While unsteady flows are widely recognized to be complex, the precise definition of their complexity remains elusive. A complex flow may contain numerous interacting structures at different scales, with highly unpredictable dynamic behavior. Therefore, some studies have described flow complexity from a multi-scale interaction perspective. However, limited effort has been made to evaluate the degree of complexity of a flow. This paper addresses this issue using a new information theory-based framework. A separated laminar boundary layer is used as an exemplary flow to test the new complexity measurement method. The flow evolution is divided into pseudo-periods, with the vorticity in each period characterized by its topological features. These topological classifications are then symbolized based on their homological group. The resulting symbol sequence allows for the evaluation of unpredictability and complexity using the well-established excess entropy method. Along the streamwise direction, the flow's unpredictability shows a rapid increase near the transition point. Downstream, the unpredictability gradually saturates. The spatial meandering of vortices and temporal unpredictability are highly correlated. The excess entropy of the vortical structure sequence, which represents the long-term coherence of the flow, remains low along the streamwise direction.
Tuna et al. (Mon,) studied this question.