A WEIGHTED-OPTIMIZATION COMPACT SCHEME FOR SHOCK-ASSOCIATED NOISE COMPUTATION AND ITS NONLINEAR EFFECT ANALYSIS
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摘要: 在超声速流动中, 激波与湍流结构的相互作用会产生高强度的激波噪声. 激波噪声的高保真计算要求激波捕捉格式具有高阶精度、低耗散和低色散特性, 同时还要尽可能地减弱格式的非线性效应. 现有的六阶精度迎风/对称混合型加权非线性紧致格式CCSSR-HW-6在基于对称模板构造网格中心处的数值通量时引入了两级加权, 且两级加权都需要构造非线性的权系数, 因而非线性效应较强. 本文以修正波数的误差积分函数为优化目标函数, 优化了CCSSR-HW-6格式的非线性特性, 建立了加权优化紧致格式WOCS. 精度验证表明WOCS格式的精度高于5阶. 谱特性分析表明, 与原方法相比, WOCS格式的耗散误差和非线性效应显著降低. 典型激波噪声问题数值实验表明: WOCS格式不仅提高了对高频波的分辨能力, 而且显著地消除了数值解中因格式的非线性效应所导致的非物理振荡.Abstract: For the supersonic flow, the shock waves interact with the turbulent structures to generate high intensity shock-associated noise. High fidelity numerical simulation of shock-associated noise requires the shock-capturing scheme to have the properties of high-order accuracy, low dissipation and low dispersion. It is also necessary to reduce the nonlinear effect caused by the nonlinear implementation of scheme as much as possible. The existing upwind/symmetric hybrid weighted non-linear compact scheme with sixth order accuracy (called by CCSSR-HW-6 scheme, Journal of Computational Physics, 2015, 284: 133-154) introduces two-stage weighting strategy to construct the numerical flux at the cell center based on the symmetric stencil. Each stage of weighting in CCSSR-HW-6 scheme must design a nonlinear function for the weighting coefficient, which makes the nonlinear effect enhanced. In this paper, a weighted optimization compact scheme (called by WOCS scheme) is established through optimizing the nonlinear characteristics of original CCSSR-HW-6 scheme. The error integral function of modified wavenumber is chosen as the optimization objective function. The accuracy verification shows that the WOCS scheme has more than fifth order accuracy. The analysis of spectral property shows that compared to original CCSSR-HW-6 scheme, the dissipation error and the nonlinear effect of WOCS scheme are significantly reduced. Numerical experiments on several typical shock-associated noise problems show that the WOCS scheme not only improves the resolving ability of high-frequency waves, but also significantly attenuates the non-physical oscillations in numerical solution caused by the nonlinear effect.
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Key words:
- shock-associated noise /
- shock-capturing scheme /
- compact scheme /
- nonlinear effect
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图 1 迎风/对称混合型加权非线性紧致格式的构造模板
Figure 1. Candidate stencils of hybrid upwind/symmetric weighted nonlinear compact scheme
图 6 激波−涡量波相互作用在t = 0.2时刻的涡量空间分布
Figure 6. The spatial distribution of vorticity in the shock-vorticity wave interaction at t = 0.2
图 7 激波−涡量波相互作用在t = 0.2时刻沿直线y = 0的涡量分布
Figure 7. The vorticity distribution along the line y = 0 in the shock-vorticity wave interaction at t = 0.2
图 8 激波−强旋涡相互作用在t = 10时刻的胀量场
Figure 8. The dilatation field of shock-strong vortex interaction at t = 10
图 9 激波−强旋涡相互作用在t = 10时刻的胀量和数值阴影沿径向的分布
Figure 9. The radial distribution of dilatation and numerical shadowgraph in shock-strong vortex interaction at t = 10
图 10 激波−强旋涡相互作用在t = 10时刻的数值阴影
Figure 10. Numerical shadowgraph of shock-strong vortex interaction at t = 10
表 1 加权优化紧致格式的数值误差和精度阶数
Table 1. Numerical errors and accuracy order of the weighted optimization compact scheme
N L1 error L1 order L∞ error L∞ order 10 2.868 × 10−3 − 4.414 × 10−3 − 20 5.357 × 10−5 5.74 8.480 × 10−5 5.70 40 1.029 × 10−6 5.70 1.619 × 10−6 5.71 80 2.156 × 10−8 5.58 3.391 × 10−8 5.58 160 5.093 × 10−10 5.40 8.003 × 10−10 5.40 
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