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The thread: The exact theory is wrong — page 13

Page 13 of 22, continuing through the 193 essays this motif runs through.

193 essays carry this thread — page 13 of 22.

$A sinusoid, distorting on its way to a shock. A finite-amplitude sound wave at four fractions of the distance to shock formation, computed by inverting the implicit simple-wave solution. Each point of the waveform travels at its own speed, so the compressions catch up with the rarefactions ahead of them and the profile leans forward. At σ = 1 the front is vertical. The linear theory says the first panel is the answer at every distance, for ever.$ Compressible flow

Every compression becomes a shock in the end

Linear acoustics has no time scale in it, which is the sign that something has been thrown away. A 120-decibel tone shocks after three hundred metres and a jet engine after twenty; the distance goes exactly as the reciprocal of the amplitude, and nothing is exempt.

A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

Ideal flow

The pressure that depends on the past

Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.

Three answers for the lift-curve slope, with three different signs. Thin-aerofoil theory has no thickness term at all and returns 2π for every section. The exact potential solution rises: 2π(1 + 0.766 t/c), measured off the Joukowski map. Real sections do the opposite, because the boundary layer thickens towards the trailing edge and decambers the section. Two of these curves are computed here; the third is what measurement says.

Circulation and lift

The half that carries nothing

Thin-aerofoil theory splits a section into a camber line that carries all the lift and a thickness distribution that carries none — at any incidence, exactly none. That is very nearly true, and what it discards decides the peak suction, the critical Mach number and where the boundary layer gives up.

The condensate, and the limit of no friction which does not exist. In a steady state the friction must remove everything the forcing puts in, so the energy is eps/(2 alpha) and the coherent velocity is sqrt(eps/alpha) — exactly a minus one half power, checked to 10⁻¹². As the friction is weakened the condensate grows without bound: the limit alpha to zero is not a flow with a weak condensate, it is a flow with no steady state at all.

Transition and turbulence

Where the inverse cascade stops

Two-dimensional turbulence sends its energy upward in scale, and the upward direction has an end: the box. Without something to remove the energy before it arrives, it accumulates there in a pair of vortices filling the domain, and the limit of no friction has no steady state at all.

So splitting a turn into N ramps costs one over N squared. A twelve-degree compression at Mach 3, done in one ramp and in up to sixty-four. The entropy is N times a cube of one Nth, so it falls as exactly the inverse square of the number of ramps — the measured exponent is −2.00 — and sixty-four ramps cost a two-hundred-and-fifty-sixth of what one costs.

Compressible flow

A compression that costs nothing in the end

Turning a supersonic stream away from itself is free and turning it into itself is not. But the price of a compression is the cube of its strength, so splitting one turn into N turns costs one over N squared — and in the limit the compression is free too.

Four bodies, four drag coefficients, and no flow was solved. Four bodies with their Newtonian drag coefficients, each computed as a quadrature over its own surface with no flow solution anywhere. The cone's answer is exactly 2sin²δ, checked against the closed form to nine decimal places; the flat disc's is exactly 2, since every element of it faces the stream; and the sphere's is 1 against the classical Newtonian value of 1. Every one of those is an integral of one expression over a shape, and none of them required knowing what the air was doing anywhere.

What is taught wrongly

The only theory simple enough to optimise

Whether Newton's sine-squared law is right has two answers — hopeless at the speeds he argued about, nearly exact behind a strong shock. This asks a different question about the same formula. Its pressure depends only on the local surface angle, so a shape's drag is a quadrature rather than a solution, and the best shape can be found by calculus.

$A swimmer that dissipates the same everywhere. Taylor's waving sheet, with the wave drawn along the bottom and the dissipation drawn against height above it. The dissipation function works out at 4μb²c²k⁶y²e^{−2ky} — with no x in it at all, so the sheet is destroying energy at the same rate under every part of the wave and at every instant of the cycle. It peaks one radian of wavelength above the sheet and is gone within about three.$ Viscosity

A swimmer that cannot go backwards

Taylor's waving sheet is the simplest self-propelled object in a viscous fluid, and its arithmetic contains a result that reads like a mistake — the work it does to travel a metre does not depend on how big its waves are. Doubling the amplitude quadruples both the speed and the power, and changes the bill for the journey by nothing at all.

A normal stress the closure makes negative. The Boussinesq closure's first normal stress in a plane strain, against the strain measured in units of the turbulence's own time scale. It crosses zero at S k/eps = 1/(3 C_mu) = 3.704 — eleven per cent above the value the constant was calibrated at — and goes on falling. A variance below zero is not a small error; it is a statement that cannot be true.

Transition and turbulence

The constant that makes a variance negative

Every engineering turbulence calculation in the world rests on one number, C-mu equals 0.09. It is not a property of turbulence. It is the assertion that a particular ratio is ten thirds, which is true in one flow — and eleven per cent above that flow the same closure reports a mean square below zero.

A compression piston, and where its characteristics first cross. Sixty C+ characteristics from an accelerating piston, drawn in the distance-time plane. Each is a straight line, because the invariant makes the state along it constant; later ones are faster, because the gas ahead of them has been compressed; so they converge, and the first crossing is the shock. The envelope formula gives 2.7529 and the first actual crossing is at 2.7510.

Compressible flow

Two numbers that do not change

One-dimensional unsteady gas flow carries two quantities that are exactly constant along two families of curves. That single fact turns a pair of coupled partial differential equations into a family of straight lines, and gives an exact speed at which a gas outruns its own expansion.

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