Turbulent-like velocity fluctuations in two-dimensional granular materials subject to cyclic shear.

Abstract

We perform a systematic experimental study to investigate the velocity fluctuations in the two-dimensional granular matter of low and high friction coefficients subjected to cyclic shear of a range of shear amplitudes, whose velocity fields are strikingly turbulent-like with vortices of different scales. The scaling behaviors of both the transverse velocity power spectra <i>E</i>_T(<i>k</i>) ∝ <i>k</i>^-<i>α</i>_T and, more severely, the longitudinal velocity power spectra <i>E</i>_L(<i>k</i>) ∝ <i>k</i>^-<i>α</i>_L are affected by the prominent peak centered around <i>k</i> ≈ 2π of the inter-particle distance due to the static structure factor of the hard-particle nature in contrast to the real turbulence. To reduce the strong peak effect to the actual values of <i>α</i>_<i>ν</i> (the subscript '<i>ν</i>' refers to either T or L), we subsequently analyze the second-order velocity structure functions of <i>S</i>(2)ν(<i>r</i>) in real space, which show the power-law scalings of <i>S</i>(2)ν(<i>r</i>) ∝ <i>r</i>^{<i>β</i>_<i>ν</i>} for both modes. From the values of <i>β</i>_<i>ν</i>, we deduce the corresponding <i>α</i>_<i>ν</i> from the scaling relations of <i>α</i>_<i>ν</i> = <i>β</i>_<i>ν</i> + 2. The deduced values of <i>α</i>_<i>ν</i> increase continuously with the shear amplitude <i>γ</i>_m, showing no signature of yielding transition, and are slightly larger than <i>α</i>_<i>ν</i> = 2.0 at the limit of <i>γ</i>_m → 0, which corresponds to the elastic limit of the system, for all <i>γ</i>_m. The inter-particle friction coefficients show no significant effect on the turbulent-like velocity fluctuations. Our findings suggest that the turbulent-like collective particle motions are governed by both the elasticity and plasticity in cyclically sheared granular materials.