Dynamical system approach and attracting manifolds in $K$-$\varepsilon$ model of turbulent jet

Abstract

We consider the $K$-$\varepsilon$ model describing an expansion of a free turbulent jet. Due to the nonlinear nature of turbulent diffusion the turbulent area has a sharp boundary. We seek solutions for the energy, dissipation and momentum as power series in spatial coordinate across the jet with time-dependent coefficients. The coefficients obey a dynamical system with clearly identifiable slow and fast variables. The system is not in a standard form, which excludes rigorous methods of analysis such as centre manifold methods. We put forward a hypothesis that there exists an attracting invariant manifold for trajectories based on a few slow variables. The hypothesis is supported numerically.