One curve for every thickness

The transonic small-disturbance equation is the only equation in ordinary aerodynamics that changes its own type inside its own answer, and the price of that is stated there in full. The equation is nonlinear, so two solutions cannot be added. It is not elliptic, so the boundary conditions cannot be applied at infinity and forgotten. There is no Prandtl–Glauert factor that survives, no thin-aerofoil theory, no panel method, and no way to take a result computed at one Mach number and read a result at another off it.

That is a long list of things that do not work, and the essay that establishes it ends at a measurement of one exponent. The exponent is the visible corner of something much larger, and the larger thing is the only general result the transonic range possesses.

Three lengths and one number #

Write the perturbation potential as $\varphi = \varepsilon\,\Phi$ and stretch the coordinate across the stream as $y = \delta Y$, leaving the coordinate along it alone. Two conditions fix the two scales. The first is that the $\varphi_{yy}$ term should be the same size as the nonlinear term, which is what “transonic” means; the second is that the body’s own boundary condition should carry no free constant. Together they give

$$\varepsilon = \left[\frac{\tau^2}{(\gamma+1)M_\infty^2}\right]^{1/3}, \qquad \delta = \frac{\varepsilon}{\tau},$$

and the equation becomes

$$\left(K - \Phi_X\right)\Phi_{XX} + \Phi_{YY} = 0, \qquad \Phi_Y(X,0) = \tfrac12 f'(X)\ \text{on the body},$$

in which neither the thickness nor the Mach number appears anywhere. What is left of them is one group:

$$K = \frac{1 - M_\infty^2}{\left[(\gamma+1)M_\infty^2\tau\right]^{2/3}}.$$

So a thin symmetric section of a given shape has a one-parameter family of flows, not a two-parameter one. Every thickness and every Mach number that produce the same $K$ produce the same $\Phi$, and therefore the same pressure distribution up to the factor $\varepsilon$.

What makes that a strong statement rather than a bookkeeping one is what the two conditions on the scales were. Neither of them was a choice. The first — that the cross-stream term should balance the nonlinear one — is the definition of the range: at Mach numbers well below one the nonlinear term is negligible and the equation is Laplace’s with a stretched coordinate, and at Mach numbers well above one it is again linear, this time hyperbolic. Transonic means precisely the band in which the two terms are comparable, and demanding that they be comparable is the transonic assumption written as an equation. The second condition — that the body’s boundary condition should carry no free constant — is the requirement that the body be in the problem rather than beside it.

Two requirements, two scales, and nothing left over. The rule is not a discovery about transonic flows; it is what the definition of the range already contained. That is worth saying plainly because a reader who has met dimensional analysis will expect a similarity parameter to be one of several possible groupings, chosen for convenience. This one is not chosen. The scaled equation has one parameter in it because there is nowhere else for the two original quantities to go.

The two-thirds power that the essay below measured is this expression evaluated at a threshold. If a pocket of supersonic flow appears at some particular value of $K$ — and it does, because $K$ is the only coordinate the scaled problem has — then the critical Mach number satisfies $1 - M_{\text{crit}}^2 \propto \tau^{2/3}$, and near one that is $1 - M_{\text{crit}} \propto \tau^{2/3}$. The exponent is a corollary, and the rule is the result.

Why the power is two-thirds #

The exponent is the part of the rule that looks arbitrary and is not, and following it through is the shortest route to understanding what the transonic range is.

Every other correction in compressible aerodynamics is a first power or a square root. The Prandtl–Glauert factor is $(1-M^2)^{-1/2}$; the Ackeret pressure is $(M^2-1)^{-1/2}$; thin-aerofoil lift is linear in incidence and in camber. All of those come from linear equations, in which the disturbance is proportional to whatever caused it, and the only question is by what factor.

The transonic equation is not linear, and the reason the exponent comes out at two-thirds is a three-way balance rather than a two-way one. The three terms are: the linear compressibility term, of size $(1-M^2)\varepsilon$; the nonlinear term, of size $\varepsilon^2$; and the cross-stream term, of size $\varepsilon/\delta^2$. In the subsonic range the first dominates the second and the balance is between the first and the third, which is a two-term balance and gives a square root. Near Mach one the first term is small, all three are the same size at once, and a three-term balance in which one term is quadratic produces a cube root.

So the exponent is a count of terms. It is two-thirds because the nonlinear term is quadratic and there are three of them, and it would be something else if the equation’s nonlinearity were cubic — which is exactly what happens in the hypersonic range, where the corresponding similarity parameter carries a different power for the same reason.

The consequence for a designer is that thickness is much more expensive near Mach one than elsewhere. A section’s critical Mach number falls as $\tau^{2/3}$, so halving the thickness buys back only $2^{2/3} = 1.59$ times as much margin rather than twice — but it also means the reverse: going from ten per cent to twelve costs $(1.2)^{2/3} = 1.13$ of the margin, a thirteen per cent loss for a twenty per cent gain in thickness. Two-thirds is the exponent that makes thinning a wing a losing battle at the top and a winning one at the bottom.

What the collapse is worth, and what would have made it worthless #

A scaling that collapses three curves is only interesting if the three curves were different to begin with.

The unscaled peak pressure coefficients are 0.514, 0.723 and 0.976: a spread of ninety per cent across the family. The scaled ones are 4.8107, 4.8110 and 4.8114, a spread of fifteen parts in a hundred thousand. Along the whole chord the worst disagreement between any two of them is 0.038 in scaled units, which on a peak of 4.81 is eight parts in a thousand and is the solver’s residual rather than the rule’s.

That is not a fit. Nothing was adjusted, the three runs solved equations whose coefficients differ by factors of order one in every term, their boundary conditions differ by a factor of two and a half, and the comparison was made after the fact.

The whole family, drawn once #

Once the parameter is the only coordinate, the transonic range can be surveyed in a way that the unscaled problem makes impossible.

At $K = 1.6$ the pressure distribution is smooth and everything is subsonic. By $K = 1.2$ there is a supersonic run of a quarter of the chord and a compression at the end of it. At $K = 0.9$ the run is half the chord and the compression has become a discontinuity a cell wide. At $K = 0.6$ the pocket covers most of the section and the shock stands near the trailing edge.

The threshold is at $K \approx 1.35$ for this section, and it is a property of the section’s shape and of nothing else. Feeding that number back through the definition of $K$ gives the critical Mach number of any thickness of this section directly, which is the whole of what the exponent measurement was reaching towards.

What it is worth to somebody with a tunnel #

The practical content is about testing, and it is the reason the rule was found when it was.

A transonic wind tunnel cannot generally run a full-size section. It runs a model, and a model is usually a scaled-down version of the real thing — same shape, same thickness ratio, smaller. That scaling leaves $K$ alone, so it presents no difficulty and needs no rule. The difficulty arises when the model cannot have the same thickness ratio: when a section is too thin to instrument at model scale, or when a tunnel cannot reach the Mach number a thin section needs.

The similarity rule says what to do. Test a thicker model at a lower Mach number, at the same $K$, and scale the answer. The pressure distribution of the real section is the model’s, multiplied by the ratio of the two values of $\varepsilon$, which is $(\tau_{\text{real}}/\tau_{\text{model}})^{2/3}$ times a factor of order one in the Mach numbers.

That substitution is exact within the small-disturbance equation and is not available anywhere else in this subject. A subsonic test at the wrong Mach number can be corrected by Prandtl–Glauert, which is a first-order statement that predicts its own failure near Mach one; a supersonic test at the wrong Mach number can be corrected by Ackeret’s linear theory. Between them there is nothing — except this.

The rule cuts in the other direction too, and that is the use it was actually put to. A tunnel near Mach one has a difficulty that has nothing to do with the model: the flow between the model and the walls chokes, the blockage correction diverges as the free stream approaches sonic, and the instrument ends up in the answer. Slotted and perforated walls were invented to deal with it and did not remove it. What the similarity rule offers is a way of avoiding the worst Mach numbers altogether: to learn what a thin section does at Mach 0.91, test a thick one at Mach 0.83, where the tunnel is well behaved and the corrections are ordinary. The three members of the family drawn above span Mach 0.83 to 0.90, and the one at 0.83 is a comfortable test while the one at 0.90 is not.

There is a third use, and it is the one that would have justified the rule on its own before computers. One solution serves a family, and solutions were extremely expensive. A relaxation that takes three seconds here took a room of people a week in the 1950s, and the arithmetic that says a single answer covers every thickness and every Mach number along a curve is the difference between a subject with a handful of computed cases and a subject with a chart. The rule’s value has fallen steadily since, for the same reason the value of every similarity rule falls: the thing it economises on has become cheap. What has not fallen is its value as a check — a solver whose answers do not collapse has a bug, and that is how the collapse above was used here.

How the solutions were obtained, which is the other half of the story #

The curves above are not analytic and cannot be. The scaled equation is still nonlinear and still changes type, so it has to be relaxed numerically, and until 1971 nobody knew how.

The difficulty is not the nonlinearity. It is the type change. Where the coefficient of $\Phi_{XX}$ is positive the equation is elliptic, every point is influenced by every other, and a central difference is right. Where it is negative the equation is hyperbolic, nothing travels upstream, and a central difference is not merely inaccurate — it lets a point be influenced by one downstream of it in a region where that is forbidden, and the iteration diverges. A backward difference everywhere is stable and throws away half the physics in the subsonic part.

Murman and Cole’s answer is to switch the difference on the sign of the coefficient, point by point, and let the solution decide which scheme it is being solved by. Written in the retarded form — the elliptic operator taken at the point, the hyperbolic one at the point upstream — that single line also produces the right behaviour at the two places where the type changes. At a sonic point neither operator survives and $\Phi_{XX}$ drops out, which is correct, because its coefficient is passing through zero. At a shock point both appear and their sum is the operator that conserves mass across the jump — the shock’s own jump conditions being what conservation alone fixes.

The shock is not put in. Nothing in the scheme knows that a discontinuity is coming; it emerges as a compression that steepens until it occupies one cell, and the jump it settles at is the one the equation requires. That is the same property a hydraulic jump has in a shallow-water calculation, and it is the reason a conservative scheme is worth the trouble.

One more property of the scheme is worth stating because it is the reason relaxation works at all here. Starting from a uniform stream and relaxing straight to a supercritical case diverges: the first sweeps produce a perturbation far larger than the converged one, the coefficient goes strongly negative over most of the chord, and the backward differencing then carries that error downstream with nothing to correct it. The cure is to approach along the family — converge a subcritical case, use it as the starting guess for a slightly faster one, and walk up in steps of a few hundredths in Mach number. Every solution in this essay was obtained that way, in ten steps from Mach 0.6.

That is not a numerical detail so much as a restatement of the physics. A transonic flow is not determined by its boundary conditions in the way a subsonic one is; the position of the shock depends on the history of how the flow was set up, in the sense that the iteration has to be led to it. The steady equation has more than one solution for some conditions, and which one a relaxation finds depends on where it started — which is the first hint, in a purely steady calculation, that the real flow may not settle down at all.

What the picture cannot show #

The equation is the small-disturbance one and the sections are thin. Twelve per cent is already at the edge of what a small-disturbance theory should be asked; the collapse holds there because the scaling is exact for the equation being solved, not because the equation is exact.

There is no viscosity anywhere. A real transonic section has a boundary layer, the shock separates it, and the separation changes the pressure distribution upstream of the shock as well as behind it. Every curve above is what the outer flow would do if the section’s surface were a streamline, and at the strong-shock end of the family that is a substantial fiction.

The section carries no lift. A symmetric section at zero incidence has the same flow above and below, which is what makes a half-plane calculation legitimate. Lift breaks that, requires a circulation and a wake, and the similarity rule then acquires a second parameter — the incidence, scaled — so the one-parameter family becomes a two-parameter one.

The shock is one cell wide. Its position is therefore known only to a cell, which is a fortieth of the chord here, and the pressure immediately behind it is an average over a region the scheme does not resolve.

And the parameter is not the only thing that matters about a section. Two sections of different shape at the same $K$ are not the same flow; the rule collapses thickness and Mach number, and the shape function stays in the problem.

Who found it, and when #

Von Kármán stated the transonic similarity rule in 1947 and Guderley gave it independently at about the same time, both working from the structure of the equation rather than from data. Murman and Cole’s switched relaxation is 1971, and the twenty-four years between them are the interval in which the rule was known to be true and could not be used for anything, because nobody could produce the one solution per family that it turns into a whole family.

The surprising connection is with a rule about a completely different equation, and it is the same kind of statement. Shallow water is a gas whose ratio of specific heats is two, and a hydraulic jump is a shock — the analogy is exact. What the transonic rule does for a family of aerofoils, the Froude number does for a family of channels: it removes two quantities from a problem and leaves one, so that a model of a spillway a hundredth of full size is a statement about the spillway rather than a picture of one. Both are the same operation — finding the group the equation actually depends on — and the transonic case is harder only because the group contains a two-thirds power and nobody guesses two-thirds.

Still open: whether a lifting family is still a family #

The rule collapses the thickness and the Mach number of a section at zero lift. What an aeroplane flies at is not zero lift, and the extension is not a formality.

At incidence the scaled problem keeps a second parameter — the incidence divided by $\tau$, roughly — so the family becomes two-parameter and the practical value of the rule drops sharply: a tunnel can match one number by choosing a Mach number and the other by choosing an angle, but the two are then no longer free to be used for anything else. Whether that is the whole of the story is a calculation rather than a guess, because the lifting problem has a third ingredient the symmetric one does not: a wake, carrying a jump in potential fixed by the Kutta condition, whose own scaling has to be checked rather than assumed.

The calculation that would settle it is the same relaxation on the full plane with a slit, a prescribed jump across the wake, and the trailing-edge loading driven to zero — then run at several thicknesses and incidences chosen to hold both scaled parameters fixed, asking whether the lift coefficients collapse as the pressures do here. The interesting possibility is that they do not collapse exactly, because the Kutta condition is a statement about a point rather than about the equation, and a point is not obviously covered by a scaling argument.