Fast Computations for the Lagrangian-averaged Vorticity Deviation Based on the Eulerian Formulations

Abstract

We propose an efficient Eulerian approach to compute the Lagrangian-averaged vorticity deviation (LAVD) of given flow fields. Traditional approaches need to solve [Formula: see text] ordinary differential equations (ODEs) for a [Formula: see text]-dimensional flow. Furthermore, if the velocity data are discrete, interpolation is required to obtain the velocity and vorticity data along the particle trace of any sampling point, which could be quite time-consuming and even affect the accuracy of the solutions. In contrast, our proposed Eulerian approach only needs to solve one single partial differential equation (PDE) in order to obtain the LAVD field and no interpolation is required. Based on the doubling technique, we also propose an efficient iterative Eulerian-type algorithm to compute the long-time LAVD for periodic flows. After that, relation between the long-time LAVD and the coherent ergodic partition is briefly discussed. Numerical examples will show the accuracy, efficiency and effectiveness of our proposed Eulerian approaches.