Quantifying the influence of vortex instability on mean velocity field statistics of wingtip vortex

Abstract

During its evolution to the far field, the wingtip vortex exhibits complex instability behaviors such as long-wave/short-wave instability and vortex wandering. However, the quantification influence of vortex instability on its velocity field statistics has not been well investigated. To this end, experimental measurements of a canonical wingtip vortex generated by an elliptical wing under various angles of attack and Reynolds numbers were conducted using particle image velocimetry. It is found that the streamwise variation of wandering amplitude presents an exponential growth within the middle-to-far wake region and asymptotically saturates to 10⁻¹b in the far wake, which differs from the previous report of a linear growth trend in the near-wake region. Further, two average methods, i.e., time average (TA) and ensemble average (EA), were adopted to compare the velocity field statistics. In both TA- and EA-obtained flow fields, the vortex radius r_c, peak vorticity \(\Omega_{x}^{p}\), and vortex circulation Γ all demonstrate a power-law scaling with respect to the streamwise location, i.e., \(r_{c}\propto x^{k_{r}},\Omega_{x}^{p}\propto x^{-k_{\omega}}\), and \(\Gamma\propto x^{-k_{\Gamma}}\), respectively. For a full rolling-up wingtip vortex in the middle-to-far wake region, the fact that k_Γ = k_ω − 2k_r demonstrates that the vortex circulation can be scaled as \(\Gamma=\Omega_{x}^{p}(r_{c})^{2}\). On the other hand, TA overestimates the decay rate of peak vorticity k_ω and the growth rate of vortex radius k_r. Furthermore, the TA-introduced bias level of the peak vorticity and vortex radius is found to be scaled with an empirical scaling between the wandering amplitude by a power law, respectively. These findings provide significant practical value for detecting wake vortex in wake vortex spacing systems.