Analysis of tip leakage flow unsteadiness in a transonic turbine cascade using data-driven modal decomposition methods

Abstract

The evolution of tip leakage flow and subsequent vortical structures is accompanied by inherent unsteadiness. This paper presents a novel characterization for the unsteady dynamics of turbine tip flow using data-driven, i.e., equation-free modal decomposition methods, which are applied to the hybrid Reynolds-averaged Navier–Stokes/large eddy simulation data at a transonic condition. By combining these techniques, the identified eigen-tuples (eigenvalues, eigenvectors, and time evolution) are well determined, and the differences between the obtained patterns (modes) are also pronounced. The snapshot proper orthogonal decomposition (POD) analysis can isolate the large-scale fluctuating structures that populate the rear part of the suction-side, which is mainly attributed to the shock-induced vortex instability. Similar to the turbulence cascade, macro-scale coherent structures that correspond to the tip leakage vortex shedding phenomena and the subsequently transitional and noisy parts closely related to the dissipation can be well derived by a quadruple reconstruction. Three dynamic mode decomposition (DMD) variants including the amplitude selecting-DMD method, the DMD with criterion method, and the sparsity promoting (SP)-DMD method are also compared in extracting dominant modes from the periodic tip flow, and the SP-DMD method which can distill modes of broadband frequencies and low dissipation is proved to be more conducive to representing and reconstructing the complex tip flow. Additional spectral-kernel-based POD (SPOD) analysis that can identify the similar primary unsteadiness frequencies as the DMD method is also encompassed in this study. Specifically, although it manifests that a physical resemblance of the pattern of pressure fluctuations to tip eddy unsteadiness can be captured by all these approaches, the behavior of small-scale vortical interaction downstream of the trailing edge can be clearly isolated with the intrinsic Karman-type vortex layer shedding process via DMD and SPOD approaches, which also demonstrates that these two techniques are more favorable to decomposing the complex tip flows into uncoupled single-frequency coherent structures compared to the conventional POD method. On this basis, resulting modes of velocity components have been accounted for verifying their contributions to the turbulent kinetic energy fields. The ensuing observations can offer a glimpse of the complex dynamics in the tip region, which also sheds light on features previously masked by conventional analysis approaches.