Scaling of turbulence-forced capillary waves

Abstract

We simulate the propagation of capillary waves at the interface between two liquid layers of equal density, viscosity and thickness, which flow in stratified conditions inside a turbulent Poiseuille channel. This setup gives us the possibility to single out the effect of inertia and surface tension – in isolation from gravity – on the propagation of interfacial capillary waves. Numerical simulations, which are based on a combined pseudo spectral/phase field method, are run keeping the shear Reynolds number constant, Reτ=300, and considering four different values of the Weber number: We=0.5, We=1.0, We=1.5 and We=2.0. In the present case, waves are naturally forced by turbulence over a broad range of scales, from the larger ones, whose size is of order of the channel height h, down to the smaller dissipative scales. We show that, by increasing the Weber number, the interface is deformed more vigorously, and is prone to sustain the propagation of waves over a broader range of scales. Upon computation of the power spectra of wave elevation, we show that the behavior of waves agrees with the theoretical predictions given by the weak wave turbulence theory, which predicts a decay Sη(k)∼k−4 in the inertial regime, followed by a steeper decay at large k, in the surface-tension-dominated regime. We clearly show that the transition between the inertial and the surface-tension-dominated regime is pushed towards larger k by increasing the Weber number. In addition, we show that the two-dimensional frequency–wavenumber spectra of the wave elevation – the dispersion relation – is in good agreement with well established theoretical prediction, ω(k)∼k3/2, and that the range of validity of such prediction broadens for increasing Weber number.