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:
The variables of space and time are fully independent. This implies that in any closure strategy not only space but also time derivatives have to be considered, hence not only allowing for a universal and consistent treatment of curvature effects but also for a universal and consistent treatment of nonstationary effects.
ii) Physical quantities as velocities or stresses always transform as tensors, irrespective of whether they are objective (frame-independent) or not. This is important when to model unclosed terms with non-objective quantities.
iii) Frame accelerations or inertial forces of any kind can be interpreted as a pure geometrical effect. This implies that inertial and non-inertial turbulence need not to be modelled separately anymore. A 4D turbulence model will describe non-inertial turbulence as rotation, swirling or curved surfaces equally well or equally bad as the corresponding inertial case.
iv) The special space-time structure of the 4D manifold allows for additional modelling constraints, which are absent in the usual (3+1)D geometrical formulation. For example, within the 4D manifold averaged and fluctuating velocities evolve differently, in the sense that the averaged 4-velocities evolve as pure time-like vectors, while the fluctuating 4-velocities as pure space-like vectors.