Length-scale distribution functions and conditional means for various fields in turbulence

Dissipation elements are identified for various direct numerical simulation (DNS) fields of homogeneous shear turbulence. The fields are those of the fluctuations of a passive scalar, of the three components of velocity and vorticity, of the second invariant of the velocity gradient tensor, turbulent kinetic energy and viscous dissipation. In each of these fields trajectories starting from every grid point are calculated in the direction of ascending and descending gradients, reaching a local maximum and minimum point, respectively. Dissipation elements are defined as spatial regions containing all the grid points from which the same pair of minimum and maximum points is reached. They are parameterized by the linear length between these points and the difference of the field variable at these points. In analysing the changes that occur during one time step in the linear length as well as in the number of grid points contained in the elements, it is found that rapid splitting and attachment processes occur between elements. These processes are much more frequent than the previously identified processes of cutting and reconnection. The model for the length-scale distribution function that had previously been proposed is modified to include these additional processes. Comparisons of the length-scale distribution function for the various fields with the proposed model show satisfactory agreement. The conditional mean difference of the field variable at the minimum and maximum points of dissipation elements is calculated for the passive scalar field and the three components of velocity. While the conditional mean difference follows the 1/3 inertial-range Kolmogorov scaling for the passive scalar field, the scaling exponent differs from the 1/3 law for each of the three components of velocity. This is thought to be due to the relatively high shear rate of the DNS calculations. The conditional mean viscous dissipation shows, differently from all other field variables analysed, a pronounced dependence on the linear length of elements. This is explained by intermittency. This finding is used to evaluate the production and the dissipation term of the empirically derived ϵ-equation that is often used in engineering calculations.