Development of novel Reynolds-averaged Navier-Stokes turbulence models based on Lie symmetry constraints

In the present work, the problem of RANS (Reynolds-Averaged Navier–Stokes) turbulence modeling is investigated from a novel angle by considering recently discovered constraints arising from Lie symmetry analysis. In this context, symmetries are defined as variable transformations that leave invariant a given equation. For equations describing physical phenomena,it is usually observed that their symmetries correspond to physical principles encoded in the equations. The key idea behind using symmetry methods for modeling tasks is that the physical principles encoded in an exact equation should also be present in a model for these equations. Lie symmetry theory establishes a mathematical framework to formalize this notion. The symmetries that govern turbulence fall into two main categories: Classical symmetries,which are present in the Navier–Stokes equations as well as in all statistical descriptions of turbulence, and statistical symmetries, which are found exclusively in statistical descriptions of turbulence and have no counterpart in the unaveraged Navier–Stokes equations. Even though the explicit use of symmetry methods in turbulence modeling is not yet prevalent, many well-established constraints imposed on turbulence models to prevent physically unreasonable behavior actually stem from symmetry arguments. This has led to a situation where the constraints implied by classical symmetries, which correspond to fundamental principles found throughout classical mechanics, have generally been taken into account when constructing turbulence models since the 1970s. Roughly speaking, two-equation eddy viscosity models are the simplest class of models to fulfill all of them. Statistical symmetries, on the other hand, are connected to special properties of turbulent statistics, and are, therefore, not as intuitive as the classical symmetries. As a result, they have so far been overlooked in turbulence modeling. The main goal of the present work is to devise a turbulence model while taking these statistical symmetries into account. This task turns out to be challenging because the combined set of classical and statistical symmetries imposes considerable restrictions on the possible form of the model equations. To overcome this challenge, a formal modeling algorithm is adapted and applied to turbulence modeling. Its results hint at the necessity for auxiliary velocity-like and pressure-like variables. With these model variables, possible model skeletons, both for an eddy-viscosity type model and for a Reynolds stress model, are developed. Subsequently, these simple base models are evolved into full turbulence models by applying them to canonical flows. Due to the complexity of the resulting Reynolds stress model, the emphasis is placed on developing a modified version of the k-ε-model that fulfills the statistical symmetries. This new model is calibrated against a wide range of canonical flows, where it performs at least equally well or better than the standard k-ε-model. Furthermore, the implementation of the standard k-ω-model in the in-house DG (Discontinuous Galerkin) solver BoSSS (Bounded Support Spectral Solver) is presented. Additionally, a special-purpose solver is developed that allows efficient numerical calculations with the modified k-ε-model for simple flows. The obtained results match well with experimental data