Evaluating the complexity of unsteady turbulent flows using excess entropy

众所周知，不稳定湍流非常复杂，但目前还没有太多努力来评估它们的复杂程度。其复杂性的原因是，在空间和时间方面的流动结构的模式包含可预测的和随机的元素。然而，预测定律或概率模型不能完全捕捉到这样的系统的行为。复杂度是衡量流场在非定常状态下的确定性或随机性的尺度。本研究将使用先进的数学方法来评估复杂程度。通过这个项目获得的知识可以应用于涉及复杂性理论的各种其他领域。本计画所探讨的研究课题将针对本科生的研究计划而量身订造，并将与K-12教师分享，以丰富他们的课堂课题。拟议的研究试图建立一个新的理论框架，以量化非定常流动的复杂性。传统的复杂性估计通常在一维数据上操作，并且不足以分析3D非稳定场结果。为了解决这一限制，拟议的研究解决了这个问题，利用大规模的流动结构作为分析的基本单位，这代表了集体运动的流动在一个定义的时间。这些结构表现出复杂的3D空间相互作用，可以分类和符号化的基础上，他们的拓扑特征，利用持久的同源算法。然后，将得到的符号序列用于使用超额熵方法来评估复杂性。这种创新的方法不同于传统的做法，通过使用经常性的时空模式作为基本的分析单元，而不是瞬时场，从而使空间相互作用的分离在一个有限的域从整个系统的长期演变。此外，所提出的方法研究了大规模结构如何自组织以塑造流动，这是一种通过短时间方法（例如，有限时间李雅普诺夫指数）或传统的统计技术（如雷诺应力和频谱）。对流型之间的时间顺序的研究也有助于深入了解相干结构的动力学，从而为诸如Navier-Stokes方程的不变解之间的关系等基本问题提供帮助。该奖项反映了NSF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而被认为值得支持。

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.