Three regularization models of the Navier-Stokes equations

AMS Citation:
Graham, J. P., D. D. Holm, P. D. Mininni, and A. Pouquet, 2008: Three regularization models of the Navier-Stokes equations. Physics of Fluids, 20, 21 pp, doi:10.1063/1.2880275.
Resource Type:article
Title:Three regularization models of the Navier-Stokes equations
Abstract: We determine how the differences in the treatment of the sub-filter-scale physics affect the properties of the flow for three closely related regularizations of Navier-Stokes. The consequences on the applicability of the regularizations as sub-grid-scale (SGS) models are also shown by examining their effects on super-filter-scale properties. Numerical solutions of the Clark–α model are compared to two previously employed regularizations, the Lagrangian-Averaged Navier-Stokes α–model (LANS–α) and Leray–α albeit at significantly higher Reynolds number than previous studies, namely Re≈3300, Taylor Reynolds number of Rλ≈790, and to a direct numerical simulation (DNS) of the Navier-Stokes equations. We derive the Kármán-Howarth equation for both the Clark–α and Leray–α models. We confirm one of two possible scalings resulting from this equation for Clark–α as well as its associated k-¹ energy spectrum. At sub-filter scales, Clark–α possesses similar total dissipation and characteristic time to reach a statistical turbulent steady-state as Navier-Stokes, but exhibits greater intermittency. As a SGS model, Clark–α reproduces the large-scale energy spectrum and intermittency properties of the DNS. For the Leray–α model, increasing the filter width, α, decreases the nonlinearity and, hence, the effective Reynolds number is substantially decreased. Therefore even for the smallest value of α studied Leray–α was inadequate as a SGS model. The LANS–α energy spectrum ~k¹, consistent with its so-called "rigid bodies," precludes a reproduction of the large-scale energy spectrum of the DNS at high Re while achieving a large reduction in numerical resolution. We find, however, that this same feature reduces its intermittency compared to Clark–α (which shares a similar Kármán-Howarth equation). Clark–α is found to be the best approximation for reproducing the total dissipation rate and the energy spectrum at scales larger than α, whereas high-order intermittency properties for larger values of α are best reproduced by LANS–α.
Peer Review:Refereed
Copyright Information:Copyright 2008 American Institute of Physics.
OpenSky citable URL: ark:/85065/d78053vz
Publisher's Version: 10.1063/1.2880275
  • Jonathan Graham - NCAR/UCAR
  • Darryl Holm
  • Pablo Mininni - NCAR/UCAR
  • Annick Pouquet - NCAR/UCAR
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