Hostname: page-component-76fb5796d-45l2p Total loading time: 0 Render date: 2024-04-25T15:14:51.072Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Anisotropic Gaussian Velocity Closure for Inertial-Particle Laden Flows

Published online by Cambridge University Press:  28 November 2014

Aymeric Vié ,
François Doisneau  and
Marc Massot
Aymeric Vié*
Affiliation:
CNRS Laboratoire EM2C – UPR 288, Grande Voie des Vignes, 92295 Chatenay-Malabry, France Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France Center For Turbulence Research, Stanford University, 488, Escondido Mall, Stanford CA 94305-3035, USA
François Doisneau
Affiliation:
CNRS Laboratoire EM2C – UPR 288, Grande Voie des Vignes, 92295 Chatenay-Malabry, France Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France ONERA, Chemin de la Hunière, 91120, Palaiseau France
Marc Massot
Affiliation:
CNRS Laboratoire EM2C – UPR 288, Grande Voie des Vignes, 92295 Chatenay-Malabry, France Ecole Centrale Paris, Grande Voie des Vignes, 92295 Chatenay-Malabry, France Fédération de Mathématiques de l'Ecole Centrale Paris, FR CNRS 3487, France
*
*Email addresses:aymeric.vie@stanford.eduaymeric.vie@ecp.fr(A. Vié), francois.doisneau@ecp.fr(F. Doisneau), marc.massot@ecp.fr(M. Massot)
Get access

Abstract

The accurate simulation of disperse two-phase flows, where a discrete particulate condensed phase is transported by a carrier gas, is crucial for many applications; Eulerian approaches are well suited for high performance computations of such flows. However when the particles from the disperse phase have a significant inertia compared to the time scales of the flow, particle trajectory crossing (PTC) occurs i.e. the particle velocity distribution at a given location can become multi-valued. To properly account for such a phenomenon many Eulerian moment methods have been recently proposed in the literature. The resulting models hardly comply with a full set of desired criteria involving: 1- ability to reproduce the physics of PTC, at least for a given range of particle inertia, 2- well-posedness of the resulting set of PDEs on the chosen moments as well as guaranteed realizability, 3- capability of the model to be associated with a high order realizable numerical scheme for the accurate resolution of particle segregation in turbulent flows. The purpose of the present contribution is to introduce a multi-variate Anisotropic Gaussian closure for such particulate flows, in the spirit of the closure that has been suggested for out-of-equilibrium gas dynamics and which satisfies the three criteria. The novelty of the contribution is three-fold. First we derive the related moment system of conservation laws with source terms, and justify the use of such a model in the context of high Knudsen numbers, where collision operators play no role. We exhibit the main features and advantages in terms of mathematical structure and realizability. Then a second order accurate and realizable MUSCL/HLL scheme is proposed and validated. Finally the behavior of the method for the description of PTC is thoroughly investigated and its ability to account accurately for inertial particulate flow dynamics in typical configurations is assessed.

Keywords

76T1076P0582C2165M0835L4052B55Particulate flowskinetic theorymoment methodparticle trajectory crossing
Type
Research Article
Information
Communications in Computational Physics , Volume 17 , Issue 1 , January 2015 , pp. 1 - 46
Copyright
Copyright © Global Science Press Limited 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Amsden, A. A., O'Rourke, P. J., and Butler, T. D.. Kiva II, a computer program for chemically reactive flows with sprays. Technical Report LA-11560-MS, Los Alamos National Laboratory, Los Alamos, New Mexico, 1989.Google Scholar
[2]Andries, P., Le Tallec, P., Perlat, J.-P., and Perthame, B.. The Gaussian-BGK model of Boltzmann equation with small Prandtl number. Eur. J. of Mech. B - Fluids, 19:813–830,2000.Google Scholar
[3]Balachandar, S. and Eaton, J.K.. Turbulent dispersed multiphase flow. Annual Review of Fluid Mechanics, 42:111–133, 2011.Google Scholar
[4]Bardos, C., Golse, F., and Levermore, D.. Fluid dynamic limits of kinetic equations. I. Formal derivations. Journal of Statistical Physics, 63, 1991.CrossRefGoogle Scholar
[5]Berthon, C.. Stability of the MUSCL schemes for the Euler equations. Commun. Math. Sci., 3(2):133–157,2005.Google Scholar
[6]Berthon, C.. Numerical approximations of the 10-moment Gaussian closure. Math. Comp., 75(256):1809–1831 (electronic), 2006.Google Scholar
[7]Bhatnagar, P. L., Gross, E.P., and Krook, M.. A model for collision proces in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev., 94:511–525, 1954.Google Scholar
[8]Bird, G. A.. Molecular gas dynamics and the direct simulation of gas flows. Oxford Science Publications, 42, 1994.CrossRefGoogle Scholar
[9]Boileau, M., Chalons, C., and Massot, M.. Robust numerical coupling of pressure and pressureless gas dynamics equations for eulerian spray DNS and LES. SIAM Journal of Scientific Computing, submitted, pages 1-23, 2013. Available on HAL: http://hal.archives-ouvertes.fr/hal-00906220.Google Scholar
[10]Bouchut, F.. Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2004.Google Scholar
[11]Bouchut, F., Jin, S., and Li, X.. Numerical approximations of pressureless and isothermal gas dynamics. SIAM J. Num. Anal., 41:135–158,2003.Google Scholar
[12]Brown, S. L.. Approximate Riemann solvers for moment models of dilute gases. PhD thesis, University of Michigan, 1996.Google Scholar
[13]Brull, S. and Schneider, J.. On the ellipsoidal statistical model for polyatomic gases. Cont. Mech. Thermodyn., 20:489–508, 2009.CrossRefGoogle Scholar
[14]Chalons, C., Fox, R. O., Laurent, F., Massot, M., and Vié, A.. A multi-Gaussian quadrature-based moment method for dispersed multiphase flows. SIAM Journal on Multiscale Modeling and Simulation, submitted, 2013.Google Scholar
[15]Chalons, C., Fox, R. O., and Massot, M.. A multi-Gaussian quadrature method of moments for gas-particle flows in a LES framework. In Proceedings of the Summer Program 2010, Center for Turbulence Research, Stanford University, pages 347–358, Stanford, 2010.Google Scholar
[16]Chalons, C., Kah, D., and Massot, M.. Beyond pressureless gas dynamics: quadrature-based velocity moment models. Communication in Mathematical Sciences, 10(4):1241–1272,2012.CrossRefGoogle Scholar
[17]Chalons, C., Massot, M., and Vié, A.. An asymptotic preserving numerical scheme for the large eddy simulation of disperse phase flows. SIAM Journal on Multiscale Modeling and Simulation, submitted, 2014. available online at http://hal.archives-ouvertes.fr/hal-00958108.Google Scholar
[18]Colin, O. and Rudgyard, M.. Development of high-order Taylor-Galerkin schemes for LES. Journal of Computational Physics, 162(2):338–371,2000.Google Scholar
[19]de Chaisemartin, S.. Eulerian models and numerical simulation of turbulent dispersion for polydisperse evaporating sprays. PhD thesis, Ecole Centrale Paris, France, 2009. Available online at http://tel.archives-ouvertes.fr/tel-00443982/en/.Google Scholar
[20]Doisneau, F.. Eulerian modeling and simulation of polydisperse moderately dense coalescing spray flows with nanometric-to-inertial droplets: application to Solid Rocket Motors. PhD thesis, Ecole Centrale Paris, 2013. Available at http://tel.archives-ouvertes.fr/tel-00966185.Google Scholar
[21]Doisneau, F., Laurent, F., Murrone, A., Dupays, J., and Massot, M.. Eulerian multi-fluid models for the simulation of dynamics and coalescence of particles in solid propellant combustion. J. Comput. Phys., 234:230–262,2013.Google Scholar
[22]Doisneau, F., Thomine, O., Laurent, F, Vié, A., Dupays, J., and Massot, M.. Eulerian modeling and simulation of small scale trajectory crossing and coalescence for moderate-stokes-number spray flows. Proceedings of the Summer Program 2012, Center for Turbulence Research, Stanford University, pages 365–374, 2012.Google Scholar
[23]Dombard, J.. Direct Numerical Simulation of non-isothermal dilute sprays using the Meso-scopic Eulerian Formalism. PhD thesis, INP Toulouse, 2011.Google Scholar
[24]Dufour, G.. Modélisation multi-fluide eulérienne pour les écoulements diphasiques à inclusions dispersées. PhD thesis, Université Paul Sabatier Toulouse III, 2005.Google Scholar
[25]Dukowicz, J. K.. A particle-fluid numerical model for liquid sprays. J. Comput. Phys., 35(2):229–253,1980.Google Scholar
[26]Einfeldt, B., Munz, C.D., Roe, P.L., and Sjögreen, B.. On Godunov-type Methods near low densities. Journal of Computational Physics, 92:273–295,1991.Google Scholar
[27]Emre, O., Laurent, F., de Chaisemartin, S., Jay, S., and Massot, M.. Numerical analysis of a two-way coupled Eulerian high order moment method for the simulation of polydisperse evaporating sprays. Journal of Computational Physics, submitted, pages 1–28, 2013.Google Scholar
[28]Ferry, J. and Balachandar, S.. A fast Eulerian method for disperse two-phase flow. Int. J. Multiphase Flow, 27:1199–1226, 2001.Google Scholar
[29]Février, P., Simonin, O., and Squires, K. D.. Partitioning of particle velocities in gas-solid turbulent flow into a continuous field and a spatially uncorrelated random distribution: theoretical formalism and numerical study. J. Fluid Mech., 533:1–46,2005.CrossRefGoogle Scholar
[30]Fox, R.O.. in ‘Computational Models for Turbulent Multiphase Reacting Flows”, Udine, July 2006, chapter Fundamental of Polydisperse Multiphase Flows. CISM Courses and Lectures. Springer Verlag, 2007.Google Scholar
[31]Giovangigli, V. and Massot, M.. Asymptotic stability of equilibrium states for multicompo-nent reactive flows. Math. Models Methods Appl. Sci., 8(2):251–297,1998.CrossRefGoogle Scholar
[32]Golse, F.. The Boltzmann equation and its hydrodynamic limits. In Evolutionary equations. Vol. II, Handb. Differ. Equ., pages 159–301. Elsevier/North-Holland, Amsterdam, 2005.Google Scholar
[33]Grad, H.. Principles of kinetic theory of gases, volume XII. Springer-Verlag, 1958.Google Scholar
[34]Graille, B., Magin, T., and Massot, M.. Kinetic theory of plasmas: Translational energy. Mathematical Models and Methods in Applied Sciences, 19:527–599,2009.CrossRefGoogle Scholar
[35]Harten, A., Lax, P.D., and van Leer, B.. On upstream differencing and Godunov-type schemes for hyperbolic conservation laws. SIAM Review, 25(1):35–61,1983.Google Scholar
[36]Holway, L.H. Jr.New statistical models for kinetic theory: methods of construction. The Physics of Fluids, 9(9):1658–1673, 1966.Google Scholar
[37]Hylkema, J. J.. Modélisation cinétique et simulation numérique d'un brouillard dense de gouttelettes. Application aux propulseurs à poudre. PhD thesis, Ecole Nat. Supérieure de l'Aéronautique et de l'Espace, 1999.Google Scholar
[38]Kah, D., Laurent, F., Fréret, L., de Chaisemartin, S., Fox, R., Reveillon, J., and Massot, M.. Eu-lerian quadrature-based moment models for dilute polydisperse evaporating sprays. Flow Turbulence and Combustion, 85(3-4):649–676, 2010.Google Scholar
[39]Kah, D., Laurent, F., Massot, M., and Jay, S.. A high order moment method simulating evaporation and advection of a polydisperse spray. J. Comput. Phys., 231(2):394–422,2012.Google Scholar
[40]Kaufmann, A., Moreau, M., Simonin, O., and Helie, J.. Comparison between Lagrangian and mesoscopic Eulerian modelling approaches for inertial particles suspended in decaying isotropic turbulence. J. Comput. Phys., 227:6448–6472,2008.CrossRefGoogle Scholar
[41]Larat, A., Vié, A., and Massot, M.. A stable, robust and high order accurate numerical method for Eulerian simulation of spray and particle transport on unstructured meshes. Annual Research Briefs of the CTR 2012, Center for Turbulence Research, Stanford University, pages 205–216, 2012.Google Scholar
[42]Laurent, F. and Massot, M.. Multi-fluid modeling of laminar poly-dispersed spray flames: origin, assumptions and comparison of the sectional and sampling methods. Combust. Theory and Modelling, 5:537–572,2001.Google Scholar
[43]Laurent, F., Vié, A., Chalons, C., Fox, R. O., and Massot, M.. A hierarchy of Eulerian models for trajectory crossing in particle-laden turbulent flows over a wide range of Stokes numbers. Annual Research Briefs of the CTR 2012, Center for Turbulence Research, Stanford University, pages 193–204, 2012.Google Scholar
[44]LeVeque, R. J.. Finite volume methods for hyperbolic problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge, 2002.Google Scholar
[45]Levermore, C.D.. Moment closure hierarchies for kinetic theories. J. Stat Phys., 83(5/6):1021–1065, 1996.Google Scholar
[46]Levermore, C.D. and Morokoff, W.J.. The Gaussian moment closure for gas dynamics. SIAM J. Appl. Math., 59(1):72–96,1998.CrossRefGoogle Scholar
[47]Levermore, C.D., Morokoff, W.J., and Nadiga, B.T.. Non-equilibrium partition function in the presence of heat flow. Physics of Fluids, 10:3214, 1998.Google Scholar
[48]Marble, F.E.. Dynamics of dusty gases. Annual Review of Fluid Mechanics, 2:397–446,1970.Google Scholar
[49]Martinez, L., Vié, A., Jay, S., Benkenida, A., and Cuenot, B.. Large Eddy Simulation of Fuel sprays using the Eulerian Mesoscopic Approach. Validations in realistic engine conditions. In ICLASS, 2009.Google Scholar
[50]Masi, E. and Simonin, . Algebraic-closure-based moment method for unsteady eulerian simulations of non-isothermal particle-laden turbulent flows at moderate stokes numbers in dilute regime. Flow Turbulence and Combustion, 92(1-2):121–145, 2014.Google Scholar
[51]Masi, E., Simonin, O., Riber, E., Sierra, P., and Gicquel, L.Y.M.. Development of an algebraic-closure-based moment method for unsteady eulerian simulations of particle-laden turbulent flows in very dilute regime. International Journal of Multiphase Flow, 58:257–278,2014.Google Scholar
[52]Massot, M.. Eulerian multi-fluid models for polydisperse evaporating sprays, volume 492 of CISM Courses and Lectures, chapter in “Multiphase Reacting Flows: Modelling and Simulation”, Udine, July 2006, pages 79–123. Springer Wien New York (2007), 2007. D.L. Marchisio and R.O. Fox Editors.Google Scholar
[53]Massot, M., Knikker, R., Péra, C., and Reveillon, J.. Lagrangian/Eulerian analysis of the dispersion of evaporating droplets in a non-homogeneous turbulent flow. In Proceedings of ICMF'04, Yokohama, Japan, May 31 - June 3, 2004.Google Scholar
[54]Massot, M., Laurent, F., Kah, D., and de Chaisemartin, S.. A robust moment method for evaluation of the disappearance rate of evaporating sprays. SIAM J. Appl. Math., 70(8):3203–3234, 2010.Google Scholar
[55]Mc Donald, J.G. and Groth, C.P.T.. Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution. Cont. Mech. Thermodyn., 25(5):573–603, 2012.Google Scholar
[56]O'Rourke, P. J.. Collective drop effects on vaporizing liquid sprays. PhD thesis, Los Alamos National Laboratory 87545, University of Princeton, 1981.Google Scholar
[57]Reeks, M.W.. On the dispersion of small particles suspended in an isotropic turbulent fluid. J. Fluid Mech., 83(03):529–546,1977.Google Scholar
[58]Refloch, A., Courbet, B., Murrone, A., Villedieu, P., Laurent, C., Gilbank, P., Troyes, J., Tessé, L., Chaineray, G., Dargaud, J.B., Quémerais, E., and Vuillot, F.. CEDRE software. Aerospace Lab, 2:1–10, 2011.Google Scholar
[59]Reveillon, J. and Demoulin, F.X.. Effects of the preferential segregation of droplets on evaporation and turbulent mixing. Journal of Fluid Mechanics, 583:273–302, 2007.Google Scholar
[60]Riber, E., Garcia, M., Moureau, V., Pitsch, H., Simonin, O., and Poinsot, T.. Evaluation of numerical strategies for les of two-phase reacting flows. Proceedings of the Summer Program 2006, Center for Turbulence Research, Stanford University, pages 197–213, 2006.Google Scholar
[61]Sabat, M., Larat, A., Vié, A., and Massot, M.. Comparison of realizable schemes for the Eulerian simulation of disperse phase flows. Proceedings of FVCA7, accepted, Berlin, Germany, pages 1–8, 2014.Google Scholar
[62]Sabat, M., Larat, A., Vié, A., and Massot, M.. On the development of high order realizable schemes for the Eulerian simulation of disperse phase flows on unstructured grids: a convex-state preserving Discontinuous Galerkin method. Journal of Computational Physics, submitted, 2014.Google Scholar
[63]Sanjosé, M., Senoner, J.-M., Jaegle, F., Cuenot, B., Moreau, S., and Poinsot, T.. Fuel injection model for Euler-Euler and Euler-Lagrange large-eddy simulations of an evaporating spray inside an aeronautical combustor. International Journal of Multiphase Flow, 37(5):514–529, 2011.Google Scholar
[64]Sierra, P.. Modeling the dispersion and evaporation of sprays in aeronautical combustion chambers. PhD thesis, INP Toulouse, 2012.Google Scholar
[65]So, R. M. C., Jin, L. H., and Gatski, T. B.. An explicit algebraic Reynolds stress and heat flux model for incompressible turbulence: Part I Non-isothermal flow. Theoretical and Computational Fluid Dynamics, 17:351–376,2004.Google Scholar
[66]Squires, K. D. and Eaton, J.K.. Preferential concentration of particles by turbulence. Physics of Fluids, 3:1169–1178,1991.Google Scholar
[67]Struchtrup, H.. Macroscopic Transport Equations for Rarefied Gas Flows. Springer, 2005.Google Scholar
[68]Toro, E.F.. Riemann solvers and numerical methods for fluid dynamics. Springer, 2009.Google Scholar
[69]Leer, B. Van. Toward the ultimate conservation difference scheme V. A second-order sequel to Godunov's method. J. Comput. Phys., 32(1):101–136,1979.Google Scholar
[70]Vasilescu, F.-H.. Hamburger and Stieltjes moment problems in several variables. Transactions of the american mathematical society, 354(3):1265–1278,2001.Google Scholar
[71]Vié, A., Chalons, C., Fox, R. O., Laurent, F., and Massot, M.. A multi-Gaussian quadrature method of moments for simulating high-Stokes-number turbulent two-phase flows. Annual Research Briefs of the CTR 2011, Center for Turbulence Research, Stanford University, pages 309–320, 2011.Google Scholar
[72]Vié, A., Jay, S., Cuenot, B., and Massot, M.. Accounting for Polydispersion in the Eulerian Large Eddy Simulation of an Aeronautical-type Configuration. Flow Turbulence and Combustion, 90(3):545–581,2013.CrossRefGoogle Scholar
[73]Vié, A., Laurent, F., and Massot, M.. A high order moment method for the simulation of polydisperse two-phase flows. Comptes Rendus Mecanique, 341(1-2):55–64, 2013.Google Scholar
[74]Vié, A., Laurent, F., and Massot, M.. Size-velocity correlations in hybrid high order moment/multi-fluid methods for polydisperse evaporating sprays: modeling and numerical issues. Journal of Computational Physics, 237:177–210,2013.Google Scholar
[75]Vié, A., Masi, E., Simonin, O., and Massot, M.. On the direct numerical simulation of moderate-stokes-number turbulent particulate flows using Algebraic-Closure-Based and Kinetic-Based Moment Methods. In Proceedings of the Summer Program 2012, Center for Turbulence Research, Stanford University, pages 355–364, 2012.Google Scholar
[76]Williams, F. A.. Spray combustion and atomization. Phys. Fluids, 1:541–545,1958.Google Scholar
[77]Williams, F. A.. Combustion Theory (Combustion Science and Engineering Series). Ed. Williams, F. A. (Reading, MA: Addison-Wesley), 1985.Google Scholar
[78]Yuan, C. and Fox, R.O.. Conditional quadrature method of moments for kinetic equations. J. Comput. Phys., 230(22):8216–8246, 2011.Google Scholar
[79]Zaichik, L.I. and Alipchenkov, V.M.. Pair dispersion and preferential concentration of particles in isotropic turbulence. Phys. Fluids, 15:1776–1787,2003.Google Scholar
[80]Zaichik, L.I., Alipchenkov, V.M., and Sinaiski, E.G.. Particles in Turbulent Flows. Wiley-VCH, Weinheim, Germany, 2008.Google Scholar
[81]Zaichik, L.I., Simonin, O., and Alipchenkov, V.M.. An Eulerian approach for large eddy simulation of particle transport in turbulent flows. Journal of Turbulence, 10(4):1–21,2009.Google Scholar