Ideal flow

The theory that solves everything

Throw away viscosity and assume nothing is spinning, and fluid mechanics collapses into a linear problem with closed-form answers. The price is one term, and the term turns out to matter more than everything kept.

d'Alembert's paradox, measured. Surface pressure round a cylinder in ideal flow, plotted against angle. The distribution is symmetric front to back, so every push on the front is matched by an equal push on the back, and the total force along the stream is exactly zero.

The exact theory says nothing has any drag

Solve the flow past a body in a fluid with no viscosity and the answer is beautiful, closed-form, and predicts that a cyclist needs no legs and an airliner no engines. This is not a small error, and it is the most useful failure in the subject.

Flows add up

The equations of ideal flow are linear, so solutions can be laid on top of one another. A uniform stream plus a doublet produces a cylinder that nobody put there, and almost every classical result is built this way.

Fast means low pressure

The trade between speed and pressure is the most useful relation in the subject and the most misused. Where it comes from, what it costs, and why the pressure over a wing is negative almost everywhere.

Flow net — a stream past a cylinder. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

One function instead of two

A velocity field carries two numbers at every point and must satisfy two constraints. Both constraints can be solved once and for all by writing the whole flow as a single scalar function — and the two families of curves that function generates cross at right angles everywhere, for reasons that have nothing to do with fluids.

Bodies made out of nothing

Put a source in a stream and a solid nose appears in front of it. Put a sink downstream of the source and the nose closes into a finite body. Nothing was placed in the fluid — the surface is a level set of a function, and it behaves like a wall because nothing crosses it.

Flow net — a free vortex. Two families of curves drawn over the same flow: the streamlines, along which the streamfunction is constant, and the equipotentials, along which the velocity potential is constant. They cross at right angles at every point, because they are the two parts of a single analytic function of position.

A wall made by reflection

Imposing a boundary condition on a plane is work. Putting a mirrored copy of everything on the far side of where the plane would be is not, and the plane then appears on its own — as a consequence of the symmetry rather than as a condition anybody enforced.

The circle plane and the aerofoil plane. A circle with a polar net around it, and the same net after the Joukowski map. Curves that crossed at right angles still cross at right angles everywhere except at the single point where the map's derivative vanishes, and that point is the sharp trailing edge.

From a circle to a wing

The flow past a circular cylinder is known exactly and is of no interest to anybody who wants to fly. A change of variable turns that circle into a wing section — and, because the change of variable preserves angles, it carries the whole solution across with it. Nothing is solved twice.

The pressure coefficient, read off the field. The closed-form surface pressure of a cylinder, and the same quantity computed from the speeds of the solved field at the same points. The coefficient is exactly one where the flow stops and exactly minus three at the shoulder, and those two numbers are properties of the shape rather than of the tunnel.

The number that does not depend on the tunnel

A pressure measured in a wind tunnel is a fact about that tunnel on that day. Divide it by the dynamic pressure and it becomes a fact about the shape — the same at any speed, in any fluid, at any scale, and equal to exactly one where the flow comes to rest.

The fluid a moving cylinder carries with it. Kinetic energy density around a cylinder moving through fluid that is at rest far away. The fluid is not dragged along in a lump: it is pushed aside in front and closes in behind, and the energy in that motion is what has to be supplied to change the body's speed.

The force of getting going

The exact theory says a body moving steadily through an ideal fluid feels no force at all. It does not say the fluid is free. Accelerating the body has to accelerate the fluid too, and the bill for that is exactly the mass of fluid the body displaces.

Two of opposite sign go somewhere. Two vortices of equal and opposite strength. Each is carried by the other's field, both are carried the same way, and the pair travels in a straight line at Γ/2πd forever, keeping its separation exactly. The speed is a consequence of one vortex's field evaluated at the other, and nothing else.

Vortices move each other

A vortex alone in an infinite fluid sits exactly still, forever — its own field is antisymmetric about it and there is nothing at its centre to be carried by. Everything a vortex does, another vortex did, and two of them already exhaust what can be written down.

The pressure, relaxed rather than quoted. The pressure field round a cylinder, obtained by relaxing ∇²p = −ρ∇·(u·∇u) on a body-fitted polar lattice with the exact pressure on the surface and on a far circle. The interior was told nothing except the velocity gradients. It agrees with Bernoulli's closed form everywhere to under two parts in ten thousand of the dynamic pressure, which is the strongest statement this site can make that the elliptic equation is the pressure's own.

Pressure has no speed

Take the divergence of the momentum equation for an incompressible flow and the time derivative disappears, the viscosity disappears, and what is left is Poisson's equation. Pressure is not carried anywhere: it is whatever satisfies an elliptic equation everywhere at once, and that is a statement about a fluid nobody has.

1.5U at the equator, and no drag at all. The exact ideal flow past a sphere, in the meridional plane, with speed contoured behind the streamlines. The fastest fluid is at the equator at 1.5U — a cylinder's is at 2U — and the field is fore-and-aft symmetric, so the pressure integral over the surface gives a drag of -1.2e-16 against a dynamic scale of order one. The streamline spacing here does not measure speed the way it does in a plane flow: the flux between two meridional streamlines depends on the distance from the axis as well, which is why the speed is contoured rather than left to be inferred.

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.

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

What survives being wound up

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

The section the pressure asked for. The designed section over the one it started from, both drawn to their own chords. Asking for 28 per cent more speed over the forward 62 per cent of the upper surface produces a section 15.0 per cent thick against the original's 10, with the extra thickness forward and the camber changed — none of which was asked for, and all of which is what that pressure distribution is. The pale outline is the baseline. The one thing the method cannot be told is where along the chord any of it happens: the speed is prescribed against the circle's parameter, and where a given station ends up is an output of the same solve that produces the shape.

Ask for the pressure, and see what shape that is

A designer knows what the pressure distribution has to do long before knowing what shape does it. Running the problem that way round is possible, it is exact, and it refuses more asks than it grants.

Three contours, one force, three different accounts of it. The same vortex, and the same total force on it, computed by a momentum balance over three contours of the same area. All three give ρUΓ to eight figures. What differs is the bookkeeping: the tall box gets 16 per cent of it from pressure and the rest from momentum flux, the wide box gets 84 per cent from pressure, and the circle gets exactly half. Neither part converges on its own as the contour is enlarged — each falls off like 1/r while the contour grows like r — so the split is a property of the shape of the limit rather than of the flow.

Where the reaction to a wing's lift is

The force on a body can be computed on any contour drawn round it, and the answer is the same every time. How much of that answer is pressure and how much is momentum flux is not — it runs from three per cent to ninety-seven, and the difference is the shape of the contour.

A body with no lift, and a moment anyway

A fuselage in ideal flow carries no lift at any incidence and still tries to turn the aeroplane over. The couple is computable in one line, it is why tails are the size they are, and the line comes from applying the wall condition to a place where there is no wall.

The two bodies the far field cannot tell apart. A circular cylinder and the Rankine oval that has the same doublet strength: 1.17 radii long against the circle's one, and 0.94 tall against its one, with a source and a sink 1.2 apart inside it. On the pale ring, one and a half radii out, the two flows differ by 19 per cent of the disturbance; at six radii by one per cent; at infinity not at all. What a far field records of a body is three numbers — its circulation, its net outflow and its doublet — and nothing else survives the journey.

What the far field remembers

Three numbers survive the journey to infinity — a circulation, a net outflow and a dipole — and nothing else about a body does. Two shapes with nothing in common can therefore make the same flow a few radii away, and the difference between them dies two orders faster than the disturbance either one makes.

Two divergence-free fields with the same boundary conditions. On the left, the exact potential flow round a cylinder moving through fluid at rest. On the right, the same flow with a divergence-free eddy added — one that has no normal velocity on the body or on the outer circle, so it changes nothing about what crosses a boundary. Both fields conserve mass, both satisfy the wall condition, and only one is the flow. Nothing in the drawing says which.

The flow with the least energy in it

Draw a flow that conserves mass and does not go through the walls, and it will look exactly like a solution. There are infinitely many of them and one is the flow. What separates it from the others is not visible anywhere in the picture — it is a number, and the number is an energy.

The mirror that is a circle

A flat wall is made by reflecting everything in it. A round one is made the same way, except that the mirror is an inversion — the image of a point at distance d sits at a²/d, and a vortex acquires a second image at the centre that nothing about the wall requires.

The Kirchhoff flow past a flat plate. A uniform stream meeting a flat plate held across it, with two streamlines leaving the edges and never returning. Between them is a wake of fluid at rest at a constant pressure. The equations solved are the same equations that give d'Alembert's paradox for a closed body, and this flow has a drag coefficient of 0.8798.

Drag in the theory that forbids it

d'Alembert's paradox is a theorem about flows that close behind the body. Stop requiring that, let two streamlines leave the edges and never come back, and the same equations — no viscosity, no vorticity — produce a drag coefficient of 0.8798.

Where the unknown boundary is the known one

A free surface is the hardest kind of boundary — its shape is part of the answer, so the region the problem is posed in is not known until the problem is solved. Draw the same flow in the plane of its own velocity and the shape becomes an arc of a circle, known in advance and exactly.

The momentum of the fluid, against the shape of the region it is added up over. Momentum of the fluid around a cylinder moving through it, divided by the body's hydrodynamic impulse, against the aspect ratio of the rectangle the integral was taken over. Every rectangle has the same area and contains the same body. A tall region gives minus the impulse, a long one gives plus it, a square gives exactly zero, and the limit of a large region is whichever of those the region was shaped like. The momentum of an unbounded ideal flow is not a number.

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.

Inviscid does not mean irrotational

Dropping viscosity gives Euler's equations. Assuming nothing is spinning gives Laplace's — one scalar, linear, unique. The second step is a separate hypothesis about the flow's history, and a flow that fails it is still an inviscid flow with exact solutions of its own.

Hill's spherical vortex. A sphere of rotating fluid travelling steadily through fluid at rest, drawn in the frame that moves with it. Outside the sphere the flow is the ordinary potential flow past a sphere; inside, the vorticity is proportional to the distance from the axis and the fluid recirculates. The two solutions match in value and in slope across the surface, and there is no body anywhere — the boundary is a streamline and nothing else.

The one rotational solution anybody can write down

A sphere of spinning fluid travelling steadily through fluid at rest, with no body anywhere in it — the boundary is a streamline and nothing else. It is exact, it is two lines long, and the reason it is the famous one turns out to be the reason it is the only one a real fluid can settle into.

Two exact solutions of the same problem. Two steady Euler flows in the same square cell with the same boundary condition, differing only in the function relating vorticity to streamfunction. On the left the vorticity is proportional to the streamfunction, which is the textbook cellular flow; on the right it is uniform. Both satisfy the equations exactly. Nothing in the ideal theory prefers either, and at the same peak streamfunction their kinetic energies differ by thirty-one per cent.

The vorticity nothing decides

A streamline that comes from upstream carries its vorticity with it. A closed one comes from nowhere, so nothing determines what it carries — the ambiguity is not one number per body but a whole function. What closes it is a limit, and setting the viscosity to zero gives a different answer from letting it go to zero.

The same flow, from two different physics. Solid lines: the depth-averaged flow between two plates a small distance apart, with an obstacle standing between them. Dashed: the ideal-flow solution for the same obstacle. They are the same field to a part in ten billion, because averaging Stokes flow across a narrow gap gives a velocity that is the gradient of a harmonic potential. The cell has no inertia at all, which is the one hypothesis the ideal theory cannot do without.

The exact theory, drawn by viscosity

Two flat plates a millimetre apart, syrup between them, an obstacle in the gap. The Reynolds number is a hundredth, inertia is absent, and the streamline pattern is the potential flow past that obstacle — exactly, to a part in ten billion. The one hypothesis ideal flow cannot do without is the one this flow most conspicuously breaks.

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.

Four vortices, and the end of prediction. The same three vortices as before with a fourth, weaker one added near the middle. Three point vortices have three independent invariants for three degrees of freedom and cannot be chaotic; four have the same three invariants and one more degree of freedom, and generically are. The energy and the impulses are conserved here to fourteen digits over the whole run, which is what makes the tangle a property of the system rather than of the arithmetic.

Three is the most that can be predicted

Point vortices are the simplest dynamical system fluid mechanics has — no cores, no viscosity, no approximations, four exactly conserved quantities. Three of them are integrable and cannot be chaotic. Add a fourth and the same equations, conserving the same quantities to fourteen digits, stop being predictable at all.

Four corners, and what the flow does in each. The local flow in corners of four different interior angles, drawn from the exact local solution ψ = r^(π/α) sin(πθ/α). The exponent of the speed is π/α − 1 and depends on nothing else: at a right angle the corner is stagnant, at a flat wall nothing happens, and at any angle greater than a straight line the speed has no bound at the corner. The last panel, at 360 degrees, is the flow round the edge of a plate.

Nothing turns a sharp corner

Near a corner the flow is fixed by the angle and by nothing else — not by the size of the corner, not by the flow far away, not by the fluid. The exponent is π/α − 1, and every sharp edge in aerodynamics is the one case where it comes out at minus one half.

A tube opened at the foot of a reservoir. Speed in a two-metre tube fed by a one-metre head, from the instant it is opened. The quasi-steady answer — Torricelli's √(2gh), which is what dropping the time derivative from Bernoulli's equation gives — is reached at the instant of opening, before any fluid has moved. The true answer is a hyperbolic tangent with a time constant of 2L/√(2gh), and it takes 2.4 seconds to come within one per cent. The whole of that transient is one term.

The pressure that depends on the past

Bernoulli's equation for an unsteady flow has a term nobody writes down and a right-hand side that is a function of time rather than a constant. The term is exactly zero once a flow has started and is the whole of the flow while it is starting, which is why it never appears in an answer and is never negligible in getting to one.

The pressure of a sum against the sum of the pressures. Ten points around a cylinder with circulation, with the pressure coefficient of the combined flow plotted against what adding the two flows' separate coefficients would give. Nothing lies on the diagonal. The gap is exactly −1 − 2u_A·u_B/U², an identity checked to the last digit at every point, and it is not small: at one of these points the two answers differ by 1.92, which is more than the whole range of a suction peak.

The one thing that does not add up

Laplace's equation is linear, so flows can be laid on top of one another and almost every classical result is built that way. The two things anybody actually wants out of a flow — the pressure and the force — are quadratic in the velocity, and neither of them adds at all.

The growth rate a discretised sheet has, at every wavelength it can carry. Kelvin–Helmholtz gives a growth rate proportional to the wavenumber and without bound. A sheet represented by N point vortices has pi m (1 − m/N) instead — the same rate at long waves and half of it at the shortest wave the grid carries, with the fastest-growing mode at the grid scale itself. Smoothing the kernel over a length delta moves that mode back to a wavelength the physics chose.

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.

Kirchhoff's rotation rate, which a point vortex does not have. A patch of uniform vorticity bounded by an ellipse turns rigidly at omega a b/(a+b)², a rate that depends on the shape alone. It is largest for a circle, where it is unobservable, and falls away as the patch is drawn out. A point vortex has no shape and therefore no entry on this axis at all.

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.

The singularity a thin aerofoil drives onto its own nose. The Joukowski map has a critical point that maps to a place inside the body, a distance 4 mu²/(1 + 2 mu) from the leading edge. Against thickness that distance is a clean square: a twelve per cent section is analytic only within eight thousandths of a chord of its own nose, and the thin-aerofoil limit is the limit in which the singularity arrives on the surface.

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

The swirl that holds a wave still

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

Five flows past one cylinder, every one of them a solution. Surface pressure round a cylinder in a stream, at five circulations. Each satisfies Laplace's equation, the tangency condition on the body and the condition at infinity, and each has a different lift. Nothing in the problem chooses between them: the domain has a hole in it, so the potential is many-valued and the circulation is a free constant.

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.

The free surface a submerged body leaves behind it, and the flat water in front. The linearised free-surface problem solved as a Fourier integral with a radiation condition. Behind the body a wave train of the wavelength that stands still relative to it, 2 pi U²/g; ahead of it, an amplitude a hundred and twenty times smaller. The asymmetry is the drag: an ideal fluid with a free surface can carry energy away.

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 patches' centroids, against the point-vortex circle. Two circular patches of uniform vorticity, advected by nothing but the velocity their own boundaries induce, over one full co-rotation. Their centroids stay within one per cent of a separation of the exact point-vortex orbit — which they must, because the exterior field of a circular patch is the point vortex's and a harmonic function's area average over a disc is its value at the centre.

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 image system of a wedge of pi/3. The vortex and its images, with the two walls. Reflection in one wall and then the other generates a dihedral group, and the group is finite exactly when the angle is pi over a whole number — here 2n vortices, alternating in sign round the circle, with both walls streamlines to a part in 10¹⁶.

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

The length the limit invents

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

The excess energy is the energy of the difference, exactly. Add any admissible perturbation to the potential flow and its kinetic energy rises by precisely the energy of the perturbation itself — not approximately, and not to leading order. The measured excess and the perturbation's own energy lie on one another to two parts in 10¹¹ across a sixty-fold range of amplitude.

How much more than the least

Of all the flows that conserve mass and stay inside the walls, the ideal one carries the least energy. That is a theorem, and the useful half of it is the part nobody quotes: it says by exactly how much every other flow misses, and the answer is the square of how wrong it looks.

A body of no volume with a finite added mass. Thin an ellipse towards a plate and the fluid it displaces goes to nothing while its broadside added mass does not move at all — it stays at πρa² to the last digit. Whatever added mass measures, it is not how much fluid a body carries with it: the ratio of the two diverges as the reciprocal of the thickness, reaching a million at a thickness ratio of a millionth.

The mass a body has to borrow

Accelerate a sphere through water and it resists as though it were half again as heavy. The half is exact, it is a rational number rather than a measurement, and almost everything a reader infers from it about carried fluid is false.

Four solved flows, and the minimum is on the surface in every one. Sampling the whole exterior of each body on a grid and comparing the lowest pressure found there with the lowest found on the surface. The surface wins by a margin that is not marginal — between 0.18 and 0.56 in pressure coefficient — and it wins for a reason rather than by luck: the pressure of an irrotational flow is superharmonic, and a superharmonic function has its minimum on a boundary.

The lowest pressure is on the body

In an ideal flow the minimum pressure is always on a surface — not usually, not for the shapes people draw, always. The proof is an identity about the velocity gradient, and the identity says exactly which flows are exempt.

One body, three interior representations. An ellipse four fifths as tall as it is long, in a uniform stream, with three sets of singularities inside it: a point doublet at the centre, a ring of them at four tenths of the semi-major axis, and a uniform disc of them. Outside the ring the three produce the identical flow to fifteen figures. Inside, they are not remotely the same field.

The inside a flow does not decide

Every body on this site is built out of singularities that are not there. The exterior flow does not merely fail to determine them — it leaves an infinite family, whose members produce the identical field to the last bit outside and are nothing alike inside, and whose coefficients span five orders of magnitude for one unit free stream.

One material loop, at five stages of being drawn out. A circle of fluid particles carried by four point vortices, drawn at equal intervals over fourteen time units. Its length grows by a factor of six and its shape becomes unrecognisable; the circulation round it does not move at all.

The drift was the instrument

Kelvin's theorem was checked on this site by carrying a loop and watching its circulation move by eight parts in a hundred thousand. That drift is not the flow forgetting. The exact number is a count of what is inside the loop, it does not move by anything at all, and the drift belongs entirely to the two instruments used to measure it.

A vortex sheet rolling up, at five stages. One period of an initially flat vortex sheet with a small perturbation on it, drawn at equal intervals. The perturbation grows, the sheet steepens, and the ends wind into a spiral. Nothing is added to the sheet after the first instant.

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.

There and back again. A blob of a hundred and twenty tracer particles at the start, after four time units of stirring, and after the same four run backwards. The third set is drawn over the first and the worst particle is 1.4·10⁻⁹ from where it began.

Reversible, and unusable

Ideal flow has no arrow of time in it. Run a stirring backwards and the dye comes back — here to 1.4 parts in a thousand million. Nudge the state by a hundred-millionth first and the same reversal returns a blob almost five hundred times further from home than the nudge was large.

Six ways of reaching one speed. Six velocity histories, all starting from rest and all reaching exactly one at the same moment. Two are ramps, two are eased, one overshoots and comes back, and one goes backwards before it goes forwards.

Everything about the start, except one vector

Six ways of accelerating a body from rest to the same speed produce six force histories with nothing in common — peaks spanning a factor of thirty-nine, two of them negative for part of the journey. The impulse left in the fluid is the same ten-figure number in every case, and so is the energy.

Six flows past one cylinder, all of them legal. The tangential speed on the surface for six values of the circulation. Every one of them solves the same equation and lets nothing through the wall; the fastest point on the surface runs from twice the free stream to eight times it.

Nothing in the present picks the flow

Six flows past one cylinder satisfy the same equation and let nothing through the wall, to the last bit of double precision. Their lifts run from zero to 37.7 and their peak suctions differ by a factor of twenty-one. The equations do not choose between them, and the thing that does is the history.

Three answers to one question: what happens after a body is jerked into motion. The force following a step change in a body's velocity, for three models. The ideal one is a spike at the instant and nothing afterwards. The viscous one falls as the inverse square root of time and never reaches zero. The compressible one holds while the signal is still crossing the body and then settles.

The theory with no memory in it

Laplace's equation has no time in it, so an ideal flow's response to a body being jerked into motion is instantaneous and complete. Its indicial kernel is a spike and nothing afterwards. Beside it sit the two kernels that are not, and the comparison says which ingredient every memory in this collection came in through.

Three lobes, then a filament. An ellipse of aspect ratio 4 with a three-lobed bump of three thousandths, as contour dynamics carries it, drawn in the frame turning with the undisturbed ellipse at t = 0, 30 and 42. By t = 30 the bump has grown to a visible three-fold asymmetry — one end fattened, the other thinned — and by t = 42, about a turn and a tenth of the ellipse, the thinned end is being drawn out into a filament. The march is stopped there, while the area is still conserved to a few parts in a thousand; resolving the filament needs a contour that adds nodes, which this one does not.

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.