A row that meets the row before it

A row is not a set of aerofoils establishes what a cascade is: blades close enough together that each one’s flow is decided by its neighbours, so the turning a row produces is not the turning its blades would produce alone.

That is a steady statement about one row. This essay is about what happens when there are two, and it is a statement about time.

A blade in the second row is going round. The blades in the first row are not — or are going round at a different speed. So the second row’s blade passes through the first row’s wakes, one after another, continuously, and every one of those wakes is a velocity deficit that arrives as an incidence change.

The blade is being loaded and unloaded by geometry that belongs to a different part of the machine.

What the blade meets #

The upstream blades each leave a wake: a narrow region of slower fluid, carried downstream and swept round by the machine’s own rotation. A blade in the second row moving relative to the first therefore sees the axial velocity dip once for each upstream blade it passes.

With thirty upstream blades that is thirty dips per revolution. Each takes the axial velocity down to 0.82 of the free stream at its centre and is 0.12 of a blade pitch wide, and they arrive in strict rotation. Since the wheel speed does not change, that dip raises the tangent of the arriving flow angle by 22.0 per cent at the centre of each wake, thirty times a turn.

A velocity deficit at fixed wheel speed is an incidence increase: the axial component has fallen and the tangential one has not, so the flow arrives at a different angle. The blade responds to that angle, so the loading follows the same comb of dips.

Nothing about the blade produced any of it. The deficit itself is the upstream row’s own drag, which is the price of a gradient delivered downstream rather than dissipated where it was made. Its own shape, its own count and its own loading decide how strongly it responds; the row in front decides what it responds to.

A comb at the upstream count #

The spectrum of that signal is a comb. All of the energy is at the upstream blade count and its multiples — thirty per revolution, sixty, ninety and so on — and there is nothing at the downstream row’s own count at all.

The first six lines carry 0.1417 between them, and the sixth is 145 times smaller than the first — so the comb has a length as well as a spacing, and both are set before any aerodynamics is done.

The amplitudes fall along the comb: the first harmonic is 0.066 of the free stream, the sixth is 4.6·10⁻⁴, a fall of 144 times. But they do not fall quickly, and that is the design problem. A narrow wake is a sharp event, and a sharp event has energy far up its harmonic series.

Two consequences follow immediately and both are counting arguments rather than aerodynamic ones.

The forcing frequency is known before anything is designed. Blade count times shaft speed, and its multiples. A blade whose natural frequency lands on one of those will be driven, and the frequency-versus-speed diagram every turbomachine is checked against — the Campbell diagram — is exactly a plot of those lines against the machine’s own modes.

And the count is a design variable. Choosing thirty upstream blades and thirty-one downstream, or counts that share no common factor, changes which harmonics can excite which modes. That is not an aerodynamic decision at all and it is one of the most consequential ones in the machine.

Why a narrow event has a long comb #

The relation between how sharp a wake is and how far its harmonics reach is the whole of the design problem, and it is worth stating as arithmetic rather than as a picture.

A periodic signal made of narrow pulses has a spectrum whose envelope extends to a frequency of order one over the pulse width. Narrow pulses, long comb; wide pulses, short comb. That is the same uncertainty relation that makes a short acoustic click broadband and a long tone narrow, and it does not depend on anything about fluids.

What the fluid decides is only the width, and the fluid widens a wake by mixing. A wake spreads as it convects, at a rate set by its own turbulence, so the width at the downstream row is set by the axial distance divided by the convection speed.

The practical form is a ratio, and five wake widths make it a number. The fourth harmonic measured against the first reads 0.691 at a width of 0.05 pitches, 0.388 at 0.08, 0.119 at 0.12 and 0.0023 at 0.2 — a fall of 294 across a four-fold widening, while the first harmonic itself only rises from 0.031 to 0.086. A wake reaching the next row at a twentieth of a blade pitch excites harmonics well past the fourth order; one reaching it at a fifth barely excites the fourth at all. That is a factor of nearly three hundred in the top of the comb bought by a factor of four in the gap, which is why the gaps in a real machine are as large as the length budget allows and no larger.

How far up the comb depends on the spacing #

The one aerodynamic lever is how much the upstream wakes have spread by the time the downstream row meets them, and the effect of it is dramatic.

A wake five per cent of a pitch wide is a nearly-square dip and its fourth harmonic is 0.53 of its first. A wake thirty per cent wide is nearly a sinusoid and its fourth harmonic is 0.0028 of its first — two orders of magnitude smaller.

The wake spreads by turbulent mixing as it convects, so the width at the downstream row is set by the axial gap between the rows. That makes the gap a noise and fatigue parameter rather than a packaging one — and it is why the gap in a real machine is a compromise between length, weight and how hard the second row is forced.

What the gap cannot change is the first harmonic, which rises with wake width rather than falling — 0.031 to 0.086 across the same four-fold widening — because a spread wake still carries the same total deficit and delivers it over more of the passage. So spacing the rows further apart moves the forcing down the comb rather than removing it: the blade is still loaded thirty times a revolution, but more gently and more smoothly.

What the solver computed, and how it was checked #

The upstream wakes are Gaussian deficits of stated depth and width, convected onto the downstream row and expressed as an incidence perturbation. That is a kinematic model: no cascade solution, no blade response, no mixing calculation, and the wake width is a parameter rather than a result.

Three checks. That the first harmonic sits at the upstream blade count rather than anywhere else, which is the essay’s central claim and is a statement about where in a spectrum to look. That the comb falls across its own length by at least a factor of three, so there is a comb rather than a single tone; it falls by 145. And that a wider wake produces a shorter comb, which is the mechanism the axial gap acts through.

The first of those is the check that would catch the most likely error, which is an indexing mistake putting the harmonics at multiples of the wrong count. A spectrum with peaks in it looks convincing whatever integer it is built on.

What a real stage adds #

The model is one row’s wakes met by another, and a real stage has more in it. Three things, and all three make the problem harder rather than easier.

Potential interaction. A blade row disturbs the flow upstream of itself as well as downstream, because pressure is elliptic — which is pressure has no speed in a machine. So the downstream row forces the upstream one too, at the downstream count, and the two combs are both present.

Wakes from two rows back. A wake is chopped by the row it meets and the pieces convect onwards, so a third row meets a signal carrying both counts and their sums and differences. The spectrum of a multistage machine is a lattice rather than a comb, and predicting which lines land where is a combinatorial exercise before it is an aerodynamic one.

And the blade responds unsteadily. The comb extends to frequencies where the reduced frequency is well above one, and there the lift deficiency is close to its high-frequency limit of a half — the lag that makes flutter possible is the machinery, and its effect here is to attenuate the top of the comb by about half in amplitude while adding a phase that varies along it.

What the blade actually does about it #

The forcing is one half of the problem and the blade’s response is the other, and separating them is what makes the design tractable.

A blade is a resonant structure. Its response to a forcing at frequency $f$ is the forcing times a transfer function which is small everywhere except near its own natural frequencies, where it is large by the reciprocal of the damping — a factor of several hundred for a blade in a compressor, which has very little damping of its own.

So the amplitude of a comb line matters enormously if it lands on a mode and hardly at all otherwise. A sixth harmonic 144 times weaker than the first is more dangerous than the first if the first misses every mode and the sixth lands on one.

That is why the whole of the practical treatment is about coincidence rather than about magnitude. The magnitudes are computed to decide how much margin is needed; the frequencies are computed to decide whether the question arises at all.

And it is why the aerodynamic lever — spacing the rows — is used to attenuate the top of the comb specifically. The high harmonics are the ones most likely to coincide with something, because a blade has many high modes and few low ones.

Why this is a memory rather than a disturbance #

The force the row is exchanging while it does this is the subject of where the reaction to a wing’s lift is, and the answer there — that the reaction is spread over a region rather than sitting on a surface — is why a blade row’s unsteady loading is not a local quantity either.

The distinction is one these essays keep making and it matters for how the problem is posed.

A disturbance is something the machine meets from outside: inlet distortion, a crosswind, atmospheric turbulence. It is unpredictable, it is characterised statistically, and the response to it is a transfer function applied to a spectrum.

This is not that. The signal a blade meets is the machine’s own output, delivered back to it after a known interval, and every feature of it is deterministic: the frequency is a blade count, the phase is a relative position, and the amplitude is the upstream row’s own loading.

That is why it is designed against rather than allowed for. The contrast with a genuinely random input is the one what a mean profile cannot tell anybody draws: a statistical description throws away exactly the phase information that a deterministic forcing consists of. A statistical disturbance is covered by a margin; a deterministic forcing at a known frequency is avoided by moving either the frequency or the mode, and that is a decision that gets made once, at the counts.

It is the same structure as a blade that flies through what it shed, where a rotor meets its own wake one blade passage later. There the interval is a fraction of a revolution and here it is a blade pitch; in both the clock is the machine’s own count.

What is done about it #

Choose the counts. Two rows whose blade counts share a large common factor put many blades in phase with one another, so the excitation is concentrated in a few nodal-diameter patterns. Counts that share no factor spread it. This is the first decision and the cheapest.

Space the rows. Increasing the axial gap spreads the wakes and truncates the comb, which reduces the high-order excitation by orders of magnitude and the first harmonic hardly at all.

Lean or sweep the blades. A blade whose leading edge is not radial meets a wake at different radii at different instants, so the forcing is smeared in time and its harmonics are reduced — the same argument as the crossing angle in a rotor’s blade-vortex encounter, and the same conclusion.

And do not put a mode on a line. The Campbell diagram exists so that no blade natural frequency crosses a multiple of the blade-passing frequency inside the operating range, and when one must, it is crossed quickly.

The same comb, heard rather than felt #

The identical forcing produces two entirely different engineering problems depending on which frequency band it is read in, and the two are worth putting together because they are usually treated by different people.

Read as a load, it is a fatigue problem. Thirty cycles a revolution at ten thousand revolutions a minute is five thousand cycles a second, which is 10⁹ cycles in a couple of days of running — so a blade lives entirely in the high-cycle regime, where the allowable stress is a small fraction of the static one and the margin is set by the alternating component alone.

Read as a pressure, it is a noise problem. The same comb radiates, and the fundamental at blade count times shaft speed is the buzz-saw tone a fan produces. Its propagation is governed by whether the corresponding spinning pattern is cut off or cut on in the duct — which depends on the nodal diameter, the duct radius and the Mach number, and is why some harmonics escape the intake and others do not.

Both are decided by the same three integers: the two blade counts and the harmonic order. What a jet keeps is a reminder that the other large noise source in a machine of this kind has nothing deterministic about it at all, which is exactly why it is treated statistically and this is not.

What the picture cannot show #

The signal is drawn as a function of angle at one radius, and a real wake is a three-dimensional object: it varies along the span, it is skewed by the radial pressure gradient, and it carries a temperature deficit as well as a velocity one in a turbine. None of that is here.

The spectrum is drawn as amplitudes with no phases, which is exactly the half of a forcing that does not determine whether it excites anything. Whether a comb line drives a particular blade mode depends on the nodal diameter it corresponds to — the pattern of phases round the annulus — and two forcings of equal amplitude at the same frequency can differ by everything in what they excite.

Where the wake came from, and why it is unavoidable #

It is worth saying plainly that the forcing cannot be designed away, because the wake is not a defect.

A blade that turns the flow has a boundary layer on both surfaces, and the two leave the trailing edge as a momentum deficit. That is not a sign of a bad blade: it is the drag, and every blade that does work has one. A perfectly designed row still leaves thirty wakes per revolution.

The only way to have no wake is to have no boundary layer, which is to have no viscosity, which is the theory that solves everything and its zero drag. So the forcing on the second row is a direct consequence of the first row being real, and its magnitude is proportional to the first row’s own loss.

That gives a small and genuine design coupling worth knowing: a more efficient upstream row forces the downstream row less, because a thinner wake is a smaller deficit. It is one of the few places in a turbomachine where the performance objective and the mechanical one point the same way.

The single encounter this is the periodic version of is the essay before it, on the same machinery.

Who found it, and when #

The recognition that a blade row’s unsteady loading comes from its neighbours is as old as axial compressors, and the systematic form of it is Kemp and Sears’, from 1953-55, who computed the response of a cascade to a convected wake and to a potential disturbance separately.

The counting arguments — which blade counts excite which nodal diameters, and therefore what to choose — are Tyler and Sofrin’s, from 1962, and were written about fan noise rather than about fatigue. The two problems are the same forcing read at two different frequencies, which is why the same rule appears in both literatures with different names.

Limits recorded rather than smoothed over #

A kinematic model. Gaussian wakes convected onto a downstream row, expressed as an incidence. No cascade solution, no blade response, no mixing, and the wake width is an input.

One radius, one row pair. A real machine is a stack of stages and the spectrum is a lattice; the comb here is the simplest possible case and is the building block rather than the answer.

The deficit is taken as constant along the wake. It is not: a wake weakens as it spreads, and the model here spreads the width without reducing the depth in proportion, so the first harmonic in the widest case is a little too strong.

No radial variation. The wake is a two-dimensional deficit at one radius. A real wake is a three-dimensional object with a radial structure of its own, and its interaction with the downstream blade varies along the span — which is the mechanism blade lean and sweep act through, and which is therefore absent from the model that motivates them.

And no phases. Everything above is amplitudes, and the quantity that decides whether a line matters is a phase pattern round the annulus, which this model has no annulus to define.