2.1. Experimental set-up

Our experiment consists of shaping a hollow in a free liquid surface by blowing a short air pulse from a nozzle: as soon as the airflow stops, the cavity relaxes under gravity to form a liquid jet. Both the depth $H$ and the width $L$ of the initial cavity can be tuned by adjusting appropriately the air pulse intensity and spreading: the higher the air pulse velocity, the deeper the cavity and the larger the height of the nozzle from the free surface, the wider the cavity. Typical heights and widths range from 3 to 10 cm and typical cavity aspect ratios $H/L$ from 0.25 to 1.1 (a hemispherical cavity would have an aspect ratio $H/L=0.5$ ). The liquid, contained in a transparent tank ( $30\ \text{cm}\times 30\ \text{cm}\times 35\ \text{cm}$ ), is a water–glycerol mixture. Depending on the proportions of water and glycerol, the liquid viscosity can be set in the range ${\it\mu}=1{-}1300\ \text{mPa}\ \text{s}$ . Note that temperature-induced variations have been accounted for throughout this study. The mixture density ranges from ${\it\rho}=1000$ to $1250\ \text{kg}\ \text{m}^{-3}$ . Pure ethanol of surface tension ${\it\sigma}=22.3\ \text{mN}\ \text{m}^{-1}$ and viscosity ${\it\mu}=1.2\ \text{mPa}\ \text{s}$ has also been used in order to vary the surface tension. The hollow relaxation and jet dynamics are analysed through ultrafast imaging. Typically, images of the back-lit scene are obtained at 2000 f.p.s. using a digital high-speed camera (Photron SA-5).

2.2. Qualitative overview of the jet dynamics

Figure 1 illustrates two typical jetting events induced by the relaxation of a large surface hollow (whose width $L$ is greater than the capillary length). The sequence in figure 1(a) has been obtained with pure water as the working liquid while the one in figure 1(b) corresponds to a water–glycerol mixture 800 times more viscous. On the first images, the depression hollowed out in the liquid by the air pulse has reached its maximum depth $H$ and is about to relax. Just after maximal expansion, the liquid starts flowing in so as to fill the entire cavity. But this filling process is not just a simple relaxation to the minimal potential energy state (i.e. flat surface): during the collapse the cavity curvature reverses as a prelude to the eruption of a violent liquid jet (Lavrentiev & Chabat Reference Lavrentiev and Chabat1980). It is noteworthy that the developing vertical jet can reach heights larger than ten times the initial depth $H$ of the cavity. This is a strong indication that the pressure built up during the collapse far exceeds the hydrostatic pressure: the gravity-driven flow focuses energy and momentum. Such focusing is generic in jet eruption and can be observed in situations as diverse as impact-driven liquid jets (Antkowiak et al. Reference Antkowiak, Bremond, Le Dizès and Villermaux2007), needle-less injection systems (Tagawa et al. Reference Tagawa, Oudalov, Visser, Peters, van der Meer, Sun, Prosperetti and Lohse2012) or even granular jets (Lohse et al. Reference Lohse, Bergmann, Mikkelsen, Zeilstra, van der Meer, Versluis, van der Weele, van der Hoef and Kuipers2004).

Insights into the role played by viscosity can readily be gained by comparing figures 1(a) and 1(b). Apart from its obvious smoothing action on the cavity, jet and lip surfaces, viscosity also proves to slow down the jet, as the time between each snapshot of figure 1 is much longer for the viscous case ( ${\rm\Delta}t_{viscous}=27.5\ \text{ms}>{\rm\Delta}t_{water}=7.5\ \text{ms}$ ). It is noteworthy that this slowing action of viscosity is achieved before jet eruption as evidenced by the jet velocity, which is seen to be constant in both sequences, at least before the final deceleration due to gravity.

A global overview of the jetting event is provided in figure 2 with a spatio-temporal diagram depicting the whole jet dynamics with the important stages in its chronology: (i) cavity formation up to maximum depth $H$ , (ii) cavity relaxation and reversal, (iii) jet eruption and development and (iv) final jet collapse and/or disruption. From the curvature of the spatio-temporal trace it can readily be seen that the collapse phase involves an acceleration twice as important as gravity. Over-accelerations are typical of ‘cumulative’ phenomena (Lavrentiev & Chabat Reference Lavrentiev and Chabat1980) and are commonly observed for example in wave impact events, where values as large as a thousand times that of gravity have been reported (Peregrine Reference Peregrine2003). Just after eruption, the jet velocity $V_{tip}$ , measured on the spatio-temporal diagram, is also seen to be important compared to the characteristic gravito-inertial velocity $\sqrt{gH}$ (here $V_{tip}=1.33\ \text{m}\ \text{s}^{-1}>\sqrt{gH}=0.53\ \text{m}\ \text{s}^{-1}$ ). Actually, in all the reported experiments, the Froude number $\mathit{Fr}=V_{tip}/\sqrt{gH}$ was systematically observed to be greater than unity. Once above the free surface, the jet tip follows a ballistic trajectory characterized by a free-fall parabola (see dotted curve figure 2), even for liquids a hundred times more viscous than water (see the supplementary material available at http://dx.doi.org/10.1017/jfm.2014.629). This ballistic behaviour demonstrates that viscosity and surface tension barely affect the jet development. The purely inertial description assumption for the jet dynamics employed in this paper results from this key observation.

Figure 2.

Spatio-temporal diagram presenting (i) the cavity formation, (ii) the cavity relaxation and reversal, (iii) the jet development and (iv) the jet collapse. These four steps are indicated in the timeline. The time origin ( $t=0$ ) is taken when the jet crosses the free surface $z=0$ , and the local slope gives the jet velocity $V_{tip}$ . The dotted curve represents the free-fall parabola corresponding to this jet eruption velocity $V_{tip}$ . In this particular case the liquid viscosity is ${\it\mu}=10\ \text{mPa}\ \text{s}$ , the jet velocity is $V_{tip}=1.33\ \text{m}\ \text{s}^{-1}$ , the cavity maximum depth and width are $H=2.9\ \text{cm}$ and $L=5.5\ \text{cm}$ .

4.1. Cavity shape similarity

Blown cavities have been the focus of a number of studies in the past, partly driven by technological applications. To quote a few of these, we might mention vertical-take-off-and-landing craft that can experience loss of visibility or damage when landing over water or a soft terrain as a consequence of the soil erosion by a gaseous jet (Barton & Edwards Reference Barton and Edwards1968), or the basic oxygen conversion process in the steel industry that makes use of a supersonic jet of oxygen impinging on molten iron to convert it into steel. Experiments modelling these phenomena have been reported for the last 50 years, with the notable contributions of Banks & Chandrasekhara (Reference Banks and Chandrasekhara1963) and Cheslak, Nicholls & Sichel (Reference Cheslak, Nicholls and Sichel1969). These studies have described how the geometrical properties of the cavity are linked to the blowing conditions (gas velocity, nozzle height and diameter). In particular, most of these works have considered steady cavity profiles to be reasonably well fitted with a parabola. Conversely, unsteady liquid cratering dynamics have been investigated mainly in the case of a point source, obtained for example by means of drop impact (Bisighini et al. Reference Bisighini, Cossali, Tropea and Roisman2010). In this type of experiment, the cavity shape is very close to an hemisphere and the dynamics of expansion–retraction is well described by the Rayleigh–Plesset equation. In our experiment, the pressure forces shaping the cavity are distributed along the cavity, i.e. not localized. (Curiously we have not been able to find a single study making use of impulsive blowing over a free surface. Stuhlman (Reference Stuhlman1932) mentioned that Professor Foulk was using such a setup to produce jets, but the corresponding paper of Foulk only alluded to this experiment (Foulk Reference Foulk1932).)

Figure 5 reports cavity profiles at their deepest – that is, just before relaxation – for eight different experiments corresponding to various values of the gas velocity (figure 5 a) and of the nozzle height (figure 5 b). We observe that by increasing the velocity of the air jet, deeper cavities can be formed keeping their width $L$ almost constant. Conversely moving the nozzle up allows wider cavities to be formed. It is notable that all these different cavities exhibit shape similarity. This is demonstrated figure 5(c) where the eight cavity profiles are plotted on the same graph after rescaling the horizontal coordinate with the cavity radius $L/2$ and the vertical coordinate with the cavity depth $H$ . All cavity contours fall onto a universal profile. The best fit for this master curve is given by $z/H=(2x/L)^{2.5\pm 0.1}-1$ , with $z$ and $x$ the vertical and radial coordinates (dashed black curve in figure 5 c). The shaded area represents the error bar taken on the exponent.

Figure 5.

Different cavity profiles with (a) almost the same width ( $L$ ) and different heights ( $H$ ) and a liquid viscosity of ${\it\mu}=219\ \text{mPa}\ \text{s}$ , and (b) the same height, different widths and ${\it\mu}=440\ \text{mPa}\ \text{s}$ . (c) The same cavity profiles after rescaling: $z/H$ versus $2x/L$ . They all fall in the grey region: $z=x^{2.5\pm 0.1}-1$ exhibiting a similar cavity shape. The black dashed line represent the curve $z=x^{2.5}-1$ .

Figure 6.

(a) Experimental jet profile evolution with time following cavity relaxation ( $H=4.9\ \text{cm}$ , $L=9.6\ \text{cm}$ , $V_{tip}=1.41\ \text{m}\ \text{s}^{-1}$ ) in a liquid of viscosity ${\it\mu}=763\ \text{mPa}\ \text{s}$ . The interval between two profiles is 5 ms. (b) The same jet profiles after rescaling of the $x$ coordinate by $\sqrt{t+t_{0}}$ . The dashed curve follows the equation $z=Cx^{-1}-z_{0}$ . (c) Time dependence of the squared radius of the jet ( $r^{2}$ ) at different locations $z$ in the jet. (d) Experimental relation between the jet radius prefactor $C$ non-dimensionalized by $C_{dim}=(gH)^{1/4}L^{3/2}$ and the cavity aspect ratio $H/L$ .

This process enables us to build similar hollows with various dimensions and aspect ratios. The further development of the jet only depends on this initial state, disregarding any prior cavity formation detail.

4.2. Jet dynamics and shape similarity

We now turn to the kinematics of jet formation, that starts with the cavity reversal. Figure 6(a) shows a typical sequence of successive jet profiles ( ${\rm\Delta}t=5\ \text{ms}$ between each) extracted from the same experiment. Here $z=0$ represents the far free surface. The roughly constant spacing between the successive jet tips is again an indication of the constant initial jet velocity (high Froude number eruption), consistent with the spatio-temporal diagram analysis (figure 2). The transverse time dependence of the jet is assessed by representing the evolution of the jet section $r_{jet}^{2}(z,t)$ (disregarding the factor ${\rm\pi}$ ) at five positions on the vertical axis $z$ (figure 6 c). The linear trend curves capture satisfyingly the evolution of the jet section for every height, revealing that the jet radius varies as the square root of time. The experimental data therefore suggest the following functional form for $r_{jet}(z,t)$ :

where the prefactor $A(z)$ is a function of the height $z$ only. This dependence of the jet shape on height is shown in the following to be completely constrained by the dynamics.

As mentioned earlier, experimental evidence suggests that the dynamics of the jet is barely affected by viscosity, capillarity or even gravity ( $\mathit{Fr}>1$ ), at least for the range of parameters studied. Exploiting both the inertia-dominated character of the jet and its slenderness, we can formulate the governing equations expressing mass and momentum conservation in a one-dimensional setting (see Eggers (Reference Eggers1997) for an historical account on one-dimensional approximations):

where $r(z,t)$ denotes the local jet radius and $u(z,t)$ the local velocity in a fluid slice.

For non-shock-forming feeding conditions, the velocity within the jet (i.e. the solution of (4.2b )) is naturally attracted by the self-similar relaxation wave velocity profile:

where $z_{0}$ and $t_{0}$ are space and time shifts (see appendix A). At the tip, the velocity field within the jet $u$ ( $z_{tip}$ ) must match the tip velocity $V_{tip}$ . This kinematic condition constrains $z_{0}$ and $t_{0}$ via the relation $V_{tip}=z_{0}/t_{0}$ .

Analogously we now investigate jet shape similarity, by looking for solutions $r(z,t)$ of the mass conservation equation (4.2a ) in the form of monomials $z^{a}t^{b}$ . A one-parameter family of solutions such that $a=-(2b+1)/2$ is readily obtained. Using the linear time dependence of the jet section established experimentally, we infer that the particular exponent pair ( $a,b$ ) = ( $-1,\frac{1}{2}$ ) is selected:

where $C$ is a constant prefactor.

This result sheds light on the experimental jet profiles reported figure 6(a). Representing these profiles as a function of the self-similar variable $r/\sqrt{t+t_{0}}$ reveals that the jet actually develops in a self-similar sheath, see figure 6(b). It is noteworthy that the jet flanks adopt a perfect hyperbolic shape, consistent with the purely inertial prediction (4.4). The experimental observations reported figure 6 therefore demonstrate not only the shape similarity of the evolving jet (square-root-of-time spreading of the radius and hyperbolic shape), but also the validity of the assumptions made in obtaining (4.3) and (4.4). Interestingly Stuhlman (Reference Stuhlman1932), performing an a posteriori analysis of prior experimental photographs obtained by Worthington, also found an hyperbolic shape for jets following a drop impact on a liquid surface.

Using the same natural scales for velocity $\sqrt{gH}$ and length $L$ as previously, we can non-dimensionalize the prefactor $C$ occurring in the expression for the radius (4.4). Without loss of generality, the resulting dimensionless number $C/C_{dim}$ (where $C_{dim}=(gH)^{1/4}L^{3/2}$ ) only depends on the cavity aspect ratio $H/L$ . This dependence is tested experimentally (figure 6 d) and it is found that $C/C_{dim}$ is insensitive to the aspect ratio. From the experimental data we therefore conclude that the prefactor $C$ is approximately equal to $0.09\,(gH)^{1/4}L^{3/2}$ .

4.3. Simple energy balance model

Insights into the particular dependence of the jet velocity on the initial geometry of the cavity can be gained using first principles. First noting that the essence of jet formation is purely inviscid, we disregard in the following viscous effects (this hypothesis amounts to considering ${\it\beta}{\it\nu}\ll {\it\alpha}\sqrt{gH}H$ in (3.4)). The initial excess gravity potential energy associated with the initial hollow is

Note that we take no notice of the cavity-shape-dependent prefactor, nor of the kinetic energy of the remnant motions, in this elementary model as we are only interested in exponent determination. We seek in the following to express the conversion of this initial energy budget into jet kinetic energy. Recalling that gravity has a trivial effect on the jet dynamics (see § 2) we disregard such effects in the following for simplicity. The total kinetic energy of the flow can be decomposed into (i) the jet kinetic energy and (ii) the energy of the (typically dipolar) feeding flow pattern. As this last part decays with time, the asymptotic kinetic energy $T$ of the jet is the total energy of the flow. This quantity strongly depends not only on the velocity field within the jet, but also on the jet shape determined earlier. $T$ therefore can be written

Expressing the conversion of an $O(1)$ fraction of the initial potential energy $U$ into jet kinetic energy $T$ finally gives

On the other hand, in the tube filling experiment, the initial potential energy has the same expression (4.5) and the kinetic energy of the erupting liquid column is $T\sim {\it\rho}V_{tip}^{2}L^{2}H$ , leading to $V_{tip}\sim \sqrt{gH}$ , as observed in experiments.