Three regularization models of the Navier-Stokes equations

Abstract

We determine how the differences in the treatment of the subfilter-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 subgrid-scale (SGS) models are also shown by examining their effects on superfilter-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 Re≈790, and to a direct numerical simulation (DNS) of the Navier-Stokes equations. We derive the de 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 subfilter 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 de 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. © 2008 American Institute of Physics.