Exact coherent states underlying chaotic falling-film dynamics

Abstract

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.

Figures (15)

• Figure 1: Streamwise and spanwise power spectra $P(k_x)$ and $P(k_y)$ for cases $(L=22,\delta=0.002)$ and $(L=30,\delta=0.65)$ corresponding to panels (a) and (b), respectively. Spectra are shown for Fourier discretizations with $N=32,64,$ and $128$.
• Figure 2: (a) Dependence of the Reynolds number as a function of the dispersion parameter in the range $\delta\in\left[10^{-3},5\right]$ for a falling film with $h_0$=0.2 mm and $\theta=1.5608$. The blue star and green point correspond to $\delta$ values of 0.002 and 0.65373, respectively, where $\text{Re}(0.002)=35.38$ and $\text{Re}(0.65373)=0.0541.$ (b) Instantaneous snapshot of the film-height field $H(x,y)$ for $\delta=0.002$ in a domain of size $L=40$, shown as an illustrative example of interfacial deformation.
• Figure 3: Power spectra of solutions representative of the different dynamical regimes identified in the regime map shown in figure \ref{['fig:regime_map']}. (a): Travelling wave solution at $L=8.57,\delta=1.14329$. (b) Bursting travelling wave solution at $L=15.71,\delta=0.53135$. (c) Chaotic solution at $L=31.43,\delta=0.85771$. Panels (d)–(f) show the temporal evolution of the energy for the cases in (a)-(c), respectively.
• Figure 4: Spatio-temporal evolution of the interfacial dynamics for the cases shown in figure \ref{['PowerSpectra']}, representative of the different dynamical regimes identified in the regime map shown in figure \ref{['fig:regime_map']}. (a-d) Travelling wave solution at $L=8.57$ and $\delta=1.14329$ at times $t=[0, 17.86, 35.73, 53.60]$. (e-h) Bursting travelling wave solution at $L=15.71,\delta=0.53135$ at times $t=[0, 11.53, 23.07, 34.61]$ respectively. (i-l) Chaotic solution at $L=31.43$ and $\delta=0.85771$ at times $t=[0, 33.33, 66.66, 100]$.
• Figure 5: Regime map of interfacial dynamics in falling films in the $(L-\delta)$ space. Here $L\in[5,40]$ denotes the domain length and and $\delta\in[0.001,1.2]$ controls the effective dissipation, corresponding to Reynolds numbers $Re \in[0.0541,35.38]$.