A study of the circular hydraulic jump: numerical simulations and experiments

Abstract

When a laminar vertical liquid jet hits a horizontal surface, the fluid spreads out radially on the surface. Initially, the spreading film is thin, however as this film spreads radially, at a particular radius (known as the jump-radius) the film thickness abruptly increases, resulting in a phenomenon known as the circular hydraulic jump. There are extensive analytical, numerical, and experimental studies which have explored how various factors, such as flow rate, gravity, viscosity, and surface tension, control the jump-radius. Most earlier studies have indicated a weak dependence of the nozzle radius and the distance from the nozzle to the disk (impinging height; both factors determine the initial momentum flux) on the jump radius. Vishwanath et al. (2015) specifically conducted experiments and simulations to examine the influence of initial momentum flux on the jump radius. They demonstrated dependence of the jump-radius on the initial momentum-flux through jet nozzle diameter and the impinging height. Here in the present work, building further on this research, we use grid-converged numerical simulations over a broader range of parameters to further investigate the effects of initial momentum flux and the inclination of the impingement surface to the horizontal. Our simulations reveal that increasing the nozzle radius significantly decreases the jump-radius, aligning with the experimental results of Vishwanath et al. (2015). Additionally, inclining the impinging surface affects the flow by providing additional momentum (conversion of potential energy into kinetic energy along the inclined surface) in the spreading direction and reducing the back pressure (due to jump), both of which increase the jump-radius. We conducted numerical simulations with inclinations from 0°to 10°(inclination from 0 to 0.174 radians) to the horizontal. Results indicate that the jump-radius systemically increases with the inclination. Moreover, the jump becomes weaker, as indicated by a lower ratio of pre-jump to post-jump film thickness compared to the horizontal case (0°inclination). We also performed scaling analysis using a generalized scaling function proposed by Vishwanath et al. (2015), which satisfactorily matched various past experimental results. Although surface tension and disk radius have a weak influence on the jump-radius, they may become significant for smaller jumps, suggesting that our current scaling may need adjustments to account for these effects. In addition to detailed numerical simulations, we conducted experiments to reconstruct the film thickness variation with radius using a non-invasive, optical technique. For this, we employed the Free-Surface Synthetic Schlieren method developed by Moisy et al. (2009). This technique relies on the apparent displacement of a doted pattern caused by changes in the refractive index at the interface between two fluids. To validate this method, we reconstructed the profile of a plano-convex lens with known dimensions. We identified a set of optimized parameters that minimized the error based on the absolute difference between the actual and reconstructed profiles. Using these parameters, we successfully reconstructed the free surface profile of a circular hydraulic jump, demonstrating the method as proof of concept.