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A Semi-Lagrangian Approach for Dilute Non-Collisional Fluid-Particle Flows

Part of: Partial differential equations, initial value and time-dependent initial-boundary value problems Time-dependent statistical mechanics (dynamic and nonequilibrium)

Published online by Cambridge University Press:  16 March 2016

Aude Bernard-Champmartin ,
Jean-Philippe Braeunig ,
Christophe Fochesato  and
Thierry Goudon
Aude Bernard-Champmartin
Affiliation:
Inria, Sophia Antipolis Méditerranée Research Centre, Project COFFEE
Jean-Philippe Braeunig
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France LRC MESO, ENS Cachan, 61, avenue du Président Wilson, 94235 Cachan cedex, France
Christophe Fochesato
Affiliation:
CEA, DAM, DIF, F-91297 Arpajon, France LRC MESO, ENS Cachan, 61, avenue du Président Wilson, 94235 Cachan cedex, France
Thierry Goudon*
Affiliation:
Inria, Sophia Antipolis Méditerranée Research Centre, Project COFFEE Univ. Nice Sophia Antipolis, CNRS, Labo J.-A. Dieudonné, UMR 7351 Parc Valrose, F-06108 Nice, France
*
*Corresponding author. Email addresses:, [email protected] (A. Bernard-Champmartin), [email protected] (J.-P. Braeunig), [email protected] (C. Fochesato), [email protected] (T. Goudon)
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Abstract

We develop numerical methods for the simulation of laden-flows where particles interact with the carrier fluid through drag forces. Semi-Lagrangian techniques are presented to handle the Vlasov-type equation which governs the evolution of the particles. We discuss several options to treat the coupling with the hydrodynamic system describing the fluid phase, paying attention to strategies based on staggered discretizations of the fluid velocity.

Keywords

Semi-Lagrangian methodsparticulate flows

MSC classification

Secondary: 82C40: Kinetic theory of gases 82C80: Numerical methods (Monte Carlo, series resummation, etc.) 65M08: Finite volume methods
Type
Research Article
Information
Communications in Computational Physics , Volume 19 , Issue 3 , March 2016 , pp. 801 - 840
Copyright
Copyright © Global-Science Press 2016 

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References

[1]Andrews, M. J. and O'Rourke, P. J.. The multiphase particle-in-cell (MP-PIC) method for dense particulate flows. InJ. Multiphase Flow, 22(2):379–402, 1996.CrossRefGoogle Scholar
[2]Arber, T. D. and Vann, R. G. L.. A critical comparison of Eulerian-grid-based Vlasov solvers. J. Comput. Phys., 180(1):339–357, 2002.CrossRefGoogle Scholar
[3]Baranger, C., Baudin, G., Boudin, L., Després, B., Lagoutière, F., Lapébie, F., and Takahashi, T.. Liquid jet generation and break-up. In Cordier, S., Goudon, T., Gutnic, M., and Sonnen-drücker, E., editors, Numerical Methods for Hyperbolic and Kinetic Equations, volume 7 of IRMA Lectures in Mathematics and Theoretical Physics. EMS Publ. House, 2005.Google Scholar
[4]Baranger, C., Boudin, L., Jabin, P.-E., and Mancini, S.. A modeling of biospray for the upper airways. ESAIM:Proc, 14:41–47, 2005.CrossRefGoogle Scholar
[5]Baranger, C. and Desvillettes, L. Coupling Euler and Vlasov equations in the context of sprays: local smooth solutions. Journal of Hyperbolic Differential Equations, 3(1):1–26, 2006.CrossRefGoogle Scholar
[6]Berres, S., Bürger, R., and Tory, E. M.. Mathematical model and numerical simulation of the liquid fluidization of polydisperse solid particle mixtures. Comput. Visual Set, 6:67–74, 2004.CrossRefGoogle Scholar
[7]Berthelin, F., Goudon, T., and Minjeaud, S.. Multifluid flows: a kinetic approach. Technical report, Univ. Nice Sophia Antipolis, CNRS, Inria, 2013. Work in preparation.Google Scholar
[8]Besse, N. and Sonnendrücker, E.. Semi-Lagrangian schemes for the Vlasov equation on an unstructured mesh of phase space. J. Comput. Phys., 191(2):341–376, 2003.CrossRefGoogle Scholar
[9]Botchorishvili, R., Perthame, B., and Vasseur, A.. Equilibrium schemes for scalar conservation laws with stiff sources. Math. Comp., 72:131–157, 2003.CrossRefGoogle Scholar
[10]Bouchut, F., Ounaissa, H., and Perthame, B.. Upwinding of the source term at interfaces for Euler equations with high friction. Computers and Math. with Appl., 53:361–375, 2007.CrossRefGoogle Scholar
[11]Boudin, L., Boutin, B., Fornet, B., Goudon, T., Lafitte, P., Lagoutière, F., and Merlet, B.. Fluid-particles flows: A thin spray model with energy exchanges. ESAIM: Proc., 28:195–210, 2009.CrossRefGoogle Scholar
[12]Boudin, L., Desvillettes, L., Grandmont, C., and Moussa, A.. Global existence of solutions for the coupled Vlasov and Navier-Stokes equations. Differential and Integral Equations, 22(11-12), 2009.CrossRefGoogle Scholar
[13]Braeunig, J.-P., Crouseilles, N., Grandgirard, V., Latu, G., Mehrenberger, M., and Sonnendrücker, E.. Some numerical aspects of the conservative PSM scheme in a 4D drift-kinetic code. Technical report, Inria, 2009. Report Inria RR 7109, available on ArXiV: 1303.2238.Google Scholar
[14]Brenier, Y. and Grenier, E.. Sticky particles and scalar conservation laws. SIAM J. Numer. Anal., 35(6):2317–2328 (electronic), 1998.CrossRefGoogle Scholar
[15]Calgaro, C., Creusé, E., and Goudon, T.. Modeling and simulation of mixture flows: Application to powder-snow avalanches. Computers and Fluids, 107:100–122, 2015.CrossRefGoogle Scholar
[16]Carpenter, R. L., Droegemeier, K. K., Woodward, P. R., and Hane, C. E.. Application of the piecewise parabolic method (PPM) to meteorological modeling. Mon. Wea. Rev., 118(3):586–612,1990.2.0.CO;2>CrossRefGoogle Scholar
[17]Carrillo, J. A. and Goudon, T.. Stability and asymptotic analysis of a fluid-particle interaction model. Comm. PDE., 31(9):1349–1379,2006.CrossRefGoogle Scholar
[18]Carrillo, J.-A., Goudon, T., and Lafitte, P.. Simulation of fluid and particles flows: asymptotic preserving schemes for bubbling and flowing regimes. J. Comput. Phys., 227(16):7929–7951, 2008.CrossRefGoogle Scholar
[19]Carrillo, J. A. and Vecil, E. Non oscillatory interpolation methods applied to Vlasov-based models. SIAM J. Sci. Comput., 29(3):1179–1206,2007.CrossRefGoogle Scholar
[20]Chalons, C., Coquel, F., Godlewski, E., Raviart, P.-A., and N. Seguin, . Godunov-type schemes for hyperbolic systems with parameter-dependent source. The case of Euler system with friction. Math. Models Methods Appl. Sci., 20(11):2109–2166,2010.CrossRefGoogle Scholar
[21]Chalons, C., Girardin, M., and Kokh, S.Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms. SIAM J. Sci. Comput., 35(6):A2874-A2902,2013.CrossRefGoogle Scholar
[22]Champmartin-Bernard, A.. Modélisation et étude numérique d'écoulements diphasiques. PhD thesis, ENS Cachan, 2011.Google Scholar
[23]Colella, P. and Sekora, M. D.. A limiter for PPM that preserves accuracy at smooth extrema. J. Comput. Phys., 227:7069–7076,2008.CrossRefGoogle Scholar
[24]Colella, P. and Woodward, P. R.. The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54(1):174–201,1984.CrossRefGoogle Scholar
[25]Crouseilles, N., Latu, G., and Sonnendrücker, E.. Hermite spline interpolation on patches for a parallel solving of the Vlasov-Poisson equation. Rapport de recherche RR-5926, INRIA, 2006.CrossRefGoogle Scholar
[26]Crouseilles, N., Mehrenberger, M., and Sonnendrücker, E.. Conservative semi-Lagrangian schemes for Vlasov equations. J. Comput. Phys., 229:1927–1953,2010.CrossRefGoogle Scholar
[27]De Luca, M.. Contribution à la modélisation de la pulvérisation d'un liquide phytosanitaire en vue de réduire les pollutions. PhD thesis, Univ. Aix-Marseille 2, 2007.Google Scholar
[28]De Vuyst, F., Fochesato, C., Loubère, R., Rouzier, P., Saas, L., Motte, R., and Ghidaglia, J.-M.. Staggered lagrange-remap schemes in conservative form. Technical report, LRC MESO, CEA, CMLA, ENS Cachan, 2012.Google Scholar
[29]Desjardins, O., Fox, R. O., and Villedieu, P.. A quadrature-based moment method for dilute fluid-particle flows. J. Comput. Phys., 227:2514–2539,2008.CrossRefGoogle Scholar
[30]Desvillettes, L.. Some new results of existence for the theory of sprays. http://www.newton.ac.uk/programmes/KIT/seminars/090710001.html, 2010. Workshop “Fluid-Kinetic Modelling in Biology, Physics and Engineering”, Isaac Newton Institute for Mathematical Sciences, Programme on PDEs in Kinetic Theories.Google Scholar
[31]Domelevo, K. and Villedieu, P.. A hierarchy of models for turbulent dispersed two–phase flows derived from a kinetic equation for the joint particle-gas pdf. Commun. Math. Sci., 5(2):331–353, 2007.CrossRefGoogle Scholar
[32]Engquist, B. and Osher, S.. Stable and entropy satisfying approximations for transonic flow calculations. Mathematics of Computation, 34(149):45–75, 1980.CrossRefGoogle Scholar
[33]Filbet, F. and Russo, G.. High order numerical methods for the space non-homogeneous Boltzmann equation. J. Comput. Phys., 186(2):457–480, 2003.CrossRefGoogle Scholar
[34]Filbet, F. and Sonnendrücker, E.. Comparison of Eulerian Vlasov solvers. Computer Physics Communications, 150(3):247–266, 2003.CrossRefGoogle Scholar
[35]Filbet, F. and Sonnendrücker, E.. Numerical methods for the Vlasov equation. In Brezzi, F., Buffa, A., Corsaro, S., and Murli, A., editors, Numerical Mathematics and Advanced Applications, pages 459–468. Springer Milan, 2003.CrossRefGoogle Scholar
[36]Filbet, F., Sonnendrücker, E., and Bertrand, P.. Conservative numerical schemes for the Vlasov equation. J. Comput. Phys., 172:166–187, 2001.CrossRefGoogle Scholar
[37]Gosse, L.. Computing Qualitatively Correct Approximations of Balance Laws : Exponential-fit, Well-balanced and Asymptotic-preserving. Springer, London, 2013.CrossRefGoogle Scholar
[38]Gosse, L. and Le Roux, A.-Y.. A well balanced scheme designed for inhomogeneous scalar conservation laws. C. R. Acad. Sci. Paris Ser. I Math., 323(5):543–546, 1996.Google Scholar
[39]Goudon, T.. Intégration. Intégrale de Lebesgue et introduction à l’analyse fonctionnelle. Références Sciences. Ellipses, 2011.Google Scholar
[40]Goudon, T., Jabin, P.-E., and Vasseur, A.. Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime. Indiana Univ. Math. J., 53(6):1495–1515, 2004.CrossRefGoogle Scholar
[41]Goudon, T., Jabin, P.-E., and Vasseur, A.. Hydrodynamic limit for the Vlasov-Navier-Stokes equations. II. Fine particles regime. Indiana Univ. Math. J., 53(6):1517–1536, 2004.CrossRefGoogle Scholar
[42]Goudon, T., Jin, S., and Yan, B.. Simulation of fluid-particles flows: heavy particles, flowing regime, and asymptotic-preserving schemes. Commun. Math. Sci., 10(1):355–385, 2012.CrossRefGoogle Scholar
[43]Goudon, T., Moussa, A., He, L., and Zhang, P.. The Navier–Stokes–Vlasov–Fokker–Planck system near equilibrium. SIAM J. Math. Anal., 42(5):2177–2202, 2010.CrossRefGoogle Scholar
[44]Goudon, T. and Poupaud, F.. On themodeling of the transport of particles in turbulent flows. M2AN Math. Model. Numer. Anal., 38(4):673–690, 2004.CrossRefGoogle Scholar
[45]Guterl, J., Braeunig, J.-P., Crouseilles, N., Grandgirard, V., Latu, G., Mehrenberger, M., and Sonnendrücker, E.. Test of some numerical limiters for the conservative PSM scheme for 4D Drift-Kinetic simulations. Rapport de recherche RR-7467, INRIA, 2010.Google Scholar
[46]Hank, S., Saurel, R., and Le Metayer, O.. A hyperbolic Eulerian model for dilute two-phase suspensions. J. Modern Physics, 2:997–1011, 2011.CrossRefGoogle Scholar
[47]Heuzé, O., Jaouen, S., and Jourdren, H.. Dissipative issue of high-order shock capturing schemes with non-convex equations of state. J. Comput. Phys., 228(3):833–860, 2009.CrossRefGoogle Scholar
[48]Hyland, K. E., McKee, S., and Reeks, M. W.. Derivation of a pdf kinetic equation for the transport of particles in turbulent flows. J. Phys. A: Math. Gen., 32:6169–6190, 1999.CrossRefGoogle Scholar
[49]Jabin, P.-E.. Large time concentrations for solutions to kinetic equations with energy dissipation. Comm. Partial Differential Equations, 25(3-4):541–557, 2000.CrossRefGoogle Scholar
[50]Lavergne, G.. Modélisation de l’écoulement multiphasique dans le propulseur à poudre P230 d’Ariane 5, 2004. Lecture Notes of the School of the Groupement Français de Combustion, Ile d’Oléron.Google Scholar
[51]Lighthill, J.. Ocean spray and the thermodynamics of tropical cyclones. J. of Engineering Math., 35:11–42, 1999.CrossRefGoogle Scholar
[52]Liu, H., Wang, Z., and Fox, R. O.. A level set approach for dilute non-collisional fluid-particle flows. J. Comput. Phys., 230(4):920–936, 2011.CrossRefGoogle Scholar
[53]Lun, C. K. K. and Savage, S. B.. Kinetic Theory for Inertia Flows of Dilute Turbulent Gas-Solids Mixtures, volume 624 of Lect. Notes Phys., pages 267–289. Springer, 2003.Google Scholar
[54]Mathiaud, J.. Etude de systèmes de type gaz–particules. PhD thesis, ENS Cachan, 2006.Google Scholar
[55]Moussa, A.. Etude mathématique et numérique du transport d’aérosols dans le poumon humain. PhD thesis, ENS Cachan, 2009.Google Scholar
[56]O’Rourke, P. J.. Collective drop effects on vaporizing liquid sprays. PhD thesis, Princeton University, NJ, 1981. Available as Technical Report #87545 Los Alamos National Laboratory.Google Scholar
[57]Patankar, N. A. and Joseph, D. D.. Lagrangian numerical simulation of particulate flows. Int. J. Multiphase Flow, 27:1685–1706, 2001.CrossRefGoogle Scholar
[58]Patankar, N. A. and Joseph, D. D.. Modeling and numerical simulation of particulate flows by the Eulerian–Lagrangian approach. Int. J. Multiphase Flow, 27:1659–1684, 2001.CrossRefGoogle Scholar
[59]Schäfer, K.et al. Influences of the 2010 Eyjafjallajökull volcanic plume on air quality in the northern alpine region. Atmos. Chem. Phys. Discuss., pages 9083–9132, 2011.Google Scholar
[60]Sircombe, N.J. and Arber, T.D.. VALIS: A split-conservative scheme for the relativistic 2D Vlasov–Maxwell system. J. Comput. Phys., 228(13):4773–4788, 2009.CrossRefGoogle Scholar
[61]Snider, D.M., O’Rourke, P. J., and Andrews, M. J.. Sediment flow in inclined vessels calculated using a multiphase particle-in-cell model for dense particle flows. Int. J. Multiphase Flow, 24:1359–1382, 1998.CrossRefGoogle Scholar
[62]Sonnendrücker, E., Barnard, J. J., Friedman, A., Grote, D. P., and Lund, S. M.. Simulation of heavy ion beams with a semi-Lagrangian Vlasov solver. Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, 464(1-3):470–476, 2001. Proc. of the 13th Int. Symp. on Heavy Ion Inertial Fusion.CrossRefGoogle Scholar
[63]Sonnendrücker, E., Roche, J., Bertrand, P., and Ghizzo, A.. The semi-Lagrangian method for the numerical resolution of the Vlasov equation. J. Comput. Phys., 149:201–220, 1999.CrossRefGoogle Scholar
[64]Sutcliffe, W. G.. BBC hydrodynamics. Technical report, CaliforniaUniv., Livermore Lawrence Livermore Lab., 1974.Google Scholar
[65]Torres, D. J. and Trujillo, M. F.. KIVA-4: An unstructured ALE code for compressible gas flow with sprays. J. Comput. Phys., 219:943–975, 2006.CrossRefGoogle Scholar
[66]Umeda, T.. A conservative and non-oscillatory scheme for Vlasov code simulations. Earth Planets Space, 60:773–779, 2008.CrossRefGoogle Scholar
[67]Umeda, T., Ashour-Abdalla, M., and Schriver, D.. Comparison of numerical interpolation schemes for one-dimensional electrostatic Vlasov code. Journal of Plasma Physics, 72:1057–1060, 2006.CrossRefGoogle Scholar
[68]Van Leer, B.. Towards the ultimate conservative difference scheme, v. a second order sequel to Godunov's method. J. Comput. Phys., 32:101–136, 1979.CrossRefGoogle Scholar
[69]Vinkovic, I.. Dispersion et mélange turbulents de particules solides et de gouttelettes par une simulation des grandes échelles et une modélisation stochastique lagrangienne. Application à la pollution de l’atmosphère. PhD thesis, Ecole Centrale de Lyon, 2005.Google Scholar
[70]Von Neumann, J. and Richtmyer, R.D.. A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys., 21:232–237, 1950.CrossRefGoogle Scholar
[71]Williams, F. A.. Combustion Theory. Benjamin/Cummings, 1985. 2nd. edition.Google Scholar
[72]Woodward, P. and Colella, P.. The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 54(1):115–173, 1984.CrossRefGoogle Scholar