On the local energy flux of turbulent flows

Abstract: We investigate the local energy flux rate $\Pi_\ell(\bf x)$ towards small scales in isotropic turbulent flows using direct numerical simulations and applying different low-pass filters. Two different filters are examined, a sharp Fourier filter and a Gaussian filter. The probability density function (pdf) of the local energy flux is calculated for the different filters and for different filtering scales. It is shown that the local energy flux is a largely fluctuating quantity taking both negative and positive values and this is more pronounced for the sharp filter. The variance, the skewness and the kurtosis of these fluctuations are shown to increase as the filtering scale is decreased. Furthermore we calculate the joint pdf of $\Pi_\ell(\bf x)$ with the local filtered strain rate $S_\ell$ and the enstrophy $\Omega_\ell$. The flux shows a good correlation with the strain but not with the enstrophy. It is shown that its conditional mean value scales like $<{\Pi_\ell}>S \propto \ell^2 S\ell^{3} $ in support to the Smagorinsky eddy viscosity model. Nonetheless strong fluctuations exist around this value that also need to be modeled. We discuss the implications of our results for subgrid scale models, and propose new modelling directions.