Exploring regular and turbulent flow states in active nematic channel flow via Exact Coherent Structures and their invariant manifolds

This work is a unified study of stable and unstable steady states of 2D active nematic channel flow using the framework of Exact Coherent Structures (ECSs). ECSs are stationary, periodic, quasiperiodic, or traveling wave solutions of the governing equations that, together with their invariant manifolds, organize the dynamics of nonlinear continuum systems. We extend our earlier work on ECSs in the preturbulent regime by performing a comprehensive study of stable and unstable ECSs for a wide range of activity values spanning the preturbulent and turbulent regimes. In the weakly turbulent regime, we compute more than 200 unstable ECSs that coexist at a single set of parameters, and uncover the role of symmetries in organizing the phase space geometry. We provide conclusive numerical evidence that in the preturbulent regime, generic trajectories shadow a series of unstable ECSs before settling onto an attractor. Finally, our studies hint at shadowing of quasiperiodic-type ECSs in the turbulent regime.

Figure 1

The three equilibria with zero or 1D flow for $R_a=3.4$.

Figure 2

Snapshots of some of the attractors. For each of the four attractors shown, the top plot shows the velocity superimposed on the (scalar) vorticity, and the bottom plot shows the director field superimposed on the nematic order parameter. The “vortical” flows (top row) and chaotic flow (bottom left) are at $R_a=3.0$ and are shown with the same colorbar scales. The modulated wave (bottom right) is at $R_a=1.1$; due to the smaller activity, it is shown with different colorbar scales. The alternating vortex lattice RPO and the modulated wave RPO both have shift-reflect symmetry ${\sigma }_x{T}_2$, the rolling vortex lattice RPO has threefold translation symmetry $T_3$, while the chaotic flow does not have any symmetry.

Figure 3

Attractors as a function of activity $R_a\equiv \alpha /A$ [Eq. (11)]. The channel dimensions are $80\ell \times 20\ell$, where $\ell$ is defined from the material constants of the nematic [Eq. (5)]. The flow alignment $\lambda$ is 0. The horizontal bars group the attractors by symmetries and other qualitative flow characteristics. A blue color indicates an equilibrium, orange an RPO, and green a PO. Some of the bars are composed of more than one ECS; for example, there are two qualitatively similar threefold rolling vortices between $R_a=3.34$ and $R_a=3.86$. Real-space flow snapshots are shown in Fig. 2, and the full data set of ECSs is given in the Supplemental Material [73].

Figure 4

A chaotic trajectory and the three attractors projected on the $(\langle U\rangle ,\langle V^2\rangle )$ (left) and $(\langle V^2\rangle ,\langle Q_{xx}\rangle )$ (right) planes, for $R_a=3.4$.

Figure 5

Left: (Top) Potential energy surface of a two degrees-of-freedom conservative dynamical system with two minima and three saddles highlighted. (Bottom) The phase space near the energy surface saddle can be locally described as a direct product of a saddle fixed point and a center fixed point on orthogonal planes. Right: Periodic orbit around the saddle in yellow has 2D stable and unstable manifolds. A trajectory transiting from the left potential well to the right potential well travels inside the stable manifold towards the PO, and inside the unstable manifold while moving away from the PO. Hence, the PO here acts like an extended saddle point.

Figure 6

Left: An unstable defectless vortex lattice steady state (equilibrium) with threefold translational symmetry $T_3$. Right: An unstable traveling wave with fourfold translational symmetry $T_4$. For both cases, the top plot shows the velocity superimposed on the (scalar) vorticity, and the bottom plot shows the director field superimposed on the nematic order parameter.

Figure 7

Trajectory “densities” at $R_a=4.5$ for the 201 unstable ECS (left) and turbulent attractor (right), projected on the $(\langle V^2\rangle ,\langle Q_{xx}\rangle )$ plane. Each figure is generated from evenly spaced time series of the corresponding trajectories, which are binned and normalized as a 2D probability density function in $\langle V^2\rangle$ and $\langle Q_{xx}\rangle$. Trajectories tend to spend more time in the yellow regions versus the violet regions.

Figure 8

Trajectory “densities” along the unstable manifold of UNI in the preturbulent regime projected on the $(\langle V^2\rangle ,\langle Q_{xx}\rangle )$ plane. The dimensionless activity is $R_a=3.4$. The full unstable manifold is 34-dimensional and impractical to compute directly. Following the method described in Sec. 3c1, we instead define an ensemble of trajectories that maps out a low-dimensional subspace. The density map (color gradient) is generated from this ensemble by binning evenly spaced time series over the corresponding interval: $0<t<100$ (top) or $100<t<{10}^4$ (bottom). The apparent overlap of high-density regions (yellow) with unstable ECSs suggests the occurrence of shadowing, which we have confirmed for selected cases (see Sec. 3c2). Note that there are three stable ECSs (or attractors): T1/9, T3/1, and T3/2, while the rest of the ECSs shown are unstable.

Figure 9

Example of shadowing along the unstable manifold of the unidirectional equilibrium in the preturbulent regime at $R_a=3.4$. The color gradient plot shows the distances [as defined by Eq. (14)] between a time-dependent trajectory and a selection of ECSs at the same value of activity. The figures illustrate shadowing of the unstable ECSs T1/5 from approximately $200\tau$ to $250\tau$. The simulation eventually approaches T1/9, which is an attractor.

Figure 10

Another example of shadowing along the unstable manifold of the unidirectional equilibrium in the preturbulent regime at $R_a=3.4$. The color gradient plot shows the distances between a time-dependent trajectory and a selection of ECSs at the same value of activity. The figures show two shadowing events: first, the unstable ECS T6/1 from about $30\tau$ to $70\tau$, followed by the unstable ECS ${\sigma }_x{\sigma }_y\left(\text{T3/3}\right)$ from $250\tau$ to $600\tau$. This simulation also eventually approaches the attractor T1/9.

Figure 11

Top: A subset of the distance plot chosen to highlight one of the closest encounters between a chaotic trajectory and an ECS in the turbulent regime at $R_a=4.5$. The trajectory visits the neighborhood of an ECS (labeled T4/1) at $t/\tau \approx 1500$. Bottom: The trajectory from $t/\tau =\text{0}\text{to}t/\tau \approx 1800$ in black along with the ECS T4/1. There is clearly no shadowing of the ECS, and the trajectory appears to be quasiperiodic. The full time series data of distances of this trajectory from all the ECSs is provided in the Supplemental Material [73].