Pressure variations in rocket nozzles. Part 3: Direct calculation of the local Mach number

Foremost amongst rocket nozzle relations is the area-Mach number expression linking the local velocity normalized by the speed of sound to the area ratio A _t/A, and the ratio of specific heats. Known as Stodola's equation, the attendant expression is transcendental and requires iteration or numerical root finding in extracting the solution under subsonic or supersonic nozzle operation. In this work, a novel analytical inversion of the problem is pursued to the extent of providing the local Mach number directly at any given cross-section. The inversion process is carried out using two unique approaches. In the first, Bürmann's Theorem of classical analysis is employed to undertake a functional reversion from which the subsonic solution may be retrieved. In the second, the Successive Approximation Approach is repeatedly applied to arrive at a closed-form representation of the supersonic root. Both methods give rise to unique recursive formulations that permit the selective extraction of the desired solution to an arbitrary level of accuracy. Results are verified numerically and the precision associated with the supersonic approximation is shown to improve with successive increases in the ratio of specific heats.