Chaotic Properties of a Turbulent Isotropic Fluid

By tracking the divergence of two initially close trajectories in phase space in an Eulerian approach to forced turbulence, the relation between the maximal Lyapunov exponent $\lambda$ and the Reynolds number Re is measured using direct numerical simulations, performed on up to $2048^3$ collocation points. The Lyapunov exponent is found to solely depend on the Reynolds number with $\lambda \propto {\mathrm{Re}}^{0.53}$ and that after a transient period the divergence of trajectories grows at the same rate at all scales. Finally a linear divergence is seen that is dependent on the energy forcing rate. Links are made with other chaotic systems.

Figure 1

The main plot shows Re against $\lambda T_0$ and the fit $0.066{\mathrm{Re}}^{0.53}$ as a solid black line. Errors for the higher wave numbers are comparable to the size of the points and are not included for clarity. The lower wave numbers have larger error. A line of ${\mathrm{Re}}^{0.5}$ fit to the data is shown in dashed red (gray). The inset shows $\lambda \tau$ against Re for the same data.

Figure 2

$E_d(k)$ in black at an intermediate time for a simulation with $\mathrm{Re}\approx 2500$ on box size $1024^3$; the main plot is logarithmic and has a dashed red (gray) line showing $k^3$, while the inset is semilogarithmic with a dashed red (gray) line showing an exponential slope.

Figure 3

$|S|$ vs Re with fit $15.1*{\mathrm{Re}}^{-0.91}$, where high-$k$ behavior of $E_d(k)$ is approximated as $\exp (-Sk)$.

Figure 4

The growth of $E_d(k)/⟨E(k)⟩$ for selected wave numbers with time for a simulation with $\mathrm{Re}\approx 2500$ and box size $1024^3$. The wave number increases upwards, the plotted wave numbers are $k=1$, 2, 5, 20, 100, in turn these are represented by crosses, empty squares, solid circles, solid squares, and empty circles. The red (gray) line follows wave number $k=20$ and shows the $t^2$ dependence for early times. The perturbation was performed at low $k$, at all wave numbers between 0 and 2.5.

Figure 5

The main plot shows $\epsilon$ against $d{E}_d(t)/dt$ at late times before saturation, with error shown on measured slope and fit $1.12\epsilon$. The inset shows $E_d$ for a run with $\mathrm{Re}\approx 800$ on the left and $\mathrm{Re}\approx 130$ on the right in black with dashed red (gray) line with slope $\epsilon$.