Periodic motion representing isotropic turbulence

Temporally periodic solutions are extracted numerically from forced box turbulence with high symmetry. Since they are unstable to small perturbations, they are not found by forward integration but can be captured by Newton-Raphson iterations. Several periodic flows of various periods are identified for the micro-scale Reynolds number $R_{\lambda}$ between $50$ and $67$. The statistical properties of these periodic flows are compared with those of turbulent flow. It is found that the one with the longest period, which is two to three times the large-eddy-turnover time of turbulence, exhibits the same behaviour quantitatively as turbulent flow. In particular, we compare the energy spectrum, the Reynolds number dependence of the energy-dissipation rate, the pattern of the energy-cascade process, and the magnitude of the largest Lyapunov exponent. This periodic motion consists of high-activity and low-activity periods, which turbulence approaches, more often around its low-activity part, at the rate of once over a few eddy-turnover times. With reference to this periodic motion the Kaplan-York dimension and the Kolmogorov-Sinai entropy of the turbulence with high symmetry are estimated at $R_{\lambda}=67$ to be $19.7$ and $0.992$ respectively. The significance of such periodic solutions, embedded in turbulence, for turbulence analysis is discussed.