A finite element BFGS algorithm for the reconstruction of the flow field generated by vortex rings

Abstract

- Vortex rings are generated in numerous practical applications (biological propulsion, internal combustion engines) by the sudden injection of a fluid. Measurements of such flows often provide partial information, generally limited to the exterior of the vortex ring. We propose in this paper a new algorithm for the reconstruction of the stream function inside a bounded domain, where a vortex ring is supposed to form.We derive a quasi-Newton Broyden- Fletcher-Goldfarb-Shanno (BFGS) algorithm for this problem, using a cost function that describes the matching between the reconstructed field and given values of normal derivatives on the boundary of the domain. The algorithm computes the best parameters of the vortex model that is used to describe the vorticity distribution inside the vortex ring. Several vortex models are considered. The method is implemented in freefem++ using a P1 finite-element spatial discretization. Validation tests using analytical models (Hill’s spherical vortex) and Navier-Stokes direct numerical simulation data show good agreement between reconstructed and reference fields.