"Direct numerical simulation of capillary waves forced by hydrodynamic turbulence"

In this thesis, the turbulent flow of two liquid layers inside a wall-bounded domain is investigated by means of numerical simulation. The main interest of this work lies on the interaction between the capillary waves that are generated at the interface between the two liquid layers and the surrounding hydrodynamic turbulence. The flow field is resolved by means of Direct Numerical Simulation (DNS) with resolution down to the smallest flow scale (Kolmogorov scale). The interface deformation is captured using a Phase-Field Method (PFM). The first case that is examined is the pressure-driven flow of two liquids occupying the same volume inside a rectangular channel. Two parametric studies are preformed in order to study, first the effects of the viscosity contrast between the two fluid layers and second the effects of the surface tension at the interface. The capillary wave regimes that emerge at the interface are analyzed by means of space-time spectral analysis and the results are compared to previous theoretical and experimental predictions of the spectral slope across different scales. Two-dimensional frequency-wavenumber spectra show that the waves propagate according to the linear dispersion relation. One-dimensional frequency and wavenumber spectra show that at larger wave scales, where turbulent forcing is not present, a scaling indicating an equipartition of energy between wave modes (Rayeleigh-Jeans distribution) takes place. This observation is in agreement with the theoretical prediction for the behaviour of the capillary wave spectrum at scales larger than the forcing scale. At smaller scales, where turbulent forcing takes place, an early departure from the theoretically predicted inertial range slope to a steeper slope indicating a sharp decrease of wave energy at the shorter waves scales occurs near the Kolmogorov-Hinze scale, which expresses the local balance between inertial and surface tension forces. The second case that is examined is that of the pressure-driven flow of a thin laminar layer over a thick turbulent layer in a rectangular channel. In this case, a single parametric study is performed to study the effect of the viscosity ratio, which is increased up to two orders of magnitude compared to the reference case with matched viscosities between the layers. This dramatic increase in the viscosity of the thin layer is catalytic to the shape and dynamics of the waves. In particular, instead of a set of waves of different wavelengths that propagate according to the linear dispersion relation that are observed in the matched viscosity case, the waves in the high viscosity cases are regular and two-dimensional, while they are purely advected with the mean velocity of the flow. Finally, a multiple resolution strategy is also presented, which allows for a selective increase of the resolution level of the phase-field transport equation. This approach is validated and tested with both a two-dimensional and a three-dimensional flow configuration and is found to significantly increase the computation efficiency in terms of time and memory usage.