Cluster-based Markov model to understand the transition dynamics of a supersonic mixing layer

Abstract

The cluster-based Markov model (CMM) is performed on a numerically simulated supersonic mixing layer at Re = 10 400 to extract physical mechanisms. The high-dimensional state space of the supersonic mixing layer is automatically partitioned into ten relatively homogeneous clusters with representative states called centroids via the cluster analysis. The transition dynamics is conceptualized as a Markov model between centroids using the cluster transition matrix from a probabilistic point of view. A comprehensive analysis of CMM’s outcomes reveals two flow regimes: the single/double-vortex interaction (SDV) and multiple-vortex interaction (MV). The SDV regime plays the dominant role in the supersonic mixing layer, although any single centroid from the MV group carries much larger energy than that from the SDV group. More complicated patterns of vortex are well captured in an intelligent way associated with triple-vortex, quadruple-vortex, and even quintuple-vortex interaction. These vortex formations transport much more energy than the double-vortex pairing/merging. The CMM reveals a complicated set of dynamics that intermittently appear in the two regimes. The inner-circulation transition inside the SDV regime is the most probable route in the supersonic mixing layer. The MV regime can only be accessed from the SDV regime; meanwhile, it inclines to move back to the SDV regime. The transitions linking two regimes undergo large energy fluctuations. The predicted distribution of future cluster probability converges to a unique stationary distribution, which approximates the statistical probability distribution of the dataset.