Proper invariant turbulence modelling within one-point statistics

Abstract

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.