Reduced Order Models of the Navier-Stokes Equations of Fluid Flows

湍流是一种明显的随机运动，在几乎所有的流体流动中都很常见。它被称为牛顿物理学中最具挑战性的未解决问题之一。我们对动荡的不完全理解给我们的社会造成的总成本是巨大的。例如，考虑到几乎所有流体热系统（如内燃机和空调）的次优性能所带来的环境和经济成本。通常不可能直接和常规地模拟湍流，因为它的多尺度特征需要过高的计算资源。即使有足够的计算资源，模拟也往往对它们产生的解决方案提供的理解太少。开发和研究（降阶）湍流模型，在保持所需的物理保真度的同时，大大减少了计算模型的尺寸和成本，这在科学和工程上有重大的好处。湍流降阶建模的最终目标是在对辅助经验模型的依赖最小的情况下提供有效和准确的解决方案。经验模型本质上是不可取的，因为它们降低了模拟的准确性和可靠性。本项目的研究目标是开发一种不需要经验闭合模型的湍流流体流动的降阶建模方法。与传统方法不同，新方法不依赖于经验湍流建模或对Navier-Stokes方程的特别修改。它所提供的空间基函数不同于通常的固有正交分解基函数，除了最优地表示解外，还提供了稳定的降阶模型。该方法通过三个测试案例进行了说明：方形盖子驱动腔内的二维流动，二维混合层和艾哈迈德体周围的三维湍流。未来的工作将把这种方法扩展到更复杂的流动，包括更高的空间维度和更高的流动速度（雷诺数）的影响。

Turbulence is the apparently random motion that occurs commonly in almost all fluid flows. It has been called one of the most challenging unsolved problems in Newtonian physics. The aggregate cost to our society resulting from our incomplete understanding of turbulence is significant. Consider, for example, the environmental and economic costs associated with sub-optimal performance of virtually every fluid-thermal system such as internal combustion engines and air-conditioners. It is usually not possible to directly and routinely simulate turbulent flow because its multi-scale characteristics require prohibitively high computational resources. Even when adequate computational resources are available, simulations often provide too little understanding of the solutions they produce. There are significant scientific and engineering benefits in developing and studying (reduced-order) models of turbulence that retain the needed physical fidelity while substantially reducing the size and cost of the computational model. The ultimate goal of reduced-order modeling of turbulence is to provide efficient and accurate solutions with minimal reliance on auxiliary empirical models. Empirical models are inherently undesirable because they degrade simulation accuracy and reliability.The research objective of this project is to develop a reduced-order modeling approach to turbulent fluid flows that is free of empirical closure models. Unlike traditional approaches, the new methodology does not rely on empirical turbulence modeling or ad hoc modification of the Navier-Stokes equations. It provides spatial basis functions different from the usual proper orthogonal decomposition basis function in that, in addition to optimally representing the solution, the new basis functions also provide stable reduced-order models. The approach is illustrated with three test cases: two-dimensional flow inside a square lid-driven cavity, two-dimensional mixing layer, and three-dimensional turbulent flow around the Ahmed body. Future work will extend this method to more complex flows including the effects of higher spatial dimensions and higher flow velocities (Reynolds numbers).