The wake that has to spin

An actuator disc has no blades. That is what makes it universal, and it is also what makes it incomplete in a way the first two rungs of this ladder could not see.

A disc with no blades cannot apply a torque. It pushes along the axis and nothing else, so the air that passes through it leaves travelling straight backwards, slower and otherwise unchanged. A real rotor spins. It takes power off a shaft, and power off a shaft is torque multiplied by angular speed — so the rotor must be applying a torque to the air, and by the same law that gives the thrust, the air must be applying an equal and opposite torque back.

Which means the air leaves rotating. And rotating air has kinetic energy that is not going anywhere useful.

Angular momentum crosses the box too #

The method has not changed. A control volume is drawn round the machine, what crosses its faces is added up, and the force follows without anybody saying what is inside. What changes is which conservation law is being read.

The first rung used mass, axial momentum and energy. The second used the same three with the signs reversed. This one adds angular momentum, and angular momentum crosses a control surface exactly as linear momentum does — as a flux, $\dot{m}\,r\,V_\theta$, integrated over the face.

The bookkeeping is then symmetric with the axial case and it is worth seeing side by side:

The last row is the whole essay. The axial slowing is not a loss — it is the extraction. The tangential spin-up is pure cost: the air leaves with rotational kinetic energy, that energy came out of the wind, and no part of it arrives at the generator.

Two induction factors, one blade #

Divide the rotor into annular rings and treat each as its own streamtube, with its own axial induction factor $a$ and its own tangential factor $a'$. Each ring sees a local speed ratio $\lambda_r = \Omega r/U$ — the blade speed there divided by the wind speed — which is small near the hub and equal to the tip-speed ratio $\lambda$ at the tip.

Maximising the power from each ring, with both factors free, gives Glauert’s pair:

$$a' = \frac{1 - 3a}{4a - 1} \qquad \lambda_r^2 = \frac{(1-a)(4a-1)^2}{1 - 3a}$$

which is a one-parameter family: $a$ runs from exactly $1/4$ at $\lambda_r = 0$ to exactly $1/3$ as $\lambda_r \to \infty$, and every ring of every optimum rotor sits somewhere on it.

The two limits are the argument in miniature. Far out on a fast rotor, $a \to 1/3$ — Betz’s value — and $a' \to 0$: the ring is doing what the bladeless disc does, and leaving no swirl. Near the hub of a slow one, $a \to 1/4$ and $a'$ is large: the ring is extracting less and spinning the air a great deal, because it is moving slowly and must push hard to take anything at all.

What was computed, and the mistake the first version made #

The power coefficient is an integral along the blade:

$$C_P = \frac{8}{\lambda^2}\int_0^{\lambda} \lambda_r^3\,a'(1-a)\,\mathrm{d}\lambda_r$$

with $a(\lambda_r)$ obtained by inverting the second Glauert relation. The solver bisects for $a$ at each node — the function is monotonic across the interval, so the bracket is safe — and integrates by Simpson’s rule.

The first version integrated in $a$ instead, and the answer came out above Betz’s limit. That is worth recording, because it is the kind of numerical error that produces a beautiful curve. Written in $a$, the integrand behaves like $(1/3 - a)^{-2}$ near the tip, so a uniform mesh in $a$ puts almost no points where almost all of the integral is, and Simpson’s rule on a mesh that under-resolves a near-singularity does not return a slightly wrong number — it returns a confidently wrong one. Nothing about the resulting curve looked wrong. It was the assertion that caught it: a power coefficient above 16/27 is a rotor taking more energy than passes through it, and the check refuses it.

Changing the integration variable to $\lambda_r$ fixes it completely, and the reason is worth stating because it generalises. In $\lambda_r$ the integrand is $\lambda_r^3 a'(1-a)$, and since $a'$ falls as $\lambda_r^{-2}$, the whole thing is very nearly linear at the outboard end. A uniform mesh is then the right mesh, and the limit comes out at 16/27 rather than above it.

Three things are asserted on every rotor curve drawn:

• No point exceeds Betz. That is what caught the mesh error.
• The curve rises monotonically with tip-speed ratio. A curve that dipped would mean the root-find had picked the wrong branch of the Glauert relation, which it can, since the relation is quadratic in $a$.
• It approaches 16/27 as the gearing goes up. Checked at $\lambda = 200$, where the shortfall is 4·10⁻⁵. That last one is what identifies the missing power as wake rotation rather than as something else the model forgot: a gap that did not vanish with gearing would be a different loss.

What it costs, in the band a real machine runs in #

The curve rises quickly and then flattens, and where a machine sits on it is a decision about the whole drivetrain.

At $\lambda = 1$, which is roughly where a traditional multi-bladed water-pumping windmill sits, $C_P$ is 0.4155: the swirl is costing 30 per cent of what is available. At $\lambda = 6$, a modern three-bladed machine, it is 0.5759 — a shortfall of 1.7 percentage points, or about three per cent of the available power. At $\lambda = 10$ it is 0.5852.

That is the argument for gearing a rotor up, and it is a strong one. It is also why the number of blades fell. Torque is what leaves swirl behind; power is torque times angular speed; so a rotor that runs fast needs less torque for the same power, and a rotor that needs less torque needs less blade area. A three-bladed machine at $\lambda = 7$ and a twenty-bladed machine at $\lambda = 1$ can have the same swept area and the same rating, and the first extracts a third more.

The counter-pressures are real and are outside this model entirely: a fast tip is a noisy tip, an eroded tip, and eventually a tip approaching the speed of sound where the blade section stops behaving. The optimum tip-speed ratio of a real machine is set by those, not by this curve, which is still climbing when they arrive.

What a tip-speed ratio is, from the blade’s point of view #

The tip-speed ratio is usually introduced as a bookkeeping quantity, and it deserves better, because a blade section experiences it directly as an angle.

A section at radius $r$ is moving sideways at $\Omega r$ and the wind is arriving at roughly $U$, so the flow it meets comes from a direction $\arctan(U/\Omega r)$ from the plane of rotation — steeply from ahead near the hub, almost edge-on out at the tip. That is why a large blade is twisted through tens of degrees along its span: every station has to be set at a sensible incidence to a flow whose direction changes continuously outward, and the twist is the geometry of the helix the blade tip traces through the air.

The angle also explains why the swirl loss is concentrated inboard. At the tip the relative flow is nearly in the plane of rotation, so a small force perpendicular to it produces a large torque about the axis and a small change in the air’s tangential velocity. At the root the relative flow is nearly along the axis, the geometry is the other way round, and the same shaft torque requires the air to be given a large tangential velocity. The energy is quadratic in that velocity, which is why the root of a slow rotor is such an expensive place to extract power and why the curve of $a'$ climbs so steeply there.

The practical consequence is one anybody who has looked at a wind farm has seen without registering it: the blades of a large modern turbine are narrow and few and turn slowly, while a Victorian farm windmill is wide and many-bladed and turns fast for its size. The two are not different tastes in engineering. They are different points on this curve, and the modern one is worth about a third more of the wind.

Where the swirl actually is #

The wake behind a rotor carries a vortex system with a definite structure, and it is the same one a finite wing sheds with the geometry wrapped round an axis.

Each blade carries bound circulation. Where that circulation changes along the span, vorticity is shed into the wake — at the tip, most strongly, where it drops to zero, and at the root. The tip vortices trail off as helices; the root vortices merge into a single line vortex along the axis. The swirl in the wake is that root vortex, viewed as a velocity field, and circulation is vorticity added up is the statement that lets one be computed from the other.

That gives the loss a second interpretation which is worth having, because it makes the size of it predictable. The rotational kinetic energy in the wake is the energy of the vortex system, and a vortex system’s energy depends on how concentrated it is. A rotor with fewer, faster blades sheds less total circulation, so its wake vortex system is weaker, so it carries less energy away.

The energy is left behind rather than destroyed, so it can be taken back #

There is a distinction hiding in the word loss that decides whether anything can be done about this one. The swirl energy is not dissipated: it is not heat, nothing has been rubbed, and no viscosity appears anywhere in Glauert’s calculation. It is ordinary kinetic energy, sitting in the wake, in an organised rotation. What makes it a loss is only that the rotor is finished with the air and has no further opportunity to take it.

Give something a further opportunity and it comes back. Anything that removes the tangential velocity from the wake recovers the energy that velocity represents, and there are three devices that do it.

A stator. A row of stationary vanes behind the rotor, turning the swirling flow back to axial, extracts the angular momentum without needing to move — the vanes take the torque to their mountings. That is why an axial compressor or turbine is built as alternating rotor and stator rows rather than as a stack of rotors: each stator exists to remove the swirl the rotor before it added, and a stage’s design is largely a statement about how much swirl leaves it.

A contra-rotating second rotor. Put a second rotor behind the first, turning the other way, and it meets a flow that is already rotating against its own direction of travel — which increases its relative velocity and lets it take both the remaining axial energy and the swirl. Contra-rotating propellers were built for exactly this reason on high-powered piston and turboprop aircraft, where the swirl loss at the disc loadings involved was worth several per cent, and the arrangement survived into service despite a gearbox nobody enjoyed.

Or a duct. A shroud constrains the wake’s expansion and, if shaped to do so, can turn some of the rotation as well.

So the ceiling in this essay is a ceiling on one rotor operating alone, and it is not a conservation limit in the way Betz’s is. Betz’s argument bounds what any device inside the control volume can take, however many rotors it contains. Glauert’s bounds what a single actuator that must apply a torque can take, and adding a second actuator that applies the opposite torque moves the bound.

Which prompts the obvious question about wind turbines, and the answer is arithmetic rather than principle. At the tip-speed ratios a modern machine runs at, the swirl loss is between one and four per cent of the available power. A contra-rotating arrangement would need a second rotor, a second bearing system, a second set of blades in the first one’s wake, and a gearbox — for a gain that starts at a few per cent and is reduced by everything the extra machinery costs. The loss is real, it is recoverable in principle, and it is smaller than the recovery mechanism, which is why wind turbines look the way they do and turboprops did not.

That inequality reverses when the disc loading rises. A propeller or a helicopter rotor works its air much harder than a wind turbine does, its swirl is correspondingly larger, and the balance can tip the other way — which is why the same idea produced a production aeroplane in one field and a curiosity in the other. Whether an organised loss is worth chasing is not decided by its mechanism; it is decided by its size against the apparatus that would undo it.

Where the model stops #

Glauert’s optimum is still an ideal. It contains no drag on the blades, no finite number of them, no tip loss, no hub, no tower and no stall, and each of those subtracts further.

The largest of the missing terms is the finite blade count. A rotor with three blades does not apply a uniform torque to an annulus; it applies three discrete impulses, and between them the flow partially recovers. Prandtl’s tip-loss factor is the standard correction and it is a fitted approximation rather than a derivation — it belongs to the same category as the turbulent friction law, a correlation doing honest work in a place where nothing exact is available.

Two further absences worth naming, since neither is small:

• The wake is assumed to keep its shape. It does not. It expands, it becomes unstable a few diameters downstream, and the helical vortices break up — a process this site’s solver cannot compute at any Reynolds number and which decides how far apart turbines in a farm must be.
• Nothing here is unsteady. The wind changes speed and direction faster than the rotor can respond, and a machine spends much of its life away from any optimum.

So the honest reading of the curve in this essay is that it is a second ceiling underneath a first one, not a prediction. A rotor cannot beat Betz because of conservation. It cannot beat Glauert because it must apply a torque. And it does not reach Glauert because it has real blades — which is the part everybody names, and the smallest of the three.

Who found it, and when #

Hermann Glauert set this out in 1935, in the aerodynamics volume of Aerodynamic Theory, applying to windmills the machinery he had built for propellers a decade earlier. Betz had the axial result in 1920 and the rotational correction was known to be missing from it almost at once; Joukowsky had attacked the same problem from the vortex side.

The interesting date is what came after: essentially nothing, for forty years. Glauert’s theory sat in the propeller literature until the 1970s, when a fuel crisis produced a wind-energy industry that needed exactly it, and blade-element momentum theory — Glauert’s rings, with real aerofoil data substituted for the ideal blade — became and remains the standard design tool. The equations in a rotor-design code today are these ones with corrections bolted on.

Where the ladder goes #

This closes the actuator-disc ladder, and it closes it on a limit rather than a result, which is the pattern the whole field follows.

A control volume does not need to know what is inside it. That is what makes Betz’s ceiling unbeatable and Froude’s efficiency universal, and it is what makes both of them silent about the things that turn out to matter — the blades, the wake’s instability, the tower. The box reports only what crosses its faces, and the art is in noticing which quantities those are. Adding one more of them, angular momentum, was enough to change the answer by three per cent and to explain the shape of every large wind turbine built.

The next rung of the field takes the same method somewhere with no rotation in it at all: a pipe, where the question is what the flow costs rather than what it yields, and where the exact result and the fitted correlation sit on the same chart looking identical.