Modal analysis of vortex rope using dynamic mode decomposition

The decelerating swirling flow in the draft tube of hydraulic turbines at part load conditions often results in a self-induced instability known as vortex rope. This phenomenon is associated with detrimental pressure pulsations in the hydropower system that need to be mitigated. A deep understanding of such instability is essential for developing effective mitigation and control strategies. The current article exploits the dynamic mode decomposition (DMD) algorithm to perform an in-depth modal analysis of the physical aspects of the vortex rope. DMD can efficiently identify distinct coherent structures with isolated frequencies. The sparsity-promoting variant of DMD is exploited to extract the most influential modes. The computational fluid dynamics (CFD) data is generated via a resolved improved delayed detached eddy simulation using OpenFOAM. Frequency analysis of the CFD data uncovered peaks at the normalized frequencies of f/fn=0.56 and 0.63, whose origins seemed initially unclear. Nevertheless, the DMD modal analysis elucidates that these excitations are associated with the rotation of the reunited vortex and fluctuations of the separated boundary layer, respectively. The non-linear dynamics of the flow field are unveiled through a modal decomposition revealing distinctive coherent structures with isolated frequencies. These include rotational and plunging modes of the vortex rope, traveling wakes of the blades, boundary layer separation due to strong adverse pressure gradient, and a reunited vortex core. The flow field reconstruction through time dynamics of DMD modes highlights while it is possible to achieve a perfect flow field reconstruction considering all recovered modes, the model typically fails to predict future behavior with an acceptable level of accuracy. The chaotic nature of the resolved turbulent flow field presents a substantial challenge for predicting the future through a model built based on prior events. The current modal analysis not only provides a more comprehensive understanding of the physics underlying the vortex rope phenomenon but also lays the groundwork for potential future applications in controlling mechanisms.