Interaction Between Capillary Waves and Hydrodynamic Turbulence in a Two-Layer Oil-Water Flow

Abstract

Abstract We use pseudo-spectral Direct Numerical Simulation (DNS), coupled with a Phase Field Method (PFM), to investigate the turbulent Poiseuille flow of two immiscible liquid layers inside a channel. The two liquid layers, which have the same thickness (h1 = h2 = h), are characterized by the same density (ρ1 = ρ2 = ρ) but different viscosities (η1 ≠ η2), so to mimick a stratified oil-water flow. This setting gives the possibility to study the interplay between inertial, viscous and surface tension forces to be studied in the absence of gravity. We focus on the role of turbulence in initially deforming the interface and on the subsequent growth of capillary waves. After an initial transient, we observe the emergence of a stationary capillary wave regime. Capillary wave propagation and interaction is studied by means of a spatiotemporal spectral analysis and compared with previous theoretical and experimental results. The computed power spectra of wave elevation are in line with previous experimental findings and can be explained in the frame of the weak wave turbulence theory. At wave scales larger than the turbulent forcing range the observed scaling of the one-dimensional wavenumber spectrum suggests an energy equipartition regime (k−1), which is predicted by theory and has been recently observed in experiments with capillary wave turbulence in microgravity. At wave scales directly forced by hydrodynamic turbulence, an initially milder slope (k−4) of the wavenumber spectrum is followed by a sharper decay (k−6) of wave energy at larger wavenumbers, with the transition taking place near the Kolmogorov-Hinze critical scale, where surface tension forces and turbulent inertial forces are balanced.