Abstract
This paper provides a preliminary study of two different methods to handle the limitations of the Lagrangian point-force approach in the context of unstructured LES solvers. Large deviations in mass, momentum, and energy exchanges between the gas and liquid phases may occur if the assumptions of the point-force approach are not verified. In particular, the point-force approach considers the particles to be subgrid-scale phenomena but the use of more and more refined grids for the carrier flow allowed by today’s computer power leads to cell sizes of the order or smaller than the particle diameters. Several methods are found in the literature to tackle this problem. However, they are usually suited for structured solvers. In the case of unstructured solvers handling several hundred thousand particles in unsteady flows, such methods are far too expensive. In this work, two original methods adapted for spray calculation in unstructured solvers are implemented and compared: the particle-bursting method, and the multigrid method. In this preliminary study, only the correction on the evaporation model is studied, with drag being neglected. Both methods greatly improve the accuracy of the evaporation model but only the multigrid method is independent of the Eulerian mesh refinement. The results presented show that the two methods are relevant to correct the evaporation source terms when the limits of the point-force model are reached. However, the study should be extended to consider the impact of the methods on the drag model calculation and the effects on the interaction with a reactive gaseous phase.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramzon, B., Sirignano, W.: Droplet vaporization model for spray combustion calculations. Int. J. Heat Mass Transf. 32(9), 1605–1618 (1989). https://doi.org/10.2514/6.1988-636
Agarwal, A.K., Park, S., Dhar, A., et al.: Review of experimental and computational studies on spray, combustion, performance, and emission characteristics of biodiesel fueled engines. J. Energy Resour. Technol., Trans. of the ASME 140(12), 120801 (2018). https://doi.org/10.1115/1.4040584
Balachandar, S., Liu, K.: A correction procedure for self-induced velocity of a finite-sized particle in two-way coupled Euler-Lagrange simulations. Int. J. Multiphase Flow 159, 104316 (2023). https://doi.org/10.1016/j.ijmultiphaseflow.2022.104316
Capecelatro, J., Desjardins, O.: An Euler - Lagrange strategy for simulating particle-laden flows. J. Comput. Phys. 238, 1–31 (2013). https://doi.org/10.1016/j.jcp.2012.12.015
Dukowicz, J.K.: A particle-fluid numerical model for liquid sprays. J. Comput. Phys. 35(2), 229–253 (1980). https://doi.org/10.1016/0021-9991(80)90087-X
Evrard, F., Denner, F., van Wachem, B.: A multi-scale approach to simulate atomisation processes. Int. J. Multiph. Flow 119, 194–216 (2019). https://doi.org/10.1016/j.ijmultiphaseflow.2019.07.005
Felden, A., Esclapez, L., Riber, E., et al.: Including real fuel chemistry in LES of turbulent spray combustion. Combust. Flame 193, 397–416 (2018). https://doi.org/10.1016/j.combustflame.2018.03.027
Hirschfelder, J., Bird, RB., Curtiss, CF.: Molecular theory of gases and liquids; rev. version. Structure of matter series, Wiley, New York, NY, (1964) https://cds.cern.ch/record/108119
Horwitz, J.A., Iaccarino, G., Eaton, J.K., et al.: The discrete green’s function paradigm for two-way coupled euler-lagrange simulation. J. Fluid Mech. 931, 882–928 (2022)
Hubbard, G., Denny, G., Mills, A.: Droplet evaporation: Effects of transients and variable properties. Int. J. Heat Mass Transf. 18(9), 1003–1008 (1975). https://doi.org/10.1016/0017-9310(75)90217-3
Ireland, P.J., Desjardins, O.: Improving particle drag predictions in Euler-Lagrange simulations with two-way coupling. J. Comput. Phys. (2017). https://doi.org/10.1016/j.jcp.2017.02.070
Khare, P., Wang, S., Yang, V.: Modeling of finite-size droplets and particles in multiphase flows. Chin. J. Aeronautics (2015). https://doi.org/10.1016/j.cja.2015.05.004
Lax, P., Wendroff, B.: Systems of conservation laws. Commun. Pure and Appl. Math. 13(2), 217–237 (1960). https://doi.org/10.1002/cpa.3160130205
Ling, Y., Zaleski, S., Scardovelli, R., et al.: Multiscale simulation of atomization with small droplets represented by a Lagrangian point-particle model. Int. J. Multiph. Flow 76, 122–143 (2015). https://doi.org/10.1016/j.ijmultiphaseflow.2015.07.002
Maxey, M.R., Patel, B.K.: Localized force representations for particles sedimenting in Stokes Flow. Int. J. Multiph. Flow 27, 1603–1626 (2001). https://doi.org/10.1016/S0301-9322(01)00014-3
Maxey, M.R., Riley, J.J.: Equation of motion for a small rigid sphere in a nonuniform flow. The Physics of Fluids 26(4), 883–889 (1983). https://doi.org/10.1063/1.864230
Paulhiac, D., Cuenot, B., Riber, E., et al.: Analysis of the spray flame structure in a lab-scale burner using large eddy simulation and discrete particle simulation. Combust. Flame 212, 25–38 (2020). https://doi.org/10.1016/j.combustflame.2019.10.013
Poustis, J.F., Senoner, J.M., Zuzio, D., et al.: Regularization of the Lagrangian point force approximation for deterministic discrete particle simulations. Int. J. Multiph. Flow 117, 138–152 (2019). https://doi.org/10.1016/j.ijmultiphaseflow.2019.04.021
Rangel, R.H., Sirignano, W.A.: An evaluation of the point-source approximation in spray calculations. Num. Heat Transfer; Part A: Appl. 16(1), 37–57 (1989). https://doi.org/10.1080/10407788908944705
Ranz, W., Marshall, W.: Evaporation from Drops. Chem. Eng. Prog. 48, 141–146 (1952)
Refloch, A., Courbet, B., Murrone, A., et al.: Cedre Software. Aerospace Lab Journal (2011) https://hal.science/hal-01182463
Riber, E., Moureau, V., García, M., et al.: Evaluation of numerical strategies for large eddy simulation of particulate two-phase recirculating flows. J. Comput. Phys. 228(2), 539–564 (2009). https://doi.org/10.1016/j.jcp.2008.10.001
Schlottke, J., Weigand, B.: Direct numerical simulation of evaporating droplets. J. Comput. Phys. 227, 5215–5237 (2008). https://doi.org/10.1016/j.jcp.2008.01.042
Sirignano, WA.: Fluid Dynamics and Transport of Droplets and Sprays, 2nd edn. Cambridge University Press, (2010) https://doi.org/10.1017/CBO9780511806728
Sontheimer, M., Kronenburg, A., Stein, O.T.: Grid dependence of evaporation rates in Euler-Lagrange simulations of dilute sprays. Combustion and Flame 232,(2021). https://doi.org/10.1016/j.combustflame.2021.111515
Zhang, J., Li, T., Ström, H., et al.: Grid-independent Eulerian-Lagrangian approaches for simulations of solid fuel particle combustion. Chem. Eng. J. 387(January), 123964 (2020). https://doi.org/10.1016/j.cej.2019.123964
Funding
The authors declare that this work does not rely on any external funding.
Author information
Authors and Affiliations
Contributions
Conceptualization: BC, ER, AP; Methodology: TL; Formal analysis and investigation: TL; Writing - original draft preparation: TL; Writing - review and editing: BC, ER, AP.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no competing interests as defined by Springer, or other interests that might be perceived to influence the results and/or discussion reported in this paper.
Ethical Approval
Not applicable
Informed Consent
Not applicable
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Lesaffre, T., Pestre, A., Riber, E. et al. Correction Methods for Exchange Source Terms in Unstructured Euler-Lagrange Solvers with Point-Source Approximation. Flow Turbulence Combust 112, 831–844 (2024). https://doi.org/10.1007/s10494-023-00487-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10494-023-00487-2