Every essay — page 14
Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search
Circulation and lift
Where lift actually comes from. The Kutta condition, the Joukowski aerofoil, and lift derived rather than assumed.
A pivoted wingtip follows the lift only with no spring stiffness and no camber
The washout a trainer's sortie wants grows in proportion to the lift coefficient, and the twist a wing gets for free follows the dynamic pressure instead. A tip panel free to pivot on a hinge ahead of its own aerodynamic centre can do better, and on one condition it does exactly what is wanted: held by a constant torque and carrying no camber moment, its own lift coefficient stays in proportion to the wing's, and every segment of the sortie flies at its own best twist. Every unit of spring stiffness gives part of that back, a cambered section's moment turns the panel into the wrong law altogether, and the free panel sheds a quarter of a gust's lift for nothing.
Sweep slows a wing root's gust cycles and leaves their excess over the lift
On a straight wing every station meets a gust at the same instant, and one wing root's bending moment crosses its mean nearly a fifth more often than the lift, because it reads the rolling half of the gust. Sweep the wing back and the outer stations meet the gust later, by their distance aft. That delay averages the gust along the flight path as a second, softer chord would: at 35° every load crosses its mean about a tenth less often, the root slightly more than the lift since its weight sits further out, and the variance hardly changes at all. The rolling excess survives almost whole.
A wing that cannot hold its suction gains little from its span
The leading-edge suction is a small force concentrated at one edge, and on a finite wing it decides what the aspect ratio is worth. With it, the drag due to lift is the induced drag, which falls as the span grows. Without it, the drag due to lift is the lift tilted back by the incidence, and that falls only as fast as the lift slope rises — which stops at the section's 2π. A slender delta that loses its suction pays exactly twice its induced drag; a wing of aspect ratio eight pays more than five times it. Without the suction no span can lift a wing's best lift-to-drag ratio past a ceiling set by the section alone.
A spar that must lift itself stops the span at a quarter of the weight
Held only to a lift and a bending moment, the least-drag wing never stops growing. An aeroplane's wing stops because its spar has weight, and the weight has to be lifted. Count it, and the best span is the one at which the spar is exactly a quarter of the lift when the wing's area is fixed, and a third when its chord is — whatever the loading. The best loading then sheds some of its tip load, but not all the way to Prandtl's bell; with the chord held and nothing else charged, it is the bell exactly, and the more of the drag the wing's own area makes, the nearer it comes back to the ellipse.
A stiffness floor keeps the span and gives the tips their load back
A spar sized by strength alone thins to nothing at the tips, and charged as lift it stops the best span where it weighs a quarter of everything the wing carries, with the tips a little unloaded. A real outer wing is sized by stiffness, and stiffness sets a floor under the spar's weight that no reduction of the tip load can lower. Put that floor in and the tips get their load back — the best loading slides from a sixth of the way to the bell towards the ellipse — the quarter does not survive, rising to two fifths as the floor spreads over the outer two thirds of the span, and the span itself barely moves. A weight per metre cannot stop a span.
A strut inside a box wing is invisible to the wake, and halves the spar
Prandtl's best box wing has, everywhere inside it, air moving straight down: no sideways flow at all. A strut joining its two wings anywhere along the span therefore meets no flow it could turn, carries no load, and leaves the induced drag exactly where it was — the wake cannot see it. The spar can. A strut two fifths of the way out, as stiff as a spar and rigidly joined, halves the flange material of both spars, though it is pushed and pulled by almost nothing: the two wings lift equally and bend together, so the strut works through its joints, and a pinned one does nothing. Its only aerodynamic price is friction, and a slim strut pays it.
Compressible flow
What changes when density stops being a constant. Shocks, expansion fans, nozzles that work backwards, and the one place on this site where an inviscid flow has drag.
What a signal travels at
Air finds out that an aeroplane is coming, and it finds out at a definite speed. That speed does not depend on how hard the air is squeezed or on how much of it there is — only on how hot it is — and every result in compressible flow is downstream of that one fact.
When the warning cannot arrive
Supersonic flow is not fast flow. It is flow in which the fluid ahead has been told nothing, because the body is outrunning its own pressure signals — and that single change turns an equation of one type into an equation of another.
Energy instead of pressure
Bernoulli's equation is a statement that pressure and speed trade against each other at fixed density. Take the fixed density away and the trade is between heat and speed instead — and what survives is not a pressure at all.
The duct that works backwards
Squeeze a pipe and the flow speeds up. Everyone knows this, it follows from continuity, and above the speed of sound it is false — a narrowing duct decelerates a supersonic stream, because the density is falling faster than the speed is rising.
The throat that stops listening
Lower the pressure downstream of a nozzle and more gas flows through it. Keep lowering it and, at a definite point, the flow stops responding — not gradually, but completely, because the news that the pressure has fallen can no longer travel upstream.
One area, two answers
The area of a duct at a station fixes the Mach number there — twice over. A ratio of 2.5 is satisfied at Mach 0.24 and again at Mach 2.44, nothing local chooses between them, and where the choice cannot be made consistently a shock appears inside the duct to join the two.