A boundary element analysis on acoustic dissipation by wall visco-thermal boundary layer

在壁面，特别是亥姆霍兹型谐振器孔固体表面的声能耗散中，壁面的粘度和热导率起主导作用。将Stokes理论应用于粘性模态，Kirchhoff理论应用于热模态，可以给出简单形状孔径（如长圆管）的声阻的显式表达式。但是对于有限长度的一般形状的n型颈，必须求解声模态波与粘热边界层之间的隐式关系。我们采用边界元数值方法和子结构技术对声模场进行分析，并通过粘热边界层的有效声模导纳实现了声模和粘热模的结合。该导纳由Stokes和Kirchhoff理论推导而来，并包含了声模边界上声压的拉普拉斯函数。对于拉普拉斯运算，我们采用了有限差分方法，采用了常数边界元。为了验证数值模拟的有效性，我们对狭缝谐振腔的耗散系数进行了一系列数值模拟和实验，并对结果进行了比较。在实验中，采用高精度尺寸的刚性试样。结果表明，在边界元尺寸小于等于0.1 mm的范围内，数值模拟结果与实验结果吻合良好。为了减少对这种精细单元尺寸的要求，正在研究使用三阶近似单元。

On acoustic energy dissipation at wall surfaces, especially at the solid surfaces of the apertures of resonators of Helmholtz type, the viscosity and thermal conductance there plays the leading role. By using the Stokes theory for the viscous mode and the Kirchhoff theory for the thermal mode, explicit expressions can been given for the acoustic resistance of the aperture of simple shape such as a long round tube. However for n neck of general shape with finite length, we have to solve implicit relationships between acoustic mode wave and the visco-thermal boundary layer.We employed a boundary element numerical approach along with the substructure techniques for the acoustic mode fields, and implemented a computer code to combine the acoustic mode and the visco-thermal mode through the effective acoustic mode admittance for the visco-thermal boundary layer. This admittance was derived from the Stokes and Kirchhoff theories, and includes the Laplacian of the acoustic pressure over the acoustic mode boundary. We took a finite difference approach for the Laplacian operation by employing the constant boundary element.To investigate the effectiveness of the numerical simulation, we conducted a series of numerical simulations and experiments for the dissipation factors of slit resonators, and compared the results. In the experiments, rigid specimens of highly precise dimensions are used. Consequently, the agreement between the dissipation factor by the numerical simulation and that by the experiment was excellent, as far as boundary element dimension was 0.1 mm or finer. To reduce the requirement of this fine element size, the use of a third order approximation element is under investigation.