Local linearity, coherent structures, and scale-to-scale coupling in turbulent flow

Turbulent and other nonlinear flows are highly complex and time dependent, but are not fully random. To capture this spatiotemporal coherence, we introduce the idea of a linear neighborhood, defined as a region in an arbitrary flow field where the velocity gradient varies slowly in space over a finite time. Thus, by definition, the flow in a linear neighborhood can be approximated arbitrarily well by only a subset of the fluid-element trajectories inside it. This slow spatiotemporal variation also allows short-time prediction of the flow. We demonstrate that these linear neighborhoods are computable in real data using experimental measurements from a quasi-two-dimensional turbulent flow and find support for our theoretical arguments. We also show that our kinematically defined linear neighborhoods have an additional dynamical significance, in that the scale-to-scale spectral energy flux that is a hallmark of turbulent flows behaves differently inside the neighborhoods. Our results add additional support to the conjecture that turbulent flows locally tend to transport energy and momentum in space or in scale but not both simultaneously.

Figure 1

(a) Schematic of the quantities involved in defining a linear neighborhood. (b) Space-time plot of several LNs found in the experimental data. The red curves show the projection of the LNs at the initial time $t_0$.

Figure 2

Scalar fields of the area of the LN associated with each point at the initial time. The color of each point shows the initial-time area of the LN associated with that point. Colors are on a logarithmic scale and the colormap is uniform for all panels. Data are shown for (a) $E=5$, (b) $E=10$, (c) $E=15$, and (d) $E=20$. In all cases, $T=T_L$. The full measurement domain is shown, measuring $12L_m\times 9L_m$.

Figure 3

Scalar fields of the area of the LN associated with each point at the initial time. Colors are on a logarithmic scale and the colormap is uniform for all panels (and the same as in Fig. 2). Data are shown for (a) $T=0.25T_L$, (b) $T=0.5T_L$, (c) $T=0.75T_L$, and (d) $T=T_L$; $E$ is fixed at 20 for all cases. The full measurement domain is shown, measuring $12L_m\times 9L_m$.

Figure 4

(a) Visualization of each LN present in the flow at a single instant of time, for $E=50$ and $T=T_L$. (b) Union of the LNs shown in (a). The full measurement domain is shown, measuring $12L_m\times 9L_m$.

Figure 5

(a) The FTLE field computed for an integration time of $T=3T_L$. (b) Okubo-Weiss parameter. Both fields are shown for the same data as in Figs. 2, 3, 4.

Figure 6

Evolution of the angle ${\mathrm{\Theta }}_{s\tau }^{\left(r\right)}$ between the turbulent stress and strain rate for trajectories that begin inside LNs. The curves have been ensemble averaged over many LNs. Each LN was computed for $T=T_L$, as indicated by the vertical dashed lines. The horizontal dashed lines show ${\mathrm{\Theta }}_{s\tau }^{\left(r\right)}=\pi /4$, for which the energy flux vanishes and which separates forward from the inverse energy transfer. In each panel, data are shown for four filter scales in the inverse energy cascade range: $r=3L$ (red), $r=3.5L$ (green), $r=4L$ (blue), and $r=4.5L$ (black). Each panel shows computations for a different value of $E$, with (a) $E=30$, (b) $E=50$, (c) $E=70$, and (d) $E=90$.

Figure 7

Evolution of ${\mathrm{\Theta }}_{s\tau }^{\left(r\right)}$, as in Fig. 6, for fixed $E=50$ but for (a) $T=0.5T_L$ and (b) $T=0.75T_L$. As in Fig. 6, the horizontal dashed lines show ${\mathrm{\Theta }}_{s\tau }^{\left(r\right)}=\pi /4$ and the vertical dashed lines show the time for which the LNs were computed.