Potential flow — where it appears
Named by 59 essays across 6 fields — each of them below, with the objects they name alongside it.
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.
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.
The number on a streamline is a flow rate
Streamlines get drawn as decoration — curves the flow follows, spaced however the plotting looked best. Each one carries a number, the difference between two of those numbers is the volume of fluid passing between them every second, and it does not matter what route the measurement takes.
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.
How much circulation is too much
Spin a cylinder faster and it lifts harder, with no limit in the equations. What does have a limit is the flow's willingness to stop anywhere on the surface — the two points where the air is at rest slide round towards each other, meet at the bottom, and leave the body altogether.
A ball that swings without spinning
A cricket ball curves in flight with no spin about any useful axis. The mechanism is not the Magnus effect; it is a seam tripping the boundary layer on one side so that side lets go later. Which way the ball then goes depends on one borrowed number, and this site's own inviscid solver supplies the value that gets it wrong.
Whether the droplet turns
The air goes round the wing. Whether what is carried in it goes round too is decided by one number — and below a critical value of that number the body collects nothing at all, however many droplets are thrown at it, because the flow turns every one of them in time.
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.
What a flap does, and what it does not
Lowering a flap gives a wing lift at an incidence where it had none. It does not make the wing more responsive to being pitched — the lift curve moves sideways and its slope does not change, and those are two quite different things to buy.
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.
How many things a flow must be told
The equations of motion do not have one answer. They have as many as the conditions on the edge allow, and the number of those is decided by the order of the equation — which is why viscosity does not make the same problem harder, it makes a different problem.
When a body tears the water
A propeller blade moving fast enough pulls the pressure at its own surface below the vapour pressure of the liquid, and the water boils at whatever temperature it happens to be. Where that happens is decided by an inviscid calculation of the pressure along the blade.
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.
The cushion that is not there
Ground effect is usually explained as air trapped under the wing and compressed into a cushion. At the heights an aeroplane actually flies at, the air under the wing is moving at four fifths of flight speed and most of what the ground gives is a drag saving the story never mentions.
The mean is not the flow
Average an unsteady flow and the result is a new object with its own properties, and it is not a solution of anything. An inviscid stream oscillating about zero has a mean velocity of exactly nothing everywhere, a mean pressure that reaches minus two dynamic pressures at the shoulders, and a missing term in its own momentum equation that can be written down in closed form.
The wing the equation is really solving
The Prandtl–Glauert rule is usually quoted as a factor on the answer. It is a change of shape — and in three dimensions the shape it changes is the aspect ratio, so a short wing is far less affected by compressibility than a long one.
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.
The drift in a wave that has none
The velocity at any fixed point under a passing wave averages to exactly zero, and every parcel of water in it moves steadily forward anyway. The orbits do not close, they miss by the same amount every time, and the missing amount is the square of the steepness times the wave speed.
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.
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.
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 effect that is real, and where it stops
The Coandă effect has a genuine mechanism, and it is the same equation that makes the suction on a wing. What separates the two is where the integral comes from: a wall jet generates all of its pressure deficit inside a layer two per cent of the radius thick, and a cylinder in a stream needs a third of a radius for half of it.
The instrument in the answer
A model in a wind tunnel is not a model in the sky. The walls are supplied by an infinite row of reflections, the stream at the model is faster than the tunnel's own instruments report, and the correction is not an empirical fudge — it is a series with a closed form.
The 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 drop that is not a tear
A falling raindrop is flattened along the direction it is going, by the pressure of the air passing it rather than by its own weight, and the group that decides is the Weber number. The teardrop of every illustration has the wrong symmetry entirely — there is no up in the problem it is drawn for.
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.
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.
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 walls are in the answer
A wind tunnel measures a wing in a box the aeroplane will never fly in, and the box is worth a fifteenth of the induced drag. The correction is exactly an eighth for a closed circular section and exactly minus an eighth for an open jet, and the sign is the whole argument.
The body the outer flow actually sees
A boundary layer lets less fluid past than an inviscid one would. The outer flow can be given exactly the same reduced flow rate by leaving the fluid inviscid and moving the wall out — so the potential flow that matters is not the flow past the body, but the flow past the body plus a thickness the boundary layer computes.
The 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 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 air that breaks a siphon nothing else can
A running siphon's heights cannot break it, and its friction only postpones the moment it is most exposed. What does break a siphon that has run for a day is the air dissolved in its water, which the crown's low pressure leaves the water carrying far more of than it can hold — and which gathers only once the flow is too slow to carry a bubble away.
The 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.
A cushion that changes its physics
A plate closing on a plane is resisted by the fluid it has to squeeze out, and the resistance is two different forces with two different laws — one from the fluid's inertia and one from its viscosity. They hand over at a gap of twenty-four kinematic viscosities per unit of closing speed, which for a wing in air is a third of a millimetre, and the two films disagree about whether the plate ever lands at all.
One formula, and it does not ask what the shape is
Kutta–Joukowski gives the lift of any two-dimensional body from one number. Six bodies with nothing else in common are put at that number here and come out with identical lifts — and with pitching moments, load distributions and suction peaks that are not even close.
The borrowed mass the boundary decides
A body accelerating near a solid wall has to squeeze out the fluid between them, and borrows more mass than it would in the open. The same body accelerating near a free surface, or inside an open-jet wind tunnel, borrows less. The fluid, the body and the speed are identical, and what reverses the answer is the one thing each boundary is allowed to tell the flow.
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.
The vorticity a clean surface cannot refuse
A clean bubble's surface cannot hold a shear stress, and it is easy to conclude that it makes no vorticity. On a curved surface it must carry exactly 2κu — three times the speed over the radius at a sphere's equator, whatever the Reynolds number. That is so much weaker than a rigid wall's that the flow stays irrotational to leading order, and the bubble's drag is the dissipation of that irrotational flow: 48/Re, three to ten times below a rigid sphere's.
The 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.
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.
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.
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.
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.
One coefficient, two errors
A closed tunnel's walls push a measurement one way and an open jet's push it the other, so a wall somewhere between them should push it neither. There is such a wall, its coefficient is a length, and it nulls the blockage at 1.184 tunnel half-heights — and the streamline curvature at 1.590.
The borrowed mass that goes negative
A body's added mass is taught as the water it carries with it, fixed by its shape. Put the body under a surface that can make waves and the amount depends on how fast it is shaken, runs from a wall's value to a free boundary's and past both, and for a circle close enough to the surface falls below zero.
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.
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.
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.
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.
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.
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 upper parcel leads by the circulation
Transit time looked like a quantity that could not be measured: a parcel released near the dividing streamline crawls past the nose for as long as one likes. The difference between two parcels' transit times, one either side of that streamline, does not crawl. It converges, and on a thin section it is the circulation divided by the square of the speed — the lift, measured in seconds. The upper parcel arrives first by that much, and behind the wing the two never close the gap.
The air shortens a jet's fastest ripple, and the drops follow
A jet in a vacuum breaks into drops nearly twice its own width, whatever its speed. A jet in air does not, and the reason is a pressure the air puts on its surface: flowing over a rippled jet it is faster over the crests and its pressure lower there, which pulls them further out. That pull lets ripples shorter than the jet's circumference grow, shortens the fastest one, shrinks the drops, and puts a ceiling on how long a fast jet can be — at the gas Weber number where the measured break-up regimes change.
A nose at incidence makes its vorticity along the flow
A wall makes vorticity along its isobars, and on a swept wing that sends part of it along the flow. A fuselage at a few degrees of incidence has no sweep anywhere, and still most of the vorticity its wall makes points along the flow — a fifth for a body twice as long as it is wide, four-fifths for one eight times — because at incidence its pressure varies round the body faster than along it. A sphere, at any incidence, makes none. The lee vortices begin at the wall.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitBoundary conditionAdded massd'Alembert's paradoxCirculationDoubletMeasurementLaplace's equationPressure coefficientConformal mapKinetic energySuperposition