Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: The exact theory is wrong — page 5

Page 5 of 22, continuing through the 193 essays this motif runs through.

193 essays carry this thread — page 5 of 22.

The second law is the only thing that forbids the other half of this curve. Entropy change across a normal shock, against the Mach number in front of it. The solid branch is the compression shock that exists. The dashed branch below Mach one is the expansion shock, and it satisfies mass, momentum and energy exactly — the residuals are zero to machine precision. It is refused by the second law alone, the one statement in the problem that no conservation residual can show. Compressible flow

The only law that forbids it

The jump conditions permit a discontinuity in either direction. An expansion shock conserves mass, momentum and energy exactly — the residuals are zero to machine precision — and it does not exist. Nothing that can be drawn rules it out.

 The inertial range, and the slope read back off it. The model energy spectrum at Re = 1e+6, with production rolling off below the integral scale and dissipation cutting it off above the Kolmogorov scale. The straight middle is the inertial range, and the number printed beside it is the slope least-squares fitted to the drawn points over the middle of that range — not the −5/3 that went in. Transition and turbulence

Where the energy goes

Energy enters a turbulent flow at the largest scale and leaves it at the smallest, and in between there is nothing for it to depend on but the rate at which it is passing through. Two quantities and one dimensional argument fix the shape of the spectrum, and the exponent is −5/3.

 Every curved streamline has a pressure gradient across it. Twelve points in the flow past a cylinder, with the arrow at each showing the pressure gradient across the streamline. It points away from the centre of curvature everywhere, and its size is ρq²κ: the fluid is being pushed round a bend, and something has to do the pushing. Both sides are computed here and they share no arithmetic — one is Bernoulli's pressure differenced across the flow, the other is the turning rate of the velocity direction along it — and they agree to 6.7e-5. This is the whole content of the effect usually named after Coandă, and it is happening on every curved streamline of every flow. What is taught wrongly

The effect that explains nothing

A jet follows a curved wall and the wall feels a suction. Both are real, both are famous, and naming them after Coandă explains neither — what is happening is the normal component of Euler's equation, and it holds on every curved streamline in every flow.

 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.

 Grid points against Reynolds number, and where a wing sits. The number of grid points needed to resolve every scale of a turbulent flow, which is Re^(9/4) — the cube of the ratio between the largest scale and the Kolmogorov scale. The line is the arithmetic and the marks are flows a reader can picture. An airliner's wing needs about 10¹⁷ points, and the largest calculations ever run are around 10¹². Transition and turbulence

The grid nobody can build

Resolving every scale of a turbulent flow needs Re to the nine-quarters grid points and Re cubed point-updates. An airliner's wing comes to 2·10¹⁷ points against the 10¹² of the largest calculation ever run, and no amount of patience closes a gap of five orders of magnitude.

 A wing at h/c = 0.4. A wing close to the ground, with the mirrored vortex row that makes the ground a streamline drawn faintly beneath it. The air between wing and ground is squeezed through a narrowing gap, and the wing carries more lift than it would in free air. Circulation and lift

The wall that pushes back

A wing near the ground carries more lift than the same wing in free air, and the usual explanation — a cushion of compressed air underneath — is not what the equations say. The ground is a mirror, and what changes is not the pressure under the wing but the circulation the wing is forced to carry.

 A 10° wedge at Mach 2.00 has two shocks that solve it. The wedge turns the flow through a fixed angle, and the θ–β–M relation offers two shock angles that achieve it. The weak solution, drawn steeply forward, leaves the flow supersonic and is what a wedge in a free stream produces. The strong solution leaves it subsonic and appears where downstream pressure forces it. Nothing local to the wedge chooses between them. Compressible flow

A shock that leans

Tilt a shock and only the velocity component across it is changed — the component along it passes through untouched. That single observation turns every oblique shock into a normal shock in disguise, and it is why a wedge at Mach 2 leaves the flow supersonic while a blunt nose does not.

 Every disturbance has its own threshold; one of them is lowest. The Rayleigh number at which a disturbance of horizontal wavenumber a becomes neutral, Ra = (π² + a²)³/a². Every wavenumber has a threshold and the layer goes unstable at the lowest of them, which a golden-section search on this curve puts at a = 2.221441 and Ra = 657.5114 — the exact π/√2 and 27π⁴/4 to fourteen digits. Below the curve the layer conducts and nothing moves. Transition and turbulence

A threshold with a closed form

A layer of fluid heated from below sits still until buoyancy overcomes both diffusions at once, and then it convects. Unlike every other threshold in this field, that one is an eigenvalue with an exact answer, and the answer is 27π⁴/4.

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