Shock Waves

, Volume 29, Issue 1, pp 73–99 | Cite as

An adaptive ALE scheme for non-ideal compressible fluid dynamics over dynamic unstructured meshes

  • B. ReEmail author
  • A. Guardone
Original Article


This paper investigates the application of mesh adaptation techniques in the non-ideal compressible fluid dynamic (NICFD) regime, a region near the vapor–liquid saturation curve where the flow behavior significantly departs from the ideal gas model, as indicated by a value of the fundamental derivative of gasdynamics less than one. A recent interpolation-free finite-volume adaptive scheme is exploited to modify the grid connectivity in a conservative way, and the governing equations for compressible inviscid flows are solved within the arbitrary Lagrangian–Eulerian framework by including special fictitious fluxes representing volume modifications due to mesh adaptation. The absence of interpolation of the solution to the new grid prevents spurious oscillations that may make the solution of the flow field in the NICFD regime more difficult and less robust. Non-ideal gas effects are taken into account by adopting the polytropic Peng–Robinson thermodynamic model. The numerical results focus on the problem of a piston moving in a tube filled with siloxane \(\mathrm {MD_4M}\), a simple configuration which can be the core of experimental research activities aiming at investigating the thermodynamic behavior of NICFD flows. Several numerical tests involving different piston movements and initial states in 2D and 3D assess the capability of the proposed adaption technique to correctly capture compression and expansion waves, as well as the generation and propagation of shock waves, in the NICFD and in the non-classical regime.


Non-ideal compressible fluid dynamics Mesh adaptation Peng–Robinson EoS Unsteady Euler equations Piston problem 



This study was partially funded by the European Research Council (Consolidator Grant No. 617603, Project NSHOCK, funded under the FP7-IDEAS-ERC scheme).


  1. 1.
    Vitale, S., Gori, G., Pini, M., Guardone, A., Economon, T.D., Palacios, F., Alonso, J.J., Colonna, P.: Extension of the SU2 open source CFD code to the simulation of turbulent flows of fuids modelled with complex thermophysical laws. 22nd AIAA Computational Fluid Dynamics Conference, Dallas, TX, AIAA Paper 2015–2760 (2015).
  2. 2.
    Gori, G., Vimercati, D., Guardone, A.: Non-ideal compressible-fluid effects in oblique shock waves. J. Phys. Conf. Ser. 821(1), 012003-1–012003-10 (2017). Google Scholar
  3. 3.
    Pini, M., Vitale, S., Colonna, P., Gori, G., Guardone, A., Economon, T., Alonso, J., Palacios, F.: SU2: the open-source software for non-ideal compressible flows. J. Phys. Conf. Ser. 821(1), 012013 (2017). Google Scholar
  4. 4.
    Ameli, A., Uusitalo, A., Turunen-Saaresti, T., Backman, J.: Numerical sensitivity analysis for supercritical \(\text{ CO }_2\) radial turbine performance and flow field. Energy Procedia 129, 1117–1124 (2017). Google Scholar
  5. 5.
    Gori, G., Zocca, M., Cammi, G., Spinelli, A., Guardone, A.: Experimental assessment of the open-source SU2 CFD suite for ORC applications. Energy Procedia 129, 256–263 (2017). Google Scholar
  6. 6.
    Head, A., Iyer, S., de Servi, C., Pini, M.: Towards the validation of a CFD solver for non-ideal compressible flows. Energy Procedia 129, 240–247 (2017). Google Scholar
  7. 7.
    Keep, J.A., Vitale, S., Pini, M., Burigana, M.: Preliminary verification of the open-source CFD solver SU2 for radial-inflow turbine applications. Energy Procedia 129, 1071–1077 (2017). Google Scholar
  8. 8.
    Colonna, P., Guardone, A.: Molecular Interpretation of nonclassical gasdynamics of dense vapors under the van der Waals model. Phys. Fluids 18(5), 56101 (2006).
  9. 9.
    Harinck, J., Guardone, A., Colonna, P.: The influence of molecular complexity on expanding flows of ideal and dense gases. Phys. Fluids 21, 086101 (2009). zbMATHGoogle Scholar
  10. 10.
    Nannan, N.R., Guardone, A., Colonna, P.: Critical point anomalies include expansion shock waves. Phys. Fluids 26, 021701 (2014). Google Scholar
  11. 11.
    Colonna, P., Rebay, S.: Numerical simulation of dense gas flows on unstructured grids with an implicit high resolution upwind Euler solver. Int. J. Numer. Methods Fluids 46(7), 735–765 (2004). MathSciNetzbMATHGoogle Scholar
  12. 12.
    Vinokur, M., Montagné, J.L.: Generalized flux-vector splitting and Roe average for an equilibrium real gas. J. Comput. Phys. 89(2), 276–300 (1990). zbMATHGoogle Scholar
  13. 13.
    Abgrall, R.: An extension of Roe’s upwind scheme to algebraic equilibrium real gas models. Comput. Fluids 19(2), 171–182 (1991). zbMATHGoogle Scholar
  14. 14.
    Arabi, S., Trépanier, J.Y., Camarero, R.: A simple extension of Roe’s scheme for real gases. J. Comput. Phys. 329, 16–28 (2017). MathSciNetGoogle Scholar
  15. 15.
    Colonna, P., Silva, P.: Dense gas thermodynamic properties of single and multicomponent fluids for fluid dynamics simulations. ASME J. Fluids Eng. 125(3), 414–427 (2003). Google Scholar
  16. 16.
    Cinnella, P., Hercus, S.: Efficient implementation of short fundamental equations of state for the numerical simulation of dense gas flows. 42nd AIAA Thermophysics Conference, Fluid Dynamics and Co-located Conferences, Honolulu, HI, AIAA Paper 2011–3947 (2011).
  17. 17.
    Pantano, C., Saurel, R., Schmitt, T.: An oscillation free shock-capturing method for compressible van der Waals supercritical fluid flows. J. Comput. Phys. 335, 780–811 (2017). MathSciNetzbMATHGoogle Scholar
  18. 18.
    Passmann, M., aus der Wiesche, S., Joos, F.: A one-dimensional analytical calculation method for obtaining normal shock losses in supersonic real gas flows. J. Phys. Conf. Ser. 821(1), 012004-1–012004-10 (2017). Google Scholar
  19. 19.
    From, C., Sauret, E., Armfield, S., Saha, S., Gu, Y.: Turbulent dense gas flow characteristics in swirling conical diffuser. Comput. Fluids 149, 100–118 (2017). MathSciNetzbMATHGoogle Scholar
  20. 20.
    Sciacovelli, L., Cinnella, P., Gloerfelt, X.: Direct numerical simulations of supersonic turbulent channel flows of dense gases. J. Fluid Mech. 821, 153–199 (2017). MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dwight, R.P.: Goal-oriented mesh adaptation for finite volume methods using a dissipation-based error indicator. Int. J. Numer. Methods Fluids 56(8), 1193–1200 (2008). MathSciNetzbMATHGoogle Scholar
  22. 22.
    Fidkowski, K.J., Darmofal, D.L.: Review of output-based error estimation and mesh adaptation in computational fluid dynamics. AIAA J. 49(4), 673–694 (2011). Google Scholar
  23. 23.
    Formaggia, L., Perotto, S.: Anisotropic error estimates for elliptic problems. Numer. Math. 94(1), 67–92 (2003). MathSciNetzbMATHGoogle Scholar
  24. 24.
    Coupez, T.: Metric construction by length distribution tensor and edge based error for anisotropic adaptive meshing. J. Comput. Phys. 230(7), 2391–2405 (2011). MathSciNetzbMATHGoogle Scholar
  25. 25.
    Kallinderis, Y., Baron, J.: Adaptation methods for a new Navier-Stokes algorithm. AIAA J. 27(1), 37–43 (1989). Google Scholar
  26. 26.
    Choi, S., Alonso, J.J., van der Weide, E.: Numerical and mesh resolution requirements for accurate sonic boom prediction of complete aircraft configurations. J. Aircr. 46(4), 1126–1139 (2009). Google Scholar
  27. 27.
    Kallinderis, Y., Lymperopoulou, E.M., Antonellis, P.: Flow feature detection for grid adaptation and flow visualization. J. Comput. Phys. 341, 182–207 (2017). MathSciNetGoogle Scholar
  28. 28.
    Huang, W., Russell, R.D.: Adaptive Moving Mesh Methods, Applied Mathematical Sciences book series, vol. 174. Springer, New York (2011).
  29. 29.
    Freitag, L.A., Ollivier-Gooch, C.: Tetrahedral mesh improvement using swapping and smoothing. Int. J. Numer. Methods Eng. 40(21), 3979–4002 (1997).;2-9
  30. 30.
    Mavriplis, D.: Adaptive meshing techniques for viscous flow calculations on mixed element unstructured meshes. Int. J. Numer. Methods Fluids 34(2), 93–111 (2000).;2-3
  31. 31.
    Dapogny, C., Dobrzynski, C., Frey, P.: Three-dimensional adaptive domain remeshing, implicit domain meshing, and applications to free and moving boundary problems. J. Comput. Phys. 262, 358–378 (2014). MathSciNetzbMATHGoogle Scholar
  32. 32.
    Babuška, I., Suri, M.: The \(p\) and \(h{-}p\) versions of the finite element method, basic principles and properties. SIAM Rev. 36(4), 578–632 (1994). MathSciNetzbMATHGoogle Scholar
  33. 33.
    Dolejší, V.: Anisotropic \(hp\)-adaptive method based on interpolation error estimates in the \(H^{1}\)-seminorm. Appl. Math. 60(6), 597–616 (2015). MathSciNetzbMATHGoogle Scholar
  34. 34.
    Guardone, A., Isola, D., Quaranta, G.: Flowmesh. (2012)
  35. 35.
    Guardone, A., Isola, D., Quaranta, G.: Arbitrary Lagrangian Eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping. J. Comput. Phys. 230(20), 7706–7722 (2011). MathSciNetzbMATHGoogle Scholar
  36. 36.
    Isola, D., Guardone, A., Quaranta, G.: Finite-volume solution of two-dimensional compressible flows over dynamic adaptive grids. J. Comput. Phys. 285, 1–23 (2015). MathSciNetzbMATHGoogle Scholar
  37. 37.
    Re, B., Dobrzynski, C., Guardone, A.: An interpolation-free ALE scheme for unsteady inviscid flows computations with large boundary displacements over three-dimensional adaptive grids. J. Comput. Phys. 340, 26–54 (2017). MathSciNetzbMATHGoogle Scholar
  38. 38.
    Colonna, P., der Stelt, T.P.: FluidProp: A Program for the Estimation of Thermophysical Properties of Fluids (2005).
  39. 39.
    der Waals, J.: On the Continuity of the Gas and Liquid State. PhD Thesis, University of Leiden, Leiden, The Netherlands (1873)Google Scholar
  40. 40.
    Peng, D.Y., Robinson, D.B.: A new two-constant equation of state. Ind. Eng. Chem. Fundam. 15(1), 59–64 (1976). Google Scholar
  41. 41.
    Martin, J.J., Hou, Y.C.: Development of an equation of state for gases. AIChE J. 1(2), 142–151 (1955). Google Scholar
  42. 42.
    Redlich, O., Kwong, J.N.S.: On thermodynamics of solutions V: an equation of state. Fugacities of gaseous solutions. Chem. Rev. 44(1), 233–244 (1949). Google Scholar
  43. 43.
    Soave, G.: Equilibrium constants from a modified Redlich–Kwong equation of state. Chem. Eng. Sci. 27(6), 1197–1203 (1972). Google Scholar
  44. 44.
    Dobrzynski, C., Dapogny, C., Frey, P., Froehly, A.: Mmg PLATFORM.
  45. 45.
    Bryson, A.E., Greif, R.: Measurements in a free piston shock tube. AIAA J. 3(1), 183–184 (1965). Google Scholar
  46. 46.
    Stalker, R.J.: The free-piston shock tube. Aeronaut. Q. 17(4), 351–370 (1966). Google Scholar
  47. 47.
    Kewley, D.J., Hornung, H.G.: Free-piston shock-tube study of nitrogen dissociation. Chem. Phys. Lett. 25(4), 531–536 (1974). Google Scholar
  48. 48.
    Hannemann, K., Itoh, K., Mee, D.J., Hornung, H.G.: Free Piston Shock Tunnels HEG, HIEST, T4 and T5. In: Igra, O., Seiler, F. (eds.) Experimental Methods of Shock Wave Research. Shock Wave Science and Technology Reference Library, vol 9, pp. 181–264. Springer, Cham (2016).
  49. 49.
    Re, B., Dobrzynski, C., Guardone, A.: Assessment of grid adaptation criteria for steady, two-dimensional, inviscid flows in non-ideal compressible fluids. Appl. Math. Comput. 319, 337–354 (2018). MathSciNetGoogle Scholar
  50. 50.
    Thompson, P.A.: A fundamental derivative in gasdynamics. Phys. Fluids 14(9), 1843–1849 (1971). zbMATHGoogle Scholar
  51. 51.
    Cramer, M.S., Best, L.M.: Steady, isentropic flows of dense gases. Phys. Fluids A 3(1), 219–226 (1991). zbMATHGoogle Scholar
  52. 52.
    Kluwick, A.: Transonic nozzle flow of dense gases. J. Fluid Mech. 247, 661–688 (1993). MathSciNetzbMATHGoogle Scholar
  53. 53.
    Cramer, M.S., Kluwick, A.: On the propagation of waves exhibiting both positive and negative nonlinearity. J. Fluid Mech. 142, 9–37 (1984). MathSciNetzbMATHGoogle Scholar
  54. 54.
    Menikoff, R., Plohr, B.J.: The Riemann problem for fluid flow of real materials. Rev. Mod. Phys. 61, 75–130 (1989). MathSciNetzbMATHGoogle Scholar
  55. 55.
    Nannan, N.R., Sirianni, C., Mathijssen, T., Guardone, A., Colonna, P.: The admissibility domain of rarefaction shock waves in the near-critical vapour–liquid equilibrium region of pure typical fluids. J. Fluid Mech. 795, 241–261 (2016). MathSciNetzbMATHGoogle Scholar
  56. 56.
    Guardone, A., Argrow, B.M.: Nonclassical gasdynamic region of selected fluorocarbons. Phys. Fluids 17(11), 116102 (2005). zbMATHGoogle Scholar
  57. 57.
    Callen, H.B.: Thermodynamics and an Introduction to Thermostatistics, 2nd edn. Wiley, New York (1985)zbMATHGoogle Scholar
  58. 58.
    Span, R., Wagner, W.: Equations of state for technical applications. I. Simultaneously optimized functional forms for nonpolar and polar fluids. Int. J. Thermophys. 24(1), 1–39 (2003). Google Scholar
  59. 59.
    Lemmon, E.W., Span, R.: Short fundamental equations of state for 20 industrial fluids. J. Chem. Eng. Data 51(3), 785–850 (2006). Google Scholar
  60. 60.
    Rinaldi, E., Pecnik, R., Colonna, P.: Exact Jacobians for implicit Navier–Stokes simulations of equilibrium real gas flows. J. Comput. Phys. 270, 459–477 (2014). MathSciNetzbMATHGoogle Scholar
  61. 61.
    Pini, M., Spinelli, A., Persico, G., Rebay, S.: Consistent look-up table interpolation method for real-gas flow simulations. Comput. Fluids 107, 178–188 (2015). MathSciNetzbMATHGoogle Scholar
  62. 62.
    Moraga, F., Hofer, D., Saxena, S., Mallina, R.: Numerical approach for real gas simulations: part I—tabular fluid properties for real gas analysis. In: Proceedings of ASME Turbo Expo 2017, 63148, pp. 1–8 (2017).
  63. 63.
    Poling, B.E., Prausnitz, J.M., O’Connell, J.P.: The Properties of Gases and Liquids, vol. 5. McGraw-Hill, New York (2001)Google Scholar
  64. 64.
    LeVeque, R.J.: Numerical Methods for Conservation Laws. Birkhäuser, Basel (1992)zbMATHGoogle Scholar
  65. 65.
    Re, B.: An Adaptive Interpolation-Free Conservative Scheme for the Three-Dimensional Euler Equations on Dynamic Meshes for Aeronautical Applications. PhD Thesis, Politecnico di Milano, Department of Aerospace Science and Technology (2016)Google Scholar
  66. 66.
    Koren, B.: Defect correction and multigrid for an efficient and accurate computation of airfoil flows. J. Comput. Phys. 77(1), 183–206 (1988). MathSciNetzbMATHGoogle Scholar
  67. 67.
    Isola, D.: An Interpolation-Free Two-Dimensional Conservative ALE Scheme over Adaptive Unstructured Grids for Rotorcraft Aerodynamics. PhD Thesis, Politecnico di Milano, Department of Aerospace Engineering (2012)Google Scholar
  68. 68.
    Carpentieri, G.: An Adjoint-Based Shape-Optimization Method for Aerodynamic Design. PhD Thesis, Technische Universiteit Delft, Netherlands (2009)Google Scholar
  69. 69.
    Harten, A.: High resolution schemes for hyperbolic conservation laws. J. Comput. Phys. 49(3), 357–393 (1983). MathSciNetzbMATHGoogle Scholar
  70. 70.
    Roe, P.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981). MathSciNetzbMATHGoogle Scholar
  71. 71.
    Guardone, A., Vigevano, L.: Roe Linearization for the van der Waals Gas. J. Comput. Phys. 175, 50–78 (2002). zbMATHGoogle Scholar
  72. 72.
    Glaister, P.: An approximate linearised Riemann solver for the Euler equations for real gases. J. Comput. Phys. 74(2), 382–408 (1988). zbMATHGoogle Scholar
  73. 73.
    Cox, C.F., Cinnella, P.: General solution procedure for flows in local chemical equilibrium. AIAA J. 32(3), 519–527 (1994). zbMATHGoogle Scholar
  74. 74.
    Toumi, I.: A weak formulation of Roe’s approximate Riemann solver. J. Comput. Phys. 102(2), 360–373 (1992). MathSciNetzbMATHGoogle Scholar
  75. 75.
    Guardone, A.: Three-dimensional shock tube flows for dense gases. J. Fluid Mech. 583, 423–442 (2007). MathSciNetzbMATHGoogle Scholar
  76. 76.
    Mottura, L., Vigevano, L., Zaccanti, M.: An evaluation of Roe’s scheme generalizations for equilibrium real gas flows. J. Comput. Phys. 138(2), 354–399 (1997). MathSciNetzbMATHGoogle Scholar
  77. 77.
    Cinnella, P.: Roe-type schemes for dense gas flow computations. Comput. Fluids 35(10), 1264–1281 (2006). zbMATHGoogle Scholar
  78. 78.
    Selmin, V.: The node-centred finite volume approach: bridge between finite differences and finite elements. Comput. Methods Appl. Mech. Eng. 102(1), 107–138 (1993). MathSciNetzbMATHGoogle Scholar
  79. 79.
    Batina, J.T.: Unsteady Euler airfoil solutions using unstructured dynamic meshes. AIAA J. 28(8), 1381–1388 (1990). Google Scholar
  80. 80.
    Venkatakrishnan, V., Mavriplis, D.: Implicit method for the computation of unsteady flows on unstructured grids. J. Comput. Phys. 127(2), 380–397 (1996). zbMATHGoogle Scholar
  81. 81.
    Degand, C., Farhat, C.: A three-dimensional torsional spring analogy method for unstructured dynamic meshes. Comput. Struct. 80(3–4), 305–316 (2002). Google Scholar
  82. 82.
    Hirt, C., Amsden, A.A., Cook, J.: An arbitrary Lagrangian–Eulerian computing method for all flow speeds. J. Comput. Phys. 14(3), 227–253 (1974). zbMATHGoogle Scholar
  83. 83.
    Donea, J., Giuliani, S., Halleux, J.: An arbitrary Lagrangian–Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Eng. 33(1), 689–723 (1982). zbMATHGoogle Scholar
  84. 84.
    Formaggia, L., Nobile, F.: Stability analysis of second-order time accurate schemes for ALE-FEM. Comput. Methods Appl. Mech. Eng. 193(39–41), 4097–4116 (2004). MathSciNetzbMATHGoogle Scholar
  85. 85.
    Mavriplis, D.J., Yang, Z.: Construction of the discrete geometric conservation law for high-order time-accurate simulations on dynamic meshes. J. Comput. Phys. 213(2), 557–573 (2006). MathSciNetzbMATHGoogle Scholar
  86. 86.
    Étienne, S., Garon, A., Pelletier, D.: Perspective on the geometric conservation law and finite element methods for ALE simulations of incompressible flow. J. Comput. Phys. 228(7), 2313–2333 (2009). MathSciNetzbMATHGoogle Scholar
  87. 87.
    Lesoinne, M., Farhat, C.: Geometric conservation laws for flow problems with moving boundaries and deformable meshes, and their impact on aeroelastic computations. Comput. Methods Appl. Mech. Eng. 134(1–2), 71–90 (1996). zbMATHGoogle Scholar
  88. 88.
    Johnson, A.A., Tezduyar, T.E.: Advanced mesh generation and update methods for 3D flow simulations. Comput. Mech. 23(2), 130–143 (1999). zbMATHGoogle Scholar
  89. 89.
    Hassan, O., Probert, E., Morgan, K., Weatherill, N.: Unsteady flow simulation using unstructured meshes. Comput. Methods Appl. Mech. Eng. 189(4), 1247–1275 (2000). zbMATHGoogle Scholar
  90. 90.
    Borouchaki, H., George, P.L., Hecht, F., Laug, P., Saltel, E.: Delaunay mesh generation governed by metric specifications. Part I. Algorithms. Finite Elem. Anal. Des. 25(1–2), 61–83 (1997). MathSciNetzbMATHGoogle Scholar
  91. 91.
    Dolejší, V.: Anisotropic mesh adaptation for finite volume and finite element methods on triangular meshes. Comput. Vis. Sci. 1(3), 165–178 (1998). zbMATHGoogle Scholar
  92. 92.
    Del Pino, S.: Metric-based mesh adaptation for 2D Lagrangian compressible flows. J. Comput. Phys. 230(5), 1793–1821 (2011). MathSciNetzbMATHGoogle Scholar
  93. 93.
    Frey, P., Alauzet, F.: Anisotropic mesh adaptation for CFD computations. Comput. Methods Appl. Mech. Eng. 194(48–49), 5068– 5082 (2005). MathSciNetzbMATHGoogle Scholar
  94. 94.
    Re, B., Guardone, A., Dobrzynski, C.: An adaptive conservative ALE approach to deal with large boundary displacements in three-dimensional inviscid simulations. 55th AIAA Aerospace Sciences Meeting, AIAA SciTech Forum, Grapevine, TX, AIAA Paper 2017–1945 (2017). doi:
  95. 95.
    Re, B., Guardone, A., Dobrzynski, C.: Numerical simulation of shock-tube piston problems with adaptive, anisotropic meshes. In: 7th International Conference on Computational Methods for Coupled Problems in Science and Engineering, Rhodes Island, Greece, pp. 1227–1238 (2017)Google Scholar
  96. 96.
    Dobrzynski, C., Frey, P.: Anisotropic Delaunay mesh adaptation for unsteady simulations. In: Proceedings of the 17th International Meshing Roundtable. Springer, Heidelberg, pp. 177–194 (2008).
  97. 97.
    Borouchaki, H., Hecht, F., Frey, P.: Mesh gradation control. Int. J. Numer. Methods Eng. 43(6), 1143–1165 (1998).;2-I MathSciNetzbMATHGoogle Scholar
  98. 98.
    Lemmon, E.W., Huber, M.L., McLinden, M.O.: NIST Reference Fluid Thermodynamic and Transport Properties—REFPROP. National Institute of Standards and Technology, Boulder, CO (2013).
  99. 99.
    Colonna, P., Nannan, N.R., Guardone, A., Lemmon, E.W.: Multiparameter equations of state for selected siloxanes. Fluid Phase Equilib. 244(2), 193–211 (2006). Google Scholar
  100. 100.
    Thompson, P.A.: Compressible-Fluid Dynamics. McGraw-Hill, New York (1972)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Aerospace Science and TechnologyPolitecnico di MilanoMilanItaly
  2. 2.Institute of MathematicsUniversity of ZürichZurichSwitzerland

Personalised recommendations