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Kármán-Howarth theorem for the Lagrangian-averaged Navier-Stokes-alpha model of turbulence. (English) Zbl 1064.76058

Summary: The Lagrangian averaged Navier-Stokes-alpha (LANS-\(\alpha\)) model of turbulence is found to possess a Kármán-Howarth (KH) theorem for the dynamics of its second-order autocorrelation functions in homogeneous isotropic turbulence. This KH result implies that alpha-filtering in the LANS-\(\alpha\) model of turbulence does not affect the exact Navier-Stokes relation between second and third moments at separation distances large compared to the model’s length scale \(\alpha\). Moreover, at separations \(r\) that are smaller than \(\alpha\), the KH scaling between energy dissipation rate and longitudinal third-order autocorrelation changes to match the scaling found in two-dimensional incompressible flow. This is consistent with the corresponding change in scaling of the kinetic energy spectrum from \(k^{-5/3}\) for larger scales with \(k\alpha< 1\), which switches to \(k^{-3}\) for smaller scales with \(k\alpha> 1\), as discovered in C. Foias, D. D. Holm and E. S. Titi [Physica D 152–153, 505–519 (2001; Zbl 1037.76022)].

MSC:

76F05 Isotropic turbulence; homogeneous turbulence
76F55 Statistical turbulence modeling

Citations:

Zbl 1037.76022
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