On the use of the finite-time Lyapunov exponent to reveal complex flow physics in the wake of a mechanical valve

Abstract

The finite-time Lyapunov exponent (FTLE) is a popular tool to extract characteristic features of flows that cannot be revealed by other criteria. However, even if the computational cost of computing particle trajectories in space and time has been reduced and optimized in a considerable number of works, the main challenge probably consists in increasing the spatial resolution of Lagrangian coherent structures locally, i.e., where the FTLE field reaches its maximum values. On the other hand, most of experimental data are obtained in planes so that the FTLE field is computed without the out-of-plane particle movements. To investigate which physics the FTLE can capture in flows that are highly three dimensional, the criterion is computed from high-speed stereoscopic particle image velocimetry measurements of the pulsatile flow that develops behind a bi-leaflet mechanical heart valve. The similitude is based on the Womersley number, and experiments are performed for a lower Reynolds number than the physiologic value to obtain sufficiently resolved data in space and time. It is found that the vortex shedding is well captured and that its development can be decomposed into four successive phases. The longest phase occurs near the peak flow rate and exhibits a break of symmetry similar to the one appearing in the wakes of two side-by-side cylinders in the regime when the separation between the cylinders is of the order of their diameter. Specifically, a vortex street with alternating vortex sheddings is observed in a narrow wake behind one of the leaflets, whereas single large vortices develop inside a wide wake downstream of the other leaflet. It appears that these patterns are difficult or even impossible to discern with classical Eulerian vortex identification techniques. The Strouhal numbers of vortex-shedding frequencies, obtained from continuous wavelet transforms and based on the apparent height of the leaflets, are also close to those found in the flow behind two cylinders. By invoking the Taylor hypothesis, an approximate three-dimensional reconstruction of the flow can be obtained and a three-dimensional FTLE field is deduced, which provides a very detailed view of the vortex structures that form and develop in the wakes of the leaflets.