Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-19T12:37:48.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Small-scale kinematics of two-phase flows: identifying relaxation processes in separated- and disperse-phase flow models

Published online by Cambridge University Press:  01 August 2019

Florence Drui Open the ORCID record for Florence Drui [Opens in a new window] ,
Adam Larat ,
Samuel Kokh  and
Marc Massot
Florence Drui*
Affiliation:
CNRS, UPR 288, Laboratoire Energétique moléculaire et macroscopique, combustion, 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France CentraleSupélec, 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France Maison de la Simulation USR 3441, Digiteo Labs, bât. 565, PC 190, CEA Saclay, 91191 Gif-sur-Yvette, France CEA/DEN/DANS/DM2S/SEMT/DYN – CEA Saclay, 91191 Gif-sur-Yvette, France
Adam Larat
Affiliation:
CNRS, UPR 288, Laboratoire Energétique moléculaire et macroscopique, combustion, 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France CentraleSupélec, 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France Fédération de Mathématiques de CentraleSupélec, FR CNRS 3487, France Université Grenoble Alpes, CNRS, Grenoble INP, LJK, 38000 Grenoble, France
Samuel Kokh
Affiliation:
Maison de la Simulation USR 3441, Digiteo Labs, bât. 565, PC 190, CEA Saclay, 91191 Gif-sur-Yvette, France CEA/DEN/DANS/DM2S/STMF/LMEC – CEA Saclay, 91191 Gif-sur-Yvette, France
Marc Massot
Affiliation:
CNRS, UPR 288, Laboratoire Energétique moléculaire et macroscopique, combustion, 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France CentraleSupélec, 8-10 rue Joliot Curie, 91190 Gif-sur-Yvette, France Fédération de Mathématiques de CentraleSupélec, FR CNRS 3487, France Centre de Mathématiques Appliquées, Ecole polytechnique, Route de Saclay, 91128 Palaiseau CEDEX, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We propose here to investigate the impact of small-scale effects on the bulk evolution of a two-phase flow system. More precisely, we choose to examine the sole influence of a small-scale (with respect to the bulk velocity) off-equilibrium velocity on the system. In order to narrow our analysis and avoid complex well-posedness issues, we choose to examine a simple barotropic 5-equation two-phase flow model that accounts for an equilibrium common bulk velocity and a small-scale off-equilibrium velocity. A full derivation of the model is presented: it is based on a variational principle which allows us to insert the two-scale kinematics into the model by considering two different kinetic energies. Additional entropy dissipation requirements allow us to add dissipative structures to the model. This system is neutral with respect to the topology of the flow structure and is equipped with parameters that can be connected to relaxation processes. When considering instantaneous relaxations, we obtain two limit systems of the literature that are used for the simulation of separated-phase flows. In this sense we obtain a hierarchy of models. We show that the parent 5-equation model is also compatible with the description of a bubbly fluid that allows small-scale vibrations for the disperse phase. This identification is verified and discussed through comparisons with experimental measurements of sound dispersion (Silberman, J. Acoust. Soc. Am., vol. 29, 1957, pp. 925–933; Leroy et al., J. Acoust. Soc. Am., vol. 123, 2008, pp. 1931–1940) and with the dispersion relations of a reference model for bubbly flows by Cheng et al. (Trans. ASME J. Heat Transfer, vol. 107, 1985, pp. 402–408). The present work is a first contribution to a larger effort that aims at unifying models that can describe both separated and disperse two-phase flows, coupling small-scale modelling with large-scale resolution.

JFM classification

Drops and Bubbles: Bubble dynamics Mathematical Foundations: Hamiltonian theory Multiphase and Particle-laden Flows: Gas/liquid flow
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 876 , 10 October 2019 , pp. 326 - 355
Copyright
© 2019 Cambridge University Press 

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

Allaire, G., Clerc, S. & Kokh, S. 2002 A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181, 577–616.Google Scholar
Andrianov, N. & Warnecke, G. 2004 The Riemann problem for the Baer–Nunziato two-phase flow model. J. Comput. Phys. 195 (2), 434–464.Google Scholar
Anez, J., Ahmed, A., Hecht, N., Duret, B., Reveillon, J. & Demoulin, F. X. 2019 Eulerian–Lagrangian spray atomization model coupled with interfacez capturing method for diesel injectors. Intl J. Multiphase Flow 113, 325–342.Google Scholar
Berdichevsky, V. L. 2009 Variational Principles of Continuum Mechanics. Springer.Google Scholar
Bernard-Champmartin, A. & De Vuyst, F. 2014 A low diffusive Lagrange-remap scheme for the simulation of violent air–water free-surface flows. J. Comput. Phys. 274, 19–49.Google Scholar
Berthon, C. & Turpault, R. 2011 Asymptotic preserving HLL schemes. Numer. Meth. Partial Differ. Equ. 27 (6), 1396–1422.Google Scholar
Bouchut, F. 2004 Nonlinear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well-Balanced Schemes for Sources. Birkhäuser.Google Scholar
Buet, C. & Cordier, S. 2007 An asymptotic preserving scheme for hydrodynamics radiative transfer models: numerics for radiative transfer. Numer. Math. 108 (2), 199–221.Google Scholar
Burman, E. & Sainsaulieu, L. 1995 Numerical analysis of two operator splitting methods for an hyperbolic system of conservation laws with stiff relaxation terms. Comput. Meth. Appl. Mech. Engng 128 (3-4), 291–314.Google Scholar
Caflisch, R. E., Jin, S. & Russo, G. 1997 Uniformly accurate schemes for hyperbolic systems with relaxation. SIAM J. Numer. Anal. 34 (1), 246–281.Google Scholar
Caro, F., Coquel, F., Jamet, D. & Kokh, S. 2006 A simple finite-volume method for compressible isothermal two-phase flows simulation. Intl J. Finite Volume 3 (1), hal-01114190.Google Scholar
Chalons, C., Girardin, M. & Kokh, S. 2013 Large time step and asymptotic preserving numerical schemes for the gas dynamics equations with source terms. SIAM J. Sci. Comput. 35 (6), A2874–A2902.Google Scholar
Chanteperdrix, G., Villedieu, P. & Vila, J. 2002 A compressible model for separated two-phase flows computations. In ASME Fluid Eng. Div. Summer Meeting 2002, Volume 1: Fora, Parts A and B: 809–816.Google Scholar
Cheng, L. Y., Drew, D. A. & Lahey, R. T.1983 An analysis of wave dispersion, sonic velocity and critical flow in two-phase mixtures. Tech. Rep. U.S. Nuclear Regulatory Commission.Google Scholar
Cheng, L. Y., Drew, D. A. & Lahey, R. T. 1985 An analysis of wave propagation in bubbly two-component, two-phase flow. Trans. ASME J. Heat Transfer 107, 402–408.Google Scholar
Chern, I.-L., Glimm, J., Mcbryan, O., Plohr, B. & Yaniv, S. 1986 Front tracking for gas dynamics. J. Comput. Phys. 62 (1), 83–110.Google Scholar
Commander, K. W. & Prosperetti, A. 1989 Linear pressure waves in bubbly liquids: comparison between theory and experiments. J. Acoust. Soc. Am. 85 (2), 732–746.Google Scholar
Cordesse, P., Kokh, S., Di Battista, R., Drui, F. & Massot, M. 2018 Derivation of a two-phase flow model with two-scale kinematics, geometric variables and surface tension by mean of variational calculus. In NASA Technical Memorandum, Proceedings of the 2018 Summer Program, NASA Ames Research Center, pp. 1–12. (Submitted).Google Scholar
Drew, D. A. 1983 Mathematical modeling of two-phase flow. Annu. Rev. Fluid Mech. 15, 261–291.Google Scholar
Drew, D. A. 1990 Evolution of geometric statistics. SIAM J. Appl. Maths 50 (3), 649–666.Google Scholar
Drew, D. A. & Passman, S. L. 1999 Theory of multicomponent fluids. In Applied Mathematical Sciences, vol. 135. Springer.Google Scholar
Dumbser, M., Enaux, C. & Toro, E. F. 2008 Finite volume schemes of very high order of accuracy for stiff hyperbolic balance laws. J. Comput. Phys. 227 (8), 3971–4001.Google Scholar
Essadki, M., de Chaisemartin, S., Laurent, F. & Massot, M. 2018 High order moment model for polydisperse evaporating sprays towards interfacial geometry. SIAM J. Appl. Maths 78, 2003–2027.Google Scholar
Essadki, M., Drui, F., de Chaisemartin, S., Larat, A., Ménard, T. & Massot, M.2019 Statistical modeling of the gas–liquid interface using geometrical variables: toward a unified description of the disperse and separated phase flows. in revision for IJMF, available on Hal https://hal.archives-ouvertes.fr/hal-01615076, pp. 1–40.Google Scholar
Fedkiw, R. P., Aslam, T., Merriman, B. & Osher, S. 1999 A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152 (2), 457–492.Google Scholar
Finlayson, B. A. 1972 The method of weighted residuals and variational principles: with application in fluid mechanics, heat and mass transfer. In Mathematics in Science and Engineering, vol. 87. Academic Press.Google Scholar
Fiorina, B., Vié, A., Franzelli, B., Darabiha, N., Massot, M., Dayma, G., Dagaut, P., Moureau, V., Vervisch, L., Berlemont, A. et al. 2016 Modeling challenges in computing aeronautical combustion chambers. Aerosp. Lab 11, 19.Google Scholar
Fleau, S., Mimouni, S., Mérigoux, N. & Vincent, S. 2015 Simulations of two-phase flows with a multifield approach. In Proceedings of CHT-15.Google Scholar
Gallice, G. 2003 Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates. Numer. Math. 94, 673–713.Google Scholar
Gallouet, T., Hérard, J. M. & Seguin, N. 2004 Numerical modeling of two-phase flows using the two-fluid two-pressure approach. Math. Models Meth. Appl. Sci. 14 (15), 663–700.Google Scholar
Gavrilyuk, S. 2012 Multiphase flow modeling via Hamilton’s principle. In Variational Models and Methods in Solid and Fluid Mechanics (ed. dell’Isola, F. & Gavrilyuk, S.), CISM Courses and Lectures, vol. 535, pp. 163–210. Springer.Google Scholar
Gavrilyuk, S. & Gouin, H. 1999 A new form of governing equations of fluids arising from Hamilton’s principle. Intl J. Engng Sci. 37, 1495–1520.Google Scholar
Gavrilyuk, S. & Saurel, R. 2002 Mathematical and numerical modeling of two-phase compressible flows with micro-inertia. J. Comput. Phys. 175, 326–360.Google Scholar
Godlewski, E. & Raviart, P.-A. 1996 Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer.Google Scholar
Gosse, L. & Toscani, G. 2004 Asymptotic-preserving and well-balanced schemes for radiative transfer and the Rosseland approximation. Numer. Math. 98 (2), 223–250.Google Scholar
Grenier, N., Vila, J.-P. & Villedieu, P. 2013 An accurate low-Mach scheme for a compressible two-fluid model applied to free-surface flows. J. Comput. Phys. 252, 1–19.Google Scholar
Hairer, E. & Wanner, G. 1996 Solving Ordinary Differential Equations II: Sti and Dierential-Algebraic Problems, 2nd edn. Springer.Google Scholar
Hérard, J.-M. & Hurisse, O. 2005 A simple method to compute standard two-fluid models. Intl J. Comput. Fluid Dyn. 19, 475–482.Google Scholar
Hirt, C. W. & Nichols, B. D. 1981 Volume of fluid (VOF) method for the dynamics of free boundaries. J. Comput. Phys. 39, 201–225.Google Scholar
Ishii, M. 1975 Thermo-Fluid Dynamics Theory of Two-Phase Flow. Eyrolles.Google Scholar
Jin, S. 1999 Efficient asymptotic-preserving (AP) schemes for some multiscale kinetic equations. SIAM J. Sci. Comput. 21 (2), 441–454.Google Scholar
Kah, D.2010 Taking into account polydispersity for the modeling of liquid fuel injection in internal combustion engines. PhD thesis, Ecole Centrale Paris, https://tel.archives-ouvertes.fr/tel-00618786/en.Google Scholar
Kedelty, D., Uglietta, J. & Herrmann, M. 2018 Sub-filter turbulent velocity reconstruction for a volume-of-fluid dual-scale approach at turbulent phase interface dynamics. In Bulletin of the American Physical Society, APS-DFD Conference.Google Scholar
Kokh, S. & Lagoutiére, F. 2010 An anti-diffusive numerical scheme for the simulation of interfaces between compressible fluids by means of a five-equation model. J. Comput. Phys. 229 (8), 2773–2809.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1966 Fluid Mechanics, 3rd edn. Course of Theoretical Physics, vol. 6. Pergamon Press.Google Scholar
Le Chenadec, V.2012 A stable and conservative framework for detailed numerical simulation of primary atomization. PhD thesis, Stanford University, Stanford, CA.Google Scholar
Le Touze, C.2015 Couplage entre modéles diphasiques á ‘phases séparées’ et á ‘phase dispersée’ pour la simulation de l’atomisation primaire en combustion cryotechnique. PhD thesis, Université Nice Sophia Antipolis, https://tel.archives-ouvertes.fr/tel-01250527/en.Google Scholar
Leroy, V., Strybulevych, A., Page, J. H. & Scanlon, M. G. 2008 Sound velocity and attenuation in bubbly gels measured by transmission experiments. J. Acoust. Soc. Am. 123 (4), 1931–1940.Google Scholar
Lhuillier, D., Chang, C.-H. & Theofanous, T. G. 2013 On the quest for a hyperbolic effective-field model of disperse flows. J. Fluid Mech. 731, 184–194.Google Scholar
Lhuillier, D., Theofanous, T. G. & Liou, M.-S. 2010 Handbook of Nuclear Engineering, vol. 1. Springer.Google Scholar
Ling, Y., Fuster, D., Zaleski, S. & Tryggvason, G. 2017 Spray formation in a quasiplanar gas–liquid mixing layer at moderate density ratios: a numerical closeup. Phys. Rev. Fluids 2, 014005.Google Scholar
Liu, T. G., Khoo, B. C. & Wang, C. W. 2005 The ghost fluid method for compressible gas–water simulation. J. Comput. Phys. 204 (1), 193–221.Google Scholar
Lord Rayleigh, O. M. F. R. S. 1917 On the pressure developed in a liquid during the collapse of a spherical cavity. Phil. Mag. 6 (34), 94–98.Google Scholar
Marchisio, D. L. & Fox, R. O. 2005 Solution of population balance equations using the direct quadrature method of moments. J. Aero. Sci. 36 (1), 43–73.Google Scholar
Masi, E., Simonin, O. & Bédat, B. 2011 The mesoscopic Eulerian approach for evaporating droplets interacting with turbulent flows. Flow Turbul. Combust. 86 (3), 563–583.Google Scholar
Masi, E., Simonin, O., Riber, E., Sierra, P. & Gicquel, L. Y. M. 2014 Development of an algebraic-closure-based moment method for unsteady Eulerian simulations of particle-laden turbulent flows in very dilute regime. Intl J. Multiphase Flow 58, 257–278.Google Scholar
Ménard, T., Tanguy, S. & Berlemont, A. 2007 Coupling level set/VOF/ghost fluid methods: validation and application to 3D simulation of the primary break-up of a liquid jet. Intl J. Multiphase Flow 33 (2), 510–524.Google Scholar
Morel, C. 2015 Mathematical Modeling of Disperse Two-Phase Flows, Fluid Mechanics and its Applications, vol. 114. Springer.Google Scholar
Mulder, W., Osher, S. & Sethian, J. A. 1992 Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100, 209–228.Google Scholar
Ndjinga, M. 2007 Influence of interfacial pressure on the hyperbolicity of the two-fluid model. C. R. Mathematique 344 (6), 407–412.Google Scholar
Osher, S. & Fedkiw, R. P. 2001 Level set methods: an overview and some recent results. J. Comput. Phys. 169 (2), 463–502.Google Scholar
Panicker, N., Passalacqua, A. & Fox, R. O. 2018 On the hyperbolicity of the two-fluid model for gas–liquid bubbly flows. Appl. Math. Model. 57, 432–447.Google Scholar
Plesset, M. S. & Prosperetti, A. 1977 Bubble dynamics and cavitation. Annu. Rev. Fluid Mech. 9 (1), 145–185.Google Scholar
Prosperetti, A. 1977 Thermal effects and damping mechanisms in the forced radial oscillations of gas bubbles in liquids. J. Acoust. Soc. Am. 61 (1), 17–27.Google Scholar
Reveillon, J. & Vervisch, L. 2005 Analysis of weakly turbulent dilute-spray flames and spray combustion regimes. J. Fluid Mech. 537, 317–347.Google Scholar
Sabat, M., Vié, A., Larat, A. & Massot, M. 2019 Statistical description of turbulent particle-laden flows in the very dilute regime using the anisotropic Gaussian moment method. Intl J. Multiphase Flow 112, 243–257.Google Scholar
Saurel, R. & Abgrall, R. 1999 A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467.Google Scholar
Saurel, R. & Pantano, C. 2018 Diffuse-interface capturing methods for compressible two-phase flows. Annu. Rev. Fluid Mech. 50 (1), 105–130.Google Scholar
Saurel, R., Petitpas, F. & Berry, R. A. 2009 Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures. J. Comput. Phys. 228 (5), 1678–1712.Google Scholar
Scardovelli, R. & Zaleski, S. 1999 Direct numerical simulation of free-surface and interfacial flow. Annu. Rev. Fluid Mech. 31, 567–603.Google Scholar
Serre, D.2007 Systems of conservation laws with dissipation. Lecture notes of the S.I.S.S.A. course, Trieste.Google Scholar
Sha, W. T. & Soo, S. L. 1979 On the effect of P𝜵𝛼 term in multiphase mechanics. Intl J. Multiphase Flow 5 (2), 153–158.Google Scholar
Silberman, E. 1957 Sound velocity and attenuation in bubbly mixtures measured in standing wave tubes. J. Acoust. Soc. Am. 29, 925–933.Google Scholar
Stuhmiller, J. H. 1977 The influence of interfacial pressure on the character of two-phase flow model equations. Intl J. Multiphase Flow 3, 551–560.Google Scholar
Sursock, J.-P. 1982 Causality violation of complex-characteristic two-phase flow equations. Intl J. Multiphase Flow 8 (3), 291–295.Google Scholar
Temkin, S. 2005 Suspension Acoustics – An Introduction to the Physics of Suspensions. Cambridge University Press.Google Scholar
Terashima, H. & Tryggvason, G. 2010 A front-tracking method with projected interface conditions for compressible multi-fluid flows. Comput. Fluids 39 (10), 1804–1814.Google Scholar
Tryggvason, G., Bunner, B., Esmaeeli, A., Juric, D., Al-Rawahi, N., Tauber, W., Han, J., Nas, S. & Jan, Y.-J. 2001 A front-tracking method for the computations of multiphase flow. J. Comput. Phys. 169, 708–759.Google Scholar
Vaudor, G., Ménard, T., Aniszewski, W., Doring, M. & Berlemont, A. 2017 A consistent mass and momentum flux computation method for two phase flows. Application to atomization process. Comput. Fluids 152, 204–216.Google Scholar
Vié, A., Laurent, F. & Massot, M. 2013 Size-velocity correlations in high order moment methods for polydisperse evaporating sprays: modeling and numerical issues. J. Comput. Phys. 237, 277–310.Google Scholar
Vincent, S., Larocque, J., Lacanette, D., Toutant, A., Lubin, P. & Sagaut, P. 2008 Numerical simulation of phase separation and a priori two-phase LES filtering. Comput. Fluids 37 (7), 898–906.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley-Interscience.Google Scholar
van Wijngaarden, L. 1972 One dimensional flow of liquids containing small gas bubbles. Annu. Rev. Fluid Mech. 4, 369–396.Google Scholar
Williams, F. A. 1958 Spray combustion and atomization. Phys. Fluids 1 (6), 541–545.Google Scholar
Williams, F. A. 1985 Combustion Theory: The Fundamental Theory of Chemically Reacting Flow Systems, 2nd edn. Combustion Science and Engineering Series. Benjamin/Cummings Pub. Co.Google Scholar
Yuan, C., Laurent, F. & Fox, R. 2012 An extended quadrature method of moments for population balance equations. Atomiz. Sprays 51 (1), 1–23.Google Scholar