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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: [email protected]
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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 

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Edstrand et al. supplementary movie 1

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

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Video 1.9 MB

Edstrand et al. supplementary movie 2

Movie of the streamwise vorticity of the temporal wake instability.

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Video 2 MB

Edstrand et al. supplementary movie 3

Movie of the streamwise vorticity of the temporal vortex instability.

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Video 1.4 MB

Edstrand et al. supplementary movie 4

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

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Video 1.5 MB

Edstrand et al. supplementary movie 5

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

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Video 1.7 MB

Edstrand et al. supplementary movie 6

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

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Video 1.3 MB

Edstrand et al. supplementary movie 7

Movie of the streamwise vorticity of the spatial wake instability.

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Video 1.4 MB

Edstrand et al. supplementary movie 8

Movie of the streamwise vorticity of the spatial vortex instability.

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Video 1.3 MB

Edstrand et al. supplementary movie 9

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

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Video 858.8 KB

Edstrand et al. supplementary movie 10

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

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Video 919.3 KB