Friction moves the sonic point past the throat

The throat that stops listening found that a nozzle chokes when its throat goes sonic: from then on no signal from downstream can reach the flow upstream of the throat, and lowering the back pressure further changes nothing. The throat was the natural place for that to happen, because it is the one place in a frictionless nozzle where the area stops changing.

Two ways to choke found a pipe that chokes with no throat at all. Friction on its own drives a subsonic flow towards Mach one and a supersonic one down to it, and a pipe long enough brings the flow to sonic at its exit, because that is where the entropy it is climbing reaches its maximum.

A real nozzle has both a throat and a wall. The widening of its divergent section is trying to accelerate a sonic flow away from Mach one, and the friction on its wall is trying to drag the flow back towards it. The question this essay settles is where, in a nozzle that has both, the flow actually goes sonic — and the answer is not the throat.

The equation with both in it #

Take the quasi-one-dimensional equations — mass, momentum with a wall shear, energy with adiabatic walls — and eliminate everything but the Mach number. What is left is

$$\frac{1}{M^2}\frac{dM^2}{dx} = \frac{2\left(1 + \frac{\gamma-1}{2}M^2\right)}{1 - M^2}\left[-\frac{1}{A}\frac{dA}{dx} + \frac{\gamma M^2}{2}\cdot\frac{4f}{D}\right],$$

with $f$ the Fanning friction factor and $D$ the local diameter. The bracket has two terms. The first is the area change, with the sign that makes a narrowing duct accelerate a subsonic flow and decelerate a supersonic one. The second is the friction, and it always has the sign of a narrowing: friction acts on a flow like a duct that is squeezing it, which is why a constant-area pipe with friction chokes.

The factor in front of the bracket is the whole difficulty. At $M = 1$ it is infinite. A flow can reach Mach one and pass through it only at a place where the bracket is zero at the same time, so that the right-hand side is zero over zero and has a finite limit.

Why friction acts like a narrowing #

The sign of the friction term is worth understanding rather than accepting, because it is what the whole essay turns on.

Friction removes momentum from the flow at the wall, and the mass flow cannot change, so the gas must find the momentum balance another way. In a subsonic flow the pressure drops to push the gas on against the drag, and a falling pressure in a subsonic flow means a rising speed — the same response a narrowing duct produces. In a supersonic flow the same loss of momentum makes the gas pile up: its pressure and density rise, and its speed falls — again what a narrowing does to a supersonic stream, which the duct that works backwards showed decelerates it.

So friction behaves, in both regimes, like an extra narrowing of the duct, growing in proportion to the square of the Mach number. A convergent section and friction push the same way; a divergent section and friction push opposite ways. In a nozzle’s divergent section, just past the throat, the widening is gentle and the friction wins, which is why the flow there is still being driven towards Mach one rather than away from it. Further down, the widening grows and takes over.

Where the bracket can vanish #

Without friction the bracket is $-(1/A)\,dA/dx$, and it vanishes where the area stops changing: at the throat. That is the whole classical result, seen from this side.

With friction the bracket vanishes at Mach one where

$$\frac{1}{A}\frac{dA}{dx} = \frac{\gamma}{2}\cdot\frac{4f}{D},$$

and the right-hand side is positive. So the area must already be growing, and the place is downstream of the throat, in the divergent section, at the point where the section’s fractional widening has grown large enough to balance the friction.

Upstream of the crossing, friction outweighs the widening and the bracket at Mach one would be positive — the flow cannot yet go sonic there, because the duct is still behaving, in effect, like a narrowing one. Downstream, the widening wins. The crossing is the only place the flow can change from subsonic to supersonic smoothly, and it moves further down the section as the friction grows.

A saddle, and why the solution is built from it #

The point where both numerator and denominator vanish is not an ordinary point of the equation. In the plane of position and Mach number it is a saddle: two special solutions cross there, one accelerating through Mach one and one decelerating, and every other solution curves away from it.

That is what makes the problem awkward to solve the obvious way. Starting from the inlet with a guess at its Mach number and integrating forwards, a guess slightly too low produces a flow that accelerates towards Mach one, falls short, and decelerates again through the divergent section — a subsonic venturi passing less gas, which is a real flow but not a choked one. A guess slightly too high reaches Mach one before the sonic point, where the bracket is not yet zero, and the equation’s slope becomes infinite. The one correct inlet Mach number sits between, at a knife-edge that no finite-precision shot from the inlet lands on.

So the solution is built from the inside. The sonic point is found from the condition above. The Mach number’s slope through it follows from applying L’Hôpital’s rule to zero over zero, which gives a quadratic whose positive root is the accelerating solution. From a small step either side of the saddle the equation is then integrated outwards, upstream to the inlet and downstream to the exit, and both halves are well behaved because they are moving away from the singular point rather than towards it.

The result is checked by a quantity that uses none of the differential equation. At fixed stagnation temperature the mass flow is proportional to the stagnation pressure times the area times a function of the Mach number, and it must be the same at every station. Along each solution it is, to seven parts in a thousand million.

The saddle’s other solution #

The quadratic that gives the Mach number’s slope through the sonic point has two roots, and only one was used. The positive root is a flow that enters subsonic and leaves supersonic. The negative root is a flow that enters supersonic and decelerates smoothly through Mach one to leave subsonic, passing the same point with the opposite slope.

That second solution is a perfect supersonic diffuser: a duct that brings a supersonic stream down to subsonic speed with no shock and no loss beyond the friction. It is a genuine solution of these equations, and it is almost never seen, because it is unstable in the same way the subsonic-to-supersonic solution is not. A small disturbance to a flow decelerating through Mach one sends the sonic point upstream, where the geometry cannot hold it, and the smooth deceleration collapses into a shock standing somewhere in the duct — the arrangement one area, two answers found for a nozzle whose back pressure is too high.

The saddle therefore picks out two flows and the world keeps one of them. The one it keeps is the one built here.

A throat that is not sonic #

For this nozzle the divergent section’s widening rate grows in proportion to the distance from the throat, so the sonic point moves down the section almost in proportion to the friction length. At a friction length of 1 it is nearly a seventh of the way to the exit, and the geometric throat — the narrowest place — carries gas at Mach 0.85.

That sentence contradicts the way choking is usually taught, and it is worth being exact about which part. The nozzle is still choked: there is a place where the flow is sonic, no signal from downstream can pass it, and the mass flow is fixed. What has changed is where that place is. The throat does the job in a frictionless nozzle because it is the only point where the bracket vanishes; with friction the job passes to the point where it vanishes now, and the throat is just the narrowest part of a subsonic flow.

Choked at a lower flow #

A choked nozzle passes a mass flow set by the stagnation pressure and area at its sonic point. Moving the sonic point downstream changes both, in opposite directions. The area there is larger than the throat’s, which on its own would pass more gas. But the gas arriving there has lost stagnation pressure to friction all the way from the inlet, which passes less.

The second wins, and the figure shows why it wins by so much. The throat is flat at its narrowest point, so moving a small distance downstream of it changes the area only to second order in the distance — 0.11 per cent for a friction length of 0.2 — while the stagnation pressure falls in proportion to the friction the gas has passed, two per cent at the same friction length. A nozzle with friction chokes at a lower flow than its throat allows, and the correction is set by the friction, not by the geometry.

The stagnation pressure’s loss is the entropy’s rise, which is the reading a duct that cannot be run backwards gave to friction in a pipe: the flow’s state records how much wall it has passed, and a nozzle’s sonic point is a place along that record, not a fixed feature of the shape.

What a micro-nozzle pays #

The friction length $4fL/D_t$ is small for a large nozzle. A rocket engine’s divergent section is a few throat diameters long, its Reynolds number is enormous, and its friction factor is a thousandth or so: a friction length of a few hundredths, a sonic point within a fraction of a per cent of the throat, and a thrust loss that the boundary layer’s displacement matters more to.

It is not small for a nozzle whose throat is a fraction of a millimetre across. A cold-gas micro-thruster etched in silicon has a throat Reynolds number of hundreds or a few thousand, laminar walls with friction factors ten times a rocket’s, and a divergent section many throat diameters long relative to its size. Its friction length can be of order one, and at a friction length of one the calculation gives an exit Mach number of 1.75 where the area ratio promises 2.44 and a thrust 16 per cent below the frictionless value. The loss is not a refinement for such a device; it is most of the difference between the design and the test.

A friction length for a real thruster #

The friction length is the one number the calculation needs, and it is worth estimating for a small device rather than leaving it as a parameter.

Take a cold-gas thruster with a throat 0.3 millimetres across, running on nitrogen from a chamber at half a bar and room temperature. At the throat the gas is sonic, at about 320 metres a second, with a density of about 0.36 kilograms per cubic metre, so the Reynolds number on the throat diameter is about 2,000 — laminar. The Fanning friction factor of fully developed laminar flow is 16 over the Reynolds number, 0.008 here, and the developing layer in a real nozzle has more drag than that, not less. A divergent section ten throat diameters long then has a friction length of about a third, and one thirty diameters long, about one.

At a friction length of a third the calculation here puts the throat at about Mach 0.94 and the thrust about six per cent below the frictionless value; at one, sixteen per cent. Raising the chamber pressure tenfold raises the Reynolds number tenfold and, while the flow stays laminar, divides the friction factor by ten — which is why small thrusters that can run at high pressure do, and why the ones that cannot are measured against the frictionless number with a gap nobody expects.

What a test stand reads #

A nozzle on a test stand is characterised by two numbers, and the calculation predicts both. The first is the discharge coefficient: the mass flow actually measured over the mass flow the throat area and the chamber conditions would pass without loss. For a nozzle with wall friction and nothing else, that is exactly the mass-flow ratio computed here — 0.98 at a friction length of a fifth, 0.91 at one. The second is the thrust coefficient’s ratio to its ideal value, and it falls faster than the discharge coefficient, because the gas that does get through leaves more slowly.

The difference between the two falls is the part worth noticing. A discharge coefficient below one is often read as a throat that is effectively smaller than it looks — a boundary layer’s displacement, a vena contracta at the inlet. Friction produces it with the throat’s area untouched, by lowering the stagnation pressure the gas arrives at its sonic point with, and a thrust coefficient that has fallen by twice as much is the evidence that the loss was pressure rather than area.

The same point, with heat added #

Friction is not the only thing that moves the sonic point. The full influence-coefficient form of the equation has a third term in the bracket, for heat added through the wall, and it has the same sign as friction’s: heating a subsonic flow drives it towards Mach one, as the Rayleigh line in two ways to choke showed. A nozzle whose walls heat the gas therefore also goes sonic downstream of its throat, and one whose walls cool it can go sonic upstream, in the convergent section, where the cooling term can balance a narrowing. The throat is the sonic point only in the one case where nothing but the area is changing the flow.

What the one-dimensional friction model leaves out #

One friction factor. The friction factor is held constant along the nozzle. In reality it depends on the local Reynolds number, which changes as the gas accelerates and its density falls, and on whether the wall layer is laminar or has become turbulent.

The boundary layer’s thickness. A one-dimensional friction term charges the wall’s drag to the whole cross-section. A real wall layer also displaces the core flow inwards, which acts like a smaller area — and in a small nozzle, where the layer can fill a large share of the throat, that displacement is as important as the drag.

One dimension. Near the throat the sonic line is curved and the flow is not uniform across the section, so “the sonic point” is really a sonic surface whose position depends on radius. The one-dimensional result is its average.

Adiabatic walls. A micro-nozzle’s walls are good conductors and are rarely at the gas’s recovery temperature, so heat crosses them — which, as above, moves the sonic point in its own right.

Influence coefficients #

The equation with area change, friction, heat and mass addition in one bracket is Shapiro’s, from his treatment of generalised one-dimensional flow in the early 1950s, where each effect enters through an influence coefficient with its own sign. The observation that a singular point is a saddle through which only one solution passes smoothly belongs to the same treatment. What it takes a calculation to show is how far the point moves, and that the mass flow it sets is governed by the pressure the gas has lost rather than by the area it has gained.

Still open: where an overexpanded nozzle lets go #

Every nozzle here runs full: the flow leaves the exit plane at the pressure its expansion produced, whatever the ambient pressure outside. One area, two answers placed a shock inside a nozzle whose back pressure is too high for a fully supersonic flow, and treated it as a clean discontinuity in an inviscid duct. In a real nozzle the shock meets the wall’s boundary layer, the pressure rise it imposes is more than the layer can climb, and the flow separates from the wall and leaves as a jet narrower than the nozzle.

Where that separation happens, and why a nozzle can switch between two separated patterns with a jump in side load, is the question that joins the jump the equations allow to the layer that friction has been thickening all the way down the section. Beside it is the loss that is pure geometry, the jet a cone sprays sideways, which the sonic point’s position does not touch.