Schematic of state variable (F x : higher order thrust force) due to fin-wake dynamics in a single, isolated, low-aspect-ratio hinged fin optimally flapping (while twisting along its span) in the transitional regime of vortex propulsion. 4,11 The oscillator bifurcation parameters (Eq. (16) ) are fixed in fin-wake modes A, B, and C, while they are bi-stable in modes A-B and B-C. The relationship of this fin-wake behavior with hydrodynamic efficiency in the fin appended body is examined in Fig. 3(a) .
(a) Schematic showing the cylinder-fin assembly (without the endcaps) and the fin motions. (b) Photograph of streamlined cylinder appended with six fins, suspended from the instrumented carriage in a tow tank. Tow direction is toward the left.
Measurements of the effects of speed (Re c ) on (a) hydrodynamic efficiency and (b) coefficient of axial thrust in the streamlined cylinder appended with six fins at ϕ0 = 45°, 9.7 × 103 < Re c < 5.8 × 104, 2.2 × 105 ⩽ Re L ⩽ 1.7 × 106. The markings in (a) indicate the mono-stable and bi-stable modes of the fin-wake dynamics. The dashed lines in the inset in (a) indicate maximum values of efficiency at a given speed.
Measurements of (a) hydrodynamic efficiency and (b) axial thrust in the single isolated fin, showing the effects of forward speed, 4.5 × 104 < Re c < 1.2 × 105 (fin-wake relationship: stable mode C).
Comparison of (a) hydrodynamic efficiency and (b) coefficient of axial thrust in rigid fins attached to a cylinder (0.40 and 0.50 m/s) and rigid fin in isolation (0.46 m/s); Re c = 45 000 (fin-wake relationship: stable mode C).
Propulsion power of the cylinders appended with six fins (NUWC BAUV and NUWC SPLINE II) compared with that of swimming animals that use vortex-based unsteady propulsion. Red muscle (RM) data source: tuna (warm blooded), 34,35 shark (cold blooded), 38 mackerel, 39 and bonito. 40,41 Solid line: Power (kW) = 0.002 + Displacement volume (m3).
Comparison of the efficiencies of single isolated rigid fins and cylinders appended with rigid fins with those of flies 42 and fish. 43 The ellipses are meant to aid graph reading for the fin-appended, streamlined cylinder during cruising: left ellipse – total electrical efficiency, which includes power input to the controller and other electronics (η e , Eq. (12) ); middle ellipse – electrical efficiency, which excludes power input to electronics (η e, propulsion , Eq. (13) ); right ellipse – hydrodynamic efficiency (η h , Eq. (11) ).
Maps of force fluctuations and their derivatives at increasing Strouhal numbers: measurements (a), (c), (e), (g); and Eq. (16) (b), (d), (f), (h). U (m/s), f (Hz), Re c , St, ϕ0, and θ0: ((a) and (b) 0.6, 0.5, 36 950, 0.23, 35°, 25°), ((c) and (d) 0.8, 0.7, 49 168, 0.24, 35°, 25°), ((e) and (f) 0.6, 0.5, 36 841, 0.30, 45°, 30°), and ((g) and (h) 0.60, 0.7, 37 010, 0.41, 45°, 45°).
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