Building Proper Invariants for Eddy-Viscosity Models

  • F. X. TriasEmail author
  • A. Gorobets
  • A. Oliva
Conference paper
Part of the Springer Proceedings in Physics book series (SPPHY, volume 165)


Direct numerical simulations of the incompressible Navier-Stokes equations are limited to relatively low-Reynolds numbers. Therefore, dynamically less complex mathematical formulations are necessary for coarse-grain simulations. Regularization and eddy-viscosity models for LES are examples thereof. They rely on differential operators that should capture well different flow configurations (laminar and 2D flows, near-wall behavior, transitional regime ...). Most of them are based on the combination of invariants of a symmetric second-order tensor that is derived from the gradient of the resolved velocity field. In the present work, they are presented in a framework where the models are represented as a combination of elements of a 5D phase space of invariants. In this way, new models can be constructed by imposing appropriate restrictions in this space.



This work has been financially supported by the Ministerio de Economía y Competitividad, Spain (ENE2014-60577-R) and a Ramón y Cajal postdoctoral contract (RYC-2012-11996). Calculations have been performed on the IBM MareNostrum supercomputer at the Barcelona Supercomputing Center. The authors thankfully acknowledge these institutions.


  1. 1.
    Verstappen, R.: When does eddy viscosity damp subfilter scales sufficiently? J. Sci. Comput. 49(1), 94–110 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Nicoud, F., Toda, H.B., Cabrit, O., Bose, S., Lee, J.: Using singular values to build a subgrid-scale model for large eddy simulations. Phys. Fluids 23(8), 085106 (2011)CrossRefGoogle Scholar
  3. 3.
    Smagorinsky, J.: General circulation experiments with the primitive equations. Mon. Weather Rev. 91, 99–164 (1963)CrossRefGoogle Scholar
  4. 4.
    Nicoud, F., Ducros, F.: Subgrid-scale stress modelling based on the square of the velocity gradient tensor. Flow Turbul. Combust. 62(3), 183–200 (1999)CrossRefzbMATHGoogle Scholar
  5. 5.
    Vreman, A.W.: An eddy-viscosity subgrid-scale model for turbulent shear flow: algebraic theory and applications. Phys. Fluids 16(10), 3670–3681 (2004)CrossRefzbMATHGoogle Scholar
  6. 6.
    Trias, F.X., Folch, D., Gorobets, A., Oliva, A.: Building proper invariants for eddy-viscosity subgrid-scale models. Phys. Fluids 27(6), 065103 (2015)CrossRefGoogle Scholar
  7. 7.
    Moser, R.D., Kim, J., Mansour, N.N.: Direct numerical simulation of turbulent channel flow up to \(Re_{\tau } = 590\). Phys. Fluids 11, 943–945 (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Trias, F.X., Gorobets, A., Oliva, A.: A simple approach to discretize the viscous term with spatially varying (eddy-)viscosity. J. Comput. Phys. 253, 405–417 (2013)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Heat and Mass Transfer Technological CenterTechnical University of Catalonia ETSEIATTerrassaSpain
  2. 2.Keldysh Institute of Applied MathematicsMoscowRussia

Personalised recommendations