Local flow structure of turbulence in three, four, and five dimensions

Probability density functions (PDF's) of the eigenvalues of the strain tensor of an incompressible isotropic turbulence in 3, 4, and 5 dimensions are computed by direct numerical simulation of Navier-Stokes equations. The PDF's of the smallest (negative) eigenvalue are found to be wider than those of the other ones in all dimensions and to be very insensitive to the dimension. In any dimension, the eigenvalues other than the lowest one increase as the lowest one decreases, so that they tend to be positive for the large magnitude of the lowest eigenvalue. In such a situation the flow comes in along a single direction and comes out in the other directions, which is consistent with the dynamics of the Burgers turbulence in d dimensions. It is suggested that a driving motor of most intermittent turbulent structure is the compression along a single direction. For the velocity 2 form the conditional averages of the enstrophy and the total squared strain in three dimensions are computed as functions of the smallest eigenvalue and found to be monotonically increasing as the magnitude of the smallest eigenvalue increases. Also, it is found that PDF of the source term of the Poisson equation for the pressure is positively skewed but tends to be symmetric with increase of the spatial dimension. Dimension effects on the dynamics of the most compressible eigenvalue are argued.