A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements

Zwick, David; Balachandar, S.

doi:10.1177/1094342019867756

Title: A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements

Abstract

Multiphase flow can be difficult to simulate with high accuracy due to the wide range of scales associated with various multiphase phenomena. These scales may range from the size of individual particles to the entire domain of interest. Traditionally, large scale systems can only be simulated using averaging approaches that filter out the locations of individual particles. In this work, the Euler–Lagrange method is used to simulate large-scale dense particle systems in which each individual particle is tracked. In order to accomplish this, the highly scalable spectral element code nek5000 has been extended to handle the multiple levels of multiphase coupling in these systems. These levels include what has been called one-, two-, and four-way coupling. Here, each level has been separated to detail the computational impact of each stage. A binned ghost particle algorithm has also been developed to efficiently handle the challenges of two- and four-way coupling in a parallel processing context. The algorithms and their implementations are then shown to scale to 65,536 Message Passing Interface (MPI) ranks in both the strong and weak limits. After this, validation is performed through simulation of a small-scale fluidized bed. Lastly, a large-scale fluidized bed is simulated with 65,536 MPImore »« less

Authors:

;

^[1]

Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, USA

Publication Date:: Mon Aug 12 00:00:00 EDT 2019

Sponsoring Org.:: USDOE

OSTI Identifier:: 1557001

Resource Type:: Published Article

Journal Name:: International Journal of High Performance Computing Applications

Additional Journal Information:: Journal Name: International Journal of High Performance Computing Applications Journal Volume: 34 Journal Issue: 3; Journal ID: ISSN 1094-3420

Publisher:: SAGE Publications

Country of Publication:: United States

Language:: English

Citation Formats


                    Zwick, David, and Balachandar, S. A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements.  United States: N. p., 2019. 
Web.  doi:10.1177/1094342019867756.

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                    Zwick, David, & Balachandar, S. A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements.  United States.  https://doi.org/10.1177/1094342019867756

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                    Zwick, David, and Balachandar, S. Mon .  
"A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements".  United States.  https://doi.org/10.1177/1094342019867756.

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@article{osti_1557001,

  title        = {A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements},

  author       = {Zwick, David and Balachandar, S.},

  abstractNote = {Multiphase flow can be difficult to simulate with high accuracy due to the wide range of scales associated with various multiphase phenomena. These scales may range from the size of individual particles to the entire domain of interest. Traditionally, large scale systems can only be simulated using averaging approaches that filter out the locations of individual particles. In this work, the Euler–Lagrange method is used to simulate large-scale dense particle systems in which each individual particle is tracked. In order to accomplish this, the highly scalable spectral element code nek5000 has been extended to handle the multiple levels of multiphase coupling in these systems. These levels include what has been called one-, two-, and four-way coupling. Here, each level has been separated to detail the computational impact of each stage. A binned ghost particle algorithm has also been developed to efficiently handle the challenges of two- and four-way coupling in a parallel processing context. The algorithms and their implementations are then shown to scale to 65,536 Message Passing Interface (MPI) ranks in both the strong and weak limits. After this, validation is performed through simulation of a small-scale fluidized bed. Lastly, a large-scale fluidized bed is simulated with 65,536 MPI ranks and is able to capture the unique physics of the onset of fluidization.},

  doi          = {10.1177/1094342019867756},

  journal      = {International Journal of High Performance Computing Applications},

  number       = 3,

  volume       = 34,

  place        = {United States},

  year         = {Mon Aug 12 00:00:00 EDT 2019},

  month        = {Mon Aug 12 00:00:00 EDT 2019}

}

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Works referenced in this record:

A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
journal, January 1998

Karypis, George; Kumar, Vipin
SIAM Journal on Scientific Computing, Vol. 20, Issue 1
DOI: 10.1137/S1064827595287997

The numerical stability of barycentric Lagrange interpolation
journal, October 2004

Higham, N. J.
IMA Journal of Numerical Analysis, Vol. 24, Issue 4
DOI: 10.1093/imanum/24.4.547

Scaling Limits for PDE-Based Simulation (Invited)
conference, June 2015

Fischer, Paul F.
22nd AIAA Computational Fluid Dynamics Conference
DOI: 10.2514/6.2015-3049

Methods for evaluating fluid velocities in spectral simulations of turbulence
journal, July 1989

Balachandar, S.; Maxey, M. R.
Journal of Computational Physics, Vol. 83, Issue 1
DOI: 10.1016/0021-9991(89)90224-6

ppiclF: A Parallel Particle-In-Cell Library in Fortran
journal, May 2019

Zwick, David
Journal of Open Source Software, Vol. 4, Issue 37
DOI: 10.21105/joss.01400

Modeling and numerical simulation of particulate flows by the Eulerian–Lagrangian approach
journal, October 2001

Patankar, N. A.; Joseph, D. D.
International Journal of Multiphase Flow, Vol. 27, Issue 10
DOI: 10.1016/S0301-9322(01)00021-0

Dynamic alignment and distribution of irregularly coupled data arrays for scalable parallelization of particle-in-cell problems
conference, January 1996

Wei-Keng Liao, ; Ranka, S.
Proceedings of International Conference on Parallel Processing
DOI: 10.1109/IPPS.1996.508039

Partitioning of unstructured problems for parallel processing
journal, January 1991

Simon, H. D.
Computing Systems in Engineering, Vol. 2, Issue 2-3
DOI: 10.1016/0956-0521(91)90014-V

Visualization of the Bulle-Effect at River Bifurcations
conference, January 2018

Dutta, Som; Moer, Mark W. Van; Fischer, Paul
Proceedings of the Practice and Experience on Advanced Research Computing - PEARC '18
DOI: 10.1145/3219104.3229433

Discrete particle simulation of two-dimensional fluidized bed
journal, October 1993

Tsuji, Y.; Kawaguchi, T.; Tanaka, T.
Powder Technology, Vol. 77, Issue 1
DOI: 10.1016/0032-5910(93)85010-7

On predicting particle-laden turbulent flows
journal, June 1994

Elghobashi, S.
Applied Scientific Research, Vol. 52, Issue 4
DOI: 10.1007/BF00936835

Determination of the normal spring stiffness coefficient in the linear spring–dashpot contact model of discrete element method
journal, September 2013

Navarro, Helio A.; de Souza Braun, Meire P.
Powder Technology, Vol. 246
DOI: 10.1016/j.powtec.2013.05.049

Particle transport in turbulent curved pipe flow
journal, March 2016

Noorani, Azad; Sardina, Gaetano; Brandt, Luca
Journal of Fluid Mechanics, Vol. 793
DOI: 10.1017/jfm.2016.136

A proof of the Kepler conjecture
journal, January 2005

Hales, Thomas
Annals of Mathematics, Vol. 162, Issue 3
DOI: 10.4007/annals.2005.162.1065

Accurate signal reconstruction for higher order Lagrangian–Eulerian back-coupling in multiphase turbulence
journal, September 2017

Zwick, D.; Sakhaee, E.; Balachandar, S.
Fluid Dynamics Research, Vol. 49, Issue 5
DOI: 10.1088/1873-7005/aa871f

An Euler–Lagrange strategy for simulating particle-laden flows
journal, April 2013

Capecelatro, Jesse; Desjardins, Olivier
Journal of Computational Physics, Vol. 238
DOI: 10.1016/j.jcp.2012.12.015

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
DOI: 10.1016/j.jcp.2016.04.034

Dynamic Load Balancing for Compressible Multiphase Turbulence
conference, January 2018

Zhai, Keke; Banerjee, Tania; Zwick, David
Proceedings of the 2018 International Conference on Supercomputing - ICS '18
DOI: 10.1145/3205289.3205304

OpenACC acceleration of the Nek5000 spectral element code
journal, March 2015

Markidis, Stefano; Gong, Jing; Schliephake, Michael
The International Journal of High Performance Computing Applications, Vol. 29, Issue 3
DOI: 10.1177/1094342015576846

Turbulent Dispersed Multiphase Flow
journal, January 2010

Balachandar, S.; Eaton, John K.
Annual Review of Fluid Mechanics, Vol. 42, Issue 1
DOI: 10.1146/annurev.fluid.010908.165243

Fluid Mechanical Description of Fluidized Beds. Equations of Motion
journal, November 1967

Anderson, T. B.; Jackson, Roy
Industrial & Engineering Chemistry Fundamentals, Vol. 6, Issue 4
DOI: 10.1021/i160024a007

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
DOI: 10.1017/jfm.2015.258

Inter-phase heat transfer and energy coupling in turbulent dispersed multiphase flows
journal, March 2016

Ling, Y.; Balachandar, S.; Parmar, M.
Physics of Fluids, Vol. 28, Issue 3
DOI: 10.1063/1.4942184

Barycentric Lagrange Interpolation
journal, January 2004

Berrut, Jean-Paul; Trefethen, Lloyd N.
SIAM Review, Vol. 46, Issue 3
DOI: 10.1137/S0036144502417715

Numerical Simulation of Dense Gas-Solid Fluidized Beds: A Multiscale Modeling Strategy
journal, January 2008

van der Hoef, M. A.; van Sint Annaland, M.; Deen, N. G.
Annual Review of Fluid Mechanics, Vol. 40, Issue 1
DOI: 10.1146/annurev.fluid.40.111406.102130

Total variation diminishing Runge-Kutta schemes
journal, January 1998

Gottlieb, Sigal; Shu, Chi-Wang
Mathematics of Computation of the American Mathematical Society, Vol. 67, Issue 221
DOI: 10.1090/S0025-5718-98-00913-2

Multiphase flow above explosion sites in debris-filled volcanic vents: Insights from analogue experiments
journal, November 2008

Ross, Pierre-Simon; White, James D. L.; Zimanowski, Bernd
Journal of Volcanology and Geothermal Research, Vol. 178, Issue 1
DOI: 10.1016/j.jvolgeores.2008.01.013

Fast Parallel Direct Solvers for Coarse Grid Problems
journal, February 2001

Tufo, H. M.; Fischer, P. F.
Journal of Parallel and Distributed Computing, Vol. 61, Issue 2
DOI: 10.1006/jpdc.2000.1676

Particle-laden turbulent flows: direct simulation and closure models
journal, October 1991

Elghobashi, S.
Applied Scientific Research, Vol. 48, Issue 3-4
DOI: 10.1007/BF02008202

A spectral element method for fluid dynamics: Laminar flow in a channel expansion
journal, June 1984

Patera, Anthony T.
Journal of Computational Physics, Vol. 54, Issue 3
DOI: 10.1016/0021-9991(84)90128-1

Turbulence modulation and drag reduction by spherical particles
journal, August 2010

Zhao, L. H.; Andersson, H. I.; Gillissen, J. J. J.
Physics of Fluids, Vol. 22, Issue 8
DOI: 10.1063/1.3478308

A scaling analysis of added-mass and history forces and their coupling in dispersed multiphase flows
journal, December 2013

Ling, Y.; Parmar, M.; Balachandar, S.
International Journal of Multiphase Flow, Vol. 57
DOI: 10.1016/j.ijmultiphaseflow.2013.07.005

Revisiting the Wen and Yu Equations for Minimum Fluidization Velocity Prediction
journal, May 2004

Delebarre, A.
Chemical Engineering Research and Design, Vol. 82, Issue 5
DOI: 10.1205/026387604323142621

Validation of a discrete element model using magnetic resonance measurements
journal, August 2009

Müller, Christoph R.; Scott, Stuart A.; Holland, Daniel J.
Particuology, Vol. 7, Issue 4
DOI: 10.1016/j.partic.2009.04.002

A generalized method for predicting the minimum fluidization velocity
journal, May 1966

Wen, C. Y.; Yu, Y. H.
AIChE Journal, Vol. 12, Issue 3
DOI: 10.1002/aic.690120343

Improving particle drag predictions in Euler–Lagrange simulations with two-way coupling
journal, June 2017

Ireland, Peter J.; Desjardins, Olivier
Journal of Computational Physics, Vol. 338
DOI: 10.1016/j.jcp.2017.02.070

Interaction of a planar shock wave with a dense particle curtain: Modeling and experiments
journal, November 2012

Ling, Y.; Wagner, J. L.; Beresh, S. J.
Physics of Fluids, Vol. 24, Issue 11
DOI: 10.1063/1.4768815

A scaling analysis for point–particle approaches to turbulent multiphase flows
journal, September 2009

Balachandar, S.
International Journal of Multiphase Flow, Vol. 35, Issue 9
DOI: 10.1016/j.ijmultiphaseflow.2009.02.013

A discrete numerical model for granular assemblies
journal, March 1979

Cundall, P. A.; Strack, O. D. L.
Géotechnique, Vol. 29, Issue 1
DOI: 10.1680/geot.1979.29.1.47

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Title: A scalable Euler–Lagrange approach for multiphase flow simulation on spectral elements

Abstract

Citation Formats

A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs journal, January 1998

The numerical stability of barycentric Lagrange interpolation journal, October 2004

Scaling Limits for PDE-Based Simulation (Invited) conference, June 2015

Methods for evaluating fluid velocities in spectral simulations of turbulence journal, July 1989

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Visualization of the Bulle-Effect at River Bifurcations conference, January 2018

Discrete particle simulation of two-dimensional fluidized bed journal, October 1993

On predicting particle-laden turbulent flows journal, June 1994

Determination of the normal spring stiffness coefficient in the linear spring–dashpot contact model of discrete element method journal, September 2013

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A proof of the Kepler conjecture journal, January 2005

Accurate signal reconstruction for higher order Lagrangian–Eulerian back-coupling in multiphase turbulence journal, September 2017

An Euler–Lagrange strategy for simulating particle-laden flows journal, April 2013

Accurate calculation of Stokes drag for point–particle tracking in two-way coupled flows journal, August 2016

Dynamic Load Balancing for Compressible Multiphase Turbulence conference, January 2018

OpenACC acceleration of the Nek5000 spectral element code journal, March 2015

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Fluid Mechanical Description of Fluidized Beds. Equations of Motion journal, November 1967

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Inter-phase heat transfer and energy coupling in turbulent dispersed multiphase flows journal, March 2016

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Numerical Simulation of Dense Gas-Solid Fluidized Beds: A Multiscale Modeling Strategy journal, January 2008

Total variation diminishing Runge-Kutta schemes journal, January 1998

Multiphase flow above explosion sites in debris-filled volcanic vents: Insights from analogue experiments journal, November 2008

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Particle-laden turbulent flows: direct simulation and closure models journal, October 1991

A spectral element method for fluid dynamics: Laminar flow in a channel expansion journal, June 1984

Turbulence modulation and drag reduction by spherical particles journal, August 2010

A scaling analysis of added-mass and history forces and their coupling in dispersed multiphase flows journal, December 2013

Revisiting the Wen and Yu Equations for Minimum Fluidization Velocity Prediction journal, May 2004

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A generalized method for predicting the minimum fluidization velocity journal, May 1966

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Interaction of a planar shock wave with a dense particle curtain: Modeling and experiments journal, November 2012

A scaling analysis for point–particle approaches to turbulent multiphase flows journal, September 2009

A discrete numerical model for granular assemblies journal, March 1979

A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
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The numerical stability of barycentric Lagrange interpolation
journal, October 2004

Scaling Limits for PDE-Based Simulation (Invited)
conference, June 2015

Methods for evaluating fluid velocities in spectral simulations of turbulence
journal, July 1989

ppiclF: A Parallel Particle-In-Cell Library in Fortran
journal, May 2019

Modeling and numerical simulation of particulate flows by the Eulerian–Lagrangian approach
journal, October 2001

Dynamic alignment and distribution of irregularly coupled data arrays for scalable parallelization of particle-in-cell problems
conference, January 1996

Partitioning of unstructured problems for parallel processing
journal, January 1991

Visualization of the Bulle-Effect at River Bifurcations
conference, January 2018

Discrete particle simulation of two-dimensional fluidized bed
journal, October 1993

On predicting particle-laden turbulent flows
journal, June 1994

Determination of the normal spring stiffness coefficient in the linear spring–dashpot contact model of discrete element method
journal, September 2013

Particle transport in turbulent curved pipe flow
journal, March 2016

A proof of the Kepler conjecture
journal, January 2005

Accurate signal reconstruction for higher order Lagrangian–Eulerian back-coupling in multiphase turbulence
journal, September 2017

An Euler–Lagrange strategy for simulating particle-laden flows
journal, April 2013

Accurate calculation of Stokes drag for point–particle tracking in two-way coupled flows
journal, August 2016

Dynamic Load Balancing for Compressible Multiphase Turbulence
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OpenACC acceleration of the Nek5000 spectral element code
journal, March 2015

Turbulent Dispersed Multiphase Flow
journal, January 2010

Fluid Mechanical Description of Fluidized Beds. Equations of Motion
journal, November 1967

Exact regularized point particle method for multiphase flows in the two-way coupling regime
journal, May 2015

Inter-phase heat transfer and energy coupling in turbulent dispersed multiphase flows
journal, March 2016

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journal, January 2004

Numerical Simulation of Dense Gas-Solid Fluidized Beds: A Multiscale Modeling Strategy
journal, January 2008

Total variation diminishing Runge-Kutta schemes
journal, January 1998

Multiphase flow above explosion sites in debris-filled volcanic vents: Insights from analogue experiments
journal, November 2008

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journal, February 2001

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journal, October 1991

A spectral element method for fluid dynamics: Laminar flow in a channel expansion
journal, June 1984

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journal, August 2010

A scaling analysis of added-mass and history forces and their coupling in dispersed multiphase flows
journal, December 2013

Revisiting the Wen and Yu Equations for Minimum Fluidization Velocity Prediction
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Validation of a discrete element model using magnetic resonance measurements
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