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

Vapour pressure — where it appears

The pressure at which a liquid boils at a given temperature, which is the lowest pressure a liquid flow can reach. Below it the liquid tears rather than continuing to accelerate, which is what limits a siphon and what cavitates a propeller.

Named by 14 essays across 2 fields — each of them below, with the objects they name alongside it.

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 heights, and only one of them is in the answer. A siphon, with the two heights that get confused. The drop from the source surface to the outlet is what drives the flow: the exit speed is √(2gΔz) = 4.43 m/s and nothing else enters it. The rise to the crown decides the pressure at the top — 72.0 kPa absolute here, against an atmosphere of 101.3 — and therefore whether the column holds together at all. A siphon over a high wall and one over a kerb, draining to the same place, flow at exactly the same rate.

The siphon that does not need the air

A siphon is explained everywhere by the atmosphere pushing the liquid over the hump. The flow rate says otherwise, in the flattest way available — the height of the hump does not appear in it at all. What the atmosphere does is hold the column together, which is a different job with a different limit.

misconceptions · Siphon Where the suction on a cylinder is actually made. The share of the surface's pressure deficit generated within a given distance of it. Twelve per cent comes from the first twentieth of a radius, half from within a third, and a tenth of it from beyond one and a half radii. There is no thin layer doing the work: the suction is made by the curvature of the whole outer field.

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.

misconceptions · Coanda 12 bar from stopping one metre per second. The head at the valve after it shuts, computed by the method of characteristics on a 600 m pipe. The rise is 122.4 m of water, which is ρaΔV/ρg to 1.4e-14 m — and the scheme was told neither ρaΔV nor anything else about the answer. The wave then runs to the reservoir and back every 2.000 s, and with no friction in the model it never decays: a real pipe damps this out in a few tens of cycles.

Stopping water costs more than moving it

Shut a valve on water running at one metre per second and the pressure that appears is twelve bar — not because the water was pushing hard, but because the only way to stop a column of fluid is to send a message back along it, and the message travels at the speed of sound in the pipe.

applied · Water hammer The one calculation in which the atmosphere really does push. How high a partial vacuum inside the tube will raise the liquid, against the absolute pressure achieved inside it. The lift is (p_atm − p_inside)/ρg and it is capped at the barometric height of 10.11 m, because below the vapour pressure the liquid boils and pulling harder buys nothing. The horizontal lines are five crown heights: a crown below a line's intersection with the curve can be primed by suction and one above it cannot, at any pump. This is the process the running siphon's argument explicitly excluded, and it is the one where the atmospheric account is the mechanism rather than a limit.

The one place the atmosphere pushes

The height of a siphon's hump is not in its flow rate, and the atmosphere holds the column together rather than driving it. That account excludes one process by name — starting. That is the process the atmospheric account describes correctly, and it is the only one in the whole device.

misconceptions · Siphon The source falls 6 m and the crown pressure does not move. The pressure at the crown of a draining siphon, against time, as the source level falls from the top of the tank to the end of the run, a drop of 6 m. It is flat to 1.5e-11 pascals — not nearly flat, exactly flat, because the two effects of a falling source cancel identically. Losing a metre of level shrinks the drop, which slows the flow and raises the crown pressure by half a velocity head; and it grows the rise, which lowers the crown pressure by ρg per metre. Those are the same number. So a siphon that starts will not break as it drains, however far the level falls, and the run ended because the level reached the outlet instead.

The siphon that does not break

A draining reservoir shrinks the drop and grows the rise at the same time, and the siphon's own coupling — a metre of extra drop costs a metre of hump — says a siphon should break as it empties. It does not. The two effects cancel exactly, and the crown pressure of a draining siphon is a constant that does not contain the source level at all.

misconceptions · Siphon 100 metres of water, −1.98 MPa absolute at the top. The absolute pressure up a transpiring column 100 metres tall, with the sap rising at 0.25 mm/s through conduits 40 µm across. It starts at 1.3 kPa at the root, falls by 9.79 kPa a metre for gravity and 10.02 for friction, and reaches −1.98 MPa at the top — below zero, which is not a low push but a pull. The pale line is the same column with nothing flowing. The floor is not the vapour pressure but the pore a gas bubble could be drawn through: −2.81 MPa for a 50 nm pore, which this column would reach at 142 metres. A suction pump lifting the same water from a free surface stops at 10.1 metres, because it offers the water somewhere to boil.

Where a liquid does pull

A fluid cannot pull, and the essays that settle what suction is are right about every gas and every liquid with a free surface near it. A liquid with nothing in it to boil on is another matter. Every tree taller than ten metres depends on the difference, and the floor under it is set by the size of a pore rather than by the vapour pressure.

misconceptions · Suction A tube 10 cm across, spun: the lowest pressure is on the axis. The absolute pressure along a water-filled tube spun about its middle, open to the air at both ends, 5 cm from the axis. In the spinning frame the water is at rest under a centrifugal pull, so the pressure falls from atmospheric at each meniscus to its lowest on the axis, as a parabola. 10 thousand rpm puts −1.3 MPa there, 20 thousand rpm puts −5.4 MPa there, 30 thousand rpm puts −12.2 MPa there and 45 thousand rpm puts −27.6 MPa there. The place the water is stretched hardest is the place furthest from both free surfaces, which is the whole merit of the method: a gas cannot reach the liquid where it is weakest.

A breaking strength that is the size of a flaw

Water can be stretched, and how far is a measurement people have made for a century and a half with instruments that agree with one another and not with the theory. Spinning a tube puts the stretch where no gas can reach it, sealing one caps the stretch at water's density maximum, and every measured number turns out to name the size of the worst cavity in the sample.

misconceptions · Suction The shaft speed is a window, and cavitation closes the top of it. Two groups against shaft speed for the same duty: the specific speed, which must be inside a band for a runner to exist, and the suction specific speed, which must be below about 3 for the impeller not to cavitate. Both rise with the shaft speed, so raising it to reach a band is also raising it towards the cavitation limit. The window here runs from 274.98 rpm to 962.31 rpm and cavitation sets its top. A duty whose window is empty needs something other than a different machine — a booster, a lower installation, or an inducer.

The group with no head in it

The number that picks a machine says nothing about whether the machine can exist. A second group formed from the same variables, with the delivered head replaced by the margin available at the inlet, decides that — and the delivered head has left the expression entirely, so how far a pump lifts is irrelevant to whether it tears the liquid apart at its own entrance.

applied · Specific speed Friction lends the crown 28.4 kPa, and the tank takes it back. The absolute pressure at the crown of a siphon with friction in its hose, through a whole drain, for the crown placed at three positions along the hose, against the frictionless constant of 23.01 kPa. With the crown 0.3 of the way it starts at 63.3 kPa, with the crown 0.5 along it starts at 51.5 kPa and with the crown 0.7 of the way it starts at 40.4 kPa. Every curve is above the constant, every curve falls towards it as the level falls, and every curve reaches it at the end — 23.06 kPa with a centimetre of level left. Friction never brings a siphon nearer to breaking; it lends a margin, and the draining tank returns it pascal by pascal, so the worst the crown ever sees is the frictionless value.

The margin friction lends a siphon

Without friction a draining siphon's crown pressure does not depend on the source level at all, and every real hose has friction. It turns out always to raise the crown pressure, by an amount the draining tank hands back pascal by pascal — and how much it lends is decided by where along the hose the crown sits, not by how rough or how narrow the hose is.

misconceptions · Siphon A litre of water at the crown carries 77 mL of gas it cannot hold. The volume of free gas a litre of air-saturated water can release at a siphon's crown, at the crown's own pressure, against that pressure, on a logarithmic scale, at three temperatures. It is zero at atmospheric and grows without limit towards the vapour pressure. At the reference siphon's starting crown pressure of 51.5 kPa it is 20.6 mL, a supersaturation of 2.02; at the frictionless floor of 23.0 kPa it is 76.7 mL, a supersaturation of 4.79. Cold water carries more: 87.4 mL at 5 °C. This is the equilibrium bound — what would come out if the water stayed long enough, which it does not.

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.

misconceptions · Siphon 121 m after the cavity closes, against 69 m from the closure. The head at a valve shut instantly on water flowing at 0.36 m/s through 600 m of 100 mm pipe, a = 1200 m/s, with a steady head of 25 m. The closure raises it to 69.1 m, the Joukowsky head; the reflection returns at one round trip, 1.00 s, and takes the head down to the vapour head, −10.1 m, where a cavity opens (shaded). It closes 2.146 round trips after the closure, and the first pulse after it reaches 121.3 m — 52.3 m above the Joukowsky head — for 146 ms. The step line is the exact solution between events; the thin line is a 240-reach grid solver that was told nothing about it and agrees with its first pulse to better than a millimetre.

Twice the margin, on top of the hammer

Shut a valve on a line whose pressure is low and the returning wave boils the water beside it. When that cavity closes, the head at the valve can pass the Joukowsky rise — by up to twice the margin that let the water boil, in a sawtooth that jumps each time one more round trip fits into the cavity's life, and for a time that is shortest exactly when the pulse is tallest.

applied · Water hammer The throat holds the hammer back only while its cavity lasts. Left, pressure at the closing valve (red) and at the upstream face of the venturi (gold); right, the throat's cavity volume; after the valve shuts with a 5 mL cavity in the throat. The valve sees the full Joukowsky rise of 14.8 bar at once. The wave reaches the venturi 33.3 ms later, and for the next 7.9 ms the upstream pipe hears nothing: its pressure stays at 5 bar while the cavity is squeezed. When the cavity closes at 41.3 ms the surge passes into the upstream pipe at 13.8 bar above its steady pressure.

A choked throat buys time, not silence

A venturi whose throat has reached vapour pressure passes a flow the downstream pressure cannot change, and it is tempting to read that as isolation: whatever happens downstream, the upstream pipe will not hear it. Slam a valve downstream and it hears it. The cavity at the throat holds the surge back only for as long as it takes to fill, and then lets 93 per cent of it through.

misconceptions · Venturi The characteristic gets a wall, and the wall does not care about the discharge. One jet pump's characteristic, at an area ratio of 0.275, with the flow ratio at which its throat entry reaches vapour pressure drawn for three values of the cavitation parameter σ = (Pₛ − pᵥ)/(Pₘ − Pₛ). Left of a wall the machine runs on its curve. At the wall no lower discharge pressure raises the flow: the head ratio can fall to zero along the vertical and the flow ratio stays where it is. The wall's position contains the nozzle and suction losses and nothing downstream of the throat entry.

The wall the suction puts in the curve

A liquid jet pump's lowest pressure is where the entrained stream enters the throat, and when that reaches vapour pressure the pump curve stops being a curve. The flow ratio freezes at a value no lower discharge pressure can move — and raising the motive pressure, the obvious cure, brings the wall closer.

applied · Ejector

Named alongside it

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

CavitationModel limitAbsolute pressureMisconceptionBarometric heightSiphonSuctionLiquid tensionColumn separationEnergy equationHydrostaticsSurface tension

All concepts