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International Journal for Multiscale Computational Engineering
IF: 1.016 5-Year IF: 1.194 SJR: 0.554 SNIP: 0.82 CiteScore™: 2

ISSN Print: 1543-1649
ISSN Online: 1940-4352

International Journal for Multiscale Computational Engineering

DOI: 10.1615/IntJMultCompEng.v6.i4.20
pages 299-307

Analytical Estimates of the Subgrid Model for Burgers Equation: Ramifications for Spectral Methods for Conservation Laws

Assad A. Oberai
Department of Mechanical Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, USA
Carlos E. Colosqui
Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215, USA
John Wanderer
Department of Aerospace and Mechanical Engineering, Boston University, Boston, MA 02215, USA

ABSTRACT

In several numerical methods used for simulating an inviscid, or nearly inviscid, nonlinear conservation law, a wavenumber-dependent viscosity is often employed as a subgrid model. In particular, in the spectral vanishing viscosity and hyperviscosity methods, the viscosity at low wavenumbers is set to zero. In this note, we verify whether this choice is consistent with the wavenumber dependence of the energy transfer to the subgrid scales. We evaluate this transfer for different choices of a desired numerical solution that are made precise by the choice of a restriction operator. We discover that, for the simple model system of Burgers equation, the exact subgrid viscosity is nonzero at low wavenumbers and, hence, the spectral vanishing viscosity and hyperviscosity methods are at odds with the exact subgrid model. We also observe that the exact subgrid viscosity is well described by a nonzero plateau at low wavenumbers, a cusp at the high wavenumbers, and is remarkably similar to the wavenumber-dependent viscosity observed in three-dimensional turbulence. We attribute this similarity to the locality of energy transfer in wavenumber space in both of these systems.


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