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

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 biglobal Linear Stability Analysis(bi-global LSA)is performed on this nearly equal-strength corotating 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,ω_(i))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 ofω_(i)>0,indicating that the wingtip vortex pair is temporally unstable.By contrast,the eigenvalues of the continuous branch are concentrated on the half-plane ofω_(i)<0 and the perturbation modes correspond to the freestream perturbation.In the primary branches of vortex-u and vortex-d,Mode P_(u) 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 P_(u) is also larger than that of Mode Pd.Moreover,the perturbation energy of Mode P_(u) 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 P_(u).By further investigating the topological structures of Mode P_(u) 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.