The effects of small-scale motions on the inertial range structure of turbulence are investigated by considering the dynamics of the velocity gradient tensor (Ãij) filtered at scale Δ. In addition to self-interactions and the filtered pressure Hessian, the evolution of Ãij is determined by the subgrid-scale stress tensor. As in the so-called restricted Euler dynamics, the evolution equations can be simplified by considering the invariants RΔ and QΔ of Ãij. The effects of the subgrid-scale stress tensor on RΔ and QΔ can be quantified unambiguously by evaluating conditional averages that appear in the evolution equation for the joint probability distribution function of these invariants. The required conditional averages are computed from three-dimensional measurements of fully developed turbulence in a square duct, at Reτ≈2360. The measurements are performed using holographic particle image velocimetry [Tao et al., Phys. Fluids 12, 941 (2000); Tao et al., J. Fluid Mech. 457, 35 (2002)]. The velocity distributions are spatially filtered in the inertial range using a box filter at about 30 Kolmogorov scales to separate large from small scales. The results show that the subgrid scale (SGS) stresses have significant effect on the evolution of filtered velocity gradients. In particular, along the so-called Vieillefosse tail at RΔ>0 and QΔ<0, they oppose the formation of a finite-time singularity that occurs in restricted Euler dynamics. Various other trends are quantified in different parts of the (RΔ,QΔ) plane. Included are the SGS dissipation rate of kinetic energy, and the effect of the SGS stress in modifying the so-called “discriminant,” which is a conserved quantity in restricted Euler dynamics. A priori tests of the Smagorinsky, nonlinear, and mixed models show that all reproduce the real SGS stress effect along the Vieillefosse tail, but that they fail in several other regions. An attempt is made to optimize the mixed model by letting the two coefficients be functions of RΔ and QΔ.

The numerical distortion of the smallest resolved-scale dynamics in large-eddy simulation may be understood in terms of the filter that is induced by the spatial discretization. At marginal subfilter resolution r = ⌬ / h, with filter width ⌬ and grid spacing h, the character of the large-eddy closure problem is strongly influenced by the numerical method. We show that additional high-pass contributions arise from the spatial discretization. The relative importance of, on the one hand, the turbulent stresses and, on the other hand, the numerically induced contributions, is quantified for general finite differencing methods. We derive and analyze the induced filters for several popular discretization methods, including higher order central and upwind methods. The application of these induced filters to small-scale turbulent flow structures gives rise to a characteristic amplitude reduction and phase shift. Their dynamic relevance is quantified in terms of the subfilter resolution. The numerical high-pass effects are found to be negligible if the subfilter resolution is large enough ͑r տ 4͒. Conversely, the numerically induced effects are comparable to, or even larger than the turbulent stresses as r =1-2.

We analyze the large-eddy equations that are obtained from the application of a spatially nonuniform filter to the Navier-Stokes equations. Next to the well-known turbulent stress tensor a second group of subgrid terms arises; the so-called commutator errors. These additional subgrid terms emerge in the large-eddy equations solely as a consequence of the nonuniformity of the filter. We compare the magnitude of the divergence of the turbulent stress tensor with that of the commutator errors and pay attention to the role of the explicit filter that is adopted. It is shown that the turbulent stress contributions and the commutator errors display the same scaling behavior on the filter width and its derivative. Correspondingly, the use of higher-order filters is shown not only to decrease the commutator errors but likewise the turbulent stresses are uniformly reduced with increasing filter order. In addition, we establish that skewness of the spatial filter has a strong influence on the magnitude of the commutator errors while leaving the turbulent stress contributions roughly unaltered. The analysis of the order of magnitude of the various terms provides an impression of the flow conditions and filter width nonuniformities which necessitate explicit commutator-error modeling next to the more traditional turbulent stress subgrid modeling. Some explicit models for the commutator errors are put forward and an a priori assessment of these commutator-error models in turbulent mixing layers is obtained. Generalized similarity models appear promising in this respect and display high correlation with the exact commutator errors.

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