A spatiotemporal theory of turbulence in terms of exact coherent structures*

The recurrent flows observed in moderate Reynolds number turbulence are shaped by close passes to unstable invariant solutions of Navier-Stokes equations. While in recent years many such solutions been computed, so far all have been confined to small computational domains.

Pipe, channel and plane flows, however, are flows on infinite spatial domains. If the Navier-Stokes equations are recast as a space-time theory, with both space and time taken to infinity, the traditional Direct Numerical Simulation codes have to be abandoned. In this theory there is no time, there is only a repertoire of admissible spatiotemporal patterns. We sketch how these are to be encoded by spatiotemporal symbol dynamics, in terms of minimal exact coherent structures. To determine these, radically different kinds of codes will have to be written, with space and time treated on equal footing.