Evaluating the complexity of unsteady turbulent flows using excess entropy

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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.