The dynamics of the k–ϵ mix model toward its self-similar Rayleigh–Taylor solution

Abstract

We propose an analysis of the non-linear system of partial differential equations for the k–ϵ model expressing the evolution of a turbulent mixing zone induced by the Rayleigh–Taylor instability. The method developed in this work is based on dynamical system theory. Our objective is to prove the global stability of the self-similar solution and at the same time to investigate the dynamics of transient phases. In fact, it is possible to show the existence of a central manifold allowing to reduce the dimension of the problem to a set of two ordinary differential equations.We establish that this simplified non-linear system globally converges toward a fixed point representing the self-similar solution by application of the Poincaré–Bendixson theorem. In addition, we shed light on the existence of a second fixed point which influences the trajectories in the phase space and leads to a non-physical enhanced growth rate in some cases explicitly detailed.

Keywords:

• turbulence mix model
• self-similar Rayleigh–Taylor regime
• central manifold
• global convergence
• reduced dynamical system
• transient phase

Acknowledgements

We wish to acknowledge J.-M. Ghidaglia and A. Llor for many interesting discussions, and O. Soulard and P. Pailhoriès for different suggestions.