Growth and spectra of gravity–capillary waves in countercurrent air/water turbulent flow

Abstract

Using direct numerical simulation of the Navier–Stokes equations, we analyse the dynamics of the interface between air and water when the two phases are driven by opposite pressure gradients (countercurrent configuration). The Reynolds number ($\mathit{Re}_{{\it\tau}}$), the Weber number ($\mathit{We}$) and the Froude number ($\mathit{Fr}$) fully describe the physical problem. We examine the problem of the transient growth of interface waves for different combinations of physical parameters. Keeping$\mathit{Re}_{{\it\tau}}$constant and varying$\mathit{We}$and$\mathit{Fr}$, we show that, in the initial stages of the wave generation process, the amplitude of the interface elevation${\it\eta}$grows in time as${\it\eta}\propto t^{2/5}$. The wavenumber spectra,$E(k_{x})$, of the surface elevation in the capillary range are in good agreement with the predictions of wave turbulence theory. Finally, the wave-induced modification of the average wind and current velocity profiles is addressed.