Author(s): Daniele Catucci; Riccardo Briganti; Valentin Heller
Keywords: Ir-water flows; Froude similarity; OpenFOAM; Scale effects; Self-similarity
Abstract: Physical modelling in the laboratory environment is widely used to investigate fluid flows at hydraulic structures. Froude similarity results in significant scale effects in small laboratory models for flows involving air-entrainment, i.e. the entrapment of air bubbles into water. Such air-water flows are essential for many hydraulic applications and physical processes including spillways, hydraulic jumps, wave breaking, plunging jets and transport processes. All these phenomena are still modelled with Froude scaling even though the Reynold and Weber numbers are not correctly scaled. This study introduces alternatives to Froude scaling laws, enabling the modelling of air-water flows without scale effects. Mathematically, this corresponds to the derivation of self-similar solutions of the governing equations for fluids, in other words, the properties are invariant with respect to the scale, i.e. no scale effects are observed. In order to derive the new scaling laws, the one-parameter Lie group of point scaling transformations is applied to the Reynolds-averaged Navier-Stokes equations, including the surface tension effect as a source term. The scaling relations are formulated for the initial-boundary conditions and for the variables, such as time, velocity, fluid properties, turbulent kinetic energy and turbulent energy dissipation. A numerical validation of the obtained scaling laws is conducted for two air-flow processes, namely (i) a dam break flow interacting with an obstacle, generating a large deformation of the free surface, and (ii) a water plunging jet based on existing experimental results. Numerical modelling based on the Volume of Fluid Method implemented in the OpenFOAM v1706 Computational Fluid Dynamics package is used. All the boundary and initial conditions, including the fluid properties, are transformed with the novel scaling laws at different geometrical scales and the results are compared with results based on Froude similarity. The results show that velocity and turbulent kinetic energy fields are self-similar for both the dam break and the plunging jet. Although details in the void fraction on the plunging jet deviate with the scale, they show overall the same distribution of bubble entrainment over time. On the other hand, the results obtained using the traditional Froude scaling laws show significant scale effects. This work shows a numerical validation of the mathematical derived relations, which provide a formal procedure to obtain self-similarity. The proposed method shows also the potential to significantly improve laboratory-modelling of air-water flows.