A distorted inflow gives back the tone more blades took away

More blades hush a propeller only while its tips are slow took a propeller’s steady thrust and torque, which are constant in the blade’s own frame and so make no sound held still, and carried them round a circle. Gutin’s formula of 1936 says how loudly they sing: harmonic $m$ of the blade-passing frequency of $B$ blades radiates through the Bessel function $J_{mB}(mBM\sin\theta)$, where $M$ is the rotating force’s Mach number and $\theta$ the angle from the axis. A Bessel function is exponentially small until its argument approaches its order, so raising the order — spreading the same thrust over more blades — silenced a slow propeller’s fundamental by twenty-five decibels and a fast one’s by one.

The essay ended on the condition that made that true: the thrust was steady. A propeller behind a wing, close to the ground, at a static test or in a crosswind does not see a uniform inflow. Each blade, as it goes round, meets faster and slower air in different parts of its circle, and its thrust rises and falls with the blade’s azimuth. That fluctuation is a force the blade carries too, and it radiates by the same mechanism — through a Bessel function whose order is not the number of blades.

A thrust that changes round the circle #

The model gives each blade a thrust that is its share of the mean, times one plus a sum of harmonics of its azimuth: $T_b(\varphi) = (T/B)\,[1 + \sum_s a_s\cos(s\varphi + \psi_s)]$. Every blade cuts the same distortion as it passes the same place, so the pattern is fixed in space and the radiated sound is still periodic at the blade-passing frequency — the distortion does not change the tone’s pitch, only its strength. Writing each harmonic of the distortion as a pair of rotating components, $e^{is\varphi}$ and $e^{-is\varphi}$, and carrying each through Gutin’s derivation, the sound at the $m$-th harmonic of the blade-passing frequency gets a contribution from distortion harmonic $s$ of

$$p_{m,s} \propto \frac{mB\Omega\cos\theta}{2\sqrt2\,\pi c r}\;\frac{T a_s}{2}\,\Big[J_{mB-s}(x),\ J_{mB+s}(x)\Big], \qquad x = mBM\sin\theta,$$

the two orders adding with phases set by the distortion’s. The steady load is the term $s = 0$, Gutin’s $J_{mB}$. When the distortion’s phases are not known — which is the case for an inflow that is turbulent rather than designed — the expected power is the sum of the squares, and that is what is drawn. The formula is checked against an independent route: a blade with a single thrust harmonic, carried round the circle and its far field integrated by retarded time, gives the same harmonic levels to six parts in a million.

Bessel functions that need no speed #

The whole effect is in the orders. Six blades put their steady load through $J_6$, and at a tip Mach number of 0.4 that is a thousandth: the rotating force’s speed towards the listener is far below the speed of sound, its successive contributions arrive almost in step, and they cancel almost perfectly — the cancellation the sound is what does not cancel made the theme of every aerodynamic noise source. A distortion harmonic $s$ goes through $J_{6-s}$. The fourth harmonic’s order is two, and $J_2$ is a third at the same argument; the sixth harmonic’s order is zero, and $J_0$ is 0.28 and rises towards one as the tip slows. A blade whose thrust varies six times round its circle, on a six-bladed propeller, makes a sound at the blade-passing frequency that does not depend on the rotation being fast at all: each blade’s thrust peaks as it passes the same six points, every blade does, and the result is a force pattern that pulses at the blade-passing frequency in place, a dipole like the wire’s, which needs no motion to radiate.

That is why the distortion matters more to six blades than to two. Two blades put their steady load through $J_2$, already a sizable number at low speed; the distortion’s harmonics add $J_1$ and $J_0$ contributions in proportion to their small amplitudes, and the steady tone stays louder at every tip speed drawn. Six blades had bought their quietness by pushing the steady load to $J_6$, and a distortion reaches around the purchase.

Where it is loudest #

The two tones go to different places. The steady load’s tone vanishes on the axis — there $\sin\theta = 0$ and a Bessel function of non-zero order is zero — and grows towards the disc’s plane, where the rotating force moves most directly towards the listener. The distortion’s tone comes mostly from the harmonic whose order is zero, and $J_0$ is largest where its argument is smallest, which is on the axis: it is a fluctuating thrust, pointing along the axis, radiating as a dipole whose lobes are fore and aft. A listener ahead of or behind a six-bladed propeller in a distorted stream hears the distortion and nothing else; one in the plane of the disc hears the steady tone if the tips are fast and the distortion if they are slow.

That directivity is also how the two are told apart in a test. A static run of a propeller on a stand is the worst case for distortion: with no forward speed the inflow contracts strongly into the disc and turbulent eddies in the surrounding air are stretched into long tubes that every blade cuts in turn, which is exactly a fixed distortion; and the result, measured on the axis, is a blade-passing tone far louder than the steady-load theory predicts. A propeller measured on a stand and then in flight can lose much of its tone without any change to its blades.

When the steady tone wins again #

The steady tone grows fast with the tip’s speed, the distortion’s slowly, so above some tip Mach number the steady one wins again. That crossing is the earlier essay’s question, and it depends on both the blade count and the distortion. With five per cent rms, four blades’ steady tone takes over above tip Mach 0.35, six blades’ above 0.59, eight blades’ above 0.72; with two per cent the crossings move down to 0.22, 0.46 and 0.60. Every extra blade pushes the steady load to a higher order and so raises the tip speed it needs, while the distortion’s harmonics always include one of order near zero.

The distortion’s scale hardly enters. Its spectrum here is turbulence’s, met along the circle at the effective radius, with an integral scale of half that radius; at a fifth of the radius or twice it, the crossings move by a few hundredths. For the fundamental of a propeller with up to eight blades, only the first eight or so harmonics of the distortion matter, and over that range a turbulence spectrum is nearly flat whatever its scale. What decides the crossing is how strong the distortion is.

What more blades still buy #

The earlier essay asked whether the many-bladed propeller’s advantage survives the turbulence it actually flies in. At tip Mach 0.5, in a clean stream, six blades are quieter than two by twenty decibels. In a two per cent distortion — the order of the turbulence in a calm atmosphere seen by a propeller in cruise — the advantage is eighteen. In five per cent, typical of a propeller behind a wing or a pylon or at a static test, it is thirteen; in ten per cent, eight. The curve flattens: in clean air each added blade buys about five decibels all the way to eight, while in a five-per-cent distortion the third and fourth blades still buy four each, the sixth a decibel and a half and the eighth nothing, because every blade added thereafter pushes the steady tone further down and leaves the distortion’s tone, now the louder, where it was.

So the advantage survives, reduced, and its size is a property of the installation rather than of the propeller. That is the familiar experience with pusher propellers behind wings and with open rotors behind pylons — configurations that are noisier than their blade counts promise, and are noisier at exactly the blade-passing frequency.

How long a distortion stays frozen #

The model’s distortion holds still for a revolution, and whether a real inflow does is a matter of two times. The propeller of the earlier essay turns at 2,700 revolutions a minute, a revolution in 22 milliseconds. On a stand, the air passes through the disc at the momentum theory’s induced speed, $\sqrt{T/2\rho A}$ — about seventeen metres a second for two kilonewtons on a disc of 2.8 square metres — so an eddy of the scale used here, about 0.38 metres, would cross the disc in one revolution if it were not stretched. It is stretched: the contraction into the disc lengthens it along the axis by the ratio of the far and near stream areas, many times over, and a stretched eddy takes many revolutions to pass. Each blade cuts it again and again, and its loading is periodic to a good approximation — the frozen distortion of the model.

In flight at sixty metres a second the same eddy crosses the disc in six milliseconds, a quarter of a revolution, and is barely stretched. Each blade then meets it once, and the fluctuating thrust it makes is a pulse rather than a harmonic: its sound spreads into a broad hump round the blade-passing frequency rather than adding to the tone. The tone figures here are therefore the static and low-speed case, and in fast flight the distortion’s contribution to the tone falls back towards zero while its contribution to the broadband noise remains — which is the more precise form of the statement that a propeller is quieter in the air than on the stand. Distortions that are fixed to the airframe, a wing’s wake or a pylon’s, are frozen at any speed, and for them the model holds in flight as well.

The same arithmetic in a jet engine’s fan #

The orders $mB - s$ have a famous relative. A turbofan’s fan is followed by a row of stator vanes, and the fan blades’ wakes sweep across those vanes, loading them at the blade-passing frequency; equivalently the vanes’ potential field is a distortion with $V$ lobes that every fan blade cuts. The interaction radiates in spinning patterns whose circumferential order is $mB - kV$ — Tyler and Sofrin’s rule of 1962, which every fan designer uses. Inside the engine’s duct a pattern of high enough order cannot propagate at the blade-passing frequency: it is cut off and decays along the duct. So fans are built with roughly twice as many vanes as blades, to make the lowest interaction order, $B - V$, large, and the fundamental’s interaction tone dies before it reaches the inlet.

The propeller has no duct, so nothing is cut off; but the Bessel function plays exactly the part the duct does. A pattern of high order $mB - s$ spinning at the blade-passing frequency has a phase speed at the blade tips below the speed of sound, and such a pattern radiates exponentially weakly — that is the free-field version of cut-off, and it is the reason the steady load, of order $mB$, is quiet. A distortion whose harmonic $s$ lands near $mB$ creates a pattern of order near zero, which is never cut off, in a duct or out of one. Tyler and Sofrin’s rule for the fan and Gutin’s for the propeller are the same statement about which spinning patterns can reach a listener; the fan designer controls $V$ and so controls the orders, and the propeller designer, whose distortion is the installation’s, cannot.

A static test, in numbers #

A propeller run on a stand in still air sees the worst inflow it will ever meet. The air drawn into the disc accelerates from rest, eddies in the surroundings are stretched into long tubes along the axis, and each blade cuts the same tube many times before it drifts away: a distortion of ten or fifteen per cent is not unusual. At tip Mach 0.6 the six-bladed propeller’s fundamental in clean air is 117 dB re 1 pW, fourteen below the two-bladed one’s 131. In a ten per cent distortion it is 124, in fifteen per cent 127 — the advantage over two blades shrinks to seven and five decibels, while the two-bladed propeller’s level has hardly moved.

On the axis the contrast is starker, because the steady tone has nothing there. In clean air the six-bladed propeller is silent five degrees off its axis, below any measurable level; in a ten per cent distortion it gives 87 dB at thirty metres, louder than the two-bladed one’s 83. A microphone placed on the axis of a stand measures almost nothing but the distortion, and a design chosen on the basis of such a test is chosen on the installation, not on the propeller. That is the case for testing propellers in flight, or in a wind tunnel with a forward speed large enough to stop the inflow contracting — and the reason flight measurements of many-bladed propellers so often come out quieter than their static tests promised.

The convention, and what the model holds fixed #

Levels are sound power in decibels relative to a picowatt, integrated over a sphere, for the fundamental only; pressures are rms at 30 metres. The propeller is the earlier essay’s — 1.88 metres, two kilonewtons, 127 kilowatts — with its thrust and shaft power held as the rotational speed is changed to set the tip Mach number, and its loads concentrated at 0.8 of the tip radius. The distortion is fixed in space over a revolution: every blade cuts the same pattern. That is the limit of an inflow whose non-uniformity changes slowly compared with a revolution — a wing’s wake, a ground vortex, or turbulence stretched into long tubes by the contraction at a static test. A turbulent inflow whose eddies pass through the disc in less than a revolution makes broadband noise around each harmonic instead of a pure tone, and the model’s tones are then the tops of humps. Only the thrust fluctuates; a real blade meeting a changing inflow angle changes its drag too, which adds a torque fluctuation through the same orders. The distortion’s phases are treated as random, so the drawn power is an expectation; one realisation can be a few decibels louder or quieter.

Who heard it #

Gutin’s steady-loading theory is of 1936. Lowson extended it to blade forces with harmonics in the 1960s, deriving the orders $mB \mp s$; Hanson’s frequency-domain theory of the 1970s and 1980s carried it to swept blades and distorted inflows, and Majumdar and Peake worked out the static-test tone from stretched turbulence in the 1990s. That the many-bladed propeller’s advantage is a clean-air property is widely known to propeller and fan designers; the crossing tip speeds here are this model’s, for this propeller and this distortion.

How it was checked #

A blade carrying only one thrust harmonic — the first, second or third, at three angles and two harmonics of the blade-passing frequency — was integrated by retarded time, averaged over two phases a quarter-turn apart to cancel the cross term between the two orders, and agrees with the formula to $6 \times 10^{-6}$. The Bessel functions of all orders are computed together by Miller’s backward recurrence and agree with the earlier essay’s series and integral to $2 \times 10^{-15}$. The distortion’s harmonics carry the stated rms to rounding, and with no distortion the model is Gutin’s. It refuses a negative rms and a scale of zero.

Still open: eddies that pass in less than a revolution #

Every distortion here is frozen for a revolution, which makes each harmonic a pure tone. Real inflow turbulence is not: an eddy passes through the disc in a time set by the stream’s speed and its own size, and each blade cuts it a finite number of times — many at a static test, where the eddy is stretched and slow; few in cruise. The next calculation would give the distortion a correlation time, so that each blade sees a thrust that is periodic only for that long, and ask how the tone broadens into a hump and how much of its energy remains at the blade-passing frequency — which decides whether a microphone, or an ear, reads it as a tone at all.