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How does the Falcon 9 v1.1 form this exhaust pattern with its nine Merlin 1D engines? Is it a bigger, rougher shock diamond? I am trying to learn more about how one could form with a cluster of engines.

Falcon 9 exhaust pattern
Aft end of the Falcon 9 v1.1 a minute or two into the launch.

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    $\begingroup$ That looks like a shot of the engine plume at high altitude and showcases an instance of nozzle under-expansion due to the fact that the atmospheric pressure is relatively low at high altitudes, thus allowing the plume to "spread out" if you will. Here's a great article explaining nozzle characteristics far better than I can: en.wikipedia.org/wiki/Rocket_engine_nozzle. At high altitudes I presume that a shock diamond would still form close to the nozzle exit, but with defuse fairly quickly due to low ambient pressure at altitude. $\endgroup$ – InquisitiveInquirer Sep 10 '14 at 6:53
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The OctaWeb design of Falcon 9 v1.1 actually forms a crude approximation of an aerospike nozzle when all engines are firing. To understand why the exhaust plume looks like that in the photo, its important to understand how rocket engine bells works.

Nozzle Expansion Description

This image shows how the nozzle expands gases to produce thrust. Rocket combustion chambers combine fuel and oxidizer to produce a high pressure gas. This gas is actually moving relatively slowly in the combustion chamber (below mach 1). The expansion nozzle takes the high pressure gas and expands it to a low pressure, high velocity gas.

The four figures shows the same nozzle with the same expansion ratio in four different operating environments (different external pressures).

The first nozzle is underexpanded, where the exhaust gas pressure > ambient air pressure. This means that when the exhaust gases leave the nozzle, some of the gas expands perpendicular from the thrust vector and thus does no useful work on the rocket.

The second figure shows the ideal expansion nozzle, where gas is expanded so that exhaust pressure=atmospheric pressure and all of its kinetic energy is heading out directly away from the rocket.

The third figure shows an overexpanded nozzle where exhaust pressure < ambient pressure. The exhaust is compressed by the ambient air and loses some efficiency.

The fourth figure shows a grossly overexpanded nozzle where exhaust pressure << ambient pressure. This can lead to instability as the exhaust is separated from the walls of the nozzle.

The ideal scenario is exhaust pressure==ambient pressure, however this ideal scenario does not happen often. Rockets are used to go to space, and as they rise in altitude the ambient air pressure decreases. Therefore a nozzle that is 'perfect' at sea level will be underexpanded as the rocket flies. Most rocket manufacturers design their first stage engines to be slightly overexpanded so that they will be in the ideal range as they gain altitude.

One way to combat this however is to use a variable geometry nozzle to stay in the ideal range for as long as possible. An aerospike is a form of variable geometry nozzle and can be considered an inverted rocket nozzle. Fuel is injected along a cone (toroidal) or a wedge (linear). Ambient pressure and the walls of the spike create a virtual nozzle to expand the gases. This engine type will be more efficient because the expansion ratio changes with ambient pressure since the ambient pressure is what is expanding the gases in the first place.

Aerospike Expansion Photo

So now back to the Falcon 9 v1.1. The Merlin 1D-Sea Level is slightly overexpanded at sea level. As the Falcon rises, it reaches ideal expansion then becomes underexpanded. The 9 engines in the OctaWeb create an aerospike-like effect where the 8 engines on the outside ring push the air away from the center engine and create a region of low pressure. At takeoff, the middle Merlin is actually producing more thrust than the rest of the Merlin's due to this effect. The plume in the photo is the underexpansion of all engines combined.

Sources:
Rocket Engine Expansion
Aerospikes

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