THEMED PAPERS ON ARTIFICIAL INTELLIGENCE IN FLUID MECHANICS
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Data-driven bayesian inference of turbulence model closure coefficients incorporating epistemic uncertainty
Daigo Maruyama1, Philipp Bekemeyer1, Stefan G?rtz1, Simon Coggon2, Sanjiv Sharma2
Daigo Maruyama , Philipp Bekemeyer , Stefan G?rtz
1. German Aerospace Center (DLR), Institute of Aerodynamics and Flow Technology
Lilienthalplatz 7, 38108 Braunschweig, Germany; 2. Airbus Operations Limited Pegasus House, Aerospace Ave, Filton, Bristol BS34 7PA
Abstract In computational fluid dynamics (CFD), various turbulence models are widely used to numerically investigate turbulent flows in academic and industrial applications. The closure coefficients in the models are typically seen as universal constants that were calibrated to match experimental results of representative, simplified flow experiments. In this article, we introduce a framework for statistical inference of the closure coefficients using machine learning methods. The objective of this framework is to quantify the epistemic uncertainty associated with the closure model by using experimental data via Bayesian statistics. The framework is tailored towards cases for which a limited amount of experimental data is available. It consists of two components. First, by treating all latent variables (non-observed variables) in the model as stochastic variables, all sources of uncertainty of the probabilistic closure model are quantified by a fully Bayesian approach. The probabilistic model is defined to consist of the closure coefficients as parameters and other parameters incorporating noise. Then, the uncertainty associated with the closure coefficients is extracted from the overall uncertainty by considering the noise being zero. The overall uncertainty is rigorously evaluated by using Markov-Chain Monte Carlo sampling, which usually requires many CFD simulations. The amount of CFD simulations required for accurate statistics is significantly reduced by using surrogate models. We apply the framework to the Spalart-Allmars one-equation turbulence model. Two test cases are considered, including an industrially relevant full aircraft model at transonic flow conditions, the Airbus XRF1, for which high-quality wind-tunnel data at flight Reynolds numbers is available. Eventually, we demonstrate that epistemic uncertainties in the closure coefficients result into uncertainties in flow quantities of interest which are prominent around, and downstream, of the shock occurring over the XRF1 wing. This data-driven approach could help to enhance the predictive capabilities of CFD in terms of reliable turbulence modeling at extremes of the flight envelope if measured data is available, which is important in the context of robust design and towards virtual aircraft certification. The plentiful amount of information about the uncertainties could also assist when it comes to estimating the influence of the measured data on the inferred model coefficients. Finally, the developed framework is flexible and can be applied to different test cases and to various turbulence models.
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