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

Superposition — the series

5 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. Ideal flow past a cylinder. A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.

    Flows add up

    The equations of ideal flow are linear, so solutions can be laid on top of one another. A uniform stream plus a doublet produces a cylinder that nobody put there, and almost every classical result is built this way.

    part 1 · inviscid
  2. Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

    Bodies made out of nothing

    Put a source in a stream and a solid nose appears in front of it. Put a sink downstream of the source and the nose closes into a finite body. Nothing was placed in the fluid — the surface is a level set of a function, and it behaves like a wall because nothing crosses it.

    part 2 · inviscid
  3. The pressure of a sum against the sum of the pressures. Ten points around a cylinder with circulation, with the pressure coefficient of the combined flow plotted against what adding the two flows' separate coefficients would give. Nothing lies on the diagonal. The gap is exactly −1 − 2u_A·u_B/U², an identity checked to the last digit at every point, and it is not small: at one of these points the two answers differ by 1.92, which is more than the whole range of a suction peak.

    The one thing that does not add up

    Laplace's equation is linear, so flows can be laid on top of one another and almost every classical result is built that way. The two things anybody actually wants out of a flow — the pressure and the force — are quadratic in the velocity, and neither of them adds at all.

    part 3 · inviscid
  4. 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.

    part 4 · inviscid
  5. On a NACA section thickness multiplies the lift by more than on a Joukowski section, by nearly one plus its thickness. The lift of a NACA four-digit section by panels over thin-aerofoil theory's, against its thickness: the 2412-type at 4°, and the symmetric section's lift slope over 2π; beside the Joukowski law 1 + 0.77 t/c. At 12 per cent the cambered section's factor is 1.115 and the symmetric slope's 1.101, against Joukowski's 1.092; at 18 per cent, 1.174 and 1.152.

    A NACA section multiplies its lift by nearly one plus its thickness, and a nose correction cannot supply it

    On Joukowski's exact sections, thickness — which lifts nothing alone — multiplies the lift of camber and incidence by 1 + 0.77 t/c. Solved by panels, the NACA four-digit sections that real wings use do more: their lift slope grows as about 1 + 0.84 t/c, and a cambered 2412 lifts 11.5 per cent more than thin-aerofoil theory says. The more of the thickness sits near the nose, the more it multiplies. Added as thin-aerofoil theory adds them, the parts lose a quarter of the suction peak and put the centre of pressure two hundredths of the chord too far aft. Riegels's nose factor, the classical repair, makes the nose speed finite and moves the peak, and adds no lift at all: the missing lift is not a nose effect.

    part 5 · inviscid

All series