Material deformation in a restricted Euler model for turbulent flows: Analytic solution and numerical tests

The restricted Euler (RE) approximation applied to the equation of the velocity gradient tensor has been shown [Cantwell, Phys. Fluids A 4, 782 (1992)] to reproduce important geometric trends concerning small scale structures in Navier-Stokes turbulence. In this paper, the Lagrangian evolution of material volume elements as they are deformed in a turbulent flow is considered. First, it is shown that the equation for the deformation tensor evolving under RE dynamics admits an analytic solution similar to that of the velocity gradient tensor. The evolution of the eigenvalues and eigenvectors of the Cauchy-Green tensor is obtained from this analytic solution. Comparison with direct numerical simulations (DNS) is performed. Results show that the analytic prediction of shapes and orientations of fluid volume elements agrees well with DNS for an evolution time scale of the order of the Kolmogorov time scale, which is also the time it takes for significant material element deformation. The analysis is repeated in the inertial range, where the deformation due to a filtered velocity field is considered. The analytic solution with filtered velocity gradients as initial conditions is evaluated. Good agreement is found for longer (turn-over) durations, of the order of the eddy turn-over time at the filter scale.