The inflow that takes time to arrive

A disc that knows no blades is this collection’s account of momentum theory: a rotor is a disc that puts a pressure jump into a stream, the thrust is the mass flow times the velocity change, and the induced velocity follows from the thrust and the disc area alone.

It is one of the most useful results in the subject and it is a steady result. It says what the induced velocity settles to; it says nothing about how long settling takes.

The answer is that it takes tens of milliseconds, and for a machine being flown by a human or a control law that is not a detail. This essay computes it, and finds that the mechanism is one this collection has already met in an entirely different setting: an added mass.

Where the lag comes from #

Momentum theory balances the thrust against the rate at which momentum leaves through the disc. In steady flow that is all there is, because nothing is changing.

In unsteady flow there is a second term. The air near the disc is being accelerated, and accelerating it takes force over and above what the steady momentum flux requires — exactly the added mass of the mass a body has to borrow, applied to a disc rather than to a body.

So the momentum balance becomes an equation of motion: the apparent mass — 168.4 kg for a six-metre disc of area 113.1 m² — times the rate of change of the induced velocity equals the thrust minus the steady momentum flux. Its steady state is momentum theory and its transient is a first-order lag.

The apparent mass used here is the classical one for a disc — a volume of air of the order of the sphere it fits in, which comes out at 0.637 times the density times the radius cubed. That is a model and it is stated as one; what the essay is about is the structure it produces rather than the exact constant.

The measurement #

Stepping a six-metre rotor from twenty to twenty-six kilonewtons at fifty milliseconds, the induced velocity climbs from 10.2 to 11.6 metres a second.

It reaches sixty-three per cent of the change after 33 milliseconds, against a first-order prediction of 31 from the apparent mass alone. The five per cent difference is the nonlinearity: the momentum flux goes as the square of the induced velocity, so the restoring term is not quite linear and the response is not quite a pure exponential.

That agreement is the essay’s first result. A rotor’s inflow behaves like a first-order system, and its time constant can be predicted from an apparent mass without solving anything unsteady.

Why momentum theory is silent about it #

It is worth being clear about why the steady result contains no hint of the transient, because the reason is general and recurs throughout this collection.

Momentum theory is a control-volume argument. It draws a box round the rotor, equates the thrust to the momentum flux across the box’s faces, and never asks what is happening inside. That is its whole strength: it works for a machine nobody has built, because it does not need to know what is in the box.

A control-volume argument in unsteady flow has an extra term — the rate of change of the momentum stored inside the box — and steady momentum theory drops it because in a steady flow it is zero. Putting it back is exactly what this essay does, and the apparent mass is a way of estimating it without solving for the field inside.

That is the same distinction a rate of change that will not hold still makes in general: the transport theorem has a storage term and a flux term, and an argument that keeps only the second is a steady argument whatever else it says.

So momentum theory’s silence is not an oversight. It is the price of not having to know what is inside the box, and the transient is precisely the part that depends on what is inside.

Which clock it is #

A time constant on its own is not informative until it is placed against the machine’s other times, and the interesting comparison is with the wake.

Three times for the same rotor:

• one blade passage, 0.05 seconds;
• the inflow time constant, 0.031 seconds;
• the time for the wake to convect one radius, 0.62 seconds.

So the lag is shorter than a blade passage and twenty times shorter than the wake’s own establishment. That is a strong statement about what it is and is not.

It is not the wake settling. A rotor that changes its thrust has a wake that is wrong for tens of revolutions afterwards — the tip vortices already laid down were laid down at the old loading, and they have to convect away before the wake matches the new state. At six metres of radius that is 0.619 seconds against the inflow’s 0.033, a factor of nineteen.

The lag computed here is much faster than that. It is the near field adjusting: the air immediately around the disc being accelerated to its new velocity, which happens locally and quickly.

The same fraction at every size #

Across a family of rotors loaded to the same disc loading, the time constant rises linearly with the radius: the apparent mass goes as the cube of the radius and the disc’s momentum flux as the square, so their ratio is a length.

A twelve-fold change in radius moves the time constant by a factor of thirteen and leaves the last column at 0.05066 in every row — the same number to four figures across the sweep, which is what makes the lag a property of the model rather than of one machine. The inflow settles in about a twentieth of the time the wake takes to convect away.

More usefully, the ratio of the lag to the wake convection time is 0.0507 at every size in the family. A dimensionless number that does not move across a sweep is either a constant of the model or a fact about the world, and here it is the first — but it is a useful first, because it means the lag can be quoted as a fraction of a quantity anybody working on rotors already knows.

What the solver computed, and how it was checked #

The momentum balance with the apparent mass in it is integrated forward through a step in thrust. The steady states at both ends are momentum theory’s, so the beginning and the end of the response are not in question; what is computed is the path between them.

Four checks. That the response has a measurable time constant, rather than being instantaneous or never settling. That the measured constant matches the first-order prediction to within a quarter — it reads 0.0330 against 0.0314, a gap of 5.2 per cent — which is loose deliberately: the equation is nonlinear and the check is asking whether the first-order picture is the right one, not whether it is exact. That the lag is shorter than the wake convection time, since a lag longer than that would mean the model was describing the wake rather than the near field. And that the ratio between them is the same across a family of sizes, to within twenty per cent; across a twelve-fold sweep it holds to the last digit of double precision — the check that says the number is a property of the model rather than of one machine.

That last check is the one that would catch the most likely mistake, which is a scaling error in the apparent mass. A constant of the wrong power in the radius would give a plausible time constant at one size and a ratio that drifted across the sweep.

Why this matters to a control law and not to a blade #

The blade’s own asymmetry, which the lag sits on top of, is the side that cannot keep up — a once-per-revolution variation that is a state of the flight condition rather than a memory of it.

The lag is thirty milliseconds, which sounds negligible, and whether it is depends entirely on what is being asked.

To a blade, it is invisible. A blade goes round in two hundred milliseconds and its own aerodynamic time scales are chords over speeds — milliseconds, against the inflow’s 33. The inflow it meets is essentially constant over anything a blade does.

To a pilot or a control law, it is not. A helicopter’s pitch and roll response to a control input is built out of the change in rotor thrust and moment, and thirty-three milliseconds of lag in the inflow is thirty-three milliseconds of lag in that response. The induced velocity here goes from 8.496 to 9.687 metres a second, and reaches 26 per cent of the way at 0.06 s, 79 per cent at 0.10 and 97 per cent at 0.16. Control bandwidths for a rotorcraft are a few radians a second, so a thirty-millisecond lag is a phase lag of several degrees at the crossover frequency — enough to matter for stability margins and more than enough to be felt.

That is why dynamic inflow models exist at all, and why they were developed by the flight-dynamics community rather than by aerodynamicists. The lag is a control problem, and it was found by people trying to match measured handling qualities rather than by anybody computing a flow.

What a measurement of it looks like #

The lag is hard to measure directly and the reason is instructive: nobody can put an instrument in the induced velocity of a manoeuvring rotor and get a number at thirty milliseconds’ resolution.

So it is measured indirectly, through the vehicle. A rotorcraft’s response to a step control input is recorded, and the inflow model’s time constants are the parameters that make the computed response match the measured one. That is a system-identification procedure rather than a fluid measurement, and it has the property that the numbers obtained are the ones that best fit the whole vehicle model — so an error elsewhere in that model appears here.

The check on it is that the identified constants should agree with the apparent-mass predictions, and they broadly do, which is the reason the potential-flow constants are used rather than fitted. When they disagree it is usually because the flight condition is one where the model’s assumptions fail — near the vortex-ring state, or in ground effect, where the apparent mass is not the free-air one.

The honest description is that the lag is predicted from potential flow and confirmed through a vehicle, which is a weaker chain of evidence than most numbers in this collection and is stated as such.

Where the model is thin, and what a real one adds #

The single-state model above captures the uniform part of the inflow and nothing else, and real dynamic inflow models carry three states rather than one.

A uniform component, which is this one.

A fore-and-aft gradient and a side-to-side one, because a rotor that is being pitched or rolled develops a non-uniform inflow across its disc, and those components have their own — and different — time constants. They are what couple the rotor’s flapping to the fuselage’s motion, and they are why the three-state Pitt-Peters model became the standard rather than a single lag.

And in forward flight, a dependence on advance ratio, since the apparent mass a disc must accelerate is different when it is being flown through the air rather than hovering.

None of that changes the structure. It is still a set of first-order lags with time constants set by apparent masses, and the single state computed here is the largest of them.

What kind of memory this is #

Placed beside the others here, the inflow lag is unusual in one respect worth naming.

Most of the memories here are stored in vorticity the flow has left somewhere — a wake, a shed sheet, a vortex the machine will meet again. This one is not: it is stored in the kinetic energy of the near field, in air that is being accelerated and has not finished.

That makes it the rotor’s version of an added-mass effect, and it therefore has the property everything about the start, except one vector establishes for added mass in general: it is a memory of the immediate past only, with no tail. A first-order lag forgets exponentially, so after a few time constants nothing of the previous state remains.

The wake’s memory is the other kind. It has no time constant of its own — the tip vortices are simply there until they convect away or decay, which is a blade that flies through what it shed — and it is the slower of the two by a factor of twenty.

So a rotor has two memories on two clocks, and which one matters depends entirely on the frequency of the question.

Where else an actuator’s output lags its own command #

The structure — a steady theory that is exactly right and takes time to become true — is not confined to rotors, and three neighbours are worth naming.

A propeller, on the same arithmetic, with a smaller radius and therefore a shorter constant. A propeller’s inflow lag is a few milliseconds and is genuinely negligible for handling qualities, which is why the topic belongs to rotorcraft.

A wind turbine, on a much longer one. A ninety-metre rotor at a low induced velocity has a lag of a sizable fraction of a second, and the wake behind it takes minutes rather than seconds to adjust — which is why wind-farm control, where one turbine is deliberately mis-set to help the one behind it, is a control problem with a long dead time in it.

And a wing, where the equivalent is the wake. The lift that arrives late is the same shape of statement for a fixed wing: the steady answer is right and takes tens of chords to arrive. The mechanism there is shed vorticity rather than an accelerated near field, and the two are the two doors of this collection’s inventory arriving in the same machine.

The same lag on a wing rather than a rotor is the first essay in this field, on the same machinery.

What the picture cannot show #

The response is drawn as a single number against time, which is the whole of the model and a small part of the flow. There is no picture of the near field being accelerated, because the model does not compute one: the apparent mass is a scalar standing in for a whole velocity field, and the figure of it would be a figure of an assumption.

Nothing here shows the wake adjusting either, and the wake is where the interesting slow behaviour is. The two would need to be drawn on axes twenty times apart in time to be shown together, which is why they are quoted as numbers instead.

Who found it, and when #

Momentum theory for a rotor is Rankine’s and Froude’s, from the 1860s and 1880s, and reached rotorcraft through Glauert in the 1920s. The unsteady form is much later: Carpenter and Fridovich measured an inflow lag in the 1950s, and the model that became standard — three states, with apparent masses from potential flow about a disc — is Pitt and Peters’, from 1981.

The apparent-mass constants in it come from an old and unrelated calculation: the added mass of a circular disc accelerating in an unbounded fluid, which is a classical potential-flow result. That is the same borrowing this collection makes in the mass a body has to borrow, and it is a good example of a nineteenth-century result being the load-bearing part of a twentieth-century flight-dynamics model.

Limits recorded rather than smoothed over #

One state, and the apparent mass is a model. The constant is the classical disc value and the equation is written by analogy rather than derived from an unsteady solution. The structure — a first-order lag with a constant proportional to the radius — is the robust part.

No blade element in it. The disc here has no blades: the thrust is imposed and the inflow responds, where a real rotor’s thrust is produced by blades whose own incidence depends on the inflow they are sitting in. That feedback tightens the loop and is what a disc that knows no blades is careful to keep separate.

Hover only. Everything above is a hovering rotor. In forward flight the momentum theory itself changes form, the apparent mass depends on the advance ratio, and the single state becomes an approximation to a set.

The nonlinearity is small here and is not always. The response is not a pure exponential because the momentum flux is quadratic, and the five per cent departure measured is for a thirty per cent thrust step. A larger step is less exponential, and a step towards zero thrust is not describable this way at all — the disc enters the vortex-ring state, where momentum theory has no valid solution.

And the 0.0507 is the model’s number. It is the ratio of two quantities both defined by this model, so it is a statement about the model’s internal consistency and about how to remember its result, not a measured property of any rotor.