The stress that picks the aerodynamics

The rung below splits a rotor’s pressure rise into a term a boundary layer limits and a term that is free — free in the sense that it costs no diffusion and has no stall in it. The free term is $(U_2^2 - U_1^2)/2$, and the way to have more of it is to move the flow outward faster.

There is no limit on that in any equation of fluid mechanics, and there is a hard one about ten centimetres away, in the metal.

The integral, and what cancels in it #

A blade rotating at $\omega$ carries its own mass outward. Take an untapered blade of constant cross-section $a$ running from a hub radius $r_h$ to a tip radius $r_t$, and integrate the centrifugal load on everything above a station down to the root:

$$F = \int_{r_h}^{r_t} \rho_b\,a\,\omega^2 r\,dr = \rho_b\,a\,\omega^2\,\frac{r_t^2 - r_h^2}{2},$$

so the stress — force over the root’s own area — is

$$\sigma = \rho_b\,\omega^2\,\frac{r_t^2 - r_h^2}{2}.$$

The cross-sectional area has cancelled. A thick blade carries more load and has more area to carry it with, in exactly the same proportion, so the stress at its root is the same as a thin one’s.

Write the annulus area $A = \pi(r_t^2 - r_h^2)$ and the shaft speed $N$ in revolutions per second, and the expression becomes

$$\sigma = 2\pi\,\rho_b\,A\,N^2.$$

That is the AN² parameter, and the list of what is not in it is the point of this essay. Not the chord. Not the blade count — thirty blades share the annulus and each carries its own weight, and the per-blade stress is unchanged. Not the aspect ratio. Not the pressure, the temperature, the density, the Mach number or the Reynolds number of the gas. Not the working fluid at all.

An aerodynamic variable capped by a metallurgist #

The annulus area is the flow area. It is what the mass flow passes through, it is what the axial velocity is computed from, it decides the flow coefficient the velocity triangles are drawn at, and it is one of the first things chosen when a machine is laid out.

Rearranged, the relation says the largest annulus a material permits at a given speed:

$$A_{\max} = \frac{\sigma_{\text{allow}}}{2\pi\,\rho_b\,N^2}.$$

So a quantity in the aerodynamicist’s first calculation is set by a ratio of two material properties, and the ratio is specific strength — allowable stress over density — rather than strength.

That ordering is the figure’s most useful content and it is not intuitive. Steel is stronger than titanium and permits a smaller annulus, because it is nearly twice as dense and carries nearly twice the load to be strong against. A hot nickel alloy in the turbine is stronger than titanium at room temperature and much weaker at its own working temperature, which is why turbine annuli are small and turbine blades are cooled — cooling being a heat-transfer problem whose driving temperature is the recovery temperature rather than the gas’s own.

There is a second reading of the same ordering that is worth having. Specific strength is exactly the group that decides how fast any rotating thing may be spun — a flywheel, a centrifuge, a turbine disc — and the reason is the same cancellation: the load a rotating body puts on itself is proportional to its own density, so what survives is a ratio. A flywheel and a fan blade are limited by the same number and it is not a fluid-mechanical one.

And it is why the fan of a large engine is titanium and not steel, which is a decision about aerodynamics — how much air the engine may swallow — arrived at entirely through a materials property.

What a designer is actually trading #

The chain runs: more pressure rise per stage wants more radius; more radius at a fixed speed wants a larger annulus or a higher tip; both raise the stress; and the stress is capped. So there is a genuine optimisation here and its variables cross a disciplinary boundary.

Fewer stages means more pressure rise from each, which means more radius or more speed, which means more stress. Every stage removed from a compressor is paid for in the metal.

A slower shaft relieves the stress as the inverse square, which is generous — but it also lowers the blade speed, which lowers the work available from Euler’s equation directly, so a slower machine needs more stages to do the same job.

A smaller hub raises the annulus for a given tip radius — and it also worsens the radial equilibrium problem, since the swirl a free-vortex design asks for grows as the radius falls and becomes unmakeable near the axis and therefore raises the stress, and it also raises the radius ratio the rung below wants. The two pull in the same direction and the metal decides where they stop.

And taper is the one free move. A blade whose tip cross-section is smaller than its root’s has less mass out at large radius to carry, so its root stress falls — by roughly $(1 + \text{taper})/2$ for a linear taper, which is a factor of 0.75 at a taper ratio of a half and is the difference between a titanium blade that works and one that does not.

The tip speed, which is the same statement in other words #

There is a second way this constraint is usually written, and setting the two side by side makes the content clearer than either does alone.

Rather than an annulus and a shaft speed, an engineer often quotes the tip speed $U_t = \omega r_t$. For a blade with a small hub relative to its tip the stress expression becomes approximately $\sigma = \rho_b U_t^2/2$, which is a remarkably clean statement: a rotating blade’s root stress is its own material’s density times the square of its tip speed, halved.

Put numbers to it. At 440 metres a second — an ordinary fan tip speed — a titanium blade carries about 440 megapascals by that estimate, which is a substantial fraction of what titanium can be asked to do, and the hub radius and the taper are what bring the real number down to something buildable.

The tip speed form is the one to carry because it makes the ceiling obvious and universal. Whatever the machine, whatever its size, whatever the gas, a blade’s root stress is set by how fast its tip is going and by what it is made of. A model engine and a fan four metres across, geometrically similar and running at the same tip speed, have identical blade stresses — which is why blade tip speeds across the whole of turbomachinery fall in such a narrow band, and why that band has moved so little.

And it explains a fact about scaling that surprises people. Making a machine bigger does not make its blades more highly stressed, provided the tip speed is held — which means the shaft speed must fall in proportion to the size. That is the same similarity the affinity laws express, arriving from the metal.

What has not cancelled #

The list of absences in the AN² relation is long enough to be worth balancing with what is present, because the cancellations can start to look like magic.

Density is present, and it is the blade’s, not the gas’s. That is the whole trick of the cancellation: the load is the blade’s own weight, so the only material in the answer is the one the blade is made of.

Radius is present, twice over. It is in the annulus area and it is in the speed, which is why the constraint bites hardest exactly where the rung below wants to go.

And geometry is present through the taper alone. Of all the shape a blade has — its camber, its stagger, its thickness distribution, its aspect ratio, its twist — only the radial distribution of cross-sectional area appears. Everything an aerodynamicist spends time on is invisible to this calculation, and everything this calculation cares about is invisible to the aerodynamics.

That mutual blindness is the reason the parameter exists. Two disciplines that shared variables would have to negotiate over each of them; two that share exactly one number can each work alone and meet over it, which is what $AN^2$ is for and why it is quoted rather than derived at every meeting.

Why the parameter is quoted rather than the stress #

Engine practice quotes $AN^2$ rather than the stress it implies, and the reason is worth stating because it is a good example of what a grouping is for.

The stress depends on the material and the taper as well as on $A$ and $N$. Splitting it into a product of things the aerodynamicist controls — the annulus and the speed — and things the stress engineer controls — the density, the allowable stress and the taper — lets the two sides negotiate over a single number. The aerodynamicist asks for an $AN^2$; the stress engineer says whether that $AN^2$ can be built and in what.

It is the same move as the specific speed in the ladder next door: a grouping chosen so that everything one party decides is on one side of it. And it has the same property, which is that the grouping is exact and the limit on it is not — where an $AN^2$ becomes unbuildable is a summary of manufacturing and material practice, which has moved substantially with single-crystal casting, with hollow fan blades and with composites.

Where the constraint has been beaten, and how #

A ceiling that has moved for seventy years is worth looking at from the direction of what moved it, because every one of the moves is legible in the expression.

Remove mass from a large radius. A hollow fan blade — a titanium sandwich with a honeycomb or truss core — has a fraction of the density of a solid one over most of its span, and the integral above is weighted by radius, so removing mass near the tip is worth several times removing it near the root. The wide-chord hollow fan blade of a modern large engine exists for exactly this term.

Or remove the metal altogether. A carbon-fibre composite blade has about a third the density of titanium at a comparable specific strength, and $\rho_b$ is in the answer to the first power. That is the largest single change available and it is why the fans of the newest large engines are composite.

Change what the root has to carry it with. The cancellation of the cross-sectional area assumed the blade’s section is constant; a blade with a fir-tree root spreads its load into the disc over a much larger area than its aerofoil section, which does not change the stress computed here but changes what the attachment has to survive. The disc, not the blade, is often the binding component.

Or cool it, which changes the allowable rather than the load. A turbine blade’s problem is that $\sigma_{\text{allow}}$ collapses with temperature, so internal cooling passages and film cooling exist to keep the metal below the gas — at the price of the cooling air, which was compressed and is then not expanded, and which is a direct and substantial loss in the cycle.

None of those is an aerodynamic improvement and all of them are aerodynamic enablements. Each one raises the annulus area or the shaft speed a designer may ask for, and what is done with that freedom is the subject of the two rungs below.

That is worth stating plainly because it inverts the usual order of a design story. The pressure rise per stage, the number of stages, the mass flow an engine can swallow, and therefore its thrust — all of them sit downstream of a stress the aerodynamics cannot influence and does not contain.

What the picture cannot show #

One stress out of several. Centrifugal tension at the root is the largest steady stress in a blade and it is not the only one. There is a gas bending load — the pressure difference across the blade, which is what Euler’s equation prices as work seen as a force — a centrifugal bending load if the sections are not stacked on a radial line, a thermal stress in a cooled turbine blade, and a vibratory stress that decides fatigue life. A blade is designed against all of them and this essay computes one.

The taper relief is an approximation. $(1 + \text{taper})/2$ is right for a linear variation of area with radius and is a fair estimate otherwise. A real blade’s area distribution is chosen deliberately and its relief factor is computed rather than estimated.

No creep and no temperature. The allowable stresses used here are single numbers, and a turbine blade’s allowable stress is a function of temperature and of how long it must last — a creep-rupture curve rather than a value. The nickel alloy’s number is illustrative of a hot condition rather than a specification.

And the annulus is treated as if it were the flow area. Blades occupy some of it, the boundary layers on the hub and casing occupy more, and the effective area for the flow is a few per cent less than the geometric one. That correction matters to the aerodynamics and not at all to the stress.

The assertion behind these figures is the one worth having: the $AN^2$ form must agree with the direct integral of the blade’s own weight to machine precision, and the annulus limit must return exactly the area that produced the stress it was given. Two expressions that agreed by construction would prove nothing; these are computed by different routes.

Who found it, and when #

The centrifugal stress in a rotating blade is an elementary result and has no discoverer. What is specific to this subject is its promotion into a design parameter — the practice of quoting $AN^2$ for a turbine and treating it as the currency in which aerodynamics and stress trade — which belongs to the gas turbine industry and dates from the 1940s and 1950s, when both disciplines first had to be reconciled at the same table.

The number’s history is a history of materials rather than of mechanics. Permitted $AN^2$ has risen steadily for seventy years and every rise came from metallurgy or from manufacturing: better alloys, directional solidification, single-crystal casting, blade cooling that let a hot section run hotter, and hollow titanium fan blades that removed mass from a large radius. Each of those is a change to the left-hand side of an equation whose right-hand side is a duty. None of it came from a better understanding of the stress, which was understood completely in 1940.

That is a slightly deflating observation and it is an honest one. The aerodynamic possibilities of a turbomachine have been advanced, for most of a century, by people who were not aerodynamicists.

Where the ladder goes next #

This anchor now has the work, the pressure and the constraint. What it does not have is the machine running at anything other than its design point.

The rung above is stage matching, and it is the problem the field’s whole method struggles with. Each stage of a multistage compressor is a control volume with its own operating point, and the stages must all be at a workable one simultaneously, at every speed the machine runs. They cannot be: a compressor designed to match at full speed is badly mismatched at part speed — a machine’s operating point sliding away from the one it was chosen at — with the front stages stalled and the rear ones choked, which is why bleed valves and variable stators exist. That is a question about a series of boxes rather than about one, and it is the one place where drawing a box round each part separately genuinely fails.

The one beside it is the vibratory problem this essay set aside. A blade passing a fixed obstruction is excited at a frequency the geometry sets, and the excitation is dangerous when it coincides with the blade’s own natural frequency — which is a resonance diagram every rotor is designed against, and is the same kind of coincidence as the lock-in a shedding body finds and which depends on the same mass and radius the steady stress does. It is the same integral asked a different question.