Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Linearisation — where it appears

The replacement of a nonlinear problem by its first-order approximation about a known state. It is exact in the limit of small disturbances and it is the source of most of what this subject can compute in closed form, including every stability result and every wave speed.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

The correction belongs to the wing, not to the air. Lift-curve slope against Mach number for three aspect ratios, each computed by solving the lifting line for the wing the transformation actually implies — one of aspect ratio βAR — and dividing by β. The dashed curve is the two-dimensional rule, which is what an aerofoil section gets. At Mach 0.7 the aspect ratio of 20 has gained 35 per cent of slope and the aspect ratio of 4 has gained 24, against the 40 per cent the section rule promises both. The β in the finite-wing term cancels the β in front of it, so the shorter the wing the less compressibility does to it.

The wing the equation is really solving

The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.

regimes · Similarity The free surface a submerged body leaves behind it, and the flat water in front. The linearised free-surface problem solved as a Fourier integral with a radiation condition. Behind the body a wave train of the wavelength that stands still relative to it, 2 pi U²/g; ahead of it, an amplitude a hundred and twenty times smaller. The asymmetry is the drag: an ideal fluid with a free surface can carry energy away.

The drag that is made of waves

D'Alembert's paradox says a body in a steady, irrotational, incompressible, inviscid flow feels no drag. Put a free surface above it and every one of those words still holds — and the drag is not zero. It is the energy walking away in the wave train behind.

inviscid · Dalembert The slow bending wave, and where the cut-off stops describing it. The frequency of the slowest bending wave on a Rankine vortex — a helical wobble of the whole core that turns against the flow — against the wavenumber times the core radius, from Kelvin's exact relation and from his long-wave formula, which the cut-off method used for vortex pairs reproduces. They agree for long waves. Past ka = 1.44 the long-wave formula reverses sign and the exact wave does not, which is why a cut-off model's instabilities at such wavelengths are not real.

A vortex is a waveguide

A spinning core is stiff in a way still fluid is not, and it carries waves along its length: an infinite family of them for every pattern round its axis, travelling at up to 0.83 of the swirl speed at its edge. The slowest is a helical bend that turns against the flow, and it is the wave every model of a bending vortex has been borrowing without saying so.

inviscid · Swirl Hull speed is not the hump. The wave-resistance coefficient of the Wigley hull, R/(½ρU²S) in thousandths, against the Froude number U/√(gL), from Michell's thin-ship integral. The curve rises through a series of humps and hollows and peaks at Fr = 0.499. The hull-speed rule's Froude number, 1/√(2π) = 0.399, where the transverse wave is as long as the hull, is neither: it lies on the steep climb between the last hollow, at 0.346, and the main hump.

Hull speed is not the hump

The hull-speed rule says a displacement vessel meets a wall where its wave is as long as its hull, at a Froude number of 0.4. Michell's thin-ship integral, computed for a standard hull, puts that speed on the steep climb between the last hollow and the main hump, and the hump itself at 0.5. Past the hump the wave resistance grows more slowly than the speed. The hollows sit where the hull's own transverse wave switches off, and a bulb that cancels at one speed multiplies the resistance at another.

regimes · Froude The integral's hull is fine-ended slow and full-ended fast. The waterline half-breadth, bow to the right, of the hull of least wave resistance with the Wigley hull's length, draught, depth profile and displacement, designed for Froude numbers of 0.25, 0.30, 0.40 and 0.50, beside the Wigley hull's parabola. At low speed the optimum pulls volume into its middle and leaves long, fine ends; at high speed it pushes volume out towards the ends. Every one is symmetric fore and aft.

The hull Michell's integral prefers

Michell's integral gives a thin ship's wave resistance as a quadratic in the hull's offsets, so the hull of least resistance at a given speed and displacement is a quadratic minimisation. Allowed only to reshape its waterline, the integral rediscovers the naval architect's oldest rule: fine ends for a slow ship, full ends for a fast one. Allowed to reshape its depth too, it drains the waterline and piles volume at the keel until the hull is not a ship. And it cannot grow a bulb at the bow, because it cannot tell the bow from the stern.

regimes · Froude Zero wave drag at one Mach number, inside a loop it cannot enter from below. The wave drag of the pair, per unit of one element's chord, against Mach number, beside a single diamond of the pair's combined thickness. The biplane's drag is zero at its design Mach number, 1.944, and rises either side. Shaded: below 1.6 the channel cannot pass the flow at all; between 1.6 and 2.15 it can, but only if it was started above the band. Accelerating from subsonic, the biplane stays choked until Mach 2.15, past its design point.

Busemann's biplane has to be flown past its design point

Two half-diamonds facing across a channel cancel each other's waves at one Mach number and carry their thickness at no wave drag. The cancellation needs the channel narrowed at its middle, and a narrowed supersonic channel will not swallow the shock that forms in it on the way up. A biplane 3 per cent thick per element, designed for Mach 1.94, stays choked until Mach 2.15, and only one that is thinner than about 2.6 per cent can start by itself at any speed.

compressible · Wave drag Added linearly, the parts give the section's suction peak a quarter too weak; carried through Bernoulli, most of it comes back. Pressure coefficient, suction upward, on the upper and lower surfaces of the 12 per cent section with 4 per cent camber at 4°: exact; the parts' disturbance speeds summed and the pressure taken linearly; and the same sum carried through Bernoulli. At a tenth of the chord on the upper surface the exact −Cp is 1.39, the linear sum 1.06 and the Bernoulli sum 1.34.

Thickness carries no load of its own and still moves the load forward

Thin-aerofoil theory builds a section's pressures from three pieces computed apart — its thickness, its camber, its incidence — and adds them. Every piece can be computed exactly on a Joukowski section, and so can the section they add up to. Added, the parts give a suction peak a quarter too weak and a loading that sits too far aft, and they miss nine per cent of the lift: thickness, which lifts nothing by itself, multiplies what camber and incidence make by 1 + 0.77 t/c. Carry the summed velocities through Bernoulli instead of adding the pressures, and most of the error goes; the part that stays is the cross term between thickness and lift, and it is first order in the lift.

inviscid · Superposition At its design Mach number the biplane's channel lifts nothing: all its lift is on the two outer faces. The lift coefficient of the pair at 2° incidence against Mach number, per unit of one element's chord, beside what the two outer faces give alone, 4α/β, and two isolated flat plates, 8α/β. At the design Mach number, 1.944, the pair's lift is 0.08378 and the outer faces' 0.08378: the channel's net lift is zero to rounding. At Mach 1.7 the channel adds 13.9 per cent, at 2.2 21.1.

Busemann's biplane lifts only on its outer faces, and its channel turns the lift into a quarter-chord load

Tilt Busemann's biplane to make it lift and the same cancellation that removed its wave drag removes the lift of its channel: at the design Mach number the waves the incidence sends across the gap meet and cancel just as the thickness's did, and every bit of lift is carried by the two outer faces — half what two separate plates would give. The channel does not go idle. It carries a couple, lift on the front half and download on the rear, which moves the centre of pressure from mid-chord, where every supersonic plate has it, to the quarter chord. The pair pays for its lift two and a half times as steeply as flat plates do, and still beats a diamond fourfold.

compressible · Wave drag

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitFroude numberGroup velocityInterferenceWave resistanceDispersionHull speedLift curve slopeWave dragArea machAspect ratioBoundary condition

All concepts