Grabowski, Wojciech W.; Wang, Lian-Ping
- Journal of Computational Physics, Vol. 225, Issue 1

##
Analysis of Classical Statistical Mechanics by Means of Collective Coordinates

journal, April 1958

- Percus, Jerome K.; Yevick, George J.
- Physical Review, Vol. 110, Issue 1

##
A new relation of drag force for high Stokes number monodisperse spheres by direct numerical simulation

journal, November 2014

- Zaidi, Ali Abbas; Tsuji, Takuya; Tanaka, Toshitsugu
- Advanced Powder Technology, Vol. 25, Issue 6

##
The lift force on a spherical body in a rotational flow

journal, October 1987

- Auton, T. R.
- Journal of Fluid Mechanics, Vol. 183

##
Effect of free rotation on the motion of a solid sphere in linear shear flow at moderate Re

journal, August 2002

- Bagchi, P.; Balachandar, S.
- Physics of Fluids, Vol. 14, Issue 8

##
Stokesian Dynamics

journal, January 1988

- Brady, J. F.; Bossis, G.
- Annual Review of Fluid Mechanics, Vol. 20, Issue 1

##
The hydrodynamic interaction of two small freely-moving spheres in a linear flow field

journal, November 1972

- Batchelor, G. K.; Green, J. T.
- Journal of Fluid Mechanics, Vol. 56, Issue 2

##
Simulation of concentrated suspensions using the force-coupling method

journal, March 2010

- Yeo, Kyongmin; Maxey, Martin R.
- Journal of Computational Physics, Vol. 229, Issue 6

##
Computational study of a single-phase flow in packed beds of spheres

journal, January 2005

- Gunjal, Prashant R.; Ranade, Vivek V.; Chaudhari, Raghunath V.
- AIChE Journal, Vol. 51, Issue 2

##
The Motion of High-Reynolds-Number Bubbles in Inhomogeneous Flows

journal, January 2000

- Magnaudet, J.; Eames, I.
- Annual Review of Fluid Mechanics, Vol. 32, Issue 1

##
Drag correlation for dilute and moderately dense fluid-particle systems using the lattice Boltzmann method

journal, January 2015

- Bogner, Simon; Mohanty, Swati; Rüde, Ulrich
- International Journal of Multiphase Flow, Vol. 68

##
Stochastic Lagrangian model for hydrodynamic acceleration of inertial particles in gas–solid suspensions

journal, January 2016

- Tenneti, Sudheer; Mehrabadi, Mohammad; Subramaniam, Shankar
- Journal of Fluid Mechanics, Vol. 788

##
Turbulent dispersion in a non-homogeneous field

journal, August 1999

- Iliopoulos, Ilias; Hanratty, Thomas J.
- Journal of Fluid Mechanics, Vol. 392

##
A new drag correlation from fully resolved simulations of flow past monodisperse static arrays of spheres

journal, October 2014

- (Yali) Tang, Y.; (Frank) Peters, E. A. J. F.; (Hans) Kuipers, J. A. M.
- AIChE Journal, Vol. 61, Issue 2

##
Immersed-boundary methods for general finite-difference and finite-volume Navier–Stokes solvers

journal, December 2010

- Pinelli, A.; Naqavi, I. Z.; Piomelli, U.
- Journal of Computational Physics, Vol. 229, Issue 24

##
History force on a sphere in a weak linear shear flow

journal, September 2005

- Wakaba, L.; Balachandar, S.
- International Journal of Multiphase Flow, Vol. 31, Issue 9

##
The force exerted on a body in inviscid unsteady non-uniform rotational flow

journal, December 1988

- Auton, T. R.; Hunt, J. C. R.; Prud'Homme, M.
- Journal of Fluid Mechanics, Vol. 197

##
Inertial flow structures in a simple-packed bed of spheres

journal, January 2003

- Suekane, Tetsuya; Yokouchi, Yasuo; Hirai, Shuichiro
- AIChE Journal, Vol. 49, Issue 1

##
The added mass, Basset, and viscous drag coefficients in nondilute bubbly liquids undergoing small‐amplitude oscillatory motion

journal, December 1991

- Sangani, A. S.; Zhang, D. Z.; Prosperetti, A.
- Physics of Fluids A: Fluid Dynamics, Vol. 3, Issue 12

##
Force variation within arrays of monodisperse spherical particles

journal, August 2016

- Akiki, G.; Jackson, T. L.; Balachandar, S.
- Physical Review Fluids, Vol. 1, Issue 4

##
Turbulent Dispersed Multiphase Flow

journal, January 2010

- Balachandar, S.; Eaton, John K.
- Annual Review of Fluid Mechanics, Vol. 42, Issue 1

##
Lagrangian Measurements of Inertial Particle Accelerations in Grid Generated Wind Tunnel Turbulence

journal, October 2006

- Ayyalasomayajula, S.; Gylfason, A.; Collins, L. R.
- Physical Review Letters, Vol. 97, Issue 14

##
Drag force of intermediate Reynolds number flow past mono- and bidisperse arrays of spheres

journal, January 2007

- Beetstra, R.; van der Hoef, M. A.; Kuipers, J. A. M.
- AIChE Journal, Vol. 53, Issue 2

##
A stochastic model for solid particle dispersion in a nonhomogeneous turbulent field

journal, March 2003

- Iliopoulos, Ilias; Mito, Yoichi; Hanratty, Thomas J.
- International Journal of Multiphase Flow, Vol. 29, Issue 3

##
Exact regularized point particle method for multiphase flows in the two-way coupling regime

journal, May 2015

- Gualtieri, P.; Picano, F.; Sardina, G.
- Journal of Fluid Mechanics, Vol. 773

##
The force on a sphere in a uniform flow with small-amplitude oscillations at finite Reynolds number

journal, November 1993

- Lovalenti, Phillip M.; Brady, John F.
- Journal of Fluid Mechanics, Vol. 256

##
Equation of motion for a small rigid sphere in a nonuniform flow

journal, January 1983

- Maxey, Martin R.
- Physics of Fluids, Vol. 26, Issue 4

##
Sphere Drag and Settling Velocity Revisited

journal, March 2003

- Brown, Phillip P.; Lawler, Desmond F.
- Journal of Environmental Engineering, Vol. 129, Issue 3

##
Sedimentation in a dilute dispersion of spheres

journal, March 1972

- Batchelor, G. K.
- Journal of Fluid Mechanics, Vol. 52, Issue 2

##
Flow past a sphere with an oscillation in the free-stream velocity and unsteady drag at finite Reynolds number

journal, April 1992

- Mei, Renwei; Adrian, Ronald J.
- Journal of Fluid Mechanics, Vol. 237

##
Shear versus vortex-induced lift force on a rigid sphere at moderate *Re*

journal, December 2002

- Bagchi, P.; Balachandar, S.
- Journal of Fluid Mechanics, Vol. 473

##
Der Widerstand gegen die Bewegung einer starren Kugel in einer zähen Flüssigkeit, die zwischen zwei parallelen ebenen Wänden eingeschlossen ist

journal, January 1922

- Faxén, Hilding
- Annalen der Physik, Vol. 373, Issue 10

##
The Resistance to Motion of a Solid Sphere in a Fluid

journal, December 1987

- Khan, A. R.; Richardson, J. F.
- Chemical Engineering Communications, Vol. 62, Issue 1-6

##
Force-coupling method for particulate two-phase flow: Stokes flow

journal, January 2003

- Lomholt, Sune; Maxey, Martin R.
- Journal of Computational Physics, Vol. 184, Issue 2

##
On the equation for spherical-particle motion: effect of Reynolds and acceleration numbers

journal, July 1998

- Kim, Inchul; Elghobashi, Said; Sirignano, William A.
- Journal of Fluid Mechanics, Vol. 367

##
Improved Formulations of the Superposition Method

journal, April 2005

- Wang, Lian-Ping; Ayala, Orlando; Grabowski, Wojciech W.
- Journal of the Atmospheric Sciences, Vol. 62, Issue 4

##
The lift on a small sphere in a slow shear flow

journal, June 1965

- Saffman, P. G.
- Journal of Fluid Mechanics, Vol. 22, Issue 2

##
Turbulent Transport of Material Particles: An Experimental Study of Finite Size Effects

journal, October 2007

- Qureshi, Nauman M.; Bourgoin, Mickaël; Baudet, Christophe
- Physical Review Letters, Vol. 99, Issue 18

##
Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres

journal, April 1963

- Wertheim, M. S.
- Physical Review Letters, Vol. 10, Issue 8

##
The Motion of Two Spheres in a Viscous Fluid

journal, May 1926

- Stimson, M.; Jeffery, G. B.
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 111, Issue 757