A Generic Self-Sustaining Process in Shear Flows

Fabian Waleffe

This nonlinear, three-dimensional Self-Sustaining Process appears to be a generic mechanism in shear flows. The mechanism has three main elements as depicted in the figure above: This process leads to self-sustained 3D traveling waves that consists of wavy streaks flanked by staggered, counter-rotating, quasi-streamwise vortices in both plane Poiseuille flow and plane Couette flow with no-slip as well as free-slip boundary conditions. This characteristic structure of the traveling waves is very similar to the coherent structures revealed by pattern eduction studies of turbulent flows, hence we refer to the traveling wave solutions of the Navier-Stokes equations as Exact Coherent Structures .

The SSP also leads to time-periodic solutions as first revealed by Waleffe and Kim (Turbulent Shear Flows conference, Munich 1991; see below), Hamilton, Kim and Waleffe, Journal of Fluid Mechanics 287 , 317-348 (March 1995) and more accurately by Kawahara and Kida, Journal of Fluid Mechanics 449 , 291-300 (2001). These authors improved on our earlier work by iterating toward an "exact" (numerically speaking) unstable periodic solution of the Navier-Stokes equations that is near the flow we had obtained by successive "squeezing" of a turbulent flow, i.e. tracking a self-sustained turbulent flow to smaller and smaller domains, and thus Reynolds numbers. Our motivation was to reveal the importance and relevance of the SSP. (Contrary to what is often written, the SSP was not discovered in Hamilton, Kim and Waleffe. That work was in fact aimed at verifying the SSP proposed earlier [W90] and demonstrate that it was the key process sustaining shear turbulence.) Squeezing the turbulent flow reduces the phase space available to the flow and eliminates some of the "non-essential" instabilities to reveal the basic process sustaining shear turbulence. That squeezing technique was an adaptation of the "minimal channel idea" due to Jimenez and Moin (JFM 1991). We focused on plane Couette flow because this flow is known to be linearly stable and our understanding of the SSP was that the process would operate within a single shear layer, so we expected it to occupy the full channel at low Reynolds number in plane Couette flow. In Poiseuille, we expected two copies of the process occuring in both half of the channel.

The SSP approach has been used by Faist and Eckhardt, PRL 91, 224502 (2003) and Wedin and Kerswell, JFM 508:333-371 (2004) to discover analogous traveling wave solutions in pipe flow. Hof, van Doorne, Nieuwstadt and Westerweel, at Delft University, were able to make direct observations of these traveling waves in their pipe flow experiments, see Science 305, 1594-1598, Sept 10, 2004 .