Hostname: page-component-76fb5796d-zzh7m Total loading time: 0 Render date: 2024-04-27T12:25:00.577Z Has data issue: false hasContentIssue false
  • English
  • Français

A parallel stability analysis of a trailing vortex wake

Published online by Cambridge University Press:  05 January 2018

Adam M. Edstrand Open the ORCID record for Adam M. Edstrand [Opens in a new window] ,
Peter J. Schmid Open the ORCID record for Peter J. Schmid [Opens in a new window] ,
Kunihiko Taira Open the ORCID record for Kunihiko Taira [Opens in a new window]  and
Louis N. Cattafesta III Open the ORCID record for Louis N. Cattafesta III [Opens in a new window]
Adam M. Edstrand*
Affiliation:
Department of Mechanical Engineering, Florida Center for Advanced Aero-Propulsion, Florida State University, Tallahassee, FL 32310, USA
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
Kunihiko Taira
Affiliation:
Department of Mechanical Engineering, Florida Center for Advanced Aero-Propulsion, Florida State University, Tallahassee, FL 32310, USA
Louis N. Cattafesta III
Affiliation:
Department of Mechanical Engineering, Florida Center for Advanced Aero-Propulsion, Florida State University, Tallahassee, FL 32310, USA
*
†Email address for correspondence: aedstrand@fsu.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

Trailing vortices are generated in aeronautical and maritime applications and produce a variety of adverse effects that remain difficult to control. A stability analysis can direct flow control designers towards pertinent frequencies, wavelengths and locations that may lead to the excitation of instabilities, resulting in the eventual breakup of the vortex. Most models for trailing vortices, however, are far-field models, making implementation of the findings from stability analyses challenging. As such, we perform a stability analysis in the formative region where the numerically computed base flow contains both a two-dimensional wake and a tip vortex generated from a NACA0012 at a $5^{\circ }$ angle of attack and a chord-based Reynolds number of $Re_{c}=1000$. The parallel temporal and spatial analyses show that at three chord lengths downstream of the trailing edge, seven unstable modes are present: three stemming from the temporal analysis and four arising in the spatial analysis. The three temporal instabilities are analogues to three unstable modes in the spatial analysis, with the wake instability dominating in both analyses. The helical mode localized to the vortex co-rotates with the base flow, which is converse with the counter-rotating $m=-1$ instabilities of a Batchelor vortex model, which may be a result of the formative nature of the base-flow vortex. The fourth spatial mode is localized to the tip vortex region. The continuous part of the spectrum contains oscillatory and wavepacket solutions prompting the utilization of a wavepacket analysis to analyse the flow field and group velocity. The structure and details of the full bi-global spectrum will help navigate the design space of effective control strategies to hasten decay of persistent wingtip vortices.

JFM classification

Vortex Flows: Vortex Flows Vortex Flows: Vortex instability Wakes/Jets: Wakes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 858 - 895
Copyright
© 2018 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Batchelor, G. K. 1964 Axial flow in trailing line vortices. J. Fluid Mech. 20, 645–658.Google Scholar
Broadhurst, M. S.2006 Vortex stability and breakdown: direct numerical simulation and stability analysis using biglobal and parabolised formulations. PhD thesis, Imperial College London.Google Scholar
Crow, S. C. 1970 Stability theory of a pair of trailing vortices. AIAA J. 8 (12), 2172–2179.Google Scholar
Devenport, W. J., Rife, M. C., Liapis, S. I. & Follin, G. J. 1996 The structure and development of a wing-tip vortex. J. Fluid Mech. 312, 67–106.Google Scholar
Edstrand, A. M., Davis, T. B., Schmid, P. J., Taira, K. & Cattafesta, L. N. 2016 On the mechanism of trailing vortex wandering. J. Fluid Mech. 801, R1.Google Scholar
Fabre, D. & Jacquin, L. 2004 Viscous instabilities in trailing vortices at large swirl numbers. J. Fluid Mech. 550, 239–262.Google Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability. J. Fluid Mech. 14 (2), 222–224.Google Scholar
Greenblatt, D. 2012 Fluidic control of a wing tip vortex. AIAA J. 50 (2), 375–386.Google Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and eigenfunctions. J. Fluid Mech. 87 (1), 33–54.Google Scholar
Ham, F. & Iaccarino, G. 2004 Energy conservation in collocated discretization schemes on unstructured meshes. In Annual Research Briefs, pp. 3–14. Center for Tubulence Research, Stanford University/NASA Ames.Google Scholar
Ham, F., Mattsson, K. & Iaccarino, G. 2006 Accurate and stable finite volume operators for unstructured flow solvers. In Annual Research Briefs, pp. 243–261. Center for Turbulence Research, Stanford University/NASA Ames.Google Scholar
Hanifi, A., Schmid, P. J. & Henningson, D. S. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8, 826–837.Google Scholar
Heaton, C. J., Nichols, J. W. & Schmid, P. J. 2009 Global linear stability of the non-parallel Batchelor vortex. J. Fluid Mech. 629, 139–160.Google Scholar
Hein, S. & Theofilis, V. 2004 On instability characteristics of isolated vortices and models of trailing-vortex systems. Comput. Fluids 33 (5–6), 741–753.Google Scholar
Herbert, T. 1997 Parabolized stability equations. Annu. Rev. Fluid Mech. 29, 245–283.Google Scholar
Jeong, J. & Hussain, F. 1995 On the identification of a vortex. J. Fluid Mech. 285, 69–94.Google Scholar
Khorrami, M. R. 1991 On the viscous modes of instability of a trailing line vortex. J. Fluid Mech. 225, 197–212.Google Scholar
Kopriva, D. A. 2009 Implementing Spectral Methods for Partial Differential Equations. Springer.Google Scholar
Lehoucq, R. B., Sorensen, D. C. & Yang, C. 1996 ARPACK Users Guide. SIAM J. Matrix Anal. 17.Google Scholar
Lessen, M. & Paillet, F. 1974 The stability of a trailing line vortex. Part 2. Viscous theory. J. Fluid Mech. 65, 769–779.Google Scholar
Lessen, M., Singh, J. P. & Paillet, F. 1974 The stability of a trailing line vortex. Part 1. Inviscid theory. J. Fluid Mech. 63, 753–763.Google Scholar
Mack, L. M. 1976 A numerical study of the temporal eigenvalue spectrum of the Blasius boundary layer. J. Fluid Mech. 73 (3), 497–520.Google Scholar
Mao, X. & Sherwin, S. 2011 Continuous spectra of the Batchelor vortex. J. Fluid Mech. 681, 1–23.Google Scholar
Margaris, P. & Gursul, I. 2010 Vortex topology of wing tip blowing. Aerosp. Sci. Technol. 14, 143–160.Google Scholar
Matalanis, C. G. & Eaton, J. K. 2007 Wake vortex alleviation using rapidly actuated segmented Gurney flaps. AIAA J. 45 (8), 1874–1884.Google Scholar
Mattingly, G. E. & Criminale, W. O. 1972 The stability of an incompressible two-dimensional wake. J. Fluid Mech. 51, 233–272.Google Scholar
Mayer, E. & Powell, K. 1992 Viscous and inviscid instabilities of a trailing vortex. J. Fluid Mech. 245, 91–114.Google Scholar
Obrist, D. & Schmid, P. J. 2003a On the linear stability of swept attachment-line boundary layer flow. Part 1. Spectrum and asymptotic behaviour. J. Fluid Mech. 493, 1–29.Google Scholar
Obrist, D. & Schmid, P. J. 2003b On the linear stability of swept attachment-line boundary layer flow. Part 2. Non-modal effects and receptivity. J. Fluid Mech. 493, 31–58.Google Scholar
Obrist, D. & Schmid, P. J. 2008 Resonance in the cochlea with wave packet pseudomodes. In Proc. Int. Congr. Theor. and Appl. Mech. (XXII ICTAM), pp. 1–2.Google Scholar
Obrist, D. & Schmid, P. J. 2010 Algebraically decaying modes and wave packet pseudo-modes in swept Hiemenz flow. J. Fluid Mech. 643, 309–332.Google Scholar
Papageorgiou, D. T. & Smith, F. T. 1989 Linear instability of the wake behind a flat plate placed parallel to a uniform stream. J. Fluid Mech. 208, 67–89.Google Scholar
Paredes, P.2014 Advances in global instability computations: from incompressible to hypersonic flows. PhD thesis, Universidad Politécnica de Madrid.Google Scholar
Paredes, P., Rodriguez, D. & Theofilis, V. 2013 Three-dimensional solutions of trailing-vortex flows using parabolized equations. AIAA J. 51 (12), 2763–2770.Google Scholar
Saric, W. S., Reed, H. L. & Kerschen, E. J. 2002 Boundary-layer receptivity to freestream disturbances. Annu. Rev. Fluid Mech. 34, 291–319.Google Scholar
Sato, H. & Kuriki, K. 1961 The mechanism of transition in the wake of a thin flat plate placed parallel to a uniform flow. J. Fluid Mech. 11, 321–352.Google Scholar
Schmid, P. & Henningson, D. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.Google Scholar
Spalart, P. R. 1998 Airplane trailing vortices. Annu. Rev. Fluid Mech. 30, 107–138.Google Scholar
Taira, K. & Colonius, T. 2009 Three-dimensional flows around low-aspect-ratio flat-plate wings at low Reynolds numbers. J. Fluid Mech. 623, 187–21.Google Scholar
Theofilis, V. 2009 Role of instability theory in flow control. In Fundamentals and Applications of Modern Flow Control (ed. Joslin, R. D. & Miller, D. N.), pp. 1–42. AIAA.Google Scholar
Theofilis, V. 2011 Global linear instability. Annu. Rev. Fluid Mech. 43, 319–352.Google Scholar
Theofilis, V., Duck, P. W. & Owen, J. 2004 Viscous linear stability analysis of rectangular duct and cavity flows. J. Fluid Mech. 505, 249–286.Google Scholar
Tisseur, F. & Meerbergen, K. 2001 The quadratic eigenvalue problem. SIAM Rev. 43 (2), 235–286.Google Scholar
Torres, G. E. & Mueller, T. J. 2004 Low-aspect-ratio wing aerodynamics at low Reynolds numbers. AIAA J. 42 (5), 865–873.Google Scholar
Trefethen, L. N. 2005 Wave packet pseudomodes of variable coefficient differential operators. Proc. R. Soc. Lond. A 461 (2062), 3099–3122.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Viola, F., Iungo, G. V., Camarri, S., Porté-Agel, F. & Gallaire, F. 2014 Prediction of the hub vortex instability in a wind turbine wake: stability analysis with eddy-viscosity models calibrated on wind tunnel data. J. Fluid Mech. 750, R1.Google Scholar
Woodley, B. M. & Peake, N. 1997 Global linear stability analysis of thin aerofoil wakes. J. Fluid Mech. 339, 239–260.Google Scholar
Wren, G. G. 1997 Detection of submerged vessels using remote sensing techniques. Australian Defence Force J. 127, 9–15.Google Scholar

Edstrand et al. supplementary movie 1

Movie of the streamwise vorticity of the temporal principal wake instability.

Download Edstrand et al. supplementary movie 1(Video)
Video 1.9 MB

Edstrand et al. supplementary movie 2

Movie of the streamwise vorticity of the temporal wake instability.

Download Edstrand et al. supplementary movie 2(Video)
Video 2 MB

Edstrand et al. supplementary movie 3

Movie of the streamwise vorticity of the temporal vortex instability.

Download Edstrand et al. supplementary movie 3(Video)
Video 1.4 MB

Edstrand et al. supplementary movie 4

Movie of the streamwise vorticity of the stable temporal vortex mode.

Download Edstrand et al. supplementary movie 4(Video)
Video 1.5 MB

Edstrand et al. supplementary movie 5

Movie of the streamwise vorticity of the stable temporal wake mode.

Download Edstrand et al. supplementary movie 5(Video)
Video 1.7 MB

Edstrand et al. supplementary movie 6

Movie of the streamwise vorticity of the spatial principal wake instability.

Download Edstrand et al. supplementary movie 6(Video)
Video 1.3 MB

Edstrand et al. supplementary movie 7

Movie of the streamwise vorticity of the spatial wake instability.

Download Edstrand et al. supplementary movie 7(Video)
Video 1.4 MB

Edstrand et al. supplementary movie 8

Movie of the streamwise vorticity of the spatial vortex instability.

Download Edstrand et al. supplementary movie 8(Video)
Video 1.3 MB

Edstrand et al. supplementary movie 9

Movie of the streamwise vorticity of the spatial primary vortex instability.

Download Edstrand et al. supplementary movie 9(Video)
Video 858.8 KB

Edstrand et al. supplementary movie 10

Movie of the streamwise vorticity of the spatial higher-order azimuthal mode.

Download Edstrand et al. supplementary movie 10(Video)
Video 919.3 KB