Generalised acoustic impedance for viscous fluids

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Abstract

A general class of acoustic impedance conditions is considered where a linear dependence is assumed between the fluid velocity and the surface traction. In addition to the classical impedance relating pressure and normal velocity, this generalised definition of the impedance includes effects such as the friction along the surface as well as coupling between the normal velocity and the tangential force on the fluid. After providing a generalised definition based on impedance and admittance matrices, various fundamental principles are considered to ensure the impedance is physically admissible. These principles include the reality, causality, stability and passivity of the surface. To illustrate the influence of the additional parameters introduced by the generalised impedance, a number of results are presented for the reflection of a plane wave on a lined surface, with or without a mean flow.

Introduction

In classical acoustics, i.e. for an inviscid fluid, the surface impedance defines a linear relation between the normal fluid velocity and pressure. In the frequency domain, this boundary condition is sufficient to obtain a well-posed problem. In the time domain, the impedance function Z(ω) should satisfy a number of admissibility constraints, see Refs. [1,2]. When considering a viscous fluid, more boundary conditions are needed. In addition to the condition on the normal velocity, a no-slip condition is generally applied, stating that the tangential velocity vanishes on the surface. An alternative is to have a zero shear stress along the surface.

Standard impedance models, such as those from Rienstra [1] or Tam & Auriault [3], do not provide any information on the behaviour of the tangential component of velocity at the surface. When applied to viscous flows, and in the absence of more information, these impedance models are generally complemented with the no-slip condition [[4], [5], [6]]. Prompted by inconsistencies observed in the impedance educed from measurements with flow [7], it has been recently suggested that additional parameters should be introduced to fully characterize an acoustic treatment under a grazing flow. One such parameter is the momentum transfer impedance relating the tangential force on the surface and the normal velocity [8]. Another option is the introduction of a friction coefficient [9]. Another mechanism under consideration is the coupling with the hydrodynamic field [10].

In this paper we consider a general class of impedance boundary conditions where a linear dependence is posited between the surface traction and the fluid velocity. In Section 2, these boundary conditions are formulated both in the time domain and the frequency domain. They provide a unified framework to describe a large group of boundary conditions, including the momentum transfer impedance. In Section 3, a number of conditions on the generalised impedance are derived in order to obtain an admissible physical model, namely the reality, causality, stability and passivity of the impedance. The two-dimensional case is considered in more details and results for the reflection of a plane wave are presented.

Section snippets

Generalised acoustic impedance

We consider a compressible, viscous fluid in contact with a surface Γ. The surface traction (i.e. the surface force) induced on the surface by the fluid is given by t = −σ ·n, where σ is the Cauchy stress tensor and n is the unit normal vector pointing into the surface. For a Newtonian fluid, we will use the constitutive relation σ = −pI + τ, where p is the pressure, I is the identity tensor and τ is the tensor of viscous stresses. It follows that t = pn −τ ·n.

Admissible impedance

A number of constraints can be imposed on the impedance and admittance matrices in order to satisfy basic physical requirements. This section discusses the reality, causality, stability and passivity of the generalised surface impedance.

Two-dimensional case

To illustrate the proposed formulation, we can consider the two-dimensional case by writingtˆntˆt=ZˆnnZˆntZˆtnZˆttûnût,andtˆntˆt=pˆ−τˆnn−τˆnt,where the subscripts n and t denote the normal and tangential components, respectively.

While the term Zˆnn is related to the traditional acoustic impedance, the term Zˆtt represents the friction generated when the fluid elements slide along the surface. The momentum transfer impedance considered by Schultz et al. [8] is represented by the cross term Zˆtn

Example

To illustrate the impact of the generalised impedance condition, we consider the reflection of an acoustic plane wave off a flat surface where the generalised impedance condition is imposed, see Fig. 1.

Summary and perspectives

We have considered a generalised definition of the acoustic impedance where a linear dependence is assumed between the fluid velocity and the force on the surface. It is possible to define admissibility conditions such as the reality, causality, stability and passivity of the surface. This last condition leads to a generalisation of the requirement that the resistance should be positive. The reflection of a plane wave by a lined surface was then used as a way to illustrate the influence of the

CRediT authorship contribution statement

Gwénaël Gabard: Conceptualization, Formal analysis, Investigation, Visualization, Writing - original draft.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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