Fractal character of isoscalar surfaces in shear free turbulence and some effects of shear on the turbulence structure

The fractal scaling of isoscalar level surfaces is examined by analyzing data from direct numerical simulations of turbulence in two and three dimensions. For the two-dimensional case, the advection-diffusion equation for the scalar is solved in a square box of size 81922 by prescribing the velocity field to be a Gaussian random variable possessing power-law scaling in space and rapid variations in time. This part of the work allows us to learn about the effects on scaling of the domain size and the concentration level of isosurfaces. These issues are central to the analysis of three-dimensional data, which are obtained by solving the scalar equation in a periodic box of size 5123 together with the Navier-Stokes equations, for the particular case of homogeneous and isotropic turbulence. The use of homogeneous and isotropic turbulence removes the complicating effects of shear on scaling. For the two-dimensional case, a fractal scaling is found for all isosurfaces. For the three-dimensional case, isosurfaces not too close to the mean concentration level possess a fractal dimension of about 2⅓. For isosurfaces closer to the mean concentration, a subset of two-dimensional sections yields constant local slopes. This observation, as well as consistency with a previous theory, suggests that those isosurfaces also may possess fractal scaling with a dimension is about 2⅔. Implications of these principal results are discussed as well as a deep analysis of fractal measuring methods. This fractal study is connected with a study of the local structure of the flow. We study the geometric transcriptions of the passive scalar level and enstrophy level set, and concluded that no rough differences in the geometry are noted between isoscalar values at the mean and far away from the mean. This supports our conclusions about the limitations of the box-counting algorithm.;We also study the effects of shear on some small-scale properties in the Kolmogorov flow for different Reynolds numbers and shear rates. Several direct numerical simulations of the Kolmogorov flow are obtained using pseudo-spectral techniques with sinusoidal forcing. In particular, we characterize the influence of shear on the second moment of the longitudinal velocity difference. The probability density function of variance of the velocity increments conditioned on the large scale velocity shows a curvature, in agreement with the behavior observed in high-Reynolds data. The PDF of the enstrophy is stretched out to higher values than for the energy dissipation. Second and fourth moments of the locally averaged fields also show a more intermittent character of enstrophy, and showing a second order effect with the shear.