Transport diffuse interface model for simulation of solid-fluid interaction

  • Li Li
  • Qian Chen
  • Baolin TianEmail author


For solid-fluid interaction, one of the phase-density equations in diffuse interface models is degenerated to a “0=0” equation when the volume fraction of a certain phase takes the value of zero or unity. This is because the conservative variables in phasedensity equations include volume fractions. The degeneracy can be avoided by adding an artificial quantity of another material into the pure phase. However, nonphysical waves, such as shear waves in fluids, are introduced by the artificial treatment. In this paper, a transport diffuse interface model, which is able to treat zero/unity volume fractions, is presented for solid-fluid interaction. In the proposed model, a new formulation for phase densities is derived, which is unrelated to volume fractions. Consequently, the new model is able to handle zero/unity volume fractions, and nonphysical waves caused by artificial volume fractions are prevented. One-dimensional and two-dimensional numerical tests demonstrate that more accurate results can be obtained by the proposed model.

Key words

solid-fluid interaction diffuse interface model phase-density equation Mie-Grüneisen equation of state (EOS) Eulerian method 

Chinese Library Classification


2010 Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



The authors are grateful to Prof. Jiequan LI from the Institute of Applied Physics and Computational Mathematics for his valuable suggestions.


  1. [1]
    LEE, C. B. and WU, J. Z. Transition in wall-bounded flows. Advances in Mechanics, 61(3), 683–695 (2009)Google Scholar
  2. [2]
    BATRA, R. C. and STEVENS, J. B. Adiabatic shear bands in axisymmetric impact and penetration problems. Computer Methods in Applied Mechanics & Engineering, 151(3-4), 325–342 (1998)zbMATHGoogle Scholar
  3. [3]
    SCHOCH, S., NIKIFORAKIS, N., and LEE, B. J. The propagation of detonation waves in nonideal condensed-phase explosives confined by high sound-speed materials. Physics of Fluids, 25(8), 452–457 (2013)Google Scholar
  4. [4]
    DIMONTE, G., TERRONES, G., CHERNE, F. J., GERMANN, T. C., DUPONT, V., KADAU, K., BUTTLER, W. T., ORO, D. M., MORRIS, C., and PRESTON, D. L. Use of the Richtmyer-Meshkov instability to infer yield stress at high-energy densities. Physical Review Letters, 107(26), 264502 (2011)Google Scholar
  5. [5]
    LEE, C. B., PENG, H. W., YUAN, H. J., WU, J. Z., ZHOU, M. D., and FAZLE, H. Experimental studies of surface waves inside a cylindrical container. Journal of Fluid Mechanics, 677(3), 39–62 (2011)MathSciNetzbMATHGoogle Scholar
  6. [6]
    LEE, C. B., SU, Z., ZHONG, H. J., CHEN, S. Y., ZHOU, M. D., and WU, J. Z. Experimental investigation of freely falling thin disks, part 2: transition of three-dimensional motion from zigzag to spiral. Journal of Fluid Mechanics, 732(5), 77–104 (2013)zbMATHGoogle Scholar
  7. [7]
    GHAISAS, N. S., SUBRAMANIAM, A., and LELE, S. K. A unified high-order Eulerian method for continuum simulations of fluid flow and of elastic-plastic deformations in solids. Journal of Computational Physics, 371(22), 452–482 (2018)MathSciNetGoogle Scholar
  8. [8]
    LI, X. L., FU, D. X., and MA, Y. W. Direct numerical simulation of hypersonic boundary layer transition over a blunt cone with a small angle of attack. Physics of Fluids, 22(2), 025105 (2010)zbMATHGoogle Scholar
  9. [9]
    BARLOW, A. J., MAIRE, P. H., RIDER, W. J., RIEBEN, R. N., and SHASHKOV, M. J. Arbitrary Lagrangian-Eulerian methods for modeling high-speed compressible multimaterial flows. Journal of Computational Physics, 322, 603–665 (2016)MathSciNetzbMATHGoogle Scholar
  10. [10]
    GHAISAS, N. S., SUBRAMANIAM, A., and LELE, S. K. High-order Eulerian methods for elasticplastic flow in solids and coupling with fluid flows. 46th AIAA Fluid Dynamics Conference, American Institute of Aeronautics and Astronautics, Washington, D. C. (2016)Google Scholar
  11. [11]
    GODUNOV, S. K. and ROMENSKII, E. I. Elements of Continuum Mechanics and Conservation Laws, Kluwer Academic/Plenum Publishers, New York (2003)zbMATHGoogle Scholar
  12. [12]
    PLOHR, B. J. and SHARP, D. H. A conservative Eulerian formulation of the equations for elastic flow. Advances in Applied Mathematics, 9(4), 481–499 (1988)MathSciNetzbMATHGoogle Scholar
  13. [13]
    HIRT, C. W. and NICHOLS, B. D. Volume of fluid method for the dynamics of free boundaries. Journal of Computational Physics, 39(1), 201–225 (1981)zbMATHGoogle Scholar
  14. [14]
    BARTON, P. T., DEITERDING, R., MEIRON, D., and PULLIN, D. Eulerian adaptive finitedifference method for high-velocity impact and penetration problems. Journal of Computational Physics, 240(5), 76–99 (2013)MathSciNetGoogle Scholar
  15. [15]
    ABGRALL, R. How to prevent pressure oscillations in multicomponent flow calculations: a quasi conservative approach. Journal of Computational Physics, 125(1), 150–160 (1994)MathSciNetzbMATHGoogle Scholar
  16. [16]
    BAER, M. R. and NUNZIATO, J. W. A two-phase mixture theory for the deflagration-todetonation transition (DDT) in reactive granular materials. International Journal of Multiphase Flow, 12(6), 861–889 (1986)zbMATHGoogle Scholar
  17. [17]
    SAUREL, R. and ABGRALL, R. A multiphase Godunov method for compressible multifluid and multiphase flows. Journal of Computational Physics, 150(2), 425–467 (1999)MathSciNetzbMATHGoogle Scholar
  18. [18]
    SAUREL, R., PETITPAS, F., and BERRY, R. A. Simple and efficient relaxation methods for interfaces separating compressible fluids, cavitating flows and shocks in multiphase mixtures. Journal of Computational Physics, 228(5), 1678–1712 (2009)MathSciNetzbMATHGoogle Scholar
  19. [19]
    KAPILA, A. K., MENIKOFF, R., BDZIL, J. B., SON, S. F., and STEWART, D. S. Two-phase modeling of deflagration-to-detonation transition in granular materials: reduced equations. Physics of Fluids, 13(10), 3002–3024 (2001)zbMATHGoogle Scholar
  20. [20]
    MURRONE, A. A five equation reduced model for compressible two phase flow problems. Journal of Computational Physics, 202(2), 664–698 (2005)MathSciNetzbMATHGoogle Scholar
  21. [21]
    FAVRIE, N., GAVRILYUK, S. L., and SAUREL, R. Solid-fluid diffuse interface model in cases of extreme deformations. Journal of Computational Physics, 228(16), 6037–6077 (2009)MathSciNetzbMATHGoogle Scholar
  22. [22]
    FAVRIE, N. and GAVRILYUK, S. L. Diffuse interface model for compressible fluid-compressible elastic-plastic solid interaction. Journal of Computational Physics, 231(7), 2695–2723 (2012)MathSciNetzbMATHGoogle Scholar
  23. [23]
    NDANOU, S., FAVRIE, N., and GAVRILYUK, S. Multi-solid and multi-fluid diffuse interface model: applications to dynamic fracture and fragmentation. Journal of Computational Physics, 295(25), 523–555 (2015)MathSciNetzbMATHGoogle Scholar
  24. [24]
    KLUTH, G. and DESPRÉS, B. Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme. Journal of Computational Physics, 229(1), 9092–9118 (2010)MathSciNetzbMATHGoogle Scholar
  25. [25]
    ABGRALL, R. and KARNI, S. Computations of compressible multifluids. Journal of Computational Physics, 169(2), 594–623 (2001)MathSciNetzbMATHGoogle Scholar
  26. [26]
    SHYUE, K. M. An efficient shock-capturing algorithm for compressible multicomponent problems. Journal of Computational Physics, 142(1), 208–242 (1998)MathSciNetzbMATHGoogle Scholar
  27. [27]
    SHYUE, K. M. Regular article: a fluid-mixture type algorithm for compressible multicomponent flow with van derWaals equation of state. Journal of Computational Physics, 156(1), 43–88 (1999)MathSciNetzbMATHGoogle Scholar
  28. [28]
    SHYUE, K. M. A fluid-mixture type algorithm for compressible multicomponent flow with Mie-Grüneisen equation of state. Journal of Computational Physics, 171(2), 678–707 (2001)MathSciNetzbMATHGoogle Scholar
  29. [29]
    MAIRE, P. H. and REBOURCET, B. A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids. Journal of Computational Physics, 235(2), 626–665 (2013)MathSciNetzbMATHGoogle Scholar
  30. [30]
    HE, Z. W., ZHANG, Y. S., LI, X. L., LI, L., and TIAN, B. L. Preventing numerical oscillations in the flux-split based finite difference method for compressible flows with discontinuities. Journal of Computational Physics, 300(5), 269–287 (2015)MathSciNetzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Applied Physics and Computational MathematicsBeijingChina

Personalised recommendations