Variationally derived closure models for large eddy simulation of incompressible turbulent flows

Arif Masud, Lixing Zhu

Research output: Contribution to journalArticlepeer-review


We present a variationally consistent method for deriving residual-based closure models for incompressible Navier–Stokes equations. The method is based on the fine-scale variational structure facilitated by the variational multiscale framework where fine scales are driven by the residuals of the Euler–Lagrange equations of the resolved scales in the balance of momentum and conservation of mass equations. A bubble-functions based approach is applied directly to the fine-scale variational equation to derive analytical expressions for the closure model. Variational consistency of the model lends itself to rigorous linearization that results in quadratic rate of convergence of the method in the iterative solution strategy for the nonlinear equations. The method is shown to work for a family of linear and quadratic hexahedral and tetrahedral elements as well as composite discretizations that are comprised of hexahedral and tetrahedral elements in the same computational domain. Numerical tests with the proposed model are presented for various classes of turbulent flow problems to show its generality and range of applicability. The test cases investigated include Taylor–Green vortex stretching, statistically stationary wall-bounded channel flows, and modeling the effects of the geometry of the leading edge of the plate on the instability of the boundary layer that leads to flow separation and flow reversal over flat plates of finite thickness.

Original languageEnglish (US)
Pages (from-to)2089-2120
Number of pages32
JournalInternational Journal for Numerical Methods in Fluids
Issue number7
StatePublished - Jul 2021
Externally publishedYes


  • hexahedral and tetrahedral elements
  • hierarchical variational multiscale method
  • large eddy simulation
  • residual-based turbulence models
  • residual-free bubbles
  • stabilized finite elements

ASJC Scopus subject areas

  • Computational Mechanics
  • Mechanics of Materials
  • Mechanical Engineering
  • Computer Science Applications
  • Applied Mathematics


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