A blade that flies through what it shed

The side that cannot keep up is this collection’s account of the rotor’s asymmetry, and it ends on the observation that vibration is the real design driver rather than the mean loading. This essay is about the largest single source of it, and about why a machine that looks axisymmetric is not.

A rotor blade sheds a tip vortex continuously. That vortex stays where it was laid down, drifting slowly, while the rotor turns. So one blade passage later — a fifth of a second, for a four-bladed rotor at three hundred revolutions a minute — the next blade arrives at the same place and flies through it.

The machine meets its own past, on a schedule set by its blade count.

The shape of the encounter #

What the blade meets is not a gust. A gust is a region of upwash, and passing through it raises the incidence for as long as the passage takes.

A vortex has upwash on one side and downwash on the other. So a blade passing one is lifted and then pressed down, or the reverse, within the width of the encounter — and the loading perturbation is a doublet rather than a bump.

That distinction decides everything downstream. A bump puts energy at low frequency; a doublet puts it at the frequency set by its own width, which is far higher — and the width here is proportional to the miss distance, so at five per cent of the radius the whole event is over in a few chords and its rate of change goes as the inverse square of the gap. A rotor’s acoustic signature is disproportionately made of these encounters for exactly that reason: the pressure a listener hears depends on the rate of change of the loading, and a doublet’s rate of change is large.

At five per cent of the radius the induced incidence reaches 5.5 degrees, which is a substantial fraction of the blade’s own working incidence and arrives and departs within a few chords.

One over the miss distance #

The size of the encounter is set by how close the blade passes, and the dependence is the simplest possible.

The peak of the pulse goes as the reciprocal of the miss distance, with a measured exponent of 1.0000. That is the vortex’s own velocity field: outside the core, the induced velocity of a line vortex falls as one over the distance, and the peak of the doublet is that velocity evaluated at the closest approach.

Six clearances, spanning a factor of forty, with the vortex strength and the blade speed held fixed:

Halving the clearance doubles the load, and the table says how nearly: from 0.4 to 0.2 the peak rises by 1.9997, from 0.2 to 0.1 by 1.9989, and from 0.02 to 0.01 by 1.9008 — the last falling short because at one per cent of the radius the blade is inside the desingularised core. The rate-of-change column spans 1,492 across the same forty-fold range, which is the square of forty to within seven per cent. There is no threshold, no saturation and no regime in which the encounter becomes unimportant — it becomes unimportant only when the blade passes far enough away, and “far enough” is set by how much load is acceptable rather than by any feature of the physics.

So blade-vortex interaction is a geometry problem. Nothing about the blade changes it; only where the vortex is — the strength here is 12 m²/s and the blade speed 200 m/s at a radius of 6 metres, and every number in the table above moves only with the gap.

Sharper as well as larger #

There is a second consequence of the same geometry and it is the one that decides the noise.

The width of the doublet — the distance between its two extremes — is proportional to the miss distance. So a close encounter is both stronger and shorter, and the rate of change of the loading, which is the peak divided by the width, goes as the inverse square.

Halving the clearance therefore doubles the load and quadruples its rate of change: 5.98 at five per cent of the radius against 36.81 at two per cent, a factor of 6.2 for a factor of 2.5 in clearance. Since the radiated sound goes with the loading’s time derivative, a rotor flown so that its blades pass its own wake a little closer is not a little noisier; it is four times as noisy for each halving, and a blade that comes in at one per cent of the radius rather than ten sees its rate of change rise by a factor of 93.

That is why blade-vortex interaction noise is a descent phenomenon. In level flight the wake is swept below the disc; in a descent the machine flies down into its own wake, the miss distances collapse, and the characteristic slapping of a helicopter on approach is the result.

Why the wake is still there to be met #

It is worth asking why the vortex has not gone away in the fifth of a second between one blade and the next, because the answer decides whether any of this is important.

A tip vortex decays by diffusing outwards, and the time scale for that is the core radius squared over the viscosity — which for a core of a few centimetres in air is minutes rather than fractions of a second. The interval between one blade laying the vortex down and the next one meeting it is a fifth of a second at four blades and 300 revolutions a minute, which is two orders of magnitude shorter. It also decays by three-dimensional instabilities, which take longer still. So on the time scale of a blade passage the vortex is essentially unchanged: what one blade laid down, the next blade meets in full strength.

That is the same statement a wake that says what made it makes about an aircraft’s wake, with the clock read at a different scale. There the useful window was a minute and the interesting time was minutes; here the interval is a twentieth of a second, so the record is fresh by a factor of a thousand. The wake-decay calculation in that essay reads a circulation of 508 m²/s falling to 89.8 over four minutes — a factor of 5.7 across 240 seconds, against 0.05 seconds here.

A rotor’s memory of itself is therefore perfect on the relevant time scale, and the design question is entirely about geometry rather than about decay. That is the opposite of the situation for a fixed wing, whose wake is somebody else’s problem by the time it matters.

When it happens #

The timing is arithmetic. A four-bladed rotor at three hundred revolutions a minute turns once every 0.2 seconds, so a blade passage is 0.05 seconds and the encounters arrive at twenty per second — with harmonics, because the pulse is narrow.

That frequency is a property of the machine’s geometry and of nothing aerodynamic. Changing the blade count changes it; changing the rotor speed changes it; changing the aerofoil, the twist, the loading or the flight condition does not. A vibration or a tone at a multiple of the blade-passing frequency is therefore diagnosable by its frequency alone, which is the first thing an instrumentation engineer uses.

What the solver computed, and how it was checked #

The vortex is treated as a straight desingularised line at a stated miss distance and the blade section as a point moving past it, feeling the vortex’s own induced velocity resolved into an incidence. That is the smallest model with the right structure in it, and it is a model rather than a solution: no blade response, no unsteady aerodynamics, no vortex deformation.

Three checks. That the pulse changes sign — it must be negative before the encounter and positive after, or it is not a doublet, and a check that only measured its peak would pass on a bump. That the peak goes as the first power of the reciprocal miss distance, to within eight per cent, since that is the claim — it reads 0.9998, so the tolerance is doing no work. And that the width scales with the miss distance too, which is what makes the rate go as the inverse square.

The first of those is the one worth pointing at. It is easy to write a check on the magnitude of an effect and much harder to write one on its shape, and the shape is what distinguishes this encounter from every other unsteady input a blade meets.

What a real interaction adds #

The model above is a stationary vortex and a point blade, and a real encounter has three things it does not.

The blade responds. A wing meeting a sharp-edged gust does not acquire its full lift immediately — that is Küssner’s function, and the collection computes it in two answers to one question. The doublet here is narrow enough that the blade’s own unsteady response smooths it substantially, which reduces the peak load and does not reduce the frequency content nearly as much.

The vortex is not straight and not stationary. It is a curved filament being convected by every other vortex in the wake, including itself, and its position at the moment of the encounter is the output of a wake calculation rather than a parameter.

And the encounter is not two-dimensional. A blade crossing a vortex at a shallow angle meets it over a long span at nearly the same instant, which is the loud case — and at 20 encounters a second with harmonics reaching well past the tenth, that is a signature in the hundreds of hertz; one crossing at a right angle meets it at one station at a time, which is far quieter. The crossing angle is the single largest factor in the noise and it is entirely absent from this model.

Why this is the machine’s memory of itself #

The vortex the blade meets is the one its own tip left, and where that wake finally goes is where the wake ends up — the far field this encounter happens in the near field of.

The vortex the blade meets is the one its own tip left, and where that wake finally goes is where the wake ends up — the far field this encounter happens in the near field of.

Placed beside the others here, the encounter is a particular kind of memory and worth naming as one.

Most of the memories here are carried by fluid that stays with the flow: a wake convecting away, a displacement accumulating, a separation point relaxing. This one is carried by fluid that the machine comes back to. The rotor is a closed device turning in place, so what it puts into the air it will meet again, at an interval it sets itself.

That makes the relevant time scale neither a convection time nor a diffusion time but a revolution, which is a property of the machine. And it makes the interaction stronger, not weaker, as the machine becomes quieter in every other respect: a rotor with a well-designed blade and low mean loading still lays down a tip vortex, and still flies through it.

The same structure appears wherever a machine’s output returns to its input. A row that meets the row before it is the compressor’s version, where the interval is set by the blade count of the row upstream rather than by a revolution, and the arithmetic of the harmonics is the same.

The two numbers that decide how bad it is #

Because the peak goes as one over the distance and the width goes as the distance, a single dimensionless statement covers both, and it is worth extracting because it is what a designer trades against.

The load is proportional to the vortex’s circulation divided by the miss distance, and the loading rate — which is what radiates — is proportional to the circulation divided by the miss distance squared, times the blade’s speed.

So there are exactly two levers: the circulation the tip sheds, and how close the following blade passes. The circulation is set by the loading at the tip, which is set by the thrust and the span loading, so reducing it costs performance directly. The clearance is set by the wake’s geometry, which is set by the flight condition, and costs nothing when the flight condition allows it.

That asymmetry is why almost everything done about blade-vortex interaction in practice is about geometry rather than about loading — and why the one loading-based fix that is used, a tip designed to diffuse rather than concentrate its vortex, is aimed at the core radius rather than at the circulation. A larger core reduces the peak velocity without reducing the circulation at all, which is the only way to get the load down without paying for it in thrust.

What is done about it #

Three things, and their relative effectiveness follows directly from the two scalings above.

Move the vortex. Since the peak goes as one over the distance, the largest available gain is geometric: a rotor flown so that its wake passes further from the following blade is quieter by the ratio of the distances. This is what is behind approach profiles designed to keep a helicopter out of its own wake.

Diffuse the vortex. A tip vortex with a larger core has a lower peak velocity, so a blade tip designed to shed a diffuse vortex rather than a tight one reduces every encounter. That is much of what a swept or tapered or slotted tip is for, and its gain is bounded by how much the core can be spread.

Break up the timing. Unequally spaced blades put the encounters at more frequencies with less energy in each, which does nothing to the loads and a great deal to how the sound is perceived.

None of the three removes the encounter, because the wake is not optional. A lifting rotor sheds vorticity by Kelvin’s theorem, which is what survives being wound up, and the shed vorticity is exactly what it will meet.

What a measurement of this looks like #

The signature is distinctive enough to be worth describing, because it is how the effect is identified in data rather than inferred from a model.

In a blade pressure trace, it is a sharp bipolar excursion at a fixed azimuth, repeating once per revolution per encounter, and its azimuth moves with the flight condition rather than with anything about the blade.

In a spectrum, it is a set of harmonics of the blade-passing frequency extending to unusually high order — a bump in the mid-frequency range rather than a decaying series, because the pulse is narrow. That mid-frequency bump is the acoustic signature people recognise as a helicopter.

And in a velocity survey, it is the vortex itself: a compact region of high vorticity at a predictable place, which is what the laser-velocimetry programmes went to measure and which settled the geometry that the models had been guessing at.

None of the three needs a model to identify, which is what makes the effect unusually well established compared with most unsteady rotor aerodynamics.

What the picture cannot show #

The pulse is drawn against distance along the blade’s path, which is the natural variable for the geometry and hides the thing an acoustician cares about: the pulse against time, whose derivative is what radiates. The two differ by the blade’s speed, which is constant here and is not on a real blade, since the section speed varies along the span and round the azimuth.

Nothing here shows the vortex either. It is a point in the drawing’s coordinates and a filament in reality, and the figure of the rotor showing where the encounter happens is a diagram rather than a computation.

Where else a machine meets its own past #

The structure — a device whose output returns to its input after a fixed interval — is not confined to rotors, and three other instances in this collection are worth putting beside it.

A propeller in its own slipstream, when the aircraft is descending or manoeuvring, meets the same geometry with a different interval.

A wind turbine in the wake of the turbine in front of it, which is a memory of a different machine and on a much longer clock — minutes rather than fractions of a second, so decay matters and the geometry does not repeat.

And a compressor stage, where a row that meets the row before it computes the same harmonics from the blade counts of two rows rather than from one rotor’s revolution.

What they share is that the forcing frequency is a counting property of the machine and not an aerodynamic one, so it can be predicted before anything is designed and cannot be tuned away by aerodynamic means.

The same encounter, made periodic rather than occasional, is what the essay next to this one is about.

Who found it, and when #

Blade-vortex interaction was identified as the source of helicopter “blade slap” in the 1960s, and the acoustic theory that connects a loading’s time derivative to the radiated pressure is Ffowcs Williams and Hawkings’, from 1969. The measurement programmes that established the geometry — how close the blades actually pass, and in which flight conditions — ran through the 1980s and needed laser velocimetry to do it.

The reason it took that long is instructive. The effect is a few per cent of the loading in a place nobody was measuring, and it dominates a quantity — noise — that was not a design requirement until it became a regulatory one.

Limits recorded rather than smoothed over #

A stationary straight vortex and a point blade. No blade response, no unsteady aerodynamics, no finite chord, no crossing angle, and no wake dynamics. The scalings are the model’s and they are the robust part; the absolute numbers are not predictions.

The desingularisation sets the peak. Inside the vortex core the induced velocity does not go as one over the distance and the model’s one-over-d law must fail. The miss distances used here are all outside a plausible core radius, and a blade passing through a core is a different and much less tractable problem.

The blade’s own response is left out. A narrow doublet is exactly the input an unsteady aerofoil theory smooths most, so the loads here are upper bounds — and putting the response in is the lag that makes flutter possible’s machinery applied at a very high reduced frequency, where the deficiency is close to its high-frequency limit of a half.

No compressibility. A rotor tip is at Mach 0.6 or above, and the acoustic radiation from these encounters is a compressible phenomenon by definition. Nothing here is.

Hover only, in effect. The timing arithmetic is a hovering rotor’s. In forward flight the wake is skewed and swept downstream, the encounters happen at particular azimuths rather than everywhere, and which azimuths depends on the advance ratio — the side that cannot keep up is where that geometry starts.

And the timing arithmetic assumes the wake stays put. It does not: it convects, contracts and descends, so the miss distance is different on each revolution and at each azimuth. The interval between encounters is the reliable number; the geometry of any one of them is not.