Chaos in wavy-stratified fluid-fluid flow

Abstract

We perform a nonlinear analysis of a fluid-fluid wavy-stratified flow using a simplified two-fluid model (TFM), i.e., the fixed-flux model (FFM), which is an adaptation of the shallow water theory for the two-layer problem. Linear analysis using the perturbation method illustrates the short-wave physics leading to the Kelvin-Helmholtz instability (KHI). The interface dynamics are chaotic, and analysis beyond the onset of instability is required to understand the nonlinear evolution of waves. The two-equation FFM solver based on a higher-order spatiotemporal finite difference scheme is used in the current simulations. The solution methodology is verified, and the results are compared with the measurements from a laboratory-scale experiment. The finite-time Lyapunov exponent (FTLE) based on simulations is comparable and slightly higher than the autocorrelation function decay rate, consistent with previous findings. Furthermore, the FTLE is observed to be a strong function of the angle of inclination, while the root mean square of the interface height exhibits a square-root dependence. It is demonstrated that this simple 1-D FFM captures the essential chaotic features of the interface dynamics. This study also adds to a growing body of work indicating that a TFM with appropriate short wavelength physics is well-behaved and chaotic beyond the KHI.