Abstract
In the framework of local volume-averaging for two-way coupling simulation of particle-laden flows, we have developed a force model that represents the particle reaction to the fluid. By considering the stress profile on the surface of finite size particles, our previous model was capable of representing the reaction force distribution even without fully resolving the boundary layer around the particle of comparable size to the grid spacing. In our method, the anisotropy of the surface stress is reasonably taken into account in the reaction force. To extend the applicability for particles smaller than the grid spacing, a smoothing method is developed in a consistent way with the discretization of the numerical simulation. The effectiveness of the smoothing method is demonstrated by applying to the following four fundamental fluid-particle interaction problems; a stationary particle in a uniform flow, a moving particle in a vortical flow, a rotating particle in fluid at rest and a particle suspension in a shear flow. The anisotropic contribution of the reaction force is reasonably represented by the smoothing method applied to the volume averaging technique particularly for rotating particle case. For the suspension in the shear flow, the non-physical effect of the grid size is suppressed by the smoothing method. Therefore, the developed reaction force model is effective for the case that the effects of shear and rotation are dominant.
Similar content being viewed by others
References
Balachandar, S., Eaton, J.K.: Turbulent dispersed multiphase flow. Ann. Rev. Fluid Mech. 42, 111–133 (2010)
Bergougnoux, L., Bouchet, G., Lopez, D., Guazzelli, E.: The motion of solid spherical particles falling in a cellular flow field at low stokes number. Phys. Fluids 26, 093302 (2014)
Eaton, J.K.: Two-way coupled turbulence simulations of gas-particle flows using point-particle tracking. Int. J. Multiph. Flow 35, 792–800 (2009)
Fröhlich, K., Schneiders, L., Meinke, M., Schröder, W.: Validation of Lagrangian two-way coupled point-particle models in Large–Eddy simulations. Flow Turbul. Combust. 101, 317–341 (2018)
Fukada, T., Takeuchi, S., Kajishima, T.: Interaction force and residual stress models for volume-averaged momentum equation for flow laden with particles of comparable diameter to computational grid width. Int. J. Multiph. Flow 85, 298–313 (2016)
Fukada, T., Fornari, W., Brandt, L., Takeuchi, S., Kajishima, T.: A numerical approach for particle-vortex interactions based on volume-averaged equations. Int. J. Multiph. Flow 104, 188–205 (2018)
Fukada, T., Takeuchi, S., Kajishima, T.: Estimation of fluid forces on a spherical particle for two-way coupling simulation based on the volume averaging. Int. J. Multiph. Flow 113, 165–178 (2019)
Good, G.H., Ireland, P.J., Bewley, G.P., Bodenschatz, E., Collins, L.R., Warhaft, Z.: Settling regimes of inertial particles in isotropic turbulence. J. Fluid Mech. 759, R3 (2014)
Happel, J., Brenner, H.: Low Reynolds Number Hydrodynamics: with Special Applications to Particulate Media. Springer, Berlin (1983)
Hwang, W., Eaton, J.K.: Homogeneous and isotropic turbulence modulation by small heavy (\(St\sim 50\)) particles. J. Fluid Mech. 564, 361–393 (2006)
Maxey, M.R.: The motion of small spherical particles in a cellular flow field. Phys. Fluids 30, 1915–1928 (1987)
Mehrabadi, M., Horwitz, J., Subramaniam, S., Mani, A.: A direct comparison of particle-resolved and point-particle methods in decaying turbulence. J. Fluid Mech. 850, 336–369 (2018)
Schiller, L., Naumann, A.: Über die grundlegende Berechnungen bei der Schwerkraft-aufbereitung. Verein Deutscher Ingenieure 77, 318–320 (1933)
Schneiders, L., Meinke, M., Schröder, W.: Direct particle-fluid simulation of Kolmogorov-length-scale size particles in decaying isotropic turbulence. J. Fluid Mech. 819, 188–227 (2017)
Acknowledgements
This work is supported by Grant-in-Aid for Young Scientists No. 18K13692 and Grant-in-Aid for Scientific Research (B) No. 17H03174 of the Japan Society for the Promotion of Science.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Rights and permissions
About this article
Cite this article
Fukada, T., Takeuchi, S. & Kajishima, T. Anisotropic Reaction Force Model in Two-way Coupling Simulation for a Smaller Particle Than Grid Spacing Based on Volume Averaging. Flow Turbulence Combust 105, 1017–1034 (2020). https://doi.org/10.1007/s10494-020-00142-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10494-020-00142-0