A disc that knows no blades

An actuator disc is one of the great simplifications of this subject. Replace a rotor by a surface across which the pressure jumps and through which mass flows, apply momentum and energy to a streamtube, and out comes the induced velocity at the disc, the fact that it is half the far-wake velocity, the ideal efficiency of a propeller and the Betz limit of 16/27 for a turbine.

None of that calculation contains a blade. There is no chord in it, no aerofoil section, no twist, no pitch angle, and above all no number of blades. The disc is a boundary condition, and a rotor is a device for approximating one.

This essay is about what the approximation costs, and the answer is measurable at fixed solidity — that is, holding the total blade area constant, so that the disc theory’s own inputs are identical and only the blade count changes.

Two accounts of the same annulus #

Blade-element momentum theory puts the blades back by demanding that two independent calculations of the same ring of the disc agree.

Momentum says the thrust on an annulus is the rate at which axial momentum is removed from the streamtube through it:

$$\mathrm dT = 4\pi\rho U^2 a(1-a)F\,r\,\mathrm dr.$$

The blade element says it is the axial component of the force the sections produce:

$$\mathrm dT = \tfrac12\rho W^2 B c\,(C_L\cos\phi + C_D\sin\phi)\,\mathrm dr,$$

with $W$ the velocity the section sees and φ the angle it arrives at. Setting the two equal, and doing the same for the torque, gives a fixed point in the pair of induction factors $(a, a')$ at every radius.

The residual between the two statements is $3\times10^{-16}$ at convergence, and it is worth reporting as a residual rather than as a convergence tolerance: the fixed point is the statement that the two accounts agree, so its residual is the physics rather than a numerical setting.

The disc’s answer, recovered rather than assumed #

If the two theories are the same theory at one end, that has to be demonstrated, and the demonstration needs a blade rather than a claim.

The ideal rotor has $a = 1/3$ at every radius. Imposing that, and taking the wake rotation from $a'\lambda_r^2 = a(1-a)$, fixes the inflow angle at every station; the equality between the two accounts then gives

$$\sigma C_L = \frac{2\sin^2\phi}{\cos\phi},$$

so choosing a section lift coefficient chooses the chord, and the twist follows.

Put that blade into the analysis, with infinitely many blades and no section drag, and the solve returns $a = 0.333327$ — a third, to five digits, from a calculation that was given the design and not the answer — and a power coefficient of 0.58632 against Betz’s 16/27 = 0.59259.

The 1.06 per cent shortfall is wake rotation, and it falls as the tip-speed ratio rises because a faster rotor extracts the same power with less swirl. That is the disc theory’s own blind spot, made visible by the theory that contains it: an actuator disc with no torque has no wake rotation and reaches 16/27 exactly, and a real rotor extracts its power through a torque and cannot.

The number the disc cannot have #

Now the residual this essay is about.

Prandtl’s tip-loss factor accounts for the fact that a finite number of blades does not load the annulus uniformly: between the blades the flow leaks round the tip, and the mean circulation in the annulus is less than the blade’s own.

$$F = \frac{2}{\pi}\arccos\left[\exp\left(-\frac{B}{2}\cdot\frac{1 - r/R}{(r/R)\sin\phi}\right)\right].$$

The blade count appears and the chord does not. What leaks round a tip is set by the gap between blades, which is $2\pi r/B$, and not by the total blade area. So two rotors of identical solidity, swept area and tip-speed ratio — identical in every variable the actuator disc has — lose different amounts.

Two blades give $C_T = 0.629$ and twenty give 0.693: a spread of 1.101 in thrust and 1.166 in power. The tip-loss factor at the outermost station is 0.281 for two blades and 0.692 for twenty.

What the loss is, physically #

It is worth being clear about what the factor represents, because “tip loss” is a name for two different things in this subject.

It is not the drag of the tip vortex in the sense a finite wing’s induced drag is: that energy is already accounted for in the momentum theory, through the induced velocity. What Prandtl’s factor accounts for is that the azimuthal average of the induced velocity is not what a blade section experiences. A blade in a three-bladed rotor spends most of its revolution in air the other two have not yet been through, and the disc theory smears that into a mean.

The correction is a model of the helicoidal wake — Prandtl’s original derivation replaces the helical vortex sheets by a stack of semi-infinite planes moving through the fluid, and computes how much flow leaks between them. It is a good model and it is a model, and Goldstein’s exact solution of the same problem in 1929 differs from it by several per cent at low blade count, which is exactly the regime where the correction is largest.

Where the blade count actually bites #

The measured spread above is at a fixed solidity, which is a fair comparison and not the comparison anybody makes when designing.

A designer chooses a blade count and then chooses the chord, so the real trade is different. Fewer blades means each must be wider to keep the solidity, which means a lower aspect ratio, a higher section Reynolds number and a more expensive structure per blade. More blades means narrower ones, more root fittings and more cost.

The aerodynamic part of that trade is what the figures compute: going from two blades to three buys about 3.6 per cent in thrust and 6 per cent in power at a fixed solidity. Beyond about five the returns are small — five to eight buys 1.6 per cent — which is why large wind turbines settled on three and propellers on two to six, and why the ones with many blades are chosen for noise or for structural reasons rather than for efficiency.

The exception is the low-tip-speed machine. A traditional water-pumping windmill has twenty blades because it runs at a tip-speed ratio near one, where the inflow angles are large, the required solidity is high and the tip loss at low blade count would be crippling.

The circulation reading of the same calculation #

There is a second way to see what the blade element adds, and it is the one that puts this essay in the circulation field rather than in applied.

A blade section produces lift by carrying a bound circulation, exactly as a wing section does. The circulation varies along the blade, so vorticity is shed into the wake at every radius — a trailing sheet, exactly as a finite wing sheds one. The difference is that the blade is rotating, so the sheet is a helix rather than a flat plane, and the induced velocity it produces at the blade is what the induction factors $a$ and $a'$ are.

That reading makes the tip loss transparent. A finite wing’s trailing sheet rolls up into two cores and the loading falls to zero at the tips; a rotor’s helical sheets do the same at each blade tip, and the gap between successive helical turns is what decides how much of the annulus is actually loaded. With infinitely many blades the turns are infinitely close, the sheet becomes a continuous cylinder, the loading is uniform round the annulus and $F = 1$.

So Prandtl’s factor is the rotor’s version of the Betz roll-up argument rather than a separate empiricism, and the blade count enters for the same reason a wing’s aspect ratio does: it sets how far apart the shed vorticity is.

And the optimum blade is Betz’s own. The design condition used above — constant induction across the disc — is the rotor equivalent of elliptic loading, and it is optimal for the same reason: a uniform induced velocity minimises the kinetic energy left in the wake for a given axial momentum change.

What the disc still gets right #

It would be wrong to leave the impression that the disc theory is superseded, because most of what it says survives every refinement.

The Betz limit is exact and is not a blade-element result. It follows from momentum and energy alone and holds for any device that extracts energy from an unbounded stream by slowing it — a fact about streamtubes rather than about rotors. The blade-element calculation approaches it and cannot beat it, which is checked here: a solve returning $C_P > 16/27$ would be a solve with an error in it.

The factor of two is exact. The induced velocity in the far wake is twice that at the disc, from the same momentum argument, and every blade-element calculation uses it in the definition of $a$.

And the optimum induction is a third. That is what the design calculation above imposes, and it is a disc result: maximising $4a(1-a)^2$ gives $a = 1/3$ with no blade anywhere in the derivation.

So the disc supplies the target and the blade element supplies the shortfall, and the honest statement is that they are one theory used at two levels of detail.

The three losses, separated #

With the machinery above the gap between an ideal disc and a real rotor separates into three parts, and it is worth naming them because they scale differently.

Wake rotation, 1.06 per cent at λ = 12 and much larger at low tip-speed ratio. It falls as $1/\lambda^2$, which is the reason modern turbines run fast.

Tip loss, 3 to 10 per cent depending on the blade count. It falls with blade number and rises at low tip-speed ratio.

And section drag, which is not computed above and is the largest of the three for a real machine: a lift-to-drag ratio of 100 at a tip-speed ratio of 7 costs roughly $\lambda/(L/D) \approx 7$ per cent of the power. That is why a wind-turbine section is chosen for its drag bucket rather than for its maximum lift, and why surface roughness — insects on the leading edge — measurably reduces the annual yield of a turbine.

Three losses, and only the first is in the disc theory at all.

What the model does not contain #

The tip-loss factor is a model. Prandtl’s is the one used here; Goldstein’s exact helicoidal solution differs from it, and both assume a lightly loaded rotor with a wake that does not expand.

The wake does not expand here. A real turbine’s wake widens as it slows, so the annuli are not cylindrical, and the streamtube through a given blade element at the disc is a different one downstream. BEM ignores that, and the error grows as the induction rises.

Above a ≈ 0.4 the momentum theory fails outright, in the way described above, and every number in this essay is from a solve that stays below it.

And each annulus is independent. There is no radial flow in this model, which is what makes it a set of one-dimensional problems rather than a two-dimensional one, and it is why BEM handles a stalled blade badly: stall on a rotating blade is delayed and modified by the radial flow the model has excluded.

Where the streamtube stops existing #

The momentum half of the theory has a hard boundary, and it is worth following because on one side of it lies a wind turbine’s ordinary operation and on the other lies a helicopter accident category.

The far-wake velocity is $U(1-2a)$. At $a = 0.5$ that is zero, and beyond it the theory says the wake runs backwards — which is not a large correction to be applied but a statement that the streamtube the whole derivation was drawn round has ceased to exist. A streamtube requires fluid to pass through it in one direction.

What happens instead is that the wake becomes unstable well before that, at around $a = 0.4$. It breaks down into turbulence and entrains ambient air from outside, so the disc is no longer processing a bounded tube of oncoming flow but is recirculating fluid that has already been through it. The measured thrust does not fall as the momentum expression’s $4a(1-a)$ says it should past its peak; it keeps rising, towards the drag of a solid disc.

Every blade-element code therefore carries an empirical thrust curve above a stated induction — a fitted replacement for a conservation argument — and a rotor operating there is being computed by correlation, in a code whose whole architecture is built on the momentum theorem.

And for a helicopter the same regime has a name and a fatality rate. A rotor descending at a rate comparable with its own induced velocity is asking for exactly this: the wake cannot get away downwards because the machine is following it down, and it recirculates through the disc as a ring of vorticity around the tips. That is the vortex ring state, the thrust becomes unsteady and falls, and the instinctive response — more collective, more induced velocity — drives the machine further into it.

The escape is to leave the regime rather than to fight it: fly forwards, so the rotor meets fresh air, or lower the collective into autorotation, which takes the rotor to the other side of the momentum curve entirely. Both are statements about a streamtube rather than about a control.

Where the theory came from #

Rankine and Froude had the actuator disc in the 1860s and 1880s, for ship propellers. Drzewiecki proposed blade-element theory in 1892 and it was wrong on its own — it took no account of the induced velocity at all, so it over-predicted thrust by a large margin. Betz derived the limit in 1920 and the tip-loss factor with Prandtl shortly after, and Glauert combined the two into the form used here in 1935.

The interesting part is the forty-three years between the blade-element idea and its combination with momentum theory. Blade-element theory alone is a calculation about aerofoils that does not know what the rotor has done to the air; momentum theory alone is a calculation about the air that does not know there are aerofoils. Neither is usable and the combination is, which is as clean an example as this collection has of two incomplete theories being exactly complementary.

Where the same argument appears elsewhere #

The structure — a smooth boundary condition that is really a set of discrete objects — is not confined to rotors, and naming two other instances shows what generalises.

A propeller and a turbine are the same calculation with the sign of the induction reversed. A propeller adds momentum, so $a < 0$ in the convention used here, the wake contracts rather than expands, and the ideal efficiency is $2/(1 + \sqrt{1 + C_T})$ rather than a Betz limit. The tip loss is the same factor with the same blade-count dependence, which is why a propeller’s blade count is chosen by the same trade.

A ducted fan is a disc with a boundary condition that has been changed. Putting a ring round the disc allows the streamtube area at the disc to be set by the duct rather than by the flow, which removes the Betz limit entirely — a diffuser-augmented turbine is not violating anything, it is processing a larger streamtube than its own disc area.

And a cascade is the two-dimensional version of the same question: a blade row replaced by a turning device that knows no blades, and a solidity that decides how nearly the device achieves what the blades were asked for.

What this leaves #

The disc collapses a rotor onto a pressure jump, and what it discards is the blade count — worth ten per cent in thrust and sixteen in power at fixed solidity — and everything to do with the sections.

The next essay keeps the blades and takes away the axisymmetry: the side that cannot keep up, where a rotor in forward flight has a loading that depends on azimuth as well as radius and would roll over without a control that hover does not need.