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

Efficiency — where it appears

The ratio of what a machine delivers to what it consumes, in whatever currency the problem is written in. Where a bound exists on it — Betz's, the propulsive one, the acoustic one — the bound is usually a conservation law rather than an engineering limit.

Named by 19 essays across 5 fields — each of them below, with the objects they name alongside it.

The tube widens because the air slows. The streamtube through an actuator disc at an induction factor of 0.333. The three radii are not drawn to taste: each is fixed by requiring the same mass to pass every station, and the slowest station is therefore the widest. The tube widening in front of a wind turbine is why some of the wind goes round it rather than through it, and it is the whole reason a disc cannot take everything.

The most a disc can take

A wind turbine cannot extract more than sixteen twenty-sevenths of the energy passing through the circle its blades sweep. That is not a limit on turbines — it is a limit on anything at all, and it follows from three conservation laws and no engineering.

applied · Actuator disc Efficiency is decided before the engine is chosen. Froude's propulsive efficiency against the ratio of jet speed to flight speed. It is 2/(1+σ) and nothing else — no engine, no fuel, no combustion. A turbojet with a jet at three times flight speed cannot exceed 50% however good its core is, and a propeller moving a great deal of air slowly is above 90% before anybody has designed anything.

A big slow push

The same thrust can be had from a lot of air moved a little or a little air moved a lot, and the two are not equivalent. One number decides which, it contains no engine, and it is why every airliner built since 1970 has a fan far larger than the machine driving it.

applied · Actuator disc Everything a normal shock does, against the Mach number in front of it. Four quantities across a normal shock, each scaled to fit one axis. Pressure and density rise without limit and without bound as the Mach number grows; the Mach number behind falls towards a floor it never passes; and the total pressure — the flow's ability to be turned back into speed — collapses. That last curve is why a supersonic intake is designed around avoiding a single strong shock.

What a shock costs

All the heat survives a shock and none of it is lost. What is lost is the ability to turn that heat back into speed — 27.9 per cent of it at Mach 2 and 93.8 per cent at Mach 5 — and every supersonic intake ever built is a scheme for paying less.

compressible · Shock Three times the wind, on a reach. The polar diagram: boat speed in every direction, as a multiple of the true wind speed, for drag angles of 14 and 6 degrees. The shaded wedge at the top is the no-go zone, whose half-angle is exactly the sum of the two drag angles. Everywhere outside about twice that angle the boat is faster than the wind, and the maximum is 2.92 times the wind at 110 degrees — which is 1/sin λ at 90° + λ, both checked.

Faster than the wind that drives it

An ice yacht in a fifteen-knot breeze does forty. That is not a trick and it does not need a special sail — it follows from two drag angles and a triangle, and the best speed a boat can reach is one over the sine of their sum.

applied · Sailing The fastest course is never the one towards the mark. Boat speed and speed made good against the course sailed. Speed rises steadily as the boat bears away, but what counts is the component along the direction wanted, and that has a maximum well off the straight line — at 55.0 degrees going upwind and 145.0 going down. Both optima are found here by search and agree with 45° + λ/2 and 135° + λ/2 to six figures.

The fastest way is not the straight one

A boat racing to a mark dead upwind sails seventy per cent further than the distance to it, and arrives first. The best angle is forty-five degrees plus half the apparent wind angle, it comes out of one line of trigonometry, and the same argument says to gybe downwind rather than run.

applied · Sailing Two triangles, and the work is the difference between them. The velocity triangles at inlet and outlet of a rotor at constant blade speed and constant axial velocity. The horizontal arrow is the blade speed; the arrow from the origin is the absolute velocity of the fluid; the arrow closing the triangle is what the blade sees. Euler's equation says the work is the blade speed times the change in the swirl component alone — the horizontal distance between the two upper corners, times U — and nothing else in the picture appears in it.

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.

applied · Turbomachine Most power at exactly half the jet speed, found by search at 0.5000. The power a bucket takes from a jet, against how fast the bucket runs, for four deflection angles. Every curve is a parabola with roots at zero — where the force is greatest and the bucket is not moving — and at the jet speed, where the bucket is running away and there is no force at all. The peak is halfway between, at U = V/2, and it is there for every angle and every flow rate. A golden-section search that knows none of the algebra puts it at 0.500000.

Half the jet speed takes everything

A bucket standing still feels the largest force and does no work; a bucket running with the jet does no work either. Between them the power peaks at exactly half the jet speed, for every bucket shape and every flow rate — and at that speed a perfect bucket leaves the water motionless.

applied · Jet Each cancellation costs two powers of the Mach number. Radiated power against compactness for three source clusters: a single monopole, two of opposite sign, and four on a square with alternating signs. The fitted slopes are 0.00, 2.00, 4.00 — zero, two and four in (kd), measured by integrating the far field over a sphere rather than assumed. A turbulent eddy turns over in about the time sound crosses it, so kd is of order the Mach number, and those exponents become the fourth, sixth and eighth powers of speed. A flow with no moving surfaces has no monopole and no dipole available to it, which is Lighthill's whole argument, and the eighth power is what is left.

The sound that only leaves

A flow is a catastrophically bad radiator, and the reason is that it has no monopole and no dipole available to it. What is left is the eighth power of speed — and the equation is equally happy with sound converging on a jet, which is ruled out by a condition imposed at infinity.

compressible · Aeroacoustics Nothing can deliver more than h/H, and here that is 10.0%. The fraction of the supply a ram can deliver, against the height it is asked to deliver to, from a supply falling 2 m. The upper curve is the exact ceiling h/H, which follows from the energy audit with every loss set to zero and can be reached by no real machine; the lower one is what a ram at 65% efficiency actually sends. Asking for twice the height halves the delivery, exactly, and there is no design that escapes it.

A pump with no engine

A hydraulic ram lifts water uphill using nothing but the water that is already falling. It has one moving part and no power supply, and everything it can and cannot do follows from an energy audit that fits on one line — including a ceiling nothing about its design can move.

applied · Water hammer A pump curve out of the momentum theorem. The pressure an ejector delivers, against how much it is entraining, at a fixed nozzle. It has the shape of every pump characteristic ever measured — a shut-off pressure with no flow, falling to no pressure at free delivery — and it was obtained from a momentum balance on a tube with nothing in it. The shut-off value here is 36.0 kPa and the machine at its best power runs at 4.56 times its own motive flow.

Mixing is a pump

Two streams at different speeds mixing in a tube destroy energy — the same Borda–Carnot expression a handbook prints beside a sudden enlargement, with two streams in it instead of one. And while they destroy it the pressure rises, which makes the loss the mechanism of a machine with no moving parts.

applied · Ejector The junction the minimisation is over. A parent vessel entering from the left and two daughters leaving to fixed points. The radii are settled by Murray's law; what is left free is where the branch point sits, and the cost of the junction depends on it. The point drawn is the one the minimisation finds.

The angle a junction chooses

Murray's law fixes the radii at a branching vessel and is where every account of it stops. The same minimisation fixes the angles completely — 74.93 degrees for a symmetric bifurcation, a right angle for a vanishing side branch — and it does so as a triangle of forces, with tensions proportional to the squares of the radii.

applied · Branching A swimmer that dissipates the same everywhere. Taylor's waving sheet, with the wave drawn along the bottom and the dissipation drawn against height above it. The dissipation function works out at 4μb²c²k⁶y²e^{−2ky} — with no x in it at all, so the sheet is destroying energy at the same rate under every part of the wave and at every instant of the cycle. It peaks one radian of wavelength above the sheet and is gone within about three.

A swimmer that cannot go backwards

Taylor's waving sheet is the simplest self-propelled object in a viscous fluid, and its arithmetic contains a result that reads like a mistake — the work it does to travel a metre does not depend on how big its waves are. Doubling the amplitude quadruples both the speed and the power, and changes the bill for the journey by nothing at all.

viscous · Swimming Same disc, same solidity, different number of blades, different answer. Thrust and power coefficients for six blade counts at a fixed solidity. They span a factor of 1.101 in thrust, and every one of those rotors is the same actuator disc: same area, same blade area, same tip-speed ratio. The disc theory cannot distinguish them because the blade count is not one of its variables.

A disc that knows no blades

Momentum theory replaces a rotor with a surface across which the pressure jumps, and gets the Betz limit, the induced velocity and the whole energy argument out of it. It has no chord, no section and no number of blades — and at one fixed solidity, two blades and twenty give thrust coefficients ten per cent apart.

circulation · Blade-element A pure number with no fluid in it. The load a plane slider bearing carries, divided by everything dimensional in it. What is left is a function of the convergence ratio alone — no viscosity, no speed, no size, no material — so the tilt that carries the most load is decided before anything about the machine is known. It is the root of one transcendental equation, at 2.1887048.

The gap that carries the most

A bearing's most consequential dimension is decided by a number with no oil in it, no speed and no size — the root of one transcendental equation, at 2.1887048. And the maximum it sits on is flat enough that the value printed in every handbook is not it.

viscous · Lubrication At 70 per cent speed the first stage runs at 0.64 of its flow coefficient and the last at 1.17. The flow coefficient of each stage of a compressor of 8 stages at 300 m/s mean blade speed, drawn for a flow coefficient of 0.5 and a work coefficient of 0.35, as a share of the value its blades were cut for, along the operating line a choked exit nozzle sets, at 110 per cent, 100 per cent, 90 per cent, 80 per cent, 70 per cent of design speed. At design speed every stage is at exactly one. Below it the front stages fall towards the stall limit (shaded below 0.82) and the rear ones rise towards the choke limit (shaded above 1.3): at 70 per cent the first stage is at 0.642 and the eighth at 1.170. Above design speed the pattern reverses, the front stages rising and the rear falling.

Matched at one speed and at no other

Every stage of a compressor passes the same mass flow, and the annulus behind each one is cut for the density the air will have reached there at design speed. Slow the shaft and the air is less dense than the metal expects, so the rear stages carry more volume than they were drawn for while the front ones starve — and below a definite speed no throttle setting keeps all of them working at once.

applied · Turbomachine A pump slowed into a system with a static lift leaves its own specific speed. The specific speed of the operating point, as a share of its value at the best point, against the fraction of the design flow delivered, for a pump drawn for 0.1 m³/s against 40 m at 1450 rpm, specific speed 0.545 at its best point. Under speed control into a system whose static lift is 0 per cent, 30 per cent, 60 per cent, 90 per cent of the design head (lines), and under a throttle at design speed into the 60 per cent system (dashed). With no static lift the speed-controlled pump stays exactly at its best point and its specific speed never moves. At half the design flow it has fallen to 1.000, 0.800, 0.708, 0.652 of the design value as the static share rises, and to 0.598 under the throttle.

The specific speed a pump spends its life at

A pump is chosen by its specific speed at its best point and then run somewhere else. Written in the pump's own coefficients the number is √φ/ψ^¾, a position along its characteristic, and a variable-speed drive keeps it there only when the system it pumps into has no static lift. Every metre of lift moves a slowed pump along its own curve, towards shut-off, and a throttle moves it further.

applied · Specific speed Six nozzles, six pump curves, and where each one is best. The head ratio a water jet pump delivers against the flow ratio it entrains, for six area ratios from a narrow nozzle to one filling four-fifths of the throat. A wide nozzle makes a tall, steep curve that is finished at a small flow; a narrow one makes a low, long curve. The dots are each curve's best-efficiency point. Nozzle, suction, throat-friction and diffuser losses are included at borrowed representative values, and the mixing loss is computed.

The nozzle that is best at one thing

Put the four losses back into a jet pump and three questions get three answers. The most head comes from a nozzle four-fifths of its throat, in closed form; the best efficiency from one a quarter of it; and the most flow from whichever nozzle is smallest, because flow has no optimum at all.

applied · Ejector 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 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.

OptimisationControl volumeModel limitDimensionlessMomentum theoremConservationDissipationMeasurementActuator discEjectorEnergy equationKinetic energy

All concepts