Spectral organization and dynamics of three-dimensional coherent structures in draft tube vortex rope of a francis turbine

Unstable vortex rope structures, characterized by the coexistence of multiple frequency components, can develop in a Francis turbine. To reveal the organization and dynamics of spectrally pure coherent structures in the vortex rope, high-fidelity numerical calculations were carried out using a Detached Eddy Simulation (DES) turbulence model, accounting for the weak compressibility of water. The dominant coherent structures were then extracted via Dynamic Mode Decomposition (DMD). Good agreement between the calculated and experimental velocity contours, as well as between the calculated and measured pressure pulsations, confirmed the accuracy and reliability of the numerical simulation. Hankel-DMD of the pressure pulsations indicated that Mode 0 dominates the spectral content, corresponding to the characteristic frequency of the vortex rope. The vortex rope was further visualized using the Δ -criterion, and its three-dimensional (3D) behavior was examined. Standard DMD decomposed the 3D flow field in DT, yielding spatial modes with high spectral purity and their associated temporal coefficients. Modes 1(15 f n ) and Mode 3 (15 f n ) were identified as paliform vortices near the DT inlet, corresponding the runner blade configuration. Mode 2 (0.27 f n ), contributing 7.5 % to the original flow field showed strong temporal coherence with pressure pulsations and aligns with vortex rope frequency. The combined mode (Mode 0 + 2) closely resembled the original vortex rope in both morphology and dynamics. The results provide a comprehensive understanding of the formation, spectral characteristics, and dynamics of coherent structures in the draft tube vortex rope.