具有物理约束的数据驱动、滤波降阶建模用于流体流动
Physically-Constrained Data-Driven, Filtered Reduced Order Modeling of Fluid Flows
AI总结：
本文提出了一种具有物理约束的数据驱动滤波降阶建模（CDDF-ROM），通过引入物理约束提高降阶模型的物理准确性，通过约束最小二乘问题提升模型精度，验证了在不同雷诺数下的有效性。
摘要：
在我们之前的工作中，我们提出了一种数据驱动滤波降阶模型（DDF-ROM）框架，用于流体流动的数值模拟，可以形式地写成 DDF-ROM = Galerkin-ROM + Correction。新的DDF-ROM通过使用ROM空间滤波和数据驱动的ROM闭合建模（用于修正项）构建，并在Re=100、Re=500和Re=1000的二维圆柱体后方通道流数值模拟中成功测试。在本文中，我们提出了一种具有物理约束的DDF-ROM（CDDF-ROM），旨在提高DDF-ROM的物理准确性。新的物理约束要求CDDF-ROM的操作符满足与流体流动方程相同的物理定律（即非线性操作符应守恒能量，ROM闭合项应耗散）。为了实现这些物理约束，在DDF-ROM的数据驱动建模步骤中，我们将无约束最小二乘问题替换为有约束最小二乘问题。我们对新的CDDF-ROM和标准DDF-ROM进行了数值研究，研究了Re=100、Re=500和Re=1000的二维圆柱体后方通道流。为此，我们考虑了再生制度以及预测（即交叉验证）制度，在其中我们使用原始训练数据的50%。数值研究清楚地表明，新的CDDF-ROM在两种制度中都比DDF-ROM更准确。
英文摘要：
In our earlier work, we proposed a data-driven filtered reduced order model (DDF-ROM) framework for the numerical simulation of fluid flows, which can be formally written as \begin{equation*} \boxed{ \text{ DDF-ROM = Galerkin-ROM + Correction } } \end{equation*} The new DDF-ROM was constructed by using ROM spatial filtering and data-driven ROM closure modeling (for the Correction term) and was successfully tested in the numerical simulation of a 2D channel flow past a circular cylinder at Reynolds numbers $Re=100, Re=500$ and $Re=1000$. In this paper, we propose a {\it physically-constrained} DDF-ROM (CDDF-ROM), which aims at improving the physical accuracy of the DDF-ROM. The new physical constraints require that the CDDF-ROM operators satisfy the same type of physical laws (i.e., the nonlinear operator should conserve energy and the ROM closure term should be dissipative) as those satisfied by the fluid flow equations. To implement these physical constraints, in the data-driven modeling step of the DDF-ROM, we replace the unconstrained least squares problem with a constrained least squares problem. We perform a numerical investigation of the new CDDF-ROM and standard DDF-ROM for a 2D channel flow past a circular cylinder at Reynolds numbers $Re=100, Re=500$ and $Re=1000$. To this end, we consider a reproductive regime as well as a predictive (i.e., cross-validation) regime in which we use as little as $50\%$ of the original training data. The numerical investigation clearly shows that the new CDDF-ROM is significantly more accurate than the DDF-ROM in both regimes.
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