The evolution of wingtip vortex wandering: a stability analysis based on stereo-PIV experiment
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Instability, as a potential mechanism for vortex flow control, is manifested typically in the phenomenon of wingtip vortex wandering. In this paper, the evolution of wingtip vortex generated by a NACA0015 rectangular wing within six chord lengths of its wake at Rec= 2.1 × 105–3.5 × 105 and AoA = 4°–10° is investigated by stereo particle image velocimetry experiment. The streamwise evolution of vortex wandering is clearly captured, which demonstrates strong anisotropy. The amplitude of vortex wandering increases under all conditions, which exhibits greater growth rates in the condition of larger AoA. To account for such phenomenon, spatial linear stability analysis is performed on the base flow obtained by SPIV experiment to investigate quantitatively the amplification of small disturbance and disturbance modes at the experiment conditions. It is discovered that the wingtip vortex in the experiment is marginally stable under all conditions. Furthermore, the spatial growth rates of instability by AoA correspond well with that of wandering amplitude, indicating that the quick enlargement of the amplitude of wandering at larger AoA condition is caused by larger spatial growth rate of disturbance. In addition, the least-stable disturbance mode by LSA reveals a non-zero directed velocity perturbation in the vortex core, which rotates periodically. Such perturbation developing in both time and space, which is stirred by vortex instability, leads to the behavior of wingtip vortex wandering.
KeywordsWingtip vortex SPIV Linear stability analysis Vortex instability
The wingtip vortex is a typical large-scale vortex structure in the wake of aircraft, which can maintain for up to 100 times the chord length, threatening the flight safety of following aircrafts and limits the airport capacity. In addition, the induced drag generated by wingtip vortex is one of the main sources of aircraft resistance . Therefore, Gerz proposed two strategies to alleviate the influence of aircraft trailing vortices: (1) low-vorticity vortex, (2) quickly decaying vortex . The circulation of wingtip vortex has been reduced effectively by facilitating wingtip devices, while for the second strategy, there have been no effective flow control principles and methods. In fact, studies have shown that vortex breakdown is closely related to its instability characteristics  and wingtip vortex manifests evident properties of instability during its evolution. Researchers have discovered that the wingtip vortex generally demonstrates a low-frequency wandering behavior, which has been proved to be caused by the inherent instability of wingtip vortex . In addition, such wandering behavior possess a dominate frequency throughout its wake . A potential flow control method can therefore be proposed to accelerate the decaying of wingtip vortex by matching the dominant frequency of the external actuation and the base flow or utilizing the long-wave or short-wave instability to amplify the growth rate of disturbance. Therefore, by investigating the wandering behavior of wingtip vortex, the mechanism of instability and its effect on wingtip vortex can be accounted, which can further guide the input excitation mode by the disturbance mode or frequency of instability.
Vortex wandering is a typical phenomenon of wingtip vortex by the effect of its inherent instability, which demonstrates a low-frequency fluctuation motion in the streamwise plane . Due to its ‘blurring’ effect on the measurement of wingtip vortex, early researches mainly focused on its correction by means of deconvolution and vortex re-centering. The instantaneous vortex center was found to conform with Gaussian distribution and displayed anisotropic feature . The amplitude of wandering was shown to grow in the streamwise direction, and decrease with increasing angle of attack [8, 9], and therefore decrease with circulation of the vortex . In Edstrand’s work , leading modes extracted from PIV-obtained flow field and most unstable modes by linear stability analysis of a fitted Batchelor vortex were compared and found to be identical, which confirmed that the origin of wandering was its instability. In short, there have been effective methods to correct the side effect of vortex wandering and instability has been taken as one of its primary factors at present. However, the mechanism underlying the spatial evolution of such wandering motion has not been fully understood, which requires further investigation.
Linear stability analysis (LSA) linearizes small disturbances in a basic flow configuration and solves its spatial/temporal growth/damping rate, frequency/wavenumber and disturbance mode quantitatively , which exactly suits the problem studied herein. Previous efforts include studies of the effect of Reynolds number, swirl parameter, and azimuthal wavenumber on the stability of a theoretical Batchelor vortex [12, 13]. In Oberleithner’s stability analysis of swirling jets, a global resonant frequency was found throughout the wake  and Bi-global LSA performed by Edstrand et al. tackled the influence of trailing wake on the stability of wingtip vortex solved by DNS computation at Re = 1000  and found the fifth wake mode with optimal attenuation effect on the wingtip vortex . However, the temporal and spatial evolution of the instability of wingtip vortex has not been fully investigated, which limits the knowledge of the underlying mechanism of vortex wandering and its development. Therefore, in this paper, linear stability analysis is performed to obtain the growth rate and also the most unstable mode of small disturbance in the wingtip vortex flow throughout the wake region covered by stereo-PIV experiment, so as to quantify the effect of instability on the wingtip vortex flow and also on the spatial development of vortex wandering.
2 Experimental setup and data processing
2.1 SPIV experiment setup
The experiment is conducted in the low-speed tunnel in Shanghai Jiao Tong University with an experimental section of 1200 mm × 900 mm, with the turbulence level of test section measured to be 0.2%. A NACA0015 rectangular wing with chord length of c = 0.203 m and an aspect ratio of 2.0 was used to generate the wingtip vortex. Three angles of attack AoA = 4°, 8°, and 10° and three wind speed conditions of 15 m/s, 20 m/s, and 25 m/s are tested in the experiment, corresponding to Rec = 2.1 × 105, 2.8 × 105, and 3.5 × 105 based on the chord length, adding up to a total of nine conditions. In addition, the wingtip vortex flow with different cross sections in the wake is measured for each condition. The measurement range is x/c = 0.6–6.0.
2.2 Data post processing
The three-dimensional vector field is processed by INSIGHT 4G commercial software with an interrogation window of 24 × 24 pixels, yielding a resolution of 0.89 mm/grid. The vorticity and each related parameter are then calculated by the three components of velocity u, v, w.
3 Wandering behavior of wingtip vortex
3.1 Instantaneous vortex centers of wingtip vortex
3.2 Wandering amplitude of wingtip vortex
4 Mechanism of wingtip vortex wandering
4.1 Linear stability analysis: methodology and code validation
Comparison of spatial stability results of Poiseuille flow in a pipe with the literature
α, present method
α, Khorrami et al. 
0.519989251732426 + 0.020835493881362i
0.51998925171 + 0.02083549388i
0.535251083108396 + 0.017227643884052i
0.53525108 + 0.01722763i
Comparison of spatial stability result of Batchelor vortex with the literature
α, present method
α, Parras et al. 
0.365174487123736 − 0.222971346495823i
0.3651762873306434 − 0.2229717123848892i
0.246667715462097 − 0.001696062822232i
0.24666715 − 0.0016967876i
4.2 Linear stability analysis of the wingtip vortex
4.3 Evolution of instability of wingtip vortex
4.4 Analysis of the mechanism of wingtip vortex wandering
In this paper, the wandering behavior of the wingtip vortex generated by a NACA0015 rectangular wing at AoA = 4°, 8°, 10°, and Rec = 2.1 × 105, 2.8 × 105, 3.5 × 105 is observed by stereo-PIV experiment in detail, which focuses on its streamwise development from x/c = 0.6–6.0. It is discovered that the distribution of instantaneous vortex center transits to an anisotropic pattern in the far-wake. The amplitude of vortex wandering increases in the streamwise direction in all conditions, but at a significantly larger amplification rate at large AoA conditions. This phenomenon is further investigated by spatial linear stability analysis, where a local LSA procedure is taken, to reveal the mechanism underlying the spatial development of vortex wandering. The stability result yields an evidently larger spatial growth rate of disturbance at the condition of large AoA, which explains the distinct characteristics of the spatial development of vortex wandering at different AoA conditions. The wingtip vortex is also found to be less stable at larger Reynolds number. Furthermore, the two-dimensional disturbance mode of the wingtip vortex measured in the stereo-PIV is obtained at each experimental condition. The crosswise and time-dependent distribution of the velocity perturbation uncovers a centralized velocity perturbation at the vortex center, which rotates periodically about the vortex center, offsetting the entire vortex. Such mechanism leads to the wandering behavior of wingtip vortex observed in the experiment. Still, there are some questions remaining open such as the anisotropic distribution of instantaneous vortex center and the origin of this unique pattern of velocity perturbation, which will be further investigated in detail in future work.
Funding was provided by National Key Basic Research Program of China (Grant no. 2014CB744802), China Postdoctoral Science Foundation (Grant no. 2018M642007). Project supported by the major research project of the National Science Foundation of China (Grant No.91952302).
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