Geometrical statistics of fluid deformation: Restricted Euler approximation and the effects of pressure

The geometrical statistics of fluid deformation are analyzed theoretically within the framework of the restricted Euler approximation, and numerically using direct numerical simulations. The restricted Euler analysis predicts that asymptotically a material line element becomes an eigenvector of the velocity gradient regardless its initial orientation. The asymptotic stretching rate equals the intermediate eigenvalue of the strain rate tensor. Analyses of numerical data show that the pressure Hessian is the leading cause to destroy the alignment between the longest axis of the material element and the strongest stretching eigendirection of the strain rate. It also facilitates the alignment between the longest axis of the element and the intermediate eigendirection of the strain rate during initial evolution, but tends to oppose the alignment later.