Adrian Bejan | Wingspan and fuselage length, from Design in Nature
The video builds a drawing of a fuselage and two wings by predicting proportions, not millimeters or centimeters, using relative sizes called proportions and aspect ratios for slenderness, the shape of the cross-section, wingspan relative to fuselage length, and wing profile features. It starts from the premise that everything that flies operates at a balance between the effort to lift itself and the effort to overcome drag as the body falls forward, so the focus is on overcoming drag. The drag is expressed as a frontal area and a friction coefficient, with contributions from the fuselage and the wings combined; the video also introduces aerodynamic loads as the wing deflects upward under lift. The best fuselage cross-section is described as roundish, like that of a dolphin or orca, and the video links this to heat loss being least for warm-blooded animals, whereas fish have a vertical oval cross-section because the whole body paddles in a horizontal plane. With these moves, the video combines the equations and concludes that the wingspan-to-fuselage-length ratio is essentially constant across airplanes, birds, and insects.
The drawing uses five degrees of freedom, including slenderness and wingspan relative to fuselage length, so you need proportions before you make it. The point is that the drawing conveys relative sizes through proportions or aspect ratios.
The analysis focuses on the drag experienced by the body as a wind approaches at a given speed, with contributions from the fuselage, the wing, and the wings together. The drag model uses a frontal-area concept and a friction coefficient defined in terms of the wetted perimeter and length.
Mechanical strength is added by looking at the wing in equilibrium in the armpit as it curls upward during takeoff. The lifting force balances weight, and the bending moment is tied to span and thickness with tension and compression over the cross-sectional area of the wing.
The fuselage cross-section is pushed toward a roundish shape, with a story about a square cross-section not being very good for flying through the air, and comparisons to transport aircraft and buildings. The video shows that a round cross-section leaks the least heat when volume is fixed for warm-blooded animals such as dolphins, orcas, manatees, and whales, whereas fish differ.
The wing is expected to be slender in cross-section when the profile area is fixed, and friction coefficients are evaluated using a Moody chart. After combining the equations, the video states that variations in body mass are negligible relative to the values you draw, so the wingspan-to-fuselage-length ratio is essentially constant across observations.
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