Drag in the theory that forbids it

d’Alembert’s paradox is the most useful failure in the subject: solve the flow past a body in a fluid with no viscosity and the drag comes out exactly zero, for every shape, at every speed. The pressure over the front is the mirror of the pressure over the back and the two cancel to the last digit the arithmetic carries.

It is worth being precise about what the theorem assumes, because one of its hypotheses is usually mistaken for a fact about the world.

It assumes the flow closes behind the body. The streamline that divides at the front stagnation point runs round the surface and rejoins at the back, and the fluid downstream is the fluid upstream with nothing missing. Drop that requirement, and the equations do not object.

What was changed, and what was not #

Nothing was added to the physics. The flow outside the wake is irrotational, inviscid and incompressible, exactly as before. What changed is the shape of the region the problem is posed in: instead of the whole plane minus a plate, it is the whole plane minus a plate and a semi-infinite cavity.

Two conditions define the cavity’s boundary, and neither is arbitrary:

• It is a streamline, because nothing crosses from the moving fluid into the still fluid.
• The speed on it is constant, by Bernoulli, because the pressure inside the cavity is constant and the pressure just outside must match it across a surface with no thickness.

That second condition is the whole trick. On an unknown curve, the speed is known exactly — and that is a boundary condition of a kind that ordinary potential-flow problems never offer.

Solving it #

The plate occupies $|y| \le a$ on $x = 0$; the stream comes from the left. Take the upper half by symmetry. Its whole boundary — the axis upstream, the plate’s front face, the free streamline — is one streamline, so $\psi = 0$ on all of it, and the flow region maps to the upper half of the $w = \phi + i\psi$ plane with no work at all.

Now write $\Omega = \ln\!\big(U/(dw/dz)\big)$. Its real part is the logarithm of the speed ratio and its imaginary part is the flow direction, and on the three pieces of boundary:

$$\operatorname{Re}\Omega = 0 \ \text{on the free streamline},\quad \operatorname{Im}\Omega = \tfrac{\pi}{2} \ \text{on the plate},\quad \operatorname{Im}\Omega = 0 \ \text{on the axis}.$$

So $\Omega$ maps the flow onto a semi-infinite strip, and the Schwarz–Christoffel map onto a strip is elementary. Composing it with the definition and simplifying gives one line that carries the whole solution:

$$\frac{dz}{dw} \;=\; \frac{\sqrt{w-b} + i\sqrt{b}}{U\sqrt{w}},$$

with $b$ the value of $\phi$ at the plate’s edge.

That expression is worth checking against, because every boundary falls out of it by inspection. For $w < 0$ both roots are imaginary and the ratio is real: motion along the axis. For $0 < w < b$ the numerator is imaginary and the denominator real: motion straight up the plate. For $w > b$ the modulus is exactly $1/U$: constant speed, which is the free streamline. Integrating the middle case across the plate returns $b(4+\pi)/2U$, which must be the plate’s half-width, and fixes

$$b = \frac{2aU}{4+\pi}.$$

The drag #

Everything after that is quadrature. On the front face the speed runs from zero at the stagnation point to exactly $U$ at the edge, so the pressure runs from the full stagnation value down to the free-stream value; behind, it is the cavity’s constant. Integrating,

$$D = \frac{2\pi\rho U^2 a}{4+\pi}, \qquad C_D = \frac{2\pi}{4+\pi} = 0.879801,$$

on the plate’s full width $2a$. The quadrature and the closed form agree to machine precision, which is the check that the map was composed correctly rather than merely plausibly.

The drag is entirely the absence of pressure recovery. There is no shear stress in this calculation, because there is no viscosity. There is no momentum deficit measured in a wake, because the wake is at rest. What there is, is a front face pushed on and a back face that nothing pushes.

The front face is right, and it was right before #

Before the theory is criticised for the back, it is worth seeing how well it does on the front, because the two solutions disagree there too and only one of them agrees with a measurement.

The attached solution puts the surface speed at $U|y|/\sqrt{a^2-y^2}$, which is zero at the centre and unbounded at the edges: a pressure coefficient running to minus infinity over the last few per cent of the plate. The free-streamline solution reaches exactly $U$ at the edge and no more, because the edge is where the fluid leaves and the speed on the departing streamline is fixed by the cavity pressure. A row of tappings across the front of a real plate reads the second curve, not the first.

That is the general pattern for a bluff body and it is worth stating as a rule. Ideal flow is accurate up to the separation point and meaningless after it — not gradually worse, but qualitatively wrong, because it is answering a question about a region the fluid has left. The whole of the error in d’Alembert’s paradox lives downstream of one point.

The number is wrong, and by a factor everybody can name #

A real flat plate held across a stream measures a drag coefficient of about 1.9. The theory gives 0.88. It is out by more than a factor of two, and the direction is worth noticing: it is too small, which is not the usual complaint about an inviscid theory.

The reason is a single assumption that has been sitting in the construction since the second paragraph. The cavity was put at free-stream pressure, and a real wake is not. Behind a bluff body the pressure is well below the free stream, with a base pressure coefficient of about $-1.2$ — measured with a tapping, not computed.

Give the cavity that pressure instead. The free-streamline family with cavitation number $\sigma = -C_{pb}$ has $C_D(\sigma) = C_D(0)(1+\sigma)$, so

$$C_D = 0.8798 \times 2.2 = 1.94,$$

against a measurement of about 1.9.

This is the least deniable case of a pattern that runs through the whole of ideal flow. The equations are exact, the boundary conditions are exact, and there is one number that neither of them contains. Supply it from outside and the answer is right.

What sets the number the theory cannot supply #

Calling $\sigma$ a missing scalar is accurate and leaves the impression that it is a constant somebody once measured. It is not. It is the outcome of a balance in the near wake, it can be shifted deliberately, and shifting it is what almost all bluff-body drag reduction consists of.

The balance runs like this. The two shear layers leaving the edges are turbulent and they entrain: each drags fluid out of the region between them, at a rate set by its own growth. That region is closed at the front by the body, so the fluid removed has to be replaced from downstream, which requires a pressure gradient pushing fluid back upstream into the base — and the base pressure falls until it is low enough to drive exactly the reflux the layers are taking away. A low base pressure is the price of feeding the entrainment, and it settles wherever the two rates match.

That immediately identifies the geometric quantity that matters, which is not the body’s shape but the formation length — how far downstream the shear layers roll up into vortices. Roll-up close behind the body means strong entrainment applied right at the base, a low base pressure and a large drag; roll-up further downstream means the entrainment is being fed largely from the free stream instead, and the base pressure recovers. Every device that reduces the drag of a blunt body works by pushing the formation length back.

A splitter plate is the cleanest demonstration. A thin plate on the centreline behind a cylinder adds essentially no wetted area and no shape change, and it prevents the two shear layers from communicating across the wake, so they cannot roll up in alternation. The formation length grows, the base pressure rises, and the drag falls by something like a quarter — from a body that has not been made more streamlined in any sense a draughtsman would recognise.

Base bleed attacks the balance directly. Inject a small mass flow into the near wake — from a slow-burning charge in the base of an artillery shell, or from a bleed slot on a vehicle — and the shear layers entrain that instead of drawing on the reflux. The pressure no longer has to fall as far, the base drag falls, and the range of a shell rises appreciably for a bleed of a per cent or two of the free-stream mass flow. It is a device that spends mass to buy pressure, and it exists because the missing scalar in this essay is purchasable.

And boat-tailing does the obvious thing, which is to make the base smaller and turn the flow into it gradually — but it is worth seeing that it works on both factors at once. A smaller base area multiplies whatever suction remains by less, and a gentler turn delays separation, which moves the formation length back as well.

So the honest form of this essay’s conclusion is slightly different from the one the flat plate suggests. The theory produces a one-parameter family and cannot choose a member; what it does not say, and what the flat plate hides by being a fixed shape with fixed separation points, is that the parameter is not a property of the fluid at all. It is a property of the near wake’s own dynamics, it varies by a factor of two between a bare cylinder and one with a splitter plate behind it, and the whole practical subject of bluff-body drag is the business of moving it. Kirchhoff’s construction is then not an incomplete theory awaiting one measurement; it is an exact map from that one design variable to the drag.

Three things the model gets wrong, none of them the drag #

The wake never closes. The free streamlines go as $y \propto \sqrt{x}$, forever. The cavity is infinitely long and infinitely wide, which is not a defect of the arithmetic but what a wake held at exactly the free-stream pressure has to do: a streamline at constant speed in a decaying disturbance field has nothing to turn it.

The separation point is an input. It was put at the plate’s sharp edges, where nobody doubts it, and that is why a flat plate is the case the theory does best. On a smooth body — a cylinder, a sphere, an aerofoil past its stall — where the flow lets go is the whole difficulty, and this construction has no way to find it. It can be told, and then it computes the consequences.

The cavity pressure is a free parameter and always was. That is not hidden — σ is written into the solution — but it is easy to read as a refinement rather than as the whole difficulty. It is the whole difficulty: the theory produces a one-parameter family of exact solutions and has no means of choosing between them, and the choice is worth a factor of two in the answer.

And nothing here is unsteady. A real wake behind a plate is not a still cavity; it is a shedding street of vortices, and the base pressure that has to be fed in is a time-average over a violently unsteady flow. This site’s solver cannot draw that street, and neither can this one.

Where the free parameter is not free #

The construction is at its weakest as a model of a wake and at its strongest where the constant pressure is a physical fact rather than an assumption, and it is worth being specific about how strong that is.

In a cavitating flow the cavity contains vapour at the liquid’s vapour pressure, so

$$\sigma = \frac{p_\infty - p_v}{\tfrac12\rho U^2}$$

is set by the depth, the temperature and the speed — three things a designer knows. The drag then follows from the geometry with nothing left to fit, and the same construction predicts the cavity’s length and width, which are what decide whether it closes on the body and erodes it or closes downstream in open water. A supercavitating hydrofoil is designed with nineteenth-century mathematics, and it is designed rather than tested into existence because in that regime the hypothesis Kirchhoff had to assume is true.

What it is good for #

Two things, and both are worth more than the flat plate.

Cavitating flows, where the assumption is not an assumption. When a body moves fast enough for the pressure to fall to the vapour pressure, the liquid tears and a real cavity of vapour forms with a genuinely constant pressure in it. The cavitation number is then a measured property of the liquid and the depth, not a fitting parameter, and free-streamline theory becomes a predictive tool rather than a one-parameter family. Supercavitating propellers and hydrofoils are designed with it.

And jets. The same construction with the plate replaced by a slot in a wall gives the shape of the emerging jet and the contraction it undergoes, which is the vena contracta and its exact π/(π+2) — a number that is computed rather than measured, from a free surface whose shape is part of the answer.

What the picture cannot show #

Nothing inside the cavity. The region between the free streamlines is drawn empty because the model says the fluid there is at rest, and in a real wake it is the most energetic part of the flow. Every figure here is honest about the outside and silent about the inside.

The pressure is not drawn as a field. It is drawn on the plate, where it is a curve, and quoted in the cavity, where it is a constant. In between it varies and could be shaded — and shading it would suggest the model knows something about the interior, which it does not.

And the sharpness of the edge is doing invisible work. The construction puts the free streamline leaving tangentially at a mathematically sharp corner. Round the corner off and the separation point becomes a question again, with an answer that depends on the boundary layer and therefore on the Reynolds number — which is how a coefficient that this theory says is independent of speed acquires a drag crisis.

The general shape of the argument is worth carrying away from this rung, because it recurs. A theory that gives an obviously wrong answer is usually being asked a question it was not given enough information to answer, and the useful response is to find out which piece is missing rather than to replace the theory. Here the missing piece is one scalar. It is not available from Laplace’s equation, it is not available from the boundary condition, and it is available from a pressure tapping.

Who found it, and when #

Helmholtz proposed discontinuous flows with free streamlines in 1868 and Kirchhoff made the construction rigorous in 1869 — one year, and the two names are usually given together. Rayleigh computed the plate at incidence shortly after; Levi-Civita generalised the hodograph method to curved bodies in 1907; and the whole subject was revived in the 1940s and 1950s for cavitating propellers, where it turned out to be the right physics rather than a rescue attempt.

The surprising connection is with what the construction was originally for. Helmholtz and Kirchhoff were not trying to model wakes. They were trying to rescue the theory of jets, where the free surface is real and visible and its shape is genuinely part of the problem. The wake application is the analogy, and it is the weaker of the two — which is the reverse of how the subject is usually taught, where free-streamline theory arrives as a failed attempt to explain drag and its successes in jets and cavities are a footnote.

Where the ladder goes next #

Above this rung is the plane the solution was found in, which is worth a rung of its own because the trick — solve in the space where the unknown boundary is the known one — outlives the problem it was invented for.

Beside it is the paradox this flow evades and the separation point it has to be told.

And below it, the reason any of it matters: what a bluff body’s drag actually is, and why the pressure part of it dominates the friction part by a factor that grows with how blunt the body is.