Evaluating the complexity of unsteady turbulent flows using excess entropy

众所周知，不稳定的湍流是非常复杂的，但没有太多的努力来评估它们到底有多复杂。其复杂性的原因在于，从空间和时间的角度来看，流结构的模式既包含可预测的元素，也包含随机元素。然而，预测定律或概率模型不能完全捕捉这些系统的行为。复杂程度是衡量流场在非定常状态下是确定的还是随机的。本研究将使用先进的数学方法评估复杂程度。通过这个项目获得的知识可以应用到涉及复杂性理论的各种其他领域。本项目所探索的研究课题将根据本科研究项目进行定制，并将与K-12教师分享，以丰富他们的课堂主题。本研究试图建立一个新的理论框架来量化非定常流动的复杂性。传统的复杂性估计方法通常在一维数据上运行，对于分析三维非定常场结果是不够的。为了解决这一限制，提出的研究通过利用大尺度流动结构作为分析的基本单位来解决这个问题，大尺度流动结构代表了流动在一定时期内的集体运动。这些结构表现出复杂的三维空间相互作用，可以利用持久同调算法根据其拓扑特征进行分类和符号化。得到的符号序列然后使用超额熵法来评估复杂性。这种创新的方法与传统的做法不同，它使用循环时空模式作为基本分析单元，而不是瞬时场，从而使有限域内的空间相互作用与整个系统的长期演变分离开来。此外，所提出的方法研究了大规模结构如何自我组织以形成流动，这是通过短时间方法（例如，有限时间李雅普诺夫指数）或传统统计技术（如雷诺兹应力和谱）实现的一项挑战。对流动模式的时间顺序的研究也使我们对连贯结构的动力学有了深入的了解，揭示了诸如纳维-斯托克斯方程不变解之间的关系等基本问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Unsteady turbulent flows are known to be very complicated, but there has not been much effort to assess just how complex they are. The reason for their complexity is that the patterns of flow structures in terms of space and time contain both predictable and random elements. However, predictive laws or probability models cannot fully capture the behavior of such systems. The degree-of-complexity is a measure of how certain or random a flow field is in an unsteady state. This research will evaluate the degree-of-complexity using advanced mathematical methods. The knowledge acquired through this project can be applied to various other fields that involve complexity theory. The research topics explored in this project will be tailored to be suitable for undergraduate research programs and will be shared with K-12 teachers to enrich their class topics.The proposed research attempts to establish a new theoretical framework for quantifying the complexity of unsteady flows. Traditional complexity estimators typically operate on 1D data and are inadequate for analyzing 3D unsteady field results. To address this limitation, the proposed research tackles the issue by utilizing large-scale flow structures as the fundamental unit of analysis, which represent the collective motion of the flow over a defined period. These structures exhibit intricate 3D spatial interactions and can be classified and symbolized based on their topological characteristics, utilizing the persistent homology algorithm. The resulting symbolic sequence is then employed to assess complexity using the excess entropy method. This innovative approach diverges from conventional practices by using recurrent spatial-temporal patterns as the fundamental analysis unit instead of instantaneous fields, thereby enabling the separation of spatial interactions within a finite domain from the long-term evolution of the entire system. Furthermore, the proposed methodology investigates how large-scale structures self-organize to shape the flow, a pursuit that is challenging to achieve through short-time methods (e.g., finite-time Lyapunov exponent) or conventional statistical techniques (such as Reynolds stresses and spectrum). The examination of temporal order among flow patterns also yields insights into the dynamics of coherent structures, shedding light on fundamental inquiries such as the relationship between invariant solutions of the Navier-Stokes equation.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.