基于直接反卷积模型的湍流大涡模拟方法

大涡模拟方法（large-eddy simulation，LES）是研究湍流问题的有效工具，其基本思想是通过滤波方法将流场按尺度做分解，对大尺度结构进行直接求解，对小尺度结构进行建模。本文将直接反卷积模型用于湍流大涡模拟，研究了直接反卷积模型的亚滤波尺度（sub-filter scale，SFS）动力学，将其扩展到各向异性滤波器的情况，并推广到离散形式。主要的研究工作如下：
（1）研究了亚滤波尺度（SFS）动力学在不同滤波器和不同滤波网格尺度比（filter-to-grid ratio，FGR）的情况下对直接反卷积模型（direct deconvolution model，DDM）精度的影响。本文考虑了多种可逆滤波器，包括高斯、Helmholtz I 型和II型、Butterworth、Chebyshev I 型和II 型、Cauchy、Pao 和快速衰减滤波器。FGR的范围是从1 到4。本文发现，FGR 对于控制误差、确保精确预测亚滤波应力至关重要。在FGR 为1 的情况下，DDM 模型无法精确地重构亚滤波应力，因为粗网格没有精确解析亚滤波动力学对亚滤波应力的影响。在FGR 为2 的情况下，除Helmholtz I 型和II 型滤波器外，大多数DDM 模型的预测能力得到了显著提高，可以相当精确地重构亚滤波应力。在FGR 为4 的情况下，所有DDM 模型都给出了非常精确的结果。此外，作者将DDM 模型与传统的亚滤波模型进行了全面比较，包括速度梯度模型（velocity gradient model，VGM）、动态Smagorinsky 模型（dynamic Smagorinsky model，DSM）、动态混合模型（dynamic mixed model，DMM）。在先验研究中，DDM 模型的亚滤波应力的相关系数远大于传统模型的相关系数。在后验研究中，DDM 模型在预测各种速度统计量和瞬时流动结构方面优于DSM 和DMM 模型。这些结果表明，具有适当FGR 的DDM 模型在大涡模拟中具有很大潜力。
（2）研究了各向异性滤波器的情况下亚滤波动力学对大涡模拟精度的影响。本文采用了不同类型的亚滤波模型，包括DSM、DMM，以及DDM。各向异性滤波器的长宽比（aspect ratio，AR）范围是从1 到16。作者发现，DDM 模型能够在滤波器各向异性很强的情况下精确地预测亚滤波应力。在先验研究中，DDM 模型重构的亚滤波应力的相关系数超过90%，远大于DSM 和DMM 模型的相关系数。在后验研究中，DDM 模型在预测各种湍流统计量方面优于DSM 和DMM 模型，包括速度谱，以及涡量、亚滤波能量流量、速度增量、应变率张量和亚滤波应力的概率密度函数（probability density function，PDF）。随着滤波器各向异性的增加，DSM 和DMM 模型的结果变得更差，但DDM 模型在各种滤波器各向异性的情况下能够给出较为精确的结果。这些结果表明，在各向异性滤波器的情况下，DDM模型是一种有效的亚滤波模型。
（3）发展了离散直接反卷积模型（discrete direct deconvolution model，D3M）。对于第一种模型D3M-1，作者用不同阶数的局部离散公式去近似高斯滤波器，并通过对离散滤波器直接求逆来重构未滤波的流场。原始高斯滤波器的逆也可以通过局部离散公式进行近似，从而得到完全局部的模型D3M-2。与传统的DSM 和DMM 相比，D3M-1 和D3M-2 模型在先验研究中具有更高的相关系数和更低的相对误差。在后验验证中，D3M-1 和D3M-2 模型都能精确地预测湍流统计量，包括速度谱，以及亚滤波应力和亚滤波能量流量的PDF。同时，能够精确预测湍流混合层中的动能谱、动量厚度和雷诺应力。此外，该模型还能很好地捕捉Q 准则等值面的空间结构。因此，D3M 模型在大涡模拟中具有很大的潜力。
综合上述情况，采用合适FGR 的直接反卷积模型具有很高精度，在各向异性滤波器的情况下仍能保持较好的预测效果，同时能够用在物理空间，在湍流大涡模拟中具有良好的应用前景。

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