2.1 Towed test models

Two different barge models, Barge I and Barge II, were used as they had different boundary-layer thickness at the location of gas injection. Schematic diagrams of the two test models are shown in figure 3. Both models had a transparent bottom to enable real-time observation of the gas injected underneath. The models were rigidly fixed with four struts to the carriage to prevent any motion relative to the carriage, and had a constant 81 mm draft.

Figure 3. Sketches of (a) Barge I and (b) Barge II. All dimensions are in metres.

Barge I had overall dimensions of 4.3 m in length, 0.7 m in width and 0.3 m in height. The spanwise uniform bow had a slope of $8.0^{\circ }$ . Particles of $150~\unicode[STIX]{x03BC}$ m mean diameter were randomly scattered and affixed across the span of the model 1.0 m from the leading edge of the model on a 0.1 m wide strip to induce turbulent boundary-layer transition upstream of the injection location. The resulting boundary-layer profile was measured at the model centreline at the gas injection location 0.4 m downstream of the beginning of the flat bottom.

Barge II had overall dimensions of 6.5 m in length, 1.5 m in width and 0.3 m in height. In order to reduce air ingestion at higher speeds, the bow was modified and consisted of a submerged flat plate with an elliptic planform fixed to a surface-piercing wedge. Particles of $150~\unicode[STIX]{x03BC}$ m mean diameter were randomly scattered and affixed across the span of the model 1.3 m from the leading edge of the model on a 0.2 m wide strip to induce turbulent boundary-layer transition ${\sim}$ 0.5 m upstream of the injection location.

Unless stated otherwise for a specific figure, the coordinate system is chosen such that the origin is at the centre of the injector, the $x$ -axis points downstream (towards stern) and the $y$ -axis is normal to the surface parallel to gravity vector (i.e. $y$ increases with depth).

2.2 The boundary layer upstream of gas injection

The boundary-layer profiles for each barge model and flow speed were measured at the location of the air injection by traversing a pitot tube (Omega Engineering PBE-H-M) in the $y$ -direction. The pitot was translated from 0 mm (flush with the bottom of the model surface) to 250 mm beneath the surface. The dynamic pressure at each height was measured using a 0–17 kPa differential-pressure transducer (Omega Engineering PX409-2.5DWU5V) with a manufacturer specified accuracy of $\pm$ 14 Pa. Boundary-layer profiles for both barge models are presented in figure 4, with $\unicode[STIX]{x1D6FF}_{1}$ and $\unicode[STIX]{x1D6FF}_{2}$ representing the boundary-layer thickness for Barges I and II, respectively. The data suggest that the boundary layers are turbulent at the location of gas injection for all conditions discussed in this paper.

Figure 4. The boundary-layer profiles measured at the location of gas injection for Barge I (grey symbols) and Barge II (black symbols); for Barge I, $15.8\leqslant \unicode[STIX]{x1D6FF}_{1}\leqslant 19.1$  mm over a speed range of $1\leqslant U_{\infty }\leqslant 2$  m s $^{-1}$ ; for Barge II, $50.7\leqslant \unicode[STIX]{x1D6FF}_{2}\leqslant 57.4$  mm over a speed range of $1\leqslant U_{\infty }\leqslant 5$  m s $^{-1}$ . The solid line shows a $1/7$ th power law.

2.3 The gas-injection system

The gas-injection system was designed to generate, regulate and measure a stable air mass flow rate. A 1.2 kW air compressor was used to supply a 0.30 m $^{3}$ reservoir maintained at 650 kPa. Airflow from the pressure reservoir was controlled by a pressure regulator that supplied the flowmeters with 101 to 308 kPa air. Multiple Omega Engineering FMA 5400/5500 series gas mass flow controllers were used to maintain accurate measurement of the air mass flow rate over a wide range. The accuracy of the mass flowmeters was $\pm$ 3 % of full scale (corresponding to at most $\pm$ 10 % of the reported value). Rotameters (Omega Engineering FL2003 and FL2001) were also used to confirm the measured flow rates with accuracies of $\pm$ 5 % of full scale.

Different gas-injection tubes were used to change both the diameter of the orifice, $D_{i}$ , and gas jet angle with respect to the free-stream flow, $\unicode[STIX]{x1D6FD}$ . Tubes were fabricated with fixed outer diameter, and various inner diameters. For Barge I, the orifice diameters were 6.0 and 10.0 mm, and for Barge II, they were 4.9, 10.2 and 19.7 mm. For simplicity, we will denote the diameters with the nominal values of $D_{i}=5$  mm, 10 mm and 20 mm. To enable the assumption of a fully developed pipe flow, the pipes had a straight lead-up section of at least 20 $D_{i}$ before the exit. Three-dimensional printing was used to fabricate angled injection tubes. For angled injectors, only the data from 5 mm inner diameter tubes are included. The injection angle, $\unicode[STIX]{x1D6FD}$ , was defined with respect to the bottom surface of the barge (see figure 1), and the injection angle varied from $22.5^{\circ }\leqslant \unicode[STIX]{x1D6FD}\leqslant 157.5^{\circ }$ . To calculate the average injected gas velocity, $U_{i}$ , the measured mass flow rate of the air was converted to a volume flow rate at the static pressure at the fixed barge draft (81 mm).

2.4 Video imaging systems

A stationary high-speed cinematography system (Phantom v710 camera in a custom watertight enclosure) was installed at the bottom of the basin, 3 m beneath the free surface to view the bottom of the model as it passed over at mid tank 49 m from the starting point of the carriage. Ten 100 W (8500 lm) light-emitting diode (LED) lights were used to illuminate the barge as it passed through the camera’s field of view. High-speed videos were recorded with $1440\times 1080$ pixel resolution at 200 Hz frame rate and 5 ms exposure time.

2.5 Experimental test conditions

The independent parameters of the experiment consisted of the free-stream flow speed (i.e. the carriage speed), $U_{\infty }$ , boundary-layer thickness at the injection location, $\unicode[STIX]{x1D6FF}$ , diameter of the gas injection orifice, $D_{i}$ , gas injection angle, $\unicode[STIX]{x1D6FD}$ , and the volume flow rate of the gas (at draft pressure), $Q_{i}$ . The ranges of these parameters are presented in table 1. The densities $\unicode[STIX]{x1D70C}_{i}=1.2$  kg m $^{-3}$ and $\unicode[STIX]{x1D70C}_{\infty }=1000$  kg m $^{-3}$ , and kinematic viscosities $\unicode[STIX]{x1D708}_{i}=1.5\times 10^{-5}$  m $^{2}$  s $^{-1}$ and $\unicode[STIX]{x1D708}_{\infty }=1.0\times 10^{-6}$  m $^{2}$  s $^{-1}$ of the gas jet and the liquid water cross-flow are assumed to be constant throughout the experiments. The water and air temperatures were $20\pm 1\,^{\circ }\text{C}$ .

3.1 Description of the numerical method

The governing equations are the conservation of mass and momentum equations for an incompressible fluid with varying mechanical properties of density and viscosity. The numerical solver is based on a formulation of the incompressible Navier–Stokes equations for multiphase flow used by Scardovelli & Zaleski (Reference Scardovelli and Zaleski1999). The volume-of-fluid (VOF) method of Hirt & Nichols (Reference Hirt and Nichols1981) is adopted here and surface tension is incorporated through the continuum surface force (CSF) approach of Brackbill, Kothe & Zemach (Reference Brackbill, Kothe and Zemach1992). In all simulations, the surface tension coefficient is taken as $S=0.07$  N m $^{-1}$ . The gas–liquid interface is defined as the contour of 50 % void fraction within the computational domain.

The discretized equations are solved on unstructured grids composed of hexahedra and prism elements. The spatial terms of the governing equations are discretized based on the generalized Gauss’s theorem and a combination of second-order linear and linear-upwind schemes. The temporal terms are handled implicitly using the second-order backwards Euler scheme. The flow quantities are stored at the cell centres in a co-located arrangements and the system of equations is solved in a segregated manner through the PISO (pressure implicit with splitting of operators) algorithm.

3.2 Boundary conditions

The rectangular computational domain has overall dimensions of 1.20 m in length, 1.00 m in width and 0.20 m in height. The air injector is located at the origin that is on the centreline and 0.20 m downstream from the upstream inlet boundary. The cylindrical injector has a diameter of 10 mm and it is modelled with a no-slip wall that extends 0.02 m above the no-slip flat bottom of the barge. The air injection rate is prescribed at the injector boundary and the water upstream inlet boundary is assigned the mean boundary-layer profiles measured in the physical experiments. The downstream outlet boundary is prescribed a fixed pressure value and the lateral and bottom boundaries are modelled as slip walls. See figure 5 for a depiction of the flow domain on the centre plane and on the wall near the injector. The reported pressure is calculated relative to the value at the outlet boundary, which produces positive and negative values.

Figure 5. Time-averaged air-cavity profiles computed on three different grid resolutions for $U_{\infty }=3.0$  m s $^{-1}$ , $Q_{i}=5.0\times 10^{-3}$  m $^{3}$  s $^{-1}$ , $\unicode[STIX]{x1D6FF}=51$  mm,  $D_{i}=10$  mm and $\unicode[STIX]{x1D6FD}=90^{\circ }$ (case A) (a) centre plane and (b) plate.

3.3 Grid resolution and solution convergence

The spatial and temporal discretization is selected based on a grid convergence study. The solution for $U_{\infty }=3.0$  m s $^{-1}$ , $Q_{i}=2.5\times 10^{-3}$  m $^{3}$  s $^{-1}$ , $\unicode[STIX]{x1D6FF}=51$  mm, $D_{i}\sim 10$  mm and $\unicode[STIX]{x1D6FD}=90^{\circ }$ (case A) is computed on a set of three geometrically similar grids. Each grid is uniform in the region where the air–water interface passes. Details of the grids are summarized in table 2. A time step size of $1.3\times 10^{-6}$  s is used to ensure that the maximum Courant number remains below 1 for all simulations. All simulations completed were performed on the Stampede cluster that is a part of the XSEDE network (Towns et al. Reference Towns, Cockerill, Dahan, Foster, Gaither, Grimshaw, Hazelwood, Lathrop, Lifka and Peterson.2014). The time-averaged air-cavity profiles obtained using the three grids are compared in figure 5. The time-averaging is performed over a time window of 1 s, after 1 s of simulation time has passed. In dimensionless time, $t^{\ast }=tU_{\infty }/D_{i}$ , the shortest averaging window is $t^{\ast }=200$ . The lowest oscillation frequency in the flow is approximately $t^{\ast }\sim 5$ , and hence the shortest averaging window contains a minimum of 40 oscillations. All three grids show a strong agreement especially in the proximity of the injector and near the reattachment region. The fine grid is used for all of the simulations and analyses presented in this paper.

4.1 The gas pocket formed near at the location of gas injection

The flow near the location of gas injection resembles that reported by Pignoux (Reference Pignoux1998) and Vigneau et al. (Reference Vigneau, Pignoux, Carreau and Roger2001a ,Reference Vigneau, Pignoux, Carreau and Roger b ) for the injection of gas into a downward flowing vertical stream. They reported that the gas injected from the circular orifice on the surface of a flat wall (i.e. the wall of the test section) would form a gas pocket for a range of free-stream velocity, boundary-layer thickness and mass-flux of gas. The shape of the gas pocket was analogous to that of a Rankine half-body, but with a nominal shape of a semi-ellipsoid. The axis of the ellipsoid in the wall-normal direction of the injected gas was typically larger than the axis in the cross-stream direction. The gas pocket, once formed, would grow in cross-sectional area with downstream distance until a critical cross-sectional area was reached. Then, the cavity would break down into a recirculating bubbly mixture, with bubbles being continually entrained in the cavity wake. These investigators measured the cavity gas–liquid interfacial profiles, cavity cross-sectional area and gas-pocket length for varying liquid flow speed, gas mass flow rate, orifice size and boundary-layer thickness.

Despite its relative simplicity, this near-injector flow is particularly difficult to scale. It is helpful to draw a comparison to the liquid flow around an axisymmetric cavitator, as discussed by Franc & Michel (Reference Franc and Michel2004). Here, a solid object placed in the liquid flow leads to the creation of a low-pressure gas-filled cavity in its wake. The dimensions of the cavity are principally related to the drag coefficient and radius of the cavitator, and the cavitation number (i.e. the under-pressure of the cavity) defined as $\unicode[STIX]{x1D70E}=(P_{\infty }-P_{C})/{\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}U_{\infty }^{2}$ . The resulting cavity is approximated by an ellipsoid far from the cavitator, and with decreasing cavitation number, the radius and length of the cavity increases. If non-condensable gas is directly injected into the liquid, the gas itself serves as the cavitator; therefore, the static pressure of the gas exiting the orifice must exceed the stagnation pressure of the incoming flow. Figure 7 shows the instantaneous cavity profile and time-traces of the normalized pressure at the injector from the CFD, and the gas exit pressure can indeed be seen to exceed the cavity pressure in the boundary layer at jet penetration depth $\bar{p}$ , where

Note that the pressure is calculated relative to the value at the outlet boundary, and hence takes values that are positive and negative. The effective drag and size of the cavitator will depend on the extent to which the gas penetrates the flow for a given mass, velocity and momentum flux of the injected gas, the injection angle, the free-stream liquid speed and, to a lesser extent, the thickness of the boundary layer compared to the size of the orifice. The unsteadiness of the cavity exit pressure shown in the figure will also be discussed below.

Figure 7. (a) Instantaneous cavity profile and (b) time history of the normalized gas pressure $\bar{p}$ at the injector exit from the computation (case A1). In reviewing (a) we should note that the gas exit pressure exceeds the static pressure in the boundary layer at jet penetration depth $y/\unicode[STIX]{x1D6FF}\cong -0.4$ and the pressure fluctuations shown in (b) exhibit a frequency of $St=0.29$ .

If the cavity pressure remains below the free-stream pressure, the cavity will reach a maximum cross-section and then close. And, the maximum cross-sectional area of the cavity is proportional to the effective cavitator drag and cavity under-pressure (a consequence of streamwise momentum conservation). If the cavity pressure rises to the static pressure of the flow (i.e. cavitation number approaches zero), the cavity will not close (i.e. it will be a super-cavity). In the case of injected non-condensable gas, the mean streamwise gas velocity within the cavity will decrease as the cavity expands, and if the average streamwise velocity of the gas reaches the free-stream velocity before the cavity has reached its maximum cross-sectional area, the pocket breaks down into a bubbly mixture. Conversely, if the average gas velocity reaches the free-stream velocity at or after the location of maximum cavity cross-sectional area, the gas pocket can persist. Therefore, it is the interaction of the relative gas volume flux and momentum flux of the gas jet that will determine the drag on the liquid at the point of injection and the downstream position where the cavity might break down into a bubbly mixture. A scaling that captures this complex phenomenon has yet to be presented; however, Vigneau et al. (Reference Vigneau, Pignoux, Carreau and Roger2001a ) demonstrated that modest changes to the thickness of the liquid boundary layer did not have a significant influence on the behaviour of the gas pocket and Vigneau et al. (Reference Vigneau, Pignoux, Carreau and Roger2001b ) scaled the cross-sectional area of the gas-pocket breakdown with the gas volume flux and the free-stream speed.

To illustrate the features of this flow, a computation was performed of gas injection into the liquid for conditions of zero buoyancy (i.e. without gravity) for $U_{\infty }=3.0$  m s $^{-1}$ and $D_{i}=10$  mm, $\unicode[STIX]{x1D6FD}=90.0^{\circ }$ , and $Q_{i}=5\times 10^{-3}$  m $^{3}$  s $^{-1}$ . Figure 8 shows the time-averaged flow field around the injector in the x–z and x–y planes. The location of the cavity interface, static pressure and flow speed with velocity vectors are presented. The stagnation region around the orifice is clearly visualized, along with the turning of the gas jet by the liquid flow. The flow speed of the gas decreases until it reaches the average free-stream speed and the cavity reaches a nearly constant cross-sectional area. This is a higher gas volume flux compared to those reported by Vigneau et al. (Reference Vigneau, Pignoux, Carreau and Roger2001b ), and thus the gas pocket persists for a longer distance from the injector.

Figure 8. The computed time-averaged cavity flow without the effect of gravity (case A2); $U_{\infty }=3.0$  m s $^{-1}$ and $D_{i}=10~\text{mm}$ , $\unicode[STIX]{x1D6FD}=90.0^{\circ }$ and $Q_{i}=5.0\times 10^{-3}$  m $^{3}$  s $^{-1}$ ; $\unicode[STIX]{x1D6FF}=51$  mm; the static pressure on the flow boundary (a) and centre plane (b); the normalized velocity magnitude on the centre plane (c) with non-scaled velocity vectors indicating direction. The white line denotes the time-averaged cavity interface location.

The steady flow described above occurs when there is equilibrium between the static pressure of the gas jet near the orifice and the stagnation pressure of incoming liquid. Pignoux (Reference Pignoux1998) noted, however, that the gas would not necessarily exit steadily, but would undergo a periodic variation in the flow rate, resulting in a puffing effect. This was also observed in the present study in both experiments (figure 9 a,b) and CFD (figures 7 and 9 c). This phenomenon results from the pinch-off of the orifice by an encroaching liquid flow at the base of the jet. If balance between the liquid stagnation pressure and the jet static pressure is perturbed, the liquid can move over the orifice, blocking the gas flow, and increasing the gas pressure upstream of the orifice. In turn, the build-up in gas pressure will eventually blow back the liquid, moving the stagnation region upstream of the orifice. Puffing results from the cyclic blockage and blowout of the gas at the orifice. The dynamics of the system depends on both the experimental flow conditions and the dynamics and control of the gas delivery system. Figure 7(b) presents a time series of gas pressure at the orifice to illustrate the puffing phenomenon captured by the simulation, where a length of the inlet pipe is included in the simulated geometry allowing for the flow to interact with the connection at the orifice. Based on the CFD, the Strouhal number ( $St=fD_{i}/U_{\infty }$ ) of this puffing was found to be ${\sim}$ 0.29. Albeit presumably due to a different physical mechanism, the gas–liquid interface with periodic structures due to the puffing has an appearance reminiscent of that seen on the interface of a cavitating jet in a cross-flow (Brandner, Pearce & de Graaf Reference Brandner, Pearce and de Graaf2015). Figure 9 also reveals that effective jet diameter $D_{E}$ at the jet exit is much larger than the orifice diameter $D_{i}$ . Figure 10 presents a plot of the $D_{E}$ as function of jet to cross-flow velocity ratio, and for all conditions $D_{E}$ has an uncertainty of $\pm$ 3 mm. We also note that the orifice diameter does not necessarily scale the geometry of the jet, even near the location of injection. There is also a clear dependence on boundary-layer thickness, as a thinner boundary layer will present higher average cross-stream flow momentum to the exiting gas stream, and the expected outcome is corroborated based on comparison of case A1 versus D results for which also the numerical results compare favourably to experimental data, as seen in table 3. It is interesting to note that the data collapse if the cross-flow velocity is scaled with the ratio of the turbulent boundary-layer thicknesses.

Figure 9. Close-up images of the gas jet showing the effective diameter, $D_{E}$ , and the puffing pattern observed during the present experiments: $D_{i}\cong 10$  mm,  $\unicode[STIX]{x1D6FD}=90^{\circ }$ , $\unicode[STIX]{x1D6FF}\cong 52$  mm; (a) $U_{\infty }=2.5$  m s $^{-1}$ , $Q_{i}=6.7\times 10^{-3}$  m $^{3}$  s $^{-1}$ ; (b) $U_{\infty }=3.0$  m s $^{-1}$ , $Q_{i}=5.0\times 10^{-3}$  m $^{3}$  s $^{-1}$ ; and computation: (c) $U_{\infty }=3.0$  m s $^{-1}$ , $Q_{i}=5.0\times 10^{-3}$  m $^{3}$  s $^{-1}$ .

Figure 10. The effective jet exit diameter, $D_{E}$ , as a function of velocity ratio. The different symbols denote various free-stream speeds, as indicated by the legend. The size of symbol indicates the orifice size, with the largest corresponding to $D_{i}=20~\text{mm}$ , mid-size to $D_{i}=10~\text{mm}$ and smallest to $D_{i}=5~\text{mm}$ . Empty markers represent data from Barge model I, with $\unicode[STIX]{x1D6FF}_{1}$ , and filled markers data from Barge model II, $\unicode[STIX]{x1D6FF}_{2}$ .

Figure 11. The computed time-averaged cavity flow with the effect of gravity (case A1); $U_{\infty }=3.0$  m s $^{-1}$ and $D_{i}=10~\text{mm}$ , $\unicode[STIX]{x1D6FD}=90.0^{\circ }$ and $Q_{i}=5.0\times 10^{-3}$  m $^{3}$  s $^{-1}$ ; $\unicode[STIX]{x1D6FF}=51$  mm; (a) static pressure, (b) normalized pressure $\bar{p}$ on the flow boundary, (c) static pressure on the $x$ – $y$ centre plane and (d) velocity magnitude with non-scaled velocity vectors indicating direction. The velocity magnitude with non-scaled velocity vectors indicating direction on the $y$ – $z$ plane at $x/\unicode[STIX]{x1D6FF}=1$ (e) and $x/\unicode[STIX]{x1D6FF}=8$ (f). The white line denotes the time-averaged cavity interface location.

4.2 Formation of the gas-filled branches

Figure 11 shows the computation with buoyancy for the conditions otherwise similar to those shown in figure 8 without buoyancy. The influence of buoyance leads to a significantly different topology of the gas pocket. The effect is evident within a few $D_{E}$ downstream of the orifice, as the liquid flowing around the cavity begins to move up toward the flow boundary, bisecting the gas pocket and creating two flow branches. Without gravity, the gas pocket forms and develops into a long slender cavity with the shape of a semi-ellipsoid. However, as seen in more detail in figure 11(b,d,f), with gravity acting normal to the flow direction, and hence buoyancy moving the gas toward the flow boundary, the gas pocket is bisected, forming the two stable branches (note the pressure peak on the surface at the bifurcation location seen in figure 11 b). This topology was observed experimentally for the wide range of flow conditions of the present study. At speeds ${\geqslant}$ 2 m s $^{-1}$ , the leading edges of the gas branches were usually well defined and straight. The chord length of the branches was also nominally constant, until entrainment of gas away from the branches led to their gradual diminution.

4.3 Sweep angle of the gas branches

The sweep angle, $\unicode[STIX]{x1D711}$ , is the angle between the plane perpendicular to the cross-flow and leading edge of the branch. The angles of both branches were measured from multiple images. The uncertainty was estimated as $\pm 1.5^{\circ }$ and is principally attributed to the fluctuations of the edge of the gas branch that is related to the puffing discussed previously. Figures 12–15 illustrate how the sweep angle, $\unicode[STIX]{x1D711}$ , changes with varying cross-flow speed $U_{\infty }$ , jet volume flow rate $Q_{i}$ , injection orifice diameter $D_{i}$ and injection angle $\unicode[STIX]{x1D6FD}$ . The sweep angle significantly varies with varying cross-flow speed, jet volume flow rate and, to a lesser extent, the injection hole diameter and injection angle. Figure 16 presents plots of $\unicode[STIX]{x1D711}$ versus $Q_{i}$ for varying $U_{\infty }$ for the three nominal orifice diameters of 5, 10 and 20 mm, for $\unicode[STIX]{x1D6FD}=90^{\circ }$ . The sweep angle is primarily a function of the free-stream speed, with the angle increasing with increasing speed. Conversely, for a fixed value of $U_{\infty }$ , the sweep angle decreases with increasing gas flow rate, $Q_{i}$ .

Figure 12. Images illustrating the change in sweep angle, $\unicode[STIX]{x1D711}$ , and chord length, $C$ , with varying flow speed, $U_{\infty }$ , with fixed $D_{i}\cong 5$  mm,  $\unicode[STIX]{x1D6FD}=90^{\circ }$ and $Q_{i}=2.0\times 10^{-3}$  m $^{3}$  s $^{-1}$ , for $U_{\infty }=2.0$  m s $^{-1}$ (a), 3.0 m s $^{-1}$ (b), and 4.0 m s $^{-1}$ (c); $\unicode[STIX]{x1D6FF}\cong 52$  mm. The resulting sweep angles and chord lengths are (a) $\unicode[STIX]{x1D711}=74.2^{\circ }$ and $C=64$  mm, (b) $\unicode[STIX]{x1D711}=79.9^{\circ }$ and $C=60$  mm, (c) $\unicode[STIX]{x1D711}=84.6^{\circ }$ and $C=38$  mm.

Figure 13. Images illustrating the change in sweep angle, $\unicode[STIX]{x1D711}$ , and chord length, $C$ , with varying volume flow rate, $Q_{i}$ , with fixed $D_{i}\cong 10$  mm,  $\unicode[STIX]{x1D6FD}=90^{\circ }$ and $U_{\infty }=2.0$  m s $^{-1}$ , $\unicode[STIX]{x1D6FF}\cong 52$  mm, for $Q_{i}=1.7\times 10^{-3}$  m $^{3}$  s $^{-1}$ (a), $3.3\times 10^{-3}$  m $^{3}$  s $^{-1}$ (b) and $6.5\times 10^{-3}$  m $^{3}$  s $^{-1}$ (c). The resulting sweep angles and chord lengths are (a) $\unicode[STIX]{x1D711}=76.7^{\circ }$ and $C=44$  mm; (b) $\unicode[STIX]{x1D711}=72.2^{\circ }$ and $C=86$  mm; (c) $\unicode[STIX]{x1D711}=68.8^{\circ }$ and $C=142$  mm.

Figure 14. Images illustrating the change in sweep angle, $\unicode[STIX]{x1D711}$ , and chord length, $C$ , with varying injection hole diameters $D_{i}$ with fixed $U_{\infty }=3$  m s $^{-1}$ , $\unicode[STIX]{x1D6FF}=51$  mm,  $\unicode[STIX]{x1D6FD}=90.0^{\circ }$ and $Q_{i}=2.5\times 10^{-3}$  m $^{3}$  s $^{-1}$ , for $D_{i}=5$  mm (a), 10 mm (b) and 20 mm (c). The resulting sweep angles and chord lengths are (a) $\unicode[STIX]{x1D711}=9.7^{\circ }$ and $C=90$  mm; (b) $\unicode[STIX]{x1D711}=81.8^{\circ }$ and $C=66$  mm; (c) $\unicode[STIX]{x1D711}=82.4^{\circ }$ and $C=40$  mm.

Variation in the orifice size, $D_{i}$ , and injection angle, $\unicode[STIX]{x1D6FD}$ , suggests that the gas exit velocity and momentum in direction of the cross-flow has a secondary, but discernible, influence on the sweep angle. A higher exit velocity (i.e. smaller orifice for fixed volume flux) and higher injection angle (i.e. increased momentum against cross-flow) led to a smaller sweep angle for a given volume flux of gas. Figure 17 shows the effect of gas injection angle, and in the extreme case of $22.5^{\circ }<\unicode[STIX]{x1D6FD}<157.5^{\circ }$ a nearly $6^{\circ }$ difference is observed, which is approximately the same as the effect of changing the gas volume flow rate by a factor of five in 3 m s $^{-1}$ liquid cross-flow.

4.4 Chord length of the gas branches

The average chord of the gas branch, $C$ , was measured along several spanwise segments across the thickest part of the branch, and the measurement uncertainty of $\pm$ 5 mm is primarily due to fluctuations of the edge of the gas branches. Figures 12–15 also illustrate the changes in $C$ for varying cross-flow speed $U_{\infty }$ , jet volume flow rate $Q_{i}$ , injection angle $\unicode[STIX]{x1D6FD}$ and injection orifice diameter, $D_{i}$ . The chord length $C$ is a strong function of the volume flow rate, $Q_{i}$ , the orifice diameter, $D_{i}$ , and the boundary-layer thickness, $\unicode[STIX]{x1D6FF}$ . Figure 18 present plots of $C$ versus $Q_{i}$ for varying $U_{\infty }$ for the three nominal orifice diameters of 5, 10 and 20 mm and for $\unicode[STIX]{x1D6FD}=90^{\circ }$ . The dominant trends are increasing $C$ with $Q_{i}$ , and figure 19 illustrates the effect of $\unicode[STIX]{x1D6FD}$ , with a clear trend of increasing chord length with increasing momentum of gas injection against cross-flow. Similar to what was observed for angle, $\unicode[STIX]{x1D711}$ , the chord length is not independent of $\unicode[STIX]{x1D6FF}$ , $D_{i}$ and $\unicode[STIX]{x1D6FD}$ , but the dominant variables are again $Q_{i}$ and $U_{\infty }$ .

Figure 15. Images illustrating the change in sweep angle, $\unicode[STIX]{x1D711}$ , and chord length, $C$ , with varying injection angles, $\unicode[STIX]{x1D6FD}$ , with fixed $D_{i}\cong 5$  mm,  $U_{\infty }=3.0$  m s $^{-1}$ , $\unicode[STIX]{x1D6FF}=53$  mm and $Q_{i}=2.5\times 10^{-3}$  m $^{3}$  s $^{-1}$ for $\unicode[STIX]{x1D6FD}=157.5^{\circ }$ (a), $90.0^{\circ }$ (b), $22.5^{\circ }$ (c). The resulting sweep angles and chord lengths are (a)  $\unicode[STIX]{x1D711}=76.0^{\circ }$ and $C=113$  mm; (b)  $\unicode[STIX]{x1D711}=79.7^{\circ }$ and $C=90$  mm; (c)  $\unicode[STIX]{x1D711}=81.2^{\circ }$ and $C=63$  mm.

Figure 16. Sweep angle, $\unicode[STIX]{x1D711}$ , versus volume flow rate, $Q_{i}$ , for varying flow speeds, $\unicode[STIX]{x1D6FD}=90^{\circ }$ : $D_{i}\cong 5$  mm (a), $\cong$ 10 mm (b) and $\cong$ 20 mm (c). Markers are such that $U_{\infty }=1.0$  m s $^{-1}$ (○), 2.0 m s $^{-1}$ (▹), 3.0 m s $^{-1}$ (◃), 4.0 m s $^{-1}$ (♢) and 5.0 m s $^{-1}$ (▫). Filled markers represent data from Barge model I, with $\unicode[STIX]{x1D6FF}_{1}$ , and empty markers data from Barge model II, $\unicode[STIX]{x1D6FF}_{2}$ .

4.5 Topology of the gas pocket at low free-stream speed

Figure 17. Sweep angle, $\unicode[STIX]{x1D711}$ , versus injection angle, $\unicode[STIX]{x1D6FD}$ , for $U_{\infty }=3.0$  m s $^{-1}$ , $Q_{i}=2.5\times 10^{-3}$  m $^{3}$  s $^{-1}$ and for $D_{i}=5$  mm. The open symbols signify Lambda, grey transitional and black filled Delta topology. Note: for the particular example data shown, the Lambda topology (open symbols) was not observed.

Figure 18. Chord length, $C$ , versus volume flow-rate, $Q_{i}$ , for varying flow speeds, $\unicode[STIX]{x1D6FD}=90^{\circ }$ : $D_{i}\cong 5$  mm (a), $\cong$ 10 mm (b) and $\cong$ 20 mm (c). Symbols are same as in figure 16.

Observation of the jet and branch topology show consistent trends except at the lowest cross-flow speed examined. Figure 20 compares the topologies for changing cross-flow speed ranging from $U_{\infty }=1$ to 3 m s $^{-1}$ . At the lowest speed, the branches barely form and have an irregular leading edge and chord length. As the speed increases, they become distinct, straight and more uniform in chord length. This phenomenon is related to the relative importance of surface tension in the flow and is discussed in § 6.4.

Figure 19. Chord length, $C$ , versus injection angle, $\unicode[STIX]{x1D6FD}$ , for $U_{\infty }=3.0$  m s $^{-1}$ , $Q_{i}=2.5\times 10^{-3}$  m $^{3}$  s $^{-1}$ and for $D_{i}=5$  mm. The open symbols signify Lambda, grey transitional and black filled Delta topology. Note: for the particular example data shown, the Lambda topology (open symbols) was not observed.

Figure 20. The topologies of the gas jet for low and higher-speed cross-flow; the speed varies from $U_{\infty }=1.0$ , 1.5, 2.0 and 3.0 m s $^{-1}$ (a–d) and 51 mm ${<}\unicode[STIX]{x1D6FF}<57$  mm, while the other conditions are fixed at $D_{i}\sim 10$  mm,  $Q_{i}=1.7\times 10^{-3}$  m $^{3}$  s $^{-1}$ and $\unicode[STIX]{x1D6FD}=90.0^{\circ }$ .

5.1 The basic cavity flow

Figure 24 presents contours of the time-averaged (a) static pressure, $p$ , and (b) the in-plane velocity magnitude, $|\boldsymbol{u}|/U_{\infty }$ , on the PX plane for cases A1 and D. Figure 24(c) is an instantaneous realization of the pressure field. When $p<0$ at the gas–liquid interface, buoyancy will tend to move the gas back toward the plate. At the jet exit, the pressure at the gas–liquid interface is such that $p\gg 0$ and matches the stagnation pressure of the incoming liquid flow. Along the interface of the cavity, $p$ decreases and the initial gas pocket begins to close and form the branches. The influence of the gas volume flux can be seen through the change in the shape of the initial gas pocket. With the higher gas flux the hump is taller and the upward momentum of the liquid following the contour of the hump is sufficient to bifurcate the gas pocket, as also seen in figure 22. Recall that the branches form as the free-stream liquid around the gas cavity returns to the flow boundary along the plane on symmetry and bifurcates the gas pocket. This is illustrated in figures 22 and 25(b,c) that show the surface pressures for the case A1 (also presented in figure 11). A high-pressure region is observed at the location of cavity bifurcation that results from the impingement of the liquid onto the wall boundary. Whether the region between the gas branches is filled with gas (Delta-type cavity) or liquid on the surface (Lambda-type) is of importance if the goal is to form an air layer evenly covering the surface. Figure 25(c) also shows the large flow structures that form in front of the gas injection location, and persist far downstream. The vortex structures are visualized with the second invariant of the velocity-gradient tensor, commonly referred to as the $Q$ -criterion (Hunt, Wray & Moin Reference Hunt, Wray and Moin1988). These structures modify the boundary layer ‘seen’ by the branch from one with a $u/U=(y/\unicode[STIX]{x1D6FF})^{1/n}$ power-law shape best fitted with $n\sim 8$ – $n\sim 5$ . This can explain how the manner in which the gas is introduced into the flow can affect the equilibrium sweep angle far downstream, and other detail discussed in § 6. Figure 25(a) also shows the mean velocity field, and in particular, the manner in which the turbulent mean profile that is applied on the inlet persists downstream until it is modified by the cavity.

Figure 24. The time-averaged (a) static pressure, $p$ , and (b) velocity magnitude, $|\boldsymbol{u}|/U_{\infty }$ , on the PX plane for cases A1 ( $Q_{i}=5.0\times 10^{-3}$  m $^{3}$  s $^{-1}$ ) and D ( $Q_{i}=1.7\times 10^{-3}$  m $^{3}$  s $^{-1}$ ); (c) is an instantaneous realization of the pressure field $\bar{p}$ . The solid line indicates the gas–liquid interface defined for 50 % void fraction.

Figure 25. Case A1 (a) velocity magnitude, $|\boldsymbol{u}|/U_{\infty }$ , on the centre plane and the cavity (defined as 50 % time-averaged void fraction) are shown as the dark grey isosurface. (b) The time-averaged normalized pressure, $\bar{p}$ , on the top boundary and the cavity. (c) The time-averaged normalized pressure, $\bar{p}$ , on the top boundary, and an isosurface of the $Q$ -criterion ( $Q=10$ ) illustrating the junction vortex formed in front of the gas branches.

Figure 26 presents a series of PY planes illustrating the time-averaged in-plane velocity magnitude for the case A1. The planes start upstream of the injection location and move downstream toward the formation of the branches, at the location where the liquid impinges on the surface. Figure 27 presents the normalized pressure, $p$ , to illustrate the influence of buoyancy on the development of the branches. Away from the gas injection site the cavity is at a constant pressure, far from the branches the pressure varies due to buoyancy as $\unicode[STIX]{x1D70C}_{\infty }gy$ . However, near the cavity the pressure is below $\unicode[STIX]{x1D70C}_{\infty }gy$ , as the flow accelerates locally.

Figure 26. A series of PY planes presenting the time-averaged in-plane velocity magnitude for case A1. The solid line indicates the gas–liquid interface defined for 50 % void fraction. The cases are for $x/\unicode[STIX]{x1D6FF}=1$ , 3, 8, 10.

Figure 27. The static pressure, $p$ , on a series of planes PY planes for case A1. The solid line indicates the gas–liquid interface defined for 50 % void fraction. The cases are for $x/\unicode[STIX]{x1D6FF}=1$ , 3, 8, 10.

5.2 Flow around and within the stable gas branches

Once the two branches have been formed, they maintain a near equilibrium topology as they extend downstream from the cavity injection location with both a fixed gas sweep angle and chord length. Figure 28 presents the flow along the branch in the plane BX, and this shows that the flow along the centre of the branch reaches equilibrium within ${\sim}5x^{\prime }/\unicode[STIX]{x1D6FF}$ distance from the injection location. The flow over the branch is reminiscent of the flow over a long swept wing, and we can use the concepts of incompressible swept wing theory. Figure 29 presents the time-averaged static pressure, $p$ , and flow speed and velocity vectors, for the case A1 at the location $x^{\prime }/\unicode[STIX]{x1D6FF}$ where the branch has reached a near equilibrium shape. The slices of figure 29 more clearly show there is only minimal variation in the branch shape until it is impinged on the exit of the computational domain. Experimentally, the variation along the branch was significant in cases where notable amounts of gas were entrained all along the closure of the branches.

Figure 28. The (a) static pressure, $p$ , and (b) in-plane velocity magnitude, $|\boldsymbol{u}|/U_{\infty }$ , for the flow along on gas branch on the plane BX for the case A1. The solid white line indicates the gas–liquid interface as 50 % time-averaged void fraction.

Figure 29. The (a) static pressure, $p$ , and (b) in-plane velocity magnitude, for the flow in the plane BY for the case A1. The solid line indicates the gas–liquid interface defined for 50 % void fraction. The planes in each panel, from left to right and top to bottom, are at $x^{\prime }/\unicode[STIX]{x1D6FF}=9$ , 10, 12, 14.

5.3 Influence of free-stream speed and boundary-layer thickness

The experimental observations suggested that the free-stream speed and boundary-layer thickness both significantly influence the topology of the gas branches, and this was explored computationally. Cases B and C were computed with $U_{\infty }=2$ and 4  m s $^{-1}$ . Figure 30 presents the (a) time-averaged static pressure, $p$ , (b) the flow speed and velocity vectors at the location $x^{\prime }/\unicode[STIX]{x1D6FF}$ where the branch has reached a near-equilibrium shape. Besides the obvious change in angle, the branches become significantly smaller in both chord, $C$ , and height, $e$ . Inspection of the average gas velocity along the branch shows good correlation with $U_{\infty }\text{sin}(\unicode[STIX]{x1D711})$ , as well as some dependence on $e$ , and this notable observation is discussed further in § 6.

Figure 30. The (a) static pressure, $p$ , (b) flow speed and velocity vectors for the cases B and C. Case B is for $U_{\infty }=2$  m s $^{-1}$ and $x^{\prime }/\unicode[STIX]{x1D6FF}=8$ , and case C is for $U_{\infty }=4$  m s $^{-1}$ and $x^{\prime }/\unicode[STIX]{x1D6FF}=8$ or 12. The solid line indicates the gas–liquid interface defined for 50 % void fraction.

The effect of incoming boundary-layer thickness was examined by computing case E with $1/3\unicode[STIX]{x1D6FF}$ of case A1. Figures 31 and 32 present the time-averaged static pressure, $p$ , and the flow speed and velocity vectors for case E, respectively. While the chord length changed modestly, a significant difference in branch thickness is observed. We should also note the difference in pressure immediately upstream of both the injection and branches, as was seen most clearly in figure 22, and is likely due to change in stagnation pressure and relative strength of large structures (i.e. junction vortex) observed in figure 25. For these cases, we also find that the average gas velocity along the branch shows dependence on $e$ , in addition to $U_{\infty }\text{sin}(\unicode[STIX]{x1D711})$ , as will be discussed in the next section.

Figure 31. The time-averaged static pressure, $p$ case E at (a) $x^{\prime }/\unicode[STIX]{x1D6FF}_{A1}=1$ , 3, 8, 10, and (b) $x^{\prime }/\unicode[STIX]{x1D6FF}_{E}=1$ , 3, 8, 10, from left to right and top to bottom, respectively. The boundary layer for case E is $1/3$ that of case A1. The solid line indicates the gas–liquid interface defined for 50 % void fraction.

Figure 32. The time-averaged flow speed and velocity vectors for case E at (a) $x^{\prime }/\unicode[STIX]{x1D6FF}_{A1}=1$ , 3, 8, 10 and (b) $x^{\prime }/\unicode[STIX]{x1D6FF}_{E}=1$ , 3, 8, 10, from left to right and top to bottom, respectively. The boundary layer for case E is $1/3$ that of case A1. The solid line indicates the gas–liquid interface defined for 50 % void fraction.

6.1 Power-law scaling

Figure 33 presents $\text{cos}\unicode[STIX]{x1D711}$ as a product of four non-dimensional groups raised to exponents derived from nonlinear regression. The scaling successfully groups the data with the following equation:

A linear regression of this function has a correlation of 0.94. From (6.1) we see that the Froude number has the largest influence on the sweep angle, with the angle decreasing with increasing $Fr$ . A similar regression is shown in figure 34 for $C/\unicode[STIX]{x1D6FF}$ :

Here, the scaling has a somewhat reduced correlation of 0.87. Based on the exponents found from regression, the cavity chord, $C/\unicode[STIX]{x1D6FF}$ , is strongly related to the rate of volume injection, as expected. However, the chord is also significantly influenced by the Froude number, injection velocity and Reynolds number. Next, we will present a physical basis for these observed scaling.

Figure 33. A plot of $\cos \unicode[STIX]{x1D711}$ versus a power law employing the Froude number, $Fr=U_{\infty }/\sqrt{g\unicode[STIX]{x1D6FF}}$ , the Reynolds number, $Re=U_{\infty }\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D708}_{\infty }$ , the scaled gas injection flow rate, $Q^{\ast }=Q_{i}/U_{\infty }\unicode[STIX]{x1D6FF}^{2}$ and the injection speed ratio, $U^{\ast }=U_{i}/U_{\infty }$ , for $U_{\infty }\geqslant 2$  m s $^{-1}$ and wall-normal injection ( $\unicode[STIX]{x1D6FD}=90^{\circ }$ ); the markers represent Delta (▵), transition (♢) and Lambda (▿) cavities; filled markers represent data from Barge model I, with $\unicode[STIX]{x1D6FF}_{1}$ , and empty markers represent data from Barge model II, $\unicode[STIX]{x1D6FF}_{2}$ ; and the size of the marker represents the orifice size, e.g. (▵) $D_{i}\sim 5$  mm, (▵) $D_{i}\sim 10$  mm, (▵) $D_{i}\sim 20$  mm.

Figure 34. A plot of $C/\unicode[STIX]{x1D6FF}$ versus a power law employing the Froude number, $Fr=U_{\infty }/\sqrt{g\unicode[STIX]{x1D6FF}}$ , the Reynolds number, $Re=U_{\infty }\unicode[STIX]{x1D6FF}/\unicode[STIX]{x1D708}_{\infty }$ , the scaled gas injection flow rate, $Q^{\ast }=Q_{i}/U_{\infty }\unicode[STIX]{x1D6FF}^{2}$ and the injection speed ratio, $U^{\ast }=U_{i}/U_{\infty }$ , for $U_{\infty }\geqslant 2$  m s $^{-1}$ and wall-normal injection ( $\unicode[STIX]{x1D6FD}=90^{\circ }$ ); the markers represent Delta (▵), transition (♢) and Lambda (▿) cavities; filled markers represent data from Barge model I, with $\unicode[STIX]{x1D6FF}_{1}$ , and empty markers represent data from Barge model II, $\unicode[STIX]{x1D6FF}_{2}$ ; and the size of the marker represents the orifice size, e.g. (▵) $D_{i}\sim 5$  mm, (▵) $D_{i}\sim 10$  mm, (▵) $D_{i}\sim 20$  mm.

6.2 A simplified model of the flow around the gas branch

A simplified model of the flow around a single gas branch can help illuminate the basic flow processes that lead to the formation of such straight and stationary gas pockets. We expect that the drag of the long gas pocket is proportional to its height, $e$ , the boundary-layer thickness, $\unicode[STIX]{x1D6FF}$ , flow static pressure, $P_{\infty }$ , and the pressure of the gas within the cavity, $P_{C}$ . If we consider the component of the drag force perpendicular to the leading edge of the gas branch, $D$ , the net force is given by

where $C_{D}$ is a drag coefficient and $U_{\infty }\cos \unicode[STIX]{x1D711}$ is the magnitude of the velocity component perpendicular the leading edge scaled by the depth the cavity penetrates the boundary layer, $e/\unicode[STIX]{x1D6FF}$ , and where $1/n$ is the exponent of the boundary-layer profile power law. If we assume that the drag coefficient is a weakly decreasing function of the Reynolds number based on the cavity thickness, then

where $m>0$ is a constant. A cavitator is formed by the leading edge of the gas pocket but it is not fixed to the surface. Therefore, in order to have a stationary gas pocket in the laboratory frame of reference, the net drag force on the gas pocket must be zero. Setting $D=0$ in (6.3) yields the force equilibrium condition:

The average cavity pressure, $P_{C}$ , will be related to the cavity height, volume rate of the injected air and the rate of gas entrainment at the local cavity closure and the terminus of the branch at the farthest downstream extent of the gas pocket. Without direct measurements of the pocket pressure, a relationship for $P_{C}$ must be prescribed. We will therefore assume that $P_{C}$ in the gas branch is of the same order of, but less than, the hydrostatic pressure in the surrounding liquid at a depth of the branch maximum thickness,  $e$ :

Finally, we need a relationship between the gas branch height, $e$ , and the volume flow rate of gas, $Q_{i}$ . We can develop such a relationship if we assume that a fraction $K_{2}$ of the injected air flows through branches $Q_{\mathit{branch}}=Q_{i}/2K_{2}$ , where $K_{2}$ would be unity for a Lambda with no gas leakage between the branches. We can also assume that the profile of the branch approximates a half ellipsoid such that the branch chord

And, finally, we will assume that the gas flow speed within the branch is equivalent to the branch-normal speed of the liquid at same location from the plate surface, $y$ , such that

The expression for the gas flux in the branch becomes:

For $\text{Re}(1/n)>-1$ this yields

where $S(n)=\unicode[STIX]{x1D6E4}((n+1)/2n)/\unicode[STIX]{x1D6E4}((4n+1)/2n)$ . (Note that for the branch thickness, $e$ , and angle, $\unicode[STIX]{x1D711}$ , found from the CFD the average gas velocity in the branch based on (6.9) with $K_{2}=1$ versus CFD match well, yielding 2.08 versus 2.08 and 2.53 versus 2.51 m s $^{-1}$ , for cases A1 and E, respectively.)

Equations (6.4)–(6.6) and (6.10), can now be combined to develop a relationship between the $Re$ , $Fr$ , $Q^{\ast }$ and $\unicode[STIX]{x1D711}$ :

where

This scaling does not include the one additional parameter associated with the wall-normal injection process, the gas injection velocity $U^{\ast }$ . It is therefore instructive to plot the scaling suggested by (6.11) as a function of $U^{\ast }=U_{i}/U_{\infty }$ . This is shown in figure 35, where the function

is plotted versus $U^{\ast }$ . The data have the best linear fit for regression constants $\hat{m}=0.40$ and $\hat{r}=-0.79$ with a correlation of 0.48. Note that a positive value of $\hat{m}$ suggests that the drag coefficient of the cavity would be decreasing with increasing Reynolds number, as we would expect. From (6.12) we would also get $n=3.9$ , which can be compared to the experimentally observed value of $n\sim 7$ . The decreased $n$ would physically translate to a more gradual rate of velocity gradient in front of the branches. Indeed, from the CFD, as discussed in conjunction with figure 25, we observed the large junction vortex-type flow structures that modified the flow seen by the branch, and in the CFD we observed these to drop $n$ from ${\sim}8$ to ${\sim}5$ , which would seem to support the idea that $n$ seen by the branches in the experiment may have well been  ${\sim}4$ .

Figure 35. A plot of $Re^{-\hat{m}}Fr^{2}\text{cos}^{2-\hat{m}}\unicode[STIX]{x1D711}Q^{\ast ^{\hat{p}}}(\text{sin}\unicode[STIX]{x1D711})^{-\hat{p}}$ versus $U^{\ast }$ for $U_{\infty }\geqslant 2$  m s $^{-1}$ and wall-normal injection ( $\unicode[STIX]{x1D6FD}=90^{\circ }$ ); and the data show the best linear fit for $\hat{m}=0.40$ and $\hat{p}=-0.79$ with a linear correlation is 0.48; the markers represent Delta (▵), transition (♢) and Lambda (▿) cavities; filled markers represent data from Barge model I, with $\unicode[STIX]{x1D6FF}_{1}$ , and empty markers represent data from Barge model II, $\unicode[STIX]{x1D6FF}_{2}$ ; and the size of the marker represents the orifice size, e.g. (▵) $D_{i}\sim 5$  mm, (▵) $D_{i}\sim 10$  mm, (▵) $D_{i}\sim 20$  mm.

It is important to note, however, that the scaling does not lead to a constant value as a function of $U^{\ast }$ , which we would expect if the force balance around the branch is independent of the gas injection process except for the gas volume flux. However, we see an approximately linear dependence on the relative injection velocity such that with increasing $U^{\ast }$ the magnitude of (6.13) increases.

To explore this further, we introduce the influence of the gas injection velocity into the scaling as

The regression for this function is shown in figure 36 as a function of $U^{\ast }$ with $\hat{m}=0.73$ , $\hat{r}=-0.82$ and $\hat{q}=0.79$ . These data show better grouping, and again the positive value of $\hat{m}$ suggests that the drag coefficient of the cavity would be decrease with increasing Reynolds number, as we would expect. This new scaling also suggests that the details of the gas injection process influence the equilibrium topology of the gas branch. We would therefore expect that the gas injection angle would also play a role. To explore this, the $U^{\ast }$ is scaled by $\sin \unicode[STIX]{x1D6FD}$

$U^{\ast }$ is shown in figure 37 as a function of (6.15) including the data for cases where $\unicode[STIX]{x1D6FD}\neq 90^{\circ }$ . These data for $\unicode[STIX]{x1D6FD}=90^{\circ }$ follow the trend as well, but there remains some noticeable scatter for the $\unicode[STIX]{x1D6FD}\neq 90^{\circ }$ data, that indicates the simple scaling does not fully capture how the method of gas injection (modifying both the hump height and large junction vortex-type flow structures) effects the overall flow, leading to a different equilibrium for the branches.

Figure 36. A plot of $Re^{-\hat{m}}Fr^{2}\text{cos}^{2-\hat{m}}\unicode[STIX]{x1D711}Q^{\ast ^{\hat{p}}}(C/\unicode[STIX]{x1D6FF}\text{sin}\unicode[STIX]{x1D711})^{-\hat{p}}U^{\ast \hat{q}}$ versus $U^{\ast }$ for $U_{\infty }\geqslant 2$  m s $^{-1}$ and wall-normal injection ( $\unicode[STIX]{x1D6FD}=90^{\circ }$ ) for $\hat{m}=0.73$ , $\hat{p}=-0.82$ and $\hat{q}=0.79$ ; the markers represent Delta (▵), transition (♢) and Lambda (▿) cavities; filled markers represent data from Barge model I, with $\unicode[STIX]{x1D6FF}_{1}$ , and empty markers represent data from Barge model II, $\unicode[STIX]{x1D6FF}_{2}$ ; and the size of the marker represents the orifice size, e.g. (▵) $D_{i}\sim 5$  mm, (▵) $D_{i}\sim 10$  mm, (▵) $D_{i}\sim 20$  mm.

Figure 37. Same as figure 36 with additional data for angles of injection. (a) All hole diameters (angles $\neq 90^{\circ }$ included only when $D_{i}=5$  mm) and (b) only $D_{i}=5$  mm. For wall-normal injection ( $\unicode[STIX]{x1D6FD}=90^{\circ }$ , grey filled symbols), downstream injection ( $\unicode[STIX]{x1D6FD}<90^{\circ }$ , black filled symbols) and upstream injection ( $\unicode[STIX]{x1D6FD}>90^{\circ }$ , empty symbols); the markers represent Delta (▵), transition (♢) and Lambda (▿) cavities; the size of the marker represents the orifice size, e.g. (▵) $D_{i}\sim 5$  mm, (▵) $D_{i}\sim 10$  mm, (▵) $D_{i}\sim 20$  mm.

The physical model discussed above presents the trends observed in the scaled data (§ 6.1). The scaling from (6.13) can be combined with that of (6.2) to create the scaling if we assume $(\text{sin}\unicode[STIX]{x1D711})^{-p}\approx 1$ . Recall that this scaling does not include the influence of $U^{\ast }$ , but it will be introduced into this scaling solely through its factor in the open regression. Combining (6.2) and (6.13), the model scaling based on the drag balance around the branch yields

This can be compared with original open regression for the sweep angle

The exponents for $Fr$ and $Q^{\ast }$ are similar, with the Froude number being the dominant parameter. The scaling based on the model over-predicts the influence of $Re$ and $U^{\ast }$ and the scaling would suggest a power-law profile of $n\sim 4$ (6.12) for the boundary layer (i.e. a more gradual boundary-layer profile). $Re$ and $U^{\ast }$ are still significant and necessary parameters for both relations.

The question, therefore, is how the process of injection could be coupled to the drag balance around the branch flow far downstream from the injection location. One possibility already discussed is that the gas injection process could cause diversion of the gas from the branches (when Delta topology is achieved) thus leading to only a fraction of the gas flowing through each branch with the remainder flowing between the branches. A second possibility is that the injection process can lead to a change in the liquid flow upstream of the gas pockets. Examination of the computed flow upstream of the branches indicates that presence of a junction vortex, visualized in figure 25, that formed at the stagnation region around the jet. The vortex flow parallel to and upstream of the gas branches and, as such, can modify the momentum balance described in (6.5), modifying the effective drag coefficient with increasing vortex strength. Then, the drag coefficient would be a function of not only the Reynolds number but also the strength of the junction vortex. The vortex strength would, in turn relate to a $U^{\ast }$ and $\unicode[STIX]{x1D6FD}$ . Note that the quantity $\unicode[STIX]{x1D70C}_{\infty }ge\approx 10^{2}$  Pa, therefore only a small modification of the pressure upstream of the branch may be sufficient to modify the drag balance.

We conducted an additional experiment to examine how the formation of a potentially stronger junction vortex would affect the formation of the gas branches if all other parameters are kept constant. To do this, we extended a portion of the outlet tube to create a solid boundary upstream of the injection location, as shown in figure 38. During the experiment, the presence of the barrier led to a stagnation flow ahead of the gas injection, and a path for the gas to extend farther into the free-stream before reattaching to the surface to form the gas pocket. Examination of the resulting flows showed that, for fixed $Fr$ , $Q^{\ast }$ and $U^{\ast }$ , the presence of the barrier led to a decrease of the sweep angle (increase in $\cos \unicode[STIX]{x1D711}$ ) and increased tendency towards a Lambda-type gas-pocket topology. This is consistent with the hypothesis that the formation of a stronger junction vortex would lead to a reduction of the drag coefficient of the branch and a resulting decrease in the equilibrium sweep angle, and with taller hump promoting Lambda topology.

Figure 38. The modified orifice for wall-normal injection that includes a solid barrier (the half of a pipe extending beyond the pipe with larger outer diameter) upstream of the gas port (a); images of the cavity with $D_{i}\sim 10$  mm with $U_{\infty }=2.0$  m s $^{-1}$ , $Q_{i}=2.1\times 10^{-3}$  m $^{3}$  s $^{-1}$ and $\unicode[STIX]{x1D6FF}\cong 19$  mm for (b) the cavity topology resulting from injection with the plain orifice and (c) the cavity topology resulting from placement of the barrier upstream on the injection port.

6.3 Stability of the gas branch

With a relationship developed between the sweep angle and the drag force on the gas branch, we can examine if our simplified model predicts a stable, straight gas branch. Given the force balance of (6.3), we can take the derivative of the drag force $D$ with $\unicode[STIX]{x1D711}$ around the point of force equilibrium:

Since $45^{\circ }<\unicode[STIX]{x1D711}<90^{\circ }$ , the sign of $(\unicode[STIX]{x2202}^{2}D/\unicode[STIX]{x2202}\unicode[STIX]{x1D711}^{2})_{D=0}$ is always greater than zero, implying that a restoring force exists that resists perturbation of the branch. Hence, when there is a positive perturbation of the sweep angle, a net negative drag force occurs, and this results in the branch returning to its equilibrium position. Similarly, if the sweep angle is reduced, the drag on the branch increases and it is pushed back. Thus, the simplified model confirms that the gas branch, once formed, will change in sweep angle until the equilibrium position is achieved and will resist perturbation.

6.4 Consideration of surface tension

The previous discussion has illustrated how the flow around the gas branches is dominated by inertia and buoyancy. However, cases with lower cross-flow speed ( $U_{\infty }\leqslant 1$  m s $^{-1}$ ) do not exhibit gas branches with strongly defined and straight leading edges. In these cases, the surface tension at the air–liquid interface may be a consideration. If we balance the dynamic pressure and pressure due to surface tension at a cross-section of the branch, we find that:

where $S=0.072$  N m $^{-1}$ is surface tension of water–air interface and $\unicode[STIX]{x1D705}$ is the curvature at the leading edge of the branch. Modifying this equation to define the Weber number, $We$ , we find the following relation:

The leading edge curvature is of the order of $\unicode[STIX]{x1D705}/\unicode[STIX]{x1D6FF}\sim 10^{-1}$ . At speeds where $U_{\infty }>3$  m s $^{-1}$ , the sweep angle varied between, $75^{\circ }<\unicode[STIX]{x1D711}<88^{\circ }$ making $0.12<\text{cos}^{2}\unicode[STIX]{x1D711}<0.3$ and therefore increasing the range of Weber numbers to $50<We<3200$ . But, at the lowest flow speed of $U_{\infty }=1$  m s $^{-1}$ , the sweep angle varied between, $55^{\circ }<\unicode[STIX]{x1D711}<70^{\circ }$ , making $0.12<\text{cos}^{2}\unicode[STIX]{x1D711}<0.33$ . Then, the range of Weber numbers is reduced to $12<We<33$ . This scaling therefore suggests that at the lowest speeds, we are approaching the conditions were the $We\approx O(1)$ , and surface tension can no longer be neglected.

6.5 Transition from Delta to Lambda topology

The data presented in figure 39 show the transition from one topology type to another is sensitive to the boundary layer, the liquid cross-flow velocity, the momentum of the gas normal to the flow and the volume flux of gas. For sufficiently high $Fr$ , increasing $Q^{\ast }$ or setting $\unicode[STIX]{x1D6FD}=90^{\circ }$ increases the degree to which the gas jet penetrates the free-stream flow, and we would expect that this would increase the likelihood that distinct gas branches (i.e. a Lambda topology) would form as the liquid flow impinges back on the plate surface. Hence, the Delta–Lambda transition, is likely to depend on the initial mechanism that bifurcates the cavity and sets the angle and chord length. Indeed, if the gas hump height is modified by changing the gas injection angle, $\unicode[STIX]{x1D6FD}$ , or is rendered quasi-independent of the gas injection flow rate by placing a solid obstacle upstream of the orifice, the Delta–Lambda transition conditions shift to different values of $Fr$ and  $Q^{\ast }$ .

Figure 39. A map of the cavity topology boundaries as a function of $Q^{\ast }$ versus $Fr$ for $\unicode[STIX]{x1D6FD}=90^{\circ }$ and for $D_{i}=5$  mm (a), 10 mm (b) and 20 mm (c) with $\unicode[STIX]{x1D6FF}\cong 52$  mm. Topology map for $Q^{\ast }$ versus $\unicode[STIX]{x1D6FD}$ with $D_{i}=5$  mm and $Fr=4.2$ . Topology map $Q^{\ast }$ versus $Fr$ for $\unicode[STIX]{x1D6FD}=90^{\circ }$ and for $D_{i}=6$  mm (e) and 10 mm (f) with $\unicode[STIX]{x1D6FF}\cong 17$  mm. The open symbols signify Lambda, grey transitional, and black filled Delta topology.