The only theory simple enough to optimise

The rung below this one settles whether Newton’s impact theory is true, and the answer has two halves: it is out by a factor of π over the angle at the speeds Newton was arguing about, and it is very nearly exact behind a strong shock in a gas whose ratio of specific heats has fallen towards one.

That leaves a question the essay does not ask. Why is a formula from 1687, known to be wrong, still the first thing a hypersonic vehicle’s shape is laid out with? Accuracy is not the answer — modified Newtonian estimates are a few per cent out at best and much worse near a shoulder. The answer is a structural property the formula has and no other aerodynamic theory does.

What locality means #

Newtonian pressure at a point is

$$C_p = C_{p,\max}\,\sin^2\theta,$$

with $\theta$ the angle between the surface at that point and the free stream. That is the entire rule, and what matters about it is what it does not contain: the rest of the body, anything upstream, the shock’s position, the Mach number except through $C_{p,\max}$, or the solution of anything.

Every other theory in this collection couples every point to every other. In potential flow, the velocity at a point is a sum over the whole boundary and moving the nose changes the pressure at the tail; a panel method’s influence matrix is dense for exactly that reason. The method of characteristics propagates information along Mach lines. A Navier–Stokes solve couples everything to everything twice over.

Newtonian theory couples nothing to anything. The consequence is arithmetic: the pressure distribution over a shape is an expression, the force is a quadrature, and — the part that matters — the derivative of the force with respect to a shape parameter can be written down.

What that buys, in one figure #

The classical result is that the minimum-drag body of revolution under Newtonian theory is a power law $r \propto x^{3/4}$, and it is one of the very few closed-form optimum shapes in aerodynamics. The sweep here finds 0.72 at this fineness ratio, and the exponent climbs towards the classical value as the body is made more slender — 0.745 at a base a tenth of the length — because 3/4 is the slender limit rather than a universal answer.

That is worth noticing because it shows the method working. The classical result is an asymptotic statement; a numerical sweep over the actual family recovers it in the limit and departs from it where the assumption behind it fails, which is what an asymptotic result is supposed to do.

No other aerodynamic theory permits that sweep at this cost. To find the minimum-drag shape under any coupled theory means solving a flow for each candidate, and the whole discipline of aerodynamic shape optimisation exists to make that affordable. Under Newtonian theory it is two hundred quadratures and takes no time at all.

The plate, and the ratio with no maximum #

The same locality gives lift and drag on a flat plate in closed form, and the answer is instructive because it is obviously wrong in a way that identifies the missing physics.

The windward face carries $C_{p,\max}\sin^2\alpha$; the leeward face is shaded and carries nothing. The force is therefore normal to the plate, and

$$C_L = C_{p,\max}\sin^2\alpha\cos\alpha, \qquad C_D = C_{p,\max}\sin^3\alpha,$$

so the lift-to-drag ratio is exactly $\cot\alpha$ — checked against the closed form at five incidences to a part in a million.

$\cot\alpha$ has no maximum. It rises without limit as the incidence goes to zero, which would say a hypersonic vehicle should fly at the smallest incidence it can and achieve any lift-to-drag ratio it likes. It does not, and the term that stops it is not in the pressure at all.

Skin friction acts along the plate rather than across it. It costs drag and contributes no lift, so its share of the total drag grows as the incidence falls and the pressure force shrinks with the cube of the sine. At a friction coefficient of 0.001 the best ratio is 5.22 at 7.2 degrees, and the whole shape of that curve is the competition between a pressure force falling as $\alpha^3$ and a friction force that does not fall at all.

That maximum is the number a hypersonic vehicle is judged against, and its smallness is the reason hypersonic flight is hard: an airliner cruises at a lift-to-drag ratio near twenty.

The rule that is not physics #

The shading — a surface turned away from the stream carries exactly zero — deserves separating from the rest, because it is a different kind of statement and it is where the method’s worst errors live.

The sine-squared law is a physical argument, however wrong: particles arrive, their normal momentum is destroyed, the pressure follows. The shading rule is geometry bolted on. It says that a surface which cannot be reached in a straight line from upstream receives nothing, and it has to be applied by hand because the formula would otherwise return a negative pressure, which is nonsense.

Two consequences follow, and both are real limitations rather than pedantry.

A leeward surface gets nothing at all — and a fluid at zero absolute pressure is not a thing, which the suction ladder settles —, so a Newtonian estimate of a body’s base drag or of the suction on a leeward flap is exactly zero rather than approximately zero. In reality the leeward side carries a small pressure — usually a fraction of the free-stream static — and neglecting it is a systematic error in a known direction.

And interference is invisible. A shock from one part of a vehicle striking another part produces a large local pressure the rule has no term for. That is not a small effect: shock–shock interaction on the X-15’s ventral fin and pylon produced local heating — a flux driven from the recovery temperature rather than the air’s — that burned through structure, and no local theory of any kind would have predicted it.

So the method’s failures are structural rather than numerical. It is wrong by a few per cent where it applies and wrong by any amount where two parts of a vehicle can see each other.

The comparison the theory gets right #

Since the claim above is that a biased theory still orders shapes correctly, it is worth testing rather than asserting — and the test is available because the rung below computed the bias.

Newtonian pressure is $2/(\gamma+1)$ of the exact strong-shock value at every wave angle, which for air is 0.833. That is a constant factor, independent of the angle, so every pressure on every shape is low by the same fraction and every drag coefficient is low by that fraction too.

A constant factor cannot change an ordering. If body A has less Newtonian drag than body B, it has less exact strong-shock drag by the same ratio, and the minimum of the family is at the same place. So the optimum exponent this essay computes is the optimum under the exact strong-shock pressure as well, and the two theories disagree about every number and agree about the answer.

That is a much stronger position than “approximately right”, and it is why the method survived. It is exactly right about the question a designer asks and exactly wrong about the number a designer quotes, and knowing which of the two is being used is the whole of using it correctly.

The caveat is the one already named. The factor is constant only where the strong-shock limit holds and only where nothing shades anything; introduce a leeward surface, a base, or a shock striking another part, and the bias stops being constant and the ordering stops being safe. The theory is trustworthy for comparisons among shapes it can see all of, which is a precise statement and a usable one.

The number every hypersonic vehicle is measured against #

The plate figure’s maximum deserves one more paragraph, because the number has a name and a famous empirical companion.

A hypersonic vehicle’s best lift-to-drag ratio is small, and it falls as the Mach number rises. The empirical rule quoted for decades is

$$\left(\frac{L}{D}\right)_{\max} \approx \frac{4(M+3)}{M},$$

which gives about 6 at Mach 6 and about 4.4 at Mach 20 — the so-called L/D barrier. It is a correlation over vehicles rather than a derivation, and its shape is exactly the competition this essay’s figure draws: a pressure force that falls with incidence and a friction force that does not.

The plate calculation gives 5.22 at a friction coefficient of 0.001, which sits inside that band and was arrived at from one algebraic pressure rule and one prescribed friction number. That agreement is worth having and it is not a validation — the barrier is a fit to flown and tested configurations, carrying every effect this theory has none of, and landing near it is a consistency check rather than a confirmation.

What the calculation does supply that the correlation does not is the reason. The barrier is not a mysterious ceiling; it is the point where a drag proportional to $\sin^3\alpha$ and a drag independent of $\alpha$ cross, and it moves with the friction coefficient exactly as the figure shows.

Where the same trade appears elsewhere #

A theory that is inaccurate and local, used because it is local, is a recognisable species and it is worth naming its relatives — because the trade is always the same and it is always worth making at the same stage of a design.

Strip theory on a wing, which treats each spanwise station as a two-dimensional aerofoil and ignores the induced flow the rest of the wing produces. It is wrong, it is local, and it is what a first pass uses before a lifting line couples the stations together.

Blade-element theory on a rotor, which is the same move on a different object.

And the momentum theorem on a control volume, which is local in a different sense — it knows the faces and refuses to look inside — and which is why a disc with no turbine in it can price every turbine.

What all of them share is that they answer the design question and not the physics question. Which shape is better, which stage count, which incidence — those are comparisons, and a theory with a consistent bias answers a comparison correctly even when every absolute number it produces is wrong. That is the strongest thing that can be said for Newtonian theory and it is quite strong: its errors are in the same direction on every candidate shape, so the ordering survives.

Why the blunt body wins, which the same rule explains #

There is one more consequence of locality worth drawing out, because it settles a question the shapes in the first figure raise and it is the most consequential design decision in the history of re-entry.

A sphere’s Newtonian drag coefficient is 1 and a slender cone’s is $2\sin^2\delta$, which at fifteen degrees is 0.134. The blunt body has seven times the drag, and it is the shape every crewed re-entry vehicle has used.

The reason is not in this essay’s rule and is in the rung next door. Drag is what dissipates the vehicle’s kinetic energy, and the energy has to go somewhere: into the gas or into the vehicle. A blunt body throws a strong detached bow shock well ahead of itself, heats an enormous volume of air, and convects most of that energy away downstream; a slender body has a shock lying close to its own surface and hands the energy to its own skin.

So the high-drag shape is the low-heating one, and the decision — Allen and Eggers, 1953 — inverted what every aerodynamicist’s instinct said. The drag coefficient the Newtonian rule computes so easily is the quantity being maximised rather than minimised.

That does not make the minimum-drag body of this essay useless; it makes it the answer to the other question. A shape is optimised for drag when the constraint is range or fuel, and for heating when the constraint is survival, and hypersonic vehicles come in both kinds — a slender cruiser and a blunt capsule — because they are solving different problems with the same rule.

And the same locality serves both. Newtonian theory computes the pressure distribution a blunt body carries as readily as a slender one’s, which is why it is the tool for laying out either — and why a formula that was wrong about a musket ball is the first thing drawn for a heat shield.

The optimisation done properly, and what it needs instead #

Since this essay’s claim is that locality makes optimisation cheap, it is worth saying what shape optimisation looks like when the theory is not local — because the contrast is what makes the point rather than the claim.

Under any coupled theory the drag of a shape is a functional of the whole shape, and the derivative of that functional with respect to a change at one point requires knowing how the change propagates everywhere. Computing it by finite differences means one flow solution per design variable, and a shape with two hundred variables is two hundred solutions per gradient.

The modern answer is the adjoint method, which computes the whole gradient in one additional solution regardless of how many variables there are — by solving a second, linear problem whose solution is exactly the sensitivity field. It is one of the genuinely large ideas in computational aerodynamics and it exists because the naive alternative is unaffordable.

Newtonian theory needs none of it. The gradient is the derivative of a quadrature with respect to a shape parameter, which is another quadrature, and it costs the same as the drag did.

That comparison is the honest measure of what locality is worth: an adjoint solver is a substantial piece of software written to recover, approximately and expensively, a property this three-hundred-year-old formula has for nothing. The formula pays for it in accuracy and the adjoint does not, which is the whole of the trade and is why both exist.

What a designer does with an inaccurate optimum #

A shape optimised under a biased theory needs handling, and the practice is worth stating because it is not “use it and hope”.

The optimum is a starting point, not an answer. A power-law body at the exponent this essay computes is the shape a coupled method is then run on — once, or a handful of times — rather than the shape that gets built. The cheap theory has narrowed a family of thousands to a neighbourhood of one, which is the expensive method’s hardest problem solved for it.

And the neighbourhood is what matters rather than the point. A drag minimum is flat near its bottom by construction, so a shape a few per cent off the optimum is a fraction of a per cent off the minimum drag — which means an inaccurate optimum in a flat basin is a very good answer, and the accuracy of the theory that found it matters much less than a reader would expect.

That flatness is a general property of optima and it is why cheap methods survive. The quantity being minimised is stationary at the minimum, so its first derivative vanishes there and the error from arriving at the wrong place is second order in how wrong. A theory that gets the location of the minimum right to within a few per cent gets the value of the minimum right to a fraction of one, and the Newtonian rule — with its constant bias and its locality — does exactly that.

What the picture cannot show #

No flow anywhere, which is the point. Nothing in this essay solves for a velocity, a shock position or a temperature. Every number is a quadrature of one algebraic expression over a surface, and a reader looking for a flow field is looking at the wrong theory.

Bodies of revolution at zero incidence. The drag figures are axisymmetric and aligned with the stream. At incidence the shading becomes a two-dimensional problem over the surface and the quadrature is a good deal more work, though still not a flow solution.

The stagnation coefficient is an input. Newton’s own value is 2; modified Newtonian uses the exact value behind a normal shock, which is about 1.83 for air at high Mach number and which the rung below computes. Every figure states which it used.

And the friction coefficient is prescribed. It is a number, not a boundary-layer solution, and a real hypersonic vehicle’s friction depends on whether the layer is laminar or turbulent — which is the least reliable calculation in the whole of high-speed aerodynamics and which moves the best lift-to-drag ratio substantially.

The assertion behind these figures is the one that could reject and it checks the quadrature against closed forms rather than against itself: a cone’s drag must equal $2\sin^2\delta$ exactly, a flat face must give the stagnation value, a sphere must give 1, and the frictionless plate’s ratio must be $\cot\alpha$ at every incidence. It also requires the shading rule to bite — a surface turned away must carry exactly zero — and refuses a search for a maximum in the frictionless case, where none exists.

Who used it, and when #

Newtonian impact theory’s second career began in the 1950s, when re-entry made hypersonic shapes an urgent problem and no method existed that could produce a pressure distribution over an arbitrary body in a hurry. Lees, Truitt and others established the modified form; the minimum-drag power-law body is usually attributed to Eggers, Resnikoff and Dennis in 1957, and it is a genuinely closed-form result of a kind hypersonics has very few of.

The historical shape of it is worth stating. A theory was written to explain a phenomenon, failed completely, was discarded for two hundred and fifty years, and then turned out to be the right first approximation in a regime its author had no notion of — and to have, in addition, a property nobody had valued in 1687 because nobody was optimising anything.

That property is locality, and it became valuable at exactly the moment somebody wanted to compare a hundred shapes before building one. The theory did not improve; the question changed.

Where the ladder goes next #

The rung above is the part of this rule that is a rule rather than a result. Shading is a geometric statement bolted onto a physical one, and the interesting rung is the case it gets wrong: two parts of a vehicle that can see each other, where a shock from one strikes the other and produces a local pressure and a local heating rate that no local theory has a term for.

That is shock–shock interference, its classification into six types is Edney’s from 1968, and the practical stakes are large — the type IV interaction produces a supersonic jet impinging on a surface, with a heating rate an order of magnitude above the undisturbed value, and it has destroyed structure on real vehicles. It is the sharpest available demonstration that locality is a property with a price.

The one beside it is the leeward side this essay assigns exactly zero. What a base or a leeward flap actually carries is a small but non-zero pressure, it is a substantial fraction of a slender vehicle’s drag, and estimating it needs a base-flow model that Newtonian theory has no room for.