Under the influence of vortex instability, wingtip vortex demonstrates a certain motion of wandering in its wake. To understand the mechanism and the evolution of vortex wandering, the instability and its development of the wingtip vortex generated by a NACA0015 rectangular wing are investigated by conducting Stereo Particle Image Velocimetry (SPIV) experiments at different Reynolds numbers and angles of attack. The results show that vortex wandering exists in the wake of 1-6 chordlength and manifests anisotropy that gradually amplifies along its streamwise position. The amplitude of wandering grows faster with streamwise distance at larger angle of attack. Based on the experimental results, the amplification ratio of the spatial/temporal instability of wingtip vortex, frequency of perturbation, and their evolution along the streamwise direction are obtained through linear stability analysis. It is found that the wingtip vortex is marginally stable within Reynolds number=2.1×105-3.5×105 with the perturbation frequency around 3-5 Hz. The fact that wingtip vortex is more unstable at larger angle of attack, shown by larger spatial/temporal growth rate, causes faster development of disturbance, which further leads to a quicker growth of wandering amplitude along the streamwise direction. Meanwhile, the transverse velocity perturbation of the most unstable mode obtained by linear stability analysis reveals strong directionality, which shifts the vortex core and causes the wandering motion of wingtip vortex. Different from the one-dimensional random oscillation, the periodic variance of velocity perturbation forces wingtip vortex to wander in each direction with certain dominate frequency. This velocity perturbation fomented by instability is the mechanism causing vortex wandering. The amplification ratio of instability controls the growth rate of its amplitude, and the directionality and periodicity of velocity perturbation accounts for the feature of this motion.
[1] GERZ T, HOLZÄPFEL F, DARRACQ D. Commercial aircraft wake vortices[J]. Progress in Aerospace Sciences, 2002, 38(3):181-208.
[2] BIRCH D, LEE T. Structure and induced drag of a tip vortex[J]. Journal of Aircraft, 2004, 41(5):1138-1145.
[3] LEIBOVICH S. Vortex stability and breakdown-Survey and extension[J]. AIAA Journal, 1984, 22(9):1192-1206.
[4] BAKER G R, BARKER J, BOFAH K K, et al. Laser anemometer measurements of trailing vortices in water[J]. Journal of Fluid Mechanics, 1974, 65(2):325-336.
[5] DEVENPORT W J, RIFE M C, LIAPIS S I, et al. The structure and development of a wing-Tip vortex[J]. Journal of Fluid Mechanics, 1996, 312(1):67-106.
[6] EDSTRAND A M, DAVIS T B, SCHMID P J, et al. On the mechanism of trailing vortex wandering[J]. Journal of Fluid Mechanics, 2016, 806(R1):1-11.
[7] OBERLEITHNER K, SIEBER M, NAYERI C N, et al. Three-dimensional coherent structures in a swirling jet undergoing vortex breakdown:Stability analysis and empirical mode construction[J]. Journal of Fluid Mechanics, 2011, 679:383-414.
[8] 薛栋. 基于流动显示的翼尖涡不稳定频率测量[J]. 北京航空航天大学学报, 2016, 42(4):837-843. XUE D. Frequency measurement of wing-tip vortex instability by flow visualization[J]. Journal of Beijing University of Aeronautics and Astronautics, 2016, 42(2):837-843(in Chinese).
[9] BAILEY S C C, TAVOULARIS S. Measurements of the velocity field of a wing-tip vortex, wandering in grid turbulence[J]. Journal of Fluid Mechanics, 2008, 601:281-315.
[10] 鲍锋, 刘锦生,朱睿,等. 飞机尾涡系Rayleigh-Ludwieg不稳定性实验研究[J]. 航空学报, 2015, 36(7):2166-2176. BAO F, LIU J S, ZHU R, et al. Experimental study on Rayleigh-Ludwieg instability of aircraft wake vortex[J]. Acta Aeronautica et Astronautica Sinica, 2015, 36(7):2166-2176(in Chinese).
[11] CROW S C. Stability theory for a pair of trailing vortices[J]. AIAA Journal, 1970, 8(12):2172-2179.
[12] MOORED K W, DEWEY P A, SMITS A J. Hydrodynamic wake resonance as an underlying principle of efficient unsteady propulsion[J]. Journal of Fluid Mechanics, 2012, 708:329-348.
[13] HEYES A L, JONES R F, SMITH D A R. Wandering of wing-tip vortices[C]//Proceedings of the 12th International Symposium on Application of Lase Techniques to Fluid Mechanics, 2004, 22(9):1192-1206.
[14] DEEM E, EDSTRAND A, REGER R, et al. Deconvolution correction for wandering in wingtip vortex flowfield data[J]. Journal of Fluid Science and Technology, 2013, 8(2):219-232.
[15] BERESH S J, HENFLING J F, SPILLERS R W. Meander of a fin trailing vortex and the origin of its turbulence[J]. Experiments in Fluids, 2010, 49:599-611.
[16] IUNGO G V, SKINNER P, BURESTI G. Correction of wandering smoothing effects on static measurements of a wing-tip vortex[J]. Experiments in Fluids, 2009, 46:435-452.
[17] KINGAN M J, PEARSE J R. Laminar boundary layer instability noise produced by an aerofoil[J]. Journal of Sound and Vibration, 2009, 322:808-828.
[18] PAREDES P. Advances in global instability computations:From incompressible to hypersonic flow[D]. Madrid:Polytechnic University of Madrid, 2014.
[19] KHORRAMI R M. On the viscous modes of instability of a trailing line vortex[J]. Journal of Fluid Mechanics, 1991, 225:197-212.
[20] KHORRAMI R M, ASH R L. Application of spectral collocation techniques to the stability of swirling flows[J]. Journal of Computational Physics, 1989, 81:206-229.
[21] BATCHELOR G K. Axial flow in trailing line vortices[J]. Journal of Fluid Mechanics, 1964, 20(4):645-658.
[22] MAYER E W, POWELL K G. Viscous and inviscid instabilities of a trailing vortex[J]. Journal of Fluid Mechanics, 1992, 245:91-114.
[23] FABRE D, JACQUIN L. Viscous instabilities in trailing vortices at large swirl numbers[J]. Journal of Fluid Mechanics, 2004, 500:239-262.
[24] PARRAS L, FERNANDES-FERIA R. Spatial stability and the onset of absolute instability of Batchelor's vortex for high swirl numbers[J]. Journal of Fluid Mechanics, 2006, 583:27-43.
[25] EDSTRAND A M, SCHMID P J, TAIRA K. A parallel stability analysis of a trailing vortex wake[J]. Journal of Fluid Mechanics, 2018:858-895.
[26] IGARASHI H, DURBIN P A, MA H, et al. A stereoscopic PIV study of a near-field wingtip vortex[C]//48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. Reston, VA:AIAA, 2010:1-13.
[27] THEOFILIS V. Advances in global linear instability analysis of nonparallel and three-dimensional flows[J]. Progress in Aerospace Sciences, 2003, 39:249-315.
[28] MACK B L M, MACK L M. A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer[J]. Journal of Fluid Mechanics, 1976, 73(3):497-520.
[29] ORSZAG S A. Accurate solution of the Orr-Sommerfeld stability equation[J]. Journal of Fluid Mechanics, 1971, 50(4):689-703.
[30] SCHMID P J, HENNINGSON D S. Stability and transition in shear flows[M]. New York:Springer, 2001.
[31] GASTER M. A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability[J]. Journal of Fluid Mechanics, 1962, 14(2):222-224.
[32] MING X, SHI S. On the instability of wakes behind stationary circular cylinders[J]. Chinese Journal of Aeronautics, 1991, 4(2):171-178.
[33] DEVENPORT W J, ZSOLDOS J S, VOGEL C M. The structure and development of a counter-rotating wing-tip vortex pair[J]. Journal of Fluid Mechanics, 1997, 332:71-104.
[34] DEVENPORT W J, VOGEL C M, ZSOLDOS J S. Flow structure produced by the interaction and merger of a pair of co-rotating wing-tip vortices[J]. Journal of Fluid Mechanics, 1999, 394:357-377.
CJA