Time-domain simulation of the acoustic nonlinear response of acoustic liners at high sound pressure level

Abstract

In the aeronautical field, numerical modeling of the acoustic response of liners at high sound pressure levels is done using impedance boundary conditions. The numerical studies are restricted to classical locally reacting liners with a known analytical expression of the impedance. In order to model numerically the acoustic response of a wider range of absorbent materials for which no impedance expression is known, this paper focuses on the numerical modeling of the acoustic response of perforated plate liners at high sound pressure levels in the time domain. To do so, a porous-based description of the perforated plate is used to represent the visco-thermal processes occurring inside the perforated plate. This is achieved through the use of the equivalent fluid model (EFM), which contains two irrational transfer functions described herein by a generic model that covers the Johnson–Champoux–Allard–Pride–Lafarge (JCAPL), the JCAL and JCA models. Nonlinear phenomena occurring at high sound pressure levels are taken into account by using Forchheimer’s correction in the time-domain EFM, which introduces a quadratic nonlinearity in the equations. The formulation of the nonlinear EFM equations in the time domain leads to an augmented system for which a proof of stability is given thanks to a Lyapunov functional. An approximate model is built for numerical simulations from the nonlinear EFM using a multipole approximation of the transfer functions. Stability conditions sufficient for the nonlinear multipole-based approximate EFM are provided. A numerical scheme using a discontinuous Galerkin method is developed to validate the model against experiments with perforated plate liners.

Appendices

Appendix A Energy analysis

The energy balance of the EFM with the generic model defined in [40] is briefly introduced in this appendix. First, the classical mechanical energy is defined and composed of the kinetic energy and the potential energy:

$$\begin{aligned} E_{\text {m}}(t) := \dfrac{\rho _0\alpha _\infty }{2}\displaystyle \int _{\Omega }\left\Vert {\textbf{u}}\right\Vert ^2\text {d}{\textbf{x}} ~+~ \dfrac{\chi _0}{2} \displaystyle \int _{\Omega }p^2\text {d}{\textbf{x}}\,. \end{aligned}$$

(A1)

Then, an energy is defined for the diffusive variables:

$$\begin{aligned} \begin{aligned} E_{\text {diff}}(t):=&~ \rho _0\,\alpha _\infty \,N\,E_{\varvec{\varphi }}(t)\\&+~\chi _0\,(\gamma -1)\,N'\,E_\psi (t)\\&+~\chi _0\,(\gamma -1)\,E_{\psi _0}(t), \end{aligned} \end{aligned}$$

(A2)

where

$$\begin{aligned}&E_{\varvec{\varphi }}(t) := \dfrac{1}{2}\displaystyle \int _{\Omega }\int _{0}^\infty \mu _{\text {g}}(\xi )\,\xi \,\left\Vert \varvec{\varphi }(\xi ;t,{\textbf{x}})\right\Vert ^2\,\text {d}\xi \,\text {d}{\textbf{x}}, \end{aligned}$$

(A3a)

$$\begin{aligned}&E_{\psi }(t) := \dfrac{1}{2}\displaystyle \int _{\Omega }\int _{L'}^\infty \nu _{\text {g}}(\xi )\,\xi \,|\psi (\xi ;t,{\textbf{x}})|^2\,\text {d}\xi \,\text {d}{\textbf{x}}, \end{aligned}$$

(A3b)

$$\begin{aligned}&E_{\psi _0}(t) := \dfrac{1}{2}\displaystyle \int _{\Omega }r_0\,(-s_0)\,|\psi (-s_0;t,{\textbf{x}})|^2\,\text {d}{\textbf{x}}. \end{aligned}$$

(A3c)

Note that the energy $E_{\psi _0}$ is similar to $E_{\psi }$ but for a fixed pole $\xi =-s_0$ with a positive residue $r_0$, and all the model parameters (M, N, L, M, $N'$, $L'$) are positive. The energy balance:

$$\begin{aligned}{\mathcal {E}}_{\text {lin}}(t):= \,E_{\text {m}}(t)\, +\, E_{\text {diff}}(t), \end{aligned}$$

is computed based on the definitions (A1) and (A2), and its derivative is found to be

$$\begin{aligned} \dfrac{\text {d}}{\text {d}t}{\mathcal {E}}(t)&= -~\rho _0\,\alpha _\infty \, M\displaystyle \int _{\Omega }\left\Vert {\textbf{u}}\right\Vert ^2\,\text {d}{\textbf{x}}\\&\quad -\rho _0\,\alpha _\infty \, N\displaystyle \int _{\Omega }\int _{L}^\infty \mu _{\text {g}}(\xi )\,\left\Vert \partial _t\varvec{\varphi }(\xi )\right\Vert ^2\,\text {d}\xi \,\text {d}{\textbf{x}}\\&\quad -\chi _0\,(\gamma -1)\, N'\displaystyle \int _{\Omega }\int _{L'}^\infty \nu _{\text {g}}(\xi )\,\left( \partial _t\psi (\xi )\right) ^2\,\text {d}\xi \,\text {d}{\textbf{x}}\\&\quad -\chi _0\,(\gamma -1) \displaystyle \int _{\Omega }r_0\,\left( \partial _t\psi (-s_0)\right) ^2\,\text {d}\xi \,\text {d}{\textbf{x}},\\&\leqslant 0, \end{aligned}$$

when there is no contribution at the boundary $\partial \Omega$ (i.e., $p=0$, or ${\textbf{u}}\cdot {\textbf{n}}=0$ on $\partial \Omega$) and the functions are defined in the appropriate functional spaces [57]. Regarding the asymptotic stability of this model, it is insufficient to prove that the extended energy ${\mathcal {E}}(t)$ serves as a Lyapunov functional for the dynamical system, i.e., that it decreases along the trajectories in the solution space. The comprehensive proof hinges upon the spectral analysis of the generator of the underlying linear semigroup, as explored in [68]. Notably, this analysis can be directly applied to our linear model.

For the nonlinear EFM, we make use of the same energy functional ${\mathcal {E}}_{\text {nl}} = {\mathcal {E}}_{\text {lin}}$. The only difference appears in the computation of the derivatives, where an additional negative term can be found for $\frac{\text {d}}{\text {d}t}{\mathcal {E}}_{\text {nl}}$ (see equation (16)). Thus, with a simple comparison principle, the asymptotic stability of the nonlinear EFM can be assessed.

Appendix B Nonlinear Semigroup

To examine the nonlinear aspect of the EFM, we will specifically consider the case where the diffusive parameters N, $M'$, $N'$, $r_0$ are set to 0. For this purpose, we will closely follow the example detailed in [82]. The nonlinear system is as follows:

$$\begin{aligned} \partial _t{\textbf{u}}= & {} -\dfrac{1}{\rho _0\,\alpha _\infty }\varvec{\nabla }p - \Psi ({\textbf{u}} ), \end{aligned}$$

(B1)

$$\begin{aligned} \partial _t p= & {} -\dfrac{1}{\chi _0}\varvec{\nabla }\cdot {\textbf{u}}\, \end{aligned}$$

(B2)

where $\Psi : {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2$, ${\textbf{u}} \mapsto \Psi ({\textbf{u}}):=\left\Vert {\textbf{u}}\right\Vert \,{\textbf{u}}$. The previous system can be seen as the abstract evolution equation $\partial _t X + {{{\mathcal {A}}}}(X) = 0$, with a nonlinear operator ${{{\mathcal {A}}}}$ acting on an appropriate Hilbert space. This operator is maximal monotone.

Maximality means that the operator $I + {{{\mathcal {A}}}}$ is onto on ${{\mathcal {H}}}$. The careful computation of this property amounts to show that the mapping $I_2+ \Psi$ is onto on ${\mathbb {R}}^2$, which happens to be the case, since $(1+ \left\Vert {\textbf{u}}\right\Vert )\,{\textbf{u}} = {\textbf{v}}$ $\Leftrightarrow$ ${\textbf{u}} = \bigg (\frac{1}{2} + \sqrt{\frac{1}{4}+ \left\Vert {\textbf{v}}\right\Vert }\bigg )^{-1}\,{\textbf{v}}$.

Monotonicity in the nonlinear case means that $({{{\mathcal {A}}}}(X) - {{{\mathcal {A}}}}(Y), X-Y)_{{{\mathcal {H}}}} \ge 0$ for any X, Y in the domain of ${{{\mathcal {A}}}}$. A careful, though classical, computation of this scalar product shows that a sufficient condition is given on the nonlinear term by:

$$\begin{aligned} ({\textbf{u}} - {\textbf{v}}) \cdot (\Psi ({\textbf{u}}) - \Psi ({\textbf{v}}) )\ge 0\, \quad \forall \,{\textbf{u}}, {\textbf{v}} \in {\mathbb {R}}^2. \end{aligned}$$

(B3)

Let us compute $({\textbf{u}} - {\textbf{v}}) \cdot (\left\Vert {\textbf{u}}\right\Vert \,{\textbf{u}} - \left\Vert {\textbf{v}}\right\Vert \,{\textbf{v}} ) = \left\Vert {\textbf{u}}\right\Vert ^3 + \left\Vert {\textbf{v}}\right\Vert ^3 - (\left\Vert {\textbf{u}}\right\Vert + \left\Vert {\textbf{v}}\right\Vert )\,{\textbf{u}}\cdot {\textbf{v}} \ge \left\Vert {\textbf{u}}\right\Vert ^3 + \left\Vert {\textbf{v}}\right\Vert ^3 - (\left\Vert {\textbf{u}}\right\Vert + \left\Vert {\textbf{v}}\right\Vert )\,\left\Vert {\textbf{u}}\right\Vert \,\left\Vert {\textbf{v}}\right\Vert = (\left\Vert {\textbf{u}}\right\Vert - \left\Vert {\textbf{v}}\right\Vert )^2\,(\left\Vert {\textbf{u}}\right\Vert +\left\Vert {\textbf{v}}\right\Vert ) \ge 0$.

Thus, the nonlinear operator ${{{\mathcal {A}}}}$ is maximal monotone and generates a nonlinear semigroup on ${{\mathcal {H}}}$; existence and uniqueness of solutions to (B1)-(B2) follow.

Appendix C General resistivity correction

In Sect. 3, the dependence of the resistivity on the velocity amplitude is taken to be linear, in agreement with the Forchheimer’s correction. However, the resistivity dependence is quadratic at low Reynolds number, leading to

$$\begin{aligned} \sigma = \sigma _0\left( 1+C_f^\star \left\Vert {\textbf{u}}\right\Vert ^2\right) . \end{aligned}$$

(C1)

A theoretical study at low velocity magnitude, where this type of correction is necessary, leads to the same theoretical results given in Theorem 1. It can even be extended to a more general case, where the total resistivity is defined as

$$\begin{aligned} \sigma = \sigma _0\Bigl (1+\Phi \bigl (\left\Vert {\textbf{u}}\right\Vert \bigr )\Bigr ). \end{aligned}$$

(C2)

with $\Phi :~{\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ of $C^1$ (e.g., quadratic at low velocity amplitude and linear at high velocity amplitude as in Fig. 21). Starting from the nonlinear EFM corrected with (C2), the method used to obtain the energy balance variations (16) leads in this case to

$$\begin{aligned} \dfrac{\text {d}}{\text {d}t}{\mathcal {E}}_{\text {nl}}(t)&=~\dfrac{\text {d}}{\text {d}t}{\mathcal {E}}(t)-\rho _0\,\alpha _\infty \, M\displaystyle \int _{\Omega }\Phi \bigl (\left\Vert {\textbf{u}}\right\Vert \bigr )\,\left\Vert {\textbf{u}}\right\Vert ^2\,\text {d}{\textbf{x}}, \end{aligned}$$

(C3)

allowing to conclude on the stability of the more general nonlinear model, since $\Phi >0$.

Fig. 21

Example of a function $\Phi :~{\mathbb {R}}^+\rightarrow {\mathbb {R}}^+$ of class $C^1$, having a quadratic behavior for small x and linear for large x

In order to apply the theory of nonlinear semigroups of Appendix B to the more general case when $\Psi ({\textbf{u}}):=\Phi (\left\Vert {\textbf{u}}\right\Vert )\,{\textbf{u}}$, the two following key ingredients must be checked:

• Maximality of the nonlinear operator ${{\mathcal {A}}}$ is ensured, as soon as the scalar function $x \mapsto x + x\,\Phi (x)$ is increasing on ${\mathbb {R}}^+$,

• Monotonicity of the nonlinear operator ${{\mathcal {A}}}$ is ensured, as soon as the scalar function $x \mapsto x\,\Phi (x)$ is increasing on ${\mathbb {R}}^+$.

The proof of the second item comes from the inequality $({\textbf{u}} - {\textbf{v}}) \cdot (\Psi ({\textbf{u}}) - \Psi ({\textbf{v}}) ) \ge (\left\Vert {\textbf{u}}\right\Vert - \left\Vert {\textbf{v}}\right\Vert )\,(\left\Vert {\textbf{u}}\right\Vert \, \Phi (\left\Vert {\textbf{u}}\right\Vert )- \left\Vert {\textbf{v}}\right\Vert \,\Phi (\left\Vert {\textbf{v}}\right\Vert ) \ge 0$. It is fulfilled by any monomial $\Phi (x)= x^p$, where $p>-1$ is not necessarily an integer.

This property can be extended in a straightforward way to positive linear combinations of such nonlinear operators, e.g., with $\Phi (x) = \phi _0\,x + \phi _1\, x^2$ where $\phi _0\ge 0$, $\phi _1\ge 0$.

Appendix D Numerical scheme

The code used in this article is based on the one used in [40] which solves the linearized Euler equations and the linear EFM in 2D, using a discontinuous Galerkin method detailed in Sect. D.1 and a Runge–Kutta method detailed in Sect. D.2

1.1 Space discretization

Given a domain $\Omega \subset {\mathbb {R}}^2$ for which there is a partition (${\mathcal {T}}_h$) where h denotes the maximum diameter of the elements in the partition. The approximation space is taken as $V_h:=\{v,|\,\forall T\in {\mathcal {T}}_h,v_{|T}\in {\mathbb {P}}^k(T)\}$ where ${\mathbb {P}}^k(T)$ is the space of polynomials of degree at most k. A basis $\{\lambda _j^i\in {\mathbb {P}}^k(T_i), j=1\}$, with $d=(k+1)(k+2)/2$, is defined for each element $T_i\in {\mathcal {T}}_h$. Thus, a scalar function q is approximated on a cell $T_i$ by

$$\begin{aligned}&q_h(t,{\textbf{x}}) := \displaystyle \sum ^{d}_{j=1}q_h^{i,j}(t)\lambda _j^i({\textbf{x}}). \end{aligned}$$

(D1)

The solution vector ${\textbf{q}}_h(t,{\textbf{x}}) = (u_h~~v_h~~p_h)$ is then defined based on (D1). Applying the DG method [83], it is possible to write:

$$\begin{aligned} \begin{aligned} 0&= \dfrac{\text {d}}{\text {d}t}\int _{T_i}{\textbf{q}}_h(t,{\textbf{x}})\,\lambda _{j}^{i}\,\text {d}\Omega \\&\quad + \int _{T_i}{\textbf{F}}({\textbf{q}}_h(t, {\textbf{x}}))\,\cdot \nabla \lambda _j^i\,\text {d}\Omega \\&\quad - \int _{\partial T_i} \mathbf {F^*}({\textbf{q}}_h^e(t,{\textbf{x}}),{\textbf{q}}_h^i(t,{\textbf{x}}))\cdot \mathbf {n_i},\lambda _j^i\,\text {d}\sigma \\&\quad + \int _{T_i}{\textbf{b}}({\textbf{q}}_h(t,{\textbf{x}}))\,\lambda _j^i\,\text {d}\Omega , \end{aligned} \end{aligned}$$

(D2)

where ${\textbf{F}}(q_h)=(A_x{\textbf{q}}_h,A_y{\textbf{q}}_h)$, ${\textbf{b}}({\textbf{q}}_h)=B{\textbf{q}}_h$, $\mathbf {n^i}=(n_x^i, n_y^i)$ is the outgoing unit normal vector with respect to the edge $\partial _tT_i$ and $\mathbf {F^*}$ is the numerical flux. The solution ${\textbf{q}}_h$ on the edge of a cell $T_i$ is denoted ${\textbf{q}}_h^i$ or ${\textbf{q}}_h^e$ depending on whether it is, respectively, the internal or the external value of $T_i$ that is taken into account. In the code, the numerical flux used is the flux vector splitting

$$\begin{aligned}&{\textbf{F}}^*({\textbf{q}}_h^i,{\textbf{q}}_h^e)\, = \, A^+{\textbf{q}}_h^i + A^-{\textbf{q}}_h^e, \end{aligned}$$

(D3)

where the incoming and outgoing flows are separated into two, respectively, associated with $A^+$ and $A^-$. The latter are matrices containing, respectively, the positive and negative eigenvalues of $A=A_xn_x+A_yn_y$. Note that this flow is an exact solution of the one-dimensional Riemann problem with constant coefficients. At the boundary of the domain $\partial \Omega$, the imposed flux is the centered flux

$$\begin{aligned}&{\textbf{F}}_{\text {BC}}^*({\textbf{q}}_h^i,{\textbf{q}}_h^e)\, := \dfrac{{\textbf{q}}_h^i + {\textbf{q}}_h^e}{2}, \end{aligned}$$

(D4)

where ${\textbf{q}}_h^e$ represents a ghost state defined with ${\textbf{q}}_h^i$. Finally, the semi-discrete equation reads:

$$\begin{aligned}&M\dfrac{\text {d}{\textbf{q}}_h}{\text {d}t}(t) := K\,{\textbf{q}}_h(t) + \widetilde{{\textbf{S}}}(t), \end{aligned}$$

(D5)

with ${\textbf{q}}_h$ the unknown and $\widetilde{{\textbf{S}}}$ the source term.

1.2 Time discretization

The inversion of the mass matrix M in (D5) is direct thanks to its diagonal structure per block. Indeed, each block is of the order of a few tens of rows, thus small enough to calculate the inverse of each small block by a direct method. Thus, the spatial discretization (D5) can be rewritten

$$\begin{aligned}&\dfrac{\text {d}{\textbf{q}}_h}{\text {d}t}(t) = \mathbf {L_h}(t,{\textbf{q}}_h(t)) = E{\textbf{q}}_h(t) + {\textbf{G}}(t), \end{aligned}$$

(D6)

with $\mathbf {\mathbf {L_h}}$ the semi-discrete operator, $E = M^{-1}K$ and ${\textbf{G}}=M^{-1}\widetilde{{\textbf{S}}}$. The Runge–Kutta method used for the spatial discretization is the fourth-order RKF84 method given by [73] and which was notably used for the results given in [84]. This method is shown to be very efficient when used in combination with a DG method for wave propagation problems in the works of [73].

Let $\{t^n\}^N_{n=0}$ be a partition of $[0,T]\subset {\mathbb {R}}^+$, $\Delta t = t^{n+1}-t^{n}$ be the time step and ${\textbf{q}}_h^{n}$ be the approximate solution at time $t^n$. The steps of the RKF84 algorithm are

$$\begin{aligned}&{{\textbf {q}}}^{(0)} = {\textbf{q}}_h^{n}, \end{aligned}$$

(D7a)

$$\begin{aligned}&{\textbf {dq}}^{(i)} = A_i{\textbf {dq}}^{(i-1)} + \mathbf {L_h}\left( t_n+c_i\Delta t,{\textbf{q}}^{(i-1)}\right) \end{aligned}$$

(D7b)

$$\begin{aligned}&{\textbf{q}}^{(i)} = {\textbf{q}}^{(i-1)} + B_i{\textbf {dq}}^{(i)}, \, \text {for } i = 1\dots 8, \end{aligned}$$

(D7c)

$$\begin{aligned}&{\textbf{q}}_h^{n+1} = {{\textbf {q}}}^{(8)}, \end{aligned}$$

(D7d)

where the coefficients $A_i$, $B_i$ and $c_i$ are given by [73, Table A.9].