Numerical study of comparison of vorticity and passive vectors in turbulence and inviscid flows.
The nonlinear vortex stretching in incompressible Navier-Stokes turbulence is compared with a linear stretching process of passive vectors (PVs). In particular, we pay special attention to the difference of these processes under long and short time evolutions. For finite time evolution, we confirm our previous finding that the stretching effect of vorticity is weaker than that of general passive vectors for a majority of the initial conditions with the same energy spectra. The above difference can be explained qualitatively by examining the Biot-Savart formula. In order to see to what extent infinitesimal time development explains the above difference, we examine the probability density functions (PDFs) of the stretching rates of the passive vectors in the vicinity of a solution of Navier-Stokes equations. It is found that the PDFs are found to have a Gaussian distribution, suggesting that there are equally many PVs that stretched less and more than the vorticity. This suggests the importance of the vorticity-strain correlation built up over finite time in turbulence. We also discuss the case of Euler equations, where the dynamics of the Jacobian matrix relating the physical and material coordinates is examined numerically. A kind of alignment problem associated with the Cauchy-Green tensor is proposed and studied using the results of numerical simulations. It is found that vorticity tends to align itself with the most compressing eigenvector of the Cauchy-Green tensor. A two-dimensional counterpart of active-passive comparison is briefly studied. There is no essential difference between stretching of vorticity gradients and that of passive scalar gradients and a physical interpretation is given to it.
Ohkitani
Koji
K
Research Institute for Mathematical Sciences, Kyoto University, Kyoto 606-8502, Japan.