Separation — where it appears
Named by 54 essays across 9 fields — each of them below, with the objects they name alongside it.
When the flow lets go
Every body asks the air behind it to slow down and climb back up to the pressure it started at. Sometimes the air cannot, and the moment it refuses is separation — the source of most drag, the cause of stall, and the reason a golf ball has dimples.
The two theories, side by side
The exact solution and the real flow, for the same body in the same stream. One is beautiful and predicts nothing has drag; the other is approximate and has a wake in it. Where they agree and where they part is the whole map of the subject.
A loss with no viscosity in it
Where a pipe suddenly widens, energy is destroyed. The amount is exact, it has been known since 1766, and the derivation never mentions viscosity, Reynolds number or roughness — because momentum does not care where the energy went, only that it left.
How much uphill a layer can take
A boundary layer running into rising pressure is climbing a hill on the last of its momentum. There is a definite steepness at which it can no longer do it, and the number is not a rule of thumb — it is where a family of solutions stops existing.
The pocket on top of the wing
An airliner cruising at Mach 0.85 has subsonic flow almost everywhere and a patch of supersonic flow over its wing. Closing that patch takes a shock, the shock separates the layer beneath it, and the correction that predicts all of this also predicts, in the plainest terms, where it stops being true.
Where the straight line stops
Ideal flow will report a lift coefficient at forty degrees of incidence without complaint. What ends the lift curve is the boundary layer refusing to follow the surface, and the estimate of when that happens joins two solvers that have nothing else in common.
The drag that falls as it speeds up
There is a band of speeds in which a smooth ball experiences less drag the faster it goes. Not a smaller coefficient — a smaller force. Dimples move that band down to where a golf ball actually flies, and they do it by making the friction worse.
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.
The gradient that does both
One line of the boundary-layer equations at the wall says the profile's curvature there equals the pressure gradient. That single sign causes separation and causes instability, and it causes the instability a long way before it causes the separation.
The cost of going turbulent
A turbulent boundary layer costs several times the friction of a laminar one, and the multiple is not a constant — it rises with Reynolds number, because the two laws have different exponents. That is why laminar flow is worth more on a long fast surface than on a short slow one.
The count a pattern cannot break
A picture of a flow has stagnation points in it, and they are not free to be arranged as anybody likes. Their kinds and their number obey an integer constraint that has nothing to do with the equations of motion — and an incompressible flow in a plane is allowed only two kinds of them in the first place.
An angle, not a speed
The number is printed in the handbook, marked in white on the airspeed indicator and used in every briefing, and the wing has no way of knowing it. A wing stalls at an angle. The speed at which an aeroplane reaches that angle is an answer with four other variables in it, and every one of them moves.
The street a steady calculation cannot draw
The alternating wake behind a cylinder is the most photographed structure in fluid mechanics, and a steady solution does not produce one. What can honestly be drawn instead is a model of it — and the model settles one thing exactly, which is the spacing.
Where vorticity comes from
Every scrap of vorticity in a flow past a body entered through its surface, and the rate at which it enters contains no viscosity at all — it is the pressure gradient along the wall. A flat plate makes none, and a closed body makes exactly as much of each sign.
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.
The eddies nobody stirs
A slow flow past the mouth of a sharp corner does not simply fail to get in. The corner fills with an infinite sequence of counter-rotating eddies, each about three times smaller and sixteen hundred times weaker than the last, and the whole structure is decided by one complex number.
Lift out of a failure
Separation is what ends an ordinary wing's lift curve. A slender delta with sharp leading edges separates on purpose, rolls the shed sheet into a pair of vortices above its upper surface, and takes most of its lift from the suction they induce — with a curve that climbs to forty-nine degrees.
Work out of a change of swirl
The work a rotor does per unit mass is the blade speed times the change in swirl, and that is all of it — no blade shape, no pressure, no efficiency, no gas properties. It is the same equation for a pump, a compressor, a turbine and a fan, and it follows from angular momentum on a box with nothing assumed about the inside.
A slot is not a nozzle
The gap between a wing and its flap is supposed to blow fast air into a tired boundary layer. A solver with no boundary layer in it at all — no viscosity, no wake, no mixing — produces most of the lift increment anyway, and produces it on the element nobody moved.
What a code says to a wall
A calculation that cannot afford to resolve the viscous sublayer has to tell the wall something else instead, and what it tells it is the law of the wall — an asymptotic result, applied at one grid point, on the assumption that the point lies in a region the calculation has not checked exists. Where it does, the answer is exact. Where it does not, the friction is out by tens of per cent, and refining the grid makes it worse.
Which part of a wing stalls first
A wing has one lift coefficient and its sections have a hundred, and no section is at the wing's. Which of them runs out first is decided by the planform, it decides whether the pilot keeps the ailerons, and on any wing that was close to elliptic the standard fix costs span efficiency for every degree of it applied.
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 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 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.
A roll that feeds itself
A spin is routinely described as a stall that got worse, and it is not a stall at all in the sense of an angle rather than a speed. It is autorotation — a roll that sustains itself because past the peak of the lift curve the down-going wing makes less lift rather than more — and the arithmetic says it begins a little past the stalling angle rather than at it.
A stall that is a place
A stall is an event, and the usual accounts treat it as one — a boundary reached, a damping lost. A high tailplane makes it something else. Swept into the wing's wake, the tail loses the download that held the nose up, and the aircraft finds a second stable trim point thirty degrees past the stall from which the elevator cannot bring it back.
A finite force from an infinite speed
Pressure on a flat plate can only act along the plate's normal. Kutta–Joukowski says the force is perpendicular to the free stream. Those two directions differ by the angle of attack, and the discrepancy is made up at a single point where the velocity is infinite and the area is zero.
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.
Long enough to make a wake
A Reynolds number cannot ask whether an oscillating flow gets round a body before it turns and comes back, because it has no time in it. The Keulegan–Carpenter number can, and it decides which of Morison's two terms is the force. What it discards is the phase — and a peak force measurement cannot recover it.
The half that carries nothing
Thin-aerofoil theory splits a section into a camber line that carries all the lift and a thickness distribution that carries none — at any incidence, exactly none. That is very nearly true, and what it discards decides the peak suction, the critical Mach number and where the boundary layer gives up.
The singularity a layer makes for itself
March Prandtl's equations into an adverse pressure gradient and the wall shear reaches zero with an infinite slope at a finite station, and the solution cannot be continued past it. The singularity is real, it is not a numerical difficulty, and it belongs to the boundary condition rather than to the equations.
The count computed on a body
The rule for a closed surface is usually quoted and seldom solved for. This solves for one, and what the computation adds is not confirmation — it is the discovery that the count survives two events in which the number of stagnation points falls, and that both of them have closed forms.
What a turning frame keeps
Euler's equation prices the work and says nothing about where the pressure comes from. In the frame turning with the blades — which is accelerating, and carries two fictitious forces — a Bernoulli-like quantity survives both of them, and it splits the pressure rise into a term a boundary layer limits and a term that is free if the radius moves.
Four profiles, one drag
The momentum integral is exact and asks nothing about the shape of the velocity profile. Four guesses at that shape span twenty-three per cent in the drag they give — and the two that satisfy the conditions the true profile satisfies are within three, which says the freedom belongs to the family rather than to the constraint.
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.
Two forces, and only one of them remembers
Morison's equation adds an inertia term to a drag term and is usually presented as an empirical patch. It is not: the two terms are two different kinds of memory, one a function of the present acceleration and one a function of the wake left by the previous half cycle.
Three numbers out of a camber line
Thin-aerofoil theory takes a whole function and returns a lift and a moment. Only three coefficients of that function survive: two camber lines matched in the first three, and eight tenths of a per cent of chord apart, have the same lift and the same moment at every incidence and load distributions thirty-seven per cent apart.
A layer that is an integral of everything upstream
Two surfaces are given external velocity distributions that agree exactly at one station — the same speed and the same gradient. The boundary layers there differ by 38 per cent in momentum thickness, and the two surfaces separate five per cent of their length apart.
Two lifts at one incidence
A wing pitched up and down through the stall does not retrace its own lift curve. At twelve degrees it carries 0.22 more lift going up than coming down, and the loop that opens between the two is the work the airstream does on it — which is where the energy for a stall flutter comes from.
One channel, one flux, two flows
Flow between two plane walls meeting at a line has an exact solution. Past a threshold that turns out to be a ratio of gamma functions, it has two — the same wedge carrying the same flux, once outward everywhere and once with the fluid running backwards along both walls, and nothing in the equations chooses.
The best a tube can do is its own radius
Taylor's dispersion coefficient rises without limit with velocity, so there is no best speed to run a tube at. Ask instead how much a slug spreads per metre travelled and an optimum appears at once — and the spreading there is the tube's radius over the square root of three, with no diffusivity and no velocity left in it.
A shock that will not stand still
A converged transonic solution puts the shock in a definite place. Over a band of conditions the real one sweeps back and forth across a fifth of the chord at a fixed frequency, and the period is set almost entirely by how nearly sonic the flow just behind it is — a quantity that is small, hard to compute, and multiplies everything.
An overexpanded nozzle lets go before its shock arrives
The textbook's overexpanded nozzle runs full until its back pressure is high enough to hold a normal shock at the exit, and then draws the shock walking inside. A real nozzle never shows that sequence. Its wall's boundary layer cannot climb the pressure rise a normal shock imposes; it separates at a wall pressure of about four-tenths of ambient, which for a rocket nozzle happens at six to eight times the pressure ratio the textbook's shock needs. The separation is not a failure: it is what keeps the nozzle's thrust, and it is why a sea-level engine can be built twice the size it expands to.
A cold wall holds its layer on longer
A compressible boundary layer's pressure gradient pushes on its density, and its density is its temperature. Solve the velocity and the enthalpy together and the wall's temperature reaches the velocity: a cold wall holds its layer on to an adverse gradient two-thirds steeper, a hot one lets it go a third sooner, the stagnation point's heat-to-friction ratio more than doubles between them, and the cold band an uncoupled layer put above a cooled nose disappears.
An adverse gradient grows the wake, not the log law
Push a turbulent boundary layer up a rising pressure and its profile changes from the outside in. The wall region keeps its law; the wake grows, the layer hollows, the shape factor climbs and the friction falls. Clauser found the layers that stay similar while this happens, and their arithmetic has a surprise in it: there is a steepest pressure rise any such layer can climb, close to a free stream falling as the inverse fourth root of the distance, and the friction approaches zero only as the layer becomes all wake.
Inside the layer the upper parcel leads by more
Two parcels parting at a wing's attachment point arrive at different times, and outside the boundary layer the difference is the circulation over the speed squared. Inside the layer one might expect the slow, sticky fluid to wash the difference out. It does the opposite: deep in the layer the upper parcel's lead is several times the inviscid one, because the suction side's layer is fuller than the pressure side's. Only the layer's outer quarter reads the inviscid lead, and on the suction side the laminar layer gives out before the trailing edge.
The side that turns turbulent first decides who arrives first near the wall
Inside a laminar layer the upper parcel's lead over the lower grows with depth. Real wings are not laminar to the trailing edge: one surface's layer turns turbulent first, and a turbulent layer is full almost to the wall. At the incidences a wing flies at, the upper side goes first and the exaggerated lead survives to the trailing edge, smaller than the laminar one and shrinking with the Reynolds number. Below about one degree the lower side goes first, and a tenth of the way into the layer the lower parcel arrives first.
A layer whose stress lags the gradient separates sooner
A turbulent boundary layer that meets an adverse pressure gradient suddenly cannot answer it at once: the large eddies that carry its shear stress were made upstream, on a flat plate, and they draw in too little fluid for the new gradient for the first twenty or so layer thicknesses. The deficit does not protect the layer. It lets the gradient pile momentum deficit into it, so the layer reaches its new equilibrium shape quickly, its friction falls through the equilibrium value on the way, and in a strong gradient it separates in little more than half the distance a calculation without the lag predicts.
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.
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.
A stagnation point can leave the flow through the wall, and the count is kept by halves
A no-slip wall is a sheet of zeros, so the index count that holds for a flow in open space has nothing to count there. Divide the velocity by the height and the wall's zeros shrink to the skin friction's nodes, foci and saddles; reflect the result through the wall and each of them is counted once while every point in the fluid is counted twice. That is why the points on a wall are halves, and an exact swirling flow onto a plate shows the halves doing their job: a stagnation point in the fluid can disappear without a partner by sinking into the wall, where the skin friction's focus changes from attachment to separation to pay for it. A section through the flow sees none of this, because in the section every one of the points is a saddle.
A stagnation point off the axis can land only on a saddle or a node, and turns it inside out
In a swirling flow onto a plate every stagnation point lived on the axis, so one sinking to the wall always met the wall's single critical point. Without the symmetry it might seem free to land anywhere — on an ordinary stretch of skin-friction line, leaving a pair of new points behind. It cannot. Divided by the height, the velocity is continuous down to the wall, where it becomes the skin friction, so a point can only reach the wall where the skin friction already vanishes. It lands on a saddle or a node, and the count does the rest: the point it lands on turns from separation to attachment. An oil-flow picture of the wall, which shows the same pattern before and after, cannot tell which it is looking at.
A layer's memory of one gradient helps it through the next, unless the next comes too soon
A turbulent layer released from an adverse gradient keeps the gradient's eddies for a dozen thicknesses, entraining more than its shape asks. Decelerate it again before they fade — behind the next compressor stage, or past a flap — and the question is whether that memory helps or hurts. Both, depending on when. A second gradient that arrives within about three thicknesses of the release meets a layer whose entrainment is still catching up, and it separates eight per cent sooner than a layer of the same shape with no history. One that arrives later meets eddies still working from the first gradient, and the layer holds on up to five per cent longer. The crossing sits at three thicknesses for every pair of gradients tried.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Model limitBoundary layerAdverse pressure gradientCirculationMeasurementShape factorStallSkin frictionWakeLift coefficientFalkner–SkanModel validity