Compact nozzle acoustic analytical solutions in compressible flow with linear equation of state

Predicting combustion instability in combustion chambers using low-order acoustic network models requires analytical solutions for nozzle acoustic responses to flow disturbances. However, most theories for acoustic and entropy transfer functions of nozzles retain the ideal gas assumption, since applying a real-gas equation of state (EoS), such as the cubic EoS, introduces significant mathematical complexity. This article aims to derive analytical expressions for wave fields generated by compact nozzles in compressible flows. The linearization of the compressibility factor Z [Sirignano, “Compressible flow at high pressure with linear equation of state,” J. Fluid Mech. 843, 244–292 (2018)] enables modification of classical relations among flow properties through several first-order coefficients (ζ1−ζ4). Building on these results and the wave equation, the relationships between wave amplitudes and parametric disturbances are established. Following the framework of Marble and Candel [“Acoustic disturbance from gas non-uniformities convected through a nozzle,” J. Sound Vib. 55(2), 225–243 (1977)], matching conditions at the inlet and outlet of the compact nozzle are derived under real-gas conditions. Numerical solutions of the linearized Euler equations are carried out to evaluate the accuracy of the analytical results for both subsonic and supersonic cases. Results for different gases are also presented, and the validity ranges of the EoS linearization are assessed in terms of temperature and pressure deviations.