In this study, we examine the consistency of a gravity-based predictive theory for a hydraulic jump, given by Kurihara [*Proceedings of the Report of the Research Institute for Fluid Engineering* (Kyusyu Imperial University, 1946), Vol. **3**, pp. 11–33]; Tani [J. Phys. Soc. Jpn. **4**, 212–215 (1949)] with the phenomenological condition at the jump given by Rayleigh [Proc. R. Soc. London, Ser. A **90**, 324–328 (1914)]; and Watson [J. Fluid Mech. **20**, 481–499 (1964)] and show that in light of experimental evidence, the gravity-based predictive theory for the kitchen sink hydraulic jump is incompatible with the phenomenological condition, which must be valid. We also examine the solution to the downstream film and its potential influence on the hydraulic jump. We show that for all practical purposes, at normal flow conditions, the downstream liquid film remains flat and does not affect the jump, and the theory given by Bhagat *et al.* [J. Fluid Mech. **851**, R5 (2018)] gives an excellent prediction of the jump radius. For high viscosity liquids, on a relatively large plate, the viscous dissipation in the downstream film could increase the jump height and, consequently, move the jump radius inward.

## I. INTRODUCTION

The circular hydraulic jump is a common phenomenon observed when we turn on the tap of a kitchen sink; the water from the tap falls as a vertical jet onto the bottom of the sink and spreads radially in a thin film. At a certain distance from the point of impact, typically a few centimeters, the film thickness increases abruptly, forming what is conventionally known as a hydraulic jump (see Fig. 1). These jumps show visual resemblance to the hydraulic jumps seen in rivers and bores, and an early mathematical model in 1914, aimed at describing bores, was built on the concept of momentum conservation,^{1} and the circular hydraulic jump was classified as a bore at small scale. From this viewpoint, the hydraulic jump is regarded as a sharp transition in water depth—a standing wave and the stationary counterpart of a tidal bore—and, at the jump, the phenomenological condition, in which the rate of change of momentum in the flowing stream should be balanced by the thrust of pressure produced by order of magnitude increase in the depth of the water, was satisfied.^{2} The first predictive model was developed in the 1940s by Tani^{3} following Kurihara.^{4} They proposed that a hydraulic jump is caused by flow separation due to an adverse gravitational pressure gradient. They modeled the flow in terms of the shallow water equation and showed that the equation becomes singular at a finite radius, where the wave speed equals the flow speed *U*. This condition was expressed in terms of the local Froude number $Fr\u2261U/gh=1$, where *g* is the acceleration due to gravity and *h* is the fluid depth. However, when compared with experiments carried out using water, the theory^{3,4} significantly over-predicted the jump radius.^{5}

In the case of the thin-film jump observed on the scale of the kitchen sink, surface tension produces a force on the spreading film. Bush and Aristoff^{6} extended the earlier model to include the influence of surface tension and applied it to tabletop scale laboratory experiments, where the jet impact occurred in a reservoir initially filled with water. On impact of the jet, the liquid spilled over a weir at the reservoir boundary that determined the height of the outer film. For a given jet flow rate, increasing the height of the weir produced a larger gravitational pressure, which moved the jump radius inwards, and at a certain height, the jump entirely disappeared. This experiment in conjunction with the phenomenological model gives an impression that, even at the scale of a tabletop, gravity is the principal force and the key to the jump dynamics.

In addition, in thin-film flows, viscosity can decelerate the fluid and, by volume conservation, increase the film depth producing a further adverse pressure gradient. Recently, Wang and Khayat^{7} rewrote the equation given by Tani^{3} and Kurihara^{4} in parametric form and concluded that the model is capable of predicting jumps in high viscosity liquids.^{5}

Recently, however, experimental evidence has cast doubt on the role of gravity in these thin-film jumps. Experiments on the normal impact of a jet onto a planar surface showed that the radius of the jump was independent of the orientation of the surface to gravity.^{8} Bhagat *et al.*^{5} provided further experimental evidence, including experiments carried out in micro-gravity on-board a “zero-gravity” flight and in a drop tower, showing the formation of jumps in micro-gravity occur at the same location as in the terrestrial experiments. It is impossible to reconcile these observations with the notion that gravity plays a central role in the formation of these thin-film jumps.

Nevertheless, the idea that gravity is central to the formation of these jumps persists. Bohr *et al.*^{12} developed a theory based on the shallow water equations, which essentially recovers the theory formulated by Tani,^{3} Kurihara,^{4} and their key result—the scaling relationship for the jump radius, $R\u223cQ5/8\nu \u22123/8g\u22121/8$—can be retrieved by imposing the jump condition, *Fr* = 1.^{13} Based on a postulation that, the exit Froude number, *Fr _{ex}*, at the jump, in the downstream film remains constant, Duchesne

*et al.*

^{14}obtained a modified version of the scaling relationship given by Bohr

*et al.*,

^{12}and apparently confirming experiments have also been reported.

^{12}So what is the issue here? Measurements of the jump radius are not unequivocal. There are experimental errors, even though the jump radius is a simple measurement. More challenging is the reconciliation of the use of fluids with different surface tensions and viscosities. Conceptually, the most difficult problem is the role of the flow downstream of the jump. As discussed above, the difference in fluid depths before and after the jump provides a pressure difference that must be matched by the change in momentum of the flow. Experiments are carried out on a plane of finite dimensions—what role, if any does that play?

In this paper, we discuss the different theories and experimental evidence that have led to this somewhat confused picture and attempt to reconcile them and to clarify the dynamics. In Sec. II, we examine the early theories and their relation to the momentum conservation constraint in order to determine the role of the downstream conditions on the jump formation. In Sec. III, we discuss the experiments and develop a consistent theory for all cases. Finally, in Sec. V, we present our conclusions.

## II. THEORY

### A. Consistency and compatibility between the phenomenological and the predictive models

The model based on the momentum theorem given by Rayleigh^{1} and Watson^{2} requires the momentum flux to be balanced by pressure difference across the jump

where *u*, *h* and *U*, *H* are the speed and depth upstream and downstream of the jump, respectively, the overline represents a depth average, and we have taken the fluid density *ρ* = 1 for convenience.

Ignoring the influence of surface tension, the gravity-based predictive model given by Tani^{3} and Kurihara,^{4} which is based on the theory that separation of the flow from the surface due to the adverse gravitational pressure gradient, produces the jump, gives for the radial gradient of the film thickness

where $u\xaf=C1us=1h\u222b0hudz,\u2009u2\xaf=C2us2=1h\u222b0hu2dz$, *u _{s}* is the surface velocity, and $Fr=u2\xafgh$; $f\u2032(0)$ arises from the velocity profile ansatz where $\tau w=\mu ushf\u2032(0)$ is the wall shear stress.

Here, we examine the consequences of applying (2) to the momentum conservation (1), noting that we expect a significant increase in the flow depth after the jump (i.e., $h\u226aH$). We consider two approximate but reasonable flow profiles. The first is laminar flow with the same velocity profile before and after the jump, and the second is turbulent flow after the jump, resulting from flow separation, with uniform velocity across the depth.

Laminar flow: Assuming that the shape factor is the same on either side of the jump, applying continuity of volume flux yields

After some algebra, we find

Turbulent flow: when the appropriate velocity profile across the jump is a turbulent profile,^{2} we write

and (5) could be written as

This equation yields three real roots, $hH=\u22120.51,0.76,1.75$, and so the only reasonable solution is $h=0.76H$.

In summary, this analysis shows that in order for inverse gravitational pressure gradient—that could cause the flow separation and consequently the jump—to build, the depth of a supercritical film has to be of the same order as that of the subcritical film, which is not experimentally observed.

### B. Viscous effects

Here, we examine the effect of viscous dissipation on the height of the jump and, consequently, the jump radius. Again consider the gravity based model (2). In the flow downstream of the jump, $Fr2\u21920$, and in this limit,

From (10), $dhdr\u21920$, at the edge of a large plate $r\u2192\u221e$, and for 2$f\u2032(0)\pi C1\nu r2HQ>1$ the slope of the film is negative and vice versa. Thus, we expect the slope of the surface downstream of the jump to be negative and to increase in magnitude as the viscosity increases.

Furthermore, (10) allows us to investigate the slope of the film near the jump, where it is expected to be the most pronounced. For typical experimental flow rates $Q(o(10\u22125\u2009m3\u2009s\u22121))$ for which $r=Rj(o(10\u22122\u2009m))$, and $h=Hj(o(10\u22123\u2009m))$, yielding 2$f\u2032(0)\pi C1\nu Rj2HjQ\u223c(o(\nu \xd7104))$, and so for a liquid with kinematic viscosity $\nu (o(10\u22126\u2009m2\u2009s\u22121))$ (for example, water), near the jump, the slope remains positive and tends asymptotically to zero, while for a liquid with viscosity $\nu >10\u22124\u2009m2\u2009s\u22121$, could have a negative slope that could asymptotically reach to zero.

### C. Effect of plate size

In order to examine the effect of a finite plate, Duchesne *et al.*^{14} formulated another theory, which also gave good agreement with their experimental data. The empirical evidence obtained from their experiments prompted them to postulate that the exit Froude number in the downstream film at the jump (see Fig. 1) is $Frex=0.33\u2009\xb1\u20090.01$,

which can be re-arranged to give an expression for the jump radius *R*,

Assuming a parabolic velocity profile and balancing gravitational pressure and viscous friction, they also formulated a simplified equation for the downstream liquid film thickness

and combining (13) with volume flux conservation $UrH=Q2\pi $ yields

Solving (14) for the boundary condition $H=H\u221e$ for $r=R\u221e$ gives

It is instructive to compare (14) and (10), the two mathematical formulations describing the downstream film. Note that balancing frictional loss with gravitational pressure ignores the terms arising from circular expansion—the first term in (10), $(Q2\pi C1)21gr3h2$—and consequently, in contrast to (10) where the slope, $dHdr$, could either be positive or negative, (14) gives a negative slope, that we will show, is not observed in experiments. They further assumed that $H\u221e4\u226a6\nu Q\pi gln\u2009R\u221er$ so that

Combining the constant exit Froude number with momentum conservation across the jump, it can be shown that the supercritical thin film Froude number at the jump also remains constant. For the laminar velocity profile on either side of the jump, *Fr* = 4.18, and for a laminar supercritical film and a turbulent subcritical film, *Fr* = 7.11.

In summary, the constant exit Froude number hypothesis^{14} gave a number of predictions that can be examined in order to test the validity of the hypothesis and the theory, namely, $R\u221dQg1/2H3/2,\u2009H\u2243(6\nu Q\pi gln\u2009R\u221er)14$ and the jump Froude number is around 5.

### D. A unified theory

A theory that includes the effects of gravity, surface tension, and viscosity was presented by Bhagat *et al.*^{8} We briefly summarize it here but do not give details. Combining Eqs. (5.1) and (5.6) of Bhagat *et al.*,^{8} (5.6) can be reformulated to obtain the relation for the slope of the surface, given by

where viscous effects are included via the wall stress *τ _{w}* and we define the Weber number and the Froude number as

respectively.

Consequently, from (18), the flow is singular, and the jump occurs when

Furthermore, Bhagat *et al.*^{8} have shown that for water and other low viscosity and high surface tension liquids, in the supercritical film at the jump, $Fr2\u226b1$ and the jump occurs when $We\u22481$. Bhagat *et al.*^{5} have shown that combining (1) and the jump condition, $We\u22481$ gives the height of the jump

Note that in the absence of gravity, the jump condition is *We* = 1, while in the absence of surface tension it is *Fr* = 1, as found before.^{3} Consequently, this unified theory of Bhagat *et al.*^{8} provides the jump condition that includes surface tension, gravity, and viscosity. In Sec. III, we discuss the experimental measurements in this context.

## III. RESULTS

We now examine data from published experiments (see Table I) and connect them to the theory presented above. We examine the jump radius, the depth of the flow downstream of the jump and its dependence on the height of the weir and the size of the plate, and the constant Froude number hypothesis.

Reference . | Liquid . | Q ($cm3\u2009s\u22121$)
. | γ ($cm3\u2009s\u22121$)
. | ν ($m$)
. | H ($m$)
. |
---|---|---|---|---|---|

16 | Water | 2.5–8.33 | 3058 | 3.83 | 3.92 |

16 | Water and surfactant | 1.66–3.33 | 807.4 | 2.75 | 2.75 |

17 | Ethylene glycol solution | 27 | 129.7 | 2.86 | 2.76 |

14 | Silicone oil-1 | 4.3 | 12.44 | 2.37 | 2.1 |

### A. Experiments with water

Figure 2 compares the measured film thickness produced by vertical impingement of a water jet falling on to a horizontal surface with (2) and (18). In the figure, the vertical line demarcates the experimentally measured location of the hydraulic jump. In the supercritical region, both models agree well with the measured thickness of the liquid film but only (18) gives an accurate prediction of the jump radius, while (2) predicts almost twice the radius. In the subcritical region, (2) and (18) are almost identical, while (21) gives excellent prediction of the observed depth. It is worth noting in this case of relatively low viscosity, the downstream film thickness is almost constant.

In Fig. 3, we show effect of changing the jet flow rate. The liquid film profiles, given in (18) for both the super and subcritical regions are plotted and compared with the observed jump radii obtained from Refs. 12 and 15 for three different flow rates. Again excellent agreement for the jump radius is observed.

A further comparison of the models and observations obtained by Mohajer and Li^{16} is shown in Fig. 4. We see, as before that (2) which ignores the influence of surface tension overpredicts the jump radius while the unified theory (18) gives excellent agreement. Both (2) and (18) give almost identical predictions for the downstream film thickness, and (21) gives excellent prediction to the experimentally measured value of the depth of the film.

### B. Experiments with other liquids

Bohr *et al.*^{17} measured the liquid film thickness for impingement of the jet of an ethylene glycol solution, which has about half the surface tension and seven times the viscosity of water. We find in Fig. 5 again that (2) overpredicts the jump radius, while (18) gives excellent predictions of both the jump and the supercritical film thickness and (21) gives $H=2.76\u2009mm$ in good agreement with observations.

The effect of viscosity is shown by experiments by Duchesne *et al.*,^{14} who measured the liquid film thickness and hydraulic jump produced by a jet of silicon oil with a viscosity of 20–100 times that of water. In this case, the experimental data (Fig. 6) show that the viscous dissipation in the downstream film plays a significant role, and, due to the increased jump height, the jump moved slightly inwards, and both theories slightly over-predict the jump; nevertheless, (18) gives better prediction. For these low surface tension and high viscosity liquids, the surface energy contribution remains relatively low while the decelerating liquid film becomes thicker, and consequently, gravity could play a significant role.

### C. Effect of a weir

So far, we have compared experimental jump radii with the theories; Bohr *et al.*^{17} measured the liquid film thicknesses produced by impingement of an ethylene glycol solution jet with different outer film thickness boundary conditions, and showed that beyond jump, the height of the liquid film remained almost constant (see Fig. 3 of Ref. 17).

In Fig. 7, we present the computed outer film thickness for water on a relatively large plate (the diameter of the plate = $1\u2009m$) and a flow rate $Q=2\u2009l\u2009min\u22121$, and one could see that irrespective of the outer film boundary condition, the films are approximately flat. Figure 8 compares the outer film thickness obtained for a flow rate $Q=1.62\u2009l\u2009min\u22121$ and different outer boundary conditions for the outer film height. Except for the minimum film thickness for which the boundary condition is taken to be $2\gamma \rho g=2.76\u2009mm$, even on a relatively large plate, the liquid films remain flat. Furthermore, the solutions obtained from (2) and (18) superimpose each other.

Figure 9 compares the liquid film thickness obtained for a liquid of surface tension $\gamma =72\u2009mN\u2009m\u22121$ and the kinematic viscosity range between $10\u22126\u2009and\u200910\u22123\u2009m2\u2009s\u22121$. For liquids with low kinematic viscosities, in this scenario, Eqs. (2) and (18) give an identical solution while for liquid with high kinematic viscosities, the solution differs; (18) including surface tension gives slightly higher values compared to (2). In the case of a liquid with high kinematic viscosity on a relatively large plate, the viscous dissipation in the downstream film is likely to move the hydraulic jump inwards. Indeed, Bhagat *et al.*^{5} found that the measured jump radii for liquid with high viscosities were slightly smaller than the predicted values. We would like to reemphasise that, in order to examine the effect of viscous dissipation, we have chosen a rather large plate, which for most practical scenarios, even for high viscosity liquids, on smaller plates viscous dissipation remains insignificant.

Figure 10 examines the effect of surface tension; increasing the surface tension gives a slightly higher value for the liquid film thickness but the effect of surface tension is minimal. In Ref. 13, we have described that for thick films, the rate of change of surface energy is low, and for flat films, it is zero, which concurs with our results here.

### D. Predictions of the constant Froude number hypothesis

In Sec. II C, we identified the testable predictions of the constant Froude number hypothesis given by Duchesne *et al.*^{14} that we examine here. Figure 11 compares the downstream film thicknesses given in (2), (15), (18), and (16) for water at a flow rate $Q=2\u2009l\u2009min\u22121$ on a target plate of diameter $1\u2009m$. Equations (2), (15), and (18) give an approximate constant liquid film thickness, except at smaller radii where they appear to be diverging. The mismatch between (15) in contrast to (2) and (18) is expected due to the assumption that frictional forces are balanced by gravity through the free surface slope, (14). Equation (16), which assumes that the depth of the film at infinity is zero, also fails to give a reasonable prediction for the liquid film thickness and the jump radius.

It is worthwhile to note that the jump relationship (17) relies upon a reasonable prediction of the jump height that Eq. (16) fails to give and, consequently, for water (17) does not predict the jump radius, and the constant exit Froude number hypothesis does not hold. Note that Duchesne *et al.*^{14} did not compare their theory with the experiments carried out using water or other low viscosity liquids.

The theory of Duchesne *et al.*^{14} also predicted that the jump radius $R\u221dQg1/2H3/2$; in the case of water and other low viscosity liquids, the downstream film and jump remain approximately constant that implies $R\u221dQ$. However, on existing experimental data, Bhagat *et al.*^{5} and Bhagat *et al.*^{8} clearly show that $R\u221d\u0338Q$. Although for a limited range of experimental data—for $Q,0\u22129\u2009cm3\u2009s\u22121$—Mohajer and Li^{16} did suggest that $R\u221dQ$ but this could be refuted by comparing the experimental data, in the same range, from other independent research groups, including Refs. 8, 12, 15, 18, and 20. Furthermore, a nonlinear relationship between *Q* and *R* also implies that the exit Froude number is not constant.

The theory of Duchesne *et al.*^{14} also implied that the jump radius is a function of the plate diameter, $R=f(R\u221e)$. However, in Ref. 5, we compared experimental data for the experiments carried out by independent research groups over more than three decades on plates of different geometries and sizes, and for water, the jump radii were found to independent of the plate geometry and sizes. Finally, the constant exit Froude number hypothesis—*Fr _{ex}* = 0.33—also implies that the supercritical film Froude number should be 4.18 or 7.11, which are arbitrary numbers, and there is no physical basis for this.

Figure 12 compares the height of the downstream film for high viscosity and low surface tension silicon oil; in this case, (16) gives a reasonable prediction of the jump height. It is instructive to note that Duchesne *et al.*^{14} compared their theory with only high viscosity liquids and showed good agreement. In Ref. 5, we have delineated why the gravity-based theory could give a reasonable prediction for high viscosity and low surface tension liquids.

## IV. DISCUSSION

It is worthwhile to consider some recent works apparently explaining why the surface tension-based theory of Bhagat *et al.*^{8} was able to predict the jump. Duchesne and Limat^{9} argued that in the scenario when the outer film is still developing, the depth of the film is dependent upon the contact line force arising due to surface tension and the wetting properties of the substrate, and so in combination with the phenomenological model, it can be shown that the jump depends upon surface tension. Essentially, the argument in the model^{9} suggests that the contact line force could influence the jump. About a decade ago, Wilson *et al.*^{10} gave virtually the same mathematical relation balancing momentum and surface forces at the jump radius on a vertical plate, but the model was incapable of explaining why these jumps are independent of the surface property and the contact angle. Furthermore, in Ref. 5, we have given experimental evidence showing why these explanations are untenable; using an experiment, we will elucidate it further. Avedisian and Zhao^{11} carried out zero-gravity experiments in a drop tower, and their experimental setup ensured complete wetting and no scope for the emergence of any contact lines. For some of their experiments, they impinged liquid jets on deep pools of liquid (see the illustrative sketch, Fig. 13, inspired from Fig. 11 of Ref. 11), where in normal gravity, the jump formation was suppressed by the large gravitational pressure; however, during free fall, the jump appeared at a location where local *We* = 1, giving excellent agreement with (18), the theory of Bhagat *et al.*^{8} We wish to reemphasise that in the scenario of an unobstructed flow on a flat plate in terrestrial experiments, and in micro-gravity experiments, for a given flow rate *Q*, the jump remained at the same location.

## V. CONCLUSIONS

In this paper, we examined the effect of downstream flows on the hydraulic jumps theoretically and compared it with experiments. We examined the consistency between the phenomenological condition at the jump, given by Watson,^{2} Taylor^{19} with Tani,^{3} and the gravity-based theory of Kurihara,^{4} which postulates that the jump occurs due to adverse pressure gradient produced due to viscous dissipation causing the liquid film to become thicker. Combining the phenomenological condition—which must be valid—and the predictive theory suggests that for the adverse pressure gradient to be significant enough to produce separation, the subcritical liquid film thickness at hydraulic jumps should be of the order of jump height, which is not experimentally observed.

We eliminated velocity and reformulated the theory of Bhagat *et al.*^{8} in the form of the film thickness, *h* and the radial coordinate, *r* and compared it with the theory of Tani^{3} and Kurihara.^{4} While the theory of Tani^{3} and Kurihara^{4} theory fails to predict the jump radius, the theory of Bhagat *et al.*^{8} gives an excellent prediction for the jump location as well as the liquid film thickness (see Figs. 2, 3, and 5).

Examining the effect of viscous dissipation in downstream films, we showed that for low viscosity liquids such as water, for all reasonable flow rates and plate dimensions, the liquid film after the jump remains flat, and both the terms arising from viscous dissipation and expansion of the film eliminate each other, giving a flat film. We also showed that in the case of high viscosity liquids, viscous dissipation in the film may become significant, subject to the dimension of the plate. As a closing remark, we would like to invoke the results recently given by Bhagat *et al.*,^{5} where they compare their theory with more than three decades of experimental data; they showed that the theory of Bhagat *et al.*^{8} gives excellent prediction to the experimental data for low surface tension liquids, while slightly over-predicting the high viscosity liquids, showing consistency. Note that these separate experiments were carried out on various plates of unknown dimensions; nevertheless, they did not affect the jump radius. Finally, we also examined the theory based on the constant exit Froude number *Fr _{ex}* hypothesis given by Duchesne

*et al.*

^{14}and showed that the theory does not predict the jump radius for water and other low viscosity liquids, and

*Fr*is not constant for such liquids.

_{ex}## ACKNOWLEDGMENTS

The authors would like to thank the Leverhulme Trust and the Issac Newton Trust (No. ECF-2021–196) and Darwin College, University of Cambridge for financial support.

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

## DATA AVAILABILITY

The data that support the findings of this study are available within the article.