The collection

Every essay — page 25

Page 25 of 39, continuing through the fields in the same order.

Flows and fields Ideal flow Circulation and lift Viscosity Regimes and numbers Compressible flow Transition and turbulence Fluids at work What is taught wrongly Series Concepts Regimes Refutations Search

What is taught wrongly

Equal transit time, Bernoulli misapplied, and the rest. Each stated fairly, then tested against a solved flow and found false.

Two integrands, and the wrong one claims 9.58 per cent more drag. The two things that get integrated across a wake, each scaled to its own peak so the shapes can be compared. The momentum integrand u(U − u)/U² is the drag; the mass integrand (U − u)/U is the displacement thickness, and it is not a drag at all. They differ by a factor of u/U inside them, so the mass one is fatter wherever the deficit is deep — and its integral here is 1.1 times the momentum one's. That ratio is decided by how deep the wake is rather than by how wide: at a twentieth of this momentum thickness it falls to 1, and at twice it rises to 1.25. The error is smallest exactly where a survey is properly done, far downstream where the wake has spread and shallowed.

Weighing what is missing

A control volume drawn round a wing gets the lift out of it, on a flow that was solved exactly. The wake survey asks the same box for the drag on a flow nobody has solved, and it is how a real aerofoil's drag is known — with three assumptions, all of which are checkable, and one integral standing next to it that is wrong.

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The one term in ground effect that really is about carried air. The added mass of a plate approaching a plane, as a multiple of its free-air value. In free air a plate borrows ρπc²/4 per unit span — exactly 0.79 for a unit chord in unit density, which is the mass of the circle its chord spans. Near a wall the fluid in the gap must leave sideways through a narrowing passage, so it moves faster than it would in the open and carries more energy: the borrowed mass grows as the inverse of the gap and diverges as the gap closes. At half a chord it is 1.21 times, at a tenth 2.06, at a twentieth 3.12. This is a cushion in the ordinary sense — fluid that has to be got out of the way and resists being — and it is the term the steady argument has none of.

The cushion that is there after all

A wing near the ground makes more lift for reasons that have no cushion in them, and that account cannot reach two cases — a wing descending, and a rotor in the hover, where there is no steady flight for the image-vortex account to be about. Those cases have a term that really is about carried air, and it grows without limit as the gap closes.

5 figures $Three instruments, three derivatives, one field. A density field — a shock, smoothed to its own thickness — and what each of the three optical techniques records across it, each scaled to its own peak so the shapes can be compared. Interferometry follows the density itself; schlieren follows its first derivative and peaks where the density is changing fastest; shadowgraph follows the second and is a light-and-dark pair straddling the same place. None of them is looking at the flow: the refractive index of a gas is linear in its density, so every one of them is a densitometer and the differences between them are differences of calculus rather than of apparatus.$

An instrument that takes a derivative

Dye, smoke and seeded particles mark the fluid and read the marks. The optical ones mark nothing — light passes through and is bent — and what a plate records is not the flow but a derivative of its density. Two of the three are therefore exactly blind to a uniform stream, at any speed it happens to have.

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A degree of temperature is worth 0.34 per cent of pressure. The error in the pressure a paint reports, against how much warmer the surface is at the test condition than at the reference, at four pressures. At 0.8 of the reference pressure the sensitivity is -0.34 per cent of pressure per kelvin, so 10 degrees is -3.45 per cent. That is not a small number against what the technique is used to measure, and a model's surface temperature is not uniform: it is warmer where the flow has been brought to rest and cooler where it has accelerated, which means the temperature error is largest exactly where the pressure gradients are. The standard remedy is a second, temperature-sensitive paint measured at the same time — an instrument added to correct an instrument.

The paint that measures the wrong field

Dye, seeded particles and the optical methods all look through the flow or at something put in it. Pressure-sensitive paint looks at the surface, reports a scalar rather than a derivative, and turns a row of taps into a field. What quenches its luminescence is oxygen, which is what makes it a pressure gauge — and temperature, which is the other field a flow is guaranteed to produce.

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100 metres of water, −1.98 MPa absolute at the top. The absolute pressure up a transpiring column 100 metres tall, with the sap rising at 0.25 mm/s through conduits 40 µm across. It starts at 1.3 kPa at the root, falls by 9.79 kPa a metre for gravity and 10.02 for friction, and reaches −1.98 MPa at the top — below zero, which is not a low push but a pull. The pale line is the same column with nothing flowing. The floor is not the vapour pressure but the pore a gas bubble could be drawn through: −2.81 MPa for a 50 nm pore, which this column would reach at 142 metres. A suction pump lifting the same water from a free surface stops at 10.1 metres, because it offers the water somewhere to boil.

Where a liquid does pull

A fluid cannot pull, and the essays that settle what suction is are right about every gas and every liquid with a free surface near it. A liquid with nothing in it to boil on is another matter. Every tree taller than ten metres depends on the difference, and the floor under it is set by the size of a pore rather than by the vapour pressure.

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A tube 10 cm across, spun: the lowest pressure is on the axis. The absolute pressure along a water-filled tube spun about its middle, open to the air at both ends, 5 cm from the axis. In the spinning frame the water is at rest under a centrifugal pull, so the pressure falls from atmospheric at each meniscus to its lowest on the axis, as a parabola. 10 thousand rpm puts −1.3 MPa there, 20 thousand rpm puts −5.4 MPa there, 30 thousand rpm puts −12.2 MPa there and 45 thousand rpm puts −27.6 MPa there. The place the water is stretched hardest is the place furthest from both free surfaces, which is the whole merit of the method: a gas cannot reach the liquid where it is weakest.

A breaking strength that is the size of a flaw

Water can be stretched, and how far is a measurement people have made for a century and a half with instruments that agree with one another and not with the theory. Spinning a tube puts the stretch where no gas can reach it, sealing one caps the stretch at water's density maximum, and every measured number turns out to name the size of the worst cavity in the sample.

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Friction lends the crown 28.4 kPa, and the tank takes it back. The absolute pressure at the crown of a siphon with friction in its hose, through a whole drain, for the crown placed at three positions along the hose, against the frictionless constant of 23.01 kPa. With the crown 0.3 of the way it starts at 63.3 kPa, with the crown 0.5 along it starts at 51.5 kPa and with the crown 0.7 of the way it starts at 40.4 kPa. Every curve is above the constant, every curve falls towards it as the level falls, and every curve reaches it at the end — 23.06 kPa with a centimetre of level left. Friction never brings a siphon nearer to breaking; it lends a margin, and the draining tank returns it pascal by pascal, so the worst the crown ever sees is the frictionless value.

The margin friction lends a siphon

Without friction a draining siphon's crown pressure does not depend on the source level at all, and every real hose has friction. It turns out always to raise the crown pressure, by an amount the draining tank hands back pascal by pascal — and how much it lends is decided by where along the hose the crown sits, not by how rough or how narrow the hose is.

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A litre of water at the crown carries 77 mL of gas it cannot hold. The volume of free gas a litre of air-saturated water can release at a siphon's crown, at the crown's own pressure, against that pressure, on a logarithmic scale, at three temperatures. It is zero at atmospheric and grows without limit towards the vapour pressure. At the reference siphon's starting crown pressure of 51.5 kPa it is 20.6 mL, a supersaturation of 2.02; at the frictionless floor of 23.0 kPa it is 76.7 mL, a supersaturation of 4.79. Cold water carries more: 87.4 mL at 5 °C. This is the equilibrium bound — what would come out if the water stayed long enough, which it does not.

The air that breaks a siphon nothing else can

A running siphon's heights cannot break it, and its friction only postpones the moment it is most exposed. What does break a siphon that has run for a day is the air dissolved in its water, which the crown's low pressure leaves the water carrying far more of than it can hold — and which gathers only once the flow is too slow to carry a bubble away.

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The cushion changes its physics 0.36 mm from the ground. The two forces on a plate 10 cm across closing on a plane at 1 m/s in air at 20 °C, per metre of span, against the gap on logarithmic axes. The viscous squeeze film, Reynolds' lubrication result μVc³/h³, rises as the cube of the closeness; the inertial one, ρV²c³/24h² from the potential flow's added mass, as the square. They are equal where the gap Reynolds number ρVh/μ is exactly 24, at 0.361 mm, where each is 3.84e+2 N/m. Above that gap the cushion is the fluid's inertia and below it the fluid's viscosity — and at the crossover neither formula is accurate, since it is where one limit hands over to the other rather than a solution of the flow between them.

A cushion that changes its physics

A plate closing on a plane is resisted by the fluid it has to squeeze out, and the resistance is two different forces with two different laws — one from the fluid's inertia and one from its viscosity. They hand over at a gap of twenty-four kinematic viscosities per unit of closing speed, which for a wing in air is a third of a millimetre, and the two films disagree about whether the plate ever lands at all.

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The same plate borrows more near a wall and less near a free surface. The added mass of a plate closing broadside on a boundary, as a multiple of its free-air value, against the gap in chords on a logarithmic axis, for a solid wall and for a boundary held at constant pressure — a free surface struck quickly, or the edge of an open jet. At a tenth of a chord the wall gives 1.966 and the free boundary 0.677; at 0.035 chords 3.97 and 0.584. The wall's value grows without limit as the gap closes, because the fluid in the gap has to be squeezed out. The free boundary's falls towards exactly one half, because a plate lying on a free surface sets in motion only the half-space below it. Same plate, same fluid, same speed — the boundary decides the sign, through the one thing it is allowed to tell the flow.

The borrowed mass the boundary decides

A body accelerating near a solid wall has to squeeze out the fluid between them, and borrows more mass than it would in the open. The same body accelerating near a free surface, or inside an open-jet wind tunnel, borrows less. The fluid, the body and the speed are identical, and what reverses the answer is the one thing each boundary is allowed to tell the flow.

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A window one core wide reads the vortex 7.5 per cent slow. The tangential velocity across a Lamb–Oseen vortex, in units of its core radius and of Γ/2π divided by it, as it is and as particle image velocimetry reports it with square interrogation windows of three widths — the average of the velocity over each window, which is what a correlation over the window returns to first order. The true peak is 0.6382 at 1.1209 core radii. A window 0.5 core radii wide reports 98.0 per cent of it, 1.021 times as far out, a window 1 core radii wide reports 92.5 per cent of it, 1.084 times as far out and a window 2 core radii wide reports 77.1 per cent of it, 1.334 times as far out. The instrument that measures velocity directly still reports a slower, fatter vortex than the one there, by an amount set entirely by the window against the core.

The window every vector is averaged over

Particle image velocimetry is the one flow-visualisation technique that reports the velocity itself, and it still applies an operator: every vector is an average over an interrogation window. A window is a filter with a transfer function, and it makes a vortex slower and fatter, a thin shear layer exactly as thick as the window, and some features smaller than the window point the wrong way.

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One per cent of noise on the image, 1.43 on the axis. The field recovered by onion peeling on 50 rings from one view carrying noise of one per cent of the instrument's own peak reading, from an interferometer's projection and from a schlieren system's deflection, against the true field. On the axis, where the truth is 1, the projection gives 1.431 and the deflection 0.876; over the whole radius their root-mean-square errors are 0.0742 and 0.0219, so the deflection is the quieter route here. Near the edge both are clean, and the error gathers towards the axis — where the field is largest and the flow usually most interesting.

One view is enough, and the axis pays for it

An axisymmetric flow — a jet, a plume, a flame — can be reconstructed from a single optical view, because Abel's integral inverts exactly. The inversion runs from the outside in, every error made on the way reaches the axis, and whether it arrives multiplied depends on which instrument took the picture.

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