Pressure variations in rocket nozzles. Part 1: Direct asymptotic predictions

We consider the one-dimensional flow equations that relate the expansion area ratio to the unique back pressures that define the operating modes of a nozzle. To eliminate guesswork and numerical root solving in deducing these unique threshold values, we apply asymptotic tools to invert their corresponding thermodynamic relations analytically. Our perturbation approach is based on the square of the reciprocal of the nozzle area expansion ratio, which does not exceed 0.3 in most applications. By extending our series approximation to higher orders, we develop a recursive expression that permits the efficient calculation of the pressure ratios to arbitrary levels of precision. In most cases, a three-term approximation entails an error of less than 1% for a nozzle expansion ratio up to 0.56. Furthermore, the error in these approximations slightly decreases as γ is decreased. All solutions are numerically verified and compared to tabulated values.