Invariant solutions of two models of evolution of turbulent bursts

Abstract

We consider two problems related to the b–l and b–ε models of propagation of turbulent bursts. We show that these equations admit some particular exact solutions which reduce to a finite-dimensional dynamical system. This makes it possible to describe a singular effect of finite-time extinction, and in particular, nonsymmetric solutions which do not exhibit symmetrization in the asymptotic behaviour. We show that in the multi-dimensional equation related to the b–l model, the nonsymmetric extinction behaviour is governed by the first-order equation. For the b–ε model with α=β=1 and γ<1, using such particular solutions, we establish that the ω-limit set of all the rescaled extinction orbits is essentially infinite-dimensional.