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Recently I asked this question about center engine performance and received an excellent answer from @OrganicMarble. The first sentence of the answer is

If anything, the flow field interactions hurt the performance of the center engine(s).

If the flow of exhaust from the outer engines cause a performance (specific impulse) loss in the center engine, then what is the purpose of including them in launch vehicles? I know that TWR is a big factor of engine count in a rocket but if there is engine-out capability then why can't a launch provider just remove the central engine to increase net efficiency?

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    $\begingroup$ Because a small performance loss doesn't offset the gain from having it? $\endgroup$ May 6 at 1:17
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    $\begingroup$ If a rocket would lift of with 5 engines on a square with center and not lift of with only 4 engines nobody would try 4. If you place the 5 engines on a pentagon with no center engine you get a larger diameter of the rocket and more aerodynamic drag. $\endgroup$
    – Uwe
    May 6 at 1:20
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    $\begingroup$ Engine-out capability only applies when the tank is half empty. If one engine goes out on the ground then the rocket won't be able to liftoff. $\endgroup$ May 6 at 17:53

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You would need to quantify the loss to find out. If the loss were something like 90%, then of course adding it would be foolhardy. If the loss were 1% on that one engine, then you might not be able to come up with a better arrangement. 8 engines could be too little for the mission and 9 engines in a ring might have to be spaced farther apart, requiring more hardware and offsetting any specific impulse gain from the engine.

While the interactions do theoretically hurt the performance, my uncalculated guess is that it's pretty minimal. The interactions are smallest at sea level and greater at altitude. But all the engines are already leaving specific impulse on the table at altitude with the small nozzle expansion. The actual number would be good to see, but I have no way to find it.

The final landing burn is done with the single center engine firing. Requiring more engines to burn (because there is no center engine), would make this process much more complex, if it's even possible at the light landing weights. So there are benefits to having the center, even if the performance is not 100% equivalent to the outer engines.

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