Reformulating self-aeration in hydraulic structures: Turbulent growth of free surface perturbations leading to air entrainment

A new formulation for the prediction of free surface dynamics related to the turbulence occurring nearby is proposed. This formulation, altogether with a breakup criterion, can be used to compute the inception of self-aeration in high velocity flows like those occurring in hydraulic structures. Assuming a simple perturbation geometry, a kinematic and a non-linear momentum-based dynamic equation are formulated and forces acting on a control volume are approximated. Limiting steepness is proposed as an adequate breakup criterion. Role of the velocity fluctuations normal to the free surface is shown to be the main turbulence quantity related to self-aeration and the role of the scales contained in the turbulence spectrum are depicted. Surface tension force is integrated accounting for large displacements by using differential geometry for the curvature estimation. Gravity and pressure effects are also contemplated in the proposed formulation. The obtained equations can be numerically integrated for each wavelength, hence resulting in different growth rates and allowing computation of the free surface roughness wavelength distribution. Application to a prototype scale spillway (at the Aviemore dam) revealed that most unstable wavelength was close to the Taylor lengthscale. Amplitude distributions have been also obtained observing different scaling for perturbations stabilized by gravity or surface tension. The proposed theoretical framework represents a new conceptualization of self-aeration which explains the characteristic rough surface at the non-aerated region as well as other previous experimental observations which remained unresolved for several decades.