A flexion-based approach for the simulation of turbulent flows

Turbulent flows at high Reynolds numbers are dominated by vortex filaments and/or sheets with sharp gradients in the vorticity field near the boundaries of the vortical structures. Numerical simulations of high Reynolds number flows are computationally demanding due to the fine grid required to accurately resolve these sharp gradient regions. In this paper, an alternative approach is proposed to improve the computational efficiency of Navier–Stokes solvers by reformulating the momentum equations as a set of equations for the time-dependent evolution of the flexion field. The flexion vector represents the curl of the vorticity field and is better able to resolve nonlinear effects in regions with large vorticity gradients. The improved resolution capabilities of the flexion-based approach are illustrated through the pseudospectral computations of the rollup of a perturbed 2D shear layer and the transition to a turbulence/viscous decay of the three-dimensional (3D) Taylor–Green vortex. The flexion-based formulation also provides further insight into the dynamics of turbulence through the evolution of the mean-square flexion or palinstrophy. Analysis of data from the Taylor–Green vortex simulations shows that the observed rapid growth of small-scale features and palinstrophy in 3D turbulent flows is primarily associated with flexion amplification by the curl of the vortex stretching vector. Consequently, we hypothesize that the primary physical mechanism responsible for energy cascade from large to small scales is the curl of the vortex stretching vector of interacting vortex tubes, as opposed to the stretching of individual vortex tubes.