弱可压缩MRT-LBM格式的灵敏度分析及自由松弛参数的确定
Sensitivity analysis and determination of free relaxation parameters for the weakly-compressible MRT-LBM schemes
摘要：
MRT-LBM中存在多个自由弛豫参数。虽然这些参数已经通过线性分析调整，这些参数和其他相关参数的灵敏度分析仍然不足以检测的行为的色散和耗散关系的MRT-LBM。以往的研究表明，在MRT LBM的体积耗散引起了显着的过阻尼的声扰动。这表明，MRT-LBM不能用于获得正确的行为的压力波动，因为固定的体弛豫参数。为了解决这一问题，提出了一种从线性化MRT-LBM恢复线性化纳维-斯托克斯方程的有效算法。恢复的L-NSE表现为矩阵形式，具有关于$δt$的任意阶的截断误差。然后，在波数空间中，矩阵特征值的一阶/二阶灵敏度分析被用来解决波数幅度的色散耗散关系的灵敏度。通过一阶灵敏度分析，首次得到了MRT-LBM群速度的数值特性。然后，在复平面上研究了MRT-LBM线性化形式对应的矩阵特征值的分布灵敏度。基于灵敏度分析和回收的L-NSE，我们提出了一些简化的优化策略来确定自由弛豫参数的MRT-LBM。同时，将最优MRT-LBM的色散和耗散关系与精确色散和耗散关系进行了定量比较。最后，一些经典的声学基准问题的数值验证，以评估新的最优MRT-LBM。
英文摘要：
It is well-known that there exist several free relaxation parameters in the MRT-LBM. Although these parameters have been tuned via linear analysis, the sensitivity analysis of these parameters and other related parameters are still not sufficient for detecting the behaviors of the dispersion and dissipation relations of the MRT-LBM. Previous researches have shown that the bulk dissipation in the MRT-LBM induces a significant over-damping of acoustic disturbances. This indicates that MRT-LBM cannot be used to obtain the correct behavior of pressure fluctuations because of the fixed bulk relaxation parameter. In order to cure this problem, an effective algorithm has been proposed for recovering the linearized Navier-Stokes equations from the linearized MRT-LBM. The recovered L-NSE appear as in matrix form with arbitrary order of the truncation errors with respect to $δt$. Then, in wave-number space, the first/second-order sensitivity analyses of matrix eigenvalues are used to address the sensitivity of the wavenumber magnitudes to the dispersion-dissipation relations. By the first-order sensitivity analysis, the numerical behaviors of the group velocity of the MRT-LBM are first obtained. Afterwards, the distribution sensitivities of the matrix eigenvalues corresponding to the linearized form of the MRT-LBM are investigated in the complex plane. Based on the sensitivity analysis and the recovered L-NSE, we propose some simplified optimization strategies to determine the free relaxation parameters in the MRT-LBM. Meanwhile, the dispersion and dissipation relations of the optimal MRT-LBM are quantitatively compared with the exact dispersion and dissipation relations. At last, some numerical validations on classical acoustic benchmark problems are shown to assess the new optimal MRT-LBM.
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