基于Mori-Zwanzig形式的大涡模拟的非马尔可夫封闭模型
Non-Markovian Closure Models for Large Eddy Simulations using the Mori-Zwanzig Formalism
摘要：
这项工作使用的Mori-Zwanzig（M-Z）形式主义，一个概念起源于非平衡统计力学，作为发展的基础粗粒度模型的湍流。广义朗之万方程（GLE）的力学被认为是从正交动力学方程获得的洞察力作为模型开发的起点。考虑一类子网格模型，其通过有限记忆近似表示非局部行为（Stinis，P.，“Mori-Zwanzig reduced models for不确定性量化I：Parameter uncertainty，”\textit{arXiv:1211.4285}，2012.），其长度使用与解析变量的雅可比矩阵的谱半径相关的试探法来确定。由此产生的模型是密切相关的基本数值分辨率，并能够近似非马尔可夫效应。Burgers方程的数值实验表明，基于M-Z的模型可以准确地预测不同网格分辨率下总动能和总耗散率的时间演化。在相空间中的每一个解决模式的轨迹准确预测的情况下，粗粒化是温和的。均质各向同性湍流和Taylor Green涡的LES表明，基于M-Z的模型能够提供出色的预测，准确地捕获子网格对能量传递的贡献。最后，充分发展的通道流的LES证明了M-Z为基础的模型的非衰减问题的适用性。值得注意的是，闭包的形式不是由建模者强加的，而是从粗粒度的数学推导出来的，突出了基于M-Z的技术定义LES闭包的潜力。
英文摘要：
This work uses the Mori-Zwanzig (M-Z) formalism, a concept originating from non-equilibrium statistical mechanics, as a basis for the development of coarse-grained models of turbulence. The mechanics of the generalized Langevin equation (GLE) are considered and insight gained from the orthogonal dynamics equation is used as a starting point for model development. A class of sub-grid models is considered which represent non-local behavior via a finite memory approximation (Stinis, P., "Mori-Zwanzig reduced models for uncertainty quantification I: Parametric uncertainty," \textit{arXiv:1211.4285}, 2012.), the length of which is determined using a heuristic that is related to the spectral radius of the Jacobian of the resolved variables. The resulting models are intimately tied to the underlying numerical resolution and are capable of approximating non-Markovian effects. Numerical experiments on the Burgers equation demonstrate that the M-Z-based models can accurately predict the temporal evolution of the total kinetic energy and the total dissipation rate at varying mesh resolutions. The trajectory of each resolved mode in phase-space is accurately predicted for cases where the coarse-graining is moderate. LES of homogeneous isotropic turbulence and the Taylor Green Vortex show that the M-Z-based models are able to provide excellent predictions, accurately capturing the sub-grid contribution to energy transfer. Lastly, LES of fully developed channel flow demonstrate the applicability of M-Z-based models to non-decaying problems. It is notable that the form of the closure is not imposed by the modeler, but is rather derived from the mathematics of the coarse-graining, highlighting the potential of M-Z-based techniques to define LES closures.
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