What the far field remembers

Stand far enough from a body in a stream and it stops being a shape. What arrives at a distant observer is a disturbance with a small number of parts, each dying at its own rate, and the parts that die fastest have taken every detail of the shape with them.

The counting is short. In two dimensions the disturbance far from any body is a vortex, plus a source, plus a dipole, plus terms that fall off faster — and the coefficients of those three are all that the distance preserves.

What survives, and at what rate #

Expand the complex potential of any two-dimensional body in a uniform stream about a point inside it, and the terms are

$$w(z) = U + \frac{-i\Gamma + Q}{2\pi z} + \frac{\mu}{z^2} + \frac{\text{octupole}}{z^3} + \cdots$$

The circulation $\Gamma$ and the net outflow $Q$ share the $1/z$ term; the dipole $\mu$ — which is what a closed body’s thickness looks like from far away — is the $1/z^2$; and everything about the shape beyond its gross size is in the terms below.

Multiplied by a perimeter that grows like $r$, only the first of these survives to infinity, which is the reason a body’s lift is a circulation and nothing else and a doublet exerts no force at any radius. That much this collection has already established. This essay is about what the higher terms take with them when they go.

Two bodies, matched deliberately #

The demonstration is a construction. A circular cylinder of radius $a$ in a stream $U$ is exactly a doublet of strength $2\pi U a^2$ — that is the standard solution, and it has no higher multipoles at all. A Rankine oval built from a source and a sink of strength $m$ separated by $s$ has a far-field dipole of $ms/2\pi$, so choosing $m = 2\pi Ua^2/s$ matches the two.

The oval that results is 1.17 radii long and 0.94 radii tall against the circle’s one and one — a visibly different body, blunter at the front and drawn out along the stream — and it is built from the same singularities every body on this site is built from.

Comparing the two flows on circles of growing radius gives the numbers:

The exponent is the result. The difference dies two full orders faster than the disturbance itself, so the fraction of the signal that carries any shape information falls as $1/r^2$. At ten radii it is a part in a thousand; at a hundred, a part in ten thousand of a signal that is itself a part in ten thousand of the free stream.

Why this is the same argument as the rest of the phase #

A limit taken at infinity is an average. It is not an average over time or over realisations, but it throws information away in exactly the same manner: many different bodies map to the same far field, the mapping cannot be inverted, and what is lost is identifiable rather than mysterious.

That gives the far-field description the same double character as every other average in this collection. It is exactly right about what it keeps — the circulation is the circulation, and Kutta–Joukowski holds on every contour to nine figures — and silent about everything else. A measurement made far away is not an imprecise measurement of the body; it is a precise measurement of three numbers.

The other body that shares a far field: a wing #

The most consequential version of this equivalence is not a cylinder and an oval. A lifting wing’s far field, at large distance, is a horseshoe vortex with the wing’s own circulation — and every detail of how that circulation is distributed along the span has gone.

That is why the span is the whole story for induced drag and the planform is worth six per cent: what the wake sees is set by the loading’s first moments, and two wings with different plans but matched loading leave nearly the same field behind them. The same statement in its most useful form is the wake plane, where the entire three-dimensional problem is replaced by a two-dimensional one drawn far downstream.

What it means for measuring things #

A wake survey measures a drag and not a body. Momentum-deficit measurements work precisely because the far field carries the force faithfully, and they cannot be inverted to say what produced it — a fact that is a convenience rather than a limitation, since the force is what was wanted.

A pressure tapping in the far field measures very little. The disturbance falls as $1/r^2$ for a closed body, so at ten radii it is one per cent of the free-stream dynamic pressure and at fifty it is four parts in ten thousand. Tunnel-interference corrections exploit the same decay from the other direction: because the disturbance dies fast, a model that is small enough compared with the working section is only weakly aware of the walls — and how weakly is a computable number.

And an inverse problem is ill-posed in a specific, quantifiable way. Recovering a body from distant measurements requires the coefficients of terms that are exponentially small in the ratio of scales — a difficulty this collection has already met in the amplification of Cauchy data, where a perturbation of size $\epsilon$ at one wavenumber grows as $e^{kL}$ when the problem is run the wrong way. Inverse design works by not doing that: it prescribes a pressure on the surface rather than inferring a surface from a distance.

The one channel that does not fade, and what it forgets too #

The list of survivors above is a list for the potential flow, and it names the wake only to set it aside. The wake deserves finishing, because it turns out to obey the same rule with a fourth number in it — and because the way it obeys the rule is the reason a drag can be measured at all.

A viscous wake carries a momentum deficit, and the deficit’s integral across the wake is fixed forever:

$$D = \rho U\!\int u_d\,dy .$$

Nothing downstream changes it. Viscosity spreads the deficit sideways and lowers it in the middle, turbulence mixes it, and the integral is the same number a metre behind the body and a kilometre behind it, because there is nothing for the momentum to be given to. So the far field does remember one thing about the viscous flow, and it is the drag.

What it forgets is everything else, and it forgets it in the same way and for a similar reason. After enough distance the wake becomes self-similar: its profile collapses onto one shape, scaled by a width and a depth, and the shape is universal — the same curve behind a cylinder, a plate, a sphere or an aeroplane. The width and depth are not free either; they are fixed by the one conserved integral. In two dimensions the width grows as $\sqrt{\theta x}$ and the centreline deficit falls as $U\sqrt{\theta/x}$; in three, the width goes as $x^{1/3}$ and the deficit as $x^{-2/3}$. In every case the only parameter that appears is $\theta = D/\rho U^2$, the momentum thickness.

Two bodies with the same drag therefore have the same far wake, in exactly the sense that two bodies with the same doublet have the same far potential field. The shape has gone; the near wake’s structure — a vortex street, a pair of standing eddies, a separation bubble, whatever the body produced — has been mixed away; and what is left is one number wearing a universal profile.

The distance it takes is worth having, because it decides where a measurement may be made. Behind a bluff body the wake is not self-similar for something like fifty to a hundred diameters, and inside that region the profile still carries the signature of how the body shed it. A wake survey taken there is measuring a wake with structure in it, and the momentum integral is still exact but the formula relating it to a measured profile is not the simple one.

Which completes the essay’s own accounting. The far field of a real body remembers four numbers: a circulation, a net outflow, a dipole, and a momentum deficit. The first three are potential and the fourth is viscous, they decay at different rates and the fourth does not decay at all, and between them they carry both of the forces anybody wants. Everything else about the body — its thickness distribution, its camber, its surface finish, whether it separated at the front or the back — is present in the near field, absent from the far one, and irrecoverable from any measurement made out where the flow has become simple.

The far field is not a blurred image of the body. It is a short list, and the list is complete.

How many terms it takes to be sure #

There is a natural follow-up question: how many multipoles would have to be measured to pin the body down? The answer is that no finite number does, and the reason is worth seeing.

Two bodies matched in the first $N$ terms differ at order $r^{-(N+2)}$, so the measurement needed to separate them at a fixed radius becomes exponentially more precise as $N$ rises. Turned round: with measurements of a given precision at a given radius, the number of terms recoverable is fixed, and everything below that level is invisible. The information available about a body from outside it is finite, and its amount is set by the ratio of the measurement radius to the body’s size.

That is not a fluid-mechanical fact. It is the same statement made about every scattering experiment in physics, and it is why an inverse problem is always accompanied by a regularisation — a decision about which of the infinitely many bodies consistent with the data is to be reported.

The three-dimensional version, briefly #

In three dimensions the counting changes and the conclusion strengthens. There is no circulation term of the two-dimensional kind; the leading disturbances are a source ($1/r^2$), a force dipole ($1/r^2$) and a doublet ($1/r^3$), and a closed body with no lift is a doublet at leading order.

The consequence used by every low-order aerodynamic method is that a slender body’s far field is a line of doublets whose strength is the cross-sectional area’s derivative, which is where slender-body theory comes from: the body has been replaced by its area distribution, and two bodies with the same area distribution have the same far field. That is not a small class of bodies, and it is why the area rule for transonic aeroplanes works on shapes that look nothing alike.

Why a body can be replaced by singularities at all #

Everything above takes for granted that a body’s exterior flow can be written as a stream plus a few singularities inside it, and that assumption deserves a sentence because it is what makes panel methods, lifting-line theory and the whole of low-order aerodynamics work.

The reason is that Laplace’s equation has no interior of its own to speak of: a harmonic function outside a closed curve is determined by its boundary values, and the family of singular solutions — sources, vortices, doublets and their higher relatives — is complete. Any exterior flow can therefore be built from them, and the only question is how many are needed and where they are put. A circle needs one. An oval needs two. An aerofoil needs a sheet of them along its camber line, which is exactly what a panel method is.

So a far-field expansion is not an approximation of the body; it is a rearrangement of the same information into an order. The terms are sorted by how far they reach, and truncating the series is the act of deciding how far away the reader is standing.

What the picture cannot show #

The comparison is inviscid. Two bodies with the same far potential field have entirely different boundary layers, wakes and drags. A wake is a defect that does not decay — it is a permanent difference in the far field, and it is the one channel through which a viscous flow does carry information about the body to a distance. Everything above concerns the potential part.

The bodies are matched at leading order only. Matching the octupole as well would take a three-singularity construction and would push the difference to $r^{-5}$; nothing here says how many terms it takes to determine a body, because the answer is that no finite number does.

Compressibility changes the counting. In a subsonic compressible flow the disturbance still decays, but the equation is not Laplace’s and the multipole rates differ; in a supersonic one the disturbance does not decay in the same way at all, since it is carried along Mach lines and reaches a distant observer as a wave rather than as a fading field. The three-numbers argument is an incompressible one, and the transonic area rule is what its careful compressible cousin looks like.

And the far field is a limit rather than a place. Every number in the table is a statement about a particular pair of bodies at a particular radius. The general statement — that a limit throws shape away — is exact; the rate at which it does so depends on which multipoles a given pair of shapes shares, and a pair chosen adversarially could disagree at any radius one liked.

One more equivalence, with a practical edge #

The result has a use that is not about measurement at all. If two bodies share a far field, then a calculation of the outer flow can use whichever of them is cheaper — and for a wing–body combination the cheaper one is a distribution of singularities on a line.

That is how a whole generation of aircraft aerodynamics was done. The fuselage becomes a line of sources and doublets, the wing becomes a lifting line or a sheet, the interference between them is computed from those representations alone, and the surface pressures are recovered afterwards from a separate near-field calculation. The method works exactly to the extent that the parts are far enough apart for their higher multipoles to have died — and it fails, famously, where they are not, which is the wing–body junction.

The junction is where the far-field argument runs out, and it is still the hardest region on an aeroplane to compute, for a reason this essay makes precise: two bodies a chord apart are in each other’s near fields, where the terms that carry the shape have not decayed at all.

Who found it, and when #

The multipole expansion of a potential flow is Laplace’s and is older than aerodynamics; its use to argue about what can be measured at a distance is the standard apparatus of every inverse problem in physics. In fluid mechanics the specific consequence — that only circulation and outflow exert a force from far away — is Kutta and Joukowski’s, and the equivalence of bodies with matched far fields is the basis of the equivalent-body arguments in transonic aerodynamics from the 1950s.

The surprising connection is with the phase’s own subject. An average is a projection, and every projection has a kernel; the useful question about any average is what lies in that kernel. For a time average of a flow it is the fluctuation, and the fluctuation turns out to matter enormously. For the limit at infinity it is every multipole beyond the third, and those turn out not to matter at all for the force — which is why the same operation is a disaster in one case and a theorem in the other. The difference is not in the averaging. It is in whether the quantity being asked for happens to survive it.

Where the ladder goes next #

The rung below is the force itself, and the fact that its split into pressure and momentum runs from three per cent to ninety-seven while the total does not move. Beside this one lies the wake — the one disturbance that does not decay — and the ladder continues into three dimensions, where the same counting supports the whole of slender-body and area-rule aerodynamics.