Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: What is conserved — page 34

Page 34 of 34, continuing through the 306 essays this motif runs through.

306 essays carry this thread — page 34 of 34.

Three strokes, and only the flat one is a theorem. Three cycles drawn in the swimmer's shape space: a square, a circle of the same width, and an out-and-back along the diagonal. The first two enclose area and carry the swimmer forward; the third encloses none and carries it exactly nowhere, which is the scallop theorem with no symmetry argument in it. What the third lacks is not a broken symmetry but an interior. Viscosity

A stroke is worth the area it encloses

The usual account of swimming without inertia is a symmetry argument about reciprocal strokes, which says what cannot work and nothing about what does. Draw the stroke in the space of the swimmer's own shapes and the displacement is a line integral — so it is an area, it does not depend on how fast the stroke is played, and the scallop theorem is Stokes' theorem.

 A drag that is nothing until there is something to be discontinuous about. The scaled wave drag of a parabolic arc against the similarity parameter. Above about 1.4 it is not small but zero to five decimal places — d'Alembert's paradox, which the small-disturbance equation inherits. Below it the drag rises by four orders of magnitude over a range of the parameter that corresponds to a few hundredths in Mach number, and the only thing that changed is that the flow now contains a shock. Compressible flow

A drag that needs a discontinuity

A closed body in a steady inviscid flow has no drag, and the transonic equation obeys that as faithfully as any other — until a shock appears inside its own answer. Then the same pressure integral that was zero to five decimal places becomes four decades larger, with no viscosity added and nothing about the body changed.

 Three speeds, and the Froude number at which two of them cross. The kinematic wave speed and the two dynamic wave speeds, all divided by the speed of a shallow-water wave on still water, against the Froude number. The kinematic wave is three halves of the water speed and the downstream dynamic wave is the water speed plus one, so they cross at Froude two — and beyond that crossing the news of a change in flux arrives before the news of a change in depth, which uniform flow cannot survive. Flows and fields

When the flux outruns the pressure

A kinematic wave is derived by throwing the momentum equation away, and the derivation never says when that is allowed. It is allowed until the kinematic wave, which travels faster than the water, overtakes the downstream dynamic wave as well — at which point uniform flow ceases to exist and a concrete chute carries a train of surges instead of a sheet.

 What an oscillation carries, against how fast it is asked. The extra axial diffusivity of a zero-mean oscillation, at fixed velocity amplitude, against the Womersley number. At low frequency it is exactly Taylor's dispersion evaluated at the mean square velocity — the profile has time to be Poiseuille's at every instant. At high frequency the shear is confined to a Stokes layer, the tracer in the core is never sheared, and the transport falls as the cube of the Womersley number. Regimes and numbers

Transport with nothing transported

Taylor's mechanism needs shear and diffusion and nothing else — and in particular it does not need a mean flow. Oscillate a tube about a fixed position and the tracer still spreads along it, by hundreds of times the molecular rate, which is how a patient is ventilated with a tidal volume smaller than the airway it goes down.

 A count of states that has a maximum in it. How many configurations of thirty point vortices have each energy, sampled from the measure their own Hamiltonian defines — which is the area measure, because a vortex's coordinates are its own conjugate pair. The count peaks at an energy of -0.054 and falls away on both sides, which no ordinary system's does. Above the peak, adding energy reduces the number of ways of arranging the fluid. Transition and turbulence

An equilibrium a three-dimensional flow cannot have

The condensate an inverse cascade ends in is treated as what is left over when the energy has nowhere further to go. It is not a remainder. A two-dimensional fluid's phase space is its own region and therefore has finite volume, so its entropy has a maximum, its temperature changes sign, and the clustered state above that point is an equilibrium.

 One coefficient, two errors, two places they vanish. The two wall-interference errors against the slot parameter, in units of the tunnel's half-height. Both are positive for an open jet at the left and negative for a solid wall at the right, so each passes through zero — the blockage at 1.184 and the streamline curvature at 1.590. A wall has one coefficient and the two zeros are a third of a tunnel height apart. Circulation and lift

One coefficient, two errors

A closed tunnel's walls push a measurement one way and an open jet's push it the other, so a wall somewhere between them should push it neither. There is such a wall, its coefficient is a length, and it nulls the blockage at 1.184 tunnel half-heights — and the streamline curvature at 1.590.

 Three aircraft, and the only thing the drag can see. The cross-sectional area against station, for a clean body, the same body with a wing added, and the body cut away where the wing is. The wave drag depends on this curve and on nothing else about the three shapes — so the middle one pays for its lump and the third does not, although the third has the same wing. Ideal flow

The drag that can only see one curve

A slender body's far field is a line of sources of strength A′(x), and nothing else about its shape reaches it. So its supersonic wave drag depends on the cross-sectional area distribution alone — and a fuselage cut away where the wing joins it can carry the wing for nothing.

 The only term that turns spin into thrust, and what it is made of. The coupling term of the propulsion matrix against the drag anisotropy, with the geometry held fixed. It is exactly proportional to the difference of the two drag coefficients, so it is zero when they are equal — not small, zero — and no helix of any pitch turned at any rate would move. A real filament sits at 1.66, which is a third of the way from useless to the unreachable limit. Viscosity

Two drags, or nothing swims

A bacterium turns a corkscrew and goes forward, and the reason is not the corkscrew. It is that a thin filament dragged broadside resists more than the same filament dragged end-on. Make the two resistances equal and the thrust is not small but exactly zero, for every pitch and every rate.

 The vapour fraction that would stop a cavity, against temperature. How much vapour a cavity would have to make, as a multiple of the liquid volume it cools, before the cooling had used up the whole driving tension. In cold water it is six hundred and the effect is unreachable; at the boiling point it is below one and the effect is the dominant thing in the problem. Four decades across the range a pump might see, out of a vapour density that rises by a factor of two hundred. Fluids at work

A cavity that cools the water it came from

The usual account takes the vapour pressure as a property of the liquid, looked up once. It is a property of the liquid at whatever temperature the cavity has left it — and making vapour costs latent heat, so a cavity suppresses itself. In cold water that is unmeasurable and in liquid hydrogen it is the dominant term.

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