Skip to main content

Numerical study of general shape particles in a concentric annular duct having inner obstacle


We have examined the behavior of solid particles in an annulus. A circular, square and elliptic shaped particle is analysed separately and in pairs. Behaviour of two circular particles moving inside two concentric moving cylinders having an internal obstacle is analysed. The interaction of particle with the fluid and circular obstacle is carried out inside a fixed circular mesh using an Eulerian approach. The coupled fluid and particles system is handled using fictitious boundary method. The hydrodynamic forces acting on the fictitious boundaries (particles) are calculated using an explicit volume integral approach. A collision model proposed by Glowinski et al. is used to prevent particle-wall, particle-particle and particle-obstacle overlapping and collision. The particulate flow is computed using multigrid finite element solver FEATFLOW.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11


  1. 1.

    Abbasi WS, Islam SU, Rahman H, Manzoor R (2018) Numerical investigation of fluid-solid interaction for flow around three square cylinders. AIP Adv 8(2):025221

    Article  Google Scholar 

  2. 2.

    Bayraktar E, Mierka O, Platte F, Kuzmin D, Turek S (2011) Numerical aspects and implementation of population balance equations coupled with turbulent fluid dynamics. Comput Chem Eng 35(11):2204–2217

    Article  Google Scholar 

  3. 3.

    Crowley TJ, Meadows Edward S, Evangelos K, Doyle III, Francis J (2000) Control of particle size distribution described by a population balance model of semibatch emulsion polymerization. J Process Control 10(5):419–432

    Article  Google Scholar 

  4. 4.

    Dyko Mark P, Kambiz V, Kader Mojtabi A (1999) A numerical and experimental investigation of stability of natural convective flows within a horizontal annulus. J Fluid Mech 381:27–61

    Article  Google Scholar 

  5. 5.

    Glowinski R, Pan TW, Hesla TI, Joseph DD (1999) A distributed lagrange multiplier/fictitious domain method for particulate flows. Int J Multiph Flow 25(5):755–794

    MathSciNet  Article  Google Scholar 

  6. 6.

    Grigull U, Hauf (1966) Natural convection in horizontal cylindrical annuli. In: The 3rd international heat transfer conference

  7. 7.

    Hackbusch W (2003) Multi-grid methods and applications. Springer series in computational mathematics. SpringerSpringer, Berlin

    MATH  Google Scholar 

  8. 8.

    Hirt CW, Amsden AA, Cook JL (1974) An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J Comput Phys 14(3):227–253

    Article  Google Scholar 

  9. 9.

    Hu HH, Joseph DD, Crochet MJ (1992) Direct simulation of fluid particle motions. Theoret Comput Fluid Dyn 3:285–306

    Article  Google Scholar 

  10. 10.

    Hu H, Patankar NA, Zhu MY (2001) Direct numerical simulations of fluid-solid systems using the arbitrary Lagrangian–Eulerian technique. J Comput Phys 169(2):427–462

    MathSciNet  Article  Google Scholar 

  11. 11.

    Hu Y, Li D, Shu S, Niu X (2015) Study of multiple steady solutions for the 2d natural convection in a concentric horizontal annulus with a constant heat flux wall using immersed boundary-lattice Boltzmann method. Int J Heat Mass Transf 81:591–601

    Article  Google Scholar 

  12. 12.

    Jabeen S, Usman K, Walayat K (2019) Numerical investigations for a chain of particles settling in a channel. Comput Part Mech 7:1–13

    Google Scholar 

  13. 13.

    John V (2002) Higher order finite element methods and multigrid solvers in a benchmark problem for the 3d Navier–Stokes equations. Int J Numeri Methods Fluids 40:775–798

    MathSciNet  Article  Google Scholar 

  14. 14.

    Kim SS, Karrila SJ (1991) Microhydrodynamics: principles and selected applications, 2nd edn. Butterworth-Heinemann, Boston

    Google Scholar 

  15. 15.

    Kuehn TH, Goldstein RJ (1976) An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders. J Fluid Mech 74(4):695–719

    Article  Google Scholar 

  16. 16.

    Lopez JM, Marques F, Mercader I, Batiste O (2007) Onset of convection in a moderate aspect-ratio rotating cylinder: Eckhaus–Benjamin–Feir instability. J Fluid Mech 590:187–208

    Article  Google Scholar 

  17. 17.

    Maury B (1996) Characteristics ale method for the unsteady 3d Navier–Stokes equations with a free surface. Int J Comput Fluid Dyn 6:175–188

    Article  Google Scholar 

  18. 18.

    Maury B (1997) A many-body lubrication model. J Comput Appl Math 325(9):1053–1058

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Mishra BK, Rajamani K (1992) The discrete element method for the simulation of ball mills. Appl Math Model 16(11):598–604

    Article  Google Scholar 

  20. 20.

    Munjiza A, Owen DRJ, Bicanic N (1995) A combined finite-discrete element method in transient dynamics of fracturing solids. Eng Comput 12:145–174

    Article  Google Scholar 

  21. 21.

    Nguyen T, Vasseur P, Robillard L (1982) Natural convection between horizontal concentric cylinders with density inversion of water for low Rayleigh numbers. Int J Heat Mass Transf 25(10):1559–1568

    Article  Google Scholar 

  22. 22.

    Osher S, Fedkiw RP (2001) Level set methods: an overview and some recent results. J Comput Phys 169(2):463–502

    MathSciNet  Article  Google Scholar 

  23. 23.

    Patankar NA, Singh P, Joseph DD, Glowinski R, Pan TW (2000) A new formulation of the distributed Lagrange multiplier/fictitious domain method for particulate flows. Int J Multiph Flow 26:1509–1524

    Article  Google Scholar 

  24. 24.

    Pillapakkam SB, Singh P (2001) A level-set method for computing solutions to viscoelastic two-phase flow. J Comput Phys 174(2):552–578

    Article  Google Scholar 

  25. 25.

    Potapov Alexander V, Hunt Melany L, Campbell Charles S (2001) Liquid-solid flows using smoothed particle hydrodynamics and the discrete element method. Powder Technol 116(2–3):204–213

    Article  Google Scholar 

  26. 26.

    Powe RE, Carley CT, Bishop EH (1969) Free convective flow patterns in cylindrical annuli. J Heat Transf 91(3):310–314

    Article  Google Scholar 

  27. 27.

    Sarrate J, Huerta A, Donea J (2001) Arbitrary Lagrangian–Eulerian formulation for fluid-rigid body interaction. Comput Methods Appl Mech Eng 190(24):3171–3188

    Article  Google Scholar 

  28. 28.

    Sarthou A, Vincent S, Caltagirone JP, Angot P (2008) Eulerian–Lagrangian grid coupling and penalty methods for the simulation of multiphase flows interacting with complex objects. Int J Numer Methods Fluids 56(8):1093–1099

    MathSciNet  Article  Google Scholar 

  29. 29.

    Singh P, Hesla TI, Joseph DD (2003) Distributed Lagrange multiplier method for particulate flows with collisions. Int J Multiph Flow 29(3):495–509

    Article  Google Scholar 

  30. 30.

    Sommerfeld M (2001) Validation of a stochastic Lagrangian modelling approach for inter-particle collisions in homogeneous isotropic turbulence. Int J Multiph Flow 27(10):1829–1858

    Article  Google Scholar 

  31. 31.

    Stefan T (1996) A comparative study of time-stepping techniques for the incompressible Navier–Stokes equations: from fully implicit non-linear schemes to semi-implicit projection methods. Int J Numer Methods Fluids 22(10):987–1011

    MathSciNet  Article  Google Scholar 

  32. 32.

    Stefan T (1997) On discrete projection methods for the incompressible Navier–Stokes equations: an algorithmical approach. Comput Methods Appl Mech Eng 143(3):271–288

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Turek S, Wan D, Rivkind L S (2003) The fictitious boundary method for the implicit treatment of Dirichlet boundary conditions with applications to incompressible flow simulations. In: Challenges in scientific computing—CISC 2002 (Eberhard Baensch, ed.), Lecture notes in computational science and engineering, vol 35. Springer, Berlin, pp 37–68 (English)

  34. 34.

    Usman K, Walayat K, Wang Z, Liu M (2017) A multigrid finite element fictitious boundary method for fluid-solid two-phase flows. In: The 8th international conference on computational mehtods (ICCM2017) (Advances in Computational Engineering Science, ed.), ScienTech Publisher

  35. 35.

    Usman K (2013) Numerical analysis of collision models in 2d particulate flow. Technische Universität Dortmund, Fakultät für Mathematik, Ph.D. Thesis

  36. 36.

    Usman K, Ali J, Mahmood R, Bilal DS, Jabeen S, Asmat J (2020) Study of a falling rigid particle passing around obstacles in a fluid channel. Int J Comput Fluid Dyn

  37. 37.

    Walayat K, Wang Z, Usman K, Liu M (2018) An efficient multi-grid finite element fictitious boundary method for particulate flows with thermal convection. Int J Heat Mass Transf 126:452–465

    Article  Google Scholar 

  38. 38.

    Walayat K, Zhang Z, Usman K, Chang J, Liu M (2019) Fully resolved simulations of thermal convective suspensions of elliptic particles using a multigrid fictitious boundary method. Int J Heat Mass Transf 139:802–821

    Article  Google Scholar 

  39. 39.

    Wan D, Turek S (2007) Fictitious boundary and moving mesh methods for the numerical simulation of rigid particulate flows. J Comput Phys 222(1):28–56

    MathSciNet  Article  Google Scholar 

  40. 40.

    Wan D, Turek S (2006) Direct numerical simulation of particulate flow via multigrid fem techniques and the fictitious boundary method. Int J Numer Methods Fluids 51(5):531–566

  41. 41.

    Wan D, Turek S (2007) An efficient multigrid-fem method for the simulation of solid-liquid two phase flows. J Comput Appl Math 203(2):561–580

    MathSciNet  Article  Google Scholar 

  42. 42.

    Wan D, Turek S, Rivkind LS (2004) An efficient multigrid fem solution technique for incompressible flow with moving rigid bodies. In: Miloslav F, Vit D, Petr K, Karel N (eds) Numerical mathematics and advanced applications. Springer, Berlin, pp 844–853 English

    Chapter  Google Scholar 

  43. 43.

    Wei Y, Wang Z, Qian Y, Guo W (2018) Study on bifurcation and dual solutions in natural convection in a horizontal annulus with rotating inner cylinder using thermal immersed boundary-lattice boltzmann method. Entropy (Basel, Switzerland) 20(10):733

    Article  Google Scholar 

  44. 44.

    Wendt John F (ed) (2009) Computational fluid dynamics. Springer, Berlin

    MATH  Google Scholar 

  45. 45.

    Yoo J (1999) Prandtl number effect on bifurcation and dual solutions in natural convection in a horizontal annulus. Int J Heat Mass Transf 42(17):3279–3290

    Article  Google Scholar 

  46. 46.

    Yoo J-S (1998) Natural convection in a narrow horizontal cylindrical annulus: \(pr \le 0.3\). Int J Heat Mass Transf 41(20):3055–3073

  47. 47.

    Zhang N, Zheng ZC (2007) A collision model for a large number of particles with significantly different sizes. J Phys D Appl Phys 40:2603–2616

    Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to K. Usman.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Jabeen, S., Usman, K. & Shahid, M. Numerical study of general shape particles in a concentric annular duct having inner obstacle. Comp. Part. Mech. (2021).

Download citation


  • Particulate flow
  • Direct numerical simulation
  • Fictitious boundary method
  • Finite element method
  • Multigrid
  • Sedimentation