Sudden Relaminarization and Lifetimes in Forced Isotropic Turbulence

We demonstrate an unexpected connection between isotropic turbulence and wall-bounded shear flows. We perform direct numerical simulations of isotropic turbulence forced at large scales at moderate Reynolds numbers and observe sudden transitions from a chaotic dynamics to a spatially simple flow, analogous to the laminar state in wall bounded shear flows. We find that the survival probabilities of turbulence are exponential and the typical lifetimes increase superexponentially with the Reynolds number. Our results suggest that both isotropic turbulence and wall-bounded shear flows qualitatively share the same phase-space dynamics.

Figure 1

Time evolution of the total energy $E(t)$ and the energy content of the small scales $E^{\prime }(t)$ for $\mathrm{Re}=76.86$ normalized by the initial energy $E_0$. Time is given in units of initial large eddy turnover time $t_0=L/U$, where $U$ is the initial rms velocity and $L$ the initial integral scale. The point around $t/t_0\approx 240$ when $E^{\prime }(t)$ vanishes and the total energy becomes constant marks the onset of the self-organized state as discussed in the main text.

Figure 2

Survival probability as a function of the dimensionless time $t/t_0$ from the beginning of a simulation.

Figure 3

Reynolds number dependence of the escape rate $t_0/\tau$. The red (gray) line is a two-parameter fit of the expression $t_0/\tau (\mathrm{Re})=0.064\exp [-\exp (a+b\mathrm{Re})]$, the black line is a two-parameter fit of the expression $t_0/\tau (\mathrm{Re})=\exp [a^{\prime }-(b^{\prime }\mathrm{Re}{)}^{5.6}])$, the dash-dotted line is a fit of an exponential, and the faint dotted line is a fit of a linear dependence of $t_0/\tau$ on Reynolds number.

Figure 4

Phase portrait $E_2$ vs $E_1$ for $\mathrm{Re}=76.86$. Each point corresponds to a particular moment in time. All energies are scaled with the initial total kinetic energy $E_0$. Inset: enlargement of the turbulent region of the main graph showing that the dynamics is organized by several points in phase space suggestive of unstable exact solutions.