Consequences of Symmetries on the Analysis and Construction of Turbulence Models

Since they represent fundamental physical properties in turbulence (conservation laws, wall laws, Kolmogorov energy spectrum, ...), symmetries are used to analyse common turbulence models. A class of symmetry preserving turbulence models is proposed. This class is refined such that the models respect the second law of thermodynamics. Finally, an example of model belonging to the class is numerically tested.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA

An analytical solution method for the unsteady, unbounded, incompressible three-dimensional Navier-Stokes equations in Cartesian coordinates using coordinate axis symmetry degeneracy

Analytical solutions are developed for the unsteady Navier-Stokes equations for incompressible fluids in unbounded flow systems with external, time-dependent driving pressure gradients using the degeneracy of the (1 1 1) axis to reduce the inherent non-linearity of the coupled partial differential equations, which is normally performed with boundary conditions. These solutions are then extended to all directions through rotation of the reference axis, yielding a general solution set. While the solutions are self-consistent and developed from a physical understanding of flow systems, they have not been proven unique or applied to experimental data

Weak Transversality and Partially Invariant Solutions

New exact solutions are obtained for several nonlinear physical equations, namely the Navier-Stokes and Euler systems, an isentropic compressible fluid system and a vector nonlinear Schroedinger equation. The solution methods make use of the symmetry group of the system in situations when the standard Lie method of symmetry reduction is not applicable.Comment: 23 pages, preprint CRM-284

Space-time decay of Navier-Stokes flows invariant under rotations

We show that the solutions to the non-stationary Navier-Stokes equations in $R^d$, $d=2,3$ which are left invariant under the action of discrete subgroups of the orthogonal group $O(d)$ decay much faster as $|x|\to\infty$ or $t \to\infty$ than in the generic case and we compute, for each subgroup, the precise decay rates in space-time of the velocity field