Scaling Laws in Turbulence - A Theoretical Approach Using Lie-Point Symmetries

In the present work, scaling laws for special turbulent flow phenomena are investigated using a mathematical method based on Lie-point symmetries. Moreover, the theoretical results are compared to available DNS data. At first, a set of governing partial differential equations (PDEs), here the multi-point correlation equations, is introduced, describing a turbulent flow of a Newtonian fluid with constant density. Lie-point symmetries, which represent transformations of functions and variables leaving the form of a differential equation unchanged, are determined for this set of differential equations. After stating general conditions for these symmetries, the classical symmetries originating from the Navier-Stokes equations are derived. Then, it is shown that even more symmetries exist for the multi-point correlation equations. If the symmetries are known, a mathematical algorithm can be applied in order to derive special solutions for the mean velocity and the components of the Reynolds stress tensor, which represent the desired scaling laws. This procedure is performed for homogeneous turbulence, channel flows at the near-wall region and the core region, boundary layers, channel flows with transpiration and rotating channel flows, where the rotational axis can lie in the spanwise or the wall-normal direction. The results are compared to solutions obtained by other theoretical approaches as well as to available numerical data. For all studied cases, a convincing agreement with the data can be obtained