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The Rao nozzle formula is an empiric formula for a parabolic nozzle used in pretty much all nozzles today. The problem is it depends on the throat and exit-angle of the nozzle, which varies with expansion-ratio and desired length. I've found many references to a graph that is shown in Sutton & Biblarz "Rocket Propulsion Elements", for example by NASA (page 15, page 29 of pdf).

However, I couldn't find any concrete mathematical formula to calculate these angles. Are they purely empirical? If so, is the data available somewhere? (i.e. what is that paper called? It is not part of Rao's original paper, where I would have expected it.)


Update

So I played around with many ideas, none of them leading to the graph mentioned. Just for completeness, some other sources where I found this specific graphic reproduced: p. 19, p. 27 of pdf, page 3, p. 15, p. 29 of pdf Followed the references as far as I could, but even my university library couldn't find this:

Rao G.V.R, "Optimum Thrust Performance of Contoured Nozzle", Bull. of first meeting JANAF Liquid Propellant Group, Johns Hopkins University, Md., Nov.1959.

Upon careful reading, Rao himself [6] mentions a formula for the exit-angle of a thrust optimised parabolic nozzle. However, plotting it for different mach numbers against their corresponding area-ratios would result in similar graphs only for $\gamma = 2.15$ (at 100%, exit-angle only), an unusually high specific-heat-ratio. (Most calculations I found are for $\gamma=1.2$ or $\gamma=1.23$)

Generally speaking, I don't trust the graph anymore. Rao claims nozzle contour would not strongly depend on $\gamma$, but any formula I have found dealing with angle in supersonic flow (Prandtl-Meyer function and ideal exit angle most prominently) heavily depend on $\gamma$.

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    $\begingroup$ I'm pretty sure those graphs were gained by calculating the optimal full length nozzle for various mach numbers (which is equal to the throat angle, keyword: prandtl-meyer fan) using the method of characteristics (see here) and truncating them at multiple points (percentages of 15 degree cone) which equals not straightening the exhaust to 0 degree but to the exhaust angle you find in this graph. $\endgroup$ – Christoph Feb 4 at 7:15
  • $\begingroup$ If simply truncated, the throat-angle would have to remain constant. I was thinking it might have to do with the maximum turn-angle of supersonic flow. It seems the total turn angle is distributed between nozzle and throat, so the total turn angle corresponds to the Prandtl-Meyer angle of the total increase in mach number. So I'm still hoping for a mathematical expression of both angles. $\endgroup$ – GammaSQ Feb 9 at 13:46

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