Model limit — where it appears
Named by 169 essays across 9 fields — each of them below, with the objects they name alongside it.
The wake that has to spin
A rotor that takes power out of the wind must apply a torque to it, and a torque applied to air is angular momentum left behind. The axial theory has nowhere to put that energy, so Betz's ceiling is unreachable at every finite tip-speed ratio — and the gap is computable.
A loss with no viscosity in it
Where a pipe suddenly widens, energy is destroyed. The amount is exact, it has been known since 1766, and the derivation never mentions viscosity, Reynolds number or roughness — because momentum does not care where the energy went, only that it left.
The depth that costs least
For a given flow there are two depths that carry it at any energy above a floor, and exactly one at the floor. That one depth is where the Froude number is one, the least energy is exactly three halves of it, and a bump in the bed that asks for more than the flow has does not thin the water — it backs it up.
A ball that swings without spinning
A cricket ball curves in flight with no spin about any useful axis. The mechanism is not the Magnus effect; it is a seam tripping the boundary layer on one side so that side lets go later. Which way the ball then goes depends on one borrowed number, and this site's own inviscid solver supplies the value that gets it wrong.
Faster than the wind that drives it
An ice yacht in a fifteen-knot breeze does forty. That is not a trick and it does not need a special sail — it follows from two drag angles and a triangle, and the best speed a boat can reach is one over the sine of their sum.
No randomness, and it mixes anyway
A steady two-dimensional flow cannot mix, however fast it is stirred, because its trajectories are its streamlines. Switch two vortices on and off alternately and the same fluid, obeying an exact map with nothing random in it, folds a patch of dye through itself until neighbouring particles separate by a factor of a thousand in six periods.
The siphon that does not need the air
A siphon is explained everywhere by the atmosphere pushing the liquid over the hump. The flow rate says otherwise, in the flattest way available — the height of the hump does not appear in it at all. What the atmosphere does is hold the column together, which is a different job with a different limit.
Where a jet stops being a jet
A tap makes drops a few centimetres down, a garden hose makes a stream that carries, a sprayer makes a mist and a diesel injector makes fog. Same liquid, same mechanism, four regimes — and the number that separates them is not the jet's inertia but the surrounding air's.
Not half a venturi
The air above a wing really is squeezed into a narrower channel, it really does speed up, and the pressure really does fall. Every step of the story is true and the whole is still not an explanation, because the channel's upper wall is a streamline — and where a streamline went is part of the answer, not part of the question.
The count a pattern cannot break
A picture of a flow has stagnation points in it, and they are not free to be arranged as anybody likes. Their kinds and their number obey an integer constraint that has nothing to do with the equations of motion — and an incompressible flow in a plane is allowed only two kinds of them in the first place.
The fastest way is not the straight one
A boat racing to a mark dead upwind sails seventy per cent further than the distance to it, and arrives first. The best angle is forty-five degrees plus half the apparent wind angle, it comes out of one line of trigonometry, and the same argument says to gybe downwind rather than run.
Three numbers left of a fluid
Saltzman truncated convection to three Fourier modes and Lorenz studied what was left. The result changed science, and it stopped being a description of a fluid at about a fifth of the way to the parameter everybody quotes it at.
Two slow things make a fast one
Shear stretches a slug of dye and mixes nothing, because it is reversible. Molecular diffusion is hopeless at any scale bigger than a hair. Put the two together in a pipe and the dye spreads along it with an effective diffusivity two million times the molecular one — which gets larger as the molecular one gets smaller.
An angle, not a speed
The number is printed in the handbook, marked in white on the airspeed indicator and used in every briefing, and the wing has no way of knowing it. A wing stalls at an angle. The speed at which an aeroplane reaches that angle is an answer with four other variables in it, and every one of them moves.
The radius that costs least
A vessel that carries a flow costs two things to own — the power to push fluid along it and the price of the tissue itself. Minimising the sum gives a best radius, the best radius makes flow proportional to radius cubed, and the rule that follows is a statement about a photograph that came out of a cost function.
Too fast for a profile
A pipe carrying a steady flow has a parabolic profile. Make the pressure oscillate and one number decides whether it still does — and above about ten the core moves as a plug, a quarter of a cycle behind the pressure, with the fastest fluid in a ring near the wall rather than on the axis.
Nothing but the shape of the gap
A machine that holds a steel shaft off its bearing with a film of oil twenty-five microns thick has no pump in it, and the pressure it generates would yield mild steel. The mechanism is not the oil and not the speed; it is that the gap narrows.
The bee that cannot fly
The claim has a traceable origin and the calculation behind it was a real calculation done with the wrong velocity. Doing it with the right one gives a number an aerofoil might plausibly produce — and still leaves a gap, and the gap is what took another sixty years to close.
The street this site cannot draw
The alternating wake behind a cylinder is the most photographed structure in fluid mechanics, and this site's solver does not produce one. What can honestly be drawn instead is a model of it — and the model settles one thing exactly, which is the spacing.
Three dimensions are kinder
Every ideal flow solved on this site so far is plane, and plane flow is the harsh case. Put the third dimension back and the fastest surface speed drops from twice the free stream to one and a half times, the disturbance dies as the cube of distance instead of the square, and the body cannot carry circulation at all.
When a body tears the water
A propeller blade moving fast enough pulls the pressure at its own surface below the vapour pressure of the liquid, and the water boils at whatever temperature it happens to be. Where that happens is decided by an inviscid calculation of the pressure along the blade.
Lift out of a failure
Separation is what ends a wing's lift curve everywhere else on this site. A slender delta with sharp leading edges separates on purpose, rolls the shed sheet into a pair of vortices above its upper surface, and takes most of its lift from the suction they induce — with a curve that climbs to forty-nine degrees.
The bubble that hammers
A vapour cavity swept into higher pressure does not deflate. It collapses, in ninety microseconds for a millimetre bubble, and the model that describes the collapse predicts a wall speed that reaches the speed of sound in water at three per cent of the original radius — which is to say it predicts its own failure, and locates it.
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.
What survives being wound up
Draw a loop of marked fluid particles and let the flow carry it. It will be stretched, folded and wound into a spiral until nothing about its shape is recognisable, and the circulation round it will not have moved at all — provided three conditions hold, each of which can be broken on purpose.
The cascade that runs backwards
Three-dimensional turbulence carries energy from large scales to small ones and dissipates it. Take away one dimension and the term that does it vanishes identically, a second quantity becomes conserved, and two conservation laws between them force the energy to go the other way — up in scale, into ever larger vortices.
What a jet keeps, and what it collects
A jet leaving a nozzle into still fluid has no boundary anywhere and one conserved quantity. Its momentum flux is exactly the same at every station downstream; its mass flux is not conserved at all and grows without limit, because a jet is a machine for acquiring fluid it did not start with.
Where lift starts
A wing at zero incidence is not a wing making no lift. The angle at which a section stops lifting is a property of its camber line and of nothing else — not of its thickness, not of its speed, not of the air — and it is an integral anybody can take.
Work out of a change of swirl
The work a rotor does per unit mass is the blade speed times the change in swirl, and that is all of it — no blade shape, no pressure, no efficiency, no gas properties. It is the same equation for a pump, a compressor, a turbine and a fan, and it follows from angular momentum on a box with nothing assumed about the inside.
Nothing but the edge
Cut an aerofoil into panels, put a singularity on each, and require the surface to be a wall. The system that comes out has one more unknown than it has equations, and the row that is missing is not a bookkeeping slip — it is the fact that ideal flow round a closed body has no unique answer at all.
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.
The wall that cancels its own waves
The area ratio of a supersonic nozzle fixes its exit Mach number and says nothing whatever about its shape. What fixes the shape is a wave-by-wave construction in which the wall turns through exactly the angle needed to absorb each expansion as it arrives — and getting it wrong leaves a stream full of oblique shocks at precisely the right Mach number.
What viscosity cannot take away
Leave a vortex alone in a viscous fluid and every local measure of it falls — the peak spin, the peak velocity, the enstrophy. The circulation round a large loop does not move at all, ever, and the far field is identical to the line vortex it started as.
A slot is not a nozzle
The gap between a wing and its flap is supposed to blow fast air into a tired boundary layer. A solver with no boundary layer in it at all — no viscosity, no wake, no mixing — produces most of the lift increment anyway, and produces it on the element nobody moved.
The picture belongs to whoever is watching
Photograph the flow past a cylinder from the tunnel and it has two stagnation points. Photograph the same flow from a frame moving with the air and it has none at all, and its surface speed is exactly the free stream at every angle. Both pictures are correct and no measurement distinguishes them.
The wind a swept wing feels
Sweeping a wing back is usually justified by saying it meets a slower wind. It does not meet a slower wind. The equations split exactly in two, and the flow along the span is a passenger that exerts no force and changes nothing — until a pressure gradient breaks the split, and then it becomes the reason a swept wing is a different problem rather than a harder one.
Every flow is two flows
Any velocity field splits into a part carrying all of the divergence and a part carrying all of the vorticity. The theorem says so and does not say which split — the two halves can be moved between each other by anything harmonic, and on a bounded region that is an infinite family.
Which part of a wing stalls first
A wing has one lift coefficient and its sections have a hundred, and no section is at the wing's. Which of them runs out first is decided by the planform, it decides whether the pilot keeps the ailerons, and the standard fix costs span efficiency in exact proportion to how much of it is applied.
A rate of change that will not hold still
Three boxes drawn in one flow at one instant give three different answers to how fast the dye inside them is changing — one falling, one falling twice as fast, one rising. All three reconcile with a single material rate, and that rate is zero.
The instrument in the answer
A model in a wind tunnel is not a model in the sky. The walls are supplied by an infinite row of reflections, the stream at the model is faster than the tunnel's own instruments report, and the correction is not an empirical fudge — it is a series with a closed form.
The loading nobody used
Elliptic loading is the least-drag answer to a question no aeroplane asks. Constrain the moment the lift makes about the wing root instead of the span, and a different curve comes out — five quarters of the span for sixty-four seventy-fifths of the drag, both exact — with an upwash over the outer wing and a yaw that turns the right way.
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.
The other branch of the same curve
Put heat into the jump conditions and the Hugoniot lifts away from the initial state, leaving a gap no wave can occupy. A burning gas has no weak waves: it must run supersonically or subsonically, and the conservation laws pick the first speed exactly and say nothing at all about the second.
A wing that leaves the plane
A winglet is not a fence and it does not block anything escaping round the tip. The induced drag of any system of lifting surfaces depends on one cross-section of its wake and on nothing else whatever, and a wake that reaches upwards is cheaper for the same reason a wake that reaches sideways is.
Where a vortex stops
Four criteria decide where a vortex ends, and in two dimensions three of them are the same criterion. The fourth is a knob. And the one that is not a knob is not objective: a co-rotating pair of vortices occupies two per cent of a window to one observer and twenty-five to another.
When a shock cannot bounce
A shock reflects off a wall until the reflected shock runs out of turning, which happens at a wedge angle well below the free stream's own limit. Between the two boundaries both configurations exist, both are stable, and which one appears depends on which direction the experiment came from.
A pump with no engine
A hydraulic ram lifts water uphill using nothing but the water that is already falling. It has one moving part and no power supply, and everything it can and cannot do follows from an energy audit that fits on one line — including a ceiling nothing about its design can move.
Two wings and it does not matter where
Move one wing of a biplane a chord forward and the induced drag does not change. Not approximately, not to a good approximation — the calculation that gives the induced drag has nowhere to put the stagger, because everything projects onto the same cross-section of the wake.
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.
The spin a shock leaves behind
A curved shock gives every streamline a different entropy rise and the same stagnation enthalpy. Crocco's theorem then forces vorticity into a flow with no viscosity anywhere — and it scales as the inverse of the shock's radius of curvature, exactly, so a straight shock makes none.
The lift beside a wing
Fly two aeroplanes with their wingtips touching and the pair costs exactly half what the two cost apart. Not approximately half — the arithmetic is a closed form, because two wings tip to tip are one wing of twice the span, and induced drag goes as the square of it.
The momentum with no value
A cylinder is pushed from rest to a steady speed. Work was done, energy went into the fluid, something was pushed. How much momentum does the fluid carry? The integral converges, the answer is finite, and it is a different finite number for every shape of region it is summed over.
The one number that runs out at three dimensions
A stream function is one function where the velocity is two, and four essays here are built on it. It exists because the divergence vanishes, it is single-valued only if nothing inside is making fluid, and in three dimensions it is not one function at all.
A shock that lies on the body
Above about Mach eight the flow round a blunt body stops depending on how fast it is going. The density ratio, the nose pressure coefficient and the shock standoff all reach limits set by γ alone — and going from a perfect gas to a dissociating one halves the standoff.
Where the wake ends up
The sheet a wing sheds rolls up into two cores within a few spans, and nobody can compute the roll-up cheaply. Nobody has to: what the cores conserve is fixed before they form, and for an elliptically loaded wing the answer contains π and comes out at 78.5 per cent of the span.
A wave nothing in it travels with
A flood crest moves at five-thirds the speed of the water it is made of, at every depth, whatever the roughness and whatever the slope. A traffic wave moves backwards through cars that are all going forwards. Neither result contains a momentum equation.
The equation that changes type inside its own answer
Near Mach one the coefficient of the streamwise second derivative depends on the perturbation velocity, which is what is being solved for. Two solutions of the linear equation no longer add — the leftover is three times the term the linear theory keeps — and the critical Mach number approaches one as the two-thirds power of thickness.
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.
Steady, three-dimensional, and mixing anyway
A steady flow that solves the Euler equations exactly, with its vorticity equal to its velocity to six parts in ten thousand million — and one of its streamlines wanders through a sixth of the box while another, started nearby, lies on a curve for ever.
When gamma stops being a number
Every compressible result on this site has used γ = 1.4, which counts the ways a nitrogen molecule can hold energy at room temperature. Behind a Mach 10 shock the gas is at 3,800 kelvin and the count is different — and the pressure barely moves while the temperature falls by fifteen per cent.
The surface in the wake
A wing has no opinion about its own incidence, which is why it needs a second surface behind it. How much that surface is worth depends on how much of the wing's downwash it is sitting in, and the factor of two everybody quotes for that is the value at infinity — where no tailplane has ever been put.
Where Darcy stops
A linear resistance law has to fail eventually, because pushing fluid into a pore and out again costs energy that has nothing to do with viscosity. Where it fails is one dimensionless number, and that number is 150/1.75 — read out of a correlation's own constants rather than measured.
The signature that forgets the shape
The pressure field near a supersonic aeroplane depends on every part of it. What reaches the ground has two parameters. Two bodies whose near-field signatures differ by fifty-five per cent in peak and by their whole shape age into the same N-wave, to two and a half per cent.
A second length at the wall
The logarithm in a turbulent wall profile exists because a region of the flow is not allowed to know about any length except the distance to the wall. Roughen the surface and there is one it does know about, which belongs neither to the fluid nor to the flow — and the slope does not change at all.
The knot a flow cannot untie
Ideal flow conserves energy, circulation and impulse, and all three are what they look like. It conserves a fourth quantity that is not: a volume integral of the velocity dotted into the vorticity, whose value counts how many times the vortex lines are linked through one another.
The one place the atmosphere pushes
The height of a siphon's hump is not in its flow rate, and the atmosphere holds the column together rather than driving it. That account excludes one process by name — starting. That is the process the atmospheric account describes correctly, and it is the only one in the whole device.
A radius that gives the energy away
Four quantities, three dimensions, one group. The radius of a strong blast must be a constant times (Et²/ρ)¹⁄⁵, and nothing about the device, the chemistry or the initial size can appear. The exponent is free and the constant is not.
The stretching rate that is not one number
A material line in a flow gets longer, and there is a theorem saying its length grows at a definite exponential rate. There is also a rate at which the average length grows, and it is nearly twice as large — and a different rate for every moment of the distribution.
What decay never forgets
Stir a box of fluid and stop. The turbulence decays, at a rate with no viscosity in it — so the rate cannot come from the fluid. It comes from an invariant of the very largest scales, fixed at the moment the stirring stops, and never revisited.
The siphon that does not break
A draining reservoir shrinks the drop and grows the rise at the same time, and the siphon's own coupling — a metre of extra drop costs a metre of hump — says a siphon should break as it empties. It does not. The two effects cancel exactly, and the crown pressure of a draining siphon is a constant that does not contain the source level at all.
The body the outer flow actually sees
A boundary layer lets less fluid past than an inviscid one would. The outer flow can be given exactly the same reduced flow rate by leaving the fluid inviscid and moving the wall out — so the potential flow that matters is not the flow past the body, but the flow past the body plus a thickness the boundary layer computes.
The least drag a volume can have
A body's supersonic wave drag depends on nothing about it except how its cross-sectional area is distributed along its length. Minimising that for a given volume gives one shape — and the answer goes as the volume squared over the fourth power of the length.
The range a real Reynolds number does not have
Kolmogorov's minus five thirds is a statement about a band of scales that has forgotten the forcing and does not feel the viscosity. Both conditions are about separation, and separation is exactly what a finite Reynolds number does not have much of.
A roll that feeds itself
A spin is routinely described as a stall that got worse, and it is not a stall at all in the sense of an angle rather than a speed. It is autorotation — a roll that sustains itself because past the peak of the lift curve the down-going wing makes less lift rather than more — and the arithmetic says it begins a little past the stalling angle rather than at it.
The discontinuity that has a thickness
The jump conditions do not contain the viscosity, which is why they are exact. The thickness is entirely viscosity — 289 nanometres at Mach 1.5, 35 at Mach 5, against a mean free path of 64. At Mach five the continuum equations have produced a structure thinner than the distance between collisions.
The frequency a wake chooses
Bluff bodies shed at Strouhal numbers from 0.145 to 0.212, and the spread is not a fact about shedding — it is a fact about which length went into the number. Change the length to the wake's own width and three bodies agree to a tenth of a per cent. Then let the body move, and the number stops deciding anything at all.
The exponents that stop being thirds
Kolmogorov's 1941 theory says every moment of the velocity difference scales with the same exponent, p over three, so the distribution keeps its shape at every scale. It does not. The exponents fall below the line, by more the higher the moment, and what the departure measures is a dimension.
A stall that is a place
A stall is an event, and the usual accounts treat it as one — a boundary reached, a damping lost. A high tailplane makes it something else. Swept into the wing's wake, the tail loses the download that held the nose up, and the aircraft finds a second stable trim point thirty degrees past the stall from which the elevator cannot bring it back.
Every compression becomes a shock in the end
Linear acoustics has no time scale in it, which is the sign that something has been thrown away. A 120-decibel tone shocks after three hundred metres and a jet engine after twenty; the distance goes exactly as the reciprocal of the amplitude, and nothing is exempt.
The number that cannot break a drop
The capillary number sets the stress that stretches a drop against the stress that holds it round, and it predicts the deformation beautifully. It cannot predict the breakup, because above a viscosity ratio of about four a drop in simple shear cannot be broken at any shear rate — and the theory that says the ratio hardly matters is the same theory that gets the deformation right.
A flux that runs both ways
The cascade is a statement about a mean. Kolmogorov's four-fifths law fixes an average and the constant flux through the inertial range is an average, and neither says anything about what the transfer is doing at any instant — which turns out to be running backwards a substantial part of the time.
A force forgets the datum, a stress cannot
Nothing sucks, because the pressure datum cancels — the normals of a closed body sum to nothing, so the lift is the same in gauge or absolute pressure. That identity is about a resultant, and it is routinely carried one step too far. The load on a panel of skin has the ambient in it as one of two numbers, not as a datum.
Long enough to make a wake
A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.
The half that carries nothing
Thin-aerofoil theory splits a section into a camber line that carries all the lift and a thickness distribution that carries none — at any incidence, exactly none. That is very nearly true, and what it discards decides the peak suction, the critical Mach number and where the boundary layer gives up.
Where the inverse cascade stops
Two-dimensional turbulence sends its energy upward in scale, and the upward direction has an end: the box. Without something to remove the energy before it arrives, it accumulates there in a pair of vortices filling the domain, and the limit of no friction has no steady state at all.
The only theory simple enough to optimise
Whether Newton's sine-squared law is right has two answers — hopeless at the speeds he argued about, nearly exact behind a strong shock. This asks a different question about the same formula. Its pressure depends only on the local surface angle, so a shape's drag is a quadrature rather than a solution, and the best shape can be found by calculus.
A disc that knows no blades
Momentum theory replaces a rotor with a surface across which the pressure jumps, and gets the Betz limit, the induced velocity and the whole energy argument out of it. It has no chord, no section and no number of blades — and at one fixed solidity, two blades and twenty give thrust coefficients ten per cent apart.
The constant that makes a variance negative
Every engineering turbulence calculation in the world rests on one number, C-mu equals 0.09. It is not a property of turbulence. It is the assertion that a particular ratio is ten thirds, which is true in one flow — and eleven per cent above that flow the same closure reports a mean square below zero.
Weighing what is missing
A control volume drawn round a wing gets the lift out of it, on a flow that was solved exactly. The wake survey asks the same box for the drag on a flow nobody has solved, and it is how a real aerofoil's drag is known — with three assumptions, all of which are checkable, and one integral standing next to it that is wrong.
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.
The side that cannot keep up
A hovering rotor is axisymmetric, so one radial distribution of circulation describes the whole disc. Move it forward and the advancing blade meets one and a half times the tip speed while the retreating one meets a half — and lift goes as the square of that. The rotor does not roll over, because the pitch is made a function of azimuth.
A sheet that cannot stay a sheet
Let a shear layer's thickness go to zero and it becomes a surface across which the velocity jumps. The model is used everywhere in this subject, it is unstable at every wavelength, and the thing it does next is worse: it develops a singularity in its own shape, at a finite time, from a smooth start.
The cushion that is there after all
A wing near the ground makes more lift for reasons that have no cushion in them, and that account cannot reach two cases — a wing descending, and a rotor in the hover, where there is no steady flight for the image-vortex account to be about. Those cases have a term that really is about carried air, and it grows without limit as the gap closes.
A speed nobody imposed
Every regime number in this collection contains a velocity somebody chose. Natural convection has none: a warm plate makes its own flow, and the Grashof number is what is left when the speed is taken out. The Reynolds number of the result — six thousand, on an ordinary radiator — is an output of the solution rather than a setting on an apparatus.
A row is not a set of aerofoils
An isolated aerofoil's incidence is measured from the free stream. A compressor blade's cannot be, because an infinite row of identical blades above and below it has a circulation that is part of its own free stream — and the velocity the theorem uses is a vector mean that exists nowhere in the machine.
The shape a vortex keeps
Outside a circular patch of uniform vorticity the flow is exactly the point vortex's — not nearly, exactly — so replacing one by the other looks free. It is not. The patch has a shape, the shape has a rotation rate of its own, and there is a strain above which no shape exists at all.
An instrument that takes a derivative
Dye, smoke and seeded particles mark the fluid and read the marks. The optical ones mark nothing — light passes through and is bent — and what a plate records is not the flow but a derivative of its density. Two of the three are therefore exactly blind to a uniform stream, at any speed it happens to have.
One group, three exponents
Lubrication theory, shallow water and slender-body theory are taught in three places and are one expansion in one group — a ratio of two lengths, with no speed, no viscosity and no fluid in it at all. The error is supposed to be second order. In three problems that look alike it is first order, second order, and an exponent that does not exist.
The condition that can be bought
Ideal flow round a closed body has one solution for every circulation, and the Kutta condition picks one. Its justification is entirely the sharp edge: every other circulation puts an infinite velocity there. Take the corner away and nothing chooses — which is not a curiosity, it is what a circulation-control aerofoil is.
The part of the flow inside the body
A potential flow outside a body is an analytic function, and an analytic function does not stop at the boundary it was defined on. It continues inward until it meets a singularity — and every body in this collection has at least one inside it, in a place that decides how the flow behaves outside.
The flow with no solution
Creeping flow past a sphere has a solution and everybody knows it. Creeping flow past a cylinder has none — not a difficult one, not one needing a clever method. The equations, the no-slip condition and the uniform stream at infinity are inconsistent, and the residual is exactly twice the free stream.
The paint that measures the wrong field
Dye, seeded particles and the optical methods all look through the flow or at something put in it. Pressure-sensitive paint looks at the surface, reports a scalar rather than a derivative, and turns a row of taps into a field. What quenches its luminescence is oxygen, which is what makes it a pressure gauge — and temperature, which is the other field a flow is guaranteed to produce.
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.
More lift than weight
An aeroplane in level flight is drawn with one arrow up and one arrow down, equal and opposite. That equation collapses the whole configuration onto one number, and it is not true of any aeroplane with a tail behind it: there are two surfaces, two equations, and the second decides the split.
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.
The singularity a layer makes for itself
March Prandtl's equations into an adverse pressure gradient and the wall shear reaches zero with an infinite slope at a finite station, and the solution cannot be continued past it. The singularity is real, it is not a numerical difficulty, and it belongs to the boundary condition rather than to the equations.
Where a liquid does pull
A fluid cannot pull, and the essays that settle what suction is are right about every gas and every liquid with a free surface near it. A liquid with nothing in it to boil on is another matter. Every tree taller than ten metres depends on the difference, and the floor under it is set by the size of a pore rather than by the vapour pressure.
One formula for both ends
Two limits, each with its own description, neither valid everywhere. The composite is the sum less the part they agree about, and it is uniformly good — but the overlap region that justifies the construction does not exist at ε = 0.01, and the composite is still accurate to two per cent there.
Where the line stops being a line
Prandtl's lifting line replaces a wing with a single bound vortex and its trailing sheet, and the formula that comes out is the most quoted in low-speed aerodynamics. Solved numerically at aspect ratio one it returns its own closed form to sixteen decimals — and the answer is forty-one per cent too high.
The constant a hole leaves behind
In a region without holes, Laplace's equation and the boundary values have exactly one solution. Cut a hole and they have a one-parameter family. Nothing in the mathematics chooses between its members, which is why the Kutta condition has to exist and why it cannot be derived.
A breaking strength that is the size of a flaw
Water can be stretched, and how far is a measurement people have made for a century and a half with instruments that agree with one another and not with the theory. Spinning a tube puts the stretch where no gas can reach it, sealing one caps the stretch at water's density maximum, and every measured number turns out to name the size of the worst cavity in the sample.
The sweep a root does not have
Simple sweep theory is one of the cleanest arguments in aerodynamics: an infinite yawed wing cannot know about the velocity along its own span, so only the normal component matters. A real wing has a root and two tips, and at the root of a thirty-five-degree wing the isobars are swept fourteen.
The drag that is made of waves
D'Alembert's paradox says a body in a steady, irrotational, incompressible, inviscid flow feels no drag. Put a free surface above it and every one of those words still holds — and the drag is not zero. It is the energy walking away in the wave train behind.
The three that never converge
Oseen's drag coefficient, the overlap layer's power-law exponent and the enstrophy range's slope are three separate disappointments in three different fields. They are one piece of arithmetic, and the arithmetic says why none of them will ever be reached.
The group with no head in it
The number that picks a machine says nothing about whether the machine can exist. A second group formed from the same variables, with the delivered head replaced by the margin available at the inlet, decides that — and the delivered head has left the expression entirely, so how far a pump lifts is irrelevant to whether it tears the liquid apart at its own entrance.
The margin friction lends a siphon
Without friction a draining siphon's crown pressure does not depend on the source level at all, and every real hose has friction. It turns out always to raise the crown pressure, by an amount the draining tank hands back pascal by pascal — and how much it lends is decided by where along the hose the crown sits, not by how rough or how narrow the hose is.
Two answers to one question
Unsteady aerofoil theory collapses a wing's whole history onto one function of one variable, and every quasi-steady gust calculation convolves something with it. There are two such functions, not one: a wing that is pitched changes its boundary condition everywhere at once, and a wing flying into a gust has not met most of the gust yet.
What a point vortex is not
Two circular patches of vorticity move exactly as two point vortices do — the centroid velocity is the point model's with no correction of any order, and that is a theorem rather than an approximation. The trouble starts the moment they stop being circular, which is immediately.
The air that breaks a siphon nothing else can
A running siphon's heights cannot break it, and its friction only postpones the moment it is most exposed. What does break a siphon that has run for a day is the air dissolved in its water, which the crown's low pressure leaves the water carrying far more of than it can hold — and which gathers only once the flow is too slow to carry a bubble away.
The control that works backwards
Divergence is the static aeroelastic failure everybody names, and a wing with its elastic axis at its aerodynamic centre cannot diverge at any speed. It can still reverse — deflect the aileron down above a certain dynamic pressure and the aeroplane rolls the other way — because the aileron's own nose-down moment is there whatever the elastic axis is doing.
The corners that can be done with mirrors
The method of images works for a wall and for a circle, and for a corner it works only when the angle is pi over a whole number. At every other angle the reflections never come back, the image set is infinite and dense, and the flow exists anyway — which says the method is a statement about symmetry rather than about fluids.
The stress that picks the aerodynamics
The free term in a rotor's pressure rise wants radius and speed, and both are capped by something with no fluid in it. Integrate a blade's own weight outward and the root stress comes out as the annulus area times the square of the shaft speed — with the chord cancelled, the blade count cancelled, and every property of the gas cancelled.
A cushion that changes its physics
A plate closing on a plane is resisted by the fluid it has to squeeze out, and the resistance is two different forces with two different laws — one from the fluid's inertia and one from its viscosity. They hand over at a gap of twenty-four kinematic viscosities per unit of closing speed, which for a wing in air is a third of a millimetre, and the two films disagree about whether the plate ever lands at all.
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.
The borrowed mass the boundary decides
A body accelerating near a solid wall has to squeeze out the fluid between them, and borrows more mass than it would in the open. The same body accelerating near a free surface, or inside an open-jet wind tunnel, borrows less. The fluid, the body and the speed are identical, and what reverses the answer is the one thing each boundary is allowed to tell the flow.
A tank that turns a hammer into a swing
Shut a turbine at the end of a two-kilometre tunnel in two seconds and the valve takes a rise of 256 metres of head. Put a shaft open to the air beside it and the rise is 51, the tunnel never carries the closure as a wave at all, and its water slows instead against a level that climbs for a minute and a half — to a height that is a closed form with the tank's area under a square root.
The thermometer that heats itself
A total-temperature probe is a wall told nothing about its temperature, made small and put on a stalk, so it obeys the same recovery arithmetic an aircraft skin does. Its recovery factor is a calibration constant near 0.98 rather than a one — and the static temperature inferred from its reading amplifies that shortfall rather than inheriting it.
The window every vector is averaged over
Particle image velocimetry is the one flow-visualisation technique that reports the velocity itself, and it still applies an operator: every vector is an average over an interrogation window. A window is a filter with a transfer function, and it makes a vortex slower and fatter, a thin shear layer exactly as thick as the window, and some features smaller than the window point the wrong way.
The vorticity a clean surface cannot refuse
A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.
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.
One view is enough, and the axis pays for it
An axisymmetric flow — a jet, a plume, a flame — can be reconstructed from a single optical view, because Abel's integral inverts exactly. The inversion runs from the outside in, every error made on the way reaches the axis, and whether it arrives multiplied depends on which instrument took the picture.
Twice as slippery along as across
A surface of alternating gas and solid stripes lets a liquid slip, and a flow far above it sees one number in place of the pattern — but the number depends on which way the flow goes. Along the stripes it is Philip's logarithm; across them it is exactly half, for a reason that takes one substitution to show. And the logarithm means that the slip is bought by the pattern's period rather than by how much of it is gas.
The jet a cone sprays sideways
The one-dimensional nozzle sends all its gas straight out along the axis. A real divergent section is a cone, and the gas leaves it as a spray of straight lines from the cone's apex. Only the axial part of that momentum pushes, and the share that does is (1 + cos α)/2 — 98.3 per cent at fifteen degrees, 93.3 at thirty — whatever the gas, the Mach number or the area ratio, and it touches the momentum and never the pressure.
A breeze the boat cannot use
The two-angle polar makes a boat's speed a fixed fraction of the wind, in any wind. A righting moment ends that at 3.7 metres a second on the beat. Past it the crew must spill force, a flattened sail's drag angle climbs, and the best course to windward moves closer to the wind rather than away from it — which 45° + λ/2 cannot say, because λ now depends on the course.
The teapot effect is a tension, not a pressure
A slow pour runs back under a spout and down its outside, and the name it is usually given is the Coandă effect. The Coandă effect borrows the ambient pressure, and a liquid in air has none to borrow. A liquid sheet is held to a lip by its own surface tension, and it lets go at the one speed that tension cannot carry — the speed of waves along the sheet — whatever the lip's radius.
A keel flies wherever the course puts it
A keel is a wing whose lift is whatever side force the rig happens to make, so its lift coefficient is chosen by the course and the wind rather than by its designer. Solved that way, the hull's drag angle stops being a property of the boat: it pulls the best beat twelve degrees closer to the wind, halves the reaching speed a fixed angle predicts, and finally charges for reefing.
Friction moves the sonic point past the throat
A choked nozzle is sonic at its throat — in a nozzle with frictionless walls. With friction the flow reaches Mach one where the section's widening rate has caught up with the friction, which is downstream of the throat, and the throat itself is subsonic. The solution through that point is a saddle that can only be found from the inside, and the mass flow it passes is less than the throat's area allows.
The truncation that cannot carry three times the heat
Lorenz's three modes give a convecting layer's heat flux in one line — one plus twice (r − 1) over r — and it can never reach three times what conduction carries. The same rolls computed with forty-two modes agree with that line exactly at onset, carry 7.1 per cent more heat at twice the critical Rayleigh number, and twice as much at thirty times it. The ceiling is not in the convection; it is in having one sine to draw the temperature with.
Hexagons remember how the heat was turned up
A layer heated from below convects in rolls, unless its top and bottom are not mirror images of each other. Then three sets of rolls at 120° can feed one another through a term the symmetry used to forbid, hexagonal cells appear before the layer is formally unstable, and there is a range of heating in which rolls and hexagons are both stable — so the pattern a layer shows depends on whether the heat was turned up or down to get there.
Twice the margin, on top of the hammer
Shut a valve on a line whose pressure is low and the returning wave boils the water beside it. When that cavity closes, the head at the valve can pass the Joukowsky rise — by up to twice the margin that let the water boil, in a sawtooth that jumps each time one more round trip fits into the cavity's life, and for a time that is shortest exactly when the pulse is tallest.
The face Newton left in shadow
Newtonian theory gives a surface turned away from the stream a pressure coefficient of exactly zero, and at hypersonic speed the rest of the theory is nearly right. The shaded face is not. Computed exactly on a flat plate, its share of the force depends on the similarity parameter K = M sin α rather than on the Mach number, it is a quarter of the force at K = 1, and it moves a hypersonic plate's best lift-to-drag ratio from 5 to 7 at Mach 10.
Matched at one speed and at no other
Every stage of a compressor passes the same mass flow, and the annulus behind each one is cut for the density the air will have reached there at design speed. Slow the shaft and the air is less dense than the metal expects, so the rear stages carry more volume than they were drawn for while the front ones starve — and below a definite speed no throttle setting keeps all of them working at once.
The drag a wake keeps however it rolls up
A plane drawn across the wake of a finite wing contains its induced drag as the kinetic energy of the swirling crossflow. The trailing sheet then rolls up into two vortices, and the energy does not change at all — roll-up moves the drag around the plane without spending any of it. What does spend it is viscosity, which turns crossflow energy into a total-pressure defect, so a plane farther back reads less induced drag, more profile drag, and the same total.
A spiral is a legible record
When a vortex sheet rolls up, the fluid in it can never change places: two points on a sheet cannot pass one another. So the arms of the spiral are a map of the initial sheet, in order, and the picture is a record of the roll-up rather than a snapshot of it.
Between hover and twice the hover inflow
A rotor's momentum balance has an answer for every climb and for every fast descent, and none for descending at anything between zero and twice its own hover inflow. There one root sends air out of both ends of its streamtube and the other root is not a real number, and at each edge of the band one end of the tube stops moving — which is where the vortex ring state lives, and where a wind turbine's thrust coefficient of one sits.
The column the chord rule cannot settle
A suspension settling in a closed column is a kinematic wave, and its flux curve bends both ways. At the top the chord rule works: clear water meets the suspension at a single falling front. At the bottom it does not, and the bed grows behind a shock that stops short of packing and a graded layer beneath it — so the interface, instead of arriving, slows for ever.
The venturi that stops listening downstream
In a venturi with real walls, the narrowing-speeds-it-up story is exact: continuity and Bernoulli run forward from the drawing. Followed far enough, the same story predicts its own limit. The throat's pressure cannot fall below the liquid's vapour pressure, and once it gets there the flow rate stops responding to anything downstream — the meter has become a limiter, and its throat is supersonic for the vapour-laden mixture passing through it.
Universal, and one of five
A turbulence that has run its Reynolds number down stops being turbulent, the equations go linear, and the decay picks up a new exponent. That exponent is quoted everywhere as 5/2 and as universal. It is neither: it is the same corner of the same spectrum deciding the answer a second time.
The specific speed a pump spends its life at
A pump is chosen by its specific speed at its best point and then run somewhere else. Written in the pump's own coefficients the number is √φ/ψ^¾, a position along its characteristic, and a variable-speed drive keeps it there only when the system it pumps into has no static lift. Every metre of lift moves a slowed pump along its own curve, towards shut-off, and a throttle moves it further.
The boom that turns back before the ground
Sound is faster in the warm air near the ground, so a sonic boom's rays bend back upward on the way down. Whether any of them arrive is one comparison — the aeroplane's speed against the fastest sound beneath it — and the ray that just grazes the ground sets the edge of the carpet, which the uniform air of the ageing calculation cannot give it.
The constant that travels
Every decay law in the subject rests on the dissipation being some constant times u³ over a length. The constant is not one. Letting it move the way grid measurements say it moves changes the decay exponent by a quarter — and lands one of the answers five per cent from another that is entirely different physics.
The solution that keeps its nonlinear term
Every exact solution before this one has been exact because the nonlinear term vanished. A rotating disc's does not vanish — at the wall it is the whole of the balance — and the reduction is exact anyway, because the radius divides out of all three momentum equations at once.
The fraction that is really four thirds
Turbulence has two exact results, not one. The second is about a scalar carried by the flow, its constant is four thirds rather than four fifths, and the difference between the two fractions has nothing in it about turbulence at all — it is the price of writing a three-component object in terms of one component.
One channel, one flux, two flows
Flow between two plane walls meeting at a line has an exact solution. Past a threshold that turns out to be a ratio of gamma functions, it has two — the same wedge carrying the same flux, once outward everywhere and once with the fluid running backwards along both walls, and nothing in the equations chooses.
The carpet an accelerating aeroplane folds
Level flight launches every boom ray with the same invariant, so they run parallel and each place hears one boom. Accelerate, and each successive ray is shorter than the last — shorter, near the cut-off, than the aeroplane's own advance — so later rays overtake earlier ones and the arrival map folds onto a line.
A choked throat buys time, not silence
A venturi whose throat has reached vapour pressure passes a flow the downstream pressure cannot change, and it is tempting to read that as isolation: whatever happens downstream, the upstream pipe will not hear it. Slam a valve downstream and it hears it. The cavity at the throat holds the surge back only for as long as it takes to fill, and then lets 93 per cent of it through.
The pulse that has to travel
In a rigid tube the Womersley number decides the shape of an oscillating flow. Make the wall elastic and the pressure pulse has to travel, a second number appears — the tube's length in wavelengths — and the first number turns out to decide something more basic than the profile: whether the tube carries a wave at all, or only a disturbance that spreads like heat.
A relation with no turbulence in it
Isotropy and incompressibility alone fix the transverse structure function from the longitudinal one. Divide the relation through and it says the ratio of the two is one plus half the local slope — so the exponent everybody measures as 0.70 and the ratio everybody measures as 1.35 are one measurement, and a model spectrum with no intermittency in it produces both.
Why the list is this long
Every textbook list of exact solutions of the Navier–Stokes equations is about a dozen long, and the usual explanation is that the equations are hard. It is not the reason. A similarity reduction is a solution invariant under a subgroup of the equations' own symmetries, so the catalogue of possible reductions is the catalogue of subgroups — and that is a finite, countable object.
The spot a local theory cannot see
Newtonian theory gives every panel of a hypersonic vehicle a pressure set by its own angle to the stream, and no panel more than the stagnation pressure behind a normal shock. Let a shock from one part cross the bow shock of another and a supersonic jet forms that reaches the surface through weaker shocks. At Mach 8 it stagnates at 8.6 times the ceiling — and the worst amplification at every Mach number is close to the Mach number itself.
The pulse that grows as it leaves the heart
The pressure pulse measured at the wrist is larger than the pulse in the aorta that drives it, and the flow pulse is smaller. Nothing downstream is pumping. A wave reflected from the end of an elastic tube arrives back in step with the outgoing wave near the end and out of step near the start, and a single number — the reflection coefficient — decides whether it is the pressure or the flow that grows.
The edge is a rumble, not a quieter bang
The rays that reach the outer half of a sonic-boom carpet arrive nearly horizontally, having travelled almost three times as far as the one under the track. Ray theory says they still carry two-thirds of the overpressure, right up to a line beyond which there is nothing. Neither half of that is what is heard — which is the useful result, because it says the edge's loudness is not a ray quantity at all.
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.
The heat the eddies do not carry
In a laminar layer the temperature and the velocity have different thicknesses in every fluid but one. In a turbulent pipe the eddies carry both, and the difference nearly vanishes — for air, water and oil alike. It does not vanish for a liquid metal, whose molecules conduct heat faster than the eddies can, and the boundary between the two behaviours is a Péclet number of about four hundred at every Reynolds number.
A boom is aged in the thin air it starts in
The rays that reach the edge of a sonic-boom carpet travel two and a half times as far as the one under the track, and it is natural to expect their signatures to have aged accordingly. They have not. A pressure wave distorts thin air far faster than dense air, so two-thirds of a boom's ageing is done in the stratosphere near the aeroplane, and the extra kilometres near the ground add little — which decides how far out a boom shaped to be quiet stays quiet.
The decay inside the four-fifths law
The four-fifths law gives the dissipation of turbulence from one measured moment with no constant in it, which makes it the obvious way to measure how the dissipation coefficient travels during a decay. But the law is exact only in a limit, and a decaying flow is not in it. The decay itself takes a quarter off the moment at the Reynolds numbers grids reach — and the bias moves as the flow decays, by a sixth, in the direction opposite to the effect being looked for.
A lighter spar turns a box wing into a biplane
Prandtl's best wing system has two-thirds of a monoplane's induced drag at a gap of a fifth of the span, and part of that saving is carried by circulation turning the corner into its fins. Ask the box to bend its root less and it pays about half what a monoplane pays — but a biplane with no fins pays nearly as little, and by the time the spar is a fifth lighter the fins carry almost nothing and the box has become the biplane it was built from.
Two rings leapfrog only if they start alike
Two coaxial smoke rings passing through each other in turn is the most famous thing vortices do, and it is not what two rings do in general. Set them off from one plane with different radii and below a definite ratio the smaller one draws ahead and never comes back. Which of the two happens is decided before either ring moves, by whether two separate rings could hold the pair's energy.
Past three, an ellipse is a shear layer
Kirchhoff's elliptical vortex turns for ever without changing shape, and Love showed in 1893 that it stops being stable at an aspect ratio of exactly three. Computed, that threshold turns out to be the first of a sequence — a new way of coming apart every one and a half aspect ratios — and the sequence ends somewhere recognisable. A long enough ellipse is a strip of vorticity, and it comes apart the way a shear layer does, at a rate Rayleigh found for the strip.
Named alongside it
The objects these essays reach for when they reach for this one.
MeasurementCirculationMisconceptionSeparationLift coefficientVorticityConservationDimensionlessOptimisationReynolds numberBoundary layerDissipation