Two ways to choke

A converging nozzle chokes because the passage narrows to a throat and the mass flow cannot be increased past the sonic condition there. That is the standard story and it is correct.

It is also not the general case. A pipe of constant area chokes too — with no throat anywhere in it — if it is long enough, or if enough heat is added to it. The mechanism has nothing to do with geometry, and it is worth seeing because it says what choking actually is.

Two different physics, two different lines #

Both lines answer the question “what states can this flow reach?”, and they differ in what is held fixed.

A Fanno line is the set of states a flow with friction and no heat can reach in a duct of constant area. Mass is conserved and energy is conserved — friction converts kinetic energy to internal energy without any of it leaving — so the stagnation temperature does not change. What is not conserved is momentum, because the wall is exerting a force.

A Rayleigh line is the set a flow with heat and no friction can reach. Mass is conserved and momentum is conserved — with no wall friction there is nothing to change $p + \rho u^2$ — while the energy is not, because heat is being added.

Two processes, each conserving mass and one other thing, each releasing the third. That is the whole of the difference between them, and it is enough to determine both curves completely.

Both peak at Mach 1, and that is what choking is #

Here is the result, and the reason it is not a coincidence.

Entropy cannot fall. Friction is dissipative, so a flow moving along its Fanno line moves in the direction of increasing entropy; heat added to a flow raises its entropy too. So in both cases the flow can only move towards the entropy maximum — and on both curves the maximum is at Mach 1.

The consequences follow immediately and are counter-intuitive in both directions:

• A subsonic flow with friction speeds up along the duct. Friction accelerates a gas, which sounds impossible until one notices that it also drops the pressure, and the flow is expanding as it goes.
• A supersonic flow with friction slows down — towards the same Mach 1.
• Heating a subsonic flow speeds it up; heating a supersonic flow slows it down. Both towards 1.
• And cooling a supersonic flow speeds it up, which is the most surprising line in the paragraph.

None of it can pass Mach 1. A duct longer than the length that brings the flow to sonic does not produce a supersonic subsonic flow or an impossible one; it reduces the mass flow until the length available is exactly the choking length. That is choking, in a pipe with no throat — and it is the same refusal a nozzle makes when its throat goes sonic, reached by a route with no geometry in it.

Why friction speeds a subsonic flow up #

The line that catches everybody deserves its own argument, because “friction accelerates the flow” reads as a contradiction and is not one.

Fix the mass flow $\rho u A$ with the area constant. Then $\rho u$ is fixed, so any increase in speed must be paid for by a fall in density — which, at roughly constant temperature, means a fall in pressure. Friction supplies exactly that: it drops the pressure along the pipe. The gas therefore expands as it goes, and an expanding gas in a constant-area duct must speed up to keep the mass flow.

What friction is removing is not speed but stagnation pressure: the flow’s capacity to be brought to rest at a high pressure. A Fanno flow loses stagnation pressure monotonically along the duct, and that loss is the entropy rise, and the entropy rise is what drives the Mach number towards one from either side.

So there is no contradiction, and there is a useful way to hold it: friction does not push the flow along, it unloads it, and the flow accelerates into the space that makes.

What the solver computed, and how it was checked #

Three things, and the first is deliberately done the long way.

The maximum is located by search. Quoting “the entropy peaks at Mach 1” from the algebra would be quoting the derivation the essay exists to test, so the entropy is swept over four thousand Mach numbers on each line and the maximum found numerically. Both come out at $1.000$ to within the sweep’s resolution.

Each line’s own invariant is recomputed from the state. For Fanno, the stagnation temperature is rebuilt as $T/T^* \cdot (1 + \tfrac{\gamma-1}{2}M^2)$ at half a dozen Mach numbers and must not move — it holds to a part in $10^9$. For Rayleigh, the same is done to the momentum flux $p(1 + \gamma M^2)$. If either drifted, the “line” being drawn would not be the process it claims to represent.

And the two lines cross where a shock lands. That is the next section, and it is the result worth the essay.

The crossing is a shock #

Draw both lines through the same state. They cross twice — once at that state, by construction, and once more.

A point on both curves conserves mass (both), energy (Fanno’s) and momentum (Rayleigh’s). Those three statements are exactly the Rankine–Hugoniot conditions, so the second crossing must be the state on the other side of a normal shock, with nothing about shocks having been mentioned anywhere in the construction.

The direction is fixed by the same second law that fixed the peaks. The crossing at higher entropy is the subsonic one, so a flow may jump from the supersonic branch to the subsonic — a compression shock — and not the other way. The pair of curves therefore contains the shock, its direction and its prohibition, without a single Rankine–Hugoniot relation being written down.

The same singularity, three times #

The three ways a one-dimensional flow can be driven — area change, friction, heat — all act on it through the same factor.

Each of them produces a governing equation of the form

$$\frac{\mathrm{d}M^2}{M^2} = \frac{(\text{something})}{1 - M^2}$$

with the numerator carrying the physics and the denominator carrying the same $1 - M^2$ every time. That factor is why every one of the surprising signs above exists: it changes sign at Mach 1, so any influence that speeds a subsonic flow up must slow a supersonic one down, and vice versa, whatever the influence is.

It is also why they all choke at the same place. The denominator vanishes at $M = 1$, so the flow cannot be driven through it by any of the three — only brought to it — and the only way to reach supersonic speed is to arrive at $M = 1$ with the numerator vanishing at exactly the same moment. That is what a nozzle throat does: the area gradient goes to zero exactly where the flow goes sonic, so the ratio stays finite.

Choking is one phenomenon, and Fanno, Rayleigh and the area relation are three views of it.

The engine that exists because of the Rayleigh limit #

Thermal choking sounds like a laboratory curiosity and it is the reason a whole class of engine is shaped the way it is. The argument runs straight off the curves above.

A ramjet decelerates the incoming air to a low subsonic Mach number, burns fuel in it, and expands the result through a nozzle. The burning is heat addition to a subsonic flow, so by the Rayleigh line it drives the Mach number up towards one, and there is a maximum stagnation-temperature ratio beyond which no solution exists at that mass flow. Exceed it and the engine does not simply perform worse: the flow chokes, the intake spills, and the shock system is expelled forward.

That ceiling is not the only one — the deceleration itself becomes ruinously expensive in total pressure at high flight speed, and the air arrives hot enough to dissociate rather than to accept heat — but it is the one that is pure Rayleigh, and together they put a ramjet’s practical ceiling somewhere around Mach six.

The way past it is to not decelerate the air. A scramjet burns in a flow that is still supersonic, at a combustor Mach number of two or three, and the same Rayleigh line then works in the engine’s favour: heat added to a supersonic flow drives the Mach number down towards one, so the flow decelerates as it burns, which is the deceleration the ramjet was paying for at the intake, obtained for free as a by-product of the combustion.

The Rayleigh limit has not gone away; it has moved. Add too much heat, or add it over too short a length, and the Mach number reaches one from above, the flow chokes, and the disturbance runs upstream and unstarts the inlet — which is the same failure the ramjet has, at a different point in the cycle and with less warning.

The engineering response is a component whose entire existence is a Fanno–Rayleigh argument. Between the inlet and the combustor sits an isolator: a constant-area duct, sometimes several diameters long, that does nothing aerodynamically useful except to hold a shock train. When the combustor’s back pressure rises, the shock train moves forward inside the isolator instead of being expelled from the inlet, and the duct’s length is a reserve of tolerance measured in exactly the friction-length units this essay uses.

So a component is sized by the two curves above. How much heat may be released is the Rayleigh limit; how much back-pressure margin the duct holds before the train reaches the inlet is a Fanno length; and the whole of a scramjet’s operating envelope is the region where neither has been exceeded.

Which one a real duct is on #

Real ducts have friction and heat, so neither line describes them exactly. What the pair is for is deciding which effect dominates and how much room is left.

A long cold pipe is essentially Fanno. The relevant number is $4fL/D$, the friction length, and the standard question is how long the pipe can be before it chokes: for air at Mach 0.3 in a smooth pipe with $f \approx 0.005$, the choking length is about $5.3\,D/4f$, which is a couple of hundred diameters. Beyond that the flow rate falls rather than the exit speed rising.

That number is worth comparing with the pipe-flow essays in the applied field, where the same friction length appears as a pressure drop rather than as a Mach number: the two are the same calculation asked in the incompressible and compressible limits, and they agree where they overlap — at low Mach number the Fanno pressure drop becomes the ordinary Darcy–Weisbach one.

A combustor is essentially Rayleigh. The relevant number is the stagnation temperature ratio, and the question is how much heat can be added before the flow chokes — which is why a ramjet’s performance is bounded by its combustion temperature in a way that has nothing to do with materials.

And a long hot pipe is neither, which is where the curves stop being answers and become a frame for a numerical integration. The same is true of anything with area change in it as well, which is most real hardware: an intake duct, a heat exchanger, a combustor with a divergence in it.

The connection to the loss with no viscosity in it is worth noticing: that essay’s sudden expansion is another case where momentum and energy accounting disagree and the disagreement is the physics. Here the same disagreement is what distinguishes the two lines.

What each process costs, in stagnation pressure #

Both processes are irreversible and both charge for it in the same currency, which makes them comparable.

The entropy rise per unit gas constant is $-\ln(p_0/p_0^*)$ on either line, so the natural way to quote either loss is as a fraction of the stagnation pressure surrendered. A Fanno flow entering a duct at Mach 0.3 and leaving it at Mach 0.6 has given up about six per cent of its stagnation pressure; heating the same flow to the same Mach number on a Rayleigh line costs a comparable amount.

That is the number an engine cares about, because stagnation pressure is what a nozzle can turn back into thrust. It is also why the two effects are usually presented together: in a real duct they compete for the same budget, and the design question is which of them is being spent deliberately.

What the picture cannot show #

Neither line has a length in it. Both are drawn against their own sonic reference, so a point on a Fanno line says how much friction length remains and not where in a particular pipe the flow is. Turning a curve into a duct requires a friction factor, which is a correlation rather than a solution.

Nothing here is two-dimensional. The whole apparatus assumes uniform properties across the duct, which is a fiction: a real pipe has a boundary layer, the friction is a wall stress rather than a body force, and the “one-dimensional flow with friction” is an average over a profile that changes shape along the pipe.

The friction factor is borrowed. Turning a Fanno line into a duct length requires $f$, and above transition that number is a correlation fitted to measurements rather than anything derived — the one imported quantity in the whole calculation, and this site marks it as such wherever it appears.

Heat addition is not combustion. A Rayleigh line adds heat to a gas of fixed composition and fixed $\gamma$. A flame changes the composition, the molecular weight and the specific heats, and a serious combustor calculation carries all three.

Where the model stops #

Both lines assume a perfect gas with constant specific heats, steady one-dimensional flow, and constant area. Relaxing the last of those gives the generalised one-dimensional flow with area change, friction and heat all at once — a set of coupled ordinary differential equations with no closed form, in which the same $1/(1 - M^2)$ factor appears in front of every term and produces the same singularity at Mach 1.

That factor is worth naming because it is the common thread. Area change, friction and heat all act on the flow through a coefficient that blows up at $M = 1$, and every one of the surprising signs above — friction accelerating a subsonic flow, cooling accelerating a supersonic one — is that factor changing sign as the Mach number crosses one. Choking is not three phenomena but one, and the three lines are three ways of arriving at it.

The models are also silent about what happens after choking. A choked pipe does not have a discontinuous flow rate; it has a flow rate set by the whole system upstream, and computing it means solving the pipe and its supply together.

Who found it, and when #

Fanno’s analysis dates from a 1904 thesis at the ETH in Zurich and is one of the earliest systematic treatments of compressible pipe flow. Rayleigh’s heat-addition line comes out of his work on aerodynamic sound and gas dynamics in the 1900s and 1910s, and both were assembled into the standard one-dimensional framework in the 1940s and 1950s, principally by Shapiro, whose two-volume treatment of compressible flow is where most engineers still meet them.

The observation that the two lines cross at the ends of a shock is older than either name in practice — it is implicit in Rankine and Hugoniot — but it became a teaching result with the h–s diagram, which is one of the few places where two entirely different physical processes can be seen to share an answer.

Where the ladder goes next #

The compressible field now has fifteen essays and every one of them is about a flow that can be solved, jumped across, or expanded through. What it has not touched is the case where the equations say a flow is stable and the flow disagrees — which is a turbulence question, and which turns out to be a statement about the angle between two eigenvectors.