## Fast method to simulate dynamics of two-phase medium with intense interaction between phases by smoothed particle hydrodynamics: gas-dust mixture with polydisperse particles, linear drag, one-dimensional tests.(English)Zbl 07506515

Summary: To simulate the dynamics of fluid with polydisperse particles on macroscale level, one has to solve hydrodynamic equations with several relaxation terms, representing momentum transfer from fluid to particles and vice versa. For small particles, velocity relaxation time (stopping time) can be much shorter than dynamical time of fluid that makes this problem stiff and thus computationally expensive.
We present a new fast method for computing several stiff drag terms in two-phase polydisperse medium with Smoothed Particle Hydrodynamics (SPH). In our implementation, fluid and every fraction of dispersed phase are simulated with different sets of particles. The method is based on (1) linear interpolation of velocity values in drag terms, (2) implicit approximation of drag terms that conserves momentum with machine precision, and (3) solution of system of $$N$$ linear algebraic equations with $$O(N^2)$$ arithmetic operation instead of $$O(N^3)$$.
We studied the properties of the proposed method on one-dimensional problems with known solutions. We found that we can obtain acceptable accuracy of the results with numerical resolution independent of short stopping time values. All simulation results discussed in the paper are obtained with open source software.

### MSC:

 76Txx Multiphase and multicomponent flows 76Mxx Basic methods in fluid mechanics 35Lxx Hyperbolic equations and hyperbolic systems

### Software:

MultiGrain; Splash
Full Text:

### References:

 [1] Klinzing, G. E.; Rizk, F.; Marcus, R.; Leung, L. S., Pneumatic Conveying of Solids. A Theoretical and Practical Approach (2010), Springer: Springer Dordrecht [2] Millán, J. M.V., Fluidization of Fine Powders. Cohesive Versus Dynamical Aggregation (2013), Springer: Springer Dordrecht [3] Varaksin, A. Y., Fluid dynamics and thermal physics of two-phase flows: problems and achievements, High Temp., 51, 377-407 (2013) [4] Vasilevskii, E. B.; Osiptsov, A. N.; Chirikhin, A. V.; Yakovleva, L. V., Heat exchange on the front surface of a blunt body in a high-speed flow containing low-inertia particles, J. Eng. Phys. Thermophys., 74, 1399-1411 (2001) [5] Haworth, T. J.; Ilee, J. D.; Forgan, D. H.; Facchini, S.; Price, D. J.; Boneberg, D. M.; Booth, R. A.; Clarke, C. J.; Gonzalez, J.-F.; Hutchison, M. A.; Kamp, I.; Laibe, G.; Lyra, W.; Meru, F.; Mohanty, S.; Panić, O.; Rice, K.; Suzuki, T.; Teague, R.; Walsh, C.; Woitke, P., Grand challenges in protoplanetary disc modelling, Publ. Astron. Soc. Aust., 33, e053 (2016), Community authors: [6] Balachandar, S.; Eaton, J. K., Turbulent dispersed multiphase flow, Annu. Rev. Fluid Mech., 42, 111-133 (2010) · Zbl 1345.76106 [7] Soo, S. L., Particulates and Continuum: Multiphase Fluid Dynamics (1989), Hemisphere, Corp.: Hemisphere, Corp. New York etc. · Zbl 0689.76001 [8] Nigmatullin, R. I., Dyn. Multiph. Media, 1 (1990) [9] Deich, M.; Filippov, G., The Gas Dynamics of Two-Phase Media (1970) [10] Gidaspow, D., Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions (1994), Academic Press: Academic Press San Diego · Zbl 0789.76001 [11] Babukha, G. L.; Shraiber, A. A., Motion of a polydisperse material in a vertical gas flow, J. Appl. Mech. Tech. Phys., 8, 30-34 (1967) [12] Marble, F. E., Dynamics of dusty gases, Annu. Rev. Fluid Mech., 2, 397-446 (1970) [13] Sternin, L. E.; Maslov, B. N.; Shraiber, A. A.; Podvysotskii, A. M., Two-Phase Mono- and Polydisperse Gas Flows Containing Particles (1980), Izdatel’stvo Mashinostroenie: Izdatel’stvo Mashinostroenie Moscow [14] Marchisio, D. L.; Fox, R. O., Computational Models for Polydisperse Particulate and Multiphase Systems, Cambridge Series in Chemical Engineering (2013), Cambridge University Press [15] Ivanenko, A. Y.; Yablokova, M. A., A mathematical simulation of the vertical pneumatic transport of a polydisperse material, Theor. Found. Chem. Eng., 53, 432-442 (2019) [16] Tukmakova, N. A.; Tukmakov, A. L., Model of the dynamics of a polydisperse vapor-droplet mixture with gas-dynamical fragmentation of droplets, J. Eng. Phys. Thermophys., 92, 1466-1474 (2019) [17] Lozhkin, Y.; Markovich, D.; Pakhomov, M.; Terekhov, V., Investigation of the structure of a polydisperse gas-droplet jet in the initial region. Experiment and numerical simulation, Thermophys. Aeromech., 21, 293-307 (2014) [18] Korolev, V. V.; Bezborodov, M. A.; Kovalenko, I. G.; Zankovich, A. M.; Eremin, M. A., Tricolor technique for visualization of spatial variations of polydisperse dust in gas-dust flows (2018) [19] Drazkowska, J.; Li, S.; Birnstiel, T.; Stammler, S. M.; Li, H., Including dust coagulation in hydrodynamic models of protoplanetary disks: dust evolution in the vicinity of a Jupiter-mass planet, Astrophys. J., 885, 91 (2019) [20] Laibe, G.; Price, D. J., Dust and gas mixtures with multiple grain species - a one-fluid approach, Mon. Not. R. Astron. Soc., 444, 1940-1956 (2014) [21] Liu, C.; Wang, Z.; Xu, K., A unified gas-kinetic scheme for continuum and rarefied flows VI: dilute disperse gas-particle multiphase system, J. Comput. Phys., 386, 264-295 (2019) · Zbl 1452.76213 [22] Chen, G.-Q.; Livermore, C. D.; Liu, T.-P., Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math., 47, 787-830 (1994) · Zbl 0806.35112 [23] Jin, S., Asymptotic preserving (ap) schemes for multiscale kinetic and hyperbolic equations: a review, Riv. Mat. Univ. Parma. New Ser., 2 (2010) [24] Degond, P.; Deluzet, F., Asymptotic-preserving methods and multiscale models for plasma physics, J. Comput. Phys., 336, 429-457 (2017) · Zbl 1375.82108 [25] Jin, S.; Livermore, C. D., Numerical schemes for hyperbolic conservation laws with stiff relaxation terms, J. Comput. Phys., 126, 449-467 (1996) · Zbl 0860.65089 [26] Stoyanovskaya, O. P.; Snytnikov, V. N.; Vorobyov, E. I., Analysis of methods for computing the trajectories of dust particles in a gas-dust circumstellar disk, Astron. Rep., 94, 1033-1049 (2017) [27] Stoyanovskaya, O. P.; Vorobyov, E. I.; Snytnikov, V. N., Analysis of numerical algorithms for computing rapid momentum transfers between the gas and dust in simulations of circumstellar disks, Astron. Rep., 62, 455-468 (2018) [28] Stoyanovskaya, O. P.; Okladnikov, F. A.; Vorobyov, E. I.; Pavlyuchenkov, Y. N.; Akimkin, V. V., Simulating dynamics of dusty gas circumstellar disks: going beyond Epstein drag mode, Astron. Rep., 64, 107-125 (2020) [29] Vorobyov, E. I.; Akimkin, V.; Stoyanovskaya, O.; Pavlyuchenkov, Y.; Liu, H. B., Early evolution of viscous and self-gravitating circumstellar disks with a dust component, Astron. Astrophys., 614, A98 (2018) [30] Miniati, F., A hybrid scheme for gas-dust systems stiffly coupled via viscous drag, J. Comput. Phys., 229, 3916-3937 (2010) · Zbl 1423.76464 [31] Li, S., An L-stable method for solving stiff hydrodynamics, AIP Conf. Proc., 1863, Article 500004 pp. (2017) [32] Benítez-Llambay, P.; Krapp, L.; Pessah, M. E., Asymptotically stable numerical method for multispecies momentum transfer: gas and multifluid dust test suite and implementation in FARGO3D, Astrophys. J. Suppl. Ser., 241, 25 (2019) [33] Monaghan, J. J.; Kocharyan, A., SPH simulation of multi-phase flow, Comput. Phys. Commun., 87, 225-235 (1995) · Zbl 0923.76195 [34] Lorén-Aguilar, P.; Bate, M. R., Two-fluid dust and gas mixtures in smoothed particle hydrodynamics: a semi-implicit approach, Mon. Not. R. Astron. Soc., 443, 927-945 (2014) [35] Stoyanovskaya, O. P.; Glushko, T. A.; Snytnikov, N. V.; Snytnikov, V. N., Two-fluid dusty gas in smoothed particle hydrodynamics: fast and implicit algorithm for stiff linear drag, Astron. Comput., 25, 25-37 (2018) [36] Monaghan, J. J., Implicit SPH drag and dusty gas dynamics, J. Comput. Phys., 138, 801-820 (1997) · Zbl 0947.76066 [37] Monaghan, J., On the integration of the SPH equations for a dusty fluid with high drag, Eur. J. Mech. B, Fluids, 79, 454-462 (2020) · Zbl 1473.76069 [38] Laibe, G.; Price, D. J., Dusty gas with smoothed particle hydrodynamics - I. Algorithm and test suite, Mon. Not. R. Astron. Soc., 420, 2345-2364 (2012) [39] Laibe, G.; Price, D. J., Dusty gas with smoothed particle hydrodynamics - II. Implicit timestepping and astrophysical drag regimes, Mon. Not. R. Astron. Soc., 420, 2365-2376 (2012) [40] Lorén-Aguilar, P.; Bate, M. R., Two-fluid dust and gas mixtures in smoothed particle hydrodynamics II: an improved semi-implicit approach, Mon. Not. R. Astron. Soc., 454, 4114-4119 (2015) [41] Booth, R. A.; Sijacki, D.; Clarke, C. J., Smoothed particle hydrodynamics simulations of gas and dust mixtures, Mon. Not. R. Astron. Soc., 452, 3932-3947 (2015) [42] Stoyanovskaya, O. P.; Akimkin, V. V.; Vorobyov, E. I.; Glushko, T. A.; Pavlyuchenkov, Y. N.; Snytnikov, V. N.; Snytnikov, N. V., Development and application of fast methods for computing momentum transfer between gas and dust in supercomputer simulation of planet formation, J. Phys. Conf. Ser., 1103, Article 012008 pp. (2018) [43] Stoyanovskaya, O.; Glushko, T.; Snytnikov, V.; Snytnikov, N., Monodisperse gas-solid mixtures with intense interphase interaction in two-fluid smoothed particle hydrodynamics (2019) [44] Yang, C.-C.; Johansen, A., Integration of particle-gas systems with stiff mutual drag interaction, Astrophys. J. Suppl. Ser., 224, 39 (2016) [45] Yang, C.-C.; Johansen, A.; Carrera, D., Concentrating small particles in protoplanetary disks through the streaming instability, A & A, 606, A80 (2017) [46] Laibe, G.; Price, D. J., Dusty gas with one fluid, Mon. Not. R. Astron. Soc., 440, 2136-2146 (2014) [47] Hutchison, M.; Price, D. J.; Laibe, G., MULTIGRAIN: a smoothed particle hydrodynamic algorithm for multiple small dust grains and gas, Mon. Not. R. Astron. Soc., 476, 2186-2198 (2018) [48] Carrillo, J.-A.; Goudon, T.; Lafitte, P., Simulation of fluid and particles flows: asymptotic preserving schemes for bubbling and flowing regimes, J. Comput. Phys., 227, 7929-7951 (2008) · Zbl 1141.76050 [49] Goudon, T.; Jin, S.; Liu, J.-G.; Yan, B., Asymptotic-preserving schemes for kinetic-fluid modeling of disperse two-phase flows, J. Comput. Phys., 246, 145-164 (2013) · Zbl 1349.82046 [50] Price, D. J.; Laibe, G., A solution to the overdamping problem when simulating dust-gas mixtures with smoothed particle hydrodynamics, Mon. Not. R. Astron. Soc., 495, 3929-3934 (2020) [51] Laibe, G.; Price, D. J., DUSTYBOX and DUSTYWAVE: two test problems for numerical simulations of two-fluid astrophysical dust-gas mixtures, Mon. Not. R. Astron. Soc., 418, 1491-1497 (2011) [52] Sod, G. A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31 (1978) · Zbl 0387.76063 [53] Price, D. J., Splash: an interactive visualisation tool for smoothed particle hydrodynamics simulations, Publ. Astron. Soc. Aust., 24, 159-173 (2007) [54] Monaghan, J. J., Smoothed particle hydrodynamics, Annu. Rev. Astron. Astrophys., 30, 543-574 (1992) [55] Epstein, P. S., On the resistance experienced by spheres in their motion through gases, Phys. Rev., 23, 710-733 (1924) [56] Fedorov, A.; Bedarev, I., The shock-wave structure in a gas-particle mixture with chaotic pressure, Math. Models Comput. Simul., 10, 1-14 (2018)
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