Field

Ideal flow

The exact theory of a fluid with no viscosity — closed-form, elegant, and predicting no drag at all. Its failure is the most useful thing in the subject.

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

The ideal flows usually drawn first are 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 in potential flow 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.

Three aircraft, and the only thing the drag can see. The cross-sectional area against station, for a clean body, the same body with a wing added, and the body cut away where the wing is. The wave drag depends on this curve and on nothing else about the three shapes — so the middle one pays for its lump and the third does not, although the third has the same wing.

The drag that can only see one curve

A slender body's far field is a line of sources of strength A′(x), and nothing else about its shape reaches it. So its supersonic wave drag depends on the cross-sectional area distribution alone — and a fuselage cut away where the wing joins it can carry the wing for nothing.

The speed past the flank, in the cell and in the ideal flow it is meant to show. The gap-averaged speed along the top of a cylinder, against the distance from its surface in cylinder radii, for three cells — the cylinder's radius 10, 35 and 100 times the Brinkman length h/√12 — and for ideal flow. Ideal flow slides along the wall at twice the stream speed. The cell's flow must stop at the wall and does so in a layer a few Brinkman lengths thick, outside which it rejoins the ideal curve.

The cell draws a larger cylinder

A Hele-Shaw cell draws the streamlines of ideal flow past an obstacle and cannot obey the one rule ideal flow breaks: the fluid must stop at the obstacle's wall. It does stop there, in a layer a third of the gap thick, and far away the whole correction amounts to one thing — the cell is drawing ideal flow past a cylinder one layer-thickness too big, and the obstacle has exactly that cylinder's drag.

The same thickness, drawn as a body of revolution and as a wing section. The surface speed along three prolate spheroids of thickness ratio a half, a quarter and a tenth, and along the plane ellipses of the same ratios, in units of the stream speed. Each plane section speeds the flow over its middle by exactly its own thickness ratio. The body of revolution of the same ratio does so by much less — at a tenth, by two per cent where the ellipse manages ten.

Thickness costs a fuselage far less than a wing

A wing section a tenth as thick as it is long speeds the air over its middle by ten per cent. A body of revolution with the same proportions speeds it by two. The difference is not a detail of shape: a plane section pays for its thickness linearly and a body of revolution pays quadratically, and that one exponent is why a fuselage reaches the speed of sound on its surface long after its wing.

Two eddies that stand still behind a cylinder in ideal flow. A uniform stream past a circular cylinder with a pair of opposite vortices standing 2 radii behind its centre and 0.859 either side of the axis, each of strength 10.31 in units of the stream speed times the radius — Föppl's equilibrium. The shading is the stream function: the fluid between the vortices and the cylinder circulates in two closed eddies and never leaves, while the stream divides round the whole assembly as if it were one longer body.

Two eddies can stand behind a cylinder, but not for long

Ideal flow, which has no viscosity and no wake, can still hold a pair of eddies standing behind a cylinder — at any distance behind it, with a strength fixed by the distance. They cost the cylinder nothing. And they cannot stay: nudged sideways by a thousandth of a radius, the pair grows its displacement exponentially and leaves, which is the first step of shedding a wake.

The slow bending wave, and where the cut-off stops describing it. The frequency of the slowest bending wave on a Rankine vortex — a helical wobble of the whole core that turns against the flow — against the wavenumber times the core radius, from Kelvin's exact relation and from his long-wave formula, which the cut-off method used for vortex pairs reproduces. They agree for long waves. Past ka = 1.44 the long-wave formula reverses sign and the exact wave does not, which is why a cut-off model's instabilities at such wavelengths are not real.

A vortex is a waveguide

A spinning core is stiff in a way still fluid is not, and it carries waves along its length: an infinite family of them for every pattern round its axis, travelling at up to 0.83 of the swirl speed at its edge. The slowest is a helical bend that turns against the flow, and it is the wave every model of a bending vortex has been borrowing without saying so.

Two cylinders side by side pull together. The ideal stream past two cylinders side by side with a gap of a fifth of a radius between them. The stream squeezes through the gap faster than round the outsides, the pressure in the gap is lower, and each cylinder is pulled towards the other with a force of 3.6 ρU²a — while the pair together feels nothing at all.

The paradox is for the pair, not for each body

D'Alembert's paradox says a body in a steady ideal stream feels no force. Put two bodies in the stream and the theorem still holds — for the pair. Each cylinder on its own is pushed: side by side they pull together, with a force that grows without limit as the gap closes, and one behind the other they push apart, so that the front cylinder is driven upstream into the stream that is flowing past it. The forces are equal and opposite at every spacing and every angle, and their sum is exactly the zero the paradox promised.

Two shapes for each strain, and one of them holds. The strain rate, over the vorticity, at which an elliptical patch of aspect ratio λ stands still — Moore and Saffman's relation — rising to its maximum of 0.1501 at λ = 2.89 and falling again. Below the maximum there are two steady shapes for each strain: a rounder one, on which every disturbance computed here stays bounded, and an elongated one, which comes apart. Above it there is no steady shape at all.

A strained vortex holds until it has no shape to hold

A patch of vorticity in a strain has two steady shapes for every strain below 0.150 of its vorticity, a rounder one and a longer one, and none above. The longer one comes apart at the slightest nudge. The rounder one, computed with disturbances of two, three, four and five lobes, never does: it nods and holds right up to the strain at which it ceases to exist. So the existence limit is the real limit, and past it a vortex is not shattered but stretched — lingering first near the shape it has lost, for a time that grows as the fourth root of how close the strain is to the limit.

Released, the side-by-side pair collides and the tandem pair parts. The distance between the centres of two cylinders released from rest in an ideal stream and free to move along the line joining them, against time in radii over the stream speed. Side by side they are drawn together and collide — neutrally buoyant ones released four radii apart after 6.5, bubbles, with no mass of their own, after 4.82. In tandem they are pushed apart and keep going.

A pair set free in a stream collides or parts

Two cylinders held in an ideal stream pull together side by side and push apart in tandem. Let them go and the forces become a motion, and the motion has a law of its own: the stream's force on each is exactly the slope of how large the pair looks from far away, so the pair moves to look bigger. Side by side that means closing, and the fluid squeezed out of the gap costs so little that nothing stops them: released four radii apart they collide in six and a half radii of stream. In tandem it means parting, for good.

$The fuselage's overspeed reaches along the wing. The extra axial speed the fuselage alone gives the air beside it, as a fraction of the flight speed, against distance out along the wing from the fuselage's axis in fuselage radii, level with its thickest section, at Mach 0, 0.6 and 0.8. At the fuselage's side it is 2.1 per cent at low speed and 2.5 at Mach 0.8; it falls only to two-thirds of that two radii out and a quarter at five — a long body's middle " + "spreads like a line of sources, not a point — and the section beside it is already running at twelve per cent over at low speed and twenty over at Mach 0.8.$

A fuselage lowers its wing root's critical Mach number

A slender fuselage barely disturbs the air: two per cent of overspeed at its side, against the twelve a wing section of ordinary thickness makes. At the wing root the two are added, and two per cent on top of twelve moves the root's critical Mach number down by more than a hundredth — as much as making the whole wing thirteen per cent thicker. The fuselage's overspeed also fades far more slowly along the span than its size suggests, because the middle of a long body spreads its disturbance like a line, not a point.

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

At thirty degrees the Earth keeps time with the sea breeze

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

A free pair turns, and forty-five degrees decides

Two cylinders in an ideal stream move to look bigger to it: side by side they close, in tandem they part. Free to turn as well, a staggered pair swings towards side by side — and does not stop there, because nothing in an ideal fluid damps the swing. Whether it then collides or flies apart is decided, far apart, by one angle and a theorem: the stream's pull on the pair falls as the inverse square of the spacing, and for such a force the sign of the energy alone decides, which changes at exactly forty-five degrees.

$A waist moves the fuselage's overspeed off the root chord. The fuselage's axial perturbation velocity at its own surface, as a fraction of flight speed, along its length at Mach 0.8, with the wing's root chord between the two rules: the plain fuselage, which adds 0.0251 everywhere along the chord; a narrow waist 6.2 per cent deep that cancels it at mid-chord and raises it at the chord's ends; and a wide one 31 per cent deep that leaves none of it positive along the chord and lifts it on the body ahead of and behind the wing.$

A waist moves the overspeed and cannot remove it

A fuselage adds two per cent of overspeed at the wing root, and a waist — a local narrowing of the body where the wing joins it — is the obvious cure. Slender-body theory says exactly what a waist can do. Its sources and sinks sum to nothing, so the velocity it adds along the body integrates to zero: it cannot remove the overspeed, only move it. Moved off the whole root chord it needs a third of the fuselage's cross-section at the wing, a sixth of what the transonic area rule would take, and it lands on the fuselage just ahead of and behind the wing, where there is room for it.

Drag makes every free pair meet

Two cylinders set free in an ideal stream swing towards side by side like a pendulum, and whether they collide or part is fixed, far apart, by forty-five degrees. Give each a drag on its motion relative to the stream and the pendulum stops swinging — but it does not stop the pair. A drag coefficient of order one turns the swing into a creep, lets no pair get far from where it started, and brings every one of them round to meet near side by side.

$The free waist is flat-bottomed, and follows the sweep aft. The cross-section each waist removes along the body, as a fraction of the fuselage's greatest, for the root strip unswept, at 30° and at 45°. The linear programme's waists have flat bottoms and steep sides: depth is spent only where the strip needs it. With sweep they lengthen aft rather than deepen — 24.8, 24.3 and 26.9 per cent. The best single dents, dashed, must widen and deepen instead: 39.8, 45.7 and 52.6 per cent.$

A free waist needs half the depth, and the sweep costs it nothing

A cosine-squared dent in a fuselage cancels its overspeed at an unswept wing's root for about a third of the fuselage's cross-section, and nothing can remove the overspeed, only move it. Let a linear programme choose the waist's shape instead, and half of that third turns out to have been the dent's fault. The free waist is flat-bottomed and steep-sided, and for a swept root it simply runs aft with the chord, where a single dent has to widen and deepen.

A wake throws a following cylinder out to the side, and the pair still meets. One cylinder's centre relative to the other's, the stream from left to right, for pairs released six radii apart at 2°, 5°, 10° and 20° from tandem at Re = 200. Without wakes (faint) the near-tandem pairs drift apart, out to 13 radii, before the drag lets the turn bring them in. With wakes the follower first drafts in, is thrown sideways out of the leader's wake, and never gets beyond 6.26 radii. Every pair touches, near side by side.

A trailing wake hurries a pair together

Two cylinders free in a stream and damped by their drag always end up meeting near side by side, and the obvious candidate for what holds real pairs apart is their wakes: a body beside another's wake is pushed towards the faster fluid, away from it. Give each cylinder a wake that carries its drag and the push is there — but it points the wrong way. It throws a following cylinder out of the leader's wake and round towards side by side, where neither is in the other's wake and nothing opposes the stream's pull. Near tandem the wakes make the pair meet seven to sixty times sooner, and the only balance they create is a saddle.

A strained vortex loses its ring at the limit of what it holds

A uniform patch of vorticity survives a strain up to 0.150 of its vorticity and no further. A real vortex is not uniform: its vorticity falls off outward, and in a strain its weak outer layers are drawn off while its core survives. The obvious way to predict how much survives is to apply the uniform limit layer by layer, each layer to its own vorticity. It is wrong by up to a factor of three. Built from a core inside a ring and followed by contour dynamics, the ring is stripped not at the limit of its own vorticity but at the limit of all the vorticity inside its edge, and a little after it.

The water climbs the body, and the wetted width outruns the drawing. The wetted half-width against penetration for a wedge of 10° deadrise and a circular cylinder of unit radius: where the body crosses the undisturbed level, von Kármán's width, and where the risen water meets it, Wagner's. For the wedge Wagner's width is π/2 = 1.571 times the geometric one at every depth; for the circle it is √2 = 1.414 times, at small penetration.

Water climbs a falling wedge, and the load comes from the climbing

A hull that strikes the sea sets the water under it moving, and the force is the rate at which it does so. The obvious estimate measures the wetted width where the hull crosses the undisturbed surface. But the water does not wait: pushed aside, it rises up the hull and wets it sooner, a factor π/2 wider for a wedge and √2 for a round bottom. The force carries that factor twice, and the peak pressure, where a thin jet leaves the hull, carries it squared on a cotangent that grows without bound as the bottom flattens.

Long waves grow, short ones are held flat, and one in between grows fastest. The growth rate of a ripple on the interface where air displaces oil in a Hele-Shaw cell, against its wavenumber, at displacement speeds of 0.5, 1 and 2 mm/s. The viscosity contrast drives every wavelength at a rate proportional to its wavenumber; surface tension holds back short ones as the cube. At 1 mm/s the cut-off is 0.245 per mm and the fastest wavenumber 0.141 per mm, a wavelength of 44.4 mm, growing at 0.0942 per second.

The fastest finger is set by the gap and one number

Push air into oil between two glass plates and the interface breaks into fingers. The averaged law of the cell makes the instability exact in its linear stage: every wavelength is driven in proportion to its wavenumber, surface tension holds back the short ones as the cube, and the fastest finger is π gap widths divided by the square root of a capillary number. A narrow channel stays flat; a wide one chooses how many fingers to make; and a growing bubble of air makes more of them the larger it gets.

$A wedge's slam peaks when the water it has set moving weighs a fifth of the hull. Slamming force per metre of a 10° wedge of half-beam 1 m entering at 5 m/s, against penetration as a fraction of the depth at which the chine wets, for hulls of 0.3, 1 and 3.73 times the added mass of their full width, and at constant speed. The constant-speed force grows to the chine, 700 kN/m; each finite hull's peaks where its recruited water is a fifth of its mass, at 99.2, 181 and 350 kN/m.$

A light hull slams softly, and a round one does not

A hull striking water slows as it sets the water moving, and the two share one momentum. For a V-shaped bottom that caps the slam: the force peaks when the water recruited weighs a fifth of the hull, and a hull as heavy as the water under its whole width feels a quarter of the force a constant-speed calculation predicts. For a round bottom it caps nothing, because the peak comes at the instant of contact, before the hull has slowed at all.

The meniscus shortens the fastest finger and its film lengthens it, and they cancel near Ca = 0.01. The fastest wavelength over the uncorrected π b/√Ca, against capillary number, with the in-plane tension reduced to π/4 alone, with the speed-dependent transverse jump alone, and with both. The reduced tension shortens every finger by √(π/4) = 0.8862; the film's jump lengthens it, more at higher capillary number; together they cancel at Ca = 0.0131, shorten the finger 5.8% at 10⁻⁴ and lengthen it 6.1% at 0.3.

The meniscus slows the finger and hardly moves it

Air pushing oil through a narrow gap fingers at a wavelength the gap and one number set, on the assumption that surface tension acts only on the interface's curve in the plane. The interface is really a meniscus across the gap, leaving a film behind, and Park and Homsy's correction for it does two things that nearly cancel in the wavelength: it cuts the tension to π/4 and adds a drag that grows with speed. At bench speeds the fastest finger barely moves; it grows a quarter more slowly, and a channel goes unstable at four-fifths of the speed.

A bottom panel overshoots its static stress, and a lighter hull's panel is loaded less. The largest bending stress anywhere on the panel against time from first contact, for a 10° wedge entering at 5 m/s at constant speed and as free hulls of the full width's added mass and three tenths of it; solid, the panel's own response, dashed, the static stress the same pressure would give a massless panel. Constant speed: 361 MPa against 287 static; Hull = full width's added mass: 245 MPa against 195 static; Hull = 0.3 of it: 132 MPa against 106 static. The water reaches the panel at 4.5 ms and covers it at 11.2 ms at constant speed; the wetted panel's period is 7.84 ms.

A hull bottom feels the slam it has time to answer

The slamming pressure on a V-bottom is a narrow ridge at the spray root, a megapascal high, that sweeps across each panel of plating in a few milliseconds — about the panel's own period. So the panel neither feels the ridge's height nor ignores its speed: it overshoots the static stress by up to half, a rule that designs for the peak pressure asks for half as much plating again as it needs, and a light hull's slower, softer slam is gentler on its plating twice over.

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

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

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

On a flat bottom the plating's own give slows the water climbing it and takes a seventh off the slam; on a deep V, almost nothing. At the impact speed that loads the one-way panel to 200 MPa at each deadrise: left, the coupled panel's peak stress over the one-way panel's, against deadrise; right, the same against the panel's largest deflection over its rise across its width, w/(ℓ tan β). 4°: 2.27 m/s, ratio 0.847, deflection 3.99 mm; 6°: 2.9 m/s, ratio 0.894, deflection 4.24 mm; 8°: 3.44 m/s, ratio 0.934, deflection 4.44 mm; 10°: 3.94 m/s, ratio 0.972, deflection 4.61 mm; 15°: 4.99 m/s, ratio 1, deflection 4.81 mm; 20°: 5.8 m/s, ratio 0.983, deflection 4.69 mm.

Plating that gives way slows the water that slams it, but only on a flat bottom

The slamming pressure on a hull's bottom is a ridge that rides at the edge of the wetted width, and how fast that edge moves depends on how far the water has to climb. A panel of plating that bends inward under the load gives the water further to climb, and while it is moving away it slows the climb itself. Fed back into Wagner's condition, that takes a seventh off the peak stress of a panel on a four-degree bottom loaded to the edge of what aluminium can bear, and almost nothing on a deep V — because what matters is the plating's deflection against the bottom's own rise across the panel.

Stalled from its leading edge, a flat plate keeps a quarter of its attached lift slope. The normal force on a flat plate over the dynamic pressure on its chord against incidence: Kirchhoff's stalled flow, by quadrature through its hodograph, with Rayleigh's closed form 2π sin α/(4 + π sin α); and the attached flat plate's 2π sin α. At 5° the stalled plate carries 0.128 against the attached 0.548; at 20°, 0.423; at 90°, 0.8798, Kirchhoff's drag.

A stalled plate keeps a quarter of its lift slope and moves its centre aft

Kirchhoff's free streamlines gave a plate held across a stream a drag with no viscosity in it. Tilt the plate, let the flow leave both its edges, and the same construction is a wing that has stalled from its leading edge. Rayleigh solved it in 1876, and the hodograph confirms him to seven figures: near zero incidence the stalled plate lifts at a quarter of the attached plate's rate, its centre of pressure sits at five sixteenths of the chord instead of a quarter, and its force is normal to it, so its lift-to-drag ratio is the cotangent of its incidence. A real stalled wing lifts more, by a factor the wake's pressure decides.

Added linearly, the parts give the section's suction peak a quarter too weak; carried through Bernoulli, most of it comes back. Pressure coefficient, suction upward, on the upper and lower surfaces of the 12 per cent section with 4 per cent camber at 4°: exact; the parts' disturbance speeds summed and the pressure taken linearly; and the same sum carried through Bernoulli. At a tenth of the chord on the upper surface the exact −Cp is 1.39, the linear sum 1.06 and the Bernoulli sum 1.34.

Thickness carries no load of its own and still moves the load forward

Thin-aerofoil theory builds a section's pressures from three pieces computed apart — its thickness, its camber, its incidence — and adds them. Every piece can be computed exactly on a Joukowski section, and so can the section they add up to. Added, the parts give a suction peak a quarter too weak and a loading that sits too far aft, and they miss nine per cent of the lift: thickness, which lifts nothing by itself, multiplies what camber and incidence make by 1 + 0.77 t/c. Carry the summed velocities through Bernoulli instead of adding the pressures, and most of the error goes; the part that stays is the cross term between thickness and lift, and it is first order in the lift.

Up a step the pressure rises to a stagnation at the foot and falls without limit at the lip, and the face feels both. The pressure coefficient along the wall of a right-angled forward-facing step of height h in ideal flow, against distance along the wall from the foot of the face in step heights: the floor ahead of it, the face, and the top beyond the lip (clipped at −3, where it goes to minus infinity). The face is pushed back by 0.6366 of a dynamic pressure times its height over its lower part and pulled forward by exactly as much near its lip.

A step's face is pushed at its foot and pulled at its lip, equally

Ideal flow running up a step has two corners in it, and the local theory of corners says what each does: the inside corner at the foot brings the flow to rest, the outside corner at the lip sends it round at infinite speed. Solved exactly, the step's face feels both — a push of 2/π of a dynamic pressure times its height spread over its lower four-fifths, and a pull concentrated at the lip — and they cancel exactly, at every ramp angle, because one reflection of the mapping exchanges the two corners. A step has no drag in ideal flow for the same reason a flat plate's lift has no drag: a finite force carried by an infinite speed.

On a NACA section thickness multiplies the lift by more than on a Joukowski section, by nearly one plus its thickness. The lift of a NACA four-digit section by panels over thin-aerofoil theory's, against its thickness: the 2412-type at 4°, and the symmetric section's lift slope over 2π; beside the Joukowski law 1 + 0.77 t/c. At 12 per cent the cambered section's factor is 1.115 and the symmetric slope's 1.101, against Joukowski's 1.092; at 18 per cent, 1.174 and 1.152.

A NACA section multiplies its lift by nearly one plus its thickness, and a nose correction cannot supply it

On Joukowski's exact sections, thickness — which lifts nothing alone — multiplies the lift of camber and incidence by 1 + 0.77 t/c. Solved by panels, the NACA four-digit sections that real wings use do more: their lift slope grows as about 1 + 0.84 t/c, and a cambered 2412 lifts 11.5 per cent more than thin-aerofoil theory says. The more of the thickness sits near the nose, the more it multiplies. Added as thin-aerofoil theory adds them, the parts lose a quarter of the suction peak and put the centre of pressure two hundredths of the chord too far aft. Riegels's nose factor, the classical repair, makes the nose speed finite and moves the peak, and adds no lift at all: the missing lift is not a nose effect.

A block's front face is pulled upstream by its own lip

A forward-facing step's face is pushed at its foot and pulled at its lip by exactly equal amounts. Give the step a back — make it a block on the floor — and the balance goes. The two lips, a short top apart, each sit in the other's fast flow, and the suction at the front lip grows while the push at the foot hardly changes: a block as long as it is tall has its front face pulled upstream by three tenths of the dynamic pressure times its height, a block a quarter as long by two thirds. The back is pulled downstream by exactly as much, so the ideal block still has no drag; but every force that ideal flow assigns is now on the wrong face for separation to undo.

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