# New Research

2006/10/02

# The skeleton of fluid turbulence

## — approach by unstable periodic motion to the problem of turbulence —

### Genta Kawahara, Professor

The states of fluid motion are generally classified into laminar and turbulent flow. The turbulent flow exhibits spatiotemporally chaotic behaviour and its details repeat neither in space nor in time. Most of the fluid motion seen in daily life, such as water flowing out of a fully-turned-on tap and vortical smoke emitted from a chimney, is turbulent.

The problem of turbulence is known as one of the great unsolved problems in physics, and its elucidation should be very significant both in science and in engineering. Difficulty in understanding turbulence might arise from the fact that we do not have any simple spatiotemporal description of complex and chaotic behaviour of turbulent flows.

We have recently discovered unstable periodic motion which is expected to be useful for resolving this problem of finding a relatively simple description. Unstable periodic motion exhibits exact recurrence; however, what we observe in reality is turbulence instead of periodic motion because of its instability. We have solved the Navier-Stokes equation, i.e. the equation of motion of fluid, numerically by an iterative method, and succeeded in obtaining a solution for unstable periodic motion embedded in turbulence.

Time evolution of spatial structures and velocity distribution are shown in the figure for periodic motion found in the flow between two parallel plates moving in the opposite directions at a constant speed (i.e., plane Couette flow) as an example of the unstable periodic motion. The corresponding data of turbulent flow are also shown for comparison. It can be seen that the spatial structures of the periodic motion resemble remarkably those for the turbulent flow at the corresponding phases. Furthermore, the mean and RMS velocities of the periodic motion are in excellent agreement with those for the turbulent flow.

The turbulent flow turns out to exhibit recurrent behaviour relevant to the periodic motion, although it has apparent incoherence. We can say that the unstable periodic motion is \\\"the skeleton of turbulence\\\" in the sense that it represents well spatiotemporal structures and statistical properties of turbulence in spite of its much simpler behaviour.

We are now working on elucidation and control of turbulent flows with the aid of the skeleton of turbulence.