论文标题
使用二尺度收敛的粘膜模型的一系列Helmholtz共振器的表面均匀化
Surface homogenization of an array of Helmholtz resonators for a viscoacoustic model using two-scale convergence
论文作者
论文摘要
我们在声学衬里中得出了线性粘多模型的弱极限，该衬里是与伸长室的周期性重复相关的腔室-Helmholtz谐振器。作为模型，我们考虑了声速和压力的时间谐波和线性化的可压缩Navier-Stokes方程。遵循Schmidt等人的方法，J。Math。 Ind 8：15，2018对于多孔板的ViscoAcoustic传输问题，将粘度缩放为$δ^{4} $，带有腔室阵列的周期$δ$，颈部的大小以及壁厚，以及$Δ^2 $，以便粘性边界层的尺寸为颈部尺寸的粘性层。应用两尺度收敛的方法，我们以稳定性假设获得的稳定性$δ\至0 $，声音压力在阻抗边界条件下满足了Helmholz方程。这些边界条件取决于谐振器的频率，长度以及通过有效的雷利电导率（可以用数值计算）在其颈部形状上。我们将极限模型与文献中的半分析模型进行了比较。
We derive the weak limit of a linear viscoacoustic model in an acoustic liner that is a chamber connected to a periodic repetition of elongated chambers -- the Helmholtz resonators. As model we consider the time-harmonic and linearized compressible Navier-Stokes equations for the acoustic velocity and pressure. Following the approach in Schmidt et al., J. Math. Ind 8:15, 2018 for the viscoacoustic transmission problem of multiperforated plates the viscosity is scaled as $δ^{4}$ with the period $δ$ of the array of chambers and the size of the necks as well as the wall thickness like $δ^2$ such that the viscous boundary layers are of the order of the size of the necks. Applying the method of two-scale convergence we obtain with a stability assumption in the limit $δ\to 0$ that the acoustic pressure fulfills the Helmholz equation with impedance boundary conditions. These boundary conditions depend on the frequency, the length of the resonators and through the effective Rayleigh conductivity -- that can be computed numerically -- on the shape of their necks. We compare the limit model to semi-analytical models in the literature.
