Skip to main content
SpringerLink
Log in

Nonlinear Finite Volume Method for the Interface Advection-Compression Problem on Unstructured Adaptive Meshes

  • GENERAL NUMERICAL METHODS
  • Published:
Computational Mathematics and Mathematical Physics Aims and scope Submit manuscript

Abstract

The paper is devoted to the nonlinear finite volume method applied for tracking interfaces on unstructured adaptive meshes. The fluid of volume approach is used. The interface location is described by the fraction of fluid in each computational cell. The interface propagation involves the simultaneous solution of the fraction advection and interface compression problems. The compression problem is solved to recover the interface (front) sharpness, which is smeared due to numerical diffusion. The problem discretization is carried out using the nonlinear monotone finite volume method. This method is applied to unstructured meshes with adaptive local refinement.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.
Fig. 11.
Fig. 12.
Fig. 13.
Fig. 14.
Fig. 15.
Fig. 16.

Similar content being viewed by others

REFERENCES

  1. M. J. Ketabdari, “Free surface flow simulation using VOF method,” Numerical Simulation: From Brain Imaging to Turbulent Flows, Ed. by Lpez-Ruiz (BoDBooks on Demand, 2016), Vol. 365.

  2. K. D. Nikitin, M. A. Olshanskii, K. M. Terekhov, et al. “An adaptive numerical method for free surface flows passing rigidly mounted obstacles,” Comput. Fluids 148, 56–68 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. V. Vassilevski, K. Nikitin, M. Olshanskii, and K. Terekhov, “CFD technology for 3D simulation of large-scale hydrodynamic events and disasters,” Russ. J. Numer. Anal. Math. Model. 27, 399–412 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  4. B. Kumar, M. Crane, and Y. Delauré, “On the volume of fluid method for multiphase fluid flow simulation,” Int. J. Model. Simul. Sci. Comput. 4 (2), 1350002 (2013).

  5. S. J. Ruuth and B. T. Wetton, “A simple scheme for volume-preserving motion by mean curvature,” J. Sci. Comput. 19, 373–384 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  6. Y. Qi, J. Lu., R. Scardovelli, et al. “Computing curvature for volume of fluid methods using machine learning,” J. Comput. Phys. 377, 155–161 (2019).

    Article  MathSciNet  Google Scholar 

  7. K. D. Nikitin, M. A. Olshanskii, K. M. Terekhov, and Y. V. Vassilevski, “A splitting method for numerical simulation of free surface flows of incompressible fluids with surface tension,” Comput. Methods Appl. Math. 15 (1), 59–77 (2015).

  8. K. D. Nikitin, K. M. Terekhov, and Y. V. Vassilevski, “Two methods of surface tension treatment in free surface flow simulations,” Appl. Math. Lett. 86, 236–242 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  9. S. McFadden and D. Browne, “A front-tracking model to predict solidification macrostructures and columnar to equiaxed transitions in alloy castings,” Appl. Math. Model. 33, 1397–1416 (2009).

    Article  MathSciNet  MATH  Google Scholar 

  10. R. Malladi and J. A. Sethian, “Level set methods for curvature flow, image enhancement, and shape recovery in medical images,” Visualization and Mathematics (Springer, 1997), pp. 329–345.

    Google Scholar 

  11. S. Popinet, “Gerris: A tree-based adaptive solver for the incompressible Euler equations in complex geometries,” J. Comput. Phys. 190, 572–600 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  12. C. Lalanne, Q. Magdelaine, F. Lequien, and J.-M. Fullana, “Numerical model using a volume-of-fluid method for the study of evaporating sessile droplets in both unpinned and pinned modes,” Eur. J. Mech. B Fluids. (2021).

  13. C. Kunkelmann and P. Stephan, “CFD simulation of boiling flows using the volume-of-fluid method within OpenFOAM,” Numer. Heat Transf. A. 56, 631–646 (2009).

    Article  Google Scholar 

  14. L. Gamet, M. Scala, J. Roenby, et al. “Validation of volume-of-fluid OpenFOAM® isoadvector solvers using single bubble benchmarks,” Comput. Fluids 213, 104722 (2020).

  15. A. Albadawi, D. Donoghue, A. Robinson, et al., “On the analysis of bubble growth and detachment at low capillary and bond numbers using volume of fluid and level set methods,” Chem. Eng. Sci. 90, 77–91 (2013).

    Article  Google Scholar 

  16. E. G. Puckett, A. S. Almgren, J. B. Bell, et al. “A high-order projection method for tracking fluid interfaces in variable density incompressible flows,” J. Comput. Phys. 130, 269–282 (1997).

    Article  MATH  Google Scholar 

  17. M. Sussman and E. G. Puckett, “A coupled level set and volume-of-fluid method for computing 3D and axisymmetric incompressible two-phase flows,” J. Comput. Phys. 162, 301–337 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Cifani, W. Michalek, G. Priems, et al., “A comparison between the surface compression method and an interface reconstruction method for the VOF approach,” Comput. Fluids. 136, 421–435 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  19. H. Rusche, Computational fluid dynamics of dispersed two-phase flows at high phase fractions, Ph.D. thesis, Imperial College London (University of London), 2003.

  20. Y. Okagaki, T. Yonomoto, M. Ishigaki, and Y. Hirose, “Numerical study on an interface compression method for the volume of fluid approach,” Fluids 6 (2), 80 (2021).

    Article  Google Scholar 

  21. M. Aboukhedr, A. Georgoulas, M. Marengo, et al., Simulation of micro-flow dynamics at low capillary numbers using adaptive interface compression," Comput. Fluids 165, 13–32 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  22. J. K. Patel and G. Natarajan, “A generic framework for design of interface capturing schemes for multi-fluid flows,” Comput. Fluids 106, 108–118 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Arote, M. Bade, and J. Banerjee, “An improved compressive volume of fluid scheme for capturing sharp interfaces using hybridization,” Numer. Heat Transf. B: Fundam. 79, 29–53 (2020).

    Article  Google Scholar 

  24. Y. Mehmani, Wrinkle-free interface compression for two-fluid flows, arXiv:1811.09744. 2018.

  25. D. J. Piro and K. Maki, An adaptive interface compression method for water entry and exit, Tech. Rep. 2013-350, University of Michigan, Department of Naval Architecture and Marine Engineering, 2013.

  26. H. Lee and S. H. Rhee, “A dynamic interface compression method for VOF simulations of high-speed planing watercraft,” J. Mech. Sci. Technol. 29, 1849–1857 (2015).

    Article  Google Scholar 

  27. J. A. Sethian and P. Smereka, “Level set methods for fluid interfaces,” Annu. Rev. Fluid Mech. 35, 341–372 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  28. D. Adalsteinsson and J. A. Sethian, “The fast construction of extension velocities in level set methods,” J. Comput. Phys. 148, 2–22 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  29. K. M. Terekhov, K. D. Nikitin, M. A. Olshanskii, and Y. V. Vassilevski, “A semi-Lagrangian method on dynamically adapted octree meshes,” Russ. J. Numer. Anal. Math. Model. 30, 363–380 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  30. K. Nikitin, M. Olshanskii, K. Terekhov, and Yu. V. Vassilevski, “Preserving distance property of level set function and simulation of free surface flows on adaptive grids,” Numerical Geometry, Grid Generation and Scientific Computing (NUMGRID-2010), 2010, pp. 25–32.

  31. R. F. Ausas, E. A. Dari, and G. C. Buscaglia, “A geometric mass-preserving redistancing scheme for the level set function,” Int. J. Numer. Methods Fluids 65, 989–1010 (2011).

    Article  MATH  Google Scholar 

  32. Z. Ge, J.-C. Loiseau, O. Tammisola, and L. Brandt, “An efficient mass-preserving interface-correction level set/ghost fluid method for droplet suspensions under depletion forces,” J. Comput. Phys. 353, 435–459 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  33. B. M. Ningegowda, Z. Ge, G. Lupo, et al., “A mass-preserving interface-correction level set/ghost fluid method for modeling of three-dimensional boiling flows,” Int. J. Heat Mass Transf. 162, 120382 (2020).

  34. J.-L. Guermond, M. Q. de Luna, and T. Thompson, “An conservative antidiffusion technique for the level set method,” J. Comput. Appl. Math. 321, 448–468 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  35. B. Leonard and S. Mokhtari, “Beyond first-order upwinding: The ultra-sharp alternative for non-oscillatory steady-state simulation of convection,” Int. J. Numer. Methods Eng. 30, 729–766 (1990).

    Article  Google Scholar 

  36. L. Silva, C. Fontes, and P. Lage, “Front tracking in recirculating flows: A comparison between the TVD and RCM methods in solving the VOF equation,” Braz. J. Chem. Eng. 22 (1), 105–116 (2005).

    Article  Google Scholar 

  37. C.-N. Lu, R.-Y. Wang, and J.-S. Sun, “WENO finite volume method for tracking moving interfaces on unstructured triangle meshes,” J. Hohai Univ. 01 (2009).

  38. S. Pirozzoli, S. Di Giorgio, and A. Iafrati, “On algebraic TVD-VOF methods for tracking material interfaces,” Comput. Fluids 189, 73–81 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  39. M. Darwish and F. Moukalled, “Convective schemes for capturing interfaces of free-surface flows on unstructured grids,” Numer. Heat Transf. B: Fundam. 49, 19–42 (2006).

    Article  Google Scholar 

  40. O. Ubbink and R. Issa, “A method for capturing sharp fluid interfaces on arbitrary meshes,” J. Comput. Phys. 153, 26–50 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  41. Y.-Y. Tsui, S.-W. Lin, T.-T. Cheng, and T.-C. Wu, “Flux-blending schemes for interface capture in two-fluid flows,” Int. J. Heat Mass Transf. 52, 5547–5556 (2009).

    Article  MATH  Google Scholar 

  42. D. Zhang, C. Jiang, D. Liang, et al., “A refined volume-of-fluid algorithm for capturing sharp fluid interfaces on arbitrary meshes,” J. Comput. Phys. 274, 709–736 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  43. E. Bertolazzi and G. Manzini, “A second-order maximum principle preserving finite volume method for steady convection-diffusion problems,” SIAM J. Numer. Anal. 43, 2172–2199 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  44. J. Droniou and C. L. Potier, “Construction and convergence study of schemes preserving the elliptic local maximum principle,” SIAM J. Numer. Anal. 49, 459–490 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  45. K. Lipnikov, D. Svyatskiy, and Y. V. Vassilevski, “Minimal stencil finite volume scheme with the discrete maximum principle,” Russ. J. Numer. Anal. Math. Model. 27, 369–386 (2012).

    Article  MathSciNet  MATH  Google Scholar 

  46. A. Chernyshenko and Y. Vassilevski, “A finite volume scheme with the discrete maximum principle for diffusion equations on polyhedral meshes,” Finite Volumes for Complex Applications VII-Methods and Theoretical Aspects (Springer, 2014), pp. 197–205.

    Google Scholar 

  47. K. M. Terekhov, B. T. Mallison, and H. A. Tchelepi, “Cell-centered nonlinear finite-volume methods for the heterogeneous anisotropic diffusion problem,” J. Comput. Phys. 330, 245–267 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  48. J. Lee and D. A. Sheen, “A parallel method for backward parabolic problems based on the Laplace transformation,” SIAM J. Numer. Anal. 44, 1466–1486 (2006).

    Article  MathSciNet  MATH  Google Scholar 

  49. R. S. Varga, Matrix Iterative Analysis, Prentice-Hall Series in Automatic Computation (Prentice-Hall, Englewood Cliffs, 1962).

    Google Scholar 

  50. A. Jameson, “Analysis and design of numerical schemes for gas dynamics, 1: Artificial diffusion, upwind biasing, limiters and their effect on accuracy and multigrid convergence,” Int. J. Comput. Fluid Dyn. 4 (3–4), 171–218 (1995).

    Article  Google Scholar 

  51. K. Lipnikov, D. Svyatskiy, and Y. V. Vassilevski, “Anderson acceleration for nonlinear finite volume scheme for advection-diffusion problems,” SIAM J. Sci. Comput. 35 (2), (2013).

  52. B. Perot, “Conservation properties of unstructured staggered mesh schemes,” J. Comput. Phys. 159, 58–89 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  53. R. Younis, H. A. Tchelepi, and K. Aziz, “Adaptively localized continuation–Newton method–nonlinear solvers that converge all the time,” Soc. Pet. Eng. J. 15, 526–544 (2010).

    Google Scholar 

  54. Y. Vassilevski, K. Terekhov, K. Nikitin, and I. Kapyrin, Parallel Finite Volume Computation on General Meshes (Springer Nature, 2020).

    Book  MATH  Google Scholar 

  55. K. Terekhov, “Parallel multilevel linear solver within INMOST platform,” Russian Supercomputing Days (Springer, 2020), pp. 297–309.

    Google Scholar 

  56. K. Terekhov “Greedy dissection method for shared parallelism in incomplete factorization within INMOST platform,” Russian Supercomputing Days (Springer, 2021), pp. 87–101.

    Google Scholar 

  57. K. Terekhov and Y. Vassilevski, “Mesh modification and adaptation within INMOST programming platform, Numerical Geometry, Grid Generation and Scientific Computing (Springer, 2019), pp. 243–255.

    Google Scholar 

  58. K. Terekhov, “Parallel dynamic mesh adaptation within INMOST platform,” Russian Supercomputing Days (Springer, 2019), pp. 313–326.

    Google Scholar 

  59. G. Karypis, K. Schloegel, and V. Kumar, Parmetis parallel graph partitioning and sparse matrix ordering library, Tech. Rep. 97-060, Univ. of Minnesota, Department of Computer Science and Engineering, 1997.

  60. G. Karypis and V. Kumar, MeTis: Unstructured Graph Partitioning and Sparse Matrix Ordering System, Version 4.0, 2009. http://www.cs.umn.edu/ metis.

  61. D. Enright, R. Fedkiw, J. Ferziger, and I. Mitchell, “A hybrid particle level set method for improved interface capturing,” J. Comput. Phys. 183, 83–116 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  62. J. Ahrens, B. Geveci, and C. Law, “Paraview: An end-user tool for large data visualization,” The Visualization Handbook, 2005, Vol. 717, no. 8.

Download references

ACKNOWLEDGMENTS

Kirill Terekhov is grateful to Mallison and Hamdi Tchelepi for discussions on the application of the nonlinear finite volume method to the advection problem. Yashar Mehmani is acknowledged for the discussion of the interface compression problem.

Funding

This work was supported by the Moscow Center of Fundamental and Applied Mathematics, project no. 075-15-2019-1624.

Author information

Authors and Affiliations

  1. Marchuk Institute of Numerical Mathematics, Russian Academy of Sciences, 119333, Moscow, Russia

    Yu. V. Vassilevski & K. M. Terekhov

  2. Sechenov University, 119991, Moscow, Russia

    Yu. V. Vassilevski

  3. Moscow Institute of Physics and Technology, 141701, Dolgoprudnyi, Moscow oblast, Russia

    K. M. Terekhov

Authors
  1. Yu. V. Vassilevski
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. K. M. Terekhov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Yu. V. Vassilevski or K. M. Terekhov.

Additional information

Translated by A. Klimontovich

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Vassilevski, Y.V., Terekhov, K.M. Nonlinear Finite Volume Method for the Interface Advection-Compression Problem on Unstructured Adaptive Meshes. Comput. Math. and Math. Phys. 62, 1041–1058 (2022). https://doi.org/10.1134/S0965542522060148

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542522060148

Keywords:

Search

Navigation