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

Eigenvalue — where it appears

The growth rate of a mode of a linear operator, whose sign decides whether that mode grows or decays. Every eigenvalue can decay while a disturbance still grows by a large factor, because the modes need not be orthogonal.

Named by 20 essays across 8 fields — each of them below, with the objects they name alongside it.

Every disturbance has its own threshold; one of them is lowest. The Rayleigh number at which a disturbance of horizontal wavenumber a becomes neutral, Ra = (π² + a²)³/a². Every wavenumber has a threshold and the layer goes unstable at the lowest of them, which a golden-section search on this curve puts at a = 2.221441 and Ra = 657.5114 — the exact π/√2 and 27π⁴/4 to fourteen digits. Below the curve the layer conducts and nothing moves.

A threshold with a closed form

A layer of fluid heated from below sits still until buoyancy overcomes both diffusions at once, and then it convects. Unlike every other threshold in this field, that one is an eigenvalue with an exact answer, and the answer is 27π⁴/4.

turbulence · Convection Stable in every mode, and 100 times larger first. The energy of the worst-case disturbance against time, on a logarithmic scale, at four Reynolds numbers and at the one the slider selects. Time is in units of Re, which is what makes the four curves the same shape; what changes with Reynolds number is the height, and it changes as the square. Every eigenvalue of this operator is negative throughout, so nothing that grows here is an instability in the sense a stability analysis reports.

Every mode decays and it grows anyway

A stability analysis asks whether any mode of a flow grows, and for pipe flow the answer is no, at every Reynolds number, which the pipe disagrees with. The missing ingredient is that the modes are not perpendicular — a disturbance made of two nearly parallel decaying pieces can grow by a factor of Re²/16 before it dies.

turbulence · Transition 45° of corner, and an infinite number of eddies in it. The creeping flow in a corner of 45 degrees, drawn from Moffatt's similarity solution. Each eddy turns the opposite way to its neighbours and is 3.17 times smaller and 1.6e+3 times weaker than the one outside it. The contour levels are rescaled inside each eddy, because they differ in strength by three orders of magnitude per step and a single set of levels would show the first and nothing else — which is itself the reason nobody has seen the third. The dividing lines between eddies are drawn where the stream function changes sign, and the dots are the centres, both found from the solved field.

The eddies nobody stirs

A slow flow past the mouth of a sharp corner does not simply fail to get in. The corner fills with an infinite sequence of counter-rotating eddies, each about three times smaller and sixteen hundred times weaker than the last, and the whole structure is decided by one complex number.

viscous · Corner Where the number says, and where it happens. Fourteen dimensionless groups on one logarithmic axis. The open circle on each row is the value at which the two terms the group compares are equal, which is one by the way the group is formed; the filled mark is the value at which the thing a reader cares about first changes by 1%. The bar between them is the distance the folklore phrase "of order one" hides, and it runs from nothing at all to a factor of 594.

What "of order one" is worth

A dimensionless group is built by comparing two terms, so it is one when the terms are equal — and that is the only thing it says. Where the behaviour actually changes is a separate question with a separate answer, and across fourteen groups on this site the two numbers differ by factors from one to five hundred and ninety-four.

regimes · Crossover Two flows with the same rate of strain, doing different things to a blob. A circle of fluid carried by two flows chosen to have exactly the same rate-of-strain magnitude, drawn at four times. Pure strain pulls it into an ellipse whose axes stay put; simple shear pulls it into an ellipse whose axes rotate as fast as they stretch. Both have zero divergence, so both preserve the area. The difference between them is not the strength of the straining but what the rotation does to the direction being stretched.

Longer, with nothing pulling it

Two flows with exactly the same rate of strain. In one a line of fluid grows by a factor of 148 in five time units; in the other it grows by 10. Turn the straining axes faster than the strain rate and no line grows at all, however hard the fluid is being strained.

kinematics · Material lines 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.

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.

turbulence · Convection The number a duct settles at is an eigenvalue. The local Nusselt number against x⁺ = x/(D·Re·Pr), from a Crank–Nicolson march that contains no eigenvalue anywhere. It settles at 3.6568, which is λ₀²/2 for the Graetz eigenvalue problem — a completely separate calculation. The mark at x⁺ = 0.05 is the entry length every textbook quotes: it delivers a Nusselt number 1.45 per cent above the developed value, which is a perfectly reasonable tolerance and is never the one stated.

How far before the heat arrives

A duct's thermal entry length is quoted everywhere as x/(D·Re·Pr) = 0.05, with no tolerance attached. Working out what it delivers gives a Nusselt number 1.45 per cent above the developed value — and the developed value itself is not a term ratio at all but an eigenvalue, 3.6568, which is also the rate at which the duct forgets its inlet.

regimes · Peclet Two roots walking towards each other, and one of them crosses. The roots of the characteristic quartic in the complex plane as the airspeed is raised from nothing to 105 metres per second — growth rate across, frequency up. At rest the two sit on the imaginary axis at the uncoupled frequencies. As the speed rises, the aerodynamic coupling drags them towards each other in frequency while pushing one left and the other right, and at 80.8 metres per second the right-hand one crosses the axis. Everything about the failure is in this picture: the coalescence, the crossing, and the fact that the flutter frequency is neither of the two the structure started with.

The shake that is not resonance

A steady airstream contains no oscillation at any frequency, so nothing is driving anything. What happens instead is that the aerodynamic forces couple two structural motions that were independent, drag their frequencies together, and turn one damping negative — and the wing takes the energy out of the air itself.

circulation · Flutter The marginal Taylor number against the axial wavenumber. The smallest Taylor number at which a disturbance of a given axial wavenumber is neutral. Its minimum is 1707.757 at a wavenumber of 3.1158, which are the critical Rayleigh number and critical wavenumber of a layer of fluid heated between two rigid walls — the same numbers, because in the narrow-gap limit the two problems are the same sixth-order eigenvalue problem. This curve is computed by that essay's own solver.

A transition that needs a second number

Fluid between rotating cylinders goes unstable at a Taylor number of 1707.762 — which is the same number, to every digit, as a layer of fluid heated between two rigid walls. It is not an analogy. And two things the number cannot carry decide whether the transition happens at all and what it looks like when it does.

regimes · Taylor Every unstable mode of every profile, inside one circle. The complex phase speeds of the unstable modes, scaled so that each profile's own semicircle is the unit one. Howard's theorem says every one of them must lie inside — the centre and the radius are the mean and half-range of the velocity profile and nothing else — and every one of them does, with the closest approach at 0.915 of the radius.

Every unstable wave is inside one circle

Before solving anything, you know where the answer is. Howard's theorem says the complex phase speed of any growing disturbance in a shear flow lies inside a circle fixed by the fastest and slowest parts of the profile — and by nothing else about it at all.

turbulence · Instability 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 The convergence exponent depends on the gas, which a dimensional exponent cannot. R ∝ (−t)^α for a converging shock, against the ratio of specific heats, for cylindrical and spherical symmetry. Guderley's exact values are marked and the agreement is to four figures. The Sedov blast's two-fifths is drawn beside them: it is the same for every gas, because it comes from dimensions and a conserved energy, and γ is dimensionless.

An exponent dimensions cannot give

A blast wave's radius goes as the two-fifths power of time, and the two-fifths is arithmetic: count the dimensions and it falls out. A shock converging on a point goes as the 0.717 power, and no amount of counting will produce that number — because it depends on the gas, and γ is dimensionless.

regimes · Similarity The two long-wave speeds, and the swirl at which one of them stops. For uniform axial velocity the wave speeds follow from the criticality condition by a Galilean boost: c = W(1 ± 2S/j), with j the first zero of J1. The upstream-running root crosses zero exactly at S = j/2 = 1.9159, and above that swirl no disturbance can travel upstream — which is what subcritical and supercritical mean here and in an open channel.

The swirl that holds a wave still

A swirling flow down a pipe carries waves, and above a certain swirl one of them stops moving. Below it, a disturbance downstream can send information upstream; above it, the flow has outrun its own waves. The words are open-channel flow's words, and they are the same words for the same reason.

inviscid · Swirl Fourteen pure numbers, and where each came from. Every one of these is dimensionless, exact and quoted as a fact about fluids. None of them comes from dimensional analysis, which says which numbers an answer may depend on and never what any of them is. They come in four kinds — algebra, an integral, a root and an optimum — and the last two are not equally knowable.

Where a pure number comes from

Sixty-four for a round pipe, sixteen twenty-sevenths for a wind turbine, 0.332 for Blasius. Counting dimensions produces none of them — it produces the list of arguments, and the function has to be solved. Which of the four ways it was solved decides how many digits are worth printing.

regimes · Dimensional The eighths nobody chose. Four physical statements — the inner layer sits in the classical one's shear, its inertia balances its own viscous stress, the pressure is of the order of that inertia, and the displacement it makes produces that pressure — are a linear system in four exponents. Solving it gives three eighths, five eighths, one eighth and a quarter, exactly.

The length the limit invents

Prandtl's equations are parabolic, so nothing at one station can depend on anything downstream of it. Every experiment shows the pressure rising ahead of a shock or a step. The resolution is a region three eighths of a power of the Reynolds number long, which the limit that produced the equations was supposed to have removed.

inviscid · Interaction Below Thoma's 6.07 m² the governed tank's swing grows; above it, it dies. The tank level after the turbine's power demand drops by two per cent, with a governor holding the power constant, for tanks of 0.7 and 1.3 times Thoma's area of 6.07 m² — a tank 2.78 m across. The smaller tank's swing grows by a factor of 1.47 every 74 s cycle and has reached −12.20 m by 427 s; the larger one's keeps 0.75 of itself every 100 s and is barely visible. Carried on, the smaller tank's run is refused at 794 s, where the head at the turbine has fallen below a quarter of its design value and the governor would be asking for a flow no turbine passes. The instability has nothing to do with the tank's height: it is the governor drawing more water as the level falls, which feeds the swing, against the tunnel's friction, which is the only thing damping it.

The better tunnel needs the bigger tank

A turbine governed to hold its power opens further when the head at it falls, and draws the tank down harder. That makes it a negative resistance, the tunnel's friction is the only thing damping the swing against it, and so the smallest stable tank grows as the friction shrinks — 2.78 metres across for five metres of friction, 6.09 for one.

applied · Water hammer The damping crosses zero once, and the crossing is the boundary. The least damping ratio of the four aeroelastic modes against speed. It falls through zero at 80.843 metres a second, and at that speed the crossing root's real part is four parts in 10¹⁸ — which is what an algebraic condition on a quartic with real coefficients looks like when it is solved numerically.

The speed where the damping is exactly zero

Flutter is an algebraic condition on a quartic: one root crosses the imaginary axis, at one speed, exactly. And the quantity that locates it is so nearly flat there that the standard way of finding it from flight test overshoots by a quarter.

circulation · Flutter A band of growing waves that opens at 5772 and narrows as the viscosity goes. The wavenumbers at which a two-dimensional wave on plane Poiseuille flow neither grows nor decays, against the Reynolds number on a logarithmic axis. Inside the tongue waves grow; outside they decay. The tongue's tip is the critical point. Both edges slope downward and towards each other in wavenumber as the Reynolds number rises, so the band of unstable waves shrinks towards long waves — the direction in which the inviscid problem, which has no growing wave at all, is reached.

The profile Rayleigh cleared and viscosity did not

Flow between two plates has no inflection point, so without viscosity no wave on it can grow. With viscosity one does, above a Reynolds number of 5772. Taking the viscosity away again slows that wave and narrows the band it grows in, because the stress that feeds it is made by viscosity in the first place.

turbulence · Instability Rolls turning side by side, with the fastest downwind water where they sink. The fastest-growing mode at a Langmuir number of 0.13, looking downwind, over two roll spacings of 2.89 decay depths and 4 decay depths down. The closed curves are streamlines of the overturning; the dashed curves are contours of the downwind velocity the rolls carry, positive under the lines where the water sinks. At the surface the cross-wind flow converges onto those lines, which is where floating foam and weed collect as windrows. The amplitude is arbitrary, as in any linear mode.

The drift that turns a current into rolls

A current carrying a Stokes drift feels a force the drift makes out of the current's own vorticity, and under a wind that force is unstable. It turns the surface layer into rolls lined up downwind, with windrows where they sink. The rolls need both the current's shear and the drift's; their growth rate sees only the product; and the split between the two decides which motion gets the energy.

kinematics · Stokes drift 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.

Linear stabilityModel limitMeasurementThresholdConvergenceDimensionlessRegimeToleranceInflection pointMixingRayleigh–Bénard convectionReynolds number

All concepts