Multivariate Gaussian Extended Quadrature Method of Moments for Turbulent Disperse Multiphase Flow


The present contribution introduces a fourth-order moment formalism for particle trajectory crossing (PTC) in the framework of multiscale modeling of disperse multiphase flow. In our previous work, the ability to treat PTC was examined with direct-numerical simulations using either quadrature reconstruction based on a sum of Dirac delta functions denoted as quadrature-based moment methods (QBMM) in order to capture large scale trajectory crossing, or by using low-order hydrodynamics closures in the Levermore hierarchy denoted as kinetic-based moment methods (KBMM) in order to capture small-scale trajectory crossing. Whereas KBMM leads to well-posed PDEs and has a hard time capturing large-scale trajectory crossing for particles with enough inertia, QBMM based on a discrete reconstruction suffers from singularity formation and requires too many moments in order to capture the effect of PTC at both the small scale and the large scale both to small-scale turbulence as well as free transport coupled to drag in an Eulerian mesoscale framework. The challenge addressed in this work is thus twofold: first, to propose a new generation of method at the interface between QBMM and KBMM with less singular behavior and the associated proper mathematical properties, which is able to capture both small-scale and large-scale trajectory crossing, and second to limit the number of moments used for applicability in two-dimensional (2-D) and 3-D configurations without losing too much accuracy in the representation of spatial fluxes. In order to illustrate its numerical properties, the proposed Gaussian extended quadrature method of moments is applied to solve 1-D and 2-D kinetic equations representing finite-Stokes-number particles in a known turbulent fluid flow.

MSC codes

  1. kinetic equation
  2. multiphase flow
  3. quadrature-based moment methods
  4. kinetic-based moment methods
  5. particle trajectory crossing
  6. hyperbolic conservation laws

MSC codes

  1. 76T10
  2. 76N15
  3. 35L65
  4. 65D32
  5. 65M08
  6. 76M12
  7. 82C40

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A. Bardow, I. V. Karlin, and A. A. Guzev, Multispeed models in off-lattice Boltzmann simulations, Phys. Rev. E, 77 (2008), 025701(R).
J.-D. Benamou, Big ray tracing: Multivalued travel time field computation using viscosity solutions of the eikonal equation, J. Comput. Phys., 128 (1996), pp. 463--474.
S. Benyahia, H. Arastoopour, T. M. Knowlton, and H. Massah, Simulation of particles and gas flow behavior in the riser section of a circulating fluidized bed using the kinetic theory approach for the particulate phase, Powder Technol., 112 (2000), pp. 24--33.
A. E. Beylich, Solving the kinetic equation for all Knudsen numbers, Phys. Fluids, 12 (2000), pp. 444--465.
G. A. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford University Press, Oxford, UK, 1994.
Z. Bouali, C. Pera, and J. Reveillon, Numerical analysis of the influence of two-phase flow mass and heat transfers on n-heptane autoignition, Combustion Flame, 159 (2012), pp. 2056--2068.
F. Bouchut, On zero pressure gas dynamics, in Advances in Kinetic Theory and Computing, Ser. Adv. Math. Appl. Sci. 22, World Scientific River Edge, NJ, 1994, pp. 171--190.
F. Bouchut, S. Jin, and X. T. Li, Numerical approximations of pressureless gas and isothermal gas dynamics, SIAM J. Numer. Anal., 41 (2003), pp. 135--158.
Y. Brenier and L. Corrias, A kinetic formulation for multibranch entropy solutions of scalar conservation laws, Ann. Inst. H. Poincaré, 15 (1998), pp. 169--190.
J. J. Brey, J. W. Dufty, C.-S. Kim, and A. Santos, Hydrodynamics for granular flow at low density, Phys. Rev. E, 58 (1998), p. 4638.
J. J. Brey, M. J. Ruiz-Montero, and D. Cubero, Homogeneous cooling state of a low-density granular flow, Phys. Rev. E, 54 (1996), pp. 3664--3671.
J. E. Broadwell, Shock structure in a simple discrete velocity gas, Phys. Fluids, 7 (1964), pp. 1243--1247.
Z. Cai, Y. Fan, and R. Li, Globally hyperbolic regularization of Grad's moment system, Comm. Pure Appl. Math., 67 (2014), pp. 464--518.
J. A. Carrillo, A. Majorana, and F. Vecil, A semi-Lagrangian deterministic solver for the semiconductor Boltzmann-Poisson system, Commun. Comput. Phys., 2 (2007), pp. 1027--1054.
C. Cercignani, The Boltzmann Equation and Its Applications, Springer, New York, 1988.
C. Cercignani, Rarefied Gas Dynamics, Cambridge University Press, Cambridge, UK, 2000.
C. Chalons, R. O. Fox, and M. Massot, 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, Stanford, CA, 2010, pp. 347--358.
C. Chalons, D. Kah, and M. Massot, Beyond pressureless gas dynamics: Quadrature-based velocity moment models, Commun. Math. Sci., 10 (2012), pp. 1241--1272.
S. Chapman and T. G. Cowling, The Mathematical Theory of Non-uniform Gases, 2nd ed., Cambridge University Press, Cambridge, UK, 1961.
G.-Q. Chen and H. Liu, Formation of $\delta$-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., 34 (2003), pp. 925--938.
Y. Cheng and J. A. Rossmanith, A class of quadrature-based moment-closure methods with application to the Vlasov-Poisson-Fokker-Planck system in the high-field limit, J. Comput. Appl. Math., 262 (2014), pp. 384--398.
S. de Chaisemartin, Eulerian Models and Numerical Simulation of Turbulent Dispersion for Polydisperse Evaporating Sprays, Ph.D. thesis, Ecole Centrale Paris, France, 2009;
S. de Chaisemartin, L. Fréret, D. Kah, F. Laurent, R. O. Fox, J. Reveillon, and M. Massot, Eulerian models for turbulent spray combustion with polydispersity and droplet crossing, C. R. Mécanique, 337 (2009), pp. 438--448.
O. Desjardins, R. O. Fox, and P. Villedieu, A quadrature-based moment method for dilute fluid-particle flows, J. Comput. Phys., 227 (2008), pp. 2514--2539.
H. Dette and W. J. Studden, The Theory of Canonical Moments with Applications in Statistics, Probability, and Analysis, Wiley Ser. Probab. Statist., John Wiley & Sons, New York, 1997.
G. Dimarco, Q. Li, L. Pareschi, and B. Yan, Numerical methods for plasma physics in collisional regimes, J. Plasma Phys., 81 (2015).
J. K. Dukowicz, A particle--fluid numerical model for liquid sprays, J. Comput. Phys., 35 (1980), pp. 229--253.
B. Engquist and O. Runborg, Multiphase computations in geometrical optics, J. Comput. Appl. Math., 74 (1996), pp. 175--192.
R. O. Fox, A quadrature-based third-order moment method for dilute gas-particle flow, J. Comput. Phys., 227 (2008), pp. 6313--6350.
R. O. Fox, Optimal moment sets for multivariate direct quadrature methods of moments, Industrial Engineering Chemistry Res., 48 (2009), pp. 6313--6350.
R. O. Fox, Large-eddy-simulation tools for multiphase flows, Annu. Rev. Fluid Mech., 44 (2012), pp. 47--76.
L. Fréret, O. Thomine, J. Reveillon, S. de Chaisemartin, F. Laurent, and M. Massot, On the role of preferential segregation in flame dynamics in polydisperse evaporating sprays, in Proceedings of the Summer Program 2010, Center for Turbulence Research, Stanford University, Stanford, CA, 2010, pp. 1--10.
U. Frisch, D. d'Humières, B. Hasslacher, P. Lallemand, Y. Pomeau, and J.-P. Rivet, Lattice gas hydrodynamics in two and three dimensions, Complex Systems, 1 (1987), pp. 649--707.
J. E. Galvin, C. M. Hrenya, and R. D. Wildman, On the role of the Knudsen layer in rapid granular flows, J. Fluid Mech., 585 (2007), pp. 73--92.
R. Garg, C. Narayanan, D. Lakehal, and S. Subramaniam, Accurate numerical estimation of interphase momentum transfer in Lagrangian--Eulerian simulations of dispersed two-phase flows, Int. J. Multiphase Flow, 33 (2007), pp. 1337--1364.
R. Garg, C. Narayanan, and S. Subramaniam, A numerically convergent Lagrangian--Eulerian simulation method for dispersed two-phase flows, Int. J. Multiphase Flow, 35 (2009), pp. 376--388.
V. Garzo and J. Dufty, Dense fluid transport for inelastic hard spheres, Phys. Rev. E, 59 (1999), pp. 5895--5911.
R. Gatignol, Théorie cinétique d'un gaz á répartition discrète de vitesses, Lecture Notes in Phys. 36, Springer, Berlin, 1975.
W. Gautschi, Orthogonal Polynomials: Computation and Approximation, Oxford University Press, Oxford, UK, 2004.
D. Gidaspow, Hydrodynamics of fluidization and heat transfer: Supercomputer modeling, Appl. Mech. Rev., 39 (1986), pp. 1--22.
D. Gidaspow, Multiphase Flow and Fluidization, Academic Press, New York, 1994.
I. Goldhirsch, Rapid granular flows, Ann. Rev. Fluid. Mech., 35 (2003), pp. 267--293.
I. Goldhirsch, S. H. Noskowicz, and O. Bar-Lev, Theory of granular gases: Some recent results and some open problems, J. Phys. Condens. Matter, 17 (2005), pp. S2591--S2608.
A. Goldshtein and M. Shapiro, Mechanics of collisional motion of granular materials, 1, General hydrodynamic equations, J. Fluid Mech., 282 (1995), p. 75.
L. Gosse, Using $k$-branch entropy solutions for multivalued geometric optics computations, J. Comput. Phys., 180 (2002), pp. 155--182.
L. Gosse, S. Jin, and X. T. Li, On two moment systems for computing multiphase semiclassical limits of the Schrödinger equation, Math. Models Methods Appl. Sci., 13 (2003), pp. 1689--1723.
H. Grad, On the kinetic theory of rarefied gases, Comm. Pure Appl. Math., 2 (1949), pp. 331--407.
N. G. Hadjiconstantinou, A. L. Garcia, M. Z. Bazant, and G. He, Statistical error in particle simulations of hydrodynamic phenomena, J. Comput. Phys., 187 (2003), pp. 274--297.
C. D. Hauck, C. D. Levermore, and A. L. Tits, Convex duality and entropy-based moment closures: Characterizing degenerate densities, SIAM J. Control Optim., 47 (2008), pp. 1977--2015.
J. T. Jenkins, Kinetic theory for nearly elastic spheres, in Physics of Dry Granular Media, H. J. Hermann, J. P. Hovi, and S. Luding, eds., Kluwer, Dordrecht, the Netherlands, 1998.
J. T. Jenkins and F. Mancini, Balance laws and constitutive relations for plane flows of a dense mixture of smooth, nearly elastic, circular disks, J. Appl. Mech., 130 (1987), pp. 187--202.
J. T. Jenkins and S. B. Savage, A theory for the rapid flow of identical, smooth, nearly elastic, spherical particles, J. Fluid Mech., 130 (1983), pp. 187--202.
S. Jin and X. T. Li, Multi-phase computations of the semiclassical limit of the Schrödinger equation and related problems: Whitham vs Wigner, Phys. D, 182 (2003), pp. 46--85.
S. Jin, H. Liu, S. Osher, and R. Tsai, Computing multi-valued physical observables for the semiclassical limit of the Schrödinger equation, J. Comput. Phys., 205 (2005), pp. 222--241.
M. Junk, Domain of definition of Levermore's five-moment system, J. Stat. Phys., 93 (1998), pp. 1143--1167.
D. Kah, F. Laurent, L. Fréret, S. de Chaisemartin, R. O. Fox, J. Reveillon, and M. Massot, Eulerian quadrature-based moment models for polydisperse evaporating sprays, Flow Turbulence Combustion, 55 (2010), pp. 1--26.
P. L. C. Lage, On the representation of QMOM as a weighted-residual method--the dual-quadrature method of generalized moments, Computers Chemical Engineering, 35 (2011), pp. 2186--2203.
P. Lallemand and L. S. Luo, Theory of the lattice Boltzmann method: Dispersion, dissipation, isotropy, Galilean invariance, and stability, Phys. Rev. E, 68 (2003), 036706.
F. Laurent, A. Vié, C. Chalons, R. O. Fox, and M. Massot, A hierarchy of Eulerian models for trajectory crossing in particle-laden turbulent flows over a wide range of Stokes numbers, in Annual Research Briefs of the Center for Turbulence Research, Stanford University, Stanford, CA, 2012, pp. 193--204.
C. D. Levermore, Moment closure hierarchies for kinetic theories, J. Stat. Phys., 83 (1996), pp. 1021--1065.
X T. Li, J. G. Wöhlbier, S. Jin, and J. H. Booske, Eulerian method for computing multivalued solutions of the Euler--Poisson equations and applications to wave breaking in klystrons, Phys. Rev. E, 70 (2004), 016502.
H. Liu, S. Osher, and R. Tsai, Multi-valued solution and level set methods in computational high frequency wave propagation, Commun. Comput. Phys., 1 (2006), pp. 765--804.
J. G. McDonald and C. P. T. Groth, Extended fluid-dynamic model for micron-scale flows based on Gaussian moment closure, in Proceedings of the 6th AIAA Aerospace Sciences Meeting and Exhibit, 2008, pp. 1--18.
J. G. McDonald and C. P. T. Groth, Towards realizable hyperbolic moment closures for viscous heat-conducting gas flows based on a maximum-entropy distribution, Contin. Mech. Thermodyn., 25 (2012), pp. 573--603.
J. G. McDonald, J. S. Sachdev, and C. P. T. Groth, Application of Gaussian moment closure to microscale flows with moving embedded boundaries, AIAA J., 52 (2014), pp. 1839--1857.
J. G. McDonald and M. Torrilhon, Affordable robust moment closures for CFD based on the maximum-entropy hierarchy, J. Comput. Phys., 251 (2013), pp. 500--523.
T. T. Nguyen, F. Laurent, R. O. Fox, and M. Massot, Solution of population balance equations in applications with fine particles: mathematical modeling and numerical schemes, J. Comput. Phys., 325 (2016), pp. 129--156.
Y. Ogata, H.-N. Im, and T. Yabe, Numerical method for Boltzmann equation with Soroban-grid CIP method, Commun. Comput. Phys., 2 (2007), pp. 760--782.
S. Ogawa, A. Umemura, and N. Oshima, On the equation of fully fluidized granular materials, Z. Angew. Math. Phys., 31 (1980), pp. 483--493.
B. Perthame, Boltzmann type schemes for compressible Euler equations in one and two space dimensions, SIAM J. Numer. Anal., 29 (1990), pp. 1--19.
S. B. Pope, Turbulent Flows, Cambridge University Press, Cambridge, UK, 2000.
J. Reveillon, DNS of Spray Combustion, Dispersion Evaporation and Combustion, CISM Courses and Lectures, 492, Springer, New York, 2007.
J. Reveillon and F. X. Demoulin, Effects of the preferential segregation of droplets on evaporation and turbulent mixing, J. Fluid Mech., 583 (2007), pp. 273--302.
O. Runborg, Some new results in multiphase geometrical optics, Math. Model. Numer. Anal., 34 (2000), pp. 53--66.
O. Runborg, Mathematical models and numerical methods for high frequency waves, Commun. Comput. Phys., 2 (2007), pp. 827--880.
M. Sabat, A. Vié, A. Larat, and M. Massot, Fully Eulerian simulation of 3D turbulent particle laden flow based on the Anisotropic Gaussian closure, in Proceedings of the 9th International Conference on Multiphase Flow, Firenze, Italy, 2016.
R. P. Schaerer and M. Torrilhon, On singular closures for the 5-moment system in kinetic gas theory, Commun. Comput. Phys., 17 (2015), pp. 371--400.
X. Shan, X. F. Yuan, and H. Chen, Kinetic theory representation of hydrodynamics: A way beyond the Navier--Stokes equation, J. Fluid Mech., 550 (2006), pp. 413--441.
J. A. Shohat and J. D. Tamarkin, The Problem of Moments, American Mathematical Society Math. Surveys, Monogr. 2, AMS, New York, 1943.
H. Struchtrup, Macroscopic Transport Equations for Rarefied Gas Flows, Springer, New York, 2005.
H. Struchtrup and M. Torrilhon, Regularization of Grad's 13-moment equations: Derivation and linear analysis, Phys. Fluids, 15 (2003), pp. 266--880.
M. Torrilhon, Characteristic waves and dissipation in the 3-moment-case, Contin. Mech. Thermodyn., 12 (2000).
M. Torrilhon, Hyperbolic moment equations in kinetic gas theory based on multi-variate Pearson---IV. Distributions, Commun. Comput. Phys., 7 (2010), pp. 639--673.
M. Torrilhon and H. Struchtrup, Regularized 13-moment equations: Shock structure calculations and comparison to Burnett models, J. Fluid Mech., 513 (2004), pp. 171--198.
A. Vié, F. Doisneau, and M. Massot, On the Anisotropic Gaussian closure for the prediction of inertial-particle laden flows, Commun. Comput. Phys., 17 (2015), pp. 1--46.
V. Vikas, Z. J. Wang, A. Passalacqua, and R. O. Fox, Realizable high-order finite-volume schemes for quadrature-based moment methods, J. Comput. Phys., 230 (2011), pp. 5328--5352.
J. C. Wheeler, Modified moments and Gaussian quadrature, Rocky Mountain J. Math, 4 (1974), pp. 287--296.
J. G. Wöhlbier, S. Jin, and S. Sengele, Eulerian calculations of wave breaking and multivalued solutions in a traveling wave tube, Phys. Plasmas, 12 (2005), 023106.
C. Yuan and R. O. Fox, Conditional quadrature method of moments for kinetic equations, J. Comput. Phys., 230 (2011), pp. 8216--8246.
C. Yuan, F. Laurent, and R. O. Fox, An extended quadrature method of moments for population balance equations, J. Comput. Phys., 51 (2012), pp. 1--23.

Information & Authors


Published In

cover image Multiscale Modeling & Simulation
Multiscale Modeling & Simulation
Pages: 1553 - 1583
ISSN (online): 1540-3467


Submitted: 1 September 2016
Accepted: 9 June 2017
Published online: 2 November 2017

MSC codes

  1. kinetic equation
  2. multiphase flow
  3. quadrature-based moment methods
  4. kinetic-based moment methods
  5. particle trajectory crossing
  6. hyperbolic conservation laws

MSC codes

  1. 76T10
  2. 76N15
  3. 35L65
  4. 65D32
  5. 65M08
  6. 76M12
  7. 82C40



Funding Information

European Union Seventh Framework Programme : 246556

Funding Information

National Science Foundation : CCF-0830214

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