A Reduced Order Kalman Filter for Computational Fluid-Dynamics Applications

In the last decade, the importance of numerical simulations for the analysis of complex engineering systems, such as thermo-fluid dynamics in nuclear reactors, has grown exponentially. In spite of the large experimental databases available for validation of mathematical models, in order to identify the most suitable one for the system under investigation, the inverse integration of such data into the CFD model is nowadays an ongoing challenge. In addition, such integration could tackle the problem of propagation of epistemic uncertainties, both in the numerical model and in the experimental data. In this framework, the data-assimilation method allows for the dynamic incorporation of observations within the computational model. Perhaps the most famous among these methods, due to its simple implementation and yet robust nature, is the Kalman filter. Although this approach has found success in fields such as weather forecast and geoscience, its application in Computational Fluid-Dynamics (CFD) is still in its first stages. In this setting, a new algorithm based on the integration between the segregated approach, which is the most common method adopted by CFD applications for the solution of the incompressible Navier-Stokes equations, and a Kalman filter modified for fluid-dynamics problems, while preserving mass conservation of the solution, has already been developed and tested in a previous work. Whereas such method is able to robustly integrate experimental data within the numerical model, its computational cost increases with model complexity. In particular, in high-fidelity realistic scenarios the error covariance matrix for the model, which represents the uncertainties associated with it, becomes dense, thus affecting the efficiency and computational cost of the method. For this reason, due to the promised reduction of computational requirements recently investigated, which combines model reduction and data-assimilation, in this work a combination of reduced order model and mass-conservative Kalman filter within a segregated approach for CFD analysis is proposed. The novelty lies in the peculiar formulation of the Kalman filter and how to construct a low-dimensional manifold to approximate, with sufficient accuracy, the high fidelity model. With respect to literature, in which the full-order Kalman filter is applied to a reduced model, the reduction is performed directly on the integrated model in order to obtain a reduced-order Kalman filter already optimised for fluid-dynamics applications. In order to verify the capabilities of this approach, this reduced-order algorithm has been tested against the lid-driven cavity test case.