Drag with nothing to rub

d’Alembert’s paradox is the most useful wrong answer in fluid mechanics. A closed body in a steady, inviscid, irrotational, incompressible flow experiences exactly zero drag, and this site computes it as $10^{-16}$ on a cylinder to make the point that the zero is real rather than approximate.

Every word in that hypothesis is load-bearing. Here is what happens when the last one is removed.

Where the proof breaks #

The classical proof of d’Alembert’s paradox works by showing that the momentum flux through a large control surface far from the body vanishes as the surface grows, because the disturbance the body makes decays fast enough.

In incompressible flow it does. A body’s disturbance falls off like a dipole field, the far-field integral converges to zero, and no momentum escapes to infinity. The fluid pushes on the body and the body pushes back and nothing is carried away.

Above Mach one the disturbance does not decay. The Mach waves the body makes run out to infinity at constant strength — in the linear theory exactly, and in reality decaying only slowly — carrying energy and momentum with them. The far-field integral does not vanish, the proof fails at its first step, and drag appears.

So wave drag is not a correction to d’Alembert’s result. It is the failure of one of its hypotheses, and the failure has a mechanism: a compressible fluid has a way of radiating energy away, and an incompressible one does not.

The energy audit #

That mechanism can be followed all the way through, and it closes.

The body does work on the fluid at a rate $D U$, where $D$ is the wave drag. That energy goes into the wave system. In the wave system it appears as entropy — because the compression waves are shocks and a shock produces entropy — which is to say it appears as a loss of total pressure in the fluid that has passed through.

So there are three descriptions of one quantity: a force on the body, an energy flux to infinity, and an entropy production rate. They are equal.

That equality is the same structure as induced drag on a finite wing, which is also a drag in a fluid with no viscosity, also paid to a disturbance carried away, and also computable exactly. The difference is what carries it: induced drag pays for the kinetic energy in the trailing vortex system, and wave drag pays for the energy in the wave system. Both are cases of a body radiating.

Solved face by face, exactly #

The section is solved by putting each face’s turn through the appropriate relation and reading off the pressure. There is no inversion and no iteration, because supersonic flow carries no information upstream.

For the flat plate at incidence $\alpha$: the upper surface turns the flow away from itself by $\alpha$, so it is a Prandtl–Meyer expansion and the pressure falls; the lower surface turns it into itself by $\alpha$, so it is an oblique shock and the pressure rises. Two calls, and the section is solved.

At Mach 2 and 5°, that gives $c_p = -0.090$ above, $+0.113$ below, a lift coefficient of 0.2021 and a drag coefficient of 0.01768.

Two routes that share no algebra #

The linear result — Ackeret’s, from 1925 — says

$$c_p = \frac{2\theta}{\sqrt{M^2-1}}, \qquad c_l = \frac{4\alpha}{\sqrt{M^2-1}}, \qquad c_d = \frac{4\alpha^2}{\sqrt{M^2-1}}$$

for a flat plate, where $\theta$ is the local surface angle to the free stream. It comes from linearising the potential equation and has no shocks or fans in it at all.

assertShockExpansionMatchesLinear requires the exact solution to agree with that as the angle shrinks, and requires the disagreement to grow with incidence. The second half is the more useful check: a first-order theory whose error does not grow with the angle is not first order, and a flat error curve would mean one of the two routes was secretly the other.

The agreement is evidence rather than tautology because the routes share nothing. One composes oblique shocks and Prandtl–Meyer fans and integrates pressures over faces; the other is a linearisation of a partial differential equation with a $\sqrt{M^2-1}$ in it. Their meeting at small angles is a real check on both.

The section, face by face, with thickness on it #

The flat plate is the clean case. A section with thickness shows the method doing something a subsonic reader will find genuinely strange.

Read the pressure coefficients round that section and compare them with a subsonic aerofoil’s. There is no suction peak near the leading edge, no gradual pressure recovery towards the trailing edge, and no relationship at all between the shape’s curvature and the pressure — because there is no curvature, and because each face’s pressure depends only on its own angle and what came before it.

The lift on this section is not produced by circulation. There is no circulation to compute, no Kutta condition to apply, and no bound vortex. The lift is the integral of a piecewise-constant pressure over four flat faces, and that is a complete account of it.

Where the drag goes at high Mach number #

The $1/\sqrt{M^2-1}$ in Ackeret’s expressions has an unexpected shape, and it is worth reading before its consequences are assumed.

At Mach 1.2 the factor is 1.51, at Mach 2 it is 0.577, at Mach 3 it is 0.354, at Mach 5 it is 0.204. So wave drag falls as the Mach number rises, for a section at fixed incidence. Faster is cheaper.

That is not the whole story, because the same factor divides the lift, so a section at fixed incidence also produces less lift as it goes faster and must fly at more incidence to hold a given load. Working at fixed lift coefficient instead, $c_d = c_l^2 \sqrt{M^2-1} / 4$, and the drag rises with Mach number.

Which of the two is relevant depends on what is held fixed, and the pair is a good example of a trap this site meets repeatedly: two correct statements pointing opposite ways, distinguished only by what the comparison holds constant. The same trap governs the least-drag and least-power speeds on a subsonic aircraft.

The drag is the lift, tilted #

For a flat plate there is a relation between the two coefficients that is exact at every incidence and is worth stating because it explains what wave drag is on a lifting surface.

$$c_d = c_l \tan\alpha$$

The plate has no thickness, so the only force on it is normal to its own surface. Resolve that normal force into the free-stream frame and the drag is the lift times the tangent of the incidence — the whole of the drag is the lift vector being tilted backwards.

assertWaveDragIsNotZero checks that identity to $10^{-9}$ on every flat-plate case it is given, and the check is a good one because it constrains the relationship between two quantities that were computed from different faces by different relations.

The same structure appears on a subsonic finite wing, where induced drag is the lift tilted back by the downwash angle. Two different mechanisms — a wave system and a vortex system — producing the same geometry of force.

What thickness costs #

A flat plate is the minimum-drag supersonic section and has no structural depth whatever, which makes the price of thickness the central design question.

Ackeret’s result for a general thin section separates cleanly:

$$c_d = \frac{4}{\sqrt{M^2-1}}\left(\alpha^2 + \overline{g^2}\right)$$

where $\overline{g^2}$ is the mean square of the surface slope due to thickness and camber. For a symmetric diamond of thickness ratio $t$ that is exactly $t^2$.

The numbers at Mach 2 and 5°: a flat plate has $c_d = 0.0177$; the same plate given 5 per cent thickness has 0.0237; at 10 per cent, 0.0417; at 12 per cent, 0.0537. Thickness has tripled the drag without contributing anything to the lift.

That is why supersonic wings are thin — 3 to 5 per cent, against 12 to 15 for a subsonic transport — and it is a genuinely different reason from the subsonic one. Subsonically, thickness costs a little friction drag and buys a great deal of structure, and the trade favours being thick. Supersonically it is charged for directly and quadratically.

Camber is a liability here #

One more inversion, and it is the one most likely to trip somebody carrying subsonic instincts across the boundary.

Camber is the cheapest lift there is at low speed: it shifts the lift curve without tilting it, giving lift at zero incidence for almost no drag. In Ackeret’s expression camber enters $\overline{g^2}$ exactly as thickness does — as a mean square surface slope — and contributes drag while contributing nothing to lift.

So the optimum supersonic section is symmetric, thin and flat, carrying its lift entirely by incidence. Every feature a subsonic aerofoil has for the sake of efficiency is, above Mach one, an expense.

The one exception is worth noting because it is the exception that founded a subject: a body’s wave drag depends on how its cross-sectional area is distributed along its length, and smoothing that distribution — Whitcomb’s area rule, 1952 — reduces the drag substantially. That is a three-dimensional argument about volume rather than a two-dimensional one about section shape, and none of it is computed here.

What the solver computes, and how it is checked #

shockExpansionSection walks the faces of a section in order, applying an oblique shock where the turn is into the flow and a Prandtl–Meyer fan where it is away, carrying the Mach number and the pressure ratio forward from face to face. It then integrates $(p - p_\infty)$ over the faces to get the force, and resolves it into the free-stream frame.

Working with $(p - p_\infty)$ rather than $p$ removes the ambient contribution, which integrates to zero round a closed body — the same discipline forceOnBody uses on the inviscid side of this site, and for the same reason.

And that is where the real bug in this calculation was found. The free stream’s angle in the chord frame was written as $-\alpha$ when it should have been $+\alpha$: a section at positive incidence has its nose up, so walking from leading edge to trailing edge walks downhill through the oncoming flow.

The consequence was not small. It made the upper front face of a diamond section an expansion and the rear face a compression — the section solved back to front — and the solver returned a wave drag of −0.023 at zero incidence. A symmetric aerofoil generating thrust in a supersonic stream.

Every conservation law in the calculation was satisfied throughout, because each individual shock and fan was computed correctly and only the order in which they were applied was wrong. Nothing in the figure looked odd. It was caught by assertWaveDragIsNotZero, whose entire content is that a supersonic section has positive drag, and which exists for exactly this.

Three drags now, and only one needs viscosity #

This site has now accounted for three distinct kinds of drag, and it is worth putting them side by side because their causes have almost nothing in common.

Friction drag requires viscosity and is what the thin layer at the surface costs. Remove viscosity and it vanishes.

Induced drag requires a finite span and is the price of having ends. It exists in a fluid with no viscosity and vanishes for an infinite wing.

Wave drag requires neither. It exists on an infinite-span wing in an inviscid fluid, and it vanishes only below Mach one.

Three mechanisms, three hypotheses that remove them, and no overlap. d’Alembert’s paradox survives all three as a statement about its own hypotheses — steady, inviscid, irrotational, incompressible, closed body, unbounded fluid — and each of these drags is a different hypothesis of that list being struck out.

Arranging for nothing to be radiated #

If the drag is the energy that escapes along the wave system, then a body that sends no waves to infinity should have no wave drag — and one can be built.

Busemann’s answer, from 1935, is two half-wedge sections mounted facing each other across a channel. The compression that leaves the first element’s leading edge arrives at the second element’s shoulder, which is exactly where the surface turns away and produces an expansion of the same strength. The two cancel. Downstream of the pair the flow is uniform and parallel again, nothing at all propagates out to the far field, and the wave drag is zero — in a steady, inviscid, supersonic flow, on a body with real thickness.

That is d’Alembert’s result restored above Mach one, obtained by removing the mechanism rather than by restoring the hypothesis. It is the strongest possible confirmation that this essay’s account of where the drag comes from is the right one.

Three things stop it being an aeroplane. The cancellation is exact at one Mach number, because the wave angles depend on it. The configuration as described is symmetric and therefore makes no lift, and any arrangement that does make lift breaks the cancellation. And the channel between the elements has a throat, so the pair chokes during the acceleration to its design speed and has a starting problem of precisely the kind a supersonic intake has.

Where the model stops #

Thin, two-dimensional and sharp-edged. Shock–expansion theory needs every turn to be within the detachment angle, so a section with a rounded leading edge is outside it entirely — that edge carries a detached shock and the marching solution has nowhere to start.

Wave drag only. Friction drag is not in these numbers and is not negligible: at Mach 2 the skin friction on a thin wing is comparable with its wave drag, and above Mach 3 the boundary layer is hot enough that its own properties change.

No wave interactions. The faces are solved in sequence with no allowance for the shock from one face crossing the fan from another, which happens downstream of any real section and modifies the flow there. It does not affect the surface pressures on a simple section, which is why the method works.

Two-dimensional, so no area rule and no wing tips. Both are three-dimensional and both matter more than anything on this page for a real aircraft.

What the picture cannot show #

The section figures draw a shock as a single line at the angle the relation gives, and a fan as a single line at the leading Mach angle, which understates the fan considerably.

More seriously, the figures stop at the body. The wave system extends to infinity, and it is where all the drag energy goes — so the figure shows the force and hides the mechanism that pays for it. Nothing in the drawing conveys that the waves are still doing something a kilometre away, which is precisely the property that makes the drag exist.

The pressure coefficients are printed as numbers on the faces rather than drawn as a distribution, because on a flat plate the distribution is constant and on a diamond it is piecewise constant. That is an honest representation of what shock–expansion theory produces, and it is worth noticing how unlike a subsonic pressure distribution it is: no suction peak, no gradual recovery, just a few plateaux.

Who found it, and when #

Jakob Ackeret published the linear supersonic aerofoil theory in 1925, at a time when the fastest aircraft in the world flew at about a third of the speed of sound. The theory had no application for twenty years.

Shock–expansion theory as the exact method is Busemann’s, in the same period, and it is the more general of the two: it holds at any angle within the detachment limit, where Ackeret’s holds only for small ones.

Whitcomb’s area rule arrived in 1952 from a wind tunnel rather than from theory, and the story is one of the good ones in aerodynamics: he was looking at transonic drag data that made no sense, worked out that the total cross-sectional area distribution was what mattered, and the result reshaped every supersonic aircraft designed afterwards into the wasp-waisted forms of the 1950s.

Where the ladder goes next #

This field has now followed the compressible story from the signal speed to the drag it produces, and one range has been avoided throughout: the one where the free stream is subsonic and part of the flow is not.

That is where nearly all fast aircraft actually operate, it is the hardest range in the subject, and what happens on top of the wing there takes the argument up one rung. It is also where the correction that predicts its own failure earns its keep.