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A Hierarchical Space-Time Spectral Element and Moment-of-Fluid Method for Improved Capturing of Vortical Structures in Incompressible Multi-phase/Multi-material Flows

  • Chaoxu Pei
  • Mehdi Vahab
  • Mark SussmanEmail author
  • M. Yousuff Hussaini
Article
  • 37 Downloads

Abstract

A novel block structured adaptive space-time spectral element and moment-of-fluid method is described for computing solutions to incompressible multi-phase/multi-material flows. The new method implements a space-time spectrally accurate method in the bulk regions of a multi-phase/multi-material flow and implements the cell integrated semi-Lagrangian moment-of-fluid method in the vicinity of mixed material computational cells. In the new method, the space-time order can be prescribed to be \(2\le p_{\ell }^{(x)}\le 16\) (space) and \(2\le p^{(t)}\le 16\) (time) respectively. \(\ell \) represents the adaptive mesh refinement level. Regardless of the space-time order, only one ghost layer of cells is communicated between neighboring grid patches that are on different compute nodes or different adaptive levels \(\ell \). The new method is first tested on incompressible vortical flow benchmark tests, then the new method is tested on the following incompressible multi-phase/multi-material problems: (i) vortex shedding past a tilted cone and (ii) atomization and spray of a liquid jet in a gas cross-flow.

Keywords

Space-time Multi-phase flow Multi-material flow Spectral accuracy Adaptive mesh refinement Scalable algorithm 

Mathematics Subject Classification

65B05 65M70 

Notes

Acknowledgements

This work and the authors were supported in part by the National Science Foundation under Contract DMS 1418983.

References

  1. 1.
    Abbassi, H., Mashayek, F., Jacobs, G.B.: Shock capturing with entropy-based artificial viscosity for staggered grid discontinuous spectral element method. Comput. Fluids 98, 152–163 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Almgren, A.S., Aspden, A.J., Bell, J.B., Minion, M.L.: On the use of higher-order projection methods for incompressible turbulent flow. SIAM J. Sci. Comput. 35(1), B25–B42 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H., Welcome, M.L.: A conservative adaptive projection method for the variable density incompressible Navier–Stokes equations. J. Comput. Phys. 142(1), 1–46 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Arienti, M., Sussman, M.: An embedded level set method for sharp-interface multiphase simulations of diesel injectors. Int. J. Multiph. Flow 59, 1–14 (2014)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bao, W., Jin, S.: Weakly compressible high-order i-stable central difference schemes for incompressible viscous flows. Comput. Methods Appl. Mech. Eng. 190(37), 5009–5026 (2001)zbMATHCrossRefGoogle Scholar
  6. 6.
    Bell, J., Berger, M., Saltzman, J., Welcome, M.: Three-dimensional adaptive mesh refinement for hyperbolic conservation laws. SIAM J. Sci. Comput. 15(1), 127–138 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bell, J.B., Colella, P., Glaz, H.M.: A second-order projection method for the incompressible Navier–Stokes equations. J. Comput. Phys. 85(2), 257–283 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Berger, M., Rigoutsos, I.: An algorithm for point clustering and grid generation. IEEE Trans. Syst. Man Cybern. 21(5), 1278–1286 (1991)CrossRefGoogle Scholar
  9. 9.
    Bourlioux, A., Layton, A.T., Minion, M.L.: High-order multi-implicit spectral deferred correction methods for problems of reactive flow. J. Comput. Phys. 189(2), 651–675 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Brown, R.E.: Rotor wake modeling for flight dynamic simulation of helicopters. AIAA J. 38(1), 57–63 (2000)CrossRefGoogle Scholar
  11. 11.
    Brown, R.E., Line, A.J.: Efficient high-resolution wake modeling using the vorticity transport equation. AIAA J. 43(7), 1434–1443 (2005)CrossRefGoogle Scholar
  12. 12.
    Chatelain, P., Curioni, A., Bergdorf, M., Rossinelli, D., Andreoni, W., Koumoutsakos, P.: Billion vortex particle direct numerical simulations of aircraft wakes. Comput. Methods Appl. Mech. Eng. 197(13–16), 1296–1304 (2008)zbMATHCrossRefGoogle Scholar
  13. 13.
    Constantin, P., Titi, E.: On the evolution of nearly circular vortex patches. Commun. Math. Phys. 119(2), 177–198 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Don, W.S., Gao, Z., Li, P., Wen, X.: Hybrid compact-weno finite difference scheme with conjugate Fourier shock detection algorithm for hyperbolic conservation laws. SIAM J. Sci. Comput. 38(2), A691–A711 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Dou, H.S.: Stability of rotating viscous and inviscid flows. arXiv preprint physics/0503083 (2005)Google Scholar
  16. 16.
    Dou, H.S.: Mechanism of flow instability and transition to turbulence. Int. J. Nonlinear Mech. 41(4), 512–517 (2006).  https://doi.org/10.1016/j.ijnonlinmec.2005.12.002 CrossRefzbMATHGoogle Scholar
  17. 17.
    Dubey, A., Almgren, A., Bell, J., Berzins, M., Brandt, S., Bryan, G., Colella, P., Graves, D., Lijewski, M., Loffler, F., O’Shea, B., Schnetter, E., Straalen, B.V., Weide, K.: A survey of high level frameworks in block-structured adaptive mesh refinement packages. J. Parallel Distrib. Comput. 74(12), 3217–3227 (2014)CrossRefGoogle Scholar
  18. 18.
    Duffy, A., Kuhnle, A., Sussman, M.: An improved variable density pressure projection solver for adaptive meshes (2012). https://www.math.fsu.edu/~sussman/MGAMR.pdf. Accessed 5 Apr 2011
  19. 19.
    Dumbser, M., Zanotti, O., Hidalgo, A., Balsara, D.S.: Ader-weno finite volume schemes with space-time adaptive mesh refinement. J. Comput. Phys. 248, 257–286 (2013).  https://doi.org/10.1016/j.jcp.2013.04.017 MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Erlangga, Y.A., Oosterlee, C.W., Vuik, C.: A novel multigrid based preconditioner for heterogeneous Helmholtz problems. SIAM J. Sci. Comput. 27(4), 1471–1492 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Fambri, F., Dumbser, M.: Spectral semi-implicit and space-time discontinuous Galerkin methods for the incompressible Navier–Stokes equations on staggered Cartesian grids. Appl. Numer. Math. 110, 41–74 (2016).  https://doi.org/10.1016/j.apnum.2016.07.014 MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Garrick, D.P., Hagen, W.A., Regele, J.D.: An interface capturing scheme for modeling atomization in compressible flows. J. Comput. Phys. 344, 260–280 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Itoh, S., Namekawa, Y.: An improvement in DS-BICGstab (l) and its application for linear systems in lattice QCD. J. Comput. Appl. Math. 159(1), 65–75 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Jacobs, G.B., Kopriva, D.A., Mashayek, F.: A conservative isothermal wall boundary condition for the compressible Navier–Stokes equations. J. Sci. Comput. 30(2), 177–192 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Jemison, M., Sussman, M., Arienti, M.: Compressible, multiphase semi-implicit method with moment of fluid interface representation. J. Comput. Phys. 279, 182–217 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  26. 26.
    Kadioglu, S.Y., Klein, R., Minion, M.L.: A fourth-order auxiliary variable projection method for zero-Mach number gas dynamics. J. Comput. Phys. 227(3), 2012–2043 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Kadioglu, S.Y., Sussman, M.: Adaptive solution techniques for simulating underwater explosions and implosions. J. Comput. Phys. 227(3), 2083–2104 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Kamkar, S., Wissink, A., Sankaran, V., Jameson, A.: Feature-driven Cartesian adaptive mesh refinement for vortex-dominated flows. J. Comput. Phys. 230(16), 6271–6298 (2011).  https://doi.org/10.1016/j.jcp.2011.04.024 MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Klaij, C.M., van der Vegt, J.J.W., van der Ven, H.: Space-time discontinuous Galerkin method for the compressible Navier–Stokes equations. J. Comput. Phys. 217(2), 589–611 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Lakehal, D.: Status and future developments of large-Eddy simulation of turbulent multi-fluid flows (leis and less). Int. J. Multiph. Flow 104, 322–337 (2018).  https://doi.org/10.1016/j.ijmultiphaseflow.2018.02.018 MathSciNetCrossRefGoogle Scholar
  31. 31.
    Lalanne, B., Rueda Villegas, L., Tanguy, S., Risso, F.: On the computation of viscous terms for incompressible two-phase flows with level set/ghost fluid method. J. Comput. Phys. 301, 289–307 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  32. 32.
    Layton, A.T.: On the choice of correctors for semi-implicit Picard deferred correction methods. Appl. Numer. Math. 58(6), 845–858 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    Layton, A.T.: On the efficiency of spectral deferred correction methods for time-dependent partial differential equations. Appl. Numer. Math. 59(7), 1629–1643 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Layton, A.T., Minion, M.L.: Conservative multi-implicit spectral deferred correction methods for reacting gas dynamics. J. Comput. Phys. 194(2), 697–715 (2004).  https://doi.org/10.1016/j.jcp.2003.09.010 MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Li, G., Lian, Y., Guo, Y., Jemison, M., Sussman, M., Helms, T., Arienti, M.: Incompressible multiphase flow and encapsulation simulations using the moment-of-fluid method. Int. J. Numer. Methods Fluids 79(9), 456–490 (2015).  https://doi.org/10.1002/fld.4062 MathSciNetCrossRefGoogle Scholar
  36. 36.
    Li, X., Soteriou, M.C.: High fidelity simulation and analysis of liquid jet atomization in a gaseous crossflow at intermediate weber numbers. Phys. Fluids 28(8), 082,101 (2016)CrossRefGoogle Scholar
  37. 37.
    Liovic, P., Lakehal, D.: Interface-turbulence interactions in large-scale bubbling processes. Int. J. Heat Fluid Flow 28(1), 127–144 (2007).  https://doi.org/10.1016/j.ijheatfluidflow.2006.03.003 CrossRefzbMATHGoogle Scholar
  38. 38.
    Liovic, P., Rudman, M., Liow, J.L., Lakehal, D., Kothe, D.: A 3d unsplit-advection volume tracking algorithm with planarity-preserving interface reconstruction. Comput. Fluids 35(10), 1011–1032 (2006).  https://doi.org/10.1016/j.compfluid.2005.09.003 MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Liu, J.G., Shu, C.W.: A high-order discontinuous Galerkin method for 2D incompressible flows. J. Comput. Phys. 160(2), 577–596 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Liu, J.G., Wang, W.C.: Energy and helicity preserving schemes for hydro-and magnetohydro-dynamics flows with symmetry. J. Comput. Phys. 200(1), 8–33 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Mcinnes, L.C., Smith, B., Zhang, H., Mills, R.T.: Hierarchical Krylov and nested Krylov methods for extreme-scale computing. Parallel Comput. 40(1), 17–31 (2014)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Miyauchi, T., Itoh, S., Zhang, S.L., Natori, M.: Dynamic selection of l for BI-CGstab (l). Trans. Jpn. Soc. Ind. Appl. Math. 11(2), 49–62 (2001)Google Scholar
  43. 43.
    Montagnac, M., Chesneaux, J.M.: Dynamic control of a BiCGSTab algorithm. Appl. Numer. Math. 32(1), 103–117 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Morinishi, Y., Lund, T., Vasilyev, O., Moin, P.: Fully conservative higher order finite difference schemes for incompressible flow. J. Comput. Phys. 143(1), 90–124 (1998).  https://doi.org/10.1006/jcph.1998.5962 MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Nonaka, A., Bell, J., Day, M., Gilet, C., Almgren, A., Minion, M.: A deferred correction coupling strategy for low mach number flow with complex chemistry. Combust. Theory Model. 16, 1053–1088 (2012)CrossRefGoogle Scholar
  46. 46.
    Palha, A., Gerritsma, M.: A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2d incompressible Navier–Stokes equations. J. Comput. Phys. 328, 200–220 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Pazner, W.E., Nonaka, A., Bell, J.B., Day, M.S., Minion, M.L.: A high-order spectral deferred correction strategy for low Mach number flow with complex chemistry. Combust. Theory Model. 20(3), 521–547 (2016)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Pei, C., Sussman, M., Hussaini, M.Y.: A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete Contin. Dyn. Syst. B 23(9), 3595–3622 (2018)MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Pei, C., Sussman, M., Hussaini, M.Y.: New multi-implicit space-time spectral element methods for advection–diffusion–reaction problems. J. Sci. Comput. 78(2), 653–686 (2019).  https://doi.org/10.1007/s10915-018-0654-5 MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Pei, C., Sussman, M., Hussaini, M.Y.: A space-time discontinuous galerkin spectral element method for nonlinear hyperbolic problems. Int. J. Comput. Methods 16(01), 1850,093 (2019)MathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Rhebergen, S., Cockburn, B., van der Vegt, J.J.W.: A space-time discontinuous Galerkin method for the incompressible Navier–Stokes equations. J. Comput. Phys. 233, 339–358 (2013)MathSciNetzbMATHCrossRefGoogle Scholar
  52. 52.
    Saad, Y.: A flexible inner–outer preconditioned GMRES algorithm. SIAM J. Sci. Comput. 14(2), 461–469 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  53. 53.
    Saye, R.: Implicit mesh discontinuous galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid-structure interaction, and free surface flow: part I. J. Comput. Phys. 344, 647–682 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  54. 54.
    Saye, R.: Implicit mesh discontinuous Galerkin methods and interfacial gauge methods for high-order accurate interface dynamics, with applications to surface tension dynamics, rigid body fluid–structure interaction, and free surface flow: part II. J. Comput. Phys. 344, 683–723 (2017)MathSciNetzbMATHCrossRefGoogle Scholar
  55. 55.
    Scardovelli, R., Zaleski, S.: Interface reconstruction with least-square fit and split Eulerian–Lagrangian advection. Int. J. Numer. Methods Fluids 41(3), 251–274 (2003)zbMATHCrossRefGoogle Scholar
  56. 56.
    Sleijpen, G.L., Fokkema, D.R.: BiCGStab (l) for linear equations involving unsymmetric matrices with complex spectrum. Electron. Trans. Numer. Anal. 1(11), 2000 (1993)MathSciNetzbMATHGoogle Scholar
  57. 57.
    Sleijpen, G.L., Van der Vorst, H.A.: Maintaining convergence properties of BiCGStab methods in finite precision arithmetic. Numer. Algorithms 10(2), 203–223 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Sollie, W.E.H., Bokhove, O., van der Vegt, J.J.W.: Space-time discontinuous Galerkin finite element method for two-fluid flows. J. Comput. Phys. 230(3), 789–817 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Srinivasan, G., McCroskey, W., Baeder, J., Edwards, T.: Numerical simulation of tip vortices of wings in subsonic and transonic flows. AIAA J. 26(10), 1153–1162 (1988)CrossRefGoogle Scholar
  60. 60.
    Steinhoff, J., Underhill, D.: Modification of the euler equations for “vorticity confinement”: application to the computation of interacting vortex rings. Phys. Fluids 6(8), 2738–2744 (1994)zbMATHCrossRefGoogle Scholar
  61. 61.
    Stewart, P., Lay, N., Sussman, M., Ohta, M.: An improved sharp interface method for viscoelastic and viscous two-phase flows. J. Sci. Comput. 35(1), 43–61 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  62. 62.
    Sussman, M., Almgren, A.S., Bell, J.B., Colella, P., Howell, L.H., Welcome, M.L.: An adaptive level set approach for incompressible two-phase flows. J. Comput. Phys. 148(1), 81–124 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  63. 63.
    Tan, F.J., Wen Wang, H.: Simulating unsteady aerodynamics of helicopter rotor with panel/viscous vortex particle method. Aerosp. Sci. Technol. 30(1), 255–268 (2013).  https://doi.org/10.1016/j.ast.2013.08.010 CrossRefGoogle Scholar
  64. 64.
    van der Vegt, J.J.W., Sudirham, J.J.: A space-time discontinuous Galerkin method for the time-dependent Oseen equations. Appl. Numer. Math. 58(12), 1892–1917 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  65. 65.
    Van der Vorst, H.A.: Bi-CGStab: a fast and smoothly converging variant of BI-CG for the solution of nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 13(2), 631–644 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  66. 66.
    Van der Vorst, H.A., Vuik, C.: GMRESR: a family of nested GMRES methods. Numer. Linear Algebra Appl. 1(4), 369–386 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  67. 67.
    Zhang, Q.: Gepup: Generic projection and unconstrained ppe for fourth-order solutions of the incompressible Navier–Stokes equations with no-slip boundary conditions. J. Sci. Comput. 67(3), 1134–1180 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  68. 68.
    Zhang, W., Almgren, A., Day, M., Nguyen, T., Shalf, J., Unat, D.: Boxlib with tiling: an adaptive mesh refinement software framework. SIAM J. Sci. Comput. 38(5), S156–S172 (2016)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA
  2. 2.Department of Mechanical EngineeringFAMU-FSU College of EngineeringTallahasseeUSA

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