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The thread: The exact theory is wrong — page 6

Page 6 of 22, continuing through the 193 essays this motif runs through.

193 essays carry this thread — page 6 of 22.

The fluid a moving cylinder carries with it. Kinetic energy density around a cylinder moving through fluid that is at rest far away. The fluid is not dragged along in a lump: it is pushed aside in front and closes in behind, and the energy in that motion is what has to be supplied to change the body's speed.

The force of getting going

The exact theory says a body moving steadily through an ideal fluid feels no force at all. It does not say the fluid is free. Accelerating the body has to accelerate the fluid too, and the bill for that is exactly the mass of fluid the body displaces.

Three equations, and the set they never leave. The Lorenz trajectory at r = 28, projected on x and z, after the transient has been discarded. It never repeats, never leaves, and never crosses itself in three dimensions. The Lyapunov exponent printed beside it is measured on this system by separating a nearby pair, so the claim of sensitive dependence is a computation.

Transition and turbulence

Three numbers left of a fluid

Saltzman truncated convection to three Fourier modes and Lorenz studied what was left. The result changed science, and it stopped being a description of a fluid at about a fifth of the way to the parameter everybody quotes it at.

Past 23.0° at Mach 2.00 there is no attached shock. The same wedge at two half-angles. On the left the θ–β–M relation has a root and the shock sits on the nose. On the right it has none, and the solver throws rather than returning the nearest thing — which matters, because a solver that quietly clamped to the maximum would draw a neat attached shock on a body that cannot carry one. The bow shock on the right is indicative: its shape is not solved here.

Compressible flow

When the wedge is too blunt

Every curve of shock angle against deflection has a maximum. Past it there is no attached shock at any angle — the solver has no root to return and must say so, rather than quietly handing back the nearest thing and drawing a picture that cannot exist.

The speed depends on the core; the slope does not. A ring's self-induced speed against the logarithm of the cutoff used to compute it. The line is a quadrature of the reduced integral; the dots are an independent sum of cross products over a hundred and twenty thousand straight segments. Both diverge as the cutoff is thinned — a filament of zero thickness would move infinitely fast — so no answer here is a ring's speed until a core model is chosen. The slope, Γ/4πR, is the same whatever is chosen, and is what the check measures.

Circulation and lift

A ring moves because it is bent

A straight vortex filament induces exactly no velocity on itself — every element's direction is parallel to the line joining it to the point being evaluated, and the cross product is zero. Bend it into a ring and it drives itself forward, at a speed that diverges logarithmically as the core is thinned.

One more condition, and the price of it. The model problem ε u″ + u′ = 0 with a condition at each end, at three values of ε. The outer solution is the flat line at one — that is the whole of the answer when ε is zero, and it is a first-order equation that can meet one condition, so it meets the one at the far end and misses the one at the wall by the whole range. Restoring ε restores the second condition and pays for it with a layer of thickness ε, inside which the gradient is of order 1/ε. The product of those two — which is what a stress is — does not depend on ε at all: it is 1.0000 at every value tried, to nine decimal places. Drag does not vanish as viscosity does. It converges.

Flows and fields

How many things a flow must be told

The equations of motion do not have one answer. They have as many as the conditions on the edge allow, and the number of those is decided by the order of the equation — which is why viscosity does not make the same problem harder, it makes a different problem.

The pressure, relaxed rather than quoted. The pressure field round a cylinder, obtained by relaxing ∇²p = −ρ∇·(u·∇u) on a body-fitted polar lattice with the exact pressure on the surface and on a far circle. The interior was told nothing except the velocity gradients. It agrees with Bernoulli's closed form everywhere to under two parts in ten thousand of the dynamic pressure, which is the strongest statement this site can make that the elliptic equation is the pressure's own.

Ideal flow

Pressure has no speed

Take the divergence of the momentum equation for an incompressible flow and the time derivative disappears, the viscosity disappears, and what is left is Poisson's equation. Pressure is not carried anywhere: it is whatever satisfies an elliptic equation everywhere at once, and that is a statement about a fluid nobody has.

The same 10° turn, taken both ways. A supersonic stream turned away from itself expands through a fan of Mach waves and keeps every bit of its total pressure. Turned into itself through the same angle it shocks, and pays. Nothing in the equations distinguishes the two cases except the sign of the angle: compression waves converge and steepen into a front, expansion waves diverge and spread.

Compressible flow

Turning the other way is free

Compression through ten degrees at Mach 2 costs 1.54 per cent of the total pressure. Expansion through the same ten degrees costs exactly nothing — not approximately nothing, nothing — and the two are the same equations with the sign of one angle changed.

Compressible flow

Drag with nothing to rub

d'Alembert's paradox says a closed body in a steady, inviscid flow has no drag, and four essays on this site argue it and none of them is wrong. Above Mach one it is false — the flow is still inviscid, still steady, and the drag is real, finite and quadratic in incidence.

One calculation is about the animal; the other is about how fast it happens to be going. The mean lift coefficient required of each animal's wings, computed two ways, on a log scale. Treating the wings as fixed and flying them at the animal's forward speed gives answers spanning a factor of 29 — from 0.88 to 26 — because the number is governed by a speed that has nothing to do with how the animal makes its lift. Doing the flapping arithmetic gives answers spanning a factor of 1.90. And in a hover the fixed-wing calculation has no answer at all: there is no dynamic pressure, and no coefficient however large will do. That is the version of the famous claim that is actually true, and it is a statement about the calculation rather than about the bee.

What is taught wrongly

The bee that cannot fly

The claim has a traceable origin and the calculation behind it was a real calculation done with the wrong velocity. Doing it with the right one gives a number an aerofoil might plausibly produce — and still leaves a gap, and the gap is what took another sixty years to close.

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