Abstract
The Froude scaling laws have been used to model a wide range of water flows at reduced size for almost a century. In such Froude scale models, significant scale effects for air–water flows (e.g. hydraulic jumps or wave breaking) are typically observed. This study introduces novel scaling laws, excluding scale effects in the modelling of air–water flows. This is achieved by deriving the conditions under which the governing equations are self-similar. The one-parameter Lie group of point-scaling transformations is applied to the Reynolds-averaged Navier–Stokes equations, including surface tension effects. The scaling relationships between variables are derived for the flow variables, fluid properties and initial and boundary conditions. Numerical simulations are conducted to validate the novel scaling laws for (i) a dam break flow interacting with an obstacle and (ii) a vertical plunging water jet. Results for flow variables, void fraction and turbulent kinetic energy are shown to be self-similar at different scales, i.e. they collapse in dimensionless form. Moreover, these results are compared with those obtained using the traditional Froude scaling laws, showing significant scale effects. The novel scaling laws are a more universal and flexible alternative with a genuine potential to improve laboratory modelling of air–water flows.
Highlights
Physical modelling at reduced size is one of the oldest and most important design tools in hydraulic engineering
The Froude scaling laws have been applied to model water flows at reduced size for almost a century
A significant disadvantage of Froude scaling is the potential for scale effects
Summary
Physical modelling at reduced size is one of the oldest and most important design tools in hydraulic engineering. Self-similar conditions for phenomena with negligible surface tension effects have previously been derived by applying the one-parameter Lie group of point scaling transformations [22]. Lie group transformations were originally used to reduce the number of independent variables of an initial boundary value problem by transforming it in a new space where the solution of the problem is the same as the original [20,23,24] This approach has been applied by [25] to derive the conditions under which various hydrological processes are self-similar through the change in size. We derive novel scaling laws by applying the Lie group transformations to the governing equations of air–water flows, including surface tension effects. This restriction leads to the well-known precise Froude scaling laws [3], as a particular case of the novel scaling laws, where g is constant and ν and ρ are scaled by keeping Re and We invariant
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