Locally isotropic pressure Hessian in a high-symmetry flow

Regions in a high-symmetry flow [J. Phys. Soc. Jpn. 54, 2132 (1985); Phys. Fluids 6, 2757 (1994)] where the pressure Hessian term remains diagonal for all times are identified. It is found that vorticity also vanishes in such regions. The components of the pressure Hessian tensor are equal only at the origin, toward which the vortices approach. For the case of isotropic pressure Hessian with no vorticity, and in the absence of viscosity, the resulting equations for the evolution of the eigenvalues and the Q invariant of the strain rate tensor are integrated numerically and analytically respectively. It is found that all the eigenvalues and the Q invariant diverge to infinity in finite time. In a recent simulation of a high-symmetry flow [Phys. Fluids 6, 2757 (1994)], it is found that the two (positive) eigenvalues of the strain rate tensor become equal as all the eigenvalues and the Q invariant show trends of divergence and as the vortices approach to the origin.