Weighing what is missing

The first rung of this ladder draws a control volume round a wing and shows that the usual momentum sum over it is wrong. The second shows a right total arrived at from a wrong picture. Both are about lift, and both are exact because the flow inside the box was solved exactly.

The same box, asked for the drag, becomes something quite different: a measurement technique, on a flow nobody has solved, that is how the drag of a real aerofoil section is known at all.

The technique, and why it exists #

A wing section in a tunnel can have its drag measured on a balance, and the measurement is bad. The tunnel itself is part of the answer — the walls are in it and the correction is a series — and a small force measured against a large one is where that shows worst. The drag of a two-dimensional section is a small force — a hundredth of the lift — and a balance measuring it has to subtract the drag of the model supports, the tare of the mounting, and the wall interference, each of which is comparable with the answer.

A wake survey avoids all of it. Traverse a rake of total-pressure probes across the wake a chord or two behind the section, measure the velocity deficit, and integrate:

$$D' = \rho\int u\,(U - u)\,dy.$$

No balance, no supports to subtract, no pressure taps on the model. The drag is what is missing from the air, weighed.

That is the same statement as a wake keeping the drag and forgetting the body: whatever produced the deficit, the deficit is the drag, and a survey does not need to know which part of the body it came from.

The integral next to it, which is wrong #

The commonest error in the technique is to integrate the wrong thing, and the two candidates look almost identical.

$$\underbrace{\rho\int u\,(U-u)\,dy}_{\text{the drag}} \qquad\text{against}\qquad \underbrace{\rho U\int (U-u)\,dy}_{\text{not a drag}}$$

The second is the mass deficit — how much less air is passing than would pass at free-stream speed — and it is $\rho U^2 \delta^*$, the displacement thickness appearing in its other role. It is always larger, because $u < U$ inside the wake makes the first integrand smaller everywhere.

How much larger is decided by the wake’s depth, and this is the interesting part. The two integrands differ by a factor of $u/U$, which is near one in a shallow wake and small in a deep one. At the momentum thickness drawn above the mass integral overstates the drag by 9.6 per cent; at a twentieth of it, by 0.4; at twice it, by 25.

That produces a conclusion nobody would guess from the arithmetic alone: the error is smallest exactly where a survey is properly done. A survey plane placed far downstream, where the wake has spread and shallowed, is a plane where the wrong integral very nearly gives the right answer — and a plane placed close in, where a hurried measurement is easiest, is where it does not.

What a finite rake misses #

The second assumption is that the traverse captures the whole wake, and a wake has no edge.

A wake profile approaches the free stream and does not reach it. The consequence is that the answer depends on where the traverse was stopped, and the amount lost is set by the profile’s tails rather than by any width that could be quoted.

The three shapes drawn carry identical momentum thicknesses and need different rakes. At two wake widths either side of the centreline, a Gaussian wake has lost about a per cent and the turbulent shape several times that, because the second’s deficit decays more slowly at large distance.

So a rake sized for one measurement is the wrong size for another, and the practical rule that follows is to traverse until the deficit is indistinguishable from noise and then keep going — which is what an experimenter does and is rarely stated as a consequence of a profile shape.

The assumption that can be removed #

The third assumption is the one everything above rests on: that the static pressure at the survey plane has recovered to the free stream’s. Close behind a body it has not, and the simple integral is then measuring something that is not the drag.

The reason the naive answer is low is worth following. A probe at a plane where the static pressure is below ambient reads a total pressure; a reduction that assumes the pressure has recovered turns that reading into a velocity that is not the local one. And downstream, as the pressure climbs back to ambient, the wake fluid decelerates further — the deficit is not finished growing — the wake is still climbing a pressure rise the way any decelerating flow does — so the drag is larger than the plane’s own velocities suggest.

Jones’ correction removes the assumption entirely. Measure the static pressure at the plane as well as the total, and the drag becomes

$$D' = \rho\int u\,(U - u_2)\,dy, \qquad u_2 = \sqrt{u^2 + \tfrac{2}{\rho}(p - p_\infty)},$$

with $u_2$ the speed that streamtube’s fluid will have once the pressure has recovered. It is exact at any station, and the figure’s check is not a rearrangement of the same algebra: the reference value is a plain momentum integral done at a recovered plane, with every streamtube carried across by its own mass, and the two agree to machine precision.

One more pressure tapping buys the freedom to survey anywhere behind the body, which is what makes the technique usable in a tunnel whose working section is a few chords long.

The one that cannot be removed #

The third assumption is that the flow is two-dimensional, and nothing in this essay checks it because nothing can. It is a property of the apparatus rather than of the reduction.

A wake survey measures the deficit in one plane and assumes it is the same in every plane along the span. A real tunnel has boundary layers on its side walls, those layers interact with the model’s own, and the flow near the walls is emphatically not the flow at the centre — so a traverse at mid-span reports the mid-span drag and is silent about what the rest of the span was doing.

And a three-dimensional wing cannot be surveyed this way at all, or rather it can and the answer means something different. The wake behind a finite wing contains the trailing vortex system, whose kinetic energy is the induced drag, and a momentum survey of one plane picks up part of it in a way that depends on where the plane is and how far the vortices have rolled up. That is a much harder measurement and it is the reason induced drag is usually computed rather than surveyed.

What the survey is actually weighing #

It is worth being explicit about what the momentum deficit is, because the technique’s name — weighing what is missing — invites a reading the first rung of this ladder has already refuted.

The deficit is not a quantity of air that the body took away. It is a momentum deficit: the air in the wake is the same air that would have been there, moving more slowly, and the amount by which its momentum falls short of the free stream’s is the impulse the body gave it.

That distinction is the whole content of what a wing carries and does not. The mass integral counts a deficit of stuff, which is the displacement thickness and is a statement about how far the outer flow was pushed aside; the momentum integral counts a deficit of motion, which is the drag. Using the first for the second is not a small numerical slip — it is the same category error, arriving from the measurement side.

And the accounting is complete rather than approximate. Everything the body did to the air is in the plane the survey crosses, provided the plane is far enough back that the pressure has recovered or Jones’ correction has been applied. There is no leakage, no term left on the body, and nothing that has to be estimated: whatever mechanism produced the drag — friction, separation, a shock — the wake carries it and the integral catches it.

That is the property the first two rungs of this ladder establish for lift and this one uses for drag. A control volume does not need to know what is inside it.

Where a survey disagrees with a balance, and who is right #

Two independent measurements of the same quantity disagreeing is the most useful thing that can happen to an experimentalist, and the disagreement here has a known catalogue of causes.

The survey reads low if the traverse was too narrow, by the amount the figure above computes.

The survey reads low if the plane was too close and the reduction was naive, by the amount the Jones figure computes.

The balance reads high by the drag of the supports, which is subtracted from a separate run and never subtracted perfectly.

And the balance reads the whole span while the survey reads one station, so on a model with any spanwise variation they are measuring different things and neither is wrong.

The catalogue is the point. Every entry has a sign that is known in advance and a size that can be estimated, so a disagreement between the two is diagnostic rather than merely troubling — it says which of the four is dominating, and the practice of running both and comparing is how a tunnel’s own corrections get calibrated.

That is a habit worth carrying past this subject. Two instruments with different systematic errors are worth more than one better instrument, because the difference between them is a measurement of the errors themselves.

Why the box is the right tool for this and not for lift #

There is an asymmetry between the two questions this ladder asks of the same control volume, and having done both it is worth stating.

For lift, the box is a way of understanding. The first rung shows that the usual momentum sum over it is wrong, and the correct sum gives a total that the circulation account gives more directly. Nobody measures a wing’s lift with a momentum survey; the box is there to settle an argument about mechanism.

For drag, the box is a way of measuring, and there is no more direct route. Drag is what viscosity did, no exact solution exists for it on any real section, and a wake survey is the way the number is known. The box is not settling an argument here — it is the instrument.

The reason for the asymmetry is what each quantity is made of. Lift on a two-dimensional section is an inviscid quantity, computable from a solution that exists; drag is a viscous one, computable only from a solution nobody has. A control volume is most valuable exactly where the interior cannot be solved, and it is a pedagogical device where the interior can.

That is the general form of what the applied field’s own summary says: a control volume does not need to know what is inside it. The two rungs below use that as a check on an argument, this one uses it as a licence to not know, and the second use is the one the method was invented for.

What the picture cannot show #

No wake was solved. Every profile here is prescribed — a Gaussian, a laminar-like shape, a turbulent-like one — and scaled so that its momentum thickness is exactly the stated number. That is deliberate: the essay is about what a reduction does to a profile, so the profile has to be a known quantity rather than a computed one.

Incompressible throughout. Jones’ form has a compressible version and it is not this one. Above about Mach 0.3 the density varies across the wake and the integrals change shape.

The static-pressure defect is modelled. Its depth is a parameter and its shape is taken as the wake’s own, which is a fair description of a near plane and is not a solution of one.

And nothing here is a real rake. A physical traverse has finite probe spacing, probe interference, a settling time per station, and a total-pressure probe whose reading depends slightly on the flow angle it sees — every one of which is a real source of error and none of which is in this model.

The assertion behind the figures is the one that could reject at several points. A full traverse must return the momentum thickness the profile was built to have; the mass integral must overstate the drag and must do so more as the wake deepens; a widening traverse must miss monotonically less; and Jones’ form must be exact where the naive one is out by a tenth, measured against a quadrature that shares no arithmetic with it.

Who found it, and when #

Betz proposed the wake-survey method in 1925 and Jones gave the form used today in 1936 — the paper is The Measurement of Profile Drag by the Pitot-Traverse Method, and its contribution is exactly the removal of the recovered-pressure assumption.

What makes Jones’ form the one that survived is its economy. Betz’s version also handles an unrecovered plane and requires a reference measurement outside the wake at the same station; Jones’ needs only the total and static pressures at each point of the traverse, which is what a rake with a static tube in it already gives. A method that needs no extra apparatus is the method that gets used.

The technique’s later history is a story about what it could not do. Two-dimensional profile drag it measures beautifully and it remains the standard; three-dimensional drag it measures ambiguously, and the modern descendants — wake integration into vortex drag, entropy drag and profile drag separately — are attempts to decompose a survey plane into terms that mean different things, and they are still argued about.

The number a survey actually produces #

It is worth putting the technique’s own precision in context, because the errors this essay computes sound large and the measurement is regarded as an accurate one.

A well-run survey of a two-dimensional section reports a drag coefficient to within about one per cent, and the figures above show errors of ten and twenty-five. The reconciliation is that every one of those errors is a mistake rather than an uncertainty: they are what happens when the wrong integral is used, the rake is too narrow, or the pressure assumption is made where it does not hold. Each has a known sign and a computable size, so each is removed rather than tolerated.

What is left after they are removed is instrument noise, probe spacing and the two-dimensionality of the apparatus — and those are the one per cent.

That is a much better position than a measurement whose errors are unknown, and it is worth stating because the essay has spent its length on things that go wrong. The technique is trusted precisely because its failure modes are enumerable and each of them is arithmetic.

A last number for scale. A section at a Reynolds number of a few million has a drag coefficient near 0.006, so a wake carrying a momentum thickness of about three thousandths of a chord — a deficit a few per cent deep, a few per cent of a chord wide, in a stream nobody would notice was disturbed. The whole of the drag of an aerofoil is a shallow dent in the air behind it, and the fact that it can be weighed at all is the achievement.

Where the ladder goes next #

This anchor has now taken the control volume through lift, through a wrong picture giving a right total, and through a measurement. What it has not done is the decomposition the last paragraph names.

The rung above is the drag breakdown from a wake plane. A survey behind a three-dimensional wing contains, in one integral, the profile drag the section made and the induced drag its trailing vortices carry — and separating them means writing the plane’s integral in terms of quantities that distinguish the two: an entropy or total-pressure defect for the first, a cross-flow kinetic energy for the second. That decomposition is standard in the modern literature, it is not unique, and different formulations give different splits of the same total, which is a genuinely interesting state for a measurement to be in.

The one beside it is the energy account this ladder keeps naming and has never done. The momentum box gives a force; the same box asked about energy gives a power, the trailing vortex system’s kinetic energy is the induced drag times the speed, and the accounting closes — with a different set of terms that are individually more physical and collectively harder to measure.