计算气动声学的最佳低色散低耗散LBM格式
Optimal low-dispersion low-dissipation LBM schemes for computational aeroacoustics
摘要：
格子Boltmzmann方法（LBM）已被证明是计算气动声学（CAA）中非常有效的方法，它已被用于捕获弱声波动的动力学。本文提出了一种减小二维多弛豫时间格子玻尔兹曼方法（MRT-LBM）色散和耗散误差的方法。通过给出一种有效的算法，我们得到了对应于MRT-LBM的线性化纳维-斯托克斯方程在波数空间中的统一形式。利用矩阵摄动理论和有限差分法的等价修正方程方法，我们提出了一类极小化问题来优化MRT-LBM中的自由参数。通过这种方法，我们得到了一个色散关系保持LBM（DRP-LBM），以规避MRT-LBM的最小色散误差。同时也提高了耗散关系的精度，保证了小体积粘度下MRT-LBM的稳定性。Von Neuman分析的线性MRT-LBM进行验证优化的色散/耗散关系，考虑单色波的解决方案。同时，定量地比较了优化MRT-LBM与原MRT-LBM的色散和耗散误差。最后，进行了一些数值模拟，以评估新的优化MRT-LBM计划。
英文摘要：
Lattice Boltmzmann Methods (LBM) have been proved to be very effective methods for computational aeroacoustics (CAA), which have been used to capture the dynamics of weak acoustic fluctuations. In this paper, we propose a strategy to reduce the dispersive and disspative errors of the two-dimensional (2D) multi-relaxation-time lattice Boltzmann method (MRT-LBM). By presenting an effective algorithm, we obtain a uniform form of the linearized Navier-Stokes equations corresponding to the MRT-LBM in wave-number space. Using the matrix perturbation theory and the equivalent modified equation approach for finite difference methods, we propose a class of minimization problems to optimize the free-parameters in the MRT-LBM. We obtain this way a dispersion-relation-preserving LBM (DRP-LBM) to circumvent the minimized dispersion error of the MRT-LBM. The dissipation relation precision is also improved.And the stability of the MRT-LBM with the small bulk viscosity is guaranteed. Von Neuman analysis of the linearized MRT-LBM is performed to validate the optimized dispersion/dissipation relations considering monochromatic wave solutions. Meanwhile, dispersion and dissipation errors of the optimized MRT-LBM are quantitatively compared with the original MRT-LBM . Finally, some numerical simulations are carried out to assess the new optimized MRT-LBM schemes.
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