Modal Analysis and Vortex Trajectory Description for Tip Leakage Flow of a Transonic Turbine Cascade

Tip leakage vortex (TLV), which develops from the clearance between the turbine blade and casing, has been studied for decades. Nevertheless, some associated phenomena, such as its unsteady behaviors, are still not well understood. In the present work, an unsteady simulation of a transonic turbine cascade was conducted by using a validated unsteady Reynolds averaged Navier-Stokes (URANS) technique with the k-omega shear stress transport (SST) turbulence model. Typical three-dimensional vortical topology in the tip region of this transonic turbine blade was depicted based on the vortex and shock wave identification. Afterwards, quantitative descriptions of TLV transient parameters, including core position, radius, intensity, wandering motion amplitude and their statistical analysis were also provided via an ellipse fitting method. Combined with the turbulent parameters in the tip region, it is recognized that the breakdown of TLV does not occur upstream of the trailing edge, and the TLV wandering, especially the spanwise motion is a dominant unsteady feature as migrating downstream. To mathematically extract underlying flow features of tip leakage flow (TLF), two data-driven modal analysis techniques, namely proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD), are presented to complement one another to reveal underlying flow feature. Observation of modes distribution allowed qualitative identification of shockwaves, vortical cluster and corresponding transient interaction. Results of POD show that the dominant unsteady structures in the tip region exhibit various morphology with moving downstream. In the front part near the leading edge, the oscillation of separation bubble and bifurcation of passage vortex paly a dominant role; while in the middle part of the tip region, the corresponding factors are the wandering of TLV and unsteady interaction between shock waves and TLF/TLV. In the vicinity of the trailing edge, the instability induced by the mixing of large-scale vortices serves as the main factor in the context of flow unsteadiness. Both the POD and DMD methods can decompose the dominant frequency of TLV evolution and its harmonic frequencies; however, the DMD method presents a superiority in segregating the high-frequency components and their corresponding unsteady structures.