Modeling Mach number and temperature distributions in supersonic nozzle flow

Stodola's area-Mach number relation is one of the most widely used expressions in compressible flow analysis. From academe to aero-propulsion, it has found utility in the design and performance characterization of numerous propulsion systems; these include rockets, gas turbines, micro-combustors, and micro-thrusters. In this study, we derive a closed form approximation for the inverted and more commonly used solution relating performance directly to the nozzle area ratio. The inverted expression provides a computationally efficient alternative to solutions based on traditional lookup tables and root finding. Here both subsonic and supersonic Mach numbers are obtained explicitly as a function of the area ratio and the ratio of specific heats. The corresponding recursive formulations enable us to specify the desired solution to any level of precision. In closing, a dual verification is achieved using a CFD simulation of a typical nozzle and through Bosley's formal approach intended to confirm the truncation error entailed in our approximations. In this process, both asymptotic and numerical solutions are compared for the Mach number and temperature distributions throughout the nozzle.