Exact coherent states underlying chaotic falling-film dynamics

Dynamical-systems approaches to spatiotemporal chaos have been developed primarily for single-phase flows, where the system state is defined by bulk velocity fields. Extending these ideas to two-phase flows remains challenging because the dynamics are intrinsically coupled to the evolution of a deforming interface. Here, we address this challenge for a two-dimensional vertical falling film by formulating the dynamics in terms of the interface evolution. Starting from the Navier–Stokes equations, we recover a classical long-wave interface evolution equation, originally derived by Topper & Kawahara (1978). Using this formulation, we perform an extensive parametric study to construct a regime map in the space of domain size and dispersion parameter. The resulting map reveals a rich range of interfacial behaviors, including travelling waves, bursting travelling waves, and fully chaotic regimes. In the chaotic falling film regime, we exploit the dissipative nature of the governing equation, which suggests that the long-time dynamics evolve onto an inertial manifold. Using a data-driven approach, we parameterize this inertial manifold and estimate its intrinsic dimension, suggesting approximately linear growth with domain size. We then construct low-dimensional models in manifold coordinates to facilitate the search for exact coherent states of the full system. Using this approach, we identify travelling waves, relative periodic orbits and equilibria embedded within the chaotic attractor. Chaotic trajectories repeatedly approach the neighbourhoods of these invariant solutions, indicating that the recurrent interfacial patterns observed in the dynamics correspond to visits to these coherent states. To the best of our knowledge, this constitutes the first identification of exact coherent structures embedded in chaotic falling-film dynamics.

💡 Research Summary

The paper addresses the long‑standing challenge of extending dynamical‑systems approaches, which have been highly successful for single‑phase turbulent flows, to two‑phase flows where the dynamics are tightly coupled to a deforming interface. Starting from the full Navier–Stokes equations for a vertical falling film, the authors recover the classical long‑wave interface evolution equation originally derived by Topper & Kawahara (1978). After nondimensionalisation, the resulting two‑dimensional dispersive thin‑film equation (Eq. 1.3) contains a nonlinear advection term, a second‑order diffusion term, and a fourth‑order hyper‑diffusion term whose relative strength is controlled by a single dispersion parameter δ, which itself can be expressed in terms of Reynolds, Froude and Weber numbers.

Numerical simulations are performed with the open‑source spectral solver Dedalus on doubly periodic square domains of size L×L. By varying L and δ, the authors construct a comprehensive regime map that identifies four distinct dynamical regimes: (i) steady or weakly perturbed states, (ii) travelling‑wave solutions, (iii) bursting travelling waves, and (iv) fully chaotic dynamics. The chaotic regime is of particular interest because the governing equation is strongly dissipative; the authors argue that the long‑time dynamics must therefore collapse onto a finite‑dimensional inertial manifold.

To extract this manifold, a two‑stage data‑driven reduction is employed. First, proper orthogonal decomposition (POD) yields an orthogonal basis that captures the bulk of the kinetic energy. Second, an implicit‑rank‑minimizing autoencoder with weight decay (IRMAE‑WD) compresses the POD coefficients into a low‑dimensional latent space. The autoencoder is designed to minimise the rank of the latent representation while preserving reconstruction accuracy, thereby providing an accurate parametrisation of the inertial manifold. The intrinsic dimension d z of the manifold is found to grow approximately linearly with the domain size L, confirming the hypothesis of a low‑dimensional attractor.

Having obtained a low‑dimensional coordinate system, the authors build reduced‑order models (ROMs) in manifold coordinates and use them to seed a Newton–Krylov solver for exact coherent states (ECS) of the full system. Symmetries of the problem (continuous translations in x and y) are handled via a first‑Fourier‑mode slice, which reduces relative periodic orbits (RPOs) to fixed points in the symmetry‑reduced space. Initial guesses for the Newton iterations are taken from near‑recurrence events identified in the DNS or from clusters in the POD/autoencoder latent space.

The search uncovers several families of ECS embedded in the chaotic attractor: (a) travelling‑wave solutions that propagate with a constant speed, (b) relative periodic orbits corresponding to bursting travelling waves, and (c) unstable equilibria. Time series from the full DNS repeatedly approach the neighbourhoods of these invariant solutions, a phenomenon known as “shadowing”. Statistical analysis shows that chaotic trajectories spend a non‑negligible fraction of time near these ECS, indicating that the complex interfacial patterns observed experimentally are organized by a skeleton of low‑dimensional unstable solutions.

This work constitutes the first identification of exact coherent structures in chaotic falling‑film dynamics, demonstrating that state‑space methods developed for single‑phase turbulence can be successfully transferred to interfacial two‑phase flows. The combination of long‑wave modeling, inertial‑manifold extraction, and data‑driven reduced‑order modeling provides a powerful framework for understanding, predicting, and potentially controlling complex thin‑film flows in industrial applications such as coating, heat exchangers, and reactors.