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

Material derivative — where it appears

The rate of change of a quantity following a fluid parcel, being the local rate plus the advective one. A steady flow has parcels accelerating in it, which is exactly the difference between the two terms.

Named by 10 essays across 2 fields — each of them below, with the objects they name alongside it.

The acceleration field of a steady flow. How hard the fluid is being accelerated at each point of a steady flow past a cylinder. The flow does not change with time anywhere in this picture, and yet almost nowhere in it is a parcel travelling at constant velocity — the pattern stands still while the fluid running through it is thrown about.

Steady does not mean nothing is happening

Photograph the flow past a cylinder twice and the two pictures are identical. Every parcel of air in them is being thrown about — braked to a dead stop, hauled round the shoulder at nearly twice the free-stream speed, braked again. Both statements are exactly true.

kinematics · Acceleration A circle of fluid, 0.90 of a gradient time later. A material circle in a uniform velocity gradient, carried by the exact matrix exponential of that gradient. It becomes an ellipse — always an ellipse, for every gradient — and the axes it stretches along are the eigenvectors of the symmetric half. In an incompressible flow the area is unchanged however extreme the distortion, which is the statement that stretching in one direction is squashing in the other.

What a parcel does in the first instant

Drop a circle of dye into a flow and it becomes an ellipse. The velocity gradient that did it splits into a stretch and a spin in exactly one way, the split is not a convention, and one half of it is the reason a fluid has any stress in it at all.

kinematics · Deformation Wound up, and worth exactly what it started with. A material loop in a steady cellular flow — an exact solution of Euler's equations — drawn at four times. Each streamline in the cell has its own period, so the loop is stretched steadily into a spiral: by the last frame its perimeter is 5.7 times what it started as. The circulation round it is 0.903741 at the start and 0.903666 at the end. Nothing about the curve survives except the number.

What survives being wound up

Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.

inviscid · Kelvin A material region, and the dye that stays inside it. The same fluid at four times, carried by an unsteady straining flow whose strain rate oscillates. The outline is a circle of the fluid at the first instant, tracked by integrating the velocity field; the shading is a blob of passive dye. The region is stretched to nearly seven to one and its area is unchanged to fifteen decimal places, because the flow is incompressible. The amount of dye inside it is unchanged to thirteen, because the dye is carried by the same fluid.

A rate of change that will not hold still

Three boxes drawn in one flow at one instant give three different answers to how fast the dye inside them is changing — one falling, one falling twice as fast, one rising. All three reconcile with a single material rate, and that rate is zero.

kinematics · Transport theorem A flow that is incompressible and carries a density that varies three to one. Ideal flow past a cylinder, shaded by a density that is constant along each streamline and runs from one to three across the field. Every parcel keeps the density it started with, so the divergence is zero — measured at 10⁻¹⁰, which is the differencing — and the flow is incompressible in the only sense the word has. The density is not uniform anywhere. Incompressible is a statement about what the flow does to a parcel's volume, not about what the fluid is made of.

Incompressible is not a property of the fluid

A flow whose density varies three to one across it, with a divergence of 10⁻¹⁰ everywhere. And a flow of air at Mach 0.1, whose divergence is three per cent of U/a and which every textbook calls incompressible. The word is about what the flow does to a parcel's volume, and about nothing else.

kinematics · Dilatation Five points, and where each one's fluid came from. Back-trajectories through six units of time in an unsteady double gyre. Each curve ends at the place the fluid now at the marked point started; nothing about that place can be read off the velocity at the marker.

A scalar is a record of where its fluid was

A conserved scalar has no value of its own. Its value at a point is whatever it was at the place that point's fluid started from, which makes a dye field a photograph of the past — and makes the map from now to then the only thing in the flow that carries the past at all.

kinematics · Advection A duct that does not change, and a parcel that does. A contraction of area ratio four, with the parcel marked at five equal intervals of time. The walls do not move, the field at every point is the same at every instant, and the spacing of the markers grows because the parcel is carrying its own history through the duct.

How long the fluid has been in there

Age is the simplest thing a flow can remember. It obeys the shortest transport equation in the subject — its material derivative is one — and no instrument pointed at a steady flow can read it, because a steady flow's every field is constant and its fluid is getting older all the time.

kinematics · Acceleration The gas flows in, against the heat. The radial velocity of the gas, in units of α/a, at the same four times — from the motion of the gas shells themselves — with the heat flux, reversed in sign and in the same units, drawn as dots. The two coincide: at constant pressure the gas moves at (γ − 1)/(γp) times the heat flux, against it. Heat leaves the spot outward and the gas comes in, everywhere.

The air flows in against the heat

A gas moving at a ten-thousandth of the speed of sound is as incompressible as anything gets, by the usual arithmetic. Heat one spot of it and let the spot cool: the gas changes its volume threefold, and it flows towards the hot spot while the heat flows out, at a speed fixed by the heat flux alone. At constant pressure, a joule is a volume.

kinematics · Dilatation Every parcel goes straight while the streamlines wave. A uniform stream across the page carrying a pattern of transverse velocity with it, v = 0.6 sin(x − t): the streamlines at one instant, which wave, and the paths of five parcels over the next six time units, which are straight lines. Each parcel keeps the transverse velocity it started with, because the pattern moves with it; the streamlines are a snapshot of a pattern that is sliding past, and no parcel ever follows one.

A parcel goes straight while the streamlines curve

In a steady flow every parcel accelerates while the picture never changes. The opposite also happens: a flow whose picture changes all the time while no parcel accelerates at all, because the local and convective halves of the acceleration cancel exactly. Such a flow has no pressure gradient anywhere, and that is a severe demand. An incompressible flow can meet it only as a pure shear at every point; a cloud that can compress meets it freely and pays later, when its straight paths cross in a caustic.

kinematics · Acceleration Moving air needs a quarter of the inertia still air does. The first crossing against Stokes number for a cloud thrown into still air — time in units of the cloud's fastest convergence rate — and for a cloud carried by air converging on a line, in units of the air's strain rate. Still air only brakes: its threshold is one. Converging air keeps pushing: its threshold is a quarter, and heavy particles in it cross in about one strain time. Dots: particles integrated in the converging sine flow; the curve is the linearised oscillator.

Drag decides whether a cloud folds, and the air decides at what

A cloud of free particles keeps its velocities, and wherever it converges its paths cross in finite time: a caustic, the density infinite along a sheet. Give each particle a drag on the air and the answer depends on what the air is doing. In still air the drag only brakes, and a cloud folds only if its Stokes number is above one. In air that is itself converging the drag keeps pushing, and a quarter is enough — the same quarter that decides whether a droplet hits a wing.

kinematics · Acceleration

Named alongside it

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

Model limitEulerian and LagrangianFlow mapMass conservationDilatationIncompressibleMeasurementVorticityCompressibilityConservationControl volumeDensity

All concepts