Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
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The thread: What is conserved — page 14

Page 14 of 33, continuing through the 292 essays this motif runs through.

292 essays carry this thread — page 14 of 33.

Tip to tip, two wings cost exactly half of what they cost apart. The induced-drag saving of a pair of wings against the gap between their tips, as a fraction of what the two pay flying alone. With the tips touching the pair is one wing of twice the span carrying twice the lift, and the arithmetic of that is exact: the drag halves, and the computation returns 0.499257 of the separate figure. Pull them apart and the saving falls away with the square of the distance. Birds fly in a V because the tips are the part worth overlapping, and the spacing that pays is a small fraction of a span rather than any distance a formation could hold by eye. Circulation and lift

The lift beside a wing

Fly two aeroplanes with their wingtips touching and the pair costs exactly half what the two cost apart. Not approximately half — the arithmetic is a closed form, because two wings tip to tip are one wing of twice the span, and induced drag goes as the square of it.

 A sphere does not go at the speed of the flow it is in. How far a force-free sphere on the axis of a round tube lags the fluid at its own centre, against its size. The lag is two-thirds of the square of the size ratio and comes from the reciprocal theorem in one line: the sphere moves at the average of the ambient flow over its own surface, and a parabolic profile is slower everywhere on that surface than at the middle. At a fifth of the tube's radius the lag is 2.7 per cent, which is the difference between a tracer and a thing being measured. Viscosity

A force without the flow that makes it

A sphere carried along by a flow does not travel at the speed of the fluid at its centre. It travels at the average of the flow over its own surface — and getting that result needs no solution of the flow around the sphere at all, only the answer to a completely different problem that everybody already knows.

 The momentum of the fluid, against the shape of the region it is added up over. Momentum of the fluid around a cylinder moving through it, divided by the body's hydrodynamic impulse, against the aspect ratio of the rectangle the integral was taken over. Every rectangle has the same area and contains the same body. A tall region gives minus the impulse, a long one gives plus it, a square gives exactly zero, and the limit of a large region is whichever of those the region was shaped like. The momentum of an unbounded ideal flow is not a number. Ideal flow

The momentum with no value

A cylinder is pushed from rest to a steady speed. Work was done, energy went into the fluid, something was pushed. How much momentum does the fluid carry? The integral converges, the answer is finite, and it is a different finite number for every shape of region it is summed over.

 Ideal flow past a sphere, drawn in a meridional plane. The Stokes stream function of ideal flow past a sphere, contoured at equal intervals. Its relations to the velocity carry factors of r sin θ that the plane stream function does not have, and differencing it reproduces the closed-form velocity to 10⁻¹¹. The surface speed at the equator is exactly one and a half times the free stream, against twice for a circular cylinder: a three-dimensional body lets the flow past in two directions rather than one. Flows and fields

The one number that runs out at three dimensions

A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.

 The density ratio and the nose pressure coefficient, against Mach number. Two quantities across a normal shock, on a logarithmic Mach axis. Both approach limits that depend on γ and on nothing else: six for the density ratio and 1.8394 for the pressure coefficient at the stagnation point. By Mach five the second is within three per cent of its limit and by Mach twenty within a tenth of a per cent. Above that the flow round a blunt body has stopped depending on how fast it is going and started depending on what the gas is. Compressible flow

A shock that lies on the body

Above about Mach eight the flow round a blunt body stops depending on how fast it is going. The density ratio, the nose pressure coefficient and the shock standoff all reach limits set by γ alone — and going from a perfect gas to a dissociating one halves the standoff.

 The exposure lengths that give the true mean, and the ones that do not. How far a finite exposure of a shedding wake lands from the long-exposure mean, against how many shedding periods the shutter was open for. It is exactly zero at every whole number of periods — an average over a complete cycle is the mean, with no error at all — and between the zeros it falls as one over the exposure. Nothing about the picture tells the reader which of these they are looking at. What is taught wrongly

The shutter is part of the answer

Flow photographs are compared as though the exposure were a detail of the camera. It is a term in the measurement: an average over exactly one shedding period returns the true mean to fourteen figures, any other length carries a residue that falls only as one over the exposure, and a two-pulse velocity reading is short by exactly the sinc of the swept angle.

 1.139 asks for a Francis. The specific-speed axis, with the four machine types on it and one duty marked: 3 m³/s at 60 m, on a shaft turning at 750 rev/min. The number is 1.1387, and the choice of runner follows from it before any blade has been drawn. What the number contains is a ratio of flow to head; what it does not contain is any size at all, which is why one axis serves a garden pump and a gigawatt turbine. Fluids at work

One number picks the machine

A flow rate, a head and a shaft speed contain exactly one dimensionless combination with no size in it. That combination decides whether a duty wants an impulse wheel, a Francis runner or a propeller — before anything has been drawn, sized, or costed.

 Three thresholds, and none of them is one. The neutral curves for a layer heated from below, computed by taking the smallest eigenvalue of the marginal-stability operator at each horizontal wavenumber. Every curve diverges at both ends — a cell wider than the layer has to carry heat sideways for ever, a narrower one loses it to conduction — so each has a minimum, and that minimum is the critical Rayleigh number. Two free surfaces give 657.5, one rigid and one free 1100.7, two rigid walls 1707.8. Nothing but the boundary condition differs, and it carries a factor of 2.6. Transition and turbulence

The threshold the walls decide

A layer heated from below convects at a Rayleigh number of 1707.762, and nothing whatever happens at one. The free–free case has a closed form and the two that do not differ from it by a factor of 2.6 — produced by nothing but what the top and bottom surfaces are permitted to do.

 Elliptic loading rolls up to πb/4, and it is π that puts it there. Three loadings on the same span carrying the same lift, with the rolled-up core positions marked below each. The core sits at the centroid of the shed vorticity, which is a quadrature over the loading and needs nothing about the roll-up itself. Elliptic loading gives πb/8 from the centreline, so the pair ends up 0.7854 of the span apart — the number every wake-separation rule is written against, and one of the few places in this subject where π turns up in an answer an engineer uses directly. The bell rolls up to 0.586 and a nearly rectangular loading to 0.978, because it sheds at the tips. Circulation and lift

Where the wake ends up

The sheet a wing sheds rolls up into two cores within a few spans, and nobody can compute the roll-up cheaply. Nobody has to: what the cores conserve is fixed before they form, and for an elliptically loaded wing the answer contains π and comes out at 78.5 per cent of the span.

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