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

Ideal flow — where it appears

The exact theory of a fluid with no viscosity, in which the flow is irrotational and the whole problem reduces to Laplace's equation. It is beautiful, closed-form and predicts no drag on anything, which is what makes its failure so instructive.

Named by 9 essays across 5 fields — each of them below, with the objects they name alongside it.

Flow past a cylinder at Re 100. A real fluid past a circular cylinder. At low Reynolds number the flow closes up behind the body much as the ideal theory says; as it rises the flow separates and a region of reversed flow appears behind, which is where drag comes from.

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.

viscous · Comparison Where the flow stops, at Γ = -9. A cylinder with circulation, with the points where the flow is at rest marked. As the circulation grows the two points slide round the surface towards each other, meet at the bottom, and then leave the body — after which there is nowhere on the surface where the air is at rest at all.

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.

circulation · Lift Every curved streamline has a pressure gradient across it. Twelve points in the flow past a cylinder, with the arrow at each showing the pressure gradient across the streamline. It points away from the centre of curvature everywhere, and its size is ρq²κ: the fluid is being pushed round a bend, and something has to do the pushing. Both sides are computed here and they share no arithmetic — one is Bernoulli's pressure differenced across the flow, the other is the turning rate of the velocity direction along it — and they agree to 6.7e-5. This is the whole content of the effect usually named after Coandă, and it is happening on every curved streamline of every flow.

The effect that explains nothing

A jet follows a curved wall and the wall feels a suction. Both are real, both are famous, and naming them after Coandă explains neither — what is happening is the normal component of Euler's equation, and it holds on every curved streamline in every flow.

misconceptions · Coanda 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.

inviscid · Dalembert 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 The jet is 61.1% of the hole. Flow out of a slot in a plane wall, solved by Kirchhoff's free-streamline method. The outer curve is not a wall and not a guess: it is the streamline on which the pressure is ambient, and where it goes is part of the solution. It leaves the edge of the slot travelling straight down the wall and turns through ninety degrees, settling to a jet whose width is π/(π+2) = 0.6110 of the opening. Every streamline drawn is a level set of the streamfunction the conformal map supplies.

The hole that halves the flow

A jet leaving a sharp-edged hole is narrower than the hole, and by an amount that is not measured but computed. One geometry gives exactly one half from momentum alone; another gives exactly π/(π+2) from a conformal map in which the shape of the free surface is part of the answer.

applied · Jet The jet divides 75% to 25%. A jet striking a plate at 60 degrees. Both sheets leave at the jet's own speed, because their surfaces are at ambient pressure and Bernoulli allows nothing else, and the plate can exert no force along itself because the fluid has no viscosity. Momentum along the plate then fixes the split at (1 + cos β)/2 = 0.7500, and the normal force at ṁV sin β = 0.8660. Nothing about the plate's material, size or roughness enters either.

What a jet cannot push sideways

A jet striking a plate divides in two, and how it divides is fixed by a single sentence — an inviscid fluid exerts no force along a surface. That one statement, plus mass, gives the split exactly — and the same sentence turns a flat plate into a bucket worth twice as much.

applied · Jet 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.

inviscid · Wedge flow Along a block's wall the pressure stagnates at each foot and plunges at each lip; between two lips close together it never recovers. The pressure coefficient along the wall unrolled, against the distance from the front foot in block heights: up the front face, along the top, down the back, for a block as long as it is tall and one six times as long; the lips' suction is infinite in ideal flow and is drawn to −6. On the cube-like block the pressure at the middle of its top is -1.89; on the long one, -0.365.

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.

d'Alembert's paradoxStagnation pointBernoulli's equationConformal mapMomentum theoremPotential flowPressure coefficientSeparationAdded massCirculationControl volumeDoublet

All concepts