Presumably the rocket engines in the center of an engine cluster would be surrounded by lower-pressure exhaust from the outer engines so essentially the "atmospheric pressure" around the center engine would be lower, resulting in a higher specific impulse. Is this correct, and has this phenomenon been observed?
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1$\begingroup$ The pressure from exhaust of peripheral engines actually inhibits expansion of exhaust from central engines. This makes the central engines see a higher “atmospheric pressure”, analogous to an aerospike engine. This effect can be adjusted by gimballing all the peripheral engines radially. I have read references to SpaceX studying this effect. Sorry, I can supply no authoritative references $\endgroup$– WoodyMay 4, 2022 at 16:06
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$\begingroup$ @Woody So that means that the inboard engine(s) have lower performance? If so, why are center engines even there in the first place if we already know there will be a performance loss? $\endgroup$– WarpPrimeMay 4, 2022 at 17:12
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$\begingroup$ @fasterthanlight ... I’m way above my pay grade with this question, so I anticipate being corrected by better informed opinions. Since bell nozzles can be optimized for only a single ambient pressure, the opportunity to adjust the ambient pressure during ascent can potentially improve launch performance. Gimballing peripheral engines towards the centerline increases the ambient pressure for center engines. As well, it asymmetrically increases the ambient pressure for the peripheral engines. Picture the exhaust of the center engines acting like the central spike of a toroidal aerospike nozzle. $\endgroup$– WoodyMay 4, 2022 at 19:34
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tl;dr If anything, the flow field interactions hurt the performance of the center engine(s).
The calculation of the flow field for a multiple rocket engine cluster is extremely complicated and the effects on the base force (see What does the Shuttle's axial forces analysis reveal? and What is the plume effect? for info on the base force) are likely, when integrating all the forces over the vehicle, to swamp any changes in engine thrust. Here's a simplified flow field example
The negative effect on center engine thrust due to an effective increase in ambient pressure is explained in the same paper (emphasis mine)
Three-dimensional simulations are carried out for the underexpanded jets with a NPR of 7.2, where NPR is defined as the ratio of nozzle exit and ambient static pressure
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The pressure values after the external shock are higher than the ambient pressure and hence the effective NPR as experienced by the jets is smaller than the NPR of 7.2.
A reduced NPR equates to an increased ambient pressure and therefore a drop in the pressure thrust term (see From the General Thrust Equation towards Tsiolkovsky, how to explain dropping these terms along the way?) The momentum thrust would be unchanged, thus a net loss in thrust results.
answered May 5, 2022 at 2:10
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$\begingroup$ Note that while increased pressure on the exterior of the nozzles reduces effective specific impulse of the individual engines, that recirculating gas also exerts pressure on the base of the rocket. $\endgroup$ May 5, 2022 at 16:24
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$\begingroup$ @ChristopherJamesHuff yep, as mentioned in the 2nd paragraph of the answer. Sometimes it's enough to help, but for the shuttle (see linked questions) the low pressure flow field at the base was a net negative. $\endgroup$ May 5, 2022 at 16:26
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$\begingroup$ What's the magnitude of this performance loss? I must have mistakenly overestimated it in space.stackexchange.com/q/59108/30164 $\endgroup$ May 6, 2022 at 2:40
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$\begingroup$ @fasterthanlight In the graph here space.stackexchange.com/a/43845/6944 you can see how much difference the change in the pressure thrust term makes for a specific engine when the ambient pressure changes by a full 15 psi. $\endgroup$ May 6, 2022 at 3:20