Effects of small-scale turbulent motions on the filtered velocity gradient tensor as deduced from holographic particle image velocimetry measurements

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.