Michael Frewer

Proper invariant turbulence modelling within one-point statistics

A new turbulence modelling approach is presented. Geometrically reformulating the averaged Navier–Stokes equations on a four-dimensional non-Riemannian manifold without changing the physical content of the theory, additional modelling restrictions which are absent in the usual Euclidean (3+1)-dimensional framework naturally emerge. The modelled equations show full form invariance for all Newtonian reference frames in that all involved quantities transform as true 4-tensors. Frame accelerations or inertial forces of any kind are universally described by the underlying four-dimensional geometry. By constructing a nonlinear eddy viscosity model within the k −ϵ family for high turbulent Reynolds numbers the new invariant modelling approach demonstrates the essential advantages over current (3+1)-dimensional modelling techniques. In particular, new invariants are gained, which allow for a universal and consistent treatment of non-stationary effects within a turbulent flow. Furthermore, by consistently introducing via a Lie-group symmetry analysis a new internal modelling variable, the mean form-invariant pressure Hessian, it will be shown that already a quadratic nonlinearity is sufficient to capture secondary flow effects, for which in current nonlinear eddy viscosity models a higher nonlinearity is needed. In all, this paper develops a new unified formalism which will naturally guide the way in physical modelling whenever reasonings are based on the general concept of invariance.

On Applying Lie-Group Symmetry Analysis To The Functional Hopf-Burgers Equation In Physical Space

The recent systematic study by Janocha et al. [Symmetry 2015, 7, 1536-1566] to determine all possible Lie-point symmetries for the functional Hopf-Burgers equation is re-examined. From a more consistent theoretical framework, however, some of the proposed symmetry transformations of the considered Hopf-Burgers equation are in fact rejected. Three out of eight proposed symmetry transformations are invalidated, while two of them should be replaced by their correct intermediate formulations, but which ultimately violate internal consistency constraints of the governing equation. It is concluded that the recently proposed symmetry analysis method for functional integro-differential equations should not be adopted when aiming at a consistent and complete approach.

An Invariant Nonlinear Eddy Viscosity Model based on a 4D Modelling Approach

Without changing the physical content the averaged Navier-Stokes equations are geometrically reformulated on a true 4D non-Riemannian space-time manifold. Its clear superiority over the usual (3+1)D Euclidean approach can be fully summarized as follows: