Effects of the sheared flow velocity profile on impedance eduction in a 2D duct

TL;DR

This paper assesses how the shape of a sheared flow velocity profile influences impedance eduction in a small 2D duct. By solving the Pridmore–Brown equation with both simplified and realistic boundary-layer profiles and comparing against a uniform-flow model with the Ingard–Myers boundary condition, it shows that realistic wall-law profiles yield better agreement with impedance eduction results, while simplified profiles can introduce significant deviations, especially in attenuation and upstream/downstream distinctions. A parametric study demonstrates that the approximation error from IMBC grows with Mach number and duct width, and experimental data corroborate that velocity-profile choice matters in impedance eduction, particularly for downstream propagation. The work supports using realistic boundary-layer representations or displacement-thickness matching in 2D impedance eduction, and suggests extending the methodology to 3D ducts and higher-order modes for broader applicability.

Abstract

Impedance eduction methods are the current standard approach to measure the impedance of acoustic liner under sheared grazing flow. The dedicated facilities for these methods consists on a waveguide with rectangular cross-section, which implies a sheared grazing flow. A current debate in the literature is the effect of this sheared flow in the impedance eduction methods. We assess the impact of the flow profile shape on acoustic propagation in a two-dimensional duct within the typical operating range of impedance eduction facilities. Firstly, a numerical experiment is proposed in which the Pridmore--Brown equation is assumed to represent the true physical behaviour, and is used with both simplified flow profiles commonly used in the literature and a realistic representation of a turbulent boundary layer using a van Driest universal law of the wall model. The data from these numerical experiments are then used with a traditional impedance eduction process, and the resulting variation in obtained impedances are investigated. Secondly, we apply a less-traditional impedance eduction method that incorporates the sheared velocity profile to data obtained from real-world experiments. The results suggest that the Ingard--Myers boundary condition remains a good approximation to a realistic boundary layer profile, such as the universal law of the wall, at least in the two-dimensional case. However, it is also shown that the simplified flow profiles often used in the literature can lead to significant deviations from the results obtained using a realistic velocity distribution.

Figures (14)

• Figure 1: Schematic duct and coordinates system assumed in this work.
• Figure 2: Reference impedances for numerical experiments. (a) SDOF-like; (b) digitalization of CT57 from roncen2020WavenumberBased.
• Figure 3: Flow velocities profiles in linear (a) and logarithmic (b) scales, and velocity gradients in linear (c) and logarithmic (d) scales, considered in the first step of this work. Hyperbolic tangent and sinusoidal flow profiles are best fitted to match the bulk Mach number and boundary layer thickness of the law of the wall.
• Figure 4: Flow velocities profiles in linear (a) and logarithmic (b) scales, and velocity gradients in linear (c) and logarithmic (d) scales, considered in the first step of this work. Hyperbolic tangent and sinusoidal flow profiles are best fitted to match the bulk Mach number and boundary layer displacement thickness of the law of the wall.
• Figure 5: Wavenumbers obtained for the SDOF-like impedance with different velocities distributions for the same $M$ and $\delta_{99.9%}$.