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The aerodynamics of revolving wings I. Model hawkmoth wings

James R. Usherwood^* and Charles P. Ellington

Department of Zoology, University of Cambridge, Downing Street, Cambridge CB2 3EJ, UK
^* Present address: Concord Field Station, MCZ, Harvard University, Old Causeway Road, Bedford, MA 01730, USA

View larger version (28K): [in a new window]   Fig. 1 . Propeller body (A) and plan and side views (B) of the complete `see-saw' propeller rig. Roman numerals identify parts of the propeller body. i, sting for attachment to wings; ii, propeller head; iii, smoke chamber (smoke in this chamber feeds into the hollow shaft and up to the propeller head); iv, cut-away section showing torque strain gauges (electrical connections run down the hollow axle); v, strain gauge bridge supply and first-stage amplification by electronics rotating with axle; vi, electrical contacts on multi-wiper slip-rings carrying power and strain gauge signal; vii, gearbox; viii, motor; ix, tachometer.

View larger version (9K): [in a new window]   Fig. 2. Standard (A) and `sawtooth' (B) hawkmoth planforms.

View larger version (17K): [in a new window]   Fig. 3 . Typical voltage signals for a single run at high angle of attack (A) and response of the vertical force transducer to the addition, and then removal, of 5, 10 and 20 g before (B) and after (C) filtering. In A, the top (green) line shows the tachometer trace, the middle (blue) line the torque signal and the bottom (red) line the vertical force signal. The wings were started after 10s. Vertical dotted lines identify five oscillations due to the lightly damped `mass-spring' system (i) inherent in the vertical transducer design, and one cycle due to a mass imbalance of the wings (ii) during a complete revolution.

View larger version (36K): [in a new window]   Fig. 4 . Averaged horizontal Ch (A) and vertical Cv (B) force coefficients plotted against angle of revolution for standard hawkmoth wings over the `abbreviated' range of angle of attack . Underlying grey panels show the averaging period for `early' (narrower panel) and `steady' (broader) pools.

View larger version (15K): [in a new window]   Fig. 5. Flow and force vectors relating to a wing element. U, velocity of wing element; Ur, relative velocity of air at a wing element; w0, vertical component of induced downwash velocity; , geometric angle of attack; r, effective angle of attack; , downwash angle; Fh' and Fv', orthogonal horizontal and vertical forces; FR', single resultant force; L' and Dpro', orthogonal lift and profile drag forces.

View larger version (11K): [in a new window]   Fig. 6. Horizontal Ch (A) and vertical Cv (B) force coefficients and the polar diagram (C) for standard hawkmoth wings under `early' and `steady' conditions. Error bars in A and B show ±1 S.E.M., N=4-10. ', angle of incidence.

View larger version (20K): [in a new window]   Fig. 7. Horizontal Ch (A) and vertical Cv (B) force coefficients and the polar diagram (C) for the `leading-edge' range under `early' and `steady' conditions. Underlying grey lines show `early' (higher) and `steady' (lower) values for standard hawkmoth wings and represent 0° twist and 0% camber. ', angle of incidence.

View larger version (18K): [in a new window]   Fig. 8. Horizontal Ch (A) and vertical Cv (B) force coefficients and the polar diagram (C) for hawkmoth wings with a range of twist under `early' and `steady' conditions. Underlying grey lines show `early' (higher) and `steady' (lower) values for standard hawkmoth wings. ', angle of incidence.

View larger version (18K): [in a new window]   Fig. 9. Horizontal Ch (A) and vertical Cv (B) force coefficients and the polar diagram (C) for hawkmoth wings with a range of camber under `early' and `steady' conditions. Underlying grey lines show `early' (higher) and `steady' (lower) values for standard hawkmoth wings. ', angle of incidence.

View larger version (22K): [in a new window]   Fig. 10. Polar diagram showing results from three models for determining the profile drag coefficient CD,pro and the lift coefficient CL from the `steady' data represented by the lower yellow line. A good model would result in values close to, or slightly above, those of the `early' conditions represented by the upper yellow line. Data are `pooled' values for all wings in the `leading-edge' series. Ch, horizontal force coefficient; Cv, vertical force coefficient; , geometric angle of attack.

View larger version (59K): [in a new window]   Fig. 11. Smoke flow over hawkmoth wings at =0° (A) and =35° (B) revolving steadily at 0.1 Hz. Smoke was released from various positions (marked with white arrows) on the leading edge and upper surface of the wings. At very low angles of attack, the smoke describes an approximately circular path as the wing revolves underneath. At higher angles of attack, a spiral leading-edge vortex and strong spanwise flow are visible. , geometric angle of attack.

View larger version (19K): [in a new window]   Fig. 12. Polar diagrams for real hawkmoth wings in steady translating flow and `pooled' model hawkmoth wings in revolution under `early' (upper grey line) and `steady' (lower grey line) conditions. Data for hawkmoth in translational flow are taken from Willmott and Ellington (1997c) for a Reynolds number of 5560, and ranges from -50 to 70° in 10° increments. Ch, horizontal force coefficient; Cv, vertical force coefficient; , geometric angle of attack.

View larger version (21K): [in a new window]   Fig. 13. Polar diagram comparing measured horizontal (Ch) and vertical (Cv) force coefficients with those predicted from the normal force relationship for the standard, flat hawkmoth planform. ranges from -20 to 100° in 5° increments. Ch, horizontal force coefficient; Cv, vertical force coefficient; , geometric angle of attack.

View larger version (16K): [in a new window]   Fig. 14. Polar diagram showing the results of dividing the resultant force coefficient into horizontal and vertical coefficients using the `normal force relationship'. The original data are for real hawkmoth wings in translational flow at a Reynolds number of 5560 (Willmott and Ellington, 1997c). ranges from -50 to 70° in 10° increments. Ch, horizontal force coefficient; Cv, vertical force coefficient; , geometric angle of attack.