Turbulent flows are commonly encountered in scientific research or engineering applications and need simulations to be resolved. The Navier-Stokes equations govern the simulations of turbulent flows. One of the most common ways to solve the Navier-Stokes equations is to analyse the Reynolds averaged form (RANS). Next to the increasing development and use of Large Eddy Simulations (LES), the RANS equations are expected to be a necessary tool for the foreseeable future in turbulence modelling. These RANS equations need a closure term to be solved. The field of research to close the RANS equations has made most of its progress during the second half of the twentieth century, but is nowadays still an active field of research. More recent efforts have been taking a data-driven approach using different machine learning frameworks.
Uncertainty quantification is an important aspect for machine learning models, as these methods can quickly become inaccurate when a test case is outside the range of the initial training data. Therefore, the aim of this research was to develop a data-driven turbulence model for the RANS equations that is able to predict and quantify the uncertainty of the model output.
Uncertainty is introduced in the anisotropic Reynolds stress tensor, which is predicted using machine learning. Two methods were developed to capture the uncertainty of the data-driven turbulence model: (1) Jackknife methods were applied to a tensor based random forest (TBRF) and (2) a Bayesian additive regression tree model (BART-TB) was developed.
Both methods converged locally to equal levels of uncertainty for different flow cases, however the Jackknife methods could not provide spatially homogeneous samples. Therefore, the BART-TB model was selected as the superior model. After reformulation, the BART-TB model was able to give more accurate predictions of the anisotropic Reynolds stress tensor. The results of the BART-TB model show great resemblance to Direct Numerical Simulations (DNS)/LES data for square duct and backwards facing step flows. Furthermore, this model shows increasing levels of uncertainty in the solution, when square duct training data is extrapolated to test cases of higher aspect ratio ducts.
Overall, the introduction of uncertainty in the anisotropic Reynolds stress fields mainly provides an additional tool to estimate the accuracy of machine learning predictions in case of the BART-TB model, however, the mean predicted solution does not seem to be a direct improvement over the TBRF. Therefore, this method is mainly beneficial to apply, when a test case is at the limits of extrapolation, outside the scope of the training data, as this marks the point that any data-driven turbulence model can become highly inaccurate.