Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: Everything happens in a thin layer — page 2

Page 2 of 10, continuing through the 84 essays this motif runs through.

84 essays carry this thread — page 2 of 10.

A wing's two drags, and the lift at which they are equal. Friction drag and induced drag plotted against lift coefficient, with their sum above them. Friction is flat, because a surface costs the same whatever the wing is doing; induced drag rises as the square of the lift. The total is least where the two are equal, and that is also the point of best glide. Viscosity

The two drags a wing pays

A wing pays for having a surface, and it pays for making lift with a finite span. One of those bills falls as it flies faster and the other rises, so there is a speed at which the total is least — and the condition for it turns out to be that the two are equal.

 The wall slope against pressure gradient, and where it runs out. How steeply the flow leaves the wall, plotted against the pressure gradient the layer is running into. A favourable gradient presses the profile against the surface and steepens it; an adverse one hollows it out. The curve reaches zero at a definite value, and beyond that there is no attached solution at all. Viscosity

How much uphill a layer can take

A boundary layer running into rising pressure is climbing a hill on the last of its momentum. There is a definite steepness at which it can no longer do it, and the number is not a rule of thumb — it is where a family of solutions stops existing.

 u⁺ = (1/κ) ln y⁺ + B, integrated rather than asserted. The velocity profile in wall units, produced by integrating the mixing-length closure outward from the wall. The straight portion is the log law and the constants beside it were least-squares fitted to the integrated curve over 50 < y⁺ < 500 — so the 1/κ is a measurement on the drawing rather than the number that was fed in. The viscous sublayer u⁺ = y⁺ comes out rather than being pasted on. Transition and turbulence

A guess with a constant in it

Prandtl's mixing length is one line — an eddy near a wall can only be as big as its distance from the wall. Integrate it and the whole structure of a turbulent wall profile falls out, sublayer and log region and all. That is a fact about the assumption, and the essay is careful about which.

 Where the flow over the section first reaches Mach one. Two curves running towards each other. The falling one is the section's peak suction, solved exactly in incompressible flow and then deepened by the Prandtl–Glauert factor as the free-stream Mach number rises. The rising one is the pressure coefficient at which the local flow would be exactly sonic. Where they cross is the critical Mach number, and above it there is a pocket of supersonic flow on a subsonic aeroplane. Regimes and numbers

The pocket on top of the wing

An airliner cruising at Mach 0.85 has subsonic flow almost everywhere and a patch of supersonic flow over its wing. Closing that patch takes a shock, the shock separates the layer beneath it, and the correction that predicts all of this also predicts, in the plainest terms, where it stops being true.

 Pressure recovery along the upper surface at 6°. Surface speed and the local Falkner–Skan pressure-gradient parameter, plotted along the upper surface from the nose. The speed peaks near the leading edge and then falls, which is the layer climbing back up to the pressure it started at, and the parameter crosses the separation value where that climb becomes too steep. Viscosity

Where the straight line stops

Ideal flow will report a lift coefficient at forty degrees of incidence without complaint. What ends the lift curve is the boundary layer refusing to follow the surface, and the estimate of when that happens joins two solvers that have nothing else in common.

 The speed for least drag is not the speed for least power. Drag and power against airspeed, in units of the speed at which drag is least. Power is drag times speed, so its minimum sits slower — at the fourth root of a third of the least-drag speed, which is 0.76 of it. Flying for range and flying for endurance are therefore different speeds, and the difference is not a rule of thumb. Viscosity

The cheapest way to stay up

There is a speed at which an aircraft's drag is least, and a different, slower speed at which its power is least. The ratio between them is the fourth root of a third — a number that does not depend on the aircraft, the altitude, or anything else about the flight.

 The cliff a rough ball reaches sooner. The drag coefficient of a sphere against Reynolds number, on log axes. The smooth curve is Morrison's correlation, which is a fit to measurements and is drawn in the colour this site reserves for a borrowed claim. The other is the same curve shifted along the Reynolds axis by a factor of 6 — a stated model of what roughness does, which is to trip the boundary layer early, and not a measurement of any real ball. Fluids at work

The drag that falls as it speeds up

There is a band of speeds in which a smooth ball experiences less drag the faster it goes. Not a smaller coefficient — a smaller force. Dimples move that band down to where a golf ball actually flies, and they do it by making the friction worse.

 Two sides of one ball, at different pressures. The surface pressure coefficient round a ball, measured from the front stagnation point, on the side the seam trips and on the side it does not. Up to separation both follow the exact potential-flow distribution 1 − (9/4)sin²θ. After it both take the same wake pressure, which is what a manometer measures rather than what the ideal theory predicts. The asymmetry is the shaded area between them, and integrating it gives a side force of 0.2949 towards the later-separating side. Fluids at work

A ball that swings without spinning

A cricket ball curves in flight with no spin about any useful axis. The mechanism is not the Magnus effect; it is a seam tripping the boundary layer on one side so that side lets go later. Which way the ball then goes depends on one borrowed number, and this site's own inviscid solver supplies the value that gets it wrong.

 Every wavelength grows, and the shortest grows fastest. Growth rate against wavenumber for a vortex sheet of zero thickness and for a layer of thickness δ. The sheet's curve rises without limit, which is the model announcing that it has no shortest scale in it; the layer's turns over and dies at kδ = 1, the neutral wavenumber computed from Rayleigh's equation rather than fitted. Viscosity

The gradient that does both

One line of the boundary-layer equations at the wall says the profile's curvature there equals the pressure gradient. That single sign causes separation and causes instability, and it causes the instability a long way before it causes the separation.

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