基于数据驱动的湍流亚格子建模与应用

大涡模拟方法是研究复杂湍流问题的重要工具，在航空航天、辐射燃烧和能源化工等领域具有广泛的应用前景。湍流大涡模拟采用较粗的计算网格求解大尺度流动结构，通过亚格子模型来表征滤波尺度以下的流动结构对大尺度流场的影响。本文将物理约束和湍流尺度相似信息嵌入到基于数据驱动的亚格子未封闭项建模中，发展了先验和后验一致的高精度亚格子模型，并对模型普适性和泛化能力做了系统性的验证。主要的研究工作如下：
（1）提出了反卷积人工神经网络（DANN）模型，建立滤波流场和原始未滤波流场的映射关系。将当地邻域的滤波变量作为神经网络的输入特征，预测当地未滤波变量，并采用尺度相似模型构造湍流亚格子未封闭项。DANN模型由于仅重构反卷积算子，因而具有对称性、可实现性条件和伽利略不变性等物理性质。DANN模型先验预测亚格子应力的相关系数达到99%以上，相对误差低于15%；后验大涡模拟预测的湍流统计量和瞬时流场结构显著优于传统动态模型，相比机器学习黑箱模型计算效率提高了接近两个数量级。DANN模型在不改变参数的情况下，可直接应用于不同滤波尺度的流动，具有较强的普适性。随后，DANN模型被成功应用于可压缩湍流，并采用马赫数外插手段验证其具有较好的泛化能力。

（2）提出了满足伽利略不变性的动态迭代近似反卷积（DIAD）模型，将当地原始未滤波变量表示为给定空间模板的滤波变量的线性加权求和。DIAD模型无需直接数值模拟数据，通过滤波物理场的多次滤波运算，动态迭代获得最佳权重系数。不同滤波尺度的亚格子应力预测结果表明：DIAD模型的先验精度显著优于传统模型，迭代收敛速度更快。为了保证DIAD模型的后验稳定性，进一步发展了基于尺度分离策略的小尺度涡粘（SSEV）模型。SSEV 模型将数值耗散仅作用于湍流小尺度，在不影响大尺度运动的前提下，准确模拟了大涡模拟无法解析的亚格子动能通量。与传统的隐式大涡模拟方法相比，SSEV 模型具有明确的物理可解释性，不需要调整经验参数。SSEV增强的DIAD模型在不增加计算成本的前提下，可以精确地预测湍流各阶统计量和瞬时流动结构，并成功应用于可压缩均匀剪切湍流。

（3）提出了采用尺度相似动态准则（SSD）自适应确定亚格子模型系数的方法，并发展了基于尺度相似准则的动态非线性代数模型。尺度相似动态准则假设原始未滤波变量和滤波变量具有相似的本构关系，因此可将亚格子未封闭项中的原始未滤波变量替换为滤波变量并动态求解模型参数。尺度相似动态准则简化了基于Germano恒等式的传统动态方法的计算流程，计算成本可降低一半，并且大涡模拟计算的先验和后验精度得到了显著提升。

（4）基于伴随同化策略，发展了变分优化亚格子模型的统一框架。通过最小化模型预测值与实际观测值之间的误差函数，建立受大涡模拟控制方程约束的目标优化问题。联立求解主控制方程和伴随反演方程，可以精确计算误差函数对模型参数的梯度，随后通过梯度优化算法在参数域开展目标寻优迭代，直至获得目标函数的最优解。在此基础上，以亚格子混合模型为基准，发展了变分优化混合（VOMM）模型。VOMM模型通过不断同化融合湍流先验统计信息，可以有效改善传统模型后验大涡模拟的稳定性，具有较高的后验精度和计算效率。VOMM模型在各向同性稳态和衰减湍流以及混合层流动中的预测结果表明：VOMM模型能够准确预测湍流统计量和空间流动结构的非定常演化过程，计算时间仅为传统模型的30%。

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