Highly turbulent solutions of the Lagrangian-averaged Navier-Stokes α model and their large-eddy-simulation potential

Abstract

We compute solutions of the Lagrangian-averaged Navier-Stokes α - (LANS α) model for significantly higher Reynolds numbers (up to Re 8300) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes inertial range followed by a second inertial range specific to the LANS α model. Both fully helical and nonhelical flows are examined, up to Reynolds numbers of ∼1300. Analysis of the third-order structure function scaling supports the predicted l3 scaling; it corresponds to a k-1 scaling of the energy spectrum for scales smaller than α. The energy spectrum itself shows a different scaling, which goes as k1. This latter spectrum is consistent with the absence of stretching in the subfilter scales due to the Taylor frozen-in hypothesis employed as a closure in the derivation of the LANS α model. These two scalings are conjectured to coexist in different spatial portions of the flow. The l3 [E(k)∼ k-1] scaling is subdominant to k1 in the energy spectrum, but the l3 scaling is responsible for the direct energy cascade, as no cascade can result from motions with no internal degrees of freedom. We demonstrate verification of the prediction for the size of the LANS α attractor resulting from this scaling. From this, we give a methodology either for arriving at grid-independent solutions for the LANS α model, or for obtaining a formulation of the large eddy simulation optimal in the context of the α models. The fully converged grid-independent LANS α model may not be the best approximation to a direct numerical simulation of the Navier-Stokes equations, since the minimum error is a balance between truncation errors and the approximation error due to using the LANS α instead of the primitive equations. Furthermore, the small-scale behavior of the LANS α model contributes to a reduction of flux at constant energy, leading to a shallower energy spectrum for large α. These small-scale features, however, do not preclude the LANS α model from reproducing correctly the intermittency properties of the high-Reynolds-number flow. © 2007 The American Physical Society.