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

Continuity — where it appears

The equation expressing conservation of mass, requiring the velocity's divergence to vanish for constant density. It is the reason a streamtube's narrowing speeds the flow up, and it holds instantaneously and on average alike.

Named by 8 essays across 3 fields — each of them below, with the objects they name alongside it.

A streamtube narrows and the flow speeds up. Two neighbouring streamlines bound a tube that no fluid crosses. Where the tube pinches, the same mass has to pass through a smaller gap every second, so it must move faster — which is mass conservation with no equations in sight.

Mass has nowhere to go

Squeeze a stream of fluid and it speeds up, not because anything pushes it but because the same amount has to get through a smaller gap every second. Almost every result in the subject is that observation with more machinery attached.

kinematics · Continuity A duct does the opposite thing above Mach one. The four cases of dA/A = (Ma² − 1) dV/V. Below the speed of sound a narrowing duct accelerates the flow, which is what continuity leads anyone to expect. Above it the sign of the bracket flips, and a narrowing duct decelerates: density is falling faster than the speed is rising, so the stream tube needs more room rather than less.

The duct that works backwards

Squeeze a pipe and the flow speeds up. Everyone knows this, it follows from continuity, and above the speed of sound it is false — a narrowing duct decelerates a supersonic stream, because the density is falling faster than the speed is rising.

compressible · Area mach The number on a streamline is a flow rate. Two streamlines and a crooked line drawn between them. The volume of fluid crossing that line every second, integrated from the velocity field, is the difference between the two streamfunction values at its ends — which is what makes a streamline a label rather than merely a curve.

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.

kinematics · Continuity There is a constriction, and it is a consequence rather than a cause. A cambered section at 5 degrees, with one streamtube traced above it and one below. The tube above narrows by 16 per cent at mid-chord and the tube below by -112 per cent, so the venturi story's premise is true: the flow over the top really is squeezed more than the flow underneath. The difficulty is that the tube's upper boundary is a streamline, not a wall. Nothing put it there but the solution of the whole flow — the same solution that already contains the lift — so the narrowing is a way of describing the answer rather than a reason for it.

Not half a venturi

The air above a wing really is squeezed into a narrower channel, it really does speed up, and the pressure really does fall. Every step of the story is true and the whole is still not an explanation, because the channel's upper wall is a streamline — and where a streamline went is part of the answer, not part of the question.

misconceptions · Venturi A duct that does not change, and a parcel that does. A contraction of area ratio four, with the parcel marked at five equal intervals of time. The walls do not move, the field at every point is the same at every instant, and the spacing of the markers grows because the parcel is carrying its own history through the duct.

How long the fluid has been in there

Age is the simplest thing a flow can remember. It obeys the shortest transport equation in the subject — its material derivative is one — and no instrument pointed at a steady flow can read it, because a steady flow's every field is constant and its fluid is getting older all the time.

kinematics · Acceleration Below a critical downstream pressure the flow rate stops listening. The flow rate through the meter against the downstream pressure, for upstream pressures of 3 bar, 5 bar, 7 bar. As the downstream pressure falls the flow rises — until the throat reaches vapour pressure, after which it is flat: 0.470 L/s from 3 bar, reached at 2.554 bar downstream; 0.608 L/s from 5 bar, reached at 4.254 bar downstream; 0.720 L/s from 7 bar, reached at 5.954 bar downstream. Everything to the left of each knee delivers the same flow, so the device holds its flow rate against any disturbance downstream.

The venturi that stops listening downstream

In a venturi with real walls, the narrowing-speeds-it-up story is exact: continuity and Bernoulli run forward from the drawing. Followed far enough, the same story predicts its own limit. The throat's pressure cannot fall below the liquid's vapour pressure, and once it gets there the flow rate stops responding to anything downstream — the meter has become a limiter, and its throat is supersonic for the vapour-laden mixture passing through it.

misconceptions · Venturi In the frame of the wave the walls stand still, and a bolus rides between them. Streamlines of a peristaltic channel of amplitude ratio 0.7 over two wavelengths, drawn in the frame moving with the wave, where the flow is steady and the walls are themselves streamlines. The time-mean flow is Θ = 0.5904 of the wave speed times the mean half-width, so the flow rate between centreline and wall in this frame is q = −0.4096 and the pressure rise per wavelength is 0.000 in units of μcλ/a². The centreline velocity changes sign at 0.106π and 0.894π, and the streamline through those points closes round a bolus holding 30.5 per cent of the fluid in each wavelength, which travels with the wave.

A wave on the wall is a pump

A channel whose wall only moves in and out, in a wave travelling along it, delivers a steady net flow with no part of the wall moving along the channel. In the frame of the wave the walls stand still and are streamlines, so continuity alone fixes how the laboratory flow rate follows the wall shape — and the momentum equation is needed only for one number, which also decides whether fluid rides along with the wave or leaks back against it.

kinematics · Continuity The best efficiency runs from nine-eighths of φ² to one. The best efficiency a peristaltic pump can reach, against the fraction of the channel its wave closes, with its two limits. For a shallow wave it is 9φ²/8, which is small — a wave closing a fifth of the channel is at best 4.5 per cent efficient. As the wave closes the channel the best efficiency tends to one, and its shortfall shrinks in proportion to the remaining gap: about 1.9(1 − φ). Nothing in between is independent of the amplitude.

The pump that is better the more it squeezes

A waving sheet swims at a cost per metre with no amplitude in it. A waving wall pumping fluid is the same mechanism turned round, and its efficiency is nothing like that: it starts at nine-eighths of the amplitude ratio squared, is exactly 2 − √3 at half closure, and rises towards one as the wave closes the tube — where the pump stops being a wave and becomes a piston.

kinematics · Continuity

Named alongside it

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

StreamtubeDivergenceIncompressibleMass conservationStreamlineBernoulli's equationDividing streamlineModel limitStreamfunctionVenturiAccelerationArea mach

All concepts