Flow reconstruction using Bayesian inference with model reduction techniques

RANS equations are nowadays widely used in industry because of their affordability in terms of computational costs. They reached a high level of complexity, as they involve systems of non-linear partial differential equations. However they still lack of generality as they are based on closure coefficients determined from fundamental real flow cases. Their accuracy drops when dealing with separating turbulent flows. There is a specific class of flows that separate after encountering a geometry-induced adverse pressure gradient. Periodic hill flows are viewed as the benchmark case of those, presenting typical gross features of this class of flows. The separation point is viewed as one of the characteristic features and its prediction is crucial for a fair downstream flow development representation. In literature it was found that different turbulence models produced different flow solutions, as both separation and reattachment points were badly reproduced. The aim of the project is therefore to reconstruct the solution under uncertainty in separation location and turbulence closure coefficients. A statistical calibration of the uncertain parameters is performed using Bayesian inference; then Markov Chain Monte Carlo Method (MCMC) is adopted to explore the a posteriori information. The separation location is controlled using a step, that, located in the area of adverse pressure gradient, forces the flow to separate. The geometry parameters of the step are considered as the uncertain parameters to be calibrated. The effect of separation forcing on the flow solution is studied when adopting different turbulence models. A strong coupling between the step influence on flow solution and closure model adopted has been found. Because of this, Launder-Sharma k-epsilon model was employed for the rest of the work. The calibration of turbulence model coefficients have been performed starting from existing works. After providing some validated data relative to a particular location of the flow field and performing parameters calibration, the entire field is reconstructed through posterior predictive distribution. In general it was found that the results of the calibration depend strictly on the location where the inverse problem is performed. The calibrated value are able to reproduce the flow solution at the inverse problem location but unable to accurately predict the solution at different locations. The calibration of the closure coefficients only, resulted in a fair prediction of the reattachment point, but a bad representation of the separation point. The inclusion of the step resulted in a slightly better representation of the flow when moving away from the inverse problem location, with particular reference to the vertical velocity profile. This proved that the influence of turbulence closure coefficients is predominant. The analysis is conducted focusing on the velocity profiles. In fact the eventual aim is to start from planar PIV data and post process them for a three-dimensional flow field reconstruction. Because of a future application in real three dimensional cases and the possible high-dimensionality of the uncertain parameter space, a mathematical tool to be used during MCMC iterations is developed. Instead of calling the CFD solver at each MCMC iteration, the flow solution is computed \textit{offline} through model reduction techniques. POD + I and Isomap + I have been tested for this purpose; the first one being used for linear spaces and the second for the non-linear ones. At the end Isomap + I was also used for examining the intrinsic metric of the flow solution space when varying the turbulence closure coefficients and the step parameters. The model reduction techniques seemed to work well and this allowed to perform the calibration at cheap computational costs.