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 the 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 numerically solved the Navier-Stokes equation, i.e. the equation of motion of fluid, by an iterative method, and succeeded in obtaining unstable periodic motion embedded in turbulence.
Time evolution of spatial structures and velocity distribution are shown in the figure for periodic motion found in 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.
Time evolution of vortical structures and temporally-averaged energy spectrum are shown in the next figure for periodic motion found in the flow in a triply-periodic box as an example of the unstable periodic motion. In the periodic motion kinetic energy is transferred from the larger scale to the smaller one through shrinking of a cluster of tubular vortices. As a consequence, the energy distributes in a wide range of length scales (wave numbers), so that the energy spectrum for the periodic motion shows excellent agreement with that for a turbulent state. It is also in good agreement with the energy spectra of homogeneous turbulence obtained from the experiment and the statistical theory.
The turbulent flow turns out to exhibit recurrent behavior 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 behavior. We are now working on elucidation and control of turbulent flows with the aid of the skeleton of turbulence.