Unstable equilibria and invariant manifolds in quasi-two-dimensional Kolmogorov-like flow

Recent studies suggest that unstable, nonchaotic solutions of the Navier-Stokes equation may provide deep insights into fluid turbulence. In this article, we present a combined experimental and numerical study exploring the dynamical role of unstable equilibrium solutions and their invariant manifolds in a weakly turbulent, electromagnetically driven, shallow fluid layer. Identifying instants when turbulent evolution slows down, we compute 31 unstable equilibria of a realistic two-dimensional model of the flow. We establish the dynamical relevance of these unstable equilibria by showing that they are closely visited by the turbulent flow. We also establish the dynamical relevance of unstable manifolds by verifying that they are shadowed by turbulent trajectories departing from the neighborhoods of unstable equilibria over large distances in state space.

Figure 1

Low-dimensional projection of the state space (cf. Appendix pp1). Black curve is a trajectory describing the temporal evolution of a turbulent flow and red spheres represent different ECSs (equilibria in this particular case). The thick black line shows the portion of a turbulent trajectory shadowing (a portion of) the unstable manifold (blue surface) of an ECS. The flow (vorticity) fields in the physical space, corresponding to a particular point on the turbulent trajectory (black ball) and the nearest ECS, are shown on the left.

Figure 2

Schematic of the experimental apparatus. (a) Top view showing a magnet array with adjacent magnets (dashed lines) having magnetization in opposite directions (up/down/up/...). The array is placed over an aluminum plate and a container is constructed using acrylic end walls and copper electrode side walls. (b) Side view showing the cross section of the apparatus holding two immiscible fluid layers: a heavy dielectric and a lighter electrolyte. A direct current density $\mathbf{J}$ passing through the electrolyte interacts with the magnetic field $\mathbf{B}$ produced by the magnet array, exerting the Lorentz force $\mathbf{F}=\mathbf{J}\times \mathbf{B}$. Spatiotemporally resolved 2D velocity fields are obtained using particle image velocimetry by imaging the tracer particles at the electrolyte-air interface with a camera (not shown) suspended above the container.

Figure 3

A sample recurrence plot used to identify signatures of ECSs in weakly turbulent flow produced by our simulations. Low (high) values of $R(t,\tau )$ indicate that flow fields at two instants $t$ and $t+\tau$ are similar (different). The dashed line corresponds to the correlation time ${\tau }_c$.

Figure 4

State space speed $s\left(t\right)$, with symbols indicating the deep local minima. Filled symbols designate a nearby unstable equilibrium was computed when the Newton-Krylov solver was initialized using the corresponding turbulent flow field. Different filled symbols represent convergence to distinct equilibria (cf. Fig. 5). Open circles indicate the solver failed to converge to an equilibrium.

Figure 5

Equilibria (a) E01 and (b) E10 (see Table 1). The left column shows initial conditions from the numerical simulation corresponding to local minima (a) $t=0$ (black circle) and (b) $t=-10$ (black diamond) in Fig. 4. The right column shows the corresponding equilibria. The normalized distances from the initial conditions to E01 and E10 are $D_0^{\mathrm{ic}}=0.20$ and 0.61, respectively.

Figure 6

Equilibrium E20: (a) Initial condition from the experiment and (b) the corresponding solution. The normalized difference between the two flow fields is $D_0^{\mathrm{ic}}=0.33$. The colormap used here and below is the same as in Fig. 5.

Figure 7

Turbulent trajectories ($T_1$, $T_2$, and $T_3$) from numerical simulation shadowing the 2D unstable manifold (blue surface) of equilibrium E01 (red sphere). Black spheres on $T_1$, $T_2$, and $T_3$ mark minima in $s\left(t\right)$. The portion of the unstable manifold shown was constructed using a one-parameter family of trajectories. Blue ribbons and red curves represent manifold extensions and reference manifold trajectories that $T_1$, $T_2$, and $T_3$ follow. The procedure used to construct this low-dimensional projection and the definition of basis vectors ${\mathbf{c}}_i$ are described in Appendix pp1. Supplemental Material video 3 compares evolution along $T_1$ and the corresponding reference trajectory.

Figure 8

The angle $\theta$ for the manifold trajectory which is closest to the points $\mathbf{u}\left(t\right)$ on a nearby turbulent trajectory.

Figure 9

The distance between the turbulent trajectory $\mathbf{u}\left(t\right)$ and the corresponding manifold trajectory ${\mathbf{u}}_{{\theta }_c}\left(t^{\prime }\right)$ (see text).

Figure 10

Flow fields for (a) the turbulent trajectory $T_1$ in Fig. 7 at $t/{\tau }_c=3.75$ and (b) the nearest point on the reference manifold trajectory.

Figure 11

Comparison between the rates of evolution along manifold and turbulent trajectories.

Figure 12

(a) A flow field from the experiment during a close pass to (b) equilibrium E03 which has a seven-dimensional unstable manifold.

Figure 13

The dominant unstable submanifold ${\mathbf{u}}_{+}\left(t^{\prime }\right)$ (solid red) and ${\mathbf{u}}_{-}\left(t^{\prime }\right)$ (dashed red) of equilibrium E03 (red sphere). Also shown are turbulent trajectories from simulation ($T_4$ in black) and experiment ($T_5$ in green) that shadow ${\mathbf{u}}_{+}\left(t^{\prime }\right)$ and ${\mathbf{u}}_{-}\left(t^{\prime }\right)$, respectively. Details of the projection coordinates are provided in Appendix pp1.

Figure 14

Distance $D_1\left(t\right)$ from turbulent trajectories $\mathbf{u}\left(t\right)$ in simulation ($T_4$) and experiment ($T_5$) to the 1D submanifold. Solid [dashed] curves correspond to distance from ${\mathbf{u}}_{+}\left(t^{\prime }\right)$ $\left[{\mathbf{u}}_{-}\left(t^{\prime }\right)\right]$. The diamond and square highlight separation between states indicated using the same symbols in Fig. 13.

Figure 15

Comparison of flow fields far from E03. States on 1D submanifold trajectories (a) ${\mathbf{u}}_{+}\left(t_m^{\prime }\right)$, marked using a red diamond in Fig. 13 and (c) ${\mathbf{u}}_{-}\left(t_m^{\prime }\right)$ (red square). Turbulent flow fields $\mathbf{u}\left(t_m\right)$ from (b) simulation (black diamond on $T_4$) and (d) experiment (green square on $T_5$).

Figure 16

Distances from turbulent trajectories to the 1D submanifold. Axes ${min}_t{D}_0\left(t\right)$ and $D_1^{\pm }$ are the minimum distances to E03 and ${\mathbf{u}}_{\pm }\left(t_m^{\prime }\right)$, respectively. Diamonds ($D_1^{+}$) and squares ($D_1^{-}$) indicate closeness to ${\mathbf{u}}_{+}$ and ${\mathbf{u}}_{-}$, respectively.

Figure 17

Eigenvectors of E01 used to construct the 3D projection of the state space shown in Fig. 7: (a) ${\widehat{\mathbf{e}}}_1$, (b) ${\widehat{\mathbf{e}}}_2$, and (c) ${\widehat{\mathbf{e}}}_5$.

Figure 18

Eigenvectors of E03 used to construct the 3D projection of the state space shown in Fig. 12: (a) ${\widehat{\mathbf{e}}}_1$, (b) $({\widehat{\mathbf{e}}}_6+{\widehat{\mathbf{e}}}_7)/2$, and (c) $({\widehat{\mathbf{e}}}_6-{\widehat{\mathbf{e}}}_7)/\left(2i\right)$.

Figure 19

Temporal autocorrelation of the velocity field at $\text{Re}=22.5$. The solid (dashed) curve corresponds to simulation (experiment).