Interaction between capillary waves and hydrodynamic turbulence in a wall-bounded oil-water flow
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 liquids are characterized by the same density (ρ1 = ρ2 = ρ), but different viscosities (µ1 ̸= µ2), so to mimic a stratified oil-water flow. This configuration represents a convenient setting to study the interplay between inertial, viscous and surface tension forces in the absence of gravity. We focus in particular, on the role of turbulence in deforming the interface, and in continuously forcing the growth of capillary waves. The capillary wave regimes at the interface are studied by means of spatiotemporal spectral analysis and are compared with previous theoretical and experimental results. The computed frequency (ω) and wavenumber (k) power spectra of wave elevation are in line with experimental find-ings and can be explained in the frame of weak wave turbulence theory. In particular, the wave spectra demonstrate a power law that is in agreement with the theoretical prediction suggesting an equiparti-tion of energy among waves of different wavelength at scales larger than the largest scale of turbulent forcing. At smaller scales, where waves are directly forced by turbulence, an early departure of the spectra from the theoretically predicted power law to a steeper decay of wave energy occurs near the critical scale corresponding to the local balance between inertial and surface tension forces. Finally, two-dimensional frequency-wavenumber spectra show that the wave propagation is in good agreement with the well-established prediction of the capillary wave dispersion relation, ω(k) ∼ k3/2.
Figure 1 : Channel geometry and flow topology of a two-layer turbulent flow simulation. Capillary wave formation can be seen on the interface between the two layers. The fluid in the upper half of the channel has a viscosity that is 50% lower than that of the fluid in the lower half of the channel. This difference is depicted in the near-wall turbulence structure - finer near the upper wall - here made visible using the value of the streamwise component of the velocity vector (ux).
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