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The science behind those pesky shock diamonds in jet engine exhaust

These are known as shock diamonds. They are caused by the interaction of a fluid's velocity and pressure. When a fluid moves quickly, its pressure decreases. You can conceive of it as the molecules moving so quickly that they don't have time to stop and push forth. You'll need to learn about the Navier-Stokes equations, which explain fluid dynamics, if you truly want to dive into it.

Photo by Niklas Jonasson on Unsplash

The crucial thing to understand about shock diamonds is that hot exhaust is the fluid (not liquid; all liquids are fluids, but gases are also fluids), and it's moving at a high speed. You want the exhaust to be rapid because equal-and-opposite-reaction means the engine will be moved very quickly in the opposite direction. That means the exhaust has low pressure, and when it expands out into the air, the ambient air pressure rises.

The air pressure forces the exhaust to collapse inward until it becomes too compressed and the air pressure is no longer strong enough to squeeze it. It's still hot and active, so it bounces back out until its pressure equals that of the air. At that point, the air pressure begins to squeeze back in, forcing the exhaust to collapse once more, and so on. The bright spots on the exhaust cone indicate where the exhaust is being pressed in owing to atmospheric pressure.

Shock diamonds, on the other hand, are a terrible thing. They occur when the ambient pressure is greater than the exhaust pressure, implying that it is pushing in on the exhaust as it exits the nozzle. That's bad because it means you're losing energy - the inward force of the atmosphere slows your exhaust, which slows your equal-opposite propulsion. 

To correct this, increase the size of the nozzle. Another portion of the Navier-Stokes equations states that when you restrict a fluid, its velocity increases (unless you constrict it too much and reduce the flow, which also relies on its viscosity...the equations are incredibly intricate and are still not entirely solved).

That's what the nozzle is doing in the first place: burning the fuel converts it from a dense liquid to a much less dense very hot gas, creating a tremendous amount of pressure inside the combustion chamber. The presence of an engine outlet converts pressure into high flow. The nozzle constricts the large volume of flow, causing it to have a very high velocity at a low pressure. When the velocity is too fast and the pressure is too low, the atmosphere wins the pressure fight and shock diamonds are formed.

If you open the nozzle too much, the exhaust will win. That's also negative because it means your exhaust will exit in a wide cone rather than straight back out of the engine. The equal-opposite response indicates that when the exhaust is going up, the engine wants to move down, and vice versa. 

That won't cause problems or cause the engine to behave strangely because the exhaust from the opposite side of the nozzle will balance it out. What it does is waste energy since the exhaust going out to the sides is now fighting itself to drive the engine in a different, sideways direction, rather of driving it forward.

What you want the nozzle to accomplish is properly balance the exhaust pressure against atmospheric pressure, so that the exhaust exiting the nozzle is constrained by the air and forced to go exactly backwards. In essence, you're using the atmosphere as a nozzle extension, turning the air into a gun barrel so that when the exhaust expands and cools, all that energy just keeps pushing right back onto the engine. That is the most effective way of utilizing the exhaust.

However, air pressure varies with altitude. It is significantly higher on the ground than it is in the air. Most jet fighters have nozzles that can adapt as they fly to compensate for pressure changes, becoming wider as they rise. 

However, the variable nozzle has a limited range of motion, therefore the engine cannot be set for all conceivable pressures from the start. Because the plane will spend the most of its time flying well above the surface, the nozzle is adjusted to be slightly inefficient at ground level in order to remain as efficient as possible when flying at mission altitude.

In this photograph, you can see an engine being tested on the ground. It's feasible that the engine is perfectly capable of managing the nozzle outflow sufficiently to prevent shock diamonds from forming on the ground. I don't know nearly enough about that engine to tell you about it. It's possible that they're specifically testing the range of motion of the nozzle, and the image captures it as it's in a position to create shock diamonds, but I'd wager that the engine will never not make shock diamonds on the ground because it's tuned to be more efficient in the air, and its current configuration is the best it can do at ground level.

In addition, when developing spacecraft, this battle poses a significant engineering problem. A nozzle that can reconfigure for changes in atmospheric pressure may not be able to withstand the extreme heat and pressure of a rocket engine. As the rocket gets closer to space, the air pressure drops to zero, and the optimal nozzle size approaches infinity. As a result, engineers must perform complex calculations to determine how large the nozzle should be for each stage of the rocket's ascension - smaller for the early stages, larger for later stages - and balance the size against the weight and qualities of the materials available.

Aero-spike style engines do not have this problem and are equally efficient at all altitudes and air pressures, but they have their own drawbacks (most notably that the very hot, probably caustic exhaust gasses are deliberately forced against the aero-spike and all the components holding and controlling it, putting a lot of stress on the material; AFAIK, engineers haven't quite figured out a suitable design that can withstand the heat and pressure in a way that is alright).

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