SPH formulation of the Finite-Time Lyapunov Exponents for the detection of Lagrangian Coherent Structures

In the present work, SPH formulae of the Finite-Time Lyapunov Exponents for the detection of Lagrangian Coherent Structures are presented for the analysis of vortical flows with violent fluid material transportation, exchanging and mixing. The Lagrangian Coherent Structures, detected by the ridges of FTLE field, reveal the hidden impenetrable inner boundaries inside the flow, which supplies a useful way for revealing the features e.g. vortices, flow trends etc. especially when moving, deforming or breaking boundaries exist. Thanks to the Lagrangian characteristic of SPH, the trajectory of each fluid particle is explicitly tracked over the whole simulation, which allows for a direct evaluation of the FTLE field implemented during the runtime or conducted as a post-processing. In this paper, ?-SPH method is applied for the numerical simulation and a particle splitting technique is adopted to improve the local accuracy and reduce the total amount of computation. With the scalar fields of FTLE, some representative complex free surface flows with strong vortical motions in naval and ocean engineering are analysis in the section of numerical results. Especially in the last case, the submerged vortical tunnels induced by the splashing bow wave are successfully captured using the proposed method.