Matrix exponential-based closures for the turbulent subgrid-scale stress tensor

Yi Li, Laurent Chevillard, Gregory Eyink, and Charles Meneveau
Phys. Rev. E 79, 016305 – Published 14 January 2009

Abstract

Two approaches for closing the turbulence subgrid-scale stress tensor in terms of matrix exponentials are introduced and compared. The first approach is based on a formal solution of the stress transport equation in which the production terms can be integrated exactly in terms of matrix exponentials. This formal solution of the subgrid-scale stress transport equation is shown to be useful to explore special cases, such as the response to constant velocity gradient, but neglecting pressure-strain correlations and diffusion effects. The second approach is based on an Eulerian-Lagrangian change of variables, combined with the assumption of isotropy for the conditionally averaged Lagrangian velocity gradient tensor and with the recent fluid deformation approximation. It is shown that both approaches lead to the same basic closure in which the stress tensor is expressed as the matrix exponential of the resolved velocity gradient tensor multiplied by its transpose. Short-time expansions of the matrix exponentials are shown to provide an eddy-viscosity term and particular quadratic terms, and thus allow a reinterpretation of traditional eddy-viscosity and nonlinear stress closures. The basic feasibility of the matrix-exponential closure is illustrated by implementing it successfully in large eddy simulation of forced isotropic turbulence. The matrix-exponential closure employs the drastic approximation of entirely omitting the pressure-strain correlation and other nonlinear scrambling terms. But unlike eddy-viscosity closures, the matrix exponential approach provides a simple and local closure that can be derived directly from the stress transport equation with the production term, and using physically motivated assumptions about Lagrangian decorrelation and upstream isotropy.

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  • Received 23 August 2008

DOI:https://doi.org/10.1103/PhysRevE.79.016305

©2009 American Physical Society

Authors & Affiliations

Yi Li1,2, Laurent Chevillard1,3, Gregory Eyink4, and Charles Meneveau1

  • 1Department of Mechanical Engineering and Center of Environmental and Applied Fluid Mechanics, The Johns Hopkins University, Baltimore, Maryland 21218, USA
  • 2Department of Applied Mathematics, The University of Sheffield H23B, Hicks Building, Hounsfield Road Sheffield, S3 7RH, United Kingdom
  • 3Laboratoire de Physique de l’École Normale Supérieure de Lyon, CNRS, Université de Lyon, 46 allée d’Italie F-69007 Lyon, France
  • 4Department of Applied Mathematics and Statistics, and Center of Environmental and Applied Fluid Mechanics, The Johns Hopkins University, Baltimore, Maryland, 21218 USA

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Vol. 79, Iss. 1 — January 2009

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