Time-reversal-symmetry Breaking in Turbulence

In three-dimensional turbulent flows, the flux of energy from large to small scales breaks time symmetry. We show here that this irreversibility can be quantified by following the relative motion of several Lagrangian tracers. We find by analytical calculation, numerical analysis, and experimental observation that the existence of the energy flux implies that, at short times, two particles separate temporally slower forwards than backwards, and the difference between forward and backward dispersion grows as $t^3$. We also find the geometric deformation of material volumes, defined by four points spanning an initially regular tetrahedron, to show sensitivity to the time reversal with an effect growing linearly in $t$. We associate this with the structure of the strain rate in the flow.

Figure 1

The difference between the backward and forward mean squared relative separation, $⟨\delta \mathbf{R}(-t{)}^2-\delta \mathbf{R}(t{)}^2⟩$, compensated using Eq. (5). The symbols correspond to experiments: circles for $R_{\lambda }=690$ ($R_0/\eta =267$, 333, 400), stars for $R_{\lambda }=350$ ($R_0/\eta =152$, 182, 212), and squares for $R_{\lambda }=270$ ($R_0/\eta =95$, 114, 133). The lines correspond to DNS at $R_{\lambda }=300$ ($R_0/\eta =19$, 38, 58, 77, 92, 123).

Figure 2

Eigenvalues of the perceived rate-of-strain tensor, ${\lambda }_{0,i}{t}_0$, $(i=1,2,3)$, defined on tetrahedra with different sizes $R_0/\eta$. Open symbols are from experiments at $R_{\lambda }=690$ and 350 and filled symbols from DNS at $R_{\lambda }=300$. The solid lines are the corresponding averages for $i=1$ (top), 2 (middle), and 3 (bottom).

Figure 3

Time-asymmetry revealed from the geometry of tetrahedra. (a) The eigenvalues of the shape tensor $\mathbf{G}$, $⟨g_i(t)⟩$, divided by $R_0^2$, such that initially regular tetrads with size $R_0$ start at $g_i(0)/R_0^2=1/2$. (b) The difference in the backward and forward growth rates of the intermediate eigenvalue, $[⟨g_2(t)-g_2(-t)⟩]/[R_0^2(t/t_0)]$, behaves as $2⟨{\lambda }_{2,0}⟩t_0$ to leading order according to Eq. (3). The horizontal thin black line indicates the value of $2⟨{\lambda }_{0,2}⟩t_0\approx 0.38$ as given by Fig. 2. The experimental results ($R_{\lambda }=690$) and the DNS at $R_{\lambda }=300$, were obtained with tetrads close to regular, with a small tolerance $\mathrm{\Delta }{R}_0$ on the edge lengths, indicated in the legend. The dashed lines (DNS, $R_{\lambda }=430$) were obtained with regular tetrads. The effect of $\mathrm{\Delta }{R}_0\ne 0$ is the strongest at small $t$ and diminishes when $\mathrm{\Delta }{R}_0/R_0$ decreases.