Three dimensions are kinder

The exact theory of ideal flow on this site is a plane theory. A cylinder, a Joukowski aerofoil, a source and a sink on a line, a row of images: all of them live in two dimensions, and the reason is that the plane has a complex potential and a circulation to play with.

Take the third dimension back and both of those go. What arrives instead is a set of numbers that are gentler in every case, and one qualitative difference that has nothing to do with arithmetic.

Stokes’ stream function is not the plane one with a coordinate added #

For a flow with an axis of symmetry there is a function ψ with

$$u_r = \frac{1}{r^2\sin\theta}\frac{\partial\psi}{\partial\theta}, \qquad u_\theta = -\frac{1}{r\sin\theta}\frac{\partial\psi}{\partial r},$$

and it does the two things a stream function is wanted for: it is constant along a streamline, and the difference between its values on two streamlines is the volume flux between them, divided by 2π.

It is a different object from the plane stream function. Its dimensions are a volume flux rather than an area flux, and — the part that matters for anybody carrying plane intuition across — it does not satisfy Laplace’s equation. There is no complex potential, no conformal mapping, and no Blasius theorem in three dimensions. What survives is superposition, which is enough to build every body in this essay.

There is a reading rule that comes with it and the figures repeat it. In a plane flow, streamlines twice as close together mean a speed twice as high. In a meridional section, the flux between two streamlines depends on the distance from the axis as well, so counting the gaps gives the wrong answer. That is why the speed is contoured in the figure above rather than left to be inferred.

Every number is smaller, and always by the same reason #

The sphere’s surface speed is $\tfrac{3}{2}U\sin\theta$ and the cylinder’s is $2U\sin\theta$. That one factor propagates into everything:

• Minimum pressure coefficient −1.25 against −3. A cylinder’s low-pressure region is more than twice as deep as a sphere’s, in the same stream.
• The cavitation number at which a body first boils the water is exactly that minimum, so a sphere cavitates at a free-stream pressure less than half a cylinder’s, in the units that question is asked in.
• The critical Mach number is higher. Sonic flow appears first at the point of lowest pressure, so a body of revolution reaches its own critical Mach number later than a plane body of the same thickness ratio, which is one of the reasons fuselages are slimmer than they look.

The reason is the same in every case and it is a statement about how much fluid has to get out of the way. A plane body forces the flow around it in one plane only; a body of revolution lets it escape in every azimuthal direction at once.

The disturbance reaches less far, too #

This is the practical half of the same fact. A wind tunnel’s walls interfere with a model by an amount that depends on how far the model’s disturbance reaches, so the blockage correction for a body of revolution is far smaller than for a plane one of the same frontal area. It is also why a submarine’s pressure signature is short-ranged and a long cylinder’s is not.

Bodies made of nothing, in space #

The plane construction — put a source in a stream and read off the dividing streamline — works unchanged, and gives different numbers.

The stagnation point sits at $\sqrt{Q/4\pi U}$ and the asymptotic radius is $\sqrt{Q/\pi U}$, exactly twice it. The corresponding plane numbers are $m/2\pi U$ and $m/2U$, whose ratio is π. Two constructions that look identical on the page have different answers because a source spreads its flux over a sphere in one case and over a circle in the other.

That closure condition is worth dwelling on, because it is the kind of thing a picture cannot show. Give the sink one per cent less strength than the source and the body no longer closes: the tail stagnation point moves, the dividing streamline runs off downstream, and the drawing looks entirely normal for several diameters before anything goes wrong. flowcheck requires the mismatch to be detected — it is the same discipline as the image system whose body leaked while its walls were perfect, found by an earlier essay here.

Half the displaced mass, and where the coefficient stops making sense #

Accelerating a body accelerates the fluid round it, and the extra force that costs is $\rho V k \,\mathrm{d}U/\mathrm{d}t$. For a sphere $k = \tfrac{1}{2}$ and for a cylinder $k = 1$ — the same factor of two, arriving for the third time.

The last row of that figure is there to spoil the pattern honestly. A flat plate accelerating edge-on displaces no volume at all and still has a finite added mass, so writing the effect as a multiple of the displaced mass is a convention that works for blunt bodies and fails for thin ones. The underlying quantity is the kinetic energy of the exterior flow, and that is what the integral above actually computes.

The difference that is not a number #

Here is the whole argument, and it takes four sentences.

Circulation round a closed loop equals the integral of the vorticity over any surface spanning it — provided that surface lies in the fluid. In the plane, the exterior of a disc is not simply connected: every surface spanning a loop that encircles the cylinder must pass through the body, so the theorem says nothing and the circulation is free to take any value at all. In space, the exterior of a sphere is simply connected: the loop can be slid up over the pole and shrunk away, the spanning surface stays in the fluid throughout, and an irrotational flow has no vorticity on it. Therefore the circulation is zero.

So there is no Kutta–Joukowski theorem for a body of revolution, and no ideal flow past a sphere can produce any lift. The measured circulation in the figure is 10⁻¹⁷, which is the arithmetic’s own noise, and the point is not that it came out small but that it could not have come out otherwise.

This is the deepest of the differences in this essay and the only one that is not about magnitudes. It also explains a fact this site has already met from the other side: lift on a finite wing costs induced drag precisely because the circulation cannot simply stop at the wingtip. Vorticity has to go somewhere, and in three dimensions the only thing a vortex line can do at the end of a wing is turn downstream. A sphere has nowhere to put it, so it has none.

What the stream function is worth as a number #

The volume-flux reading of ψ is the thing that makes these pictures quantitative rather than decorative, and it is worth one worked case.

For the half-body, the source’s whole output has to end up inside the body far downstream, because the body is the dividing streamline and nothing crosses a streamline. So the asymptotic radius is fixed by a flux balance rather than by a shape: $\pi R_\infty^2 U = Q$, giving $R_\infty = \sqrt{Q/\pi U}$, which is the number the figure marks. No differential equation is needed for it at all — the stream function has already done the bookkeeping.

That last sentence is the axisymmetric version of a fact this site keeps meeting: a construction with one free parameter has one length, and everything else about the picture is fixed. The half-body has no shape parameter at all, which is why the ovoid, with two singularities and a separation between them, is the first body in this family that can be made fat or thin.

What the plane has that space does not #

Three dimensions are kinder about magnitudes and much poorer in tools, and the trade is worth setting out, because it explains why this site’s ideal-flow essays are nearly all plane.

The complex potential is gone. In the plane, φ and ψ are the real and imaginary parts of one analytic function, and every theorem about analytic functions is therefore a theorem about ideal flow. That is not a convenience; it is the reason a circle can be mapped into a wing and the reason the force on any plane body can be written as a single contour integral. There is no analytic function of three variables in that sense, and none of it survives.

Conformal mapping is gone with it. The plane trick of solving one easy shape and transforming it into a hard one has no three-dimensional analogue: the only conformal maps of space are inversions and similarities, by Liouville’s theorem, which is a much shorter list than the plane’s.

Circulation is gone, as the previous section showed. Everything on this site about lift is therefore a plane argument extended by lifting-line theory — which is not a three-dimensional solution but a stack of two-dimensional ones with an induced angle between them.

What is left is superposition, and it turns out to be enough for a surprising amount: a sphere, a half-body, an ovoid, and, by distributing sources along an axis rather than placing two, any slender body of revolution one likes. That is the method that designed airship hulls, and it is the ancestor of the panel methods that solve aircraft today.

The honest summary is that the third dimension makes the physics gentler and the mathematics harder, and that the subject’s classical theory looks the way it does because of the second half rather than the first.

The pattern across the site #

Collecting the comparisons in one place makes a point that no individual one does.

A sphere’s fastest surface speed is 1.5U against a cylinder’s 2U; its minimum pressure coefficient is −1.25 against −3; its disturbance dies as 1/r³ against 1/r²; its added mass is half the displaced fluid against all of it. Four different quantities, four different derivations, and the same direction every time.

The reason they agree is that all four are consequences of one thing: how much fluid must be displaced sideways for a body to pass. A plane body forces the flow to go round it within a plane; a body of revolution lets it escape in every azimuthal direction at once, so less fluid has to be moved and less of it has to move quickly.

That is worth having as an intuition, because it makes the results predictable rather than a list. A quantity that measures how much the body disturbs the flow will be smaller in three dimensions, and one that measures how far the disturbance reaches will fall off faster. The exceptions — where three dimensions is worse — are all about tools rather than magnitudes, and they are the subject of the section above.

No lift, and a moment anyway #

The topological argument settles the force and says nothing about the moment, and the two answers are different in a way that decides how a whole class of vehicle is built.

Put a slender body of revolution at a small incidence. Slender-body theory treats each cross-section as a two-dimensional problem — a circle of the local radius, being accelerated sideways by the component of the free stream normal to the axis — and the force on a slice is then the rate of change of the transverse momentum it carries. That is the added-mass argument applied strip by strip, and it gives a lift per unit length proportional to $\alpha\,\mathrm dS/\mathrm dx$: the body lifts where its cross-section is growing and pushes down where it is shrinking.

Integrate along a body that begins and ends at zero area and the total is $S(L) - S(0) = 0$. The lift vanishes, exactly, in agreement with the topology.

The moment does not. Weighting the same distribution by $x$ and integrating by parts turns $\int x\,\mathrm dS$ into $-\int S\,\mathrm dx$, which is the body’s volume. So a slender body at incidence carries a pitching moment of order $\rho U^2 V\alpha$ with no net force at all — a pure couple, produced by lift on the forebody and an equal download on the afterbody.

And its sign is the wrong one. The couple acts to increase the incidence: nose up when the nose is already up. A streamlined body of revolution is statically unstable in pitch and in yaw, in an ideal fluid, with no viscosity and no separation needed to make it so.

That is the Munk moment, and it is why every airship, submarine and torpedo carries fins. The fins do not exist to make lift; they exist to move the aerodynamic centre far enough aft that the destabilising couple from the hull is beaten by a restoring one. The couple is proportional to volume, so making the hull more slender at fixed capacity does not reduce it, and a larger vessel needs proportionally larger surfaces.

It is a satisfying complement to the result above. The exterior of a sphere is simply connected, so the circulation is zero and the force must be too. Nothing in that argument constrains where the pressure sits on the surface — and a distribution that integrates to zero can still have a first moment.

What the model does not contain #

Viscosity, and therefore drag. The computed drag is zero, and a real sphere at a Reynolds number of 10⁵ has a drag coefficient of about 0.5. That gap is not small and it is not a correction: it is the whole of what the ideal theory leaves out, and a sphere is the shape on which the failure is most spectacular, because the flow separates at the widest section and leaves a wake as broad as the body.

Any body that is not a body of revolution. Everything here uses one axis of symmetry. An ellipsoid at incidence, a wing, a hull with a keel — none of them is axisymmetric, and the tool that solves them is a panel method rather than a closed form.

Angle of attack. A body of revolution at incidence has a flow that is not axisymmetric, and the Stokes stream function does not exist for it. What slender-body theory says about that case is described above and computed nowhere here — no figure on this page carries an incidence.

Anything unsteady except through the added mass. The energy integral assumes the flow is the instantaneous potential solution, which is exact for an inviscid fluid and says nothing about the history a real fluid keeps.

The far field of the ovoid is not checked against an exact closed form, because the ovoid does not have one — its surface is defined implicitly by ψ = 0 and traced numerically here. What is asserted is that the trace closes and that the two stagnation points exist.

Who found it, and when #

The sphere in a stream was solved by Stokes in the 1840s, along with the stream function that carries his name, and the same paper contains the creeping-flow solution that gives a sphere its low-Reynolds drag — so the same person supplied both the answer with no drag in it and the answer that is all drag.

Rankine’s construction of bodies from sources and sinks dates from 1871 and was a working method rather than a curiosity: it is how airship hulls were designed, by choosing a distribution of sources along an axis and seeing what body came out, for forty years before anything better existed. The method survives as the source-panel method, which is the same idea with the distribution solved for rather than chosen.

Where the ladder goes next #

The topological argument above says that the circulation round a loop in an irrotational flow is zero. It says nothing about what happens to the circulation round a loop that is carried along by the fluid, and that turns out to be the single most useful theorem in the subject: the circulation round a material loop does not change at all, however violently the loop is stretched and folded, provided three conditions hold. Each of them can be broken on purpose, and the breaking is what produces every vortex on this site.