On "verifiability" of models of the motion of large eddies in turbulent flows

Abstract

If the Navier-Stokes equations are averaged by a local, spacial filter (denoted by an over-bar) the following system results: $$\nabla \cdot {\overline{u}} = 0 \ \text{and}\ {\overline{u}}_t + {\overline{u}} \cdot \nabla {\overline{u}} - Re^{-1} \ \Delta {\overline{u}} + \nabla \cdot \mathbb R(u) + \nabla {\overline{p}} = {\overline{f}},$$ where $\mathbb R(u): = {\overline{u \ u}} - {\overline{u}} \ {\overline{u}}$ denotes the Reynold's stresses. To close this system, various models for the Reynold's stresses of the form $\mathbb R(u) \cong \tilde {\mathbb R}({\overline{u}})$ are used. The desire is that the solution to the resulting system (call it $w$) be close to ${\overline{u}}$. However, models are often validated instead by calculating $u$ and explicitly checking $||\mathbb R(u) - \tilde {\mathbb R}({\overline{u}})||$. This report studies conditions under which the latter being small implies $w$ is close to ${\overline{u}}$. (we call this {\it verifiability} of a model.) Since ${\overline{u}} \rightarrow u$ as $\delta \rightarrow 0$, we also study when it can be proven that $w\rightarrow u$ as $\delta \rightarrow 0$.