Uncertainty Quantification in Flows with Discontinuities: A Probabilistic Approach on Nonlinear Manifolds

Abstract

Uncertainty quantification (UQ) in transonic and supersonic flows is difficult due to the presence of strong nonlinearities and discontinuities. This challenge is further exacerbated by limited training data and the high cost of computational fluid dynamics simulations. This study presents a non-intrusive, nonlinear UQ method called proISOMAP to efficiently propagate uncertainty in problems with high-dimensionality and shockwaves. The objective is to identify a low-dimensional manifold that best captures the dynamics of the high-dimensional data and nonlinear features of the problem. This manifold is then parameterized by mapping the variations in the uncertain inputs to changes in the projected data in the latent space. Specifically, proISOMAP combines a data-driven global manifold learning procedure with an adaptive polynomial chaos expansion (PCE) to develop a probabilistic approach on manifolds for rapid UQ. The performance of the proposed methodology is compared to state-of-the-art linear and local manifold learning approaches for UQ on examples using wedges and nozzles under uncertain geometric and flow conditions. The accuracy and robustness of the method is assessed as polynomial chaos order and the number of training samples are varied. Several scalar and field-level error metrics are introduced to measure both global and local predictive performance.