A Hyperbolic Eulerian Model for Dilute Two-Phase Suspensions

ABSTRACT

Conventional modeling of two-phase dilute suspensions is achieved with the Euler equations for the gas phase and gas dynamics pressureless equations for the dispersed phase, the two systems being coupled by various relaxation terms. The gas phase equations form a hyperbolic system but the particle phase corresponds to a hyperbolic degenerated one. Numerical difficulties are thus present when dealing with the dilute phase system. In the present work, we consider the addition of turbulent effects in both phases in a thermodynamically consistent way. It results in two strictly hyperbolic systems describing phase’s dynamics. Another important feature is that the new model has improved physical capabilities. It is able, for example, to predict particle dispersion, while the conventional approach fails. These features are highlighted on several test problems involving particles jets dispersion and are compared against experimental data. With the help of a single parameter (a turbulent viscosity), excellent agreement is obtained for various experimental configurations studied by different authors

Conventional modeling of two-phase dilute suspensions is achieved with the Euler equations for the gas phase and gas dynamics pressureless equations for the dispersed phase, the two systems being coupled by various relaxation terms. The gas phase equations form a hyperbolic system but the particle phase corresponds to a hyperbolic degenerated one. Numerical difficulties are thus present when dealing with the dilute phase system. In the present work, we consider the addition of turbulent effects in both phases in a thermodynamically consistent way. It results in two strictly hyperbolic systems describing phase’s dynamics. Another important feature is that the new model has improved physical capabilities. It is able, for example, to predict particle dispersion, while the conventional approach fails. These features are highlighted on several test problems involving particles jets dispersion and are compared against experimental data. With the help of a single parameter (a turbulent viscosity), excellent agreement is obtained for various experimental configurations studied by different authors

Cite this paper

nullS. Hank, R. Saurel and O. Metayer, "A Hyperbolic Eulerian Model for Dilute Two-Phase Suspensions,"*Journal of Modern Physics*, Vol. 2 No. 9, 2011, pp. 997-1011. doi: 10.4236/jmp.2011.29120.

nullS. Hank, R. Saurel and O. Metayer, "A Hyperbolic Eulerian Model for Dilute Two-Phase Suspensions,"

References

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[3] P. D. Lax, “Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation,” Commu-nications on Pure and Applied Mathematics, Vol. 7, No. 1, 1954, pp. 159-193.

[4] R. Saurel, E. Daniel and C. Loraud, “Two-Phase Flows: Second-Order Schemes and Boundary Conditions,” AIAA Journal, Vol. 32, No. 6, 1994, pp. 1214-1221. doi:10.2514/3.12122

[5] Y. Brenier and E Grenier, “Sticky Particles and Scalar Conservation Laws,” SIAM Journal on Numerical Analy-sis, Vol. 35, No. 6, 1998, pp. 2317-2328. doi:10.1137/S0036142997317353

[6] A. Chertock, A. Kurganov and Y. Rykov, “A New Sticky Particle Method for Pressureless Gas Dynamics,” SIAM Journal on Numerical Analysis, Vol. 45, No. 6, 2007, pp. 2408-2441. doi:10.1137/050644124

[7] R. Saurel, S. Gavrilyuk and F. Renaud, “A Multiphase Model with Internal Degrees of Freedom: Application to Shock-Bubble Interaction,” Journal of Fluid Mechanics, Vol. 495, No. 1, 2003, pp. 283-321. doi:10.1017/S002211200300630X

[8] G. Rudinger, “Some Effects of Finite Particle Volume on the Dynamics of Gas-Particle Mixtures (Gas Particle Mixture with Finite Particle Volume Affecting Frozen and Equilibrium Flows Behind Shock Wave),” AIAA Journal, Vol. 3, 1965, pp. 1217-1222.

[9] R. Saurel, A. Chinnayya and F. Renaud, “Thermodynamic Analysis and Numerical Resolution of a Turbulent-Fully Ionized Plasma Flow Model,” Shock Waves, Vol. 13, No. 4, 2004, pp. 283-297. doi:10.1007/s00193-003-0216-z

[10] E. F. Toro, “Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction,” Springer Verlag, Berlin, 1997.

[11] L. Schiller and Z. Naumann, “A Drag Coefficient Corre-lation,” VDI Zeitung, Vol. 77, 1935, pp. 318-320.

[12] E. F. Toro, M. Spruce and W. Speares, “Restoration of the Contact Surface in the HLL-Riemann Solver,” Shock Waves, Vol. 4, No. 1, 1994, pp. 25-34. doi:10.1007/BF01414629

[13] K. Hishida, K. Kaneko and M. Maeda, “Turbulence Structure of a Gas-Solid Two-Phase Circular Jet,” Trans. JSME, Vol. 51, 1985, pp. 2330-2337. doi:10.1299/kikaib.51.2330

[14] Y. Tsuji, Y. Morikawa, T. Tanaka, K. Karimine and S. Nishida, “Measurement of an Axisymmetric Jet Laden with Coarse Particles,” International Journal of Multi-phase Flow, Vol. 14, No. 5, 1988, pp. 565-574. doi:10.1016/0301-9322(88)90058-4

[1] F. E. Marble, “Dynamics of Dusty Gases,” Annual Review of Fluid Mechanics, Vol. 2, No. 1, 1970, pp. 397-446. doi:10.1146/annurev.fl.02.010170.002145

[2] Y. B. Zeldovich, “Gravitational Instability: An Approx-imate Theory for Large Density Perturbations,” Astron & Astrophys, Vol. 5, No. 84, 1970, pp. 168.

[3] P. D. Lax, “Weak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computation,” Commu-nications on Pure and Applied Mathematics, Vol. 7, No. 1, 1954, pp. 159-193.

[4] R. Saurel, E. Daniel and C. Loraud, “Two-Phase Flows: Second-Order Schemes and Boundary Conditions,” AIAA Journal, Vol. 32, No. 6, 1994, pp. 1214-1221. doi:10.2514/3.12122

[5] Y. Brenier and E Grenier, “Sticky Particles and Scalar Conservation Laws,” SIAM Journal on Numerical Analy-sis, Vol. 35, No. 6, 1998, pp. 2317-2328. doi:10.1137/S0036142997317353

[6] A. Chertock, A. Kurganov and Y. Rykov, “A New Sticky Particle Method for Pressureless Gas Dynamics,” SIAM Journal on Numerical Analysis, Vol. 45, No. 6, 2007, pp. 2408-2441. doi:10.1137/050644124

[7] R. Saurel, S. Gavrilyuk and F. Renaud, “A Multiphase Model with Internal Degrees of Freedom: Application to Shock-Bubble Interaction,” Journal of Fluid Mechanics, Vol. 495, No. 1, 2003, pp. 283-321. doi:10.1017/S002211200300630X

[8] G. Rudinger, “Some Effects of Finite Particle Volume on the Dynamics of Gas-Particle Mixtures (Gas Particle Mixture with Finite Particle Volume Affecting Frozen and Equilibrium Flows Behind Shock Wave),” AIAA Journal, Vol. 3, 1965, pp. 1217-1222.

[9] R. Saurel, A. Chinnayya and F. Renaud, “Thermodynamic Analysis and Numerical Resolution of a Turbulent-Fully Ionized Plasma Flow Model,” Shock Waves, Vol. 13, No. 4, 2004, pp. 283-297. doi:10.1007/s00193-003-0216-z

[10] E. F. Toro, “Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction,” Springer Verlag, Berlin, 1997.

[11] L. Schiller and Z. Naumann, “A Drag Coefficient Corre-lation,” VDI Zeitung, Vol. 77, 1935, pp. 318-320.

[12] E. F. Toro, M. Spruce and W. Speares, “Restoration of the Contact Surface in the HLL-Riemann Solver,” Shock Waves, Vol. 4, No. 1, 1994, pp. 25-34. doi:10.1007/BF01414629

[13] K. Hishida, K. Kaneko and M. Maeda, “Turbulence Structure of a Gas-Solid Two-Phase Circular Jet,” Trans. JSME, Vol. 51, 1985, pp. 2330-2337. doi:10.1299/kikaib.51.2330

[14] Y. Tsuji, Y. Morikawa, T. Tanaka, K. Karimine and S. Nishida, “Measurement of an Axisymmetric Jet Laden with Coarse Particles,” International Journal of Multi-phase Flow, Vol. 14, No. 5, 1988, pp. 565-574. doi:10.1016/0301-9322(88)90058-4