Improved Multipole Broadband Time-Domain Impedance Boundary Condition

Abstract

ACOUSTIC liners are widely used in industry for noise reduction. They are indispensable noise-suppression devices for jet engines. Acoustic liners are effective over a wide band of frequencies. To account for such characteristics, broadband impedance models are used for time-domain computation. The acoustic impedance of a surface, Z ! , is originally defined in the frequency domain. Problems involving noise propagation over impedance surfaces may be solved in the time domain. Time-domain methods have obvious advantages over frequency-domain methods for broadband and nonlinear problems. However, because an impedance boundary condition is set up for use in the frequency domain, an equivalent time-domain impedance boundary condition is needed. Mathematically, the time-domain equivalent may be derived by taking its inverse Fourier transform. For the equivalent time-domain impedance condition to be physical, the impedance function must be causal, real, and passive [1]. In general, the inverse Fourier transform of a frequency-domain impedance boundary condition would lead to a convolution integral. Computation of the convolution integral is time consuming. However, if the impedance can be represented by a sum of certain special functions, the evaluation of the convolution may be carried out in a simple systematic manner. Tam and Auriault [2] introduced a three-parameter time-domain impedancemodel. Thismodel can provide accurate representation of liner impedance for single discrete frequency problems. But the three-parameter model cannot offer a general representation of liner impedance over awide band of frequencies. Rienstra [1] proposed an extended Helmholtz resonator model. The free parameters of the model are the geometrical dimensions of the resonator. The impedance of the model is derived by the response of the Helmholtz resonator to periodic acoustic input. A comparison study by Richter et al. [3] indicated that a numerical instability could occur in some cases. A filtering technique has to be applied for avoiding this problem. Recently, Reymen et al. [4] developed a new and efficient time-domain impedance boundary condition based on the use of recursive convolution. The key to their method is the representation of the impedance by a series of simple poles in the complex plane. For the method to work efficiently, the value of the function in the convolution integral within a time step must be approximated by a linear or quadratic function. This impedance model has been proven to be physical, causal, and real. The primary objective of this paper is to develop an improved multipole broadband time-domain impedance boundary condition. A secondary objective is to derive an analytical solution for problems involving damping of broadband acoustic input by liners with broadband impedance characteristics. These solutions may serve for benchmarking numerical computations involving broadband sound. This paper is organized as follows. Section II discusses the improved multipole broadband impedance model. The implementation of the time-domain impedance boundary condition is presented in Sec. III. Section IV presents analytical solutions of twodimensional problems with broadband incident acoustic waves. These solutions are then used to validate numerical results with a liner having broadband impedance properties. Section IV concludes this work.