Catucci, Daniele
(2023)
Analytical derivation and numerical test of novel scaling laws for air-water flows.
PhD thesis, University of Nottingham.
Abstract
Physical modelling at reduced size is one of the oldest and most important design tools in hydraulic engineering. To predict the behaviour of the prototype, a physical model needs to incorporate all involved forces. The Froude scaling laws are being used to model a wide range of water flows at reduced size for almost a century. They are based on the invariance of the Froude number, i.e. the square root of the ratio between the inertial and gravity force between the model and its prototype. However, scale effects are observed due to forces excluded from the Froude number, for example in air-water flows, such as hydraulic jumps or wave breaking, the viscous, surface tension and air compressibility forces are scaled incorrectly at reduced size.
This study introduces novel scaling laws (NSLs) to exclude scale effects in the modelling of air-water flows. This is achieved by deriving the conditions under which the governing equations are self-similar with respect to the geometric scale, i.e. they are scale-invariant. To this end, the one-parameter Lie group of point-scaling transformations are applied to the governing equations for air-water flows. These scaling laws involve relationships for all the flow parameters, fluid properties and initial and boundary conditions. First, the scaling laws are derived under the assumption of air incompressibility, subsequently, this assumption is removed. For the former, the Reynolds-averaged Navier–Stokes equations, including surface tension effects, are used. For the case in which the compressibility of air is taken into account, the equations of heat transfer and perfect gas are added to the system of governing equations.
The NSLs are validated by numerically simulating different phenomena at different scales, i.e. i) a plunging water jet for incompressible air-water flows, ii) a Taylor bubble where the air is considered compressible and iii) a dam break flow impacting an obstacle for both cases. To this end, the governing equations are computed by two-phase flows solvers based on the volume of fluid method available in OpenFOAM v.1706, namely \textit{interFoam} and \textit{interIsoFoam} for incompressible air-water flows and \textit{compressibleInterIsoFoam} when air is considered compressible. When the NSLs are used, some restrictions are applied by still maintaining self-similarity. For the incompressible air case, the gravitational acceleration is kept invariant between the model and its prototype, while the temperature is kept invariant through the scales for compressible air. Further, it is shown that the precise Froude scaling laws, i.e. when the properties of fluids are strictly scaled, are a particular configuration of the NSLs.
Results for the void fraction, turbulent kinetic energy, velocity, pressure and temperature are shown to be self-similar at different scales for i) to iii), i.e. they collapse in dimensionless form. These results are compared with those obtained using the traditional Froude scaling laws, i.e. when ordinary water and air are used in the model, which, on the other hand, show significant scale effects. Furthermore, the importance of modelling air compressibility is analysed by comparing the results of the dam break flows in which the pressure on the obstacle shows an oscillation when the compressibility of air is taken into account. Finally, the implications of the NSLs for the laboratory experiments are discussed and an easy-to-use MATLAB app to obtain suitable scaling configurations satisfying self-similarity is presented. The NSLs are shown to be more universal and flexible than the Froude scaling laws. Their key advantage is that fluids of different densities from water (at 293 K) and with increased gravitational acceleration now also qualify as potential candidates for laboratory experiments.
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