# Why do rocket nozzles flare?

Why do rocket nozzles open wider at the end than, let's say, get narrower?

Let me explain:

A jet engine works by having this amazing thing called a combustion chamber. The combustion chamber ignites the air to expand it, thus, causing more thrust. And most jet engines (such as turbofan engines) increase the amount of thrust by simply having the engine get narrower as you get behind. (i.e hole getting smaller).

But for rockets, the nozzle gets larger and larger. Such as below.

Wouldn't it be theoretically more efficient to have the nozzle end get slightly narrower similar to a jet engine, to gain more thrust? I'm assuming people have already thought of this idea, because it seems like such a basic concept. Yet, I can't find/think of an answer why...

• en.m.wikipedia.org/wiki/De_Laval_nozzle – Russell Borogove Nov 2 '16 at 3:47
• As I understand it, without the expanding portion of the nozzle, the exhaust would expand in all directions unconstrained as it exited the throat. The energy of the portion of the flow that was moving sideways instead of downward would be wasted rather than contributing to thrust. The expanding nozzle directs all the accelerating exhaust gas in the same direction, maximizing thrust yield. – Russell Borogove Nov 2 '16 at 5:12
• @RussellBorogove Makes sense.. How do you know so much about space? – Frank Nov 2 '16 at 5:24
• For the last four years I've been playing Kerbal Space Program, aggressively Googling everything I get curious about, and answering questions right at the edge of my understanding on this site. – Russell Borogove Nov 2 '16 at 5:31
• @Frank What RussellBorogove is talking about are divergence losses, which are important in designing efficient nozzles, but not the reason for the convergent-divergent shape. – Rikki-Tikki-Tavi Nov 2 '16 at 12:56

The purpose of this nozzle is to achieve maximum acceleration of the flow to obtain the highest possible exit velocity.

The shape of convergent / divergent (de Laval) nozzles is dictated by the thermodynamic properties of gases.

For a subsonic gas flow, a converging passage accelerates the flow. The physics are opposite for supersonic flows: they are accelerated by a diverging passage. So the relatively short converging portion of the nozzle you see is where the flow accelerates to supersonic speed in the throat, then the long diverging portion accelerates the supersonic flow.

Where you cut the nozzle off depends on what you want the nozzle exit plane pressure to be.

Reference: The Dynamics and Thermodynamics of Compressible Fluid Flow, Ascher Shapiro, Volume I (1953). See Converging-Diverging Nozzles p.93 and especially the discussion of figure 4-12 Operation of converging-diverging nozzle at various back pressures

• Would this be a different question, but how do you ensure a shock isn't created inside the nozzle? As a rocket rises, and the backpressure is reduced, wouldn't that create a series of shock waves that approach the exit plane? What prevents them from entering and degrading the nozzle performance? – Mark Nov 2 '16 at 22:48
• You are correct that if the nozzle exit plane pressure is not matched to the ambient pressure there will be disturbances at the exit. If the flow is underexpanded (exit plane pressure too high) there will be a pattern of shock waves around the nozzle exit as the flow expands to match the ambient pressure. If the flow is overexpanded (exit plane pressure too low) it can detach from the nozzle walls or in a less severe case, contract when it leaves the nozzle. But disturbances can't propagate upstream in supersonic flow so the effects are limited to the interface of the jet flow and ambient. – Organic Marble Nov 2 '16 at 22:59
• I did not know they couldn't propagate upstream. That's very interesting. Thanks! – Mark Nov 2 '16 at 23:37
• @Mark The speed of sound is the average velocity of the air particles. In supersonic flow, the expected velocity is greater than the distribution. Therefore none of the air particles will move upstream. – Aron Nov 3 '16 at 4:56
• @supercat It seems you are very mistaken. High speed fluid flow LOWERS pressure, not increases it. 1st thing you learn in aerodynamics. – Aron Nov 3 '16 at 16:14

The gas at the narrowest part (the throat) of a convergent-divergent nozzle used in a rocket engine is ideally moving at the Mach 1, the speed of sound. This creates a choked flow condition. After the throat, the gas expands, the temperature drops, and because of the Venturi effect, it speeds to beyond Mach 1. A convergent-divergent nozzle thusly converts some of the thermal energy in the exhaust into kinetic energy.

There are limits to the extent to which a rocket can convert that thermal energy into kinetic energy. A rocket operating in vacuum could, in theory, convert almost all of that thermal energy into kinetic energy. The exhaust would leave the nozzle at close to zero Kelvin, and with the exhaust being almost perfectly columnated. This would however require an infinitely long nozzle. At some point, adding to the nozzle becomes a net detriment rather than a benefit. A rocket operating in the atmosphere ideally has the exit pressure equal to ambient. Increasing the nozzle beyond this ideal results in back pressure by the atmosphere against the rocket, while decreasing the nozzle below this ideal results in decreased exit velocity.

What about aircraft? Subsonic aircraft don't use a convergent-divergent nozzle because the flow isn't choked. There's a performance penalty to be paid for flows that exceed the speed of sound. That is a price that must be paid to escape the Earth's atmosphere and go into orbit. Some hypersonic aircraft do use a convergent-divergent nozzle, but it's often hidden inside the engine. The nozzle is a conical shape that is widest at the throat and tapers toward the rear of the engine. Instead of flowing through the nozzle, the exhaust flows around the nozzle in a jet engine.

• Hypersonic aircraft? – Organic Marble Nov 2 '16 at 18:41
• Yeah, don't any supersonic aircraft with top speeds of less than Mach 5 have nozzles like you describe? – Peter Cordes Nov 3 '16 at 12:44
• @OrganicMarble en.wikipedia.org/wiki/NASA_X-43 – SnakeDoc Nov 3 '16 at 19:37
• @SnakeDoc Note that the X-43 does not use a de Laval nozzle in its engine. It has supersonic flow throughout the engine. – Organic Marble Nov 3 '16 at 19:43
• @PeterCordes I believe the variable geometry nozzles on most afterburning jets give some level of con-di at fully open, hence fighter jet exhaust exists supersonic with shock cones visible when they have the burners lit – Talisker Apr 9 '20 at 12:59

The combustion of propellants is an exothermic process, it mainly provides heat. Initial velocity (think of the turbo pumps) and the changed specific gas constant of the combustion product are negligible. Heat also translates into pressure via the gas law.

Heat and pressure are somewhat useless once the exhaust gas does not interact with our rocket any more. They are lost when the gas mixes with the atmosphere or simply expands in empty space. The fact that these processes are irreversible is expressed as an increase in entropy.

Only momentum of the gas provides thrust and that is proportional to velocity. The task of combustion chamber, choke and nozzle is to convert pressure into velocity efficiently in an adiabatic expansion.

For good thermodynamic efficiency, the gas flow should be isentropic (but the stagnation point shifts depending on ambient pressure and speed, so it's always a compromise for ascent stages). This design goal dictates the form of the exhaust duct. For a subsonic flow, a narrowing duct will generate backpressure, effectively reducing the pressure at the choke and increasing velocity (Benoulli's principle).

Backpressure does however not work for supersonic flows. The pressure will simply not propagate backwards, instead shockwaves would build up. The optimum profile of a duct does therefore have its narrowest point where the flow reaches Mach 1 and does widen after that choke. A formal derivation of this fact from the state equation for a compressible fluid flow can be found here. Physics.SE also has an answer on the theoretical background.

Aircraft capable of supersonic cruise also have convergent divergent nozzles, because they need to generate a supersonic exhaust flow to maintain their speed. Note that supersonic nozzles can be used for subsonic movement (as is the case for a rocket at liftoff) but not the other way around.

You're missing one thing: the combustion chamber (hidden behind the piping at the top of the photo).

Here's a cross-section of a rocket engine:

F = qVe +(Pe-Pa)Ae

F = thrust force
q = the amount of mass going out
Ve = exhaust velocity
Pe = pressure at the nozzle end
Pa = ambient pressure
Ae = area of the nozzle end

Edit to remove an incorrect statement: Borrowing from OrganicMarble's answer: the flow is accelerated by a diverging passage, so you want a large nozzle diameter. The nozzle diameter is limited by ambient pressure, though, roughly because you need the pressure in the nozzle to be higher than ambient. It's a bit more complicated than that, apparently Pe can be lower than ambient

Pc is large, so if you want to make Pe equal to Pa, you need a nozzle opening much larger than the throat (the spot where the combustion chamber meets the nozzle).

• You are linking to an excellent source for rocket propulsion. It does however not explain why jet engines are different. – Andreas Nov 2 '16 at 13:00
• We see the exhaust pushing back the nozzle on your drawing! The long arrows. – curiousguy Jul 21 '19 at 22:23

The role of the combustion chamber is to burn as much of the fuel+oxidizer as possible; never discard any unburnt fuel or oxidizer because it had to be carried there at huge expense of fuel and oxidizer earlier - every gram counts.

Jets carry only fuel, they have air available in abundance, so as long as all fuel is burnt, surplus of air not having reacted with the fuel doesn't hurt - and actually helps; heated it expands and provides thrust, without need for huge exhaust velocity which would be hard on the turbines; more gas expanding by less, instead of a small amount of gas expanding by a huge factor - in rockets a tiny amount of mass provides a lot of thrust. In jets, the amount of mass carried by the plane is even smaller, but the mass providing thrust - intake air - is much larger, the airplane over a single flight pushing many times its own weight in air through the engines.

And then there's aerodynamics. Refer to this question.

The two engines on the left have nozzles for atmospheric use. By the time the exhaust gas reaches the opening, its pressure isn't much higher than atmospheric, and it can't provide much more thrust.

And this is the bell nozzle attached to the third of the above engines - meant for vacuum.

Every last bit of momentum is squeezed from the exhaust gas, which would otherwise escape uselessly sideways.

Installing such a thing on an airplane would be completely counter-productive because the huge nozzle itself would introduce so much air drag (through its outside in the air stream) it would completely nullify all the benefits.

Although, your question does have a significant merit. Nothing beats the bell nozzle in void; it's the most efficient way to harvest momentum out of gas expanding into void. But bell nozzles for atmospheric engines are a crutch, an unoptimal reduction of the void bell nozzle problem to border conditions of the atmospheric pressure. They work, they work well, but they don't work optimally.

The counterpart of the nozzle of an airplane jet engine in rocketry is the Aerospike engine.

The aerospike is definitely superior to bell nozzle in atmospheric conditions. The problem though, is that large-scale implementation of aerospike engines would require a lot of new research, while bell nozzles are 'tried and true', tested, well known and readily available. And so, because nobody wants to pay for "being the first", we're still stuck with bell nozzles for atmospheric rocketry.

• Aerospikes are heavy and hard to cool. If we are going to see any advanced nozzles in the near future, I think it will be dual bell nozzles. – Rikki-Tikki-Tavi Nov 3 '16 at 12:42
• @Rikki-Tikki-Tavi: Engineering problems. The shape is optimal, but the secondary practical problems crop up to a level that makes them lose to bell nozzles. Though I'm pretty sure with advances in material engineering we may see actually practical aerospikes. I could imagine one with the spike made from a high-temperature ablator; extending to retain the shape as the ablating surface recedes. Or just made from a material that can withstand the few minutes of atmospheric flight. – SF. Nov 3 '16 at 15:11
• "optimal" is a very complicated term. Few things in engineering are ever optimal in every way. It always comes down to a trade-off. The fact that you can imagine a solution doesn't mean that one will readily present itself. – Rikki-Tikki-Tavi Nov 4 '16 at 10:06

As explained there, the exhausted mass is constant but its parameters aren't. At the entry of the nozzle, the gas is hot and has a very high pressure. At the output of the nozzle, the pressure is a lot smaller (as it's the one from the atmosphere or space).

The nozzle is used to expand the exhaust gas from that high pressure to the very small external pressure.

Pressure and temperature are related to velocity. Hence, reducing the pressure allows to gain in exhaust velocity and thus in thrust.

You will also notice that the nozzle is bigger on second stages than on first stages because of the difference of pressure of the environment (atmosphere versus space).

The equation that you really want to look at is called the Area-Mach Relation. It’s an equation derived from 1D isentropic flow assumptions with varying cross-sectional area. Without going through the entire derivation, we can skip to the end result and interpret its implications.

Area-Mach Relation:

$$\frac{dA}{A} = (M^2-1)\frac{du}{u}$$

This one equation tells you everything you want to know about how changes in area (dA) affect changes in velocity (du) at various mach numbers (M).

For subsonic flow M<1, then $$M^2-1<0$$. In other words, the coefficient on the right hand side is negative. This means that dA and du have opposite behavior. If we make the cross section of the nozzle smaller (dA negative), the change in flow velocity (du) has to be positive. This is the effect we commonly observe when we put a thumb on the end of a garden hose: a smaller opening results in faster flow. Conversely, it also means that flow velocity decreases when the change in area increases (e.g. taking your thumb off of a garden hose).

The exact opposite is true when we have supersonic flow. If M>1, then the coefficient $$M^2-1$$ is positive, which means dA and du have similar behavior. Thus, if we have a positive change in area (i.e. we increase the nozzle area), we also increase the speed of the flow (du is positive). This is not a phenomenon that we commonly experience in day to day life, but a rather surprising result derived from the physics and mathematics.

Another surprising result from this equation: at the point where we reach the speed of sound (M=1), we need the change in area to be zero. That means that the slope of the tangent of the nozzle at the sonic point must be horizontal. Now combine this with the other two cases in a nozzle:

When the pressure builds up inside a rocket combustion chamber, it starts out at a slow, subsonic speed. To make it go faster, initially, we have to contract the cross-sectional area of the nozzle.

That is until we reach the speed of sound, Mach 1. At this point, the nozzle must reach its most contracted area.

After this point, the equation tells us that we have to start increasing the nozzle cross-sectional area to continue driving the flow to faster, supersonic speed.

The end result of this is a nozzle that contracts initially up to the sonic point, then expands afterwards. This is the classic De Laval nozzle design.

## Because it's supersonic.

Jets of gas that reach supersonic speed (which includes, in many cases, spraying air from an air compressor in a blow gun) have a bit of interesting behavior.

When the gas starts out at high pressure and low speed, making the channel it moves in narrower will make it faster. (the area where it's getting narrower is covered up by tubes and pipes in that engine picture, but it is there.)

At the narrowest point, the gas flow reaches the speed of sound. (Assuming the flow rate and/or the ratio of ambient to supply pressure are high enough)

If you want to keep it speeding up, you need to make the channel wider. This converts pressure (and heat) into velocity -- the gas cools, expands, and accelerates. The engine nozzle doesn't have to be bell-shaped -- in the early days they were just cones, usually with around 30 degrees included angle -- but that's a pretty efficient shape to make them.

If you are in an atmosphere, then eventually the pressure of the gas will get closer to the atmospheric pressure, and the rocket nozzle had better end at or before that point to avoid ruining its efficiency (too short is better than too long). In a vacuum, it can theoretically go on forever -- this is why vacuum-only rocket engines have such large nozzle extensions.

These same rules apply to jet engines as long as the jet flow is supersonic, which is definitely true for engines that propel planes at supersonic speed -- if you look closely at the nozzles of a fighter jet engine running in afterburner you can see that it's expanding. When the fluid flowing is either not a gas or is not going supersonic and therefore not acting as a compressible gas, then your intuitive understanding of nozzles is accurate. However, I note that just because a jet engine nozzle looks like it's all converging does not mean it actually is.

As I know, the spacecrafts nozzles are convergent divergent, I explain, it starts with a big chamber and it be getting smaller until a point that the preassure converts from subsonic to supersonic, that´s the point the nozzle gets again bigger propelling the rocket at supersonic speed.