Numerical and analytical modelling of entropy noise in a supersonic nozzle with a shock

Analytical and numerical assessments of the indirect noise generated through a nozzle are presented. The configuration corresponds to an experiment achieved at DLR by Bake et al. [The entropy wave generator (EWG): a reference case on entropy noise, Journal of Sound and Vibration 326 (2009) 574–598] where an entropy wave is generated upstream of a nozzle by an electrical heating device. Both 3-D and 2-D axisymmetric simulations are performed to demonstrate that the experiment is mostly driven by linear acoustic phenomena, including pressure wave reflection at the outlet and entropy-to-acoustic conversion in the accelerated regions. Moreover, the spatial inhomogeneity of the upstream entropy fluctuation has no visible effect for the investigated frequency range (0–100 Hz). Similar results are obtained with a purely analytical method based on the compact nozzle approximation of Marble and Candel [Acoustic disturbances from gas nonuniformities convected through a nozzle, Journal of Sound and Vibration 55 (1977) 225–243] demonstrating that the DLR results can be reproduced simply on the basis of a low-frequency compact-elements approximation. Like in the present simulations, the analytical method shows that the acoustic impedance downstream of the nozzle must be accounted for to properly recover the experimental pressure signal. The analytical method can also be used to optimize the experimental parameters and avoid the interaction between transmitted and reflected waves.