The equation that changes type inside its own answer

The small-disturbance equation for a thin body in a compressible stream is

$$\Big[(1 - M_\infty^2) - (\gamma+1)M_\infty^2\,\frac{u}{U}\Big]\phi_{xx} + \phi_{yy} = 0,$$

where $u = \phi_x$ is the perturbation the body makes. Subsonic theory throws the second term in the bracket away, which is legitimate while $M_\infty$ is well below one, and what is left is Laplace’s equation with a stretched $x$ — elliptic, linear, and solvable by superposition.

Those three words are what the whole circulation field rests on. Near Mach one all three go, and not by degrees.

The type is part of the answer #

Where the bracket is positive the equation is elliptic and the flow is subsonic. Where it is negative the equation is hyperbolic and the flow is supersonic. And which it is at a given point depends on $u$, which is what is being solved for.

There is no way to know the type of the problem before solving it.

Evaluated along a chord at Mach 0.85 with a stated suction distribution, forty-two per cent of the chord is hyperbolic. The small-disturbance criterion and the exact local-Mach criterion put the sonic point 0.65 per cent of a chord apart — a real difference, and one worth reporting, since a figure that draws one and captions it as the other is wrong about the size of the supersonic pocket.

The picture that goes with it is the one mach draws: a pocket of supersonic flow sitting on the section, bounded on both sides by subsonic flow, with the boundary between them a free boundary whose position is part of the answer.

Superposition fails, and by how much #

The nonlinear term makes the residual of a sum

$$-K\big(u_A\phi_{Bxx} + u_B\phi_{Axx}\big),$$

which is not zero. Take two thin sections, each an exact solution of the linearised equation, add them, and substitute into the transonic one: at Mach 0.9, adding a ten per cent section to a six per cent one leaves a residual three times the size of the term the linear theory keeps.

Superposition is not slightly wrong there. It is the wrong operation, and everything built on it stops being available: thin-aerofoil theory, panel methods, the decomposition of a wing’s loading into a camber part and an incidence part, and the habit of adding a flap’s increment to a clean-wing result.

The Prandtl–Glauert factor’s own report #

$1/\sqrt{1-M^2}$ is 2.29 at Mach 0.9 and unbounded at one. That divergence is usually described as the correction “breaking down”, which understates it: the factor is the linear theory reporting that its own assumption has failed. A correction that multiplies a small perturbation by a large number has produced a perturbation that is not small.

The sharpest way to see that it is not merely a matter of accuracy is to compare it with another linear correction of the same order. Kármán–Tsien is derived from a different linearisation of the same equations, agrees with Prandtl–Glauert to 1.5 per cent at Mach 0.3, and differs from it by 37 per cent at Mach 0.85.

Both are correct to first order and they disagree at second, which is exactly the signature of a second order that has stopped being small.

The exponent that gives it away #

If transonic flow were subsonic flow with a large correction, the distance of the critical Mach number from one would go as the thickness — one perturbation, one linear response.

It does not. Sweeping the thickness down to an eighth of a per cent and fitting:

$$1 - M_{\mathrm{crit}} \propto \tau^{2/3},$$

with a fitted exponent of $0.6617$ at the thin end against $2/3 = 0.6667$, and $0.5751$ at the thick end where the finite-thickness corrections are still present.

Two thirds is neither one nor a half, and that is the plainest sign available that the governing equation near Mach one is not the one the rest of the subject uses. It comes from balancing the two terms in the bracket, which requires $1 - M^2 \sim (\gamma+1)M^2 u/U$ with $u/U \sim \tau$ — the transonic similarity scaling, and the reason transonic results collapse onto one curve when plotted against $(1-M^2)/\tau^{2/3}$ and onto nothing when plotted against anything else.

Where the nonlinear term comes from #

The bracket’s second term deserves a derivation, because seeing where it comes from explains why it is unavoidable rather than an artefact of a particular expansion.

The full potential equation for compressible flow has a coefficient $a^2 - u^2$ on the streamwise second derivative, where $a$ is the local speed of sound. Both $a$ and $u$ vary through the field: the flow accelerates over the body, and accelerating a gas cools it, so the local sound speed falls at the same time as the speed rises. The coefficient therefore falls twice as fast as a constant-$a$ argument would suggest.

Linearising about the free stream gives $(1 - M_\infty^2)$ for the constant part and $-(\gamma+1)M_\infty^2 u/U$ for the first correction — the $(\gamma+1)$ being the sum of the two effects, one from the velocity and one from the sound speed. At low Mach number the correction is small compared with $(1-M_\infty^2)$ and is dropped. Near Mach one the constant part vanishes and the correction is all there is.

So the nonlinearity is not a higher-order refinement that happens to matter; it is the leading term of the coefficient wherever the leading term of the linear theory has gone to zero. That is why no amount of care with a linear method recovers it, and why the transonic equation retains exactly one nonlinear term and drops everything else of the same order — the one it keeps is the one multiplying a derivative whose coefficient would otherwise vanish.

What the two-thirds law is worth in practice #

The scaling is not only a diagnostic; it is the organising principle of transonic data, and it is worth saying how it is used.

Von Kármán’s transonic similarity rule says that pressure coefficients on affinely related sections collapse when $C_p/\tau^{2/3}$ is plotted against $(1-M_\infty^2)/[(\gamma+1)M_\infty^2\tau]^{2/3}$. That is a two-parameter family collapsed to one curve, and it is what made systematic transonic testing possible before computation: a handful of sections tested over a handful of Mach numbers gives the whole map.

It also says which changes are equivalent. Thickening a section by ten per cent and increasing the Mach number by the amount that keeps the similarity parameter fixed produce the same scaled pressure distribution — so a designer trading thickness against cruise Mach number is moving along a line the rule draws. The famous consequence is that a one per cent reduction in thickness-to-chord buys a definite increment of cruise Mach number, and the exchange rate is $2/3$ rather than $1$.

And it explains supercritical sections. Flattening the upper surface reduces the peak suction without reducing the thickness, which moves the section along a different line than the similarity rule’s — which is the whole design idea, and the reason it took until the 1960s to find.

Two things the equation is not #

Two confusions worth heading off, because both are common and both matter.

It is not that the flow is transonic because the aeroplane is near Mach one. A wing at Mach 0.75 has a supersonic pocket on it, and an aeroplane at Mach 1.4 has subsonic regions behind its shocks. The regime is set by the local Mach number reaching one somewhere in the field, and the free-stream Mach number is only a proxy. Which speed goes in the number is the essay this collection has on exactly that.

And it is not a stability or a turbulence problem. The nonlinearity here is in the steady equation. It produces a mixed-type boundary-value problem, not a time-dependent instability, and the difficulty is present in a perfectly steady laminar flow with no disturbances anywhere. That distinguishes it from almost everything else in this collection that the word nonlinear attaches to.

What is left when the linear apparatus goes #

Three things survive and they are what transonic aerodynamics is made of.

Characteristics, in the supersonic pocket. The equation is hyperbolic there, so the method of characteristics applies inside the pocket even though it does not outside — and the pocket’s boundary is where the two descriptions have to be matched.

Conservation. The shock jump conditions do not care about the type of the equation, so the shock terminating the pocket is described exactly by the same relations as any other shock. That is why the drag rise can be estimated from the shock strength without solving the field.

And the area rule, which is the one large design result in the regime and is a statement about the integral of the geometry rather than about the flow. It is the same slender-body wave-drag integral the next essay is about, applied at Mach one where the whole aeroplane is one slender body.

The shock that ends the pocket #

The pocket has to close, and how it closes is the reason transonic flow costs money.

Inside the pocket the flow is supersonic and expanding. It cannot rejoin the subsonic flow downstream smoothly, because an expansion cannot decelerate a supersonic stream and a compression through Mach one from above is a shock. So the pocket is terminated by a shock standing on the section — usually nearly normal near the surface and curving forward at its top.

That shock does three things, and only the first is usually named.

It costs total pressure, which appears as wave drag and is the drag rise.

It puts an adverse pressure gradient on the boundary layer, over a very short distance, which is the worst possible thing to do to one. If the layer separates the shock moves, the separation moves, and the result can be a self-sustaining oscillation — buffet — which is a flight limit rather than a performance one.

And it is curved, so by Crocco’s theorem it leaves vorticity in the flow behind it. The flow downstream of a transonic wing is rotational for that reason alone, before any viscosity is considered.

None of the three is in the small-disturbance equation, which has no shock in it until one is put there by the numerical scheme’s own dissipation. That is the honest position: the equation predicts that a pocket forms and predicts nothing about the shock that ends it.

The design answer, which is not to avoid the pocket #

A regime with no linear apparatus has to be designed in by trial, and the one large idea that came out of it is worth stating because it inverts the obvious strategy.

The obvious strategy is to keep the flow subsonic: make the section thin, keep the suction peak shallow, stay below the critical Mach number. That works, it is what pre-1960s sections do, and it is expensive — thinness costs structural depth, which costs weight and fuel volume, and a thin wing at a given weight must be smaller or more heavily loaded.

Whitcomb’s supercritical section does the opposite. It accepts the supersonic pocket and manages the shock that ends it. The upper surface is deliberately flattened, so that instead of a sharp suction peak over a short run the flow reaches a modest supersonic Mach number and holds it over much of the chord. The pocket is larger and the shock terminating it is weaker.

That trade pays extravagantly, and the reason is a result this collection has already computed: the entropy a shock produces goes as the cube of its strength. Halving the Mach number excess at the shock cuts the loss by a factor of eight, so spreading the same amount of supersonic flow over a longer, gentler pocket is worth far more than the extra pocket costs. The weaker shock is also gentler on the boundary layer beneath it, which pushes the buffet boundary up as well as the drag rise.

And the lift has to be found somewhere else. Flattening the top removes most of the section’s suction, so the loading is recovered by strong aft camber — a cusped, downward-curved trailing region carrying load where a conventional section carries almost none. That is why a supercritical section looks upside down at the back, and it is why the idea carries a real cost: rear loading produces a large nose-down pitching moment, which the tail must trim out, at a drag of its own.

The payoff, at constant thickness, is something like five to ten hundredths of Mach number before the drag rises — or, taken the other way, a wing ten to fifteen per cent thicker at the same cruise speed. Every swept-wing transport since the 1970s has taken the second option.

Why the band is narrow and unavoidable #

A transport aeroplane cruises at Mach 0.78 to 0.85. That is not a coincidence and not a preference: it is the top of the band in which the drag has not yet risen, and the band’s top is set by the thickness the structure needs.

So the most economically important flow regime in aviation is the one with the fewest analytical tools, and it stayed that way until computers could solve the nonlinear equation directly. Murman and Cole’s mixed difference scheme in 1970 — central differences where the flow is subsonic, upwind where it is supersonic, switched pointwise on the sign of the bracket — was the first method that worked, and the switch is this essay’s opening observation implemented as three lines of code.

Reading a transonic result #

Two habits follow, and they are worth stating because they are unlike the habits the rest of this collection encourages.

Do not add anything. An increment computed at one condition cannot be applied at another, and a result decomposed into contributions has been decomposed illegitimately. The whole configuration has to be computed as one — which is why the area rule is stated for the complete aeroplane including its nacelles and tail, and why removing one of them and adding its increment back is not a legitimate move.

And do not trust a correction factor. A subsonic result multiplied by a compressibility factor is reliable while the factor is near one and becomes unreliable exactly where it becomes interesting. Two factors disagreeing is the diagnostic, and it is cheap to run.

What is not in the flow #

The solution itself.

Every essay gathered with this one names something outside the flow that the answer needs: an observer, a boundary, a gas, a history. This one names the answer. The equation’s type — elliptic or hyperbolic, which decides what kind of problem it is, what boundary conditions it needs, and which numerical method applies — is a function of the perturbation velocity, which is the unknown.

That is what nonlinearity is, stated at its most consequential. Not that the answer is harder to compute, but that the character of the question depends on the answer, so the problem cannot be classified before it is solved.

The model limit #

Three.

The pressure distribution used here is a shape rather than a solution. A stated smooth suction curve is enough to demonstrate that the bracket changes sign and that the two sonic criteria differ, and it is not a transonic solution. Nothing in this essay computes a transonic flow field; what it computes are properties of the equation.

The small-disturbance equation is itself an approximation, valid for thin bodies at small incidence, and it drops terms that matter for a real wing at cruise. Its virtue is that it is the simplest equation that has the type change in it.

And the critical Mach sweep uses Prandtl–Glauert to get there — which is a linear correction, being used to locate the point at which linear corrections fail. That is legitimate for the scaling, which is what the two-thirds exponent is, and it is not legitimate as a prediction of any particular section’s critical Mach number. The exponent is the result; the numbers on the axis are illustrative.

There is a fourth limit worth naming because it is where the whole framework ends. Everything here is inviscid. At the conditions where a transport wing actually operates, the shock terminating the pocket sits on a turbulent boundary layer and the interaction between the two changes the shock’s position, its strength and the pressure distribution ahead of it. A transonic calculation without that interaction gets the drag rise qualitatively right and the buffet boundary — which is the number a certification depends on — not at all.