Realistic Sheared Flow Profile Effects on Acoustic Impedance Eduction in Small Three-Dimensional Ducts

We investigate the influence of realistic sheared grazing flow on acoustic propagation in three-dimensional rectangular ducts. We show that the conclusions reached in the literature about the effects of sheared grazing flow on acoustic propagation in lined ducts are dependent on the flow profiles used in those studies and that significantly different conclusions are reached once a realistic flow profile is used. We particularly focus on small ducts typical of most experimental impedance eduction facilities, for which velocity gradients are relevant in a significant fraction of the duct cross section. We assess the effect of simplifying the velocity distribution in the cross section to either a one-dimensional (two-dimensional, spanwise-infinite duct) or a uniform flow profile. Three flow profiles are considered, namely, i) the tensorized hyperbolic tangent, ii) the law of the wall, and iii) one obtained from a RANS simulation. These flow profiles are used as input in numerical simulations, based on the solution of the three-dimensional Pridmore–Brown equation, to perform in silico impedance eduction experiments. Results show that realistic flow profiles can be well approximated for acoustic wave propagation in ducts by uniform or one-dimensional flow profiles, provided that the bulk Mach number is correctly accounted for, which contrasts with previous findings considering more simplistic flow profiles. The key conclusion of this work is that if viscous effects are negligible and acoustic impedance is a good representation of a lined wall with grazing flow, then the simplification to a uniform flow is a reasonable approximation, and traditional eduction methods are sufficiently accurate.