Fluid Flow & Aerodynamics
why things fly, and why the usual explanation is wrong
Concept

Pressure coefficient — where it appears

Pressure measured against the free stream and divided by the dynamic pressure. It is exactly one at a stagnation point and negative wherever the flow is faster than the stream, and it is the same number at every speed for an incompressible flow.

Named by 27 essays across 6 fields — each of them below, with the objects they name alongside it.

Ideal flow past a cylinder. A uniform stream past a circular cylinder in a fluid with no viscosity. The solution is exact and closed-form: streamlines part at a stagnation point, run round the surface and close up perfectly behind, and the pressure recovers to exactly what it was in front.

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.

inviscid · Pressure The lift curve that made flight look impossible. Lift coefficient against incidence, by Newton's impact theory and by thin-aerofoil theory. One is quadratic in the angle and the other linear, so at small incidence — which is where aircraft fly — they differ by more than an order of magnitude. Newton's version says a wing large enough to carry a man would need an engine nobody could build, and for a century that arithmetic was taken as settling the question.

The theory that forbade flight

Newton treated air as a hail of particles that give up their normal momentum on impact, and got a lift coefficient of 2sin²α cos α. At five degrees that is a thirty-sixth of what a wing actually makes, and the quadratic is why powered flight looked arithmetically impossible for two centuries. The same formula is exact at Mach twenty.

misconceptions · Newtonian What a barometer on the wing would read at 60 m/s. The absolute pressure along an aerofoil's surface, in kilopascals, with the atmosphere put back. Nothing on the wing is near zero: the deepest point of the so-called suction peak is a few per cent below a hundred kilopascals, and the air there is still pushing on the wing hard enough to hold up a column of itself eight kilometres tall. The lift is the difference between two large pushes, not a pull.

Nothing sucks

The upper surface of a wing is universally described as being under suction, which sounds like a pull. A fluid cannot pull. The lowest absolute pressure on a wing at sixty metres a second is 96 kilopascals — a five per cent dip in a hundred — and the force does not depend on where zero was put, because the normals of a closed body sum to nothing.

misconceptions · Suction Two sides of one ball, at different pressures. The surface pressure coefficient round a ball, measured from the front stagnation point, on the side the seam trips and on the side it does not. Up to separation both follow the exact potential-flow distribution 1 − (9/4)sin²θ. After it both take the same wake pressure, which is what a manometer measures rather than what the ideal theory predicts. The asymmetry is the shaded area between them, and integrating it gives a side force of 0.2949 towards the later-separating side.

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.

applied · Sport ball 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.

inviscid · Pressure The stretch of surface that is boiling. The pressure coefficient along both surfaces of a section at 4 degrees, computed from the same potential-flow solution as the other ideal-flow aerofoil figures. The horizontal line is the vapour pressure at a cavitation number of 1: wherever the suction curve is above it, the liquid there has been pulled below its vapour pressure and is boiling at whatever temperature it happens to be. This section cavitates at any σ below 1.428.

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.

applied · Cavitation 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.

inviscid · Axisymmetric What the ground actually gives a wing. Induced drag near the ground as a fraction of the same wing's in free air, against height in spans, at constant lift. The ground is a plane of symmetry, so the wake is joined by a mirrored wake of opposite circulation below it, and the upwash from that image is what takes the drag away. At a tenth of a span the wing keeps 0.516 of its induced drag — a saving of 48 per cent — and by a span and a half the effect is 1.3 per cent and going. The lift is held fixed all the way along this curve: what the ground gives here is not more lift, it is the same lift for less drag, and the two are different claims about a landing aeroplane.

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.

misconceptions · Ground cushion 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.

inviscid · Inverse design A falling drop is a bun, not a tear. The shape of a drop pressed on by the air it is falling through, computed by matching the Legendre component of a sphere's own potential-flow pressure distribution against the change in curvature it produces. The result is oblate — flattened along the direction of travel — because the pressure is high at the poles and low round the equator. The teardrop of every illustration has the wrong symmetry entirely: a drop in free fall has no up, and one at terminal speed is being pressed on from in front.

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.

regimes · Drop shape Which speed the number is formed on. The fractional change in air density at three places on a body, against the free-stream Mach number. At a stagnation point the density rises, and it reaches five per cent at M = 0.314 — which is where the familiar 0.3 comes from, and it is a five per cent tolerance rather than a physical boundary. At the suction peak the density falls instead, and how fast depends on the body: a lightly loaded section is milder than its own nose, and one working at cp₀ = −2 reaches five per cent at M = 0.22 and is at Mach 0.55 over its shoulder while the free stream is at 0.3.

Which speed goes in the number

The most quoted threshold in the subject — air is incompressible below Mach 0.3 — is a five per cent tolerance on the density at a stagnation point wearing a physical boundary's clothes. A wing working for its living is at Mach 0.55 over its shoulder while the free stream is still at 0.3.

regimes · Mach The density ratio and the nose pressure coefficient, against Mach number. Two quantities across a normal shock, on a logarithmic Mach axis. Both approach limits that depend on γ and on nothing else: six for the density ratio and 1.8394 for the pressure coefficient at the stagnation point. By Mach five the second is within three per cent of its limit and by Mach twenty within a tenth of a per cent. Above that the flow round a blunt body has stopped depending on how fast it is going and started depending on what the gas is.

A shock that lies on the body

Above about Mach eight the flow round a blunt body stops depending on how fast it is going. The density ratio, the nose pressure coefficient and the shock standoff all reach limits set by γ alone — and going from a perfect gas to a dissociating one halves the standoff.

compressible · Shock layer The coefficient of the equation's second derivative, along a chord. The bracket multiplying the streamwise second derivative in the transonic small-disturbance equation, along a chord at Mach 0.85. Where it is positive the equation is elliptic and the flow is subsonic; where it is negative the equation is hyperbolic and the flow is supersonic. Which it is at a given point depends on the perturbation velocity there, which is the thing being solved for. Forty-two per cent of this chord is hyperbolic, and no amount of inspecting the problem beforehand could have said so.

The equation that changes type inside its own answer

Near Mach one the coefficient of the streamwise second derivative depends on the perturbation velocity, which is what is being solved for. Two solutions of the linear equation no longer add — the leftover is three times the term the linear theory keeps — and the critical Mach number approaches one as the two-thirds power of thickness.

compressible · Transonic Identical lift at every altitude, and the skin load nearly doubles. The same wing at the same dynamic pressure at seven altitudes. The lift is identical at every one of them — the flat line, computed from the absolute pressures rather than assumed, because a net force cannot depend on where the pressure datum is set. The load on a vented panel is identical too, at 4.86 kPa, because both sides of it moved together. The load on a sealed panel with 75.26 kPa inside is not: it is 28.27 kPa at sea level and 60.84 kPa at twelve kilometres. The datum cancelled in the force and it is one of the two numbers in the stress.

A force forgets the datum, a stress cannot

Nothing sucks, because the pressure datum cancels — the normals of a closed body sum to nothing, so the lift is the same in gauge or absolute pressure. That identity is about a resultant, and it is routinely carried one step too far. The load on a panel of skin has the ambient in it as one of two numbers, not as a datum.

misconceptions · Suction Four bodies, four drag coefficients, and no flow was solved. Four bodies with their Newtonian drag coefficients, each computed as a quadrature over its own surface with no flow solution anywhere. The cone's answer is exactly 2sin²δ, checked against the closed form to nine decimal places; the flat disc's is exactly 2, since every element of it faces the stream; and the sphere's is 1 against the classical Newtonian value of 1. Every one of those is an integral of one expression over a shape, and none of them required knowing what the air was doing anywhere.

The only theory simple enough to optimise

Whether Newton's sine-squared law is right has two answers — hopeless at the speeds he argued about, nearly exact behind a strong shock. This asks a different question about the same formula. Its pressure depends only on the local surface angle, so a shape's drag is a quadrature rather than a solution, and the best shape can be found by calculus.

misconceptions · Newtonian 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.

inviscid · Superposition A degree of temperature is worth 0.34 per cent of pressure. The error in the pressure a paint reports, against how much warmer the surface is at the test condition than at the reference, at four pressures. At 0.8 of the reference pressure the sensitivity is -0.34 per cent of pressure per kelvin, so 10 degrees is -3.45 per cent. That is not a small number against what the technique is used to measure, and a model's surface temperature is not uniform: it is warmer where the flow has been brought to rest and cooler where it has accelerated, which means the temperature error is largest exactly where the pressure gradients are. The standard remedy is a second, temperature-sensitive paint measured at the same time — an instrument added to correct an instrument.

The paint that measures the wrong field

Dye, seeded particles and the optical methods all look through the flow or at something put in it. Pressure-sensitive paint looks at the surface, reports a scalar rather than a derivative, and turns a row of taps into a field. What quenches its luminescence is oxygen, which is what makes it a pressure gauge — and temperature, which is the other field a flow is guaranteed to produce.

misconceptions · Visualisation The six bodies, drawn at the same scale. Two circles, two ellipses and two Joukowski sections, each at the incidence that gives it a circulation of exactly two. There is no family resemblance and no common parameter; what they share is one number, and the theorem needs nothing else.

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.

circulation · Lift 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.

inviscid · Ideal flow The shaded face's share of the force is set by K, and it is not small until K is. The fraction of a flat plate's normal force carried by its leeward face against K = M sin α, at Mach 3, 5, 10, 20, with the hypersonic small-disturbance value and the share the leeward face would have at vacuum (dashed). Newtonian theory puts it at zero. The curves collapse on K: the small-disturbance share is 35.7 per cent at K = 0.5, 24.2 at 1, 11.2 at 2 and 3.4 at 4, and the vacuum bound is 28.8, 11.4 and 3.4 per cent at 1, 2 and 4. The zero is a good approximation only where K is large — which is also the only place the Newtonian windward pressure is itself accurate.

The face Newton left in shadow

Newtonian theory gives a surface turned away from the stream a pressure coefficient of exactly zero, and at hypersonic speed the rest of the theory is nearly right. The shaded face is not. Computed exactly on a flat plate, its share of the force depends on the similarity parameter K = M sin α rather than on the Mach number, it is a quarter of the force at K = 1, and it moves a hypersonic plate's best lift-to-drag ratio from 5 to 7 at Mach 10.

misconceptions · Newtonian The worst jet amplifies the stagnation pressure by about the Mach number. The largest amplification of the stagnation pressure, over every incident turn, against the free-stream Mach number, on a logarithmic axis: the type IV jet, the best single turning shock followed by a normal shock, and the lossless ceiling. The jet's peak runs close to the line equal to the Mach number itself, from 3.5 at Mach 4 to 12.4 at Mach 12. The ceiling grows as the Mach number to the power of three and a half and is never approached. The estimate with one turning shock falls further behind the jet as the Mach number rises.

The spot a local theory cannot see

Newtonian theory gives every panel of a hypersonic vehicle a pressure set by its own angle to the stream, and no panel more than the stagnation pressure behind a normal shock. Let a shock from one part cross the bow shock of another and a supersonic jet forms that reaches the surface through weaker shocks. At Mach 8 it stagnates at 8.6 times the ceiling — and the worst amplification at every Mach number is close to the Mach number itself.

misconceptions · Newtonian 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.

inviscid · Axisymmetric 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.

inviscid · Dalembert 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.

inviscid · Superposition

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.

inviscid · Wedge flow

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.

inviscid · Superposition

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.

inviscid · Wedge flow

Named alongside it

The objects these essays reach for when they reach for this one.

Model limitPotential flowMisconceptionSuctionSuperpositionConformal mapStagnation pointd'Alembert's paradoxBernoulli's equationCirculationDimensionlessHypersonic

All concepts