Instability characteristics of a co-rotating wingtip vortex pair based on bi-global linear stability analysis

Abstract

The Stereo Particle Image Velocimetry (SPIV) technology is applied to measure the wingtip vortices generated by the up-down symmetrical split winglet. Then, the temporal bi-global Linear Stability Analysis (bi-global LSA) is performed on this nearly equal-strength co-rotating vortex pair, which is composed of an upper vortex (vortex-u) and a down vortex (vortex-d). The results show that the instability eigenvalue spectrum illustrated by (ωr, ωi) contains two types of branches: discrete branch and continuous branch. The discrete branch contains the primary branches of vortex-u and vortex-d, the secondary branch of vortex-d and coupled branch, of which all of the eigenvalues are located in the unstable half-plane of ωi > 0, indicating that the wingtip vortex pair is temporally unstable. By contrast, the eigenvalues of the continuous branch are concentrated on the half-plane of ωi < 0 and the perturbation modes correspond to the freestream perturbation. In the primary branches of vortex-u and vortex-d, Mode Pu and Mode Pd are the primary perturbation modes, which exhibit the structures enclosed with azimuthal wavenumber m and radial wavenumber n, respectively. Besides, the results of stability curves for vortex-u and vortex-d demonstrate that the instability growth rates of vortex-u are larger than those of vortex-d, and the perturbation energy of Mode Pu is also larger than that of Mode Pd. Moreover, the perturbation energy of Mode Pu is up to 0.02650 and accounts for 33.56% percent in the corresponding branch, thereby indicating that the instability development of wingtip vortex is dominated by Mode Pu. By further investigating the topological structures of Mode Pu and Mode Pd with streamwise wavenumbers, the most unstable perturbation mode with a large azimuthal wavenumber of m = 5–6 is identified, which imposes on the entire core region of vortex-u. This large azimuthal wavenumber perturbation mode can suggest the potential physical-based flow control strategy by manipulating it.