An application of the multiple-scales theory to acoustic double transition in ducts with flow

A theoretical model based on the multiple-scales technique is proposed to study the propagation of an acoustic mode that encounters multiple turning points in a hardwalled duct with slowly varying geometry and in the presence of mean flow. The case where two of such turning points arise in the duct due to radius variations is studied for both cut-off/cut-on/cut-off and cut-on/cut-off/cut-on transitions. The acoustic field through the whole duct is determined by writing the solution for a single transition around each turning point and by matching the two solutions. These two turning points provide a mechanism by which the upstream and downstream travelling waves interact and modify the overall energy balance. For the cut-off/cut-on/cut-off case, one of the possible consequences of such an interaction is that the mode is trapped by these turning points, which can lead to acoustic resonances only controlled by the geometry. As for the cut-on/cut-off/cut-on case, this interaction makes the mode not to vanish at the first transition but to tunnel through the cut-off region. Several test cases, including at least two transitions, are studied to validate the model and understand its limitations with respect to two other propagation models (finite difference and multimodal methods). The agreement between the different methods is excellent in most cases. However, when the modal scattering becomes important, the model starts to show some deviations. Nevertheless, it allows, in a preliminary design of an engine, to provide an estimate of the frequencies where an amplification occurs and thus to avoid creating resonators that can have dire implications on the sound emission.

Automated summary

A theoretical model based on the multiple-scales technique is proposed to study the propagation of an acoustic mode that encounters multiple turning points in a hardwalled duct with slowly varying geometry and in the presence of mean flow. The model provides insights into the interaction between upstream and downstream travelling waves and its effect on the overall energy balance.