Circular Hydraulic Jumps Research Articles

Jet impingement and the circular hydraulic jump on adverse-sloped beds

Hydraulic jump type stilling basins are the most common hydraulic structures for decreasing the excess kinetic energy of flow and are divided into three types: classical, radial and circular basins. Previous studies have reported that the hydraulic characteristics of flow are more favourable in circular basins than in the other types. Therefore, theoretical/experimental modelling was applied to examine the characteristics of circular hydraulic jumps on adverse-sloped beds in circular stilling basins. The present theoretical model applies the momentum and energy equations to derive equations for the sequent depth ratio and relative energy loss of circular hydraulic jumps and classical hydraulic jumps (CHJs). Analysing the effectiveness of the dominant independent parameters showed that increasing the sequent radius ratio up to 2 and the bed slope up to −5% decrease the sequent depth ratio and length of the circular hydraulic jumps by about 30% and 60%, respectively, whereas the relative energy loss increases by about 30%. In addition to the correlation coefficient, different error functions were determined to analyse the precision of the derived semi-theoretical equations. The present circular stilling basin acts like a plunge pool, having a small water depth for low discharges to have a submerged jet condition, downstream of large dams.

Averaging theory for heat transfer in circular hydraulic jumps with a separation bubble

Analytical investigations of heat transfer during the vertical impingement of an unsubmerged axisymmetric liquid jet on a horizontal plate have been limited to the regions ahead of the jump. This limitation is due to the complex flow physics in the jump region arising from sudden changes in the flow field. This is addressed in here by extending the averaging theory (AT) introduced by Bohr et al. (Phys. Rev. Lett., vol. 79, issue 6, 1997, pp. 1038–1041) which was further developed by Watanabe et al. (J. Fluid Mech., vol. 480, 2003, pp. 233–265), to describe the heat transfer problem in circular hydraulic jumps including separation. The applicability of the resulting theory to determine the temperature field in the jump region is evaluated using the data available in the literature and also by means of fully resolved numerical solutions. Good agreement is observed for moderate Prandtl numbers. However, for sufficiently high Prandtl numbers, deviations become notable. The reasons for the deviations according to their relevance are (i) monotonically decreasing temperature profile inherent to the AT, whereas the fully resolved numerical solutions exhibit a local maximum in the temperature profile away from the plate; and (ii) inapplicability of the concept of dividing the flow field into a region affected and a region unaffected by heat transfer according to the thermal boundary layer thickness. This concept leads to the overestimation of the temperature close to the wall and to the existence of a threshold Prandtl number, for which the thermal boundary layer thickness does not meet the free surface anymore. Around this threshold Prandtl number, the temperature field shows a discontinuous behaviour.

Oscillations and Cavity Modes in the Circular Hydraulic Jump.

We report spontaneous oscillations of circular hydraulic jumps created by the impact of a submillimeter water jet on a solid disk. The jet flow rate is shown to condition the occurrence of the oscillations while their period is independent of this parameter. The period, however, varies linearly with the disk radius. This dependency is rationalized by investigating surface gravity wave modes in the cavity formed by the disk. We show that the jump oscillation frequency systematically matches one of the surface wave disk-cavity eigenfrequencies, leading to the conclusion that the oscillations are self-induced by the interaction between the jump and surface wave eigenmodes.

Effects of Circular Hydraulic Jump on the Material Removal Characteristics of Jet Electrochemical Machining

Jet electrochemical machining (Jet-ECM) is a potential precision machining method. Due to the advantages of non-stress-free machining and high flexibility, it is widely used in surface finishing and complex structural machining. However, the “w” shaped removal profile and wide removal range caused by inappropriate process parameters reduce the stability and localizability of Jet-ECM. In this paper, the material removal characteristics of Jet-ECM were investigated both theoretically and experimentally. A multi-physics field simulation model for Jet-ECM was established, and the overall shape of the circular hydraulic jump (CHJ), current density distribution and workpiece deformation were calculated. The effects of CHJ on the material removal characteristics were investigated by experiments with jet flow rate and inter-electrode gap (IEG) distance. The results reveal the qualitative relationship between CHJ states and process parameters and give the range of parameters that can form a stable CHJ and ensure a Gaussian shaped removal profile and a narrower removal range. This study provides theoretical guidance on the process parameters of Jet-ECM to improve its processing stability and localizability, which is significant for the accuracy of surface finishing and the controllability of complex structural machining, promoting the further application of Jet-ECM.

Circular undular hydraulic jumps in turbulent free-surface flows

Near-critical turbulent axisymmetric free-surface flow over a horizontal bottom is investigated. Assuming that undular hydraulic jumps occur at large radii, an asymptotic analysis of the governing equations is performed in the double limit of very large Reynolds numbers and Froude numbers close to the critical value 1. The results are kept free of turbulence modelling due to a specific coupling of the two limiting processes. The final result of the asymptotic analysis is a new steady-state version of an extended Korteweg–de Vries (KdV) equation for the free-surface elevation. The extended KdV equation is derived as a uniformly valid differential equation, describing the flow near the origin of the undular jump as well as far downstream. Numerical solutions of the extended KdV equation show that circular undular jumps can develop if the reference state is located in the region where the effect due to axisymmetry prevails over the effect of friction. In this case, the solution oscillates over a very long distance until the accumulating friction effects force a breakdown. The comparison between the theories of undular jumps in turbulent and inviscid flows shows that friction is of minor importance near the development of the undular jump. However, friction has to be taken into account to describe the flow further downstream. Remarkably, the extended KdV equation is valid and yields undular solutions for both turbulent source and sink flow.

Circular hydraulic jumps: where does surface tension matter?

Recently, an unusual scaling law has been observed in circular hydraulic jumps and has been attributed to a supposed missing term in the local energy balance of the flow (Bhagat et al., J. Fluid Mech., vol. 851, 2018, R5). In this paper, we show that – though the experimental observation is valuable and interesting – this interpretation is presumably not the right one. When transposed to the case of an axial sheet formed by two impinging liquid jets, the assumed principle leads in fact to a velocity distribution in contradiction with the present knowledge for this kind of flow. We show here how to correct this approach by maintaining consistency with surface tension thermodynamics: for Savart–Taylor sheets, when adequately corrected, we recover the well-known $1/r$ liquid thickness with a constant and uniform velocity dictated by Bernoulli's principle. In the case of circular hydraulic jumps, we propose here a simple approach based on Watson's description of the flow in the central region (Watson, J. Fluid Mech., vol. 20, 1964, pp. 481–499), combined with appropriate boundary conditions on the circular front formed. Depending on the specific condition, we find in turn the new scaling by Bhagat et al. (2018) and the more conventional scaling law found long ago by Bohr et al. (J. Fluid Mech., vol. 254, 1993, pp. 635–648). We clarify here a few situations in which one should hold rather than the other, hoping to reconcile the observations of Bhagat et al. with the present knowledge of circular hydraulic jump modelling. However, the question of a possible critical Froude number imposed at the jump exit and dictating logarithmic corrections to scaling remains an open and unsolved question.

ASSESSMENT OF THE INTERFACE COMPRESSION SCHEME IN THE VOLUME-OF-FLUID MODELING OF CIRCULAR HYDRAULIC JUMPS

ASSESSMENT OF THE INTERFACE COMPRESSION SCHEME IN THE VOLUME-OF-FLUID MODELING OF CIRCULAR HYDRAULIC JUMPS

Experimental Study of Stable Circular Hydraulic Jumps

The geometry of a downstream obstacle is one of the major parameters affecting the stability range of circular hydraulic jumps and, yet, it has not been adequately addressed by researchers. The present study investigates the effects of the downstream obstacle geometry and the parameters like the flow rate, the jet diameter, and the downstream obstacle height on the stability range of circular hydraulic jumps. Its findings indicate that an increase in the fluid jet diameter leads to the narrowing of the stability range of circular jumps. In addition, an increase in the downstream obstacle height produces a reduction of the hydraulic jump radius and its stability range. The results also show that in the presence of a square downstream obstacle the stability range of circular jumps is less than that in the case of a triangular downstream obstacle and greater than in the case of a circular downstream obstacle.

Role of gravity and capillary waves in the origin of circular hydraulic jumps

A study shows that hydraulic jumps are governed by both gravitational and capillary forces. The jump location results from a competition of supercritical flow with gravity-capillary waves traveling in the upstream direction. Fluid properties, flow conditions, and the state of development scale the respective forces.

Experimental and numerical study of circular hydraulic jumps on convex and flat target plates

Abstract In this research, the effect of curvature radius of convex target plates on the radius of formed circular hydraulic jump is investigated using experimental investigation and numerical modeling. Ethylene glycol with constant density and viscosity is used as the working fluid. In order to simulate the hydraulic jump, the volume of fluid (VOF) method is applied using a continuous surface force model (CSF) and geometric reconstruction for determining the interface of the two fluids. The results of experimental studies and numerical simulation indicate that the hydraulic jump is a function of three important parameters, including the vertical jet radius, the curvature radius of the target plate and the volumetric flow rate. Subsequently, an empirical relation for the hydraulic jump radius formed on convex target plates is proposed. Comparing the radius of the hydraulic jump on the convex and flat target plates it is observed that the jump radius increases with increasing the volumetric flow rate and significantly decreases with increasing the curvature radius of the convex target plate and impinging jet radius.

Numerical and experimental investigations on a circular hydraulic jump due to normal impinging free liquid jet on a flat horizontal target plate

Circular hydraulic jump (CHJ) due to normal impingement of a free liquid jet of water and ethylene glycol on stationary horizontal plate is numerically simulated for the prediction of radius of the CHJ and for the film thickness of the radially spreading flow by solving continuity and momentum equation using volume-of-fluid method. The Froude number measured just after the jump is observed to be constant and independent of the flow rate and kinematic viscosity of the liquid. CHJ is also investigated both experimentally and numerically by making a small hole in the impingement region of the target plate such that the central core of the impinging jet drains out through the hole. The effect of this modified impingement region flow is observed to be insignificant as far as the radius of the jump and outer Froude number is concerned.

Experimental measurement of radius of circular hydraulic jumps: Effect of radius of convex target plate

Hydraulic Jump is a phenomenon that occurs in free surface flows during transition from supercritical to subcritical flow. During the impingement of a vertical fluid jet with a flat surface, a circular hydraulic jump forms, which has been studied by several researchers; but no study has been performed on the effect of convexity of the target surface on the jump radius and its form yet. Therefore, in the present study the hydraulic jump phenomenon on convex surfaces and comparison with the flat surface are presented, experimentally. The tests are performed using 3 nozzles in various volumetric flow rates. The results showed that convex surfaces significantly increased the jump radius up to about 30% compared to flat plates for about 50% increase in convex target plate radius. In addition, an experimental relation for the hydraulic jump radius in the convex target plates is proposed.

The role of gravity in the prediction of the circular hydraulic jump radius for high-viscosity liquids

The free-surface flow formed by a circular jet impinging on a stationary disk is analysed theoretically. We develop a simple and coherent model to predict the location and height of the jump for high-viscosity liquids. The study explores the effect of gravity in the supercritical flow. The formulation reduces to a problem, involving only one parameter: $\unicode[STIX]{x1D6FC}=Re^{1/3}Fr^{2}$, where $Re$ and $Fr$ are the Reynolds and Froude numbers based on the flow rate and the jet radius. We show that the jump location coincides with the singularity in the thin-film equation when gravity is included, suggesting that the jump location can be determined without the knowledge of downstream flow conditions such as the jump height, the radius of the disk, which corroborates earlier observations in the case of type I circular hydraulic jumps. Consequently, there is no need for a boundary condition downstream to determine the jump radius. Our results corroborate well existing measurements and numerical simulation. Our predictions also confirm the constancy of the Froude number $Fr_{J}$ based on the jump radius and height as suggested by the measurements of Duchesne et al. (Europhys. Lett., vol. 107, 2014, 54002). We establish theoretically the conditions for $Fr_{J}$ to remain independent of the flow rate. The subcritical flow and the height of the hydraulic jump are sought subject to the thickness at the edge of the disk, comprising contributions based on the capillary length and minimum flow energy. The thickness at the edge of the disk appears to be negligibly small for high-viscosity liquids.

On the origin of the circular hydraulic jump in a thin liquid film

This study explores the formation of circular thin-film hydraulic jumps caused by the normal impact of a jet on an infinite planar surface. For more than a century, it has been believed that all hydraulic jumps are created due to gravity. However, we show that these thin-film hydraulic jumps result from energy loss due to surface tension and viscous forces alone. We show that, at the jump, surface tension and viscous forces balance the momentum in the liquid film and gravity plays no significant role. Experiments show no dependence on the orientation of the surface and a scaling relation balancing viscous forces and surface tension collapses the experimental data. A theoretical analysis shows that the downstream transport of surface energy is the previously neglected critical ingredient in these flows, and that capillary waves play the role of gravity waves in a traditional jump in demarcating the transition from the supercritical to subcritical flow associated with these jumps.

Computational simulation of radially asymmetric hydraulic jumps and jump–jump interactions

Circular hydraulic jump (CHJ) and its deviation from the azimuthal symmetry due to non-orthogonal impingement of liquid jet on a solid surface are studied using finite volume based numerical framework. Asymmetric interfacial jump has been modelled from volume of fluid (VOF) prediction for a different angle of inclination of a jet from normal of the surface. The extent of asymmetry is found to be increasing as the jet becomes more and more horizontal, deviation from orthogonal impingement. To understand the role of wall adhered jet on jump, flow physics has been studied which reveals back flow in the jump region causing circulation. Moreover, equipotential and heterogeneous jump–jump interactions are studied to predict fluidic features like fountain formation and upwash. Numerical simulations reveal that strength of interacting wall jets play a major role in the formation of a pattern as consequence of neighbouring twin jet impingement. A vertical flat liquid protrusion is observed at equidistant stagnation line from the impingement points of equipotential jets. On the other hand, a curvilinear upwash film has been found out as characteristics of the interaction between heterogeneous strengths of interacting jets. At a higher ratio of jet strengths, complete engulfment of a smaller hydraulic jump as an eye of larger one has been obtained from careful numerical simulation. Physical insights of these rich fluidic phenomena are described using well-resolved velocity vectors in the wall jet and upwash.

Harmonic solutions for polygonal hydraulic jumps in thin fluid films

This article contains numerical and theoretical results on the circular and polygonal hydraulic jumps in the framework of inertial lubrication theory. The free surface and velocity fields are computed along with cross-sections of the vorticity and pressure, in agreement with experimental data. The forces that drive and resist the instability are identified with the radial shear force, the azimuthal surface tension and the hydrostatic azimuthal force, in addition to a nonlinear term in the radial coordinate. Periodic solutions are obtained from the first orders of a perturbation theory by considering azimuthal symmetries. The thresholds of the instability are defined at closed jumps for discontinuous solutions and at one-sided hydraulic jumps for continuous curves that conserve fluid mass density.

The Effect of Initial Momentum Flux on the Circular Hydraulic Jump

Earlier studies on the circular hydraulic jump have shown that the radial position of the hydraulic jump depends on the flow rate, gravity, and fluid viscosity. In this study, results from numerical simulations and experiments on circular hydraulic jumps are presented and through analysis, it is shown that the momentum flux is an additional controlling parameter in determining the jump location. Apart from the jump location, the variation of the film thickness with flow parameters is also obtained from experiments and numerical simulations. By including the dependence of the momentum flux and dissipation in the film along with other controlling parameters, the data on jump radius obtained from experiments and simulation (including the present study) covering a wide range of parameters reported in the literature can be collapsed on to a single curve.

Detection of liquid–metal, free-surface flow using the DLP measurement technique

Novel accelerator applications favor free-surface liquid–metal flows, in which the liquid acts both as a target producing secondary particles but also to remove efficiently the heat deposited. A crucial aspect for the operation is the continuous monitoring of both shape and position of the liquid’s surface. This demands, in a nuclear environment, a non-intrusive measurement technique with high temporal and spatial resolution. In this context, the double-layer projection (DLP) technique based on geometric optics has been developed, allowing one to detect either point-wise or area-wise the shape and position of the nearly totally reflecting liquid–metal surface. The DLP technique employs a laser beam projected through a coplanar glass plate to the surface from which it is reflected to the plate again. Beam locations captured by means of a camera permit the position and shape of the surface to be reconstructed. The parameters affecting the resolution and performance of the DLP technique are discussed. Additionally, validation studies using static and moving objects of pre-defined shape are conducted, exhibiting spatial and temporal resolutions of 300 μm and 100 Hz, respectively. Finally, the DLP system is applied to perform measurements of a circular hydraulic jump (CHJ) in a liquid metal. The DLP system has proved the capability to measure the jump both qualitatively and quantitatively. Additionally, the experiments identified, at high Reynolds numbers, the existence of a two-step jump. The analysis of spectral data of the DLP surface measurements shows clearly that, at the outer radius, gravity waves occur. Also, contributions from the pump oscillations were found, demonstrating the high performance of the DLP system.

Two Dimensional Laminar Flow of a Liquid with Circular Hydraulic Jump

We examine two-dimensional laminar flow of a liquid with circular hydraulic jump using boundary layer approach, but with the inclusion of a velocity profile approximated by a quadratic function. Our motivation is due to an earlier work of * on radial spread of a liquid over a horizontal plate. We obtain a new relation for the displacement thickness, momentum thickness and position of the jump. Our approximate values based on ** are compared with the exact values due to ***. The comparison shows the error of about 9% in the shear rate relation on the plate and the error of about 3.5% in the thickness ratio. Our values agree to a large extent with the exact values and also show improvement of our work upon that of * with respect to the thickness ratio.

Numerical Study of Circular Hydraulic Jump Using Volume-of-Fluid Method

When a vertical liquid jet impacts on a solid and horizontal surface, the liquid starts spreading radially on the surface, until a sudden increase in the fluid height occurs and a circular hydraulic jump (CHJ), easily seen in the kitchen sink, is formed. In this study, the formation of CHJ is numerically simulated by solving the flow governing equations, continuity and momentum equations, along with an equation to track the free surface advection using the volume-of-fluid (VOF) method and Youngs’ algorithm. The numerical model is found to be capable of simulating the jump formation and its different types. Extensive comparisons are performed between the model results and those of the available experiments and modified Watson’s theory. The model is shown to accurately predict the jump location and its behavior. Also a parametric study for the effects of different parameters including volumetric flow rate, downstream height, viscosity and gravity on the jump radius, and its characteristics is carried out. Compared with previous works on CHJ available in the literature, employing the VOF method considering the surface tension effects and performing a full parametric study and a complete comparison with experiments and theory are new in this paper. The simulations are performed for two different liquids, water and ethylene glycol, where it is found that the jump is more stable and its location is less sensitive to the downstream height for the more viscous liquid (ethylene glycol). When the downstream height is increased, the radius of the circular hydraulic jump reduces up to a certain limit after which there would be no stable jump. If the gravity is decreased, the radius of the jump and the length of the transition zone will both increase. The radius of the jump in microgravity conditions is less sensitive to the downstream height than it is in normal gravity.