2 Previous studies

We are concerned here with the initiation of the jump shown by the schematic in figure 2(a), so we consider the impact of a jet normally on an infinite plane. In practice, all experiments involve a plane of finite dimensions and eventually the liquid drains from the edges of the surface. Consequently, we restrict ourselves to considerations of the flow before it reaches the edge of the surface, and ignore the changes that occur once the downstream boundary condition changes (see supplementary movie 2).

Figure 2. (a) Schematic of the liquid film and hydraulic jump created by an impinging jet. (b) Control volume of the film element on which the energy balance is applied.

Watson (Reference Watson1964) developed a similarity solution for radial flow in a thin liquid film, which we describe further below. He also proposed the first description of a thin-film circular hydraulic jump incorporating viscous friction in the film, ignoring the tangential stress due to surface tension, and balanced the momentum and hydrostatic pressure across the jump. Watson’s solution, which involves gravity, requires experimental measurement of the film thickness at the jump location to predict the jump radius, and overpredicts the radius for smaller jumps by as much as 50 %. Bush & Aristoff (Reference Bush and Aristoff2003) added surface tension to Watson’s theory but stated that its influence was small, as they argued its effect was confined to the hoop stress associated with the increase in circumference of the jump. They recognised that addition of a surfactant substantially (20 %) increased the jump radius but they did not pursue this aspect further, owing to the complications involved with surfactants. Mathur et al. (Reference Mathur, DasGupta, Selvi, John, Kulkarni and Govindarajan2007) presented results for hydraulic jumps in liquid metals where the jump radii are in range of micrometres. They recognised that jumps on this scale are created due to surface tension but are associated with very high curvature and small jump radii. However, they remarked that gravity is the key to all jumps on the scale of the kitchen sink hydraulic jump.

Earlier analysis by Kurihara (Reference Kurihara1946) and Tani (Reference Tani1949) used thin-film boundary layer equations including gravity to model circular hydraulic jumps. This analysis was critiqued by Bohr, Dimon & Putkaradze (Reference Bohr, Dimon and Putkaradze1993), who solved the axisymmetric shallow water equations, which again, naturally, include gravity. Bohr et al. (Reference Bohr, Dimon and Putkaradze1993) found that the outer solution of the equations became singular at a finite radius. Consequently, they solved the equations inwards from the edge of the plate or the boundary from where liquid drains due to gravity, and connected the inner and the outer solutions for radial flow through a shock. This analysis gave a scaling relation for the jump radius $R\sim Q^{5/8}\unicode[STIX]{x1D708}^{-3/8}g^{-1/8}$ , where $Q,\unicode[STIX]{x1D708}$ and $g$ are the jet volume flux, the kinematic viscosity of the fluid and the acceleration due to gravity, respectively. They argued that the jump could be understood qualitatively in terms of the interplay between gravity, viscosity and the momentum of the liquid. A similar axisymmetric shallow water model was also proposed by Kasimov (Reference Kasimov2008). Again this study did not consider the initial formation of the jump, but connected the thin film with a deeper flow established as a result of a downstream boundary condition at the edge of the domain.

Two studies considered the case where gravity is unimportant. Godwin (Reference Godwin1993) assumed the jet diameter was an important parameter and showed that the jump radius, when independent of gravity, scales as $R\sim Q^{1/3}d^{2/3}\unicode[STIX]{x1D708}^{-1/3}$ , where $d$ is the jet diameter. We argue that, since $R\gg d$ , the jet diameter is not a relevant parameter. Avedisian & Zhao (Reference Avedisian and Zhao2000) studied the circular hydraulic jump at low gravity in a drop tower. In these experiments, a horizontal plate was submerged in a pool of liquid to impose a constant downstream liquid thickness condition and was impacted by a liquid jet to create a hydraulic jump. They found that, at normal gravity, the jump radius decreased when the depth of the downstream liquid film increased. However, at low gravity, the downstream liquid film height had no effect, and the jump radius increased compared to its value under normal gravity. Capillary waves were also observed, and they concluded that at low gravity the jumps were dominated by viscosity and surface tension.

Hansen et al. (Reference Hansen, Hørlück, Zauner, Dimon, Ellegaard and Creagh1997) studied surface waves in circular hydraulic jumps. They reported that the hydraulic jump radius scales as $R\propto Q^{0.77}$ for water, and $R\propto Q^{0.72}$ for less viscous oils. They concluded that, on a flat plate without any reflectors, the waves are gravity–capillary waves. Further, in the limit of zero surface tension, Rojas, Argentina & Tirapegui (Reference Rojas, Argentina and Tirapegui2013) reported that $R\sim Q^{3/4}\unicode[STIX]{x1D708}^{-1/4}H^{-1/2}g^{-1/4}$ , where $H$ is the height of the film downstream of the hydraulic jump.

In the experiments described below we observed that, under the same flow conditions, normal impingement of a liquid jet gives a circular hydraulic jump with the same initial radius irrespective of the orientation of the surface (figure 1). On a vertical plate, where the spreading liquid film and gravity are coplanar, an approximately circular hydraulic jump is still formed (figure 1 b). The thicker liquid film beyond the hydraulic jump then drains downwards due to gravity and above the point of impingement the location of the jump remains constant in time. Our experiments on a surface inclined at $45^{\circ }$ also produced circular jumps. A similar lack of dependence of the radius on both vertical and inclined surfaces has also been observed by Wilson et al. (Reference Wilson, Le, Dao, Lai, Morison and Davidson2012), Wang, Davidson & Wilson (Reference Wang, Davidson and Wilson2015), Bhagat & Wilson (Reference Bhagat and Wilson2016). Similarly, when a jet impinges onto a horizontal surface from below, an abrupt increase in film thickness is also observed (figure 1 c). Under the influence of gravity, the thicker liquid film falls as droplets or as a continuous film forming a water bell (Jameson et al. Reference Jameson, Jenkins, Button and Sader2010). In our experiments we changed the surface tension of the liquid by preparing homogeneous water–alcohol solutions and a surfactant. Supplementary movie 3 shows the change in a kitchen-sink-scale hydraulic jump by changing the surface tension $\unicode[STIX]{x1D6FE}$ when $Q$ and $\unicode[STIX]{x1D708}$ are kept constant. The existing treatments are unable to explain this behaviour, as they hold that surface tension only becomes significant for much smaller jump radii.

3 Experiments

Circular hydraulic jumps were produced by impinging a liquid jet normally onto a planar surface. Both a vertical jet impinging on a horizontal plate from above and below, and normal impingement on a vertical plate and a plate inclined at $45^{\circ }$ to the horizontal were studied. For most experiments the jet nozzle diameter was 2 mm and the jet flow rate $Q$ varied from $0.49{-}2~\text{L}~\text{min}^{-1}$ . For low flow rates ( $Q<1.3~\text{L}~\text{min}^{-1}$ ), liquid was supplied from a constant-head apparatus to glass Pasteur pipettes. For higher $Q$ a centrifugal pump and a brass nozzle were used (Wang et al. Reference Wang, Davidson and Wilson2015). Target plates were Perspex^TM, glass or Teflon^® sheets. The horizontal plate was a 0.25 m diameter circular disk; horizontal and inclined planes consisted of a $1\times 0.4~\text{m}^{2}$ rectangular plate. It was found that the jump radius was independent of the plate material and we will not consider this factor further. Nozzle diameters of 1 and 3 mm were also used, and no significant difference in the jump radius was observed when measured from the edge of the resulting jet.

The viscosity and surface tension were varied by using mixtures of water with 1-propanol (5 and 20 w/w %, labelled as $\text{WP95}/5$ and $\text{WP80}/20$ , respectively) and with glycerol (70 and 90 w/w %, labelled as $\text{WG30}/70$ and $\text{WG10}/90$ , respectively). We also used a $3~\text{mmol}~\text{L}^{-1}$ solution of sodium dodecyl benzene sulphonate (SDBS). The surface tension was varied by an approximate factor of three and the kinematic viscosity by a factor of nearly 100. The fluid properties are listed in table 1.

Table 1. Properties of the liquids used.

A Photron Fastcam SA3 was used to acquire images of the liquid film and the hydraulic jump at up to 2000 f.p.s. These were subsequently processed using MATLAB and ImageJ to obtain the jump radius $R$ as a function of the jet and fluid properties. In total over 150 experiments were conducted. Error bars were determined by the standard deviation of repeated experiments.

4 Scaling analysis

We consider the axisymmetric flow shown schematically in figure 2. The jump is characterised by its radius $R$ and depth $h$ , along with the radial velocity $u$ of the film. Based on our observations we assume that gravity is unimportant and hence the flow depends on the jet flow rate  $Q$ , the jet diameter  $d$ , the film thickness  $h$ and the fluid properties (i.e. the density  $\unicode[STIX]{x1D70C}$ , viscosity $\unicode[STIX]{x1D708}$ and the surface tension  $\unicode[STIX]{x1D6FE}$ ). Since observations (e.g. figure 1) show that the jump radius $R\gg d$ and we observed no dependence on  $d$ , we will ignore the jet diameter. We then have six parameters; $R,h,\unicode[STIX]{x1D70C},\unicode[STIX]{x1D708},\unicode[STIX]{x1D6FE}$ and $Q$ (or equivalently  $u$ ) and three dimensions, giving three dimensionless parameters: the Reynolds number $Re$ , the dimensionless film thickness, $\unicode[STIX]{x1D6FC}$ , and the Weber number, $We$ , given by

(4.1a-c ) $$\begin{eqnarray}Re=\frac{uh}{\unicode[STIX]{x1D708}},\quad \unicode[STIX]{x1D6FC}=\frac{h}{R},\quad We=\frac{\unicode[STIX]{x1D70C}u^{2}h}{\unicode[STIX]{x1D6FE}}.\end{eqnarray}$$

We assume that the radial flow is balanced by viscous drag $u/R\sim \unicode[STIX]{x1D708}/h^{2}$ which implies $\unicode[STIX]{x1D6FC}Re=O(1)$ . Then, if we further assume that surface tension is important and that, at the jump $We\sim 1$ , and use the fact that continuity implies $Q\sim uhR$ , we find $R/R_{0}=\text{const.}$ , where the characteristic length $R_{0}$ is given by

(4.2) $$\begin{eqnarray}R_{0}=\frac{Q^{3/4}\unicode[STIX]{x1D70C}^{1/4}}{\unicode[STIX]{x1D708}^{1/4}\unicode[STIX]{x1D6FE}^{1/4}}.\end{eqnarray}$$

We plot the measurements of the dimensionless jump radius $R/R_{0}$ against the jet flow rate $Q$ in figure 3. The data from experiments covering the full range of  $Q$ , surface material and orientation and fluid properties (table 1) all collapse on to the line $R/R_{0}=0.289\pm 0.015$ . This collapse of the data, consistent with (4.2), implies that the dominant balance in the formation of thin-film jumps is associated with surface tension and viscous drag, and that gravity is irrelevant. In § 5, we will develop a more quantitative estimate of the jump radius.

Figure 3. Dimensionless jump radius plotted against the flow rate for all our experiments with different liquids and surface orientation.

5 Theory

We consider a cylindrical coordinates $r$ and $z$ , the radial and jet-axial coordinates, respectively, $u$ and $w$ , the associated velocity components (figure 2), and assume circular symmetry about the jet axis. In order to analyse the jump we use the ansatz developed by Watson (Reference Watson1964) for the velocity within the thin film. We write the radial velocity as $u=u_{s}\,f(\unicode[STIX]{x1D702})$ , $\unicode[STIX]{x1D702}\equiv z/h(r)$ ( $0\leqslant \unicode[STIX]{x1D702}\leqslant 1$ ), where $\unicode[STIX]{x1D702}$ is the dimensionless thickness of the film and $u_{s}$ is the velocity at the free surface. Using continuity we define the flux-average velocity $\bar{u}\equiv C_{1}u_{s}$ by

(5.1) $$\begin{eqnarray}\int _{0}^{h}ur\,\text{d}z=u_{s}rh\int _{0}^{1}f(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}=C_{1}u_{s}rh\equiv \bar{u}rh=\frac{Q}{2\unicode[STIX]{x03C0}}=\text{const.},\end{eqnarray}$$

where $C_{1}=\int _{0}^{1}f(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}=0.6137$ is a shape factor determined from the similarity solution.

We now balance the flux of mechanical energy across an annular control volume shown in figure 2(b), from $r$ to $r+\unicode[STIX]{x0394}r$ and from 0 to  $h$ (where we have cancelled out the common factor  $2\unicode[STIX]{x03C0}$ ),

(5.2) $$\begin{eqnarray}\displaystyle & & \displaystyle \left.\frac{(\unicode[STIX]{x1D70C}\bar{u^{2}}\bar{u}rh)}{2}\right|_{r}-\left.\frac{(\unicode[STIX]{x1D70C}\bar{u^{2}}\bar{u}rh)}{2}\right|_{r+\unicode[STIX]{x0394}r}-(\unicode[STIX]{x1D6FE}\bar{u}r)|_{r}+(\unicode[STIX]{x1D6FE}\bar{u}r)|_{r+\unicode[STIX]{x0394}r}\nonumber\\ \displaystyle & & \displaystyle \quad +\,p\bar{u}rh|_{r}-p\bar{u}rh|_{r+\unicode[STIX]{x0394}r}+\left.\frac{\unicode[STIX]{x1D70C}g\bar{u}rh^{2}}{2}\right|_{r}-\left.\frac{\unicode[STIX]{x1D70C}g\bar{u}rh^{2}}{2}\right|_{r+\unicode[STIX]{x0394}r}-r\unicode[STIX]{x1D70F}_{w}\bar{u}\unicode[STIX]{x0394}r=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D70F}_{w}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}(u_{s}/h)f^{\prime }(0)$ is the wall shear stress and, from the velocity profile ansatz, we have $f^{\prime }(0)=1.402$ .

In these jump conditions, the first term is the flux of kinetic energy which is balanced by pressure, gravity and viscous work, given by the third, fourth and fifth terms, respectively. These are standard and the new term is the second term $\unicode[STIX]{x1D6FE}\bar{u}r$ , which represents the flux of surface energy that has been neglected in previous studies. This term results from the increase of surface area across the control volume as a result of the increase in the circumference from $r$ to $r+\unicode[STIX]{x0394}r$ .

Dividing (5.2) by $\unicode[STIX]{x0394}r$ , taking the limit $\unicode[STIX]{x0394}r\rightarrow 0$ and using the fact that $\bar{u}rh=\text{const.}$ (see (5.1)) yields

(5.3) $$\begin{eqnarray}\frac{1}{2}\frac{\text{d}(\unicode[STIX]{x1D70C}\bar{u^{2}})\bar{u}rh}{\text{d}r}-\frac{\text{d}(\unicode[STIX]{x1D6FE}\bar{u}r)}{\text{d}r}=-\bar{u}rh\frac{\text{d}p}{\text{d}r}-\frac{1}{2}\unicode[STIX]{x1D70C}g\bar{u}rh\frac{\text{d}h}{\text{d}r}-\unicode[STIX]{x1D70F}_{w}r\bar{u}.\end{eqnarray}$$

From the boundary layer velocity profile ansatz we write $\bar{u^{2}}=\int _{0}^{1}u^{2}\,\text{d}\unicode[STIX]{x1D702}\equiv C_{2}u_{s}^{2}$ , where $C_{2}\equiv \int _{0}^{1}f^{2}(\unicode[STIX]{x1D702})\,\text{d}\unicode[STIX]{x1D702}=0.4755$ is a second shape factor. Then (5.3) implies

(5.4) $$\begin{eqnarray}C_{2}\unicode[STIX]{x1D70C}u_{s}^{2}hr\frac{\text{d}u_{s}}{\text{d}r}-\unicode[STIX]{x1D6FE}r\frac{\text{d}u_{s}}{\text{d}r}-\unicode[STIX]{x1D6FE}u_{s}=-u_{s}rh\frac{\text{d}p}{\text{d}r}-\frac{1}{2}\unicode[STIX]{x1D70C}gu_{s}rh\frac{\text{d}h}{\text{d}r}-\unicode[STIX]{x1D70F}_{w}ru_{s}.\end{eqnarray}$$

Note that the shape factor $C_{1}$ cancels out in this equation, and so plays no role. Again using (5.1), we write $u_{s}r(\text{d}h/\text{d}r)=-h(\text{d}(u_{s}r)/\text{d}r)$ in (5.4), to obtain

(5.5) $$\begin{eqnarray}\left(C_{2}\unicode[STIX]{x1D70C}u_{s}^{2}h-\unicode[STIX]{x1D6FE}-\frac{1}{2}\unicode[STIX]{x1D70C}grh^{2}\right)r\frac{\text{d}u_{s}}{\text{d}r}=-u_{s}rh\frac{\text{d}p}{\text{d}r}+\unicode[STIX]{x1D6FE}u_{s}+\frac{1}{2}\unicode[STIX]{x1D70C}gu_{s}h^{2}-\unicode[STIX]{x1D70F}_{w}ru_{s}.\end{eqnarray}$$

Finally, rearranging (5.5) we find that the gradient of the radial velocity satisfies

(5.6) $$\begin{eqnarray}\frac{\text{d}u_{s}}{\text{d}r}=\frac{-u_{s}rh{\displaystyle \frac{\text{d}p}{\text{d}r}}+\unicode[STIX]{x1D6FE}u_{s}+{\displaystyle \frac{1}{2}}\unicode[STIX]{x1D70C}gu_{s}h^{2}-\unicode[STIX]{x1D70F}_{w}ru_{s}}{\left(1-{\displaystyle \frac{1}{We}}-{\displaystyle \frac{1}{Fr^{2}}}\right)(C_{2}\unicode[STIX]{x1D70C}u_{s}^{2}hr)}.\end{eqnarray}$$

Here, we define the Weber number and Froude number as, respectively,

(5.7a,b ) $$\begin{eqnarray}We\equiv \frac{C_{2}\unicode[STIX]{x1D70C}u_{s}^{2}h}{\unicode[STIX]{x1D6FE}},\quad Fr\equiv \sqrt{\frac{2C_{2}u_{s}^{2}}{gh}}.\end{eqnarray}$$

It is clear that (5.6) is singular when

(5.8) $$\begin{eqnarray}We^{-1}+Fr^{-2}=1,\end{eqnarray}$$

and that the hydraulic jump occurs where this condition is satisfied.

In order to obtain quantitative results, (5.6) was solved for $u_{s}$ using the similarity velocity profile and the initial condition obtained from Watson (Reference Watson1964). The boundary layer first occupies the full depth of the film at $r_{b}$ , given by $r_{b}/d=0.1833Re_{j}^{1/3}$ , where the jet Reynolds number $Re_{j}=4Q/\unicode[STIX]{x03C0}\unicode[STIX]{x1D708}d$ . At this location, which for the present values of $Re_{j}\sim 10^{4}$ , $r_{b}\ll R$ , $u_{s}$ is set equal to the mean jet velocity, and (5.6) provides its subsequent radial values. The solution of (5.6) is not sensitive to the initial condition obtained using Watson’s similarity profile, as other boundary layer velocity profiles yield similar values of  $r_{b}$ . We then calculate $R$ as the location where $We^{-1}+Fr^{-2}=1$ , and (5.6) becomes singular. This condition provides a more precise estimate and a physical basis for the scaling argument (4.2), and also includes the effect of gravity.