What is refuted here, and by what — page 6

The logarithmic velocity profile near a wall is a consequence of Prandtl's mixing-length hypothesis, and stands or falls with it.

What decides it: The mixing length does imply the logarithm — integrating it gives the profile, and κ comes back out of the integration — but so does every other closure that has replaced it, and the law was known before any of them. What actually produces it is the requirement that an overlap region exist in which neither the viscous length ν/u_τ nor the layer thickness δ may appear, which leaves y du/dy with nothing to depend on and forces it to be a constant. The diagnostic function computed here is flat over two decades at Re_τ = 100,000 and over nothing at all at Re_τ = 180, which is a statement about the limit rather than about the model.

Tested in The layer with no length in it, at the figure it turns on.

A measured energy spectrum with a −5/3 slope over a decade or more is evidence that a Kolmogorov cascade is operating in the flow.

What decides it: A field synthesised with exactly those amplitudes and independent phases is built here, and its spectrum is the Kolmogorov one by construction while its energy flux is zero. Every second-order statistic of the two fields agrees — spectrum, correlation, second-order structure function, integral scale, Taylor microscale — because all of them are the same information. What distinguishes a cascade from a coincidence is the third moment, which the synthesised field has only as sampling noise: one realisation gives 0.18, and sixty-four average to 0.019 and keep falling as one over the square root of the ensemble.

Tested in The moment a spectrum cannot hold, at the figure it turns on.

Air can be treated as incompressible below Mach 0.3, so a body in a stream at Mach 0.3 or less may be analysed with the incompressible equations.

What decides it: It is a five per cent tolerance on the density at a stagnation point, and it is exact for that: the stagnation density rises by five per cent at M = 0.314. It says nothing about the body, and every body has a suction peak as well as a nose. The two exchange places at exactly cp₀ = −1, where the local speed is root two times the free stream; a section at cp₀ = −2 reaches the same five per cent at M = 0.22, and at a free-stream Mach of 0.3 its shoulder is at Mach 0.55 with the density down by nine and a half per cent.

Tested in Which speed goes in the number, at the figure it turns on.

A body's drag is the rate at which it is heating the fluid, so measuring the heat made in a region around it measures the drag.

What decides it: The equality holds only over the whole of the fluid. A creeping sphere's dissipation inside a radius R falls short of the drag power by exactly 3a/2R — a seventh at ten radii — and at Reynolds number 100 a box five diameters long behind a cylinder has not caught most of it. Two things leave a control volume that are not heat: work done by the tractions on its surface, and kinetic energy carried out through it.

Tested in Where the heat of a drag is made, at the figure it turns on.

Geostrophic balance holds when the Rossby number is small compared with one, so a weather system with Ro well below unity may be analysed by setting the Coriolis force against the pressure gradient.

What decides it: Keeping the centripetal term gives V_g/V = 1 ± Ro exactly for circular flow, so the fractional error in the geostrophic wind IS the Rossby number. "Ro is small" and "the balance is accurate" are therefore the same statement with the same number in it — which makes the folklore threshold of one a hundred per cent error rather than a boundary. And the anticyclonic branch is a quadratic whose roots vanish at a geostrophic Rossby number of exactly a quarter: past that there is no balanced wind at all, which no reading of the geostrophic approximation can suggest.

Tested in The balance that is its own error, at the figure it turns on.

A flow arranges itself to dissipate as little as it can, so of all the ways fluid could move between two walls, nature picks the cheapest.

What decides it: True for creeping flow and false as soon as there is inertia. Turbulent pipe flow at Reynolds number 10^5 dissipates roughly three times as much as laminar flow at the same flow rate between the same walls, and the laminar solution still exists there — it is merely unstable. The theorem is a property of the Stokes equations, whose solutions are unique and minimising; it says nothing whatever about which of several solutions a real flow chooses.

Tested in The cheapest shape the walls allow, at the figure it turns on.

The bumblebee story is a story about arithmetic: somebody used the forward flight speed instead of the wing speed, and doing the blade-element sum properly settles it.

What decides it: Doing the sum properly is necessary and is not sufficient. A quasi-steady sum assumes the wing has reached its steady circulation and that it flies in still air. Wagner's function averaged over each of four insects' own half-strokes gives 0.69 to 0.76, and momentum theory puts the animal's own induced velocity at 30 to 45 per cent of the wing's stroke speed. Both errors go the same way, and the mechanism that closes the gap — a leading-edge vortex that has not had time to shed — is a memory too.

Tested in A calculation with no memory in it, at the figure it turns on.

A jet pump works by suction — the fast jet has a low pressure by Bernoulli's equation, and that low pressure pulls the secondary fluid in.

What decides it: The low pressure at the nozzle is real and it is what starts the secondary flow, but Bernoulli does not explain the machine, because Bernoulli forbids what the machine does. Between the mixing tube's inlet and its outlet the pressure RISES while the flow keeps moving, which no Bernoulli argument permits: it is a momentum result, and it comes out of Cauchy-Schwarz — a single uniform stream always carries less momentum flux than the two it was made from, and the difference has to appear as pressure. The energy that pays for it is destroyed by the mixing, in exactly the Borda-Carnot expression a sudden enlargement produces, and the audit closes here to one part in ten billion. A device explained by Bernoulli alone would be a device that cannot raise the pressure of anything.

Tested in Mixing is a pump, at the figure it turns on.

A small neutrally buoyant particle released in a flow moves with the fluid, so a particle image measures the velocity at the particle's position.

What decides it: Faxén's first law says the particle moves at the surface average of the ambient flow, which differs from the value at its centre by a sixth of its radius squared times the Laplacian. On the axis of a round tube that is a lag of two-thirds of (a/R)² — 0.7 per cent for a particle a tenth of the radius, 6 per cent for one three-tenths — and the sign is always the same, so it does not average out over many particles.

Tested in A force without the flow that makes it, at the figure it turns on.

A body moving steadily through an ideal fluid carries a definite quantity of momentum with it, equal to its added mass times its speed.

What decides it: The momentum of the fluid is ∫ρu dA, which for a translating cylinder is conditionally convergent: it is exactly zero over a disc, and over rectangles of equal area it runs from −ρπa²U for a tall one to +ρπa²U for a long one, taking every value in between. The added mass times the speed is a real and unique quantity — it is the hydrodynamic impulse, a surface integral over the body — but it is not the momentum of the fluid, and the fluid's momentum has no value at all until the shape of the limit is named.

Tested in The momentum with no value, at the figure it turns on.

Streamlines drawn at equal intervals of the stream function show the speed: where they crowd together the flow is fast.

What decides it: True in a plane flow, where |∇ψ| is the speed exactly. In an axisymmetric meridional plot |∇ψ| = q·r sin θ, so the same reading is wrong by the distance from the axis — a factor running from 0.77 to 4.00 across one figure of ideal flow past a sphere, measured against the closed-form velocity to 10⁻¹⁰. The crowding of streamlines in a meridional plane is a statement about the volume flux in an annulus, and the annulus grows with radius.

Tested in The one number that runs out at three dimensions, at the figure it turns on.

Faster means a stronger shock and a bigger effect, so hypersonic flow is supersonic flow with the numbers turned up.

What decides it: The pressure ratio and the temperature ratio do grow without bound as M². The pressure coefficient at the nose, the density ratio across the shock and the shock standoff distance do not: they reach limits that are functions of γ alone. Measured, Cp at the nose is 1.8089 at Mach 5 and 1.8389 at Mach 40, against a limit of 1.8394, and the standoff changes by 8.5 per cent between Mach 8 and Mach 40 while changing by a factor of three between Mach 2 and Mach 8. Above Mach eight the question stops being about the speed and becomes a question about the molecule.

Tested in A shock that lies on the body, at the figure it turns on.

Choosing between a Pelton wheel, a Francis runner and a Kaplan propeller is an engineering judgement based on experience with similar installations — there is no principle that decides it.

What decides it: There is a principle and it is dimensional. A rotating machine's variables — shaft speed, diameter, flow, specific energy gH, density and power — have a dimension matrix of rank three, so three independent groups exist, and exactly one combination of them has a diameter exponent of zero. That combination, omega root Q over (gH) to the three quarters, is therefore a statement about a machine's SHAPE that is independent of its size, and this site checks that it survives rescaling to a machine four times the diameter at a third of the speed to nine decimal places. What IS a matter of experience is where the band boundaries sit — 0.35 or 0.4 for the Pelton-Francis crossover — and the bands are drawn here in the colour reserved for a borrowed claim.

Tested in One number picks the machine, at the figure it turns on.

Two flat plates with a film of liquid between them are held together by the surface tension of the liquid, which is why a wet glass sticks to a table.

What decides it: Surface tension supplies a force that does not depend on how fast the plates are being pulled apart, and the observed force does. A 40 mm disc separated at a millimetre a second across a 0.1 mm film of light oil resists with 75 newtons; the capillary force on the same rim is about 0.01 newtons. Pull slowly and the film gives way; pull fast and it does not, which is a viscous signature and not a capillary one.

Tested in The last of the oil, at the figure it turns on.

The Prandtl–Glauert factor corrects a subsonic result for compressibility, so it works up to the critical Mach number and stops there.

What decides it: Prandtl–Glauert and Kármán–Tsien are both linear corrections of the same order applied to the same incompressible peak. They agree to 1.5 per cent at Mach 0.3 and differ by 37 per cent at Mach 0.85 — which is inside the range a transport aeroplane cruises in and below the critical Mach number of a thin section. Where two approximations of the same order disagree by a third, the failure is not that the correction ran out but that the equation being linearised stopped being linear.

Tested in The equation that changes type inside its own answer, at the figure it turns on.

A duct's thermal entry length is x/(D·Re·Pr) = 0.05, past which the flow is thermally developed and the Nusselt number is 3.66.

What decides it: Marching the energy equation to that station gives a local Nusselt number of 3.710 against a developed 3.657 — a 1.45 per cent excess. That is a perfectly sensible tolerance and it is never stated, so nobody can say whether their own problem needs a longer duct or a shorter one. The entry length at ten per cent is x⁺ = 0.024 and at a tenth of one per cent it is 0.085, a factor of three and a half, and the quoted 0.05 is one unlabelled point on that curve.

Tested in How far before the heat arrives, at the figure it turns on.

A steady flow cannot mix, because its trajectories are its streamlines and a particle released on one stays on it.

What decides it: Both halves of that sentence are true in any number of dimensions and the conclusion follows only in the plane. A curve in a plane separates the plane; a curve in a volume does not, and can come arbitrarily close to every point of it. The ABC flow is steady, incompressible to machine precision and an exact solution of Euler's equations, and a single streamline of it covers 18 per cent of a Poincaré section and is still finding new territory after four thousand crossings.

Tested in Steady, three-dimensional, and mixing anyway, at the figure it turns on.

A real-gas correction changes everything about a shock, so a perfect-gas calculation at high Mach number is simply wrong.

What decides it: The jump conditions do not contain the heat capacity. Mass and momentum give ρ₁u₁ = ρ₂u₂ and p₁ + ρ₁u₁² = p₂ + ρ₂u₂², with no thermodynamics in them at all, and the caloric behaviour of the gas enters only through the energy equation. Measured across Mach 2 to 15, the two models' pressure ratios differ by at most 4.4 per cent while their temperature ratios differ by 18 and their density ratios by 27. A perfect-gas calculation gets the force on the body nearly right and the heat into it badly wrong.

Tested in When gamma stops being a number, at the figure it turns on.

A sonic boom carries information about the aeroplane that made it, so a quieter shape can be designed by measuring what arrives and working backwards.

What decides it: The far-field N-wave has two parameters, an overpressure and a duration, so at most two things about the aeroplane can be read from it — and they arrive mixed with the altitude and the atmosphere in between. Two bodies whose F-functions share only the maximum of their running integral, and differ by fifty-five per cent in peak and by their whole shape, produce signatures converging from seventy-three per cent apart to two and a half. Everything else the near field contained is destroyed by the propagation, which is why quiet supersonic design works on preventing the ageing rather than on making the wave smaller.

Tested in The signature that forgets the shape, at the figure it turns on.

Every dimensionless group's threshold is really a rule of thumb with a tolerance hidden in it, so the value one is never exactly where anything happens.

What decides it: It is true of every group formed by comparing two term sizes, which is most of them. It is false of a group formed from a characteristic condition. At Fr = 1 the flow speed equals the speed of a surface wave, so the upstream signal speed U − √(gh) changes sign — exactly, with no tolerance in the statement — the specific-energy curve has its minimum at the same place, and the governing equations change from elliptic to hyperbolic. Asking for the threshold at a tenth of a per cent or at ten per cent returns the same number.

Tested in The number that really is one, at the figure it turns on.

A shear-thinning fluid flows more easily, so pumping it through a pipe costs less than pumping a Newtonian fluid of the same viscosity.

What decides it: At the same pressure gradient the wall shear stress is GR/2 for every fluid, whatever its flow index — it is a momentum balance on the whole pipe and knows nothing about the fluid in it. What shear-thinning changes is where the shearing happens: at n = 0.2 the profile is flat in the middle and steep at the wall, and 96 per cent of the dissipation is made outside half the radius against 94 per cent for a Newtonian fluid. Nothing has been made easier; the cost has been moved.

Tested in A viscosity that depends on the question, at the figure it turns on.

A photograph of a Hele-Shaw cell shows what an inviscid flow looks like, so it is a demonstration of potential flow.

What decides it: The depth-averaged velocity in a cell is the gradient of a harmonic potential, so the streamline pattern is the ideal one to a part in ten billion — and the flow itself satisfies no-slip on the obstacle, which the averaged model violates at 2U, and its profile across the gap is a parabola, so a tracer at mid-gap travels at 1.5 times the average and one near a plate barely moves. The pattern is exact. Everything about the speeds is wrong.

Tested in The exact theory, drawn by viscosity, at the figure it turns on.

Dimensional analysis gives the answer to the strong-blast problem.

What decides it: It gives the exponent and cannot give the constant. Four quantities in three dimensions leave one group, so R = C(Et²/ρ₀)¹⁄⁵ with C undetermined. Getting C needs a solution: a thin-shell energy balance gives 0.907 against Sedov's exact 1.033 at γ = 1.4, and across γ from 1.2 to 5/3 the model stays between 12.1 and 12.3 per cent low — a steady bias that only a sweep across γ could show to be steady.

Tested in A radius that gives the energy away, at the figure it turns on.

A chaotic flow stretches material lines at a rate given by its Lyapunov exponent, so the length of a line after time t is its initial length times e to the lambda t.

What decides it: That is the rate of the typical element and not of the average one. Measured over eight hundred elements in a steady three-dimensional flow, the mean of the logarithm of the length grows at 0.157 and the logarithm of the mean length grows at 0.273 — a ratio of 1.74 — and the rate keeps rising with the moment, reaching 0.504 at the fourth. Half the total length is carried by the longest 0.75 per cent of the elements.

Tested in The stretching rate that is not one number, at the figure it turns on.

A turbulent spectrum has a minus five thirds range, so fitting a power law to the middle of a measured spectrum measures the Kolmogorov exponent.

What decides it: The band over which the local slope is within 0.05 of −5/3 is 0.04 decades at a Taylor Reynolds number of 30, 0.33 at 100 and 0.72 at 200 — it does not reach one decade until the high hundreds. A one-decade least-squares fit at the most favourable band returns −1.82 at 30, −1.62 at 100 and −1.67 at 3,200, so its error changes sign on the way. A laboratory watching its fitted exponent improve as it raises the Reynolds number may be watching two errors cancel.

Tested in The range a real Reynolds number does not have, at the figure it turns on.

A system with no viscosity, no approximation and exactly conserved energy is deterministic, so its future can be computed for as long as anybody is willing to keep integrating.

What decides it: Four point vortices conserve their energy and both impulses to fourteen digits over two hundred time units here, and two runs from initial conditions a billionth apart separate by a factor of 140,000 in the same interval — an exponential rate of 0.033 per time unit. Three vortices, with the same integrator and the same invariants, separate by a factor of 180. Determinism is intact and prediction is not, and the difference between three and four is one degree of freedom.

Tested in Three is the most that can be predicted, at the figure it turns on.

A shock is thin but finite, and the Navier–Stokes equations describe what is inside it.

What decides it: They describe it at weak strengths and not at interesting ones. Becker's exact solution gives a thickness of 3,154 nanometres at Mach 1.05 — fifty mean free paths, comfortably a continuum — and 35 nanometres at Mach 5, which is half of one. A structure thinner than the distance a molecule travels between collisions is outside the hypothesis that produced it, and measurements give a thickness several times larger. The end states either side remain exact however thin it gets, because the jump conditions contain no transport coefficient at all.

Tested in The discontinuity that has a thickness, at the figure it turns on.

The Strouhal number of a bluff body is about 0.2, and that is a constant of the shedding process.

What decides it: Three bodies measured in the same range give 0.212, 0.200 and 0.145 — a spread of 1.462 — so it is not one number across shapes. It is not a constant across Reynolds number either: Roshko's own fit is 0.212 − 2.7/Re, which is 0.137 at Re = 60, thirty-five per cent below the asymptote, and only comes within one per cent of 0.212 above Re = 1274. What is nearly constant is the number formed with the wake's width and the speed on the streamline bounding it: 0.1628 for all three bodies, to a tenth of a per cent.

Tested in The frequency a wake chooses, at the figure it turns on.

The log-normal model of intermittency, with mu about 0.25, describes the measured scaling exponents.

What decides it: It describes the low moments and it cannot be right about the high ones, because its exponent turns over: the slope of p/3 − mu·p(p−3)/18 goes negative at p = 3/2 + 3/mu = 13.5, so beyond that it says a higher moment scales with a smaller exponent. Novikov's constraint forbids that — the velocity difference is bounded, so the exponents must be non-decreasing — and 13.5 is inside the range routinely measured, not out at some academic moment.

Tested in The exponents that stop being thirds, at the figure it turns on.

In a closed wave tank the Stokes drift is cancelled by a return current, so there is no net motion.

What decides it: The depth-integrated transport is exactly cancelled, because the water has nowhere to go. The motion is not. Three return currents with exactly the required transport — a uniform slab, a parabola with no slip at the bed, and one concentrated near the surface — put the depth at which the drift reverses at 23, 98 and 100 per cent of the water column, and disagree about the sign of the drift at the bed. The constraint fixes an integral; where the water actually goes needs the dynamics.

Tested in The drift a closed box will not allow, at the figure it turns on.

An infinite velocity in a potential-flow solution is a sign that the solution has broken down and nothing quantitative can be taken from it.

What decides it: The square-root singularity at a sharp leading edge carries a finite force, ½πρC², and rounding the edge off to a parabola of radius r returns exactly the same number for every r over four decades — the peak speed rising as r^(−1/2) while the region it acts over shrinks as r^(1/2). The force is a limit rather than an artefact, and without it a flat plate in an inviscid fluid would have drag.

Tested in A finite force from an infinite speed, at the figure it turns on.

Requiring the flow to be finite and physically sensible is enough to pick out the right solution near a corner.

What decides it: Every mode r^(mπ/α) with m a non-zero integer satisfies Laplace's equation with both walls as streamlines, and the finite-energy condition selects exactly the positive m — the energy integral converges for m > 0 and grows without limit as the cut-off falls for m < 0, measured here at 0.785 against 7.9×10⁵. It is a physical requirement imposed from outside the equations, and even after it the surviving solution at a 360° corner still has an unbounded velocity.

Tested in Nothing turns a sharp corner, at the figure it turns on.

Sound is a small disturbance that travels at the speed of sound and keeps its shape.

What decides it: Linear acoustics contains no time scale at all, which is what says it has thrown something away. Each point of a waveform travels at a₀ + ½(γ+1)u rather than at a₀, so the profile distorts and becomes vertical after a distance a₀²/(βωu₀). For a 120-decibel tone at 1 kHz that is 307 metres; for a jet engine at a metre it is 20. The distance goes exactly as the reciprocal of the amplitude, from any amplitude whatever, and the second harmonic appears in direct proportion to the distance travelled with no threshold at all.

Tested in Every compression becomes a shock in the end, at the figure it turns on.

A drop breaks when the capillary number exceeds a critical value of about a half.

What decides it: In simple shear that is true for viscosity ratios between about 0.1 and 2, where the measured critical number runs from 0.61 to 0.70. Outside that window it is false in both directions: at a ratio of 10⁻³ the measured value is 8.7, fourteen times larger, and above 4.08 there is no critical value at all because a drop in simple shear cannot be broken at any shear rate. The first-order theory cannot see this — its own critical number spans a factor of 1.14 over the entire range — and the reason is that simple shear's rotation is exactly as strong as its strain, so the drop turns out of the stretching direction before it can be pulled apart.

Tested in The number that cannot break a drop, at the figure it turns on.

Energy passes steadily from large scales to small ones through the inertial range at a constant rate equal to the dissipation.

What decides it: That is true of the average and of nothing else. In a shell model whose nonlinear term conserves energy to 2·10⁻¹⁶, the mean transfer is flat to ten per cent across seven shells and equals the dissipation — and the instantaneous transfer through those shells is negative between two and fifteen per cent of the time depending on the shell, nine per cent averaged across the band, with a standard deviation between two and five times its own mean.

Tested in A flux that runs both ways, at the figure it turns on.

A scheme that is second-order accurate in space and time makes second-order errors in every quantity, so refining it enough makes any particular error negligible.

What decides it: A second-order face rule on a moving mesh changes a uniform scalar in a fluid at rest by 2.2·10⁻⁴ over two time units at a step of 0.05 — an error in a quantity that is not changing, of a size that would be read as a physical transient. It does fall as the square of the step, 1.977 measured, so refinement removes it. What refinement cannot do is tell anybody it was there, because a smooth excursion in a conserved scalar is exactly what a real flow produces.

Tested in The mesh that makes its own mass, at the figure it turns on.

A porous medium's permeability can be estimated from its porosity and its specific surface, which is what the Kozeny–Carman relation is for.

What decides it: Forty-nine bundles of capillaries are built with exactly the same porosity and exactly the same specific surface — matched to eight figures — and nothing else in common. Their permeabilities span a factor of eighty-two. The permeability of a bundle is the fourth moment of its radius distribution over the second; its specific surface is the first over the second. They are different moments of one distribution and neither determines the other.

Tested in A permeability that is only the geometry, at the figure it turns on.

The dissipation in a two-dimensional flow is the viscosity times the integral of the squared vorticity, so a region with no vorticity in it is destroying no energy.

What decides it: The identity holds over the whole plane and not over a region. Outside a Lamb–Oseen vortex's core the fluid is irrotational and contributes nothing to the enstrophy — and it is being sheared, so it contributes plenty to the dissipation. Inside a circle of radius R the two differ by exactly μΓ²/πR², measured here and matching the closed form, and the difference is the work the fluid outside is doing on the fluid inside.

Tested in The energy a vortex cannot have, at the figure it turns on.

Opening a valve at the bottom of a tank produces Torricelli's velocity √(2gh) at the outlet.

What decides it: Torricelli's result is the steady answer, and dropping the ∂φ/∂t term from Bernoulli's equation gives it at every instant including the first — full speed before any fluid has moved. The unsteady equation gives U = U∞ tanh(U∞t/2L), a time constant of 2L/√(2gh) set by the length of the tube and by nothing else, and for a two-metre tube under a one-metre head it takes 2.4 seconds to come within one per cent.

Tested in The pressure that depends on the past, at the figure it turns on.

Two-dimensional turbulence has a k^(−3) enstrophy range, so a measured spectrum should show a slope of −3.

What decides it: Kraichnan's own correction multiplies it by the logarithm of the wavenumber ratio to the power minus one third, because a cubic spectrum makes the strain rate the same at every scale and the cascade non-local. The local slope is therefore steeper than −3 by one third of the reciprocal of that logarithm — it is −3.060 eight octaves above the forcing and −3.0075 at sixty-four, and reaching −3.01 requires forty-eight octaves. No measurement and no computation will show −3.

Tested in Where the inverse cascade stops, at the figure it turns on.

A supersonic compression is irreversible, so an isentropic inlet is an approximation that gets closer to reversible as more ramps are added.

What decides it: It is exact arithmetic rather than an approximation, and the rate is the interesting part: the entropy across a weak shock is third order in its strength, so N equal ramps cost N times a cube of one Nth, which is exactly one over N squared. Sixty-four ramps cost a two-hundred-and-fifty sixth of what one costs, and the limit is genuinely isentropic. What the limit costs instead is length — the Mach waves from a smooth turn converge on a focus, and making the turn gentler brings the focus nearer rather than pushing it away.

Tested in A compression that costs nothing in the end, at the figure it turns on.

The mean velocity of a turbulent flow is a well-defined quantity, so an experiment's mean and a simulation's mean are the same thing and can be compared directly.

What decides it: In a flow whose density varies there are two means. On an intermittent field with a density ratio of seven — a flame — the time-weighted mean is 60 metres a second and the mass-weighted one is 30. Both are correct averages of the same signal; a hot wire returns the first and a compressible model computes the second; and the two turbulence intensities that go with them differ by a third.

Tested in Two averages of one flow, at the figure it turns on.

A microscopic swimmer goes faster by beating harder, and beating harder costs more per unit distance travelled, so there is an economical stroke and a wasteful one.

What decides it: For a waving sheet the power goes as the square of the amplitude and the speed goes as the square of the amplitude, so the work per unit distance travelled is 2μkc and contains no amplitude at all. What does change the cost per metre is the wave speed, linearly. A large slow undulation and a small one of the same wave speed cost exactly the same to travel a given distance, and the small one merely takes longer.

Tested in A swimmer that cannot go backwards, at the figure it turns on.

Solving unsteady gas dynamics needs a numerical method, because the equations are nonlinear and coupled.

What decides it: In a simple wave it needs no method at all. One invariant is the same everywhere in a region reached by characteristics from a uniform state, which reduces the pair of equations to a one-parameter family of straight lines whose slopes are known from the boundary alone — so the whole solution is arithmetic. Computed for a piston-driven fan, the constant invariant holds to four parts in ten to the sixteenth, and substituting the resulting field back into Euler's equations leaves a residual of a part in ten million.

Tested in Two numbers that do not change, at the figure it turns on.

A flowmeter's discharge coefficient accounts for the approach flow, so a calibrated meter reads correctly whatever the profile upstream of it.

What decides it: The inference from pressure drop to flow rate contains the approach profile's kinetic-energy coefficient, which is exactly one for a uniform profile, 1.058 for a fully developed turbulent one and exactly two for a laminar one. At a contraction ratio of 0.6 that is an error of 0.4 per cent, 7.7 per cent and 17.6 per cent for an annular jet — and two profiles built to share both coefficients differ by 48 per cent of the mean velocity and read identically.

Tested in The profile a meter cannot see, at the figure it turns on.

Taylor–Couette flow becomes unstable when the Taylor number exceeds about 1,708.

What decides it: Two conditions have to hold first and neither is in the Taylor number. If the outer cylinder turns faster than η² times the inner, the angular momentum increases outwards everywhere and no Taylor number is large enough — Rayleigh's criterion, found here by bisection at μ = η² to a part in 10¹⁶, contains no viscosity and no Taylor number at all. And 1,708 is a minimum over a continuum of axial wavenumbers, which a column of finite height does not have: at an aspect ratio of 2.5 the threshold is 5.66 per cent higher, and the vortex count that appears is decided by the height of the apparatus.

Tested in A transition that needs a second number, at the figure it turns on.

A helicopter's forward speed is limited by the advancing blade going supersonic.

What decides it: That is a limit and it is not usually the binding one. The trimmed solve here shows the retreating side running out first: at μ = 0.5 the section at 0.7R on the advancing side carries fifty-eight times what the retreating one does, the retreating section is at −0.09° of incidence and carrying nothing, and 2.07 per cent of the disc is at its lift-coefficient cap. The advancing tip at that advance ratio is at Mach 0.9 or so on a typical machine — high, and behind the retreating blade's problem, which is that lift goes as the square of a velocity that has fallen to a half.

Tested in The side that cannot keep up, at the figure it turns on.

Whether a shear flow is unstable, and how fast, depends on the details of its velocity profile.

What decides it: Whether it is unstable does. How fast is bounded, before any calculation, by two numbers taken from the ends of the profile: every unstable mode has a complex phase speed inside a circle of radius half the velocity range centred on the mean, so the growth rate cannot exceed the wavenumber times half the range whatever the shape between the ends. Across six profiles with different inflection structures the bound holds with the closest approach at 0.915 of the radius, and the actual maxima come in at 30 to 55 per cent of what it allows.

Tested in Every unstable wave is inside one circle, at the figure it turns on.

A surge calculation is specified by the closure time: state how long the valve takes to shut and the peak pressure follows.

What decides it: Three closure laws of exactly six seconds on the same 1,200 m line give peak rises of 50.2, 84.4 and 99.2 metres of head — 97 per cent apart on the smallest. The peak is a functional of the whole opening curve, and the sensitivity to it is measured here: bumping the valve at 5.6 s moves the peak by 345 metres per unit of opening, and bumping it before 0.45 s moves it by less than a hundredth of that.

Tested in The part of the closure a pipe cannot see, at the figure it turns on.

A concentrated vortex may always be replaced by a point vortex of the same circulation, since the exterior fields agree.

What decides it: They agree exactly for a circular patch and only for a circular patch, and a patch in anything but a uniform flow does not stay circular. The exterior field of an elliptical patch of the same circulation differs by 27 per cent at two radii and falls as the inverse square thereafter, because the first moment the two do not share is the quadrupole. More seriously, the point vortex has no strain limit: a patch is torn out into a filament above a strain-to-vorticity ratio of 0.150142, and a point vortex survives any strain whatever.

Tested in The shape a vortex keeps, at the figure it turns on.

At a low enough Reynolds number, inertia may be neglected and the Stokes equations solved instead.

What decides it: Near the body, yes; far from it, never. The neglected inertial term exceeds the retained viscous one beyond a radius of the body size divided by the Reynolds number, whatever the Reynolds number is, so the limit of vanishing Reynolds number does not commute with the limit of large distance. In two dimensions the consequence is that no solution exists at all: with the far-field conditions imposed there is one constant left for two wall conditions, and the tangential velocity that survives is exactly twice the free stream, at every radius and every speed.

Tested in The flow with no solution, at the figure it turns on.

The dissipation can be computed either as twice the viscosity times the strain-rate squared or as the viscosity times the enstrophy, since the two are equivalent.

What decides it: Their averages are equal, exactly, because their difference is a divergence and a divergence integrates to nothing over a homogeneous domain. Their values are not. On six synthetic incompressible fields the two correlate between minus 0.18 and plus 0.24 — uncorrelated — with a root-mean-square difference of 150 to 185 per cent of the mean they share, and their maxima 0.46 apart on a box of side two pi. So "where is the dissipation" has two answers and only the integral is well posed.

Tested in Equal on average, and nothing else, at the figure it turns on.

Cavitation begins when the local pressure falls below the vapour pressure, so inception is decided by a pressure and the cavitation number is the number that decides it.

What decides it: A 5 micron air bubble is given a rectangular tension pulse and asked whether it grows past the radius at which its own gas can no longer hold it. At 100 microseconds it needs 1.05 bar; at 0.3 microseconds it needs 43.2, a factor of 41 for the same bubble in the same liquid. The threshold is a curve in amplitude and duration, and the duration that separates its two ends is the time the bubble needs to grow to its critical radius.

Tested in A threshold that is also a duration, at the figure it turns on.

In level flight an aeroplane's lift equals its weight.

What decides it: The *total* lift equals the weight and the wing's does not. With a quarter-chord static margin and an ordinary zero-lift pitching moment, the trim solve gives a wing load of 11,672 N against a weight of 11,000 and a tail load of −672 N: the wing is carrying 106.1 per cent of the weight, and the excess is the tail's download. The two-equation solve residual is 1.6 × 10⁻¹⁷, so this is arithmetic rather than an approximation, and the excess grows as the centre of gravity moves forward.

Tested in More lift than weight, at the figure it turns on.

Vortex breakdown occurs above a swirl number of about 1.9, so a swirl number is what decides whether a vortex will break down.

What decides it: 1.9158530 is the critical swirl of one profile — uniform axial velocity with solid-body rotation filling the pipe — and it is exact for that one. Shrink the core to a quarter of the radius and the critical value computed from the same eigenvalue problem is 4.917; measure the same family's swirl at the wall instead and it is 0.307; a q-vortex with its swirl concentrated near the axis gives 0.403. Across twelve ordinary columnar vortices the critical swirl number spans a factor of sixteen, and the radial distribution decides.

Tested in The swirl that holds a wave still, at the figure it turns on.

Matched asymptotic expansions work because there is a region in which both the inner and the outer solution are valid, and the matching is done there.

What decides it: The overlap region is measured here, as the range of x over which both approximations are within five per cent of the exact solution. At ε = 10⁻⁴ it is 1.67 decades wide. At ε = 10⁻² it does not exist at all — there is no x at which both are within five per cent — and the composite built by the same recipe is nevertheless accurate to 2.4 per cent everywhere. The construction works well before its own justification does, which is the honest state of the method rather than a criticism of it.

Tested in One formula for both ends, at the figure it turns on.

The tensile strength of water is a property of water, so a careful enough measurement will find one number for it.

What decides it: The careful measurements find two. Spinning tubes and most other methods break water near −26 to −28 MPa; water sealed in quartz inclusions holds near −140, and classical nucleation theory for a pure liquid gives −148 MPa for a cubic millimetre watched for a second — a number that twenty-one decades of volume and time move only between −175 and −126, so the sample size cannot explain the gap. Read through the Laplace threshold, −27.7 MPa is a cavity of 5.25 nm and −140 MPa one of 1.04 nm: each measurement reports the worst flaw its sample contained.

Tested in A breaking strength that is the size of a flaw, at the figure it turns on.

An inviscid, irrotational, steady, incompressible flow past a body produces no drag, so any drag a ship pays must come from viscosity or from separation.

What decides it: Every one of those four words holds for a body under a free surface, and the drag is not zero. A submerged doublet leaves a wave train behind whose amplitude is four pi times the wavenumber times the doublet moment, decaying exponentially with depth, and the resistance is one quarter of rho·g·A² — the energy the train carries away at the group velocity, which in deep water is half the phase velocity. The paradox's hypothesis includes the domain, and a free surface is a boundary through which energy can leave.

Tested in The drag that is made of waves, at the figure it turns on.

The optimum convergence ratio for a plane slider bearing is 2.1889.

What decides it: The maximum of the load bracket is at 2.1887048, found by bisecting the derivative rather than by searching the load — which is flat to parts in ten to the thirteenth over that range and cannot locate its own maximum. The difference matters not at all: 2.1889 costs eight parts in a hundred billion of load, and every convergence ratio from 1.99 to 2.43 is within one per cent of the best. Both halves are the point. The number is exact, and it is worth about as much as its neighbours.

Tested in The gap that carries the most, at the figure it turns on.

An asymptotic result becomes accurate once the small parameter is small enough.

What decides it: Only if the correction is a power. When the leading and next terms differ by a logarithm the correction is 1/ln(1/eps), whose local exponent is the correction itself — so at any scale the result looks like a power law with a small exponent, and at the next scale down it looks like a different one with a smaller exponent. Reaching one per cent takes forty-three decades of the small parameter and a tenth of a per cent takes four hundred and thirty-four; a square-root correction reaches one per cent in four.

Tested in The three that never converge, at the figure it turns on.

Point-vortex dynamics is an approximation whose error grows smoothly with the size of the vortices, so a calculation with small enough cores is always safe.

What decides it: The error is not smooth in the size and it is not an approximation while the patches are round: each component of a point vortex's velocity is harmonic, so its average over a disc is its value at the centre, and a circular patch's centroid moves at exactly the point-vortex speed — 1.3·10⁻¹³ by quadrature, at every separation tried. What breaks the model is the strain deforming the patches, and that has a threshold: at five radii the orbit is followed to 0.9 per cent and at three and a half to 14, a fifteenfold change for a thirty per cent move.

Tested in What a point vortex is not, at the figure it turns on.

Dimensional analysis tells you what the answer to a fluid-mechanics problem is, up to a constant.

What decides it: It tells you which dimensionless groups the answer may depend on, and it never produces a value. Every pure number in this collection is the answer to a problem that had to be solved — an algebraic identity, an integral over a solved field, a root of a computed function, or its stationary point — and the fourth kind is systematically the worst known: with an objective known to a part in a million, a root is located to five parts in ten million and an optimum only to one part in a thousand, an exponent of exactly one against exactly a half.

Tested in Where a pure number comes from, at the figure it turns on.

A siphon that has started and has not broken will run until its source is empty, since nothing about its heights changes as it drains.

What decides it: Nothing about its heights does, and something about its water does. Saturated with air at one atmosphere, each litre arriving at a crown at 23 kPa carries 76.7 mL more free gas than that pressure can hold. While the flow is faster than a long bubble climbs the falling leg — 0.351√(gD), 0.123 m/s in a 12.5 mm hose — the gas is carried out. The reference drain falls below that speed after 30.9 hours with 4.7 cm of level left, and an 18.4 mL pocket then fills in 2 to 74 minutes across a twenty-fold range of how much gas actually comes out.

Tested in The air that breaks a siphon nothing else can, at the figure it turns on.

The Karman momentum integral is insensitive to the assumed velocity profile, so any reasonable guess gives the drag.

What decides it: Four reasonable guesses — a straight line, a parabola, a cubic and a quarter sine — give skin-friction coefficients from thirteen per cent below Blasius to ten per cent above, a span of twenty-three. The two that also satisfy the conditions the true profile satisfies are within 1.4 and 2.7 per cent, and the two that satisfy neither are the two that are ten and thirteen per cent out. The insensitivity is a property of the family, not of the integral.

Tested in Four profiles, one drag, at the figure it turns on.

A boundary layer is parabolic, so the flow at a station cannot be affected by anything downstream of it.

What decides it: It is parabolic only when the pressure is prescribed, which is the leading-order limit and not the flow. Let the pressure respond to the layer's own displacement and the problem admits a free interaction growing exponentially in the streamwise direction, at a rate of 0.8272 times the wall shear to the five-fourths power — a disturbance appearing out of nothing upstream. At a Reynolds number of a million it decays over 27 boundary-layer thicknesses, which is what every measurement ahead of a shock has always shown.

Tested in The length the limit invents, at the figure it turns on.

A capillary viscometer measures a viscosity, since it measures a shear stress and a shear rate.

What decides it: It measures a shear stress and a flow rate. The wall stress follows from a force balance and is exact for every fluid; the wall shear rate does not follow from the flow rate at all unless the fluid is Newtonian. For a power-law fluid the true wall shear rate exceeds the apparent one by (3n+1)/4n — a factor of two at n = 0.2 — and recovering it needs the local slope of the flow curve, which takes runs at neighbouring pressure drops.

Tested in The stress a pipe knows, at the figure it turns on.

An elliptic planform is the efficient one, so a wing that is not elliptic is paying for it.

What decides it: It is paying quadratically, which at any sensible taper ratio means it is paying almost nothing. The induced-drag penalty is a sum of squares of the loading's departure from elliptic, so a loading eight per cent away from elliptic costs under two per cent in induced drag, and every straight taper ratio between 0.25 and 0.51 sits within half a per cent of the best one. The exact optimum is real; choosing it rather than its neighbour buys nothing a wing can measure, and the choice is made on root bending moment instead.

Tested in The optimum that does not matter, at the figure it turns on.

Particle image velocimetry measures the velocity field itself, so a PIV vector is the flow's velocity at the point where it is drawn.

What decides it: A PIV vector is, to first order, the average of the velocity over its interrogation window, and an average over a window of width W is a filter whose response to a wave of wavenumber k is sin(kW/2)/(kW/2). A window one core radius wide reads a Lamb–Oseen vortex's peak velocity at 92.5 per cent, places it 1.084 times too far out, and gives 76.7 per cent of the central vorticity. A wave three quarters of a window long is reported at 20.7 per cent of its amplitude with its sign reversed, and a shear layer thinner than the window is reported as exactly as thick as the window.

Tested in The window every vector is averaged over, at the figure it turns on.

Flutter is found in flight test by measuring the damping at increasing speeds and extrapolating it to zero.

What decides it: It is, and the extrapolation is unreliable in the unsafe direction for a computable reason. The damping falls towards zero with a small slope — it is still at 0.19 per cent of critical two per cent below the boundary — so a straight line fitted to three sub-critical points between sixty and eighty per cent of the flutter speed predicts 101.8 metres a second where the true boundary is 80.8. That is twenty-six per cent high, from a curve that looks perfectly straight over the range fitted.

Tested in The speed where the damping is exactly zero, at the figure it turns on.

Viscous diffusion has a time scale, so the fluid a given distance from a wall responds to what the wall did about one diffusion time ago.

What decides it: There is a most likely delay and it is y²/6ν, and it is a poor summary of the distribution around it. The kernel that carries a wall's motion into the fluid is a probability density in the delay whose tail falls as the delay to the power −3/2. At two millimetres in water the mode is 0.67 seconds, the median is 4.40 — 6.6 times later — a tenth of the response is older than 127 seconds, and the mean does not exist: computed over a window it grows as that window's square root and does not converge.

Tested in The wall the fluid is listening to, at the figure it turns on.

One photograph of an axisymmetric flow determines the flow, and a sharper photograph determines it better.

What decides it: The first half is true: Abel's integral inverts exactly, and onion peeling on 200 rings recovers a Gaussian field from its exact projection to four parts in ten thousand. The second is false for an interferogram. With noise on each ray the recovered field's noise on the axis grows exactly in proportion to the number of rings — a gain of 4.75 at 12 rings, 19.8 at 50 and 79.1 at 200 — while the same relative noise on a schlieren deflection gives a gain of 0.94 at every resolution.

Tested in One view is enough, and the axis pays for it, at the figure it turns on.

Cavitation starts wherever the flow is fastest, so it starts at the shoulder of the body.

What decides it: In an irrotational flow, yes, and the reason is stronger than the observation: the pressure of an incompressible flow satisfies a Poisson equation whose source is twice the density times Q, and Q is never positive when the vorticity is zero, so the pressure is superharmonic and attains its minimum on a boundary. Where the flow has vorticity, Q can be positive and the minimum moves off every surface — which is why cavitation in a propeller wake begins in the tip-vortex core, several diameters behind the blade that made it.

Tested in The lowest pressure is on the body, at the figure it turns on.

A wake sheds at the Strouhal frequency, so a body oscillating at a different frequency produces a wake with both frequencies in it.

What decides it: It does, outside a band. Inside the band the wake abandons its own frequency entirely and adopts the forcing's, which is lock-in, and the width of that band is proportional to how hard the body is forced — from 0.093 at a forcing amplitude of 0.05 to 0.65 at 0.4, in units of the natural frequency. Outside the band the two frequencies coexist and the amplitude beats at their difference. Near the edge the capture is slow: the settling time rises from 14 time units at small detuning to 86 near the boundary.

Tested in A wake told what to do, at the figure it turns on.

A seeding particle small enough to follow the flow measures the flow, so the only question is whether it is small enough.

What decides it: "Small enough" is a statement about a frequency, not about the flow, because a particle is a first-order filter and its response depends on which frequency it is being asked about. Its magnitude is one over the root of one plus the Stokes number squared, so at a Stokes number of one it follows 71 per cent and lags by 45 degrees. Following to within one per cent needs a Stokes number below 0.14 at every frequency of interest — and the same particle is faithful at a hundred hertz and useless at ten kilohertz.

Tested in A particle is a low-pass filter, at the figure it turns on.

Induced drag is a fixed property of the wing, so a survey plane anywhere behind it reads the same induced drag as the kinetic energy of the trailing vortices.

What decides it: The inviscid half is exact: a smoothed trailing sheet rolling up into two vortices keeps its crossflow energy to 10⁻⁹, so roll-up changes nothing. Viscosity does. Two Gaussian vortices holding the sheet's energy keep 81 per cent of it as crossflow energy at a viscous age 4νt/b₀² of 0.01, 43 per cent at 0.1 and 9 per cent at 1; the rest is still in the plane, as a total-pressure defect, and a survey that separates the two reports it as profile drag.

Tested in The drag a wake keeps however it rolls up, at the figure it turns on.

A photograph of a rolled-up shear layer shows the state of the flow at the moment it was taken.

What decides it: It shows a state, and it also shows a history, because the material points of a sheet cannot change their order: the sheet is a one-dimensional object carried by the flow, so the parameter along it is a label the fluid keeps for ever. In the computation here 200 points wind through a full turn without a single pair ever changing places, so each arm of the spiral is a known interval of the initial sheet, and the number of turns is a clock. What the picture cannot say on its own is how thin the sheet was taken to be, and three smoothing lengths give arclengths spanning 76 per cent.

Tested in A spiral is a legible record, at the figure it turns on.

A duct's development length is the distance at which its flow becomes fully developed, so it is a property of the duct and the Reynolds number.

What decides it: It is a property of the duct, the Reynolds number and a threshold nobody usually states. A disturbance to the developed profile decays mode by mode at rates going as the square of the mode number, so what is left after a short distance is one mode decaying exponentially — and the distance at which it falls below a stated fraction is the logarithm of that fraction divided by one rate. Sweeping the fraction from a tenth to a thousandth moves the length from 1.02 m to 5.69, a factor of 5.6, with nothing about the duct or the flow having changed.

Tested in A duct that forgets everything but one number, at the figure it turns on.

The dimensionless groups of fluid mechanics are a long list to be learned, each belonging to its own corner of the subject.

What decides it: A large family of them is one group. Eight produced by this collection alone — Deborah, reduced frequency, Stokes, Keulegan-Carpenter, Womersley squared, Damköhler, the inverse root Ekman and a turnover-to-strain ratio — are each a memory time divided by a process time, and their representative values span a factor of 394 while the construction is identical. All eight govern the same first-order response: unity below about a tenth, a reciprocal above about ten, and two decades in between where the answer depends on the history and nothing simpler will do.

Tested in Every memory number is one time over another, at the figure it turns on.

A flow whose equations are reversible has kept all the information about where its fluid came from, so that information can be recovered.

What decides it: The information is kept and recovering it is a different question. Reversing a point-vortex stirring exactly returns 120 tracer particles to within 1.4·10⁻⁹ of their starting places. Displacing one vortex by 10⁻⁸ at the turning point — a perturbation to the state rather than to the tracers — returns the worst of them 4.9·10⁻⁶ away, a gain of 486, and the gain grows as the 2.87 power of the window. Below a nudge of about 10⁻⁹ the round trip's own arithmetic dominates at 2.2·10⁻⁷, so there is a floor beneath which no improvement in the state is visible at all.

Tested in Reversible, and unusable, at the figure it turns on.

Langmuir circulation is driven by the waves: the rolls draw their energy from the Stokes drift.

What decides it: The rolls need two shears of the same sign — the wind-driven current's and the drift's — and do not grow with either removed. Their growth rate depends only on the product of the two, so it is identical whether the current or the waves supply more of that product. What the split decides is where the energy goes: the current's shear powers the downwind jet under each windrow and the drift's shear powers the overturning. For an 8-second, one-metre swell under a moderate wind, with an eddy viscosity of 0.01 m²/s, the current supplies 85 per cent of the fastest roll's energy.

Tested in The drift that turns a current into rolls, at the figure it turns on.

The Womersley number describes the shape of the velocity profile in a pulsating vessel, and wave propagation is a separate matter governed by the wall's stiffness.

What decides it: In an elastic tube the same function of the Womersley number that shapes the profile, F₁₀(α), multiplies the wave speed: c = c₀√(1 − F₁₀). At the aorta's α of 14.7 the wave travels at 96 per cent of the Moens–Korteweg speed and keeps 73 per cent of its amplitude per wavelength; at α = 2, 73 per cent of the speed and 3 per cent per wavelength; at small α the speed tends to α/2 of the inviscid speed and the disturbance diffuses rather than travels. The wall sets the scale of the speed. The Womersley number decides whether there is a wave.

Tested in The pulse that has to travel, at the figure it turns on.

Ideal flow is a good approximation for short times, because viscosity has not had time to act yet.

What decides it: It is a good approximation at short times for the *magnitude* of the force and a bad one for its *character*, and the difference is what the indicial kernel shows. An ideal flow delivers the whole of its response at the instant of a step and exactly nothing afterwards. The viscous response falls as the inverse square root of time — it has no time constant, so there is no moment at which it has finished, and its tail carries most of its own total. Over six crossing times the ideal kernel has delivered all of its impulse and the viscous one is still delivering.

Tested in The theory with no memory in it, at the figure it turns on.

The no-slip condition is a law of fluid mechanics, so a liquid in contact with a solid moves with it exactly and a slip length is a correction for exotic surfaces.

What decides it: Nothing in the Navier–Stokes equations produces it; it is one of the conditions the equations have to be given, and the alternative — a slip length relating wall velocity to wall shear — is equally admissible. What decides between them is measurement, and the measurement that can do it is a drainage experiment, because the shortfall in the drainage force is twice the slip length over the gap and the gap can be made small. For a wetting liquid on a smooth solid the bound is under two nanometres; for a rarefied gas, a superhydrophobic surface and a polymer melt it is not.

Tested in A length you can only measure by squeezing, at the figure it turns on.

Averaging a gust across a wing's span matters because it reduces the gust the wing feels.

What decides it: It barely reduces it: a span of a tenth of the turbulence scale keeps 0.948 of the gust's root-mean-square, because the variance lives in wavelengths much longer than the wing. What the average does is remove the wavenumbers shorter than the span, one power of wavenumber steeper, and that is the difference between a load crossing rate that grows without limit as the spectrum's cut-off is raised — as its sixth root, local exponent 0.19 — and one that settles at 2.29 crossings per scale, exponent 0.004.

Tested in The span takes away the infinity, not the gust, at the figure it turns on.

By the Taylor–Proudman theorem, flow in a rotating fluid cannot pass over an obstacle on the floor: it goes round it, as though a rigid column of fluid extended above the obstacle to the top of the layer.

What decides it: The theorem is the limit of zero Rossby number and says nothing about how small the number must be. Conserving potential vorticity through a layer of depth H over a hill of height h₀ gives a flow that depends only on δ = (h₀/H)/Ro. Below a threshold every streamline crosses the hill, only bent; above it a closed region forms. The threshold is the reciprocal of the hill's peak swirl — exactly 2 for a flat-topped cylinder, 3.134 for a Gaussian hill and 16/3 for a cone — so a hill a hundredth of the depth stops a current only if its Rossby number is below about 0.003, and the fluid held is not a column over the summit but an eddy beside it.

Tested in The hill a slow current will not climb, at the figure it turns on.

A displacement hull has a hull speed, at a Froude number of about 0.4 where its bow wave is as long as the hull, beyond which its wave-making resistance climbs faster than any plausible engine can supply.

What decides it: Michell's integral for the Wigley hull puts Fr 0.399 on the steepest part of the curve — the wave resistance there grows as speed to the power 7.6 — but the peak of the resistance coefficient is at Fr 0.499, and past it the resistance grows as speed to the power 0.75 at Fr 0.6 and 0.5 at Fr 0.8. Per unit weight it rises sixfold between Fr 0.30 and 0.50, and by three-fifths more while the speed doubles again. What stops a heavy displacement hull near hull speed is the power needed to climb to the hump, and the sinkage and trim that linear theory leaves out, not a wave resistance that keeps accelerating.

Tested in Hull speed is not the hump, at the figure it turns on.

Washing a filter cake adds little to a filtration cycle, because the wash is only a pore volume or two and the cake's pores hold a small fraction of the liquid that was filtered through it.

What decides it: The wash runs at the cake's final rate, which for a cake-dominated filtration is half the mean rate of the filtration, so each cubic metre of wash costs twice the time of an average cubic metre of filtrate. Two pore volumes cost 2.4% of the output on a slurry of 50 kg of silica per cubic metre, where the pores do hold little, and 15.6% on one of 400 kg per cubic metre, where they hold a tenth of the filtrate.

Tested in A washed filter works as long as it rests, at the figure it turns on.

Birds in a V take turns at the front because the leader works hardest; a formation in which every bird paid the same would have to give up some of the flock's saving.

What decides it: The leader of a straight V does pay most — 0.554 of flying alone at the fairest straight angle, against 0.34 for the middle ranks. But a curved V, its arms swept 1.6° to 32° from apex to tips, gives every one of nine birds exactly 0.424, which is the straight V's average: by Munk's stagger theorem the fore-and-aft positions can redistribute the saving and cannot change it. Fairness is free; only the lateral spacing sets the total.

Tested in A fair V is a curved V, at the figure it turns on.

Arteries branch according to Murray's law, and a branching network built that way is efficient for the pulsing flow the heart actually delivers.

What decides it: Murray's law minimises the cost of a steady flow. For a pressure wave, a Murray junction reflects −11.5 per cent of the pressure when the wave speed does not change with radius, and a six-generation Murray tree with non-reflecting ends returns 30 per cent of the pressure at the heart rate. The branching exponent that minimises the tree's reflection over the pulse's harmonics is 2.15 for a constant wave speed and 2.60 for one that rises as smaller arteries stiffen — between area-preserving and Murray, not at Murray's three.

Tested in Murray's law is not the rule for a pulse, at the figure it turns on.

The bow shock of a blunt body at hypersonic speed is a Mach cone at its sharpest, so the faster the body flies the narrower its shock hugs it.

What decides it: Behind a blunt nose the shock grows as the square root of the distance, R/d = 0.795 C^¼ (x/d)^½, C the drag coefficient, with no Mach number in it: the width is set by the drag, which is the energy the body leaves in each metre of air. The Mach number decides only how far back that holds — the analogy's own shock stays strong for 1.7 diameters at Mach 10 and 27 at Mach 40 — and far behind that the shock does fall back towards a Mach cone.

Tested in A hypersonic body leaves a line explosion behind it, at the figure it turns on.

A bird in a V formation saves energy simply by flying in the upwash of the bird ahead, whatever it does with its own wings.

What decides it: A flapping leader's tip vortex lies along the path its wingtip traced, and the upwash beside it is narrow: with a tip-vortex core a twentieth of a span across, a gliding follower's saving falls from 0.84 to 0.52 of its own induced drag a tenth of a span above or below the vortex. A follower whose tip swings a fifth of a span each way keeps 0.84 when it lags its leader by the time the wake takes to arrive, and only 0.19 half a beat away from that phase.

Tested in The follower beats in time with the wake, not the bird, at the figure it turns on.

The restricted Euler equation's finite-time blow-up shows that a parcel's velocity gradient is driven to singularity by its own nonlinearity, so something other than pressure — viscosity — must be what keeps real gradients finite.

What decides it: The blow-up comes from discarding the anisotropic part of the pressure Hessian. Supplying it as a pressure that is isotropic in the coordinates the parcel had one Kolmogorov time earlier — with the matching viscous term — turns a gradient that restricted Euler takes to infinity at 0.83 large-eddy times into one that rises by a twelfth and decays, and no parcel in any forced ensemble blows up. The pressure's memory of the deformation, not viscosity acting alone, is what the restricted equation was missing.

Tested in The pressure a parcel remembers keeps it finite, at the figure it turns on.

Richardson's t³ law and his diffusion equation describe turbulent pair separation, so a measured separation cloud should have Richardson's shape, with a kurtosis of 3.76, and his constant fixes the growth.

What decides it: The t³ law is dimensional and survives any memory; the diffusion equation assumes none. Giving each pair a relative velocity correlated over β turnover times of its own separation, at the same diffusivity, lowers the kurtosis from Richardson's 3.76 to 2.55 at β = 0.1 and 1.83 at β = 1, and the t³ constant from 14.1 to 5.2 and 0.58. A cloud's shape is a measurement of the pairs' memory, and a memoryless fit reads the lower constant as a smaller diffusivity.

Tested in A pair's memory shapes the cloud it spreads into, at the figure it turns on.

Once a box wing's root bending moment is held to a monoplane's, its fins buy nothing, so a box wing is a biplane with extra wetted area.

What decides it: The moment held was the lift's, summed over both wings. The fins join the spars at their tips into a frame that carries part of that moment as an axial couple across the gap, which needs flange material in proportion to the spar's depth over the gap rather than to the moment itself. At a gap of a fifth of the span, with fins as stiff as the spars, the couple carries 29.5% of the root moment and the structural moment falls to 0.72 of the lift's; the free box then needs 0.78 of the elliptic monoplane's spar at 0.671 of its drag.

Tested in A box wing's fins earn their keep in the spar, at the figure it turns on.

A broken gas line empties at the rate its choked outlet allows, since a choked break passes gas at the speed of sound whatever happens downstream.

What decides it: The choked rate sets the blowdown only while friction can deliver gas to the break at that rate. Marched with the unsteady equations, a line a hundred friction lengths long stays choked through its whole half-time, but one of 1,000 is choked for the first 13% of it and one of 3,000 for 3%; thereafter friction sets the pace, and the half-time grows as the square root of fL/D — 4.2 transit times at 100, 20.2 at 3,000.

Tested in A broken line empties at the pace of its friction, at the figure it turns on.

Murray's law, which sizes blood vessels for the cheapest steady flow, is simply the wrong rule for the pulsatile arterial tree, whose junctions need a different exponent to pass the pulse.

What decides it: The transparent exponent depends on the parent artery's Womersley number and nothing else: 2.06 for a 12 mm parent, near area-preserving, but 2.96 for a 1 mm parent and 3.00 at 0.3 mm — Murray's cube — because below a Womersley number of about two the pulse is carried by viscosity as the steady flow is. Murray's law is wrong for the aorta's pulse and exactly right for the pulse in the small arteries, at any asymmetry of the split.

Tested in Murray's law passes the pulse where the pulse is viscous, at the figure it turns on.

A sea breeze grows until friction balances the pressure force that drives it, so the breeze's strength is set by the land–sea temperature contrast and the friction.

What decides it: The push reverses every day, and the breeze's size is set first by how long the push lasts: with no friction at all it is the push divided by the daily frequency, about 3.8 hours of push, and it peaks six hours after the strongest heating. A three-hour spin-down reduces it only to 0.62 of that and brings the peak to 2.5 hours after the heating. Near 30° latitude the Earth's rotation, not friction, decides it: there the inertial period is a day and a weakly damped breeze is 3.4 times stronger.

Tested in At thirty degrees the Earth keeps time with the sea breeze, at the figure it turns on.

Once a cloud of particle pairs has forgotten its release separation, its kurtosis depends on the pairs' memory alone, so a measured cloud's shape is a reading of how long pairs keep their relative velocity.

What decides it: Give the relative velocity the flatness measured in the inertial range, about 4, by letting its amplitude vary as a lognormal remembered over α turnover times, with the mean diffusivity unchanged. At a direction memory of 0.7 turnover times the kurtosis rises from 1.89 to 3.13 at α = 1 and 3.41 at α = 3, against Richardson's 3.76 for no memory at all. The shape reads a combination of two memories and a flatness, not one memory.

Tested in Bursts and memory pull a cloud both ways, at the figure it turns on.

A wake with four vortices, as behind a wing with its flaps down, breaks up faster than a clean wing's two, because the extra vortices make it far more unstable.

What decides it: The four-vortex wake does grow faster: with a flap vortex of 0.3 of the tip's strength at 0.4 of its station, a wave 1.8 spacings long grows at 1.57 e-folds per descent time against Crow's 0.83. But in that wave the tip vortices carry 1.3 per cent of the displacement; it is the flap vortex being bent as it orbits. The long wave in which tips and flap vortices move together, the one that links the two sides, grows at 0.79 — slightly slower than Crow's.

Tested in Four vortices bend faster, and bend the wrong one, at the figure it turns on.

A tailored shock tube holds its reservoir until the driver gas arrives, so tailoring always multiplies the test time.

What decides it: In one dimension the tailored reservoir ends when the head of the driver's expansion, reflected from the driver's closed end, reaches the end wall, and that arrival is proportional to the driver's length: 0.69 L/a₁ per driven length of driver for helium into air. A tube running at nine-tenths of the tailored Mach number hears its echo from the contact after 0.113 L/a₁; a tailored helium tube matches that only with a driver 0.29 of the driven length long, and with a shorter one tailoring gains nothing.

Tested in A tailored tube buys its test time with its driver, at the figure it turns on.