Abstract
Symmetries are among the most successfully employed concepts in science and mathematics. They form a cornerstone of various branches of physics, such as classical and quantum mechanics, particle physics and relativity. The governing equations of hydrodynamics generally possess wide symmetry
groups and therefore there is a great potential to exploit these symmetries so as to derive similarity solutions, conservation laws and invariants or to study the effects of symmetry breaking due to the
presence of boundaries or additional body-forces. However, to date,symmetries are often used in a non-explicit or indirect way in hydrodynamics and turbulence theory. On the other hand, there exist
powerful and general methods introduced in the field of group analysis of differential equations which, when suitably adapted, can be readily applied to the aforementioned fields. In this talk we will introduce an algorithmic method which allows associating with a given object its invariant counterpart. The object
under consideration can be, e.g., a turbulence closure model or a finite-difference discretization of a differential equation, which can then be invariantized to yield a turbulence model or a finite-difference
discretization that is invariant under the same Lie point symmetry group as admitted by the original governing equations of hydrodynamics. This method can therefore be used to correct artificial symmetry breaking due to non-appropriately designed turbulence models. As an example it is shown that classical hyperdiffusion as used in two-dimensional (decaying) turbulence simulations violates the symmetries of the incompressible Euler equations. Invariantization of these hyperdiffusion terms yields symmetry-preserving but nonlinear diffusion-like terms. Using the notion of differential invariants it is
demonstrated that the invariantized hyperdiffusion models can be modified with quite some flexibility while still preserving their desired invariance characteristics. First numerical tests show that the invariant hyperdiffusion schemes which can be obtained by this method might be able to reproduce the -3 slope of the energy spectrum in the enstrophy inertial range.
