Despite the increasing availability of high-performance computational resources, Reynolds-Averaged Navier-Stokes (RANS) closure models are projected to remain a widely used option for the prediction of turbulent flows. However, it is well known that the resulting predictions are potentially sensitive to parametric and model-form uncertainty. The former concerns imperfectly known closure coefficients, while the latter deals with the uncertainty due to assumptions made in the mathematical formulation of the model itself.
We will investigate various means in which Bayesian data assimilation can be used to combine (experimental) data with the RANS models, with the goal of obtaining of obtaining flow predictions with quantified uncertainty. As a first step we will outline the various components required for a Bayesian calibration of a single closure model, which results in a posterior probability density function of the closure coefficients. This includes a short discussion on Bayes’ theorem, likelihood specification and Markov-Chain Monte Carlo sampling.
Once the calibration is finished, a critical next step is the use of the obtained posterior distribution in a predictive setting, when no reference data is available. However, performing a calibration on a single model, using only one data set introduces a bias which might not extrapolate well to flow scenarios with a very different topology. Ensemble methods can be used to reduce the bias introduced by the choice of model and calibration scenario. We will discuss a Bayesian method able to combine multiple closure models and data sets into a single stochastic model. While we focus on turbulence models in this lecture, the presented methods are general and can in principle be applied to all simulation codes subject to uncertainty.