Turbulent Hele-Shaw flow: a paradigm for large-scale model extraction

Most direct numerical simulations of wall-bounded turbulence (DNS) take place in a doubly infinite plane channel, approximated as a rectangular periodic box. But a plane channel also has significant dynamics in its two homogeneous directions, on a scale typically larger than the channel's thickness, which is often overlooked. In the laminar flow this is evident in the noted Hele-Shaw cell with its governing 2D diffusion equation. It makes sense then to ask what dynamics does a turbulent plane channel, seen as a Hele-Shaw cell, exhibit; or in other words what kind of differential, local (or perhaps integral, nonlocal) equation asymptotically describes it. This question only involves the flow's behaviour in two statistically homogeneous directions, and is therefore in some sense easier to answer in an objective way than other fundamental questions concerning turbulence, yet it does not appear to have received attention. In its simpler form it can be answered theoretically, but to a much deeper extent it can be answered by extracting a large-scale linear response, or transfer function, from the DNS itself.

This presentation will display some results concerning the dynamics of turbulent Hele-Shaw flow, which exhibit a previously unexplored example of the anomalous diffusion that is sometimes predicted to be associated with turbulence. They also open the way to more advanced (non-homogeneous) applications of objective transfer-function extraction.