A unified approach coupling linearized Navier-Stokes equations and Helmholtz equations to predict sound propagation with viscothermal losses in acoustic liners

Preprint, 2023

This paper introduces a unified approach for coupling the classical linearized Navier-Stokes equations (LNSE), the original Helmholtz equation and a modified form of the Helmholtz equation for porous material modelling in frequency domain. This approach is termed the unified LNSE (ULNSE). It accounts for viscothermal losses of sound propagation in acoustic liners with small-scale perforation. This method also provides a possibility to switch from LNSE to the classical and modified Helmholtz equations depending on local acoustic properties. The ULNSE is validated and applied to simulation of conventional and hybrid liners in three dimensions (3D), and also compared to a semi-empirical model and experiments. The hybrid acoustic liner consists of a perforated plate, a porous foam layer, and a rectangular cavity. Unlike the hybrid liner, the porous foam is not mounted in the conventional liners. In the perforated plate where the viscothermal effect plays a role, sound waves are solved using the LNSE. The modified Helmholtz equation is formulated with a complex wavenumber to model the porous foam as an equivalent fluid model. In the liner cavity and external duct, the original Helmholtz equation is utilized. The ULNSE is cheaper than the classical LNSE since the Helmholtz equations, which consume less computational resources, are used locally where the viscothermal losses are negligible. Meanwhile, the ULNSE can maintain high numerical accuracy. The liners are analyzed in terms of critical parameters such as the porous foam material, and perforated plate thickness and porosity. The porous foam proves to be effective in sound absorption.