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

Stratification — where it appears

A variation of density with height, which gives a fluid a restoring force that no uniform flow has. It supports internal waves, resists vertical mixing, and turns the question of whether a flow goes over an obstacle or round it into a comparison of two time scales.

Named by 11 essays across 5 fields — each of them below, with the objects they name alongside it.

A cross at 30.0° for every wavelength it makes. A body oscillating at ω = 0.5N in a stratified fluid, and the four beams along which its energy leaves. The angle is arccos(ω/N) from the vertical — 30.0 degrees from the horizontal here — and it is the same for every wavelength the body excites, because the dispersion relation has no length in it. The short strokes are the crests, which lie along the beams rather than across them: the phase advances perpendicular to the energy, and the two are exactly at right angles. Raise the frequency and the cross closes towards the vertical; reach ω = N and it shuts entirely, because nothing above the buoyancy frequency propagates.

The number that stops the mixing

A fluid whose density falls with height resists being stirred, and the resistance has a threshold at exactly one quarter, from an energy balance with no fluid mechanics in it. The waves such a fluid carries are stranger still — their frequency decides the direction they travel in and says nothing about their wavelength.

turbulence · Stratification 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 numbers, one flow, one name. A hyperbolic-tangent shear layer with a matching density profile, measured five ways, each of which appears in the literature as "the Richardson number". The minimum gradient value is what Miles' and Howard's theorem is about and is the only one the quarter belongs to. The bulk numbers depend on which thickness and which velocity difference were used; the depth-averaged one depends on how far from the layer the measurement extended, and grows without limit as it extends further.

Five numbers, one name

The Richardson number has a threshold at a quarter, and the quarter belongs to one of the five quantities that go by the name. On a single tanh shear layer they run from J to seventeen J, and the largest of them grows without limit as the measurement is extended further from the layer.

turbulence · Stratification Instability up to a quarter, and none past it. The fastest growth rate of a stratified shear layer against its Richardson number, on a profile whose gradient Richardson number is the same at every height. It falls smoothly towards zero and reaches it at a quarter: at Ri = 0.2499 the fastest mode still grows at 0.00106, and at 0.26 the solver finds no unstable mode at all.

Sufficient, and not necessary

A stratified shear layer whose Richardson number exceeds a quarter everywhere cannot go unstable. That is a theorem with an exact number in it. What it does not say — and what it is constantly read as saying — is that a layer below a quarter will.

turbulence · Stratification Over a day the breeze turns right round. The tip of the surface wind vector over one day, onshore to the right and along the coast upwards, at 15°, 30° and 45° north, with a spin-down time of twelve hours, in units of the push over the daily frequency. Without rotation the wind would swing on and off shore along one line. The Coriolis force turns it clockwise through the day into an ellipse; at 30°, where the Earth's inertial period is exactly a day, the turning keeps pace with the push and the ellipse is nearly a circle, of radius 1.8 against the ellipses' 1.2 at 15° and 1.2 at 45°.

At thirty degrees the Earth keeps time with the sea breeze

A sea breeze is Kelvin's circulation theorem failing: where warm air over land meets cool air over the sea, pressure and density surfaces cross and a circulation grows out of still air. Bjerknes' theorem prices the push, and the textbook's numbers give twenty-six metres a second in an hour — far more than any sea breeze blows. What stops it first is not friction but the evening: the push reverses before the wind can grow, and the day, not the drag, sets the breeze's size. Friction sets its hour, the Earth's rotation turns it through the day, and at thirty degrees, where the inertial period is a day, the turning keeps time with the push.

inviscid · Kelvin Stratification makes a hill easier to block. The strength at which a current over a Gaussian hill first stops and traps fluid, δc = (h₀/H)/Ro at onset, against the stratification B = NH/fa — the depth over the height fa/N to which a rotating stratified flow feels the hill. Homogeneous, it is the 3.134 of the unstratified layer. As B grows the hill's anticyclone gathers at the bottom, where it is stronger, and the threshold falls; in deep water it falls as 2.243/B, which is a Froude number: the current is blocked when N h₀/U exceeds 2.243, and the rotation has dropped out.

A stratified sea blocks a current sooner and traps less

A slow current in a rotating layer stops over a hill once the hill's height over the depth, divided by the Rossby number, passes 3.134, and a column of water over the hill is trapped from floor to surface. In a stratified sea the hill's anticyclone gathers near the floor and fades upward over a height of fa/N. It is stronger there, so the current stops sooner; and it is confined there, so what it traps is a cap, not a column. In deep water the threshold loses the rotation altogether and becomes a Froude number: a current is blocked when N h₀/U exceeds 2.24.

regimes · Geostrophy A thin interface stays unstable far past a quarter. The fastest growth rate of any disturbance against the bulk Richardson number J, on a logarithmic scale, for a density interface as thick as the shear and two, two and a half and three times thinner. The matched layer's billow dies at a quarter, as Miles' theorem requires. Two times thinner, the billow dies sooner, its last growth at J = 0.12, and nothing replaces it. Three times thinner, the billow's last is at 0.08 and a travelling wave takes over: 0.0335 at a quarter, 0.0109 at one and 0.00629 at 1.3, falling steadily with no threshold in the range solved. At 2.5 the waves are weaker and reach 0.001 at 1.3.

A thin interface keeps its waves past a quarter

Miles' quarter rules a shear layer whose density changes over the same depth as its velocity. Make the density interface three times thinner and the stationary billow dies early, but a pair of travelling waves takes its place and is still growing at five times the quarter. No theorem is broken: at the edges of the shear, where the waves draw their energy, the local Richardson number has fallen to nothing.

turbulence · Stratification A free interface's thickness ratio heads to the root of its Prandtl number. The ratio of the shear layer's thickness to the density interface's against time, for an interface starting four times thinner than its shear, heat and salt in water. A free shear layer thickens by viscosity faster than the interface by conduction, so the ratio heads to √Pr: 2.64 for heat, 25.8 for salt. Held at its thickness, the shear lets the ratio fall through 2 — at about 50 s for heat and 4725 s for salt.

What holds the shear decides whether the waves last

Holmboe's travelling waves need a density interface about half as thick as the shear around it, and diffusion thickens both. If the shear layer is held at its thickness, the interface catches up — in under a minute for heat in a laboratory tank, over an hour for salt — and the waves get two e-folds or a hundred and seventy. If the shear layer is free it thickens faster than the interface, the ratio heads to the root of the Prandtl number, and the waves last until the Richardson number ends them instead.

turbulence · Stratification In stable air a wide-body's wake sinks a few hundred metres and stops tens of kilometres behind it. The depth the trailing vortex pair of a wide-body airliner has sunk, against distance behind the aeroplane, for buoyancy frequencies of 0.005, 0.01 and 0.02 per second and in neutral air: weight 3 MN, span 60 m, 250 m/s, air density 0.4 kg/m³, so the pair starts down at 2.15 m/s. It stops 638, 319 and 160 m down, 117, 58.3 and 29.2 km behind; in neutral air it would go on sinking.

Stable air takes back the momentum a wing pushes down

Behind a wing in neutral air the trailing vortices carry the downward momentum the wing gave the air for ever, and a face across the wake keeps its share of the weight however far back it is drawn. In stably stratified air the column the pair carries down is lighter than the air around it, and buoyancy pushes back. The pair stops a few hundred metres down, its momentum returned, after a time that is the same for every aeroplane — 2.33 divided by the buoyancy frequency, about four minutes — and from there on the whole weight crosses the box's upper and lower faces as pressure.

misconceptions · Momentum lift At the equator the front runs as long as the land is warmer; at forty-five degrees it stops hours before. The sea-breeze front's distance inland against the hours since the land first became warmer than the sea, at the equator and at 30°, 45° and 60°. At 0° it reaches 185 km. At 30° it reaches 133 km, 99 per cent of that by hour 11. At 45° it reaches 99.2 km, 99 per cent of that by hour 9.1. At 60° it reaches 80.3 km, 99 per cent of that by hour 8. The quasi-steady estimate, the head speed of a current fed from a deep reservoir at each hour's contrast, integrated, is 189 km.

The sea-breeze front runs till evening only near the equator

A real sea breeze arrives: the cool sea air runs inland under the warm air as a gravity current, its head moving at a speed set by the density contrast and the depth of the cool layer. How far it gets depends on the latitude. Near the equator the head keeps running as long as the land stays warmer, and the length of the afternoon decides the reach. In mid-latitudes the Earth's rotation turns the current along the coast and stops the head hours before the contrast ends, a third further inland than the Rossby radius, and a longer afternoon adds nothing.

inviscid · Kelvin An oval that sheds its fluid is braked less, so the pair sinks further and stops later, until it sheds too fast to stop at all. The depth of a trailing vortex pair below its starting level, in spacings, against time in spacings over its initial speed, in air of N = 0.22: Saffman's oval, which keeps its fluid, and one detraining with α = 0.05. Saffman's stops at depth 6.75 after 10.6; this one at 6.93 after 12.4.

A wake that sheds its fluid sinks further, and past a point does not stop

In stable air the pair of trailing vortices behind a wing carries a column of lighter air down with it, and buoyancy pushes the column back until the pair has given up its momentum and stops. That picture keeps every parcel of the column. A real pair sheds the light air as it sinks, and shedding does two things at once: it takes away part of the momentum the buoyancy had to return, and it takes away the buoyancy that was returning it. The second wins. A pair that sheds a little stops later and deeper; one that sheds faster than a threshold loses its column before buoyancy can stop it. The shed air keeps a quarter of the momentum swinging up and down behind the aeroplane.

misconceptions · Momentum lift

Named alongside it

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

Model limitRichardson numberBuoyancyLinear stabilityShear layerBuoyancy frequencyMixingThresholdBaroclinicCoriolisDimensionlessDownwash

All concepts