Dina Razafindralandy

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.

Time integration algorithm based on divergent series resummation, for ordinary and partial differential equations

Borels technique of divergent series resummation is transformed into a numerical code and used as a time integration scheme. It is applied to the resolution of regular and singular problems arising in fluid mechanics. Its efficiency is compared to those of classical discretization schemes.

Symmetry invariant subgrid models

Navier–Stokes equations have fundamental properties such as the invariance under some transformations, called symmetries, which play an important role in the description of the physics of the equations (conservation laws, wall laws, . . . ). It is essential that turbulent models respect these properties. Unfortunately, the analysis reveals that it is not the case of most of LES models. A new way of deriving a class of symmetry consistent models is then proposed. This class is refined such that the models also conform to the second law of thermodynamics. Finally, a simple model of the class is numerically tested to the configuration of a ventilated room. It gives better results than those provided by Smagorinsky and dynamic models.

The Symmetry Group of the Non-Isothermal Navier-Stokes Equations and Turbulence Modelling

In this work, the non-isothermal Navier–Stokes equations are studied from the group theory point of view. The symmetry group of the equations is presented and discussed. Some standard turbulence models are analyzed with the symmetries of the equations. A class of turbulence models which preserve the physical properties contained in the symmetry group is built. The proposed turbulence models are applied to an illustrative example of natural convection in a differentially heated cavity, and the results are presented.

A review of some geometric integrators

Some of the most important geometric integrators for both ordinary and partial differential equations are reviewed and illustrated with examples in mechanics. The class of Hamiltonian differential systems is recalled and its symplectic structure is highlighted. The associated natural geometric integrators, known as symplectic integrators, are then presented. In particular, their ability to numerically reproduce first integrals with a bounded error over a long time interval is shown. The extension to partial differential Hamiltonian systems and to multisymplectic integrators is presented afterwards. Next, the class of Lagrangian systems is described. It is highlighted that the variational structure carries both the dynamics (Euler–Lagrange equations) and the conservation laws (Nœther’s theorem). Integrators preserving the variational structure are constructed by mimicking the calculus of variation at the discrete level. We show that this approach leads to numerical schemes which preserve exactly the energy of the system. After that, the Lie group of local symmetries of partial differential equations is recalled. A construction of Lie-symmetry-preserving numerical scheme is then exposed. This is done via the moving frame method. Applications to Burgers equation are shown. The last part is devoted to the Discrete Exterior Calculus, which is a structure-preserving integrator based on differential geometry and exterior calculus. The efficiency of the approach is demonstrated on fluid flow problems with a passive scalar advection.

Numerical divergent series resummation in fluid flow simulation

The perturbation theory has proved to be an efficient tool for the numerical resolution of non-linear problems in mechanics. However, it is not suitable for singular problems, for which the series solution is divergent. We propose to use the Borel-Laplace series resummation method for the resolution of such a problem. The resulting algorithm is applied to some model problems in fluid mechanics.