The skin that lags the flight

The rung below this one computes the temperature a wall settles at when it is told nothing: most of the way to the stagnation temperature, at a fraction — the recovery factor — that comes out of the energy equation and is exactly one at a Prandtl number of one. That is a real number and every high-speed structural calculation contains it.

It is also, on its own, an answer to a question nobody asks. An aircraft skin is not insulated and it is not held at a temperature. It is a sheet of metal with heat capacity, sitting in a stream whose condition is changing, and what it does is lag.

What the fluid actually supplies #

The right way to see the previous rung’s result is not as a temperature but as half of a boundary condition, and it is worth being explicit about which half.

What crosses the wall is a flux, and the layer’s own answer for it is

$$q = h\,(T_{aw} - T_w),$$

with $T_{aw}$ the adiabatic wall temperature and $h$ the heat transfer coefficient. Both come from the fluid. Neither is a temperature the wall has.

The first is the rung below’s whole subject, and its importance is that the driving temperature is $T_{aw}$ rather than the free stream’s — which is why a wall hotter than the air can be being heated by it, and why the sign of the flux is not what it looks like. The heat came from the flow’s own kinetic energy, which is what the compressible energy equation is about and is the same accounting that gives a shock its temperature rise. The second comes from the Reynolds analogy on the same layer: $\mathrm{St} = (C_f/2)\, \mathrm{Pr}^{-2/3}$, which that essay measures against the solved layer and finds good to a few per cent.

So the fluid hands the structure two numbers and no temperature. Everything after that is an ordinary differential equation with no fluid mechanics in it at all, and the whole of what makes a real skin’s history interesting is in the solution of that equation.

The equation, and the time constant in it #

A thin metal skin has a small Biot number — it conducts through its own thickness far faster than it exchanges with the air — so it is a lumped mass at one temperature, and its energy balance is one line:

$$\rho_s c_s \tau \frac{dT_w}{dt} = h\,(T_{aw} - T_w) - \varepsilon\sigma\left(T_w^4 - T_{surr}^4\right).$$

With the radiation term dropped and the flight condition held, that is linear and its solution is an exponential with time constant

$$\tau_{th} = \frac{\rho_s c_s \tau}{h}.$$

For two millimetres of aluminium — $\rho_s c_s \tau = 4{,}860$ J m⁻² K⁻¹ — against the $h$ a turbulent layer at $\mathrm{Re}_x = 10^7$ delivers at Mach 2, that is about twelve seconds. Six millimetres is thirty-six. Twelve is eighty. Those are memories in the sense this collection uses the word — a present state that depends on a history rather than on a condition — and the width of the memory is the time constant.

Those are not small numbers on the timescale of a manoeuvre, and they are the reason the rest of this essay exists. A calculation that returns $T_{aw}$ has computed where the skin is going; it has said nothing about whether the flight lasts long enough to get there.

The dash #

This is the arithmetic behind a design decision that looks like bravado and is not. An aircraft can be flown to a Mach number its structure cannot sustain, provided it does not stay there — because the temperature the structure reaches depends on the integral of the heating rather than on its instantaneous value, and a short exposure integrates to less.

The figure puts numbers on it. The adiabatic wall temperature peaks at 590 K. A two-millimetre skin, with a time constant of a few seconds at that condition, gets to within seven degrees of it and is effectively at the steady answer. A twelve-millimetre skin — a spar cap, a leading-edge member, a fitting — reaches 448 K and never comes close.

So thickness is not only structure, it is thermal protection, and the two roles pull in opposite directions when the load is thermal: a thicker member is stronger and also colder, but the thermal gradient through a thick member is what produces the stress. That is a structural argument rather than a fluid one and it is where this essay hands the problem over.

There is a second feature in that figure worth naming because it is easy to miss and is a genuine consequence of having a memory. The thin skin’s peak temperature comes after the air’s, so for a few seconds during the deceleration the skin is hotter than the adiabatic wall temperature at that instant — and is therefore losing heat to a flow that was heating it a moment earlier. The heat flux reverses sign in the middle of the manoeuvre, at a time no steady calculation contains.

Where the steady answer stops being an upper bound #

The other thing a real skin does that an insulated one does not is radiate, and the effect is negligible where it is usually mentioned and dominant where it is usually skipped.

The steady state of a radiating skin is not where the convected flux vanishes but where the two fluxes balance:

$$h\,(T_{aw} - T_w) = \varepsilon\sigma\left(T_w^4 - T_{surr}^4\right),$$

which has exactly one root between the surroundings and $T_{aw}$, because the left side falls across that interval and the right side rises. The root is found here by bisection rather than by a correlation, and the interesting quantity is the margin $T_{aw} - T_w$.

At Mach 2 that margin is under two degrees in three hundred and sixty, and ignoring radiation is correct. At Mach 6 it is a hundred and fifty in fifteen hundred, and ignoring it is a ten per cent error in the temperature a material is being selected against.

The insulated-wall calculation is therefore a worse upper bound the faster the vehicle goes, which is the opposite of the impression its ubiquity gives. It is quoted most in the regime where it is least accurate, and the reason is that the fourth power beats the square.

The one number the structure cannot get from the fluid #

There is an asymmetry in this problem that is worth stating plainly, because it decides how the two disciplines talk to each other.

The fluid side is nearly certain about $T_{aw}$ and nearly uncertain about $h$. The recovery factor is 0.84 laminar and 0.89 turbulent, so a wrong guess about transition moves the driving temperature by three per cent. The heat transfer coefficient between the same two states differs by a factor of three to five, which is the whole cost of going turbulent arriving on the thermal side, so the same wrong guess moves the flux — and therefore the time constant, and therefore everything in this essay — by several hundred per cent.

And predicting transition on a real surface is the least reliable calculation in high-speed aerodynamics, depending on roughness, on the wall’s own temperature, on free-stream noise and on shapes no criterion captures. So the structural answer inherits the fluid’s worst uncertainty rather than its best one.

That is not a counsel of despair; it is a statement about where effort belongs. Refining the recovery factor from 0.8417 to 0.8419 changes nothing. Deciding whether the layer is laminar or turbulent at the station in question changes the answer by a factor.

Why the exponential is the right shape, and when it is not #

The relaxation above is exponential, and it is worth saying why rather than assuming it, because the reason is also the boundary of the model.

The equation is $C\,\dot{T}_w = h\,(T_{aw} - T_w)$ with $C$ and $h$ constants: linear, first-order, one time constant. Its solution is one exponential, so a skin has one number of memory and no more. Give it a step in flight condition and its whole future is determined by where it is now and where it is going, with no dependence on how it got here.

That is a strong claim and it is exactly the kind this collection is suspicious of. It holds here for two reasons, both of which fail in identifiable circumstances.

It holds because the skin is thin. A lumped mass has one degree of freedom. A thick member has a temperature field through it, which has infinitely many modes with a spectrum of time constants, and its response to a step is a sum of exponentials rather than one — so its late behaviour is set by the slowest mode and its early behaviour by all of them. The lumped model is the slowest mode surviving, and it is right after the fast ones have gone and wrong immediately after a step.

And it holds because radiation was dropped. With the fourth-power term restored the equation is nonlinear, its approach to equilibrium is not an exponential, and — the part that matters — it is faster than exponential from above and slower from below, because the radiated flux grows steeply with temperature. So a hot skin cools quickly at first and then lingers, which is a shape a single time constant cannot produce and which the figures here do integrate.

The practical consequence is a warning about time constants quoted in isolation. A number like “twelve seconds” is a property of a linear model at one condition; it changes with altitude because $h$ does, it changes with Mach number because $h$ does, and it stops being a single number at all once radiation matters. It is a useful order of magnitude and it is not a constant of the aircraft.

The heating a manoeuvre integrates #

There is a reformulation worth carrying because it is how the calculation is usually done in practice and because it makes the dash argument obvious rather than surprising.

Write the linear case in terms of the integral. The skin’s temperature at time $t$ is

$$T_w(t) = T_0 e^{-t/\tau_{th}} + \frac{1}{\tau_{th}}\int_0^{t} T_{aw}(s)\,e^{-(t-s)/\tau_{th}}\,ds,$$

which says that the skin reports a weighted average of the flight’s history, with the weight falling off exponentially into the past over one time constant. It is a convolution, and the time constant is the width of the kernel.

Everything then follows without any further calculation. A condition held for much longer than $\tau_{th}$ is reported at full weight and the skin reaches the steady answer. A condition held for much less is reported at a weight of roughly its duration over $\tau_{th}$, so a ten-second dash on a sixty-second skin collects about a sixth of the temperature rise. And two profiles with the same peak Mach number and different durations give different peak temperatures, which is the design freedom the dash exploits.

The kernel’s width is the whole of the model. Everything the fluid contributes is in one number, and once that number is known the structure’s thermal behaviour is a filter applied to a flight profile.

What a designer does with this #

The chain from a flight condition to a material choice is short enough to write out, and writing it out is the clearest statement of what the two rungs of this ladder are for.

Pick the condition. A Mach number and an altitude give a free-stream temperature and a density.

Get the driving temperature. The recovery factor turns the stagnation rise into the adiabatic wall temperature, and it is 0.84 if the layer is laminar and 0.89 if it is turbulent — a three per cent question with a clear answer either way.

Get the coefficient. The Reynolds analogy turns the friction into a heat transfer coefficient, which needs the friction, which needs to know whether the layer is laminar. This is the step that carries the uncertainty.

Get the time constant. Divide the member’s heat capacity per unit area by that coefficient. This is where the structure enters and the fluid leaves.

Compare it with the profile. If the condition is held for many time constants, the member reaches the adiabatic wall temperature, less whatever radiation buys. If it is held for a fraction of one, it reaches a fraction of the rise.

Then choose the material against the answer, and iterate, because a material chosen for temperature has a density and a specific heat, which change the time constant, which changes the temperature. The loop converges quickly and it does have to be run.

The thing worth noticing about that list is where the disciplines meet. Steps one to three are fluid mechanics and produce two numbers. Steps four to six are structures and produce a material. Nothing crosses the boundary except $T_{aw}$ and $h$, which is why the two calculations are done by different people in different offices — and why an error in $h$ is so expensive: it arrives on the structural side with no way to notice it there.

The steady state nobody reaches #

The whole of this essay’s arithmetic is about approaching an equilibrium, and it is worth asking how often an aircraft is actually at one — because the answer decides whether the transient is a detail or the whole story.

A subsonic transport is. It climbs for half an hour and cruises for six, and its time constants are tens of seconds, so every structural temperature has settled long before anything matters. The steady calculation is the right one and this essay is unnecessary.

A supersonic transport is, and barely. Concorde’s cruise was two or three hours against a skin time constant of minutes, so it too reached equilibrium — and its cruise Mach number was chosen so that the equilibrium was survivable. The transient mattered on the acceleration and the deceleration and not in between.

A military aircraft on a supersonic dash is not, which is the case the figures draw and the reason the manoeuvre exists.

And a re-entering vehicle is emphatically not. Re-entry lasts a few minutes, the heating rate varies by orders of magnitude within it, and no part of the structure is at equilibrium at any point. The peak temperature reached is an integral over the trajectory, and the trajectory is designed against that integral rather than against any instantaneous condition.

So the fraction of high-speed flight that the steady calculation describes is smaller than the literature’s emphasis suggests, and the reason for the emphasis is that the steady answer is the one that can be written down. It is an upper bound, it is easy, and it is quoted — and for two of the four cases above it is also the answer.

What the vehicle does about it #

The chain from a flight condition to a material was set out above, and there is a step in it a designer takes that the arithmetic makes legible: change the time constant rather than the temperature.

The time constant is $\rho_s c_s \tau / h$, and three of those four are choices.

Thickness. A thicker skin has a longer time constant, so it reaches less of the adiabatic wall temperature in a given exposure. Adding metal is a thermal decision as well as a structural one, and on a dash aircraft the two point the same way.

Heat capacity. $\rho_s c_s$ per unit volume varies less between structural metals than their strengths do — aluminium and steel are within about a third of each other — so this is a weak lever unless something exotic is used. A heat sink is exactly a strong version of it: the X-15’s early flights relied on the Inconel structure’s own capacity to absorb the heat of a short exposure and radiate it afterwards, which is this term used deliberately.

And $h$, which is the flow’s. Anything that keeps the layer laminar lengthens the time constant by a factor of three to five, which is the largest lever of the three and the least reliable.

The fourth quantity, $T_{aw}$, is the one that cannot be changed at all — it is set by the flight condition and the recovery factor, and the recovery factor is a property of the gas. That is the division this ladder keeps arriving at: the fluid supplies a temperature that is fixed and a coefficient that is uncertain, and everything a designer can act on is on the other side of the equation.

What the picture cannot show #

The skin is one temperature. The lumped model has no gradient through the thickness, which is the right approximation for a thin metal sheet and the wrong one for an insulator or an ablator — and it is precisely the through-thickness gradient that produces thermal stress, so this model computes the temperature and cannot compute the load.

No conduction along the skin. A real structure conducts sideways into ribs, spars and fuel, and those are large heat sinks. The model here is a square metre of skin exchanging with the air and with nothing else, which makes every temperature it reports an overestimate.

The flight profile is invented. The dash is a ramp, a hold and a ramp, chosen to be legible rather than to be any aircraft’s. Nothing here is a simulation of a vehicle.

And $h$ comes from a correlation, which is a different kind of statement from the solved profiles the layer itself is built on. The Reynolds analogy with a one-seventh-power friction law is a fit, good to a few per cent on a flat plate and worse on anything with a pressure gradient, a shock interaction or a three-dimensional corner. Every time constant in this essay carries that.

The assertion behind the figures is the one that could have caught an error and does: with radiation off and the condition held, the integrator must reproduce the exact exponential at every step, and the radiating case must land on the bisected root rather than near it. A model that drifted from its own closed form would have been found there.

Who computed it, and when #

The lumped-capacity treatment of a skin is old enough to have no single author — it is the standard first calculation in any text on aerodynamic heating, and it dates from the period when supersonic flight made the question urgent, in the late 1940s and 1950s.

What changed the subject was the thermal barrier being reached in practice rather than in principle. The gas itself stops being a perfect one somewhere above this range — gamma stops being a number once the air behind the shock is hot enough to change chemically — so the arithmetic here is honest to about Mach 5 and increasingly not beyond it. The X-15 flew with a skin of Inconel X selected for its strength at temperature rather than for its strength; the Concorde cruised at Mach 2 with an aluminium skin at about 400 K, a temperature chosen to sit below where aluminium’s long-term creep becomes a problem — and the aircraft’s cruise Mach number was set by that material limit rather than by anything aerodynamic. Its fuselage grew about three hundred millimetres in flight, and the gap that opened at the flight engineer’s station was famously filled with a cap at the end of the last flight.

That is the clearest statement of what this rung is about. The Mach number of a supersonic transport was chosen by a structural material’s temperature limit, and the chain from the flight condition to that limit runs through the recovery factor, the heat transfer coefficient, and an ordinary differential equation with the skin’s own heat capacity in it.

Where the ladder goes next #

The rung above is the instrument, and it is the same arithmetic pointed at something much smaller. A total-temperature probe in a stream is a stagnation region with a thermocouple in it, so it has its own recovery factor — near 0.98 rather than 1 — and everything in this essay applies to it at a scale of millimetres and milliseconds. Measuring the air’s temperature in flight turns out not to be measuring the air’s temperature, and what is inferred from the reading amplifies the probe’s error rather than inheriting it.

The one beside it is the through-thickness problem this essay declined: a member thick enough to have a gradient in it, where the temperature is a field rather than a number and the quantity of interest is the stress that gradient causes. That is a heat-conduction problem with the fluid supplying a flux at one face, and the fluid-side input is exactly what has been computed here.