Low Rank Education of Cascade Loss Sensitivity to Unsteady Parameters by Proper Orthogonal Decomposition

Abstract

Abstract In this study, proper orthogonal decomposition (POD) has been applied to a large dataset describing the profile losses of low-pressure turbine (LPT) cascades, thus allowing (i) the identification of the most influencing parameters that affect the loss generation; (ii) the identification of the minimum number of requested conditions useful to educate a model with a reduced number of data. The dataset is constituted by the total pressure loss coefficient distributions in the pitchwise direction. The experiments have been conducted varying the flow Reynolds number, the reduced frequency, and the flow coefficient. Two cascades are considered: the first for tuning the procedure and identifying the number of really requested tests, and the second for the verification of the proposed model. They are characterized by the same axial chord but different pitch-to-chord ratio and different flow angles, hence two Zweifel numbers. The POD mode distributions indicate the spatial region where losses occur, the POD eigenvectors provide how such losses vary for different design conditions and the POD eigenvalues provide the rank of the approximation. Since the POD space shows an optimal basis describing the overall process with a low-rank representation (LRR), a smooth kernel is educated by means of least-squares method (LSM) on the POD eigenvectors. Particularly, only a subset of data (equal to the rank of the problem) has been used to generate the POD modes and related coefficients. Thanks to the LRR of the problem in the POD space, predictors are low-order polynomials of the independent variables (Re, f+, and ϕ). It will be shown that the smooth kernel adequately estimates the loss distribution in points that do not participate to the education. In addition, keeping the same steps for the education of the kernel on another cascade, loss distribution and magnitude are still well captured. Thus, the analysis show that the rank of the problem is much lower than the tested conditions, and consequently, a reduced number of tests are really necessary. This could be useful to reduce the number of hi-fidelity simulations or detailed experiments in the future, thus further contributing to optimize LPT blades.