NUMERICAL STUDIES OF TWO-PHASE HYPERELASTIC MODEL

Cover Page

Cite item

Full Text

Open Access Open Access
Restricted Access Access granted
Restricted Access Subscription Access

Abstract

The paper is devoted to numerical studies of two-phase hyperbolic model describing the dynamics of hyperelastic media. The considered model is a generalization of the well known model of multivelocity fully nonequilibrium Baer–Nunziato model, widely used for description of shock-wave and detonation processes in multiphase media. The equations of the model are given both in the general and in the spatially one-dimensional case, and its properties are described. For numerical study, the pathconservative HLL method is applied. The numerical study is carried using a number of Riemann problems.

About the authors

I. M Ermakov

Keldysh Institute of Applied Mathematics of RAS

Email: 11503ermakov@gmail.com
Moscow, Russia

R. R Polekhina

Keldysh Institute of Applied Mathematics of RAS

Email: polekhina@keldysh.ru
Moscow, Russia

E. B Savenkov

Keldysh Institute of Applied Mathematics of RAS

Email: savenkov@keldysh.ru
Moscow, Russia

References

  1. Baer, M.R. A two-phase mixture theory for the deflagration-to-detonation transition (DDT) in reactive granular materials / M.R. Baer, J.W. Nunziato // Int. J. Multiph. Flow. — 1986. — V. 12, № 6. — P. 861–889.
  2. Truesdell, C. The mechanical foundations of elasticity and fluid dynamics / C. Truesdell // J. of Rational Mechanics and Analysis. — 1952. — V. 1. — P. 125–300.
  3. Truesdell, C. Rational Thermodynamics / C. Truesdell. — New York : Springer, 1969. — 578 p.
  4. Coleman, B.D. The thermodynamics of elastic materials with heat conduction and viscosity / B.D. Coleman, W. Noll // The Foundations of Mechanics and Thermodynamics: Selected Papers. — 1974. — P. 145–156.
  5. Noll, W. A mathematical theory of the mechanical behavior of continuous media / W. Noll // The Foundations of Mechanics and Thermodynamics: Selected Papers. — 1974. — P. 1–30.
  6. Bedford, A. Theories of immiscible and structured mixtures / A. Bedford, D.S. Drumheller // Int. J. Eng. Sci. — 1983. — V. 21, № 8. — P. 863–960.
  7. Evaluation of an Eulerian multi-material mixture formulation based on a single inverse deformation gradient tensor field / N.S. Ghaisas, A. Subramaniam, S.K. Lele, A.W. Cook // Annual Research Briefs — 2017. — Stanford : Center for Turbulence Research, 2017. — P. 377–390.
  8. Ghaisas, N.S. High-order eulerian methods for elastic-plastic flow in solids and coupling with fluid flows / N.S. Ghaisas, A. Subramaniam, S.K. Lele // 46th AIAA Fluid Dynamics Conference. — Reston, Virginia : American Institute of Aeronautics and Astronautics, 2016.
  9. Subramaniam, A. High-order eulerian simulations of multimaterial elastic–plastic flow / A. Subramaniam, N.S. Ghaisas, S.K. Lele // J. Fluids Eng. — 2018. — V. 140, № 5. — Art. 050904.
  10. Ndanou, S. Multi-solid and multi-fluid diffuse interface model: applications to dynamic fracture and fragmentation / S. Ndanou, N. Favrie, S. Gavrilyuk // J. Comput. Phys. — 2015. — V. 295. — P. 523–555.
  11. Drumheller, D.S. On theories for reacting immiscible mixtures / D.S. Drumheller // Int. J. Eng. Sci. — 2000. — V. 38, № 3. — P. 347–382.
  12. Schumacher, S.C. Generalized continuum mixture theory for multi-material shock physics / S.C. Schumacher, M.R. Baer // Int. J. Multiph. Flow. — 2021. — V. 144. — Art. 103790.
  13. Алексеев, М.В. Математическая модель двухфазной гиперупругой среды. “Скалярный” случай / М.В. Алексеев, Е.Б. Савенков // Препринты ИПМ имени М.В. Келдыша. — 2022. — № 40. — 63 с.
  14. LeFloch, P. Shock waves for nonlinear hyperbolic systems in nonconservative form / P. LeFloch // IMA Preprint Series. — 1989. — № 593.
  15. Dal Maso, G. Definition and weak stability of nonconservative products / G. Dal Maso, P.G. LeFloch, F. Murat // J. de math´ematiques pures et appliqu´ees. — 1995. — V. 74, № 6. — P. 483–548.
  16. LeFloch, P.G. Why many theories of shock waves are necessary: kinetic functions, equivalent equations, and fourth-order models / P.G. LeFloch, M. Mohammadian // J. Comput. Phys. — 2008. — V. 227, № 8. — P. 4162–4189.
  17. Saurel, R. Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures / R. Saurel, F. Petitpas, R.A. Berry // J. Comput. Phys. — 2009. — V. 228, № 5. — P. 1678–1712.
  18. Меньшов, И.С. Численная модель многофазных течений на основе подсеточного разрешения контактных границ / И.С. Меньшов, А.А. Серёжкин // Журн. вычислит. математики и мат. физики. — 2022. — Т. 62, № 10. — С. 1740–1760.
  19. In-cell discontinuous reconstruction path-conservative methods for non conservative hyperbolic systems — second-order extension / E. Pimentel-Garc´ıa, M.J. Castro, C. Chalons [et al.] // J. Comput. Phys. — 2022. — V. 459. — P. 111–152.
  20. Годунов, С.К. Элементы механики сплошных сред и законы сохранения / С.К. Годунов, Е.И. Роменский. — Новосибирск : Научная книга, 1998. — 280 с.
  21. Exact and approximate solutions of Riemann problems in non-linear elasticity / P.T. Barton, D. Drikakis, E. Romenski, V.A. Titarev // J. Comput. Phys. — 2009. — V. 228, № 18. — P. 7046– 7068.
  22. Why many theories of shock waves are necessary: convergence error in formally path-consistent schemes / M.J. Castro, P.G. LeFloch, M.L. Mun˜oz-Ruiz, C. Par´es // J. Comput. Phys. — 2008. — V. 227, № 17. — P. 8107–8129.
  23. Par´es, C. Numerical methods for nonconservative hyperbolic systems: a theoretical framework / C. Par´es // SIAM J. on Numerical Analysis. — 2006. — Vol. 44, № 1. — P. 300–321.
  24. Par´es, C. On some difficulties of the numerical approximation of nonconservative hyperbolic systems / C. Par´es, M.L. Mun˜oz-Ruiz // SeMA J.: Bolet´ın de la Sociedad Espan˜ola de Matema´tica Aplicada. — 2009. — № 47. — P. 23–52.
  25. Dumbser, M. A new efficient formulation of the HLLEM Riemann solver for general conservative and non-conservative hyperbolic systems / M. Dumbser, D.S. Balsara // J. Comput. Phys. — 2016. — V. 304. — P. 275–319.

Supplementary files

Supplementary Files
Action
1. JATS XML
Download

Copyright (c) 2025 Russian Academy of Sciences

Согласие на обработку персональных данных

 

Используя сайт https://journals.rcsi.science, я (далее – «Пользователь» или «Субъект персональных данных») даю согласие на обработку персональных данных на этом сайте (текст Согласия) и на обработку персональных данных с помощью сервиса «Яндекс.Метрика» (текст Согласия).