Correcting Dynamical Models from Sparse Observations using Data Assimilation and Machine Learning : Application to Unsteady Turbulence Modeling

Overview & Synopsis

Turbulence modeling is an essential tool to make the numerical prediction of industrial flows affordable. However, it may suffer from deficiencies in capturing complex flow phenomena, such as separation and stall. In recent years, the use of Machine Learning to identify corrections to turbulence models has been increasingly investigated. Despite of significant progress in this research field, the sparse character of data as available for aerodynamic applications may still hinder the application of Machine Learning techniques. In this context, this PhD thesis investigates the possibilities in learning corrections to turbulence models from limited and possibly noisy data. To this end, a methodology that merges Data Assimilation, Machine Learning and the Expectation-Maximiazation Algorithm in a Bayesian formulation is exposed and assessed here. In this framework, it may be justified to alternate between Data Assimilation stages, where full flow descriptions are inferred from the limited observations, which is here achieved through ensemble Kalman Filter techniques, and Machine Learning stages, where these full states are employed to update the considered model correction. Two versions of this methodology are elaborated that account or not for residual modeling errors. The described methodology is first applied to an one-dimensional Ginzburg-Landau model. Numerical experiments indicate that taking into account residual modeling errors may favor the quality of the inferred model and its robustness with respect to the time period of the observations, while both versions of the methodology exhibit low sensitivity with respect to the spatial density of the observations. These results confirm the efficacy of the developed methodologies and highlight the benefit of accounting for modeling uncertainties in the Data Assimilation phases. The present approach is then applied to the URANS modeling of the turbulent flow past a circular cylinder at Re=3900. Pointwise observations of the velocity field are considered to learn corrective terms in the momentum equations and in the transport equation that is associated to the employed turbulence model. The application of the present methodology successfully improves the quality of the URANS results compared to the baseline model. This enhancement is shown to persist at other values of the Reynolds number for which data have been extracted. The robustness of these findings with respect to the spatial density of the data is also investigated. Overall, a promising strategy that optimally combines the strengths of Data Driven and Machine Learning techniques to infer model corrections from limited observations is presented and successfully applied to unsteady turbulence modeling in particular. The robustness of the discussed methodology with respect to the observation sparsity along with its non-intrusive character should favor its application to more complex and industrial flow configurations.