Two parallel Gaussian vortices of circulations Γ1 and Γ2 radii a1 and a2, separated by a distance b may become unstable by the elliptical instability due the elliptic deformation of their cores. The goal of the paper is to analyse this occurrence theoretically in a general framework. An explicit formula for the temporal growth rate of the elliptical instability in each vortex is obtained as a function of the above global parameters of the system, the Reynolds number Γ1/v and the non-dimensionalized axial wavenumber kzb of the perturbation. This formula is based on a known asymptotic expression for the local instability growth rate at an elliptical stagnation point which depends on the local characteristics of the elliptical flow and the inclination angle of the local perturbation wavevector at this point. The elliptical flow characteristics are estimated by considering each Gaussian vortex alone in a weak uniform external strain field whose properties are provided by a point vortex modelling of the vortex pair. The inclination angle is obtained from the dispersion relation for the Gaussian vortex normal modes and the local expression near each vortex centre for the two helical modes of azimuthal wavenumber m = 1 and m = −1 which constitute the elliptical instability global mode. Both the final formula and the hypotheses made for its derivation are tested and validated by direct numerical simulations and large-eddy simulations.
The objective of this study is to perform direct numerical simulations (DNS) of the three-dimensional short-wavelength elliptic instability developing in a counter-rotating vortex pair, and to reproduce numerically a water-tank experiment. The main features of the elliptic instability are recovered by the simulations. In particular, the spatial structure and the temporal evolution of the most amplified perturbation mode during the linear regime correspond to both experimental measurements and theoretical predictions. The long-term evolution is also simulated, and the stages leading to transition to turbulence are described. Some elements resulting from simulations related to the interaction between the short-wavelength elliptic instability and the long-wavelength Crow instability are provided.
This study describes large eddy simulations of the interaction between an exhaust jet and a trailing vortex, in the near-field of an aircraft wake. Two cases are analyzed: in the first one, typical of cruise flight, the jet and the vortex axes are sufficiently well separated to study first the jet dynamics before considering its interaction with the vortex. Dynamics and mixing are controlled both by the jet diffusion and its entrainment around the vortex core. In the second case the jet partially blows in the vortex core, making the flow similar to a Batchelor vortex. The strong perturbations injected into the core cause an instability of the system which is continuously fed by the jet elements wrapping around the core. This leads to a strong decay of angular momentum and diffusion of the core. Global mixing properties, such as plume area and global mixedness evolutions, are analyzed and two applications to environmental problems are finally discussed.
The propagation of pressure waves in a Lamb-Oseen vortex has been investigated both by three-dimensional direct numerical simulations as well as a set of large-eddy simulations. The pressure wave is initiated by locally increasing the core radius of a Lamb-Oseen vortex at its edge. This wave travels along the vortex axis towards the region with a thinner core radius. Behind the wave the axial velocity increases so that sufficient swirl may trigger the helical instability. An abrupt change of flow structure in the vortex core is observed in the case of intersecting pressure waves: this phenomenon is known as vortex bursting. A vortex system composed of two symmetric counter-rotating vortices, which is similar to that of an aircraft has also been investigated by means of large-eddy simulations. In the far-field region the system develops linear instabilities such as the Crow instability, which is characterized by a large-scale symmetric sinusoidal deformation resulting in the reconnection of the two vortices. It also demonstrates the occurrence of helical instability and vortex bursting.
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