Computational Mechanics

, Volume 58, Issue 6, pp 1051–1069 | Cite as

A pseudo-compressible variational multiscale solver for turbulent incompressible flows

Original Paper


In this work, we design an explicit time-stepping solver for the simulation of the incompressible turbulent flow through the combination of VMS methods and artificial compressibility. We evaluate the effect of the artificial compressibility on the accuracy of the explicit formulation for under-resolved LES simulations. A set of benchmarks have been solved, e.g., the 3D Taylor–Green vortex problem in turbulent regimes. The resulting method is proven to be an effective alternative to implicit methods in some application ranges (in terms of problem size and computational resources), providing comparable results with very low memory requirements. As an example, with the explicit approach, we are able to solve accurately the Taylor-Green vortex benchmark in a fine mesh with \(512^3\) cells on a 12 cores 64 GB ram machine.


Matrix-free Artificial compressibility method Variational multiscale method (VMS) Explicit time stepping Turbulent incompressible flows 



S. Badia’s work has been partially funded by the European Research Council under the FP7 Program Ideas through the Starting Grant No. 258443—COMFUS: Computational Methods for Fusion Technology and the FP7 NUMEXAS project under grant agreement 611636. S. Badia and R. Codina gratefully acknowledge the support received from the Catalan Government through the ICREA Acadèmia Research Program.


  1. 1.
    Badia S, Codina R (2008) Algebraic pressure segregation methods for the incompressible Navier–Stokes equations. Arch Comput Methods Eng 15(3):343–369MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Badia S, Codina R (2009) On a multiscale approach to the transient Stokes problem: dynamic subscales and anisotropic spacetime discretization. Appl Math Comput 207(2):415–433MathSciNetMATHGoogle Scholar
  3. 3.
    Badia S, Martín AF, Principe J (2014) A highly scalable parallel implementation of balancing domain decomposition by constraints. SIAM J Sci Comput 36(2):C190–C218MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Badia S, Martín AF, Principe J (2015) Balancing domain decomposition at extreme scales. SIAM J Sci Comput (in press) Google Scholar
  5. 5.
    Bazilevs Y, Calo VM, Cottrell JA, Hughes TJR, Reali A, Scovazzi G (2007) Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows. Comput Methods Appl Mech Eng 197(1–4):173–201MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Boersma BJ, Kooper MN, Nieuwstadt TTM, Wesseling P (1997) Local grid refinement in Large-Eddy simulations. J Eng Math 32(2/3):161–175MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brachet ME (1991) Direct simulation of three-dimensional turbulence in the Taylor–Green vortex. Fluid Dyn Res 8(1):1–8MathSciNetCrossRefGoogle Scholar
  8. 8.
    Brachet ME, Meiron D, Orszag S, Nickel BG, Morf RH, Frisch U (1983) Small-scale structure of the Taylor–Green vortex. J Fluid Mech 130:411–452CrossRefMATHGoogle Scholar
  9. 9.
    Chorin AJ (1967) A numerical method for solving incompressible viscous flow problems. J Comput Phys 2(1):12–26CrossRefMATHGoogle Scholar
  10. 10.
    Codina R (2002) Stabilized finite element approximation of transient incompressible flows using orthogonal subscales. Comput Methods Appl Mech Eng 191:4295–4321MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Codina R, Blasco J (2002) Analysis of a stabilized finite element approximation of the transient convection-diffusion-reaction equation using orthogonal subscales. Comput Vis Sci 4:167–174MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Codina R, Principe J, Baiges J (2009) Subscales on the element boundaries in the variational two-scale finite element method. Comput Methods Appl Mech Eng 198:838–852MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Codina R, Principe J, Guasch O, Badia S (2007) Time dependent subscales in the stabilized finite element approximation of incompressible flow problems. Comput Methods Appl Mech Eng 196:2413–2430MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Codina R, Soto O (2004) Approximation of the incompressible Navier–Stokes equations using orthogonal subscale stabilization and pressure segregation on anisotropic finite element meshes. Comput Methods Appl Mech Eng 193:1403–1419MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Codina R, Zienkiewicz OC (2002) CBS versus GLS stabilization of the incompressible Navier-Stokes equations and the role of the time step as stabilization parameter. Commun Numer Methods Eng 18:99–112CrossRefMATHGoogle Scholar
  16. 16.
    Colomés O, Badia S, Codina R, Principe J (2015) Assessment of variational multiscale models for the large eddy simulation of turbulent incompressible flows. Comput Methods Appl Mech Eng 285:32–63MathSciNetCrossRefGoogle Scholar
  17. 17.
    DeBonis J (2013) Solutions of the Taylor–Green vortex problem using high-resolution explicit finite difference methods. AIAA Paper 382:2013Google Scholar
  18. 18.
    Ding H, Shu C, Yeo KS, Xu D (2006) Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method. Comput Methods Appl Mech Eng 195(7):516–533CrossRefMATHGoogle Scholar
  19. 19.
    Ghia U, Ghia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J Comput Phys 48(3):387–411CrossRefMATHGoogle Scholar
  20. 20.
    Gravemeier V (2006) The variational mulstiscale method for laminar and turbulent flow. Arch Comput Mech State Art Rev 13:249–324MathSciNetMATHGoogle Scholar
  21. 21.
    Guasch O, Codina R (2013) Statistical behavior of the orthogonal subgrid scale stabilization terms in the finite element large eddy simulation of turbulent flows. Comput Methods Appl Mech Eng 261–262:154–166MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Hughes TJR, Feijoo GR, Mazzei L, Quincy JB (1998) The variational multiscale method—a paradigm for computational mechanics. Comput Methods Appl Mech Eng 166:3–24MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hughes TJR, Mazzei L, Jansen KE (2000) Large eddy simulation and the variational multiscale method. Comput Vis Sci 3:47–59CrossRefMATHGoogle Scholar
  24. 24.
    John V (2005) An assessment of two models for the subgrid scale tensor in the rational LES model. J Comput Appl Math 173(1):57–80MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Lesieur M, Staquet C, Le Roy P, Comte P (1988) The mixing layer and its coherence examined from the point of view of two-dimensional turbulence. J Fluid Mech 192:511–534MathSciNetCrossRefGoogle Scholar
  26. 26.
    Malan AG, Lewis RW, Nithiarasu P (2002) An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part II. Application. Int J Numer Methods Eng 54(5):715–729CrossRefMATHGoogle Scholar
  27. 27.
    Michalke A (1964) On the inviscid instability of the hyperbolic tangent velocity profile. J Fluid Mech 19:543–556MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Moin P, Mahesh K (1998) Direct numerical simulation: a tool in turbulence research. Ann Rev Fluid Mech 30(1):539–578MathSciNetCrossRefGoogle Scholar
  29. 29.
    Nithiarasu P (2003) An efficient artificial compressibility (AC) scheme based on the characteristic based split (CBS) method for incompressible flows. Int J Numer Methods Eng 56(13):1815–1845CrossRefMATHGoogle Scholar
  30. 30.
    Rudi J, Malossi ACI, Isaac T, Stadler G, Gurnis M, Staar PWJ, Ineichen Y, Bekas C, Curioni A, Ghattas O (2015) An extreme-scale implicit solver for complex pdes: highly heterogeneous flow in earth’s mantle. In: Proceedings of the international conference for high performance computing, networking, storage and analysis, p 5. ACMGoogle Scholar
  31. 31.
    Wilcox DC (2006) Turbulence modeling for CFD. DCW Industries, IncorporatedGoogle Scholar
  32. 32.
    Yang J-Y, Yang S-C, Chen Y-N, Hsu C-A (1998) Implicit weighted eno schemes for the three-dimensional incompressible Navier–Stokes equations. J Comput Phys 146(1):464–487CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.School of Mathematics, Computer Science and EngineeringCity, University of LondonLondonUK
  2. 2.Centre Internacional de Metodes Numerics en Enginyeri (CIMNE)Universitat Politècnica de Catalunya (UPC)CastelldefelsSpain
  3. 3.Universitat Politècnica de Catalunya (UPC)BarcelonaSpain

Personalised recommendations