A Modification of The Large-Particle Method to a Scheme Having The Second Order of Accuracy in Space and Time for Shockwave Flows in a Gas Suspension

We develop the previously proposed approach of constructing difference schemes for solving stiff problems of shockwave flows of heterogeneous media using an implicit non-iterative algorithm for calculating interactions between the phases. The large particle method is modified to a scheme having the second order of accuracy in time and space on smooth solutions. At the first stage, we use the central differences with artificial viscosity of TVD type. At the second stage, we implement TVD-reconstruction by weighted linear combination of upwind and central approximations with flow limiters. The scheme is supplemented by a two-step Runge-Kutta method in time. The scheme is K-stable, i.e. the time step does not depend on the intensity of interactions between the phases, but is determined by the Courant number for a homogeneous system of equations (without source terms). We use test problems to confirm the monotonicity, low dissipation, high stability of the scheme and convergence of numerical results to the exact self-similar equilibrium solutions in a gas suspension. Also, we show the scheme capability for numerical simulation of physical instability and turbulence. The method can be used for flows of gas suspensions having complex structure.

Keywords

gas-suspension flow; stiff problem; difference scheme; stability; accuracy.

References

1. Belotserkovskii O.M., Davydov Yu.M. A Non-Stationary "Coarse Particle" Method for Gas-Dynamical Computations. USSR Computational Mathematics and Mathematical Physics, 1971, vol. 11, no. 1, pp. 241-271. DOI: 10.1016/0041-5553(71)90112-1
2. Gubaidullin A.A., Ivandaev A.I., Nigmatulin R.I. A Modified "Coarse Particle" Method for Calculating Non-Stationary Wave Processes in Multiphase Dispersive Media. USSR Computational Mathematics and Mathematical Physics, 1977, vol. 16, no. 7, pp. 180-192. DOI: 10.1016/0041-5553(77)90183-5
3. Nigmatulin R.I. Dinamika mnogofaznyh sred [Dynamics of Multiphase Media]. Moscow, Nauka, 1987. (in Russian)
4. Sadin D.V., A Modified Large-Particle Method for Calculating Unsteady Gas Flows in a Porous Medium. Computational Mathematics and Mathematical Physics, 1996, vol. 36, no. 10, pp. 1453-1458.
5. Sadin D.V., A Method for Computing Heterogeneous Wave Flows with Intense Phase Interaction. Computational Mathematics and Mathematical Physics, 1998, vol. 38, no. 6, pp. 987-993.
6. Sadin D.V. Stiffness Problem in Modeling Wave Flows of Heterogeneous Media with a Three-Temperature Scheme of Interphase Heat and Mass Transfer. Journal of Applied Mechanics and Technical Physics, 2002, vol. 43, no. 2, pp. 286-290. DOI: 10.1023/A:1014714012032
7. Sadin D.V. TVD Scheme for Stiff Problems of Wave Dynamics of Heterogeneous Media of Nonhyperbolic Nonconservative Type. Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 12, pp. 2068-2078. DOI: 10.1134/S0965542516120137
8. Sadin D.V. Schemes with Customizable Dissipative Properties as Applied to Gas-Suspensions Flow Simulation]. Mathematical Models and Computer Simulations, 2017, vol. 29, no 12, pp. 89-104. (in Russian)
9. Saurel R., Abgrall R. A Multiphase Godunov Method for Compressible Multifluid and Multiphase Flows. Journal of Computational Physics, 1999, vol. 150, no. 2, pp. 425-467. DOI: 10.1006/jcph.1999.6187
10. Saurel R., Le Metayer O., Massoni J., Gavrilyuk S. Shock Jump Relations for Multiphase Mixtures with Stiff Mechanical Relaxation. Shock Waves, 2007, vol. 16, no. 3, pp. 209-232. DOI: 10.1007/s00193-006-0065-7
11. Saurel R., Petitpas F., Berry R.A. Simple and Efficient Relaxation Methods for Interfaces Separating Compressible Fluids, Cavitating Flows and Shocks in Multiphase Mixtures. Journal of Computational Physics, 2009, vol. 228, no. 5, pp. 1678-1712. DOI: 10.1016/j.jcp.2008.11.002
12. Grishchenko D.S., Kovalev Yu.M., Kovaleva E.A. Modifcation of Method of Large Particles for Research of Currents of Gas-Suspensions. Bulletin of the South Ural State University. Series: Mathematical Modelling, Programming and Computer Software, 2015, vol. 8, no. 2, pp. 36-42. DOI: 10.14529/mmp150203 (in Russian)
13. Sadin D.V. Application of Scheme with Customizable Dissipative Properties for Gas Flow Calculation with Interface Instability Evolution. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2018, vol. 18, no. 1, pp. 153-157. DOI: 10.17586/2226-1494-2018-18-1-153-157 (in Russian)
14. Gottlieb S., Shu C.-W. Total Variation Diminishing Runge-Kutta Schemes. Mathematics of Computation, 1998, vol. 67, no. 221, pp. 73-85. DOI: 10.1090/S0025-5718-98-00913-2
15. Sadin D.V., Odoev S.A. Comparison of Difference Scheme with Customizable Dissipative Properties and WENO Scheme in the Case of One-Dimensional Gas and Gas-Particle Dynamics Problems. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2017, vol. 17, no. 4, pp. 719-724. DOI: 10.17586/2226-1494-2017-17-4-719-724 (in Russian)
16. Ivanov A.S., Kozlov V.V., Sadin D.V. Unsteady Flow of a Two-Phase Disperse Medium from a Cylindrical Channel of Finite Dimensions Into the Atmosphere. Fluid Dynamics, 1996, vol. 31, no. 3, pp. 386-391. DOI: 10.1007/BF02030221
17. Sadin D.V. Stiff Problems of a Two-Phase Flow with a Complex Wave Structure. Physical-Chemical Kinetics in Gas Dynamics, 2014, vol. 15, no. 4, pp. 1-17. (in Russian)
18. Haas J.-F., Sturtevant B. Interaction of Weak Shock Waves with Cylindrical and Spherical Gas Inhomogeneities. Journal of Fluid Mechanics, 1987, vol. 181, pp. 41-76. DOI: 10.1017/S0022112087002003
19. Shyue K.-M., Xiao F. An Eulerian Interface Sharpening Algorithm for Compressible Two-Phase Flow: the Algebraic THINC Approach. Journal of Computational Physics, 2014, vol. 268, pp. 326-354. DOI: 10.1016/j.jcp.2014.03.010
20. Quirk J.J., Karni S. On the Dynamics of a Shock-Bubble Interaction. Journal of Fluid Mechanics, 1996, vol. 318, pp. 129-163. DOI: 10.1017/S0022112096007069