计算气动声学的精确色散和自适应耗散四阶差分格式
A fourth-order finite difference scheme with accurate dispersion and adaptive dissipation for computational aeroacoustics
摘要：
对于计算声学，方案需要具有低色散和低耗散特性，以便正确捕获波的振幅和相位。为了改善该方案的频谱特性，作者以前提出了一个尺度传感器，自动调整数值耗散。因此，提出了一种具有最小色散和自适应耗散（MDAD）的四阶有限差分格式[1]。在这项研究中，我们进一步研究这种方法的高保真数值模拟的声学问题，并提出了一种新的色散控制方法，这是不同于传统的色散关系保持（DRP）的方法。首先，尺度传感器，它量化的局部长度尺度的解决方案作为有效的缩放波数，被修改为更好的性能上的复合波。然后，采用尺度传感器控制方案的色散和耗散。分别用解析和人工方法建立了色散/耗散参数与有效标度波数之间的关系。因此，构造了一个具有精确色散和自适应耗散的四阶有限差分格式。近似色散关系（ADR）表明，ADAD方案在k < 2.5时具有精确的色散特性。耗散在低波数时可以忽略不计，并在k = 1后逐渐增加，以抑制非物理振荡。计算声学的几个基准案例，以验证所提出的方案相比，传统的谱优化方案的高分辨率。
英文摘要：
For computational acoustics, schemes need to have low-dispersion and low-dissipation properties in order to capture the amplitude and phase of the wave correctly. To improve the spectral properties of the scheme, the authors have previously proposed a scale sensor to automatically adjust the numerical dissipation. In consequence, a fourth-order finite difference scheme with minimized dispersion and adaptive dissipation (MDAD) has been proposed [1]. In this study, we further investigate this method for the high-fidelity numerical simulation of the acoustic problems and a new dispersion control method is proposed which is different from the traditional dispersion relation preserving (DRP) approach. Firstly, the scale sensor, which quantifies the local length scale of the solution as the effective scaled wavenumber, is modified for better performance on composite waves. Then the scale sensor is applied to control both the dispersion and dissipation of the scheme. The relationships between the dispersion/dissipation parameter and the effective scaled wavenumber are analytically and artificially constructed respectively. Thus, a fourth-order finite difference scheme with accurate dispersion and adaptive dissipation (ADAD) is constructed. The approximate dispersion relation (ADR) shows that the ADAD scheme achieves accurate dispersion property at k < 2.5. The dissipation is negligible at low wave number and gradually increases after k = 1 to suppress non-physical oscillations. Several benchmark cases of computational acoustics are presented to verify the high resolution of the proposed scheme compared with the conventional spectral optimized schemes.
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