PROPAGATION OF ACOUSTIC WAVES USING THE PARABOLIZED STABILITY EQUATIONS

The Parabolized Stability Equations (PSE) constitute a technique in which a normal-mode Ansatz is considered for the solution of the linearized Navier-Stokes Equations. The Ansatz is decomposed into a rapidly and a slowly varying part in the direction of the mean-flow. While PSE have long been used for calculating disturbances in boundary layers focusing on the determination of hydrodynamic stability, their application to ducts is very restricted. This work aims at demonstrating the capabilities of PSE to compute the propagation of acoustic waves in ducts. PSE is an interesting tool since it can provide accurate results with a low computational cost compared to standard computational methods, thanks to the use of an Ansatz from the method of multiple scales that allows parabolization of the linearized equations. The focus is given to bypass ducts of turbofans where there is propagation of disturbances generated by the fan. PSE proves its value when axial variation of properties is present. In this work, the wall impedance and the mean-flow temperature were chosen to be varied along the cylindrical duct axis. These variables are defined, respectively, by the presence of acoustic liners and cooling/heating processes on the flow. Contours and axial distributions of PSE pressure fluctuations for the cases with axial gradient of temperature were compared with analytical results and showed good agreement. For varying impedance, an analytical solution is not available; however, the contours and axial distributions of PSE pressure fluctuations were in accordance with expected qualitative trends. For all cases tested, the real part of the axial wavenumber obtained with the PSE was in close agreement with the analytical, locally-parallel result, which can be obtained using classical acoustics. Hence, PSE is able to correctly describe the pressure disturbances field in cylindrical ducts with axially-varying impedance and temperature, with a reduced computational cost.