In computational fluid mechanics, the direct numerical solution of the Navier-Stokes equations is extremely time-consuming, and can only be performed on very specific flow geometries and characteristics. To overcome this limitation, fluid mechanics develop closure models such as RANS, where the Navier-Stokes equations are averaged in time. This averaging operation gives rise to an unknown term characteristic of the flow's turbulence: the Reynolds stress tensor. Determining this tensor is crucial if the turbulence of the flow studied is to be representative of physical reality. The proposed thesis concerns the development of a methodology for quantifying uncertainties in the Reynolds tensor. Two main lines of research have been identified. The first involves modeling the spatial field of the Reynolds tensor as a Gaussian random field, where advanced methods for learning and sampling such a random field will be investigated. The second research axis concerns the development of advanced mathematical tools for the statistical description of the Reynolds tensor field. Indeed, statistics such as quantiles do not admit simple extension in dimensions greater than 1. A new notion of multivariate quantile based on optimal transport theory will be considered, as well as the development of efficient estimation algorithms.