Experiments on the elliptic instability in vortex pairs with axial core flow

摘要

Results are presented from an experimental study on the dynamics of pairs of vortices, in which the axial velocity within each core differs from that of the surrounding fluid. Co- and counter-rotating vortex pairs at moderate Reynolds numbers were generated in a water channel from the tips of two rectangular wings. Measurement of the three-dimensional velocity field was accomplished using stereoscopic particle image velocimetry, revealing significant axial velocity deficits in the cores. For counter-rotating pairs, the long-wavelength Crow instability, involving symmetric wavy displacements of the vortices, could be clearly observed using dye visualisation. Measurements of both the axial wavelength and the growth rate of the unstable perturbation field were found to be in good agreement with theoretical predictions based on the full experimentally measured velocity profile of the vortices, including the axial flow. The dye visualisations further revealed the existence of a short-wavelength core instability. Proper orthogonal decomposition of the time series of images from high-speed video recordings allowed a precise characterisation of the instability mode, which involves an interaction of waves with azimuthal wavenumbers m = 2 and m = 0. This combination of waves fulfils the resonance condition for the elliptic instability mechanism acting in strained vortical flows. A numerical three-dimensional stability analysis of the experimental vortex pair revealed the same unstable mode, and a comparison of the wavelength and growth rate with the values obtained experimentally from dye visualisations shows good agreement. Pairs of co-rotating vortices evolve in the form of a double helix in the water channel. For flow configurations that do not lead to merging of the two vortices over the length of the test section, the same type of short-wave perturbations were observed. As for the counter-rotating case, quantitative measurements of the wavelength and growth rate, and comparison with previous theoretical predictions, again identify the instability as due to the elliptic mechanism. Importantly, the spatial character of the short-wave instability for vortex pairs with axial flow is different from that previously found in pairs without axial flow, which exhibit an azimuthal variation with wavenumber m = 1.