Construction of a three-dimensional model of the convection of aggregating particles




aggregation, spatial heterogeneity, OpenFOAM


This paper discusses technical aspects related to modeling aggregation processes in a heterogeneous medium with unsteady velocities. Smoluchowski operators are added to the model to account for aggregation. Spatial heterogeneity is modeled by advection and diffusion operators. The velocity field was obtained using OpenFOAM — the framework for modeling hydrodynamic systems.

Author Biography

Rishat R. Zagidullin


  1. V. A. Galkin, Smoluchowski Equation (Fizmatlit, Moscow, 2001) [in Russian].
  2. S. A. Matveev, D. A. Zheltkov, E. E. Tyrtyshnikov, and A. P. Smirnov, “Tensor Train Versus Monte Carlo for the Multicomponent Smoluchowski Coagulation Equation,” J. Comput. Phys. 316, 164-179 (2016).
    doi 10.1016/
  3. D. A. Stefonishin, S. A. Matveev, A. P. Smirnov, and E. E. Tyrtyshnikov, “Tensor Decompositions for Solving the Equations of Mathematical Models of Aggregation with Multiple Collisions of Particles,” Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie) 19 (4), 390-404 (2018).
    doi 10.26089/NumMet.v19r435
  4. S. A. Matveev, E. E. Tyrtyshnikov, A. P. Smirnov, and N. V. Brilliantov, “A Fast Numerical Method for Solving the Smoluchowski-Type Kinetic Equations of Aggregation and Fragmentation Processes,” Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie) 15 (1). 1-8 (2014).
  5. S. A. Matveev, “A Parallel Implementation of a Fast Method for Solving the Smoluchowski-Type Kinetic Equations of Aggregation and Fragmentation Processes,” Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie) 16 (3), 360-368 (2015).
    doi 10.26089/NumMet.v16r335
  6. R. R. Zagidullin, A. P. Smirnov, S. A. Matveev, and E. E. Tyrtyshnikov, “An Efficient Numerical Method for a Mathematical Model of a Transport of Coagulating Particles,” Vestn. Mosk. Univ., Ser. 15: Vychisl. Mat. Kibern. 41, No. 4, 28-34 (2017)[Moscow Univ. Comput. Math. Cybern. 41 (4), 179-186 (2017)].
    doi 10.3103/S0278641917040082
  7. A. E. Aloyan, V. O. Arutyunyan, A. A. Lushnikov, and V. A. Zagaynov, “Transport of Coagulating Aerosol in the Atmosphere,” J. Aerosol Sci. 28 (1), 67-85 (1997).
    doi 10.1016/S0021-8502(96)00043-2
  8. G. Falkovich, A. Fouxon, and M. G. Stepanov, “Acceleration of Rain Initiation by Cloud Turbulence,” Nature 419, 151-154 (2002).
    doi 10.1038/nature00983
  9. G. Falkovich, M. G. Stepanov, and M. Vucelja, “Rain Initiation Time in Turbulent Warm Clouds,” J. Appl. Meteorol. Climatol. 45 (4), 591-599 (2006).
    doi 10.1175/JAM2364.1
  10. V. J. Anderson and H. N. W. Lekkerkerker, “Insights into Phase Transition Kinetics from Colloid Science,” Nature 416, 811-815 (2002).
    doi 10.1038/416811a
  11. R. W. Samsel and A. S. Perelson, “Kinetics of Rouleau Formation. I. A Mass Action Approach with Geometric Features,” Biophys. J. 37 (2), 493-514 (1982).
    doi 10.1016/S0006-3495(82)84696-1
  12. M. Anand, K. B. Rajagopal, and K. R. Rajagopal, “A Model for the Formation and Lysis of Blood Clots,” Pathophysiol. Haemost. Thromb. 34 (2), 109-120 (2005).
    doi 10.1159/000089931
  13. C. Stein, T. Dannemann Purnat, N. Fietje, et al., WHO European Health Report 2018 (World Health Organization, Geneva, 2018), Vol. 164.
  14. J. E. Bennett, H. Tamura-Wicks, R. M. Parks, et al., “Particulate Matter Air Pollution and National and County Life Expectancy Loss in the USA: A Spatiotemporal Analysis,” PLoS Med. 16 (7), Article Number e1002856 (2019).
    doi 10.1371/journal.pmed.1002856
  15. K. Semeniuk and A. Dastoor, “Current State of Atmospheric Aerosol Thermodynamics and Mass Transfer Modeling: A Review,” Atmosphere 11 (2), Article Number 156 (2020).
    doi 10.3390/atmos11020156
  16. J. Garratt, The Atmospheric Boundary Layer (Cambridge Univ. Press, Cambridge, 1992).
  17. Y. Han, M. Stoellinger, and J. Naughton, “Large Eddy Simulation for Atmospheric Boundary Layer Flow over Flat and Complex Terrains,” J. Phys.: Conf. Ser. 753 (3), Article Id. 032044 (2016).
    doi 10.1088/1742-6596/753/3/032044
  18. R. Stoll, J. A. Gibbs, S. T. Salesky, et al., “Large-Eddy Simulation of the Atmospheric Boundary Layer,” Boundary-Layer Meteorol. 177 (2-3), 541-581 (2020).
    doi 10.1007/s10546-020-00556-3
  19. M. Diebold, C. Higgins, J. Fang, et al., “Flow over Hills: A Large-Eddy Simulation of the Bolund Case,” Boundary-Layer Meteorol. 148 (1), 177-194 (2013).
    doi 10.1007/s10546-013-9807-0
  20. R. Zagidullin, A. P. Smirnov, S. Matveev, et al., “Aggregation in Non-Uniform Systems with Advection and Localized Source,” J. Phys. A: Math. Theor. 55 (26), Article Id. 265001 (2022).
    doi 10.1088/1751-8121/ac711a
  21. L. Gallen, A. Felden, E. Riber, and B. Cuenot, “Lagrangian Tracking of Soot Particles in LES of Gas Turbines,” Proc. Combust. Inst. 37 (4), 5429-5436 (2019).
    doi 10.1016/j.proci.2018.06.013
  22. OpenFOAM. . Cited November 3, 2023.
  23. S. A. Matveev, A. P. Smirnov, and E. E. Tyrtyshnikov, “A Fast Numerical Method for the Cauchy Problem for the Smoluchowski Equation,” J. Comput. Phys. 282, 23-32 (2015).
    doi 10.1016/
  24. The Computational Geometry Algorithms Library (CGAL). . Cited November 3, 2023.
  25. R. Zagidullin, A. Smirnov, S. Matveev, and E. Tyrtyshnikov, “Supercomputer Modelling of Spatially-Heterogeneous Coagulation Using MPI and CUDA,” in Communications in Computer and Information Science (Springer, Cham, 2019), Vol. 1129, 403-414.
    doi 10.1007/978-3-030-36592-9_33
  26. R. R. Zagidullin, “Solving the Transport-Coagulation Problem in a Two-Dimensional Spatial Region,” Prikl. Mat. Inform. No. 62, 27-33 (2019) [Comput. Math. Model. 31 (1), 19-24 (2020)].
    doi 10.1007/s10598-020-09473-z
  27. M. S. Darwish and F. Moukalled, “TVD Schemes for Unstructured Grids,” Int. J. Heat Mass Transf. 46 (4), 599-611 (2003).
    doi 10.1016/S0017-9310(02)00330-7
  28. F. Denner and B. G. M. van Wachem, “TVD Differencing on Three-Dimensional Unstructured Meshes with Monotonicity-Preserving Correction of Mesh Skewness,” J. Comput. Phys. 298, 466-479 (2015).
    doi 10.1016/
  29. C. Le Touze, A. Murrone, and H. Guillard, “Multislope MUSCL Method for General Unstructured Meshes,” J. Comput. Phys. 284, 389-418 (2015).
    doi 10.1016/



How to Cite

Загидуллин Р.Р. Construction of a Three-Dimensional Model of the Convection of Aggregating Particles // Numerical Methods and Programming (Vychislitel’nye Metody i Programmirovanie). 2023. 24. 430-439. doi 10.26089/NumMet.v24r429



Methods and algorithms of computational mathematics and their applications