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Relationship of roll and pitch oscillations in a fin flapping at transitional to high Reynolds numbers

  • Promode R. Bandyopadhyay (a1), David N. Beal (a1), J. Dana Hrubes (a1) and Arun Mangalam (a2)

Hydrodynamic effects of the relationship between the roll and pitch oscillations in low-aspect-ratio fins, with a laminar section and a rounded leading edge, flapping at transitional to moderately high Reynolds numbers, are considered. The fin is hinged at one end and its roll amplitude is large. Also examined is how this relationship is affected by spanwise twist, which alters the pitch oscillation amplitude and its phase relative to the roll motion. Force, efficiency and surface hot-film-anemometry measurements, and flow visualization are carried out in a tow tank. A fin of an abstracted penguin-wing planform and a NACA 0012 cross-section is used, and the chord Reynolds number varies from 3558 to 150 000 based on total speed. The fin is forced near the natural shedding frequency. Strouhal number and pitch amplitude are directly related when thrust is produced, and efficiency is maximized in narrow combinations of Strouhal number and pitch amplitude when oscillation of the leading-edge stagnation point is minimal. Twist makes the angle of attack uniform along the span and enhances thrust by up to 24 %, while maintaining high efficiency. Only 5 % of the power required to roll is spent to pitch, and yet roll and pitch are directly related. During hovering, dye visualization shows that a diffused leading-edge vortex is produced in rigid fins, which enlarges along the span; however, twist makes the vortex more uniform and the fin in turn requires less power to roll. Low-order phase maps of the measurements of force oscillation versus its derivative are modelled as due to van der Pol oscillators; the higher-order maps show trends in the sub-regimes of the transitional Reynolds number. Fin oscillation imparts a chordwise fluid motion, yielding a Stokes wave in the near-wall vorticity layer. When the roll and pitch oscillations are directly related, the wave is optimized: causing vorticity lift-up as the fin is decelerated at the roll extremity; the potential energy at the stagnation point is converted into kinetic energy; a vortex is produced as the lifted vorticity is wrapped around the leading edge; and free-stream reattachment keeps the vortex trapped. When the twist oscillation is phased along the span, this vortex becomes self-preserving at all amplitudes of twist, indicating the most stable (low-bandwidth) tuned nature.

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Type Description Title

Bandyopadhyay et al. supplementary movie
The formation of the LEV in twisted and untwisted fins is shown by injecting blue and red dyes near the leading edge from two sides of the fin. The nature of the wake vortices and thrust are also shown

 Video (8.8 MB)
8.8 MB

Bandyopadhyay et al. supplementary movie
The formation of the LEV in twisted and untwisted fins is shown by injecting blue and red dyes near the leading edge from two sides of the fin. The nature of the wake vortices and thrust are also shown

 Video (32.2 MB)
32.2 MB
Supplementary Materials

Bandyopadhyay et al. supplementary material
Supplementary data and figures

 PDF (1.8 MB)
1.8 MB

Bandyopadhyay et al. supplementary movie
The fin is driven by a van der Pol oscilaltor. The roll versus pitch angles are plotted for the case shown in Movie 2.

 Video (1.0 MB)
1.0 MB

Bandyopadhyay et al. supplementary movie
The fin is given a square pulse of external disturbance. The disturbance rejection property is demonstrated. See Movie 1 for pitch versus roll angles

 Video (2.5 MB)
2.5 MB

Bandyopadhyay et al. supplementary movie
The fin is given a square pulse of external disturbance. The disturbance rejection property is demonstrated. See Movie 1 for pitch versus roll angles

 Video (2.0 MB)
2.0 MB

Bandyopadhyay et al. supplementary movie
The fin is driven by a van der Pol oscilaltor. The roll versus pitch angles are plotted for the case shown in Movie 2.

 Video (334 KB)
334 KB


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