Velocity Fluctuations in Isotropic Turbulence and Their Statistical Dependence

Abstract

While velocity differences in turbulence have attracted much interest, velocity fluctuations themselves are also fundamental in describing turbulence. Even though there is a large literature on the non-Gaussian nature of the probability density functions (PDFs) of the turbulent velocity gradients, the PDFs of the velocity fluctuations in homogeneous turbulence is often assumed to be Gaussian. In fact, many experiments yield a sub-Gaussian PDF, which has a less pronounced tail than a Gaussian PDF. Systematic examination of grid turbulence has shown that at small distances from the grid, where the turbulence is still developing, the PDF is sub-Gaussian. At intermediate distances, where the turbulence is fully developed, the PDF is Gaussian. At large distances, where the turbulence decays, the PDF is hyper-Gaussian. Turbulence is induced by supplying kinetic energy at some scale L. This energy could be transferred to both the larger and the smaller scales. However, the energy is on average transferred to smaller scales because it is eventually dissipated into heat at the smallest scales, described by the Kolmogorov length η. The energy transfer from largest scales L to smallest scales η consists of many random steps, each of which occurs preferentially between neighboring scales. Τurbulence exhibits significant velocity fluctuations even at scales much larger than the scales of the energy supply. These large-scale fluctuations have many degrees of freedom and are thereby analogous to thermal fluctuations studied in the statistical mechanics. We perform Direct Numerical Simulations (DNS) in the presence of homogenous isotropic turbulence to show that the deviation from the Gaussian behavior is a natural consequence of the steepness of the energy spectrum, and of the properties of the energy-containing eddies. A clear dependence is observed based on the ratio of the integral length varies as a function of the periodic-box size for different Reynolds numbers.