Hybrid Finite-Volume-Particle Method for Dusty Gas Flows
The SMAI Journal of computational mathematics, Volume 3 (2017), pp. 139-180.

We first study the one-dimensional dusty gas flow modeled by the two-phase system composed of a gaseous carrier (gas phase) and a particulate suspended phase (dust phase). The gas phase is modeled by the compressible Euler equations of gas dynamics and the dust phase is modeled by the pressureless gas dynamics equations. These two sets of conservation laws are coupled through source terms that model momentum and heat transfers between the phases. When an Eulerian method is adopted for this model, one can notice the obtained numerical results are typically significantly affected by numerical diffusion. This phenomenon occurs since the pressureless gas equations are nonstrictly hyperbolic and have degenerate structure in which singular delta shocks are formed, and these strong singularities are vulnerable to the numerical diffusion.

We introduce a low dissipative hybrid finite-volume-particle method in which the compressible Euler equations for the gas phase are solved by a central-upwind scheme, while the pressureless gas dynamics equations for the dust phase are solved by a sticky particle method. The obtained numerical results demonstrate that our hybrid method provides a sharp resolution even when a relatively small number of particle is used.

We then extend the hybrid finite-volume-particle method to the three-dimensional dusty gas flows with axial symmetry. In the studied model, gravitational effects are taken into account. This brings an additional level of complexity to the development of the finite-volume-particle method since a delicate balance between the flux and gravitational source terms should be respected at the discrete level. We test the proposed method on a number of numerical examples including the one that models volcanic eruptions.

Published online:
DOI: 10.5802/smai-jcm.23
Classification: 65M08, 76M12, 76M28, 86-08, 76M25, 35L65
Keywords: Two-phase dusty gas flow model, three-dimensional axial symmetry, compressible Euler equations, pressureless gas dynamics, finite-volume-particle methods, central-upwind schemes, sticky particle methods, operator splitting methods

Alina Chertock 1; Shumo Cui 2; Alexander Kurganov 3

1 Department of Mathematics, North Carolina State University, Raleigh, NC 27695, USA
2 Department of Mathematics, Temple University, Philadelphia, PA 19122, USA
3 Department of Mathematics, Southern University of Science and Technology of China, Shenzhen, 518055, China and Mathematics Department, Tulane University, New Orleans, LA 70118, USA
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{SMAI-JCM_2017__3__139_0,
     author = {Alina Chertock and Shumo Cui and Alexander Kurganov},
     title = {Hybrid {Finite-Volume-Particle} {Method} for {Dusty} {Gas} {Flows}},
     journal = {The SMAI Journal of computational mathematics},
     pages = {139--180},
     publisher = {Soci\'et\'e de Math\'ematiques Appliqu\'ees et Industrielles},
     volume = {3},
     year = {2017},
     doi = {10.5802/smai-jcm.23},
     zbl = {1416.76143},
     mrnumber = {3707729},
     language = {en},
     url = {https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.23/}
}
TY  - JOUR
AU  - Alina Chertock
AU  - Shumo Cui
AU  - Alexander Kurganov
TI  - Hybrid Finite-Volume-Particle Method for Dusty Gas Flows
JO  - The SMAI Journal of computational mathematics
PY  - 2017
SP  - 139
EP  - 180
VL  - 3
PB  - Société de Mathématiques Appliquées et Industrielles
UR  - https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.23/
DO  - 10.5802/smai-jcm.23
LA  - en
ID  - SMAI-JCM_2017__3__139_0
ER  - 
%0 Journal Article
%A Alina Chertock
%A Shumo Cui
%A Alexander Kurganov
%T Hybrid Finite-Volume-Particle Method for Dusty Gas Flows
%J The SMAI Journal of computational mathematics
%D 2017
%P 139-180
%V 3
%I Société de Mathématiques Appliquées et Industrielles
%U https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.23/
%R 10.5802/smai-jcm.23
%G en
%F SMAI-JCM_2017__3__139_0
Alina Chertock; Shumo Cui; Alexander Kurganov. Hybrid Finite-Volume-Particle Method for Dusty Gas Flows. The SMAI Journal of computational mathematics, Volume 3 (2017), pp. 139-180. doi : 10.5802/smai-jcm.23. https://smai-jcm.centre-mersenne.org/articles/10.5802/smai-jcm.23/

[1] N. Botta; R. Klein; S. Langenberg; S. Lützenkirchen Well balanced finite volume methods for nearly hydrostatic flows, J. Comput. Phys., Volume 196 (2004) no. 2, pp. 539-565 | DOI | MR | Zbl

[2] F. Bouchut On zero pressure gas dynamics, Advances in kinetic theory and computing (Ser. Adv. Math. Appl. Sci.), Volume 22, World Sci. Publ., River Edge, NJ, 1994, pp. 171-190 | MR | Zbl

[3] F. Bouchut; F. James Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. Partial Differential Equations, Volume 24 (1999) no. 11-12, pp. 2173-2189 | DOI | MR | Zbl

[4] F. Bouchut; S. Jin; X. Li Numerical approximations of pressureless and isothermal gas dynamics, SIAM J. Numer. Anal., Volume 41 (2003), pp. 135-158 | DOI | MR | Zbl

[5] Y. Brenier; E. Grenier Sticky particles and scalar conservation laws, SIAM J. Numer. Anal., Volume 35 (1998), pp. 2317-2328 | DOI | MR | Zbl

[6] S. Carcano; L. Bonaventura; T. Esposti Ongaro; A. Neri A semi-implicit, second order accurate numerical model for multiphase underexpanded volcanic jets, Geosci. Model Dev. Discuss., Volume 6 (2013) no. 1, pp. 399-452 | DOI

[7] P. Chandrashekar; C. Klingenberg A second order well-balanced finite volume scheme for Euler equations with gravity, SIAM J. Sci. Comput., Volume 37 (2015) no. 3, p. B382-B402 | DOI | MR | Zbl

[8] G.-Q. Chen; H. Liu Formation of δ-shocks and vacuum states in the vanishing pressure limit of solutions to the Euler equations for isentropic fluids, SIAM J. Math. Anal., Volume 34 (2003), pp. 925-938 | DOI | MR | Zbl

[9] A. Chertock; S. Cui; A. Kurganov; Ş. N. Özcan; E. Tadmor Well-balanced central-upwind schemes for the Euler equations with gravitation (Submitted)

[10] A. Chertock; S. Cui; A. Kurganov; T. Wu Well-balanced positivity preserving central-upwind scheme for the shallow water system with friction terms, Internat. J. Numer. Meth. Fluids, Volume 78 (2015), pp. 355-383 | DOI | MR

[11] A. Chertock; A. Kurganov On a hybrid finite-volume particle method, M2AN Math. Model. Numer. Anal, Volume 38 (2004), pp. 1071-1091 | DOI | Numdam | MR | Zbl

[12] A. Chertock; A. Kurganov On a practical implementation of particle methods, Appl. Numer. Math., Volume 56 (2006), pp. 1418-1431 | DOI | MR | Zbl

[13] A. Chertock; A. Kurganov; G. Petrova Finite-volume-particle methods for models of transport of pollutant in shallow water, J. Sci. Comput., Volume 27 (2006), pp. 189-199 | DOI | MR | Zbl

[14] A. Chertock; A. Kurganov; Yu. Rykov A new sticky particle method for pressureless gas dynamics, SIAM J. Numer. Anal., Volume 45 (2007), pp. 2408-2441 | DOI | MR | Zbl

[15] G.-H. Cottet; P. D. Koumoutsakos Vortex methods, Cambridge University Press, Cambridge, 2000

[16] S. Dartevelle; W. Rose; J. Stix; K. Kelfoun; J.W. Vallance Numerical modeling of geophysical granular flows: 2. Computer simulations of plinian clouds and pyroclastic flows and surges, Geochem. Geophys. Geosyst., Volume 5 (2004) no. 8 | DOI

[17] V. Desveaux; M. Zenk; C. Berthon; C. Klingenberg A well-balanced scheme for the Euler equation with a gravitational potential, Finite volumes for complex applications. VII. Methods and theoretical aspects (Springer Proc. Math. Stat.), Volume 77, Springer, Cham, 2014, pp. 217-226 | MR | Zbl

[18] F. Dobran; A. Neri; G. Macedonio Numerical simulation of collapsing volcanic columns, J. Geophys. Res., Volume 98 (1993), pp. 4231-4259 | DOI

[19] J. Dufek; G. W. Bergantz Dynamics and deposits generated by the Kos Plateau Tuff eruption: Controls of basal particle loss on pyroclastic flow transport, Geochem. Geophys. Geosyst., Volume 8 (2007) no. 12 | DOI

[20] W. E; Yu. G. Rykov; Ya. G. Sinai Generalized variational principles, global weak solutions and behavior with random initial data for systems of conservation laws arising in adhesion particle dynamics, Comm. Math. Phys., Volume 177 (1996), pp. 349-380 | DOI | MR | Zbl

[21] B. Einfeld On Godunov-type methods for gas dynamics, SIAM J. Numer. Anal., Volume 25 (1988), pp. 294-318 | DOI | MR

[22] P. Glaister Flux difference splitting for the Euler equations with axial symmetry, J. Engrg. Math., Volume 22 (1988) no. 2, pp. 107-121 | DOI | MR | Zbl

[23] S. Gottlieb; D. Ketcheson; C.-W. Shu Strong stability preserving Runge-Kutta and multistep time discretizations, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011, xii+176 pages | Zbl

[24] S. Gottlieb; C.-W. Shu; E. Tadmor Strong stability-preserving high-order time discretization methods, SIAM Rev., Volume 43 (2001), pp. 89-112 | DOI | MR | Zbl

[25] M. Gurris; D. Kuzmin; S. Turek Finite element simulation of compressible particle-laden gas flows, J. Comput. Appl. Math., Volume 233 (2010) no. 12, pp. 3121-3129 | DOI | MR | Zbl

[26] S. Hank; R. Saurel; O. Le Metayer A hyperbolic Eulerian model for dilute two-phase suspensions, Journal of Modern Physics, Volume 2 (2011), pp. 997-1011 | DOI

[27] A. Harlow; A. A. Amsden Numerical calculation of multiphase fluid flow, J. Comput. Phys., Volume 17 (1975), pp. 19-52 | DOI | Zbl

[28] A. Harten; P. Lax; B. van Leer On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., Volume 25 (1983), pp. 35-61 | DOI | MR | Zbl

[29] A. Kurganov; C.-T. Lin On the reduction of numerical dissipation in central-upwind schemes, Commun. Comput. Phys., Volume 2 (2007), pp. 141-163 | MR | Zbl

[30] A. Kurganov; S. Noelle; G. Petrova Semi-discrete central-upwind scheme for hyperbolic conservation laws and Hamilton-Jacobi equations, SIAM J. Sci. Comput., Volume 23 (2001), pp. 707-740 | DOI

[31] A. Kurganov; G. Petrova A second-order well-balanced positivity preserving central-upwind scheme for the Saint-Venant system, Commun. Math. Sci., Volume 5 (2007), pp. 133-160 | DOI | MR | Zbl

[32] A. Kurganov; E. Tadmor New high resolution central schemes for nonlinear conservation laws and convection-diffusion equations, J. Comput. Phys., Volume 160 (2000), pp. 241-282 | DOI | MR | Zbl

[33] A. Kurganov; E. Tadmor Solution of two-dimensional Riemann problems for gas dynamics without Riemann problem solvers, Numer. Methods Partial Differential Equations, Volume 18 (2002), pp. 584-608 | DOI | MR | Zbl

[34] M.-C. Lai; C. S. Peskin An immersed boundary method with formal second-order accuracy and reduced numerical viscosity, J. Comput. Phys., Volume 160 (2000), pp. 705-719 | DOI | MR | Zbl

[35] R. J. LeVeque The dynamics of pressureless dust clouds and delta waves, J. Hyperbolic Differ. Equ., Volume 1 (2004), pp. 315-327 | DOI | MR | Zbl

[36] R. J. LeVeque; D. S. Bale Wave propagation methods for conservation laws with source terms, Hyperbolic problems: theory, numerics, applications, Vol. II (Zürich, 1998) (Internat. Ser. Numer. Math.), Volume 130, Birkhäuser, Basel, 1999, pp. 609-618 | MR | Zbl

[37] G. Li; Y. Xing High order finite volume WENO schemes for the Euler equations under gravitational fields, J. Comput. Phys., Volume 316 (2016), pp. 145-163 | DOI | MR | Zbl

[38] K.-A. Lie; S. Noelle On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws, SIAM J. Sci. Comput., Volume 24 (2003) no. 4, pp. 1157-1174 | DOI | MR | Zbl

[39] J. Luo; K. Xu; N. Liu A well-balanced symplecticity-preserving gas-kinetic scheme for hydrodynamic equations under gravitational field, SIAM J. Sci. Comput., Volume 33 (2011) no. 5, pp. 2356-2381 | DOI | MR | Zbl

[40] H. Miura; I. I. Glass On a dusty-gas shock tube, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Volume 382 (1982) no. 1783, pp. 373-388 | DOI

[41] A. Neri; T. E. Ongaro; G. Macedonio; D Gidaspow Multiparticle simulation of collapsing volcanic columns and pyroclastic flow, J. Geophys. Res, Volume 108 (2003), pp. 1-22 | DOI

[42] H. Nessyahu; E. Tadmor Nonoscillatory central differencing for hyperbolic conservation laws, J. Comput. Phys., Volume 87 (1990) no. 2, pp. 408-463 | DOI | MR | Zbl

[43] B. Nilsson; O. S. Rozanova; V. M. Shelkovich Mass, momentum and energy conservation laws in zero-pressure gas dynamics and δ-shocks: II, Appl. Anal., Volume 90 (2011) no. 5, pp. 831-842 | DOI | MR | Zbl

[44] B. Nilsson; V. M. Shelkovich Mass, momentum and energy conservation laws in zero-pressure gas dynamics and delta-shocks, Appl. Anal., Volume 90 (2011) no. 11, pp. 1677-1689 | DOI | MR | Zbl

[45] M. Pelanti; R. J. Leveque High-resolution finite volume methods for dusty gas jets and plumes, SIAM J. Sci. Comput., Volume 28 (2006), pp. 1335-1360 | DOI | MR | Zbl

[46] C. S. Peskin The immersed boundary method, Acta Numer., Volume 11 (2002), pp. 479-517 | DOI | MR | Zbl

[47] P.-A. Raviart An analysis of particle methods, Numerical methods in fluid dynamics (Como, 1983) (Lecture Notes in Math.), Volume 1127, Springer, Berlin, 1985, pp. 243-324 | MR

[48] Yu. G. Rykov Propagation of singularities of shock wave type in a system of equations of two-dimensional pressureless gas dynamics, Mat. Zametki, Volume 66 (1999), p. 760-769 (Russian); translation in Math. Notes 66 (1999), pp. 628–635 (2000) | MR | Zbl

[49] Yu. G. Rykov On the nonhamiltonian character of shocks in 2-D pressureless gas, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., Volume (8) 5 (2002), pp. 55-78 | MR | Zbl

[50] T. Saito Numerical analysis of dusty-gas flows, J. Comput. Phys., Volume 176 (2002), pp. 129-144 | DOI | Zbl

[51] B. Shotorban; G. B. Jacobs; O. Ortiz; Q. Truong An Eulerian model for particles nonisothermally carried by a compressible fluid, Int. J. Heat Mass Transfer, Volume 65 (2013), pp. 845-854 | DOI

[52] C.-W. Shu; S. Osher Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., Volume 77 (1988), pp. 439-471 | DOI

[53] G. Strang On the construction and comparison of difference schemes, SIAM J. Numer. Anal., Volume 5 (1968), pp. 506-517 | DOI | MR | Zbl

[54] P. K. Sweby High resolution schemes using flux limiters for hyperbolic conservation laws, SIAM J. Numer. Anal., Volume 21 (1984) no. 5, pp. 995-1011 | DOI | MR | Zbl

[55] C. T. Tian; K. Xu; K. L. Chan; L. C. Deng A three-dimensional multidimensional gas-kinetic scheme for the Navier-Stokes equations under gravitational fields, J. Comput. Phys., Volume 226 (2007) no. 2, pp. 2003-2027 | DOI | MR | Zbl

[56] R. Touma; U. Koley; C. Klingenberg Well-balanced unstaggered central schemes for the Euler equations with gravitation, SIAM J. Sci. Comput., Volume 38 (2016) no. 5, p. B773-B807 | DOI | MR | Zbl

[57] G. Valentine; K. Wohletz Numerical models of Plinian eruption columns and pyroclastic, J. Geophys. Res, Volume 94 (1989), pp. 1867-1887 | DOI

[58] B. van Leer Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys., Volume 32 (1979) no. 1, pp. 101-136 | DOI | Zbl

[59] K. H. Wohletz; T. R. McGetchin; M. T. Sandford II; E. M. Jones Hydrodynamic aspects of caldera-forming eruptions: numerical models, J. Geophys. Res, Volume 89 (1984), pp. 8269-8285 | DOI

[60] Y. Xing; C.-W. Shu High order well-balanced WENO scheme for the gas dynamics equations under gravitational fields, J. Sci. Comput., Volume 54 (2013) no. 2-3, pp. 645-662 | DOI | MR | Zbl

[61] K. Xu; J. Luo; S. Chen A well-balanced kinetic scheme for gas dynamic equations under gravitational field, Adv. Appl. Math. Mech., Volume 2 (2010), pp. 200-210 | MR

[62] M. Yuen Some exact blowup solutions to the pressureless Euler equations in N , Commun. Nonlinear Sci. Numer. Simul., Volume 16 (2011) no. 8, pp. 2993-2998 | DOI | MR | Zbl

Cited by Sources: