Rao Nozzle Contour Angles

A Rao-style bell nozzle replaces a longer conical nozzle with a curved contour. The contour begins at the end of the throat blend with wall angle \(\theta_n\) and reaches the nozzle exit with wall angle \(\theta_e\). Both angles depend on the nozzle expansion ratio and the selected nozzle length.

The expansion ratio is

\[ \varepsilon = \frac{A_e}{A_t}, \]

where \(A_e\) is the nozzle-exit area and \(A_t\) is the throat area. The percentage shown beside each curve is the bell-nozzle length expressed as a fraction of the length of a 15-degree conical nozzle with the same expansion ratio. Shorter nozzles save length and mass, but require different initial and exit angles to form the bell contour.

Pyskyfire uses \(\theta_n\) and \(\theta_e\) as the endpoint slopes of a parabolic bell. For values between the plotted curves, it first interpolates each angle in expansion ratio and then interpolates between the two surrounding nozzle length fractions.

How the chart is generated

tools/generate_engineering_charts.py creates the chart with pyskyfire.viz.PlotThetaVsEpsilon. The plotting class reads the digitised \(\theta_n\) and \(\theta_e\) curves in src/pyskyfire/regen/data/theta_n.json and src/pyskyfire/regen/data/theta_e.json, then plots the data for nozzle length fractions from 60% to 100%. These are the same data tables used by Pyskyfire’s Rao contour generator.

Source

The angle curves were digitised from page 15 of Liquid Rocket Engine Nozzles, NASA Space Vehicle Design Criteria (Chemical Propulsion), NASA SP-8120, July 1976. The original report is available from the NASA Technical Reports Server.