Combining deterministic and statistical approaches to compute two-dimensional turbulent flows with walls

Abstract

We are searching for the best approximation to compute fully-developed two-dimensional turbulent flows. Fluid mechanics is governed by the Navier-Stokes equations, which are entirely deterministic. Fully-developed turbulence corresponds to very large Reynolds number flows (for which the micro-scale Reynolds number Re is larger than 104) and is the regime where the nonlinear advective term of Navier-Stokes equations strongly dominates the linear dissipative term. In this limit, the solutions to the Navier-Stokes equations are highly chaotic and we are unable to integrate them. Therefore, in order to compute fully-developed turbulent flows we need to combine a deterministic numerical integration with a statistical model. In this paper we propose a possible solution to this problem, based on the wavelet representation. ‘Although this may seem a paradox, all exact science is dominated by the idea of approximation’ (Bertrand Russell).