Uncertainty transmission of fluid data upon proper orthogonal decompositions

Abstract

Proper orthogonal decomposition (POD) serves as a principal approach for modal analysis and reduced-order modeling of complex flows. The method works robustly with most types of fluid data and is fundamentally trusted. While, in reality, one has to discern the input spatiotemporal data as passively contaminated globally or locally. To understand this problem, we formulate the relation for uncertainty transmission from input data to individual POD modes. We incorporate a statistical model of data contamination, which can be independently established based on experimental measurements or credible experiences. The contamination is not necessarily a Gaussian white noise, but a structural or gusty modification of the data. Through case studies, we observe a general decaying trend of uncertainty toward higher modes. The uncertainty originates from twofold: self-correlation and cross correlation of the contamination terms, where the latter could become less influential, given the narrow correlation width measured in experiments. Mathematically, the self-correlation is determined by the inner product of the data snapshot and the mode itself. Therefore, the similarity between the input data and the resulting POD modes becomes a critical and intuitive indicator when quantifying the uncertainty. A scaling law is shown to be applicable for self-correlation that promotes fast quantification on sparse grids.