As part of a larger effort on data-driven turbulence modeling, this paper investigates machine learning models in their capability to reconstruct the functional forms of spatially distributed quantities extracted from high fidelity simulation and experimental data. Such datasets typically involve very high dimensional feature spaces with sparsely populated and noisy data. A new multiscale Gaussian process regression technique is described and is compared to 'conventional' Gaussian process regression and artificial neural networks. All these techniques are applied to the reconstruction of functions arising from Bayesian inference applied to turbulent channel flow and bypass transition. The efficiency, accuracy and effectiveness of each learning algorithm as well as factors that influence their output is assessed. The results highlight the potential of machine learning as an enabling tool in data-driven turbulence modeling.

While several innovative ideas for turbulence modeling have been proposed over the past three decades, it can be argued that the improvement of modeling accuracy in complex flows has not been consistent or significant. Our view is that experimental and high-fidelity data, combined with, and informed by knowledge of the physical processes could be an essential link to improve turbulence closures. In this work, we report on a data-driven paradigm for turbulence model augmentation, which we refer to as Field Inversion and Machine Learning (FIML). This approach uses inverse modeling to extract the spatial distribution of model discrepancies, and machine learning to transform the discrepancy information from a large number of inverse problems into corrective model forms. This reconstructed discrepancy is embedded within a predictive solver. To further elucidate the methodology, a tutorial is presented on the use of the framework in model development with the goal of addressing two different types of model discrepancy-in time scales and Reynolds stress anisotropy. In problems involving flow separation, the inference process is shown to assimilate sparse data while improving the solution over the entire computational domain. When applied in a predictive setting to turbulent flows over airfoils, the data-augmented model is shown to provide significant improvement in predicting surface and integral quantities, including in airfoil shapes and flow conditions that were not part of the training set.