Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Visualisation — where it appears

Any technique that makes a flow visible — smoke, dye, tufts, an oil film, a spark trace. Each records a different quantity and each integrates it over a window the experimenter set, so a picture is a measurement with two parameters rather than a view.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

Streamlines and pathlines are not the same curve. In an unsteady flow the line tangent to the velocity everywhere at one instant, and the track a single particle actually follows, are different curves. They coincide only when the flow is steady, which is the hypothesis most figures forget to state.

Streamlines are not the paths particles take

Three different curves get drawn through a flow and they are routinely treated as one. In steady flow they coincide, which is why the confusion survives; in unsteady flow they are as different as a photograph and a long exposure.

kinematics · Flow curves 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.

misconceptions · Visualisation 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.

misconceptions · Visualisation 4 nodes and 2 saddles, and the difference is the shape. The skin-friction pattern on a body, drawn as the streaks a film of oil would leave, looking at the nose. Every critical point of the field is marked: 4 nodes, where the streaks converge or diverge, and 2 saddles, where two streaks cross. Their indices sum to 2, which is the Euler characteristic of a sphere, and nothing about the flow enters that number — the stream is at 0 and the crossflow at 0, and neither may change it. What the flow decides is where the points are and what they look like; what the body decides is how many there are of each.

The count computed on a body

The rule for a closed surface is usually quoted and seldom solved for. This solves for one, and what the computation adds is not confirmation — it is the discovery that the count survives two events in which the number of stagnation points falls, and that both of them have closed forms.

kinematics · Topology The line the dye actually draws. A streakline in an oscillating uniform stream: everything released from the origin over the last 3.4 units of time, drawn where it has got to at one instant. The parameter along it is release time, not distance and not time of flight, which is why it is a record rather than a curve of the field. The mean speed of 1 exceeds the amplitude of 0.6, so the fluid never reverses and the filament is single-valued in x.

The line the dye actually draws

The streakline is the curve most photographs really show, and of the three curves it is the hardest to compute, because it needs the whole history of the flow. This draws it, in a flow where all four curves have closed forms, and the third curve turns out to be a different kind of object from the other two rather than a third example of the same one.

kinematics · Flow curves The one curve of the four that measures a gradient. A row of particles released along a line at one instant, seen at 4 later times, in a layer profile. Each line has been displaced by the local velocity times the elapsed time and by nothing else, so dividing the displacement back out returns the profile exactly — checked here to machine precision at every one of two hundred heights. A streamline, a pathline and a streakline each report where fluid went; this one reports how fast neighbouring fluid was going relative to its neighbour, which is the quantity a boundary layer is made of and the one the other three never show.

The curve that measures a gradient

Three of the four curves drawn through a flow answer the same question — where did the fluid go. The fourth answers a different one. A line of particles released together is displaced by the local velocity and by nothing else, so its shape is the velocity profile, and dividing the elapsed time back out returns that profile exactly rather than approximately.

kinematics · Flow curves 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.

misconceptions · Visualisation 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.

misconceptions · Visualisation A vortex sheet rolling up, at five stages. One period of an initially flat vortex sheet with a small perturbation on it, drawn at equal intervals. The perturbation grows, the sheet steepens, and the ends wind into a spiral. Nothing is added to the sheet after the first instant.

A spiral is a legible record

When a vortex sheet rolls up, the fluid in it can never change places: two points on a sheet cannot pass one another. So the arms of the spiral are a map of the initial sheet, in order, and the picture is a record of the roll-up rather than a snapshot of it.

inviscid · Vortex sheet The part of a field 4 cameras cannot see. Middle, a field with no symmetry on a 16 × 16 grid. Left, the part of it that projects to exactly nothing in every one of 4 views, shaded one way above zero and the other below: a pattern of streaks and hollows that cancels along every ray. Right, the field with that part taken away. The middle and right fields give identical pictures in all 4 views, to 2e-13 of the largest ray, and no reconstruction from those views can tell them apart. The invisible part is one combination of 177 independent patterns the views cannot see.

Four cameras and a field they cannot see

An axisymmetric flow can be rebuilt from one photograph because its symmetry supplies every other view. A flow without an axis has to be photographed from several directions, and a few directions do not merely give a noisy answer — they leave whole patterns of density that every camera records as nothing. Four views of a 16 × 16 field see 79 of its 256 independent patterns and are exactly blind to the rest.

misconceptions · Visualisation

Named alongside it

The objects these essays reach for when they reach for this one.

InstrumentMeasurementModel limitMisconceptionMaterial lineDensityIll-posedInitial conditionInverse problemParticle image velocimetryPathlineRefractive index

All concepts