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Shape, flapping and flexion: wing and fin design for forward flight

S. A. Combes^* and T. L. Daniel

Department of Zoology, University of Washington, Seattle, WA 98195, USA

View larger version (57K): [in a new window]   Fig. 1. (A) Two-dimensional strip oscillating with amplitude h0 and moving forward at velocity U while a wave passes rearward at velocity c. The amplitude changes from the leading to the trailing edge by a factor , the ratio of h to h0. The instantaneous location of a point (x) on the strip is described by h(x,t), where t is time. (B) A ratfish with a wave (highlighted) traveling backwards on its pectoral fin at wave speed c. (C) Diagram of a ratfish illustrating the angle () subtended by a flapping fin and tip amplitude (H).

View larger version (19K): [in a new window]   Fig. 2. Chord length distributions of theoretical wings and a ratfish pectoral fin. (A) Chord length distribution of the pectoral fin of the adult ratfish. Total area 0.0069m2; aspect ratio 2.2. (B) Sample chord length distributions generated with a first-degree polynomial. (C) Chord length distributions generated with a second-degree polynomial. (D) Chord length distributons generated with a third-degree polynomial. (E) Chord length distributions generated with a beta distribution. (F) Chord length distributions generated with an exponential function. Chord length distributions in B–F represent wings of the same aspect ratio (2.5) and area (0.0069m2). See Appendix 2 for the mathematical equations used to generate the theoretical wings.

View larger version (25K): [in a new window]   Fig. 3. Performance of theoretical wings generated with the five shape equations versus the proportion of area in the outer one-fifth of the wing. Each point represents the performance of one theoretical wing shape. The area of each wing tested is 0.0069m2 (multiplied by 2 to give the total performance of an animal). U=0.15ms-1, c=0.3ms-1, H=0.058m and f=2.27s-1. Wing area was fixed, and span was adjusted to create wings with the four specified aspect ratios (AR). (A) Thrust versus the proportion of area in the outer one-fifth of the wing. (B) Efficiency versus the proportion of area in the outer one-fifth of the wing. U, velocity; c, wave speed; H, fin tip amplitude; f, flapping frequency; lavg, average chord length.

View larger version (24K): [in a new window]   Fig. 4. Thrust and efficiency versus average chord length for theoretical wings with different shape, but the same proportion of area in the outer one-fifth of the wing. Wings were generated with three different shape equations: a beta distribution, a first-degree polynomial and a third-degree polynomial. All wings have 9% of area in the outer one-fifth of the wing. Chord length distributions are also shown. Sp=0.0069m2, U=0.15ms-1, c=0.3ms-1, H=0.058m and f=2.27s-1. Sp, total area of one pectoral fin; U, velocity; c, wave speed; H, fin tip amplitude; f, flapping frequency.

View larger version (32K): [in a new window]   Fig. 5. Performance of theoretical wings at varying flapping frequencies with Sp=0.0069m2, U=0.15ms-1, c=0.3ms-1 and H=0.058m. All wings were generated with first-degree polynomials. (A,B) Thrust and efficiency versus average chord length for triangular wings that have 4% of the wing area in the outer one-fifth of the wing. (C,D) Thrust and efficiency versus average chord length for wings that have 12% of the wing area in the outer one-fifth of the wing. (E,F) Thrust and efficiency versus average chord length for rectangular wings that have 20% of the wing area in the outer one-fifth of the wing. Sp, total area of one pectoral fin; U, velocity; c, wave speed; H, fin tip amplitude; f, flapping frequency.

View larger version (23K): [in a new window]   Fig. 6. Performance of theoretical rectangular wings with varying wave speeds. Wings were generated with a first-degree polynomial (20% of the wing area in the outer one-fifth). Sp=0.0069m2, U=0.15ms-1, H=0.058m and f=2.27s-1. (A) Thrust versus average chord length. (B) Efficiency versus average chord length. Sp, total area of one pectoral fin; U, velocity; c, wave speed; H, fin tip amplitude; f, flapping frequency.

View larger version (23K): [in a new window]   Fig. 7. (A) Non-dimensional power and thrust versus strip length. Non-dimensional power and thrust are calculated by multiplying the power or thrust coefficient by strip length and unit width (w=1) and dividing by amplitude squared. (B) Efficiency versus strip length. (C) Total thrust produced versus average chord length for rectangular wings. (D) Whole-wing efficiency versus average chord length for rectangular wings. Each point in parts C and D represents the performance of a single wing with the given average chord length. For all graphs, U=0.15ms-1, c=0.3ms-1, H=0.058m and f=2.27s-1. U, velocity; c, wave speed; H, fin tip amplitude; f, flapping frequency; CP, power coefficient; CT, thrust coefficient; h0, amplitude; l, strip length.