Reduction of non-linearity in turbulent flows — further lines of research and their use in the development of the theory of turbulence

Abstract

This is a subject that was not actually covered explicitly in the discussions of the last three days. I think I should explain first what we mean by reduction of non-linearity because there are different interpretations of this term. If we adopt a dynamical systems viewpoint, then a turbulent flow can be thought of in terms of a trajectory in the function space of all solenoidal vector fields satisfying certain weak constraints (e.g. boudedness of |u|). The fixed points in this function space are then fields ^uE(x) that are steady solutions of the Euler equations

$$ \frac{{\partial u}} {{\partial t}} = {\text{u x }}\omega - \nabla P,{\text{ }}\nabla \cdot {\text{ u}} = 0 $$

((1))

where ω= curl u, P =p/< + (1/2) ^u2. By solving the Poisson equation ∇²P=∇ ·(u x ω) for P, and substitutinig back in (1), this may equally be written

$$ \frac{{\partial u}} {{\partial t}} = (u{\text{ x }}\omega {\text{)}}_S $$

((2))

where the suffix S represents solenoidal projection. Just as Beltrami flows satisfy the condition u x ω ≡ 0, so Euler flows satisfy the weaker conditon (_{u x ω)S} ≡ 0.