Physical systems are commonly described by a set of partial differential equations. In multidimensional, time-dependent systems (such as transient or chaotic ones), the behavior of these equations leads to the formation of distinct regions with different dynamics, which can be clustered accordingly. In this study, we propose a hybrid approach to flow clustering. It consists of characterizing each sample point of the system with equation-based features, i.e., features are budgets that represent the contribution of each term from the original governing equation to the local dynamics at each sample point. This was achieved by applying the a sparse regression algorithm pointwise to time evolution data. The method proceeds with equation-based clustering using the Girvan-Newman algorithm. This allows the detection of communities that share the same physical dynamics underlined by the active terms of the governing equation of desired fidelity. The algorithm is implemented in both Eulerian and Lagrangian frameworks. In the Lagrangian, i.e., dynamic approach, the clustering is performed on the trajectory of each point. Our results show that while the dynamics of the regions remain the same (with the same active terms in the regression process), the location of the clusters moves with time when applied to transient or turbulent data. The performance of the algorithm is first tested on a flow around a cylinder. The construction of the dynamic clusters in this test case clearly shows the evolution of the wake from the steady-state solution through the transient to the oscillatory solution. Dynamic clustering was then successfully tested on turbulent flow data. Two distinct and well-defined clusters were identified and their temporal evolution was reconstructed.

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• Cluster models
• Eulerian flow field
• Lagrangian flow field
• Machine learning models