Parameterization of Small Scales of Three-Dimensional Isotropic Turbulence Utilizing Spectral Closures

TLDR

A spectral equation from two‑point closures for 3‑D isotropic turbulence is examined as a subgrid‑scale model with a cutoff wavenumber kc in the inertial range. The goal is to predict large‑scale statistics (k<kc) using a boundary condition at kc without needing small‑scale information. Kraichnan eddy‑viscosity concepts are used to parameterize subgrid transfer, applying a boundary condition at kc, and the same eddy‑viscosity is incorporated into DNS and predictability studies across resolutions. The approach recovers a k−5/3 Kolmogorov spectrum up to kc without artificial dissipation and is valid for both forced stationary and freely decaying turbulence, as shown by DNS and spectral averaging.

Abstract

A spectral equation derived from two-point closures applied to three-dimensional isotropic turbulence is studied from the subgrid-scale modeling point of view, with a cutoff wavenumber kc located in the inertial range of turbulence. Ideas of Kraichnan concerning eddy viscosities are then used to evaluate the parameterized subgrid-scale transfer. This, together with a suitable boundary condition at kc, allows us to predict statistically the large scales (k<kc) without further information about the small scales (k>kc). A k−5/3 energy spectrum extending to kc is recovered without any artificial dissipation range in the neighborhood of kc. This procedure is valid both for forced stationary turbulence and for freely decaying turbulence. The same eddy-viscosity is then introduced in a direct numerical simulation of three-dimensional homogeneous isotropic turbulence without external forcing. Again, the energy spectrum, evaluated by averaging on a spherical shell of radius k, follows the Kolmogorov law up to the cutoff wavenumber. Finally, predictability studies calculating the correlation between two fields with different resolutions are performed.