3.1 Flow visualization

Figure 5.

The large field of view and close-up view near the cavitator of the internal flow patterns inside the supercavity with (a) RJ closure (supplementary movies S1 for far view, S2 for near view) and (b) TV closure (supplementary movies S3 for far view, S4 for near view). The scale bars are applied to both TV and RJ cases correspondingly.

Flow visualization based on fog patterns provides a qualitative view of detailed gas flows inside the ventilated supercavity. As shown in figure 5(a), for the ventilated supercavity with RJ closure, the internal flow exhibits three main features: (i) a ventilation influence region observed behind the cavitator; (ii) a flow region of high shear extended from the liquid–gas interface (referred to as internal boundary layer); and (iii) a reverse flow region occurring in the centre of the supercavity. In addition, supplementary movies S1–S4, available at https://doi.org/10.1017/jfm.2018.1006, are provided to offer better presentation on the instantaneous flow structure inside the supercavity. Specifically, in the ventilation influence region, the air is entrained by the moving liquid–gas interface shortly after it exits the ventilation holes on the cavitator. Note that quantitative determination of the streamwise extent of the ventilation influence region is based on the mean velocity field, as will be discussed in § 3.2. Here the fog visualization only provides a qualitative description on the extent of this region, which is estimated from the cavitator to the location where the dense fog region has a minimum vertical extent. As it shows, in this region, a portion of air is entrained from the internal boundary layer to the reverse flow region and circulated back to the boundary layer. The ventilation influence region has a vertical extent close to $d_{C}$ , and the horizontal extent of the region (visualized by the fog pattern) fluctuates at approximately $2d_{C}$ downstream from the cavitator. Downstream of the ventilation influence region, the development of an internal boundary layer with appreciable air entrainment towards the interface can be clearly observed. The internal boundary layer is similar to the one developed from a moving foil that moves at the same velocity as the local water flow (Spurk Reference Spurk2002). Due to the strong disturbance caused by impingement of gas jets from the ventilation holes at the liquid–gas interface, the boundary layer transitions directly to a turbulent boundary layer. The boundary layer gradually grows thicker up to a location slightly downstream of the location of maximum supercavity diameter. Then the trend of boundary layer growth slows down as the supercavity reduces its diameter in the downstream. The presence of this internal boundary layer and its important role on air entrainment is consistent with the theory from Spurk (Reference Spurk2002), and a similar trend of boundary layer thickness variation was observed in the simulation from Kinzel et al. (Reference Kinzel, Lindau and Kunz2009). However, these prior studies did not specifically comment on the impact of the ventilation influence region on the general characteristics and the development of the boundary layer. The reverse flow region is observed throughout the entire streamwise span of the flow visualization (i.e. till ${\sim}80\,\%\,L$ downstream of the cavitator) in the centre portion of supercavity. Downstream of the ventilation influence region, the reverse flow covers the majority of the supercavity and travels with an almost uniform streamwise velocity, which is much slower than that of the moving liquid–gas interface. Besides, coherent spanwise vortical structures are advected upstream with a slight deformation in the reverse flow region, indicating the flow in this region is more uniform compared to that of the internal boundary layer. When the reverse flow approaches the location approximately $0.5d_{C}$ downstream of the cavitator, a clear deceleration is observed. It is worth noting that neither Spurk (Reference Spurk2002) nor Kinzel et al. (Reference Kinzel, Lindau and Kunz2009) indicated the presence of reverse flow for the supercavity with RJ closure even though Kinzel et al. noted such flow could occur for TV type supercavities. In addition, some subtle flow features can be observed from our flow visualization. For example, in the ventilation influence region, groups of tiny vortices with both clockwise and counter-clockwise rotation appear behind the cavitator and immediately above the liquid–gas interface, as shown from the close-up view in figure 5(a). These vortical structures are produced by ventilation jets and are likely to depend on the configuration of the ventilation schemes according to some of our preliminary tests using cavitators with modified ventilation. Moreover, at the outer extent of the internal boundary layer, a series of hairpin-like vortical structures emerge and are entrained into the reverse flow region. Note that the orientation of these coherent structures is opposite to that of the hairpin vortices observed in the wall-bounded flows due to the reversed direction of the shear stress associated with the moving liquid–gas interface.

As shown in figure 5(b), the supercavity with TV closure displays main flow features similar to those in the RJ supercavity. Specifically, the ventilation influence region has a similar vertical extent (i.e. ${\sim}d_{C}$ ) and streamwise extent (i.e. developing from 0 to ${\sim}2d_{C}$ downstream of the cavitator). However, some apparent discrepancies on the detailed characteristics of the reverse flow and the internal boundary layer are observed. In particular, the reverse flow shows a wider extent, almost filling the entire lateral span of the supercavity, and moves faster in the streamwise direction. Moreover, there seems to be more occurrences of coherent vortical structures being rolled-up from the boundary layer and entrained into the reverse flow region. Additionally, as shown in the close-up view in figure 5(b), more tiny vortices with enlarged size are produced in the vicinity of the location where the ventilation jet impinges the liquid–gas interface, due to the increase in the flow rate of ventilation. Furthermore, the close-up view of the supercavity with TV closure presents a series of coherent vortices rolled-up along the shear interface between the reverse flow and the ventilation jet flow. These vortices may be induced by enhanced Kelvin–Helmholtz (KH) instability, which is caused by the increased shear rate at the shear interface.

3.2 Mean flow field

To provide quantitative characterization of the main internal flow features observed from the flow visualization, the time-averaged internal flow fields are examined for supercavities of both the RJ and TV closure modes. Figure 6 shows the mean internal flow field of the RJ supercavity obtained by compiling ensemble-averaged flow fields (after the distortion correction) measured at three streamwise locations. Note that vectors in the gaps ( ${\sim}3.5$  mm) between adjacent PIV flow fields are interpolated based on five columns of interrogation windows (corresponding to ${\sim}5$  mm) on either side of each gap to ensure a smooth transition. The streamwise location ( $x$ ) and vertical location ( $y$ ) of the mean velocity field are normalized using the supercavity half-length ( $L_{Dmax}$ ), i.e. the distance from the cavitator to the location of the maximum supercavity diameter $D_{max}$ , and the cavitator diameter ( $d_{C}$ ), respectively (i.e. $\hat{x}=x/L_{Dmax}$ , ${\hat{y}}=y/d_{C}$ ).

Figure 6.

Ensemble-averaged (a) internal streamwise velocity contour, (b) streamlines and (c) velocity profile at $\hat{x}=0.25$ , 1.00, 1.75 for the supercavity with RJ closure. The bold solid curve and the open circles in (a) mark the liquid–gas interface and the locations of zero streamwise velocity, respectively. The circles and asterisks in (c) present the velocity distribution in the reverse flow and the internal boundary layer, respectively. The dashed curve in (c) presents the interpolated velocity profile close to the liquid–gas interface. Note that we do not add the interpolated segment for the velocity profile at $\hat{x}=0.25$ since it is situated in the ventilation influence region.

As shown in figure 6(a,b), the internal streamwise velocity field $U_{G}$ (the subscript ‘ $G$ ’ stands for ‘Gas’) exhibits three flow regions (the ventilation influence region, the reverse flow region and the internal boundary layer), consistent with the flow visualization results. In particular, the liquid–gas interface is extracted using a fourth-order polynomial fit and plotted as solid curve beneath the velocity field in figure 6(a). A layer of zero streamwise velocity (marked using open circles) can be determined using the mean flow field, which separates the forward flow in the developing internal boundary layer and the reverse flow in the bulk centre region of the supercavity.

The contour of mean streamwise velocity field shows the presence of the ventilation influence region behind the cavitator, consisting of a large portion of forward flow and the clear shear layer in figure 6(a). The streamwise extent of the ventilation influence region is estimated to be approximately $0.8L_{Dmax}$ ( $\hat{x}\approx 0.8$ ), measured from the cavitator (i.e. $\hat{x}=0$ ) to the location where the forward flow reaches its minimum vertical extent from the interface (i.e. $\hat{x}\approx 0.8$ ). Moreover, based on the streamline patterns of the internal flow (figure 6 b), a circulation region with a vertical extent of approximately $0.6d_{c}$ (estimated behind the cavitator at $\hat{x}\approx 0.2$ ), i.e. the ventilation influence region, can be clearly observed. As shown in figure 6(a), the reverse flow covers the centre portion of the supercavity and the speed of the reverse flow first increases as it moves upstream from the closure region towards the location of  $D_{max}$ . This trend may be associated with the streamwise adverse pressure gradient inside the supercavity, which has been shown in numerical studies of a ventilated supercavity (for example, Kirschner et al. Reference Kirschner, Fine, Uhlman and Kring2001; Cao et al. Reference Cao, Karn, Arndt, Wang and Hong2017; Kinzel et al. Reference Kinzel, Krane, Kirschner and Moeny2017). Around the location of $D_{max}$ (i.e. $\hat{x}\approx 1$ ), the reverse flow reaches its maximum speed of $1.26~\text{m}~\text{s}^{-1}$ , corresponding to 35 % of the velocity of the liquid–gas interface ( $U_{WG}$ ). The speed of the reverse flow remains relatively unchanged as it moves further upstream and decreases sharply as it approaches the cavitator (i.e. $\hat{x}\approx 0.5$ ) due to the blockage effect of the cavitator. In comparison, the internal boundary layer, identified as the region between the location of zero velocity and the liquid–gas interface downstream of the ventilation influence region, starts with a minimum thickness of 2.2 mm at $\hat{x}\approx 0.8$ . The boundary layer grows thicker downstream and the trend of boundary layer growth slows down after $\hat{x}\approx 1.7$ . It is also worth noting that at $\hat{x}\approx 1.7$ , the reverse flow speed reaches its maxima at a location slightly below the centreline of the cavitator (figure 6 c). Such a pattern is consistent with a recent numerical simulation by Cao et al. (Reference Cao, Karn, Arndt, Wang and Hong2017) under similar flow conditions (i.e. 30 mm cavitator, $4~\text{m}~\text{s}^{-1}$ incoming flow speed and TV type closure). It is likely to be a result of the asymmetric profile of the supercavity due to the buoyancy effect at low $Fr$ number.

To further quantify the velocity distribution at different regions inside the supercavity, velocity profiles at three selected streamwise locations (i.e. $\hat{x}=0.25$ , 1.00 and 1.25) are presented in figure 6(c). The streamwise internal flow velocity $U_{G}$ is normalized using the velocity of the liquid–gas interface $U_{WG}$ (i.e. $\hat{U} =U_{G}/U_{WG}$ ), where $U_{WG}$ can be derived from the equation below according to Logvinovich (Reference Logvinovich1972):

(3.1) $$\begin{eqnarray}U_{WG}(y)=\sqrt{U_{W,\infty }^{2}(1+\unicode[STIX]{x1D70E})+2gy}.\end{eqnarray}$$

In practice, in our experiments, $U_{W,\infty }^{2}(1+\unicode[STIX]{x1D70E})\gg 2gy$ , so we calculate $U_{WG}$ using $U_{WG}\approx U_{W,\infty }\sqrt{1+\unicode[STIX]{x1D70E}}$ , and use it to scale the internal flow velocity. The normalized velocity distribution of the reverse flow and the internal boundary layer are marked using circles and asterisks, respectively. Moreover, note that due to size limits in the interrogation window size for PIV, there is a region of ${\sim}1$  mm immediately above the liquid–gas interface where the velocity cannot be accurately resolved. Therefore, we interpolate the velocity profile in this region (outside the ventilation influence region) using the turbulent boundary layer model presented in Spurk (Reference Spurk2002). Specifically, according to the model, the streamwise velocity $U_{G}(r)$ at a location at distance $r$ from the liquid–gas interface (inside the boundary layer) is calculated using the following equation:

(3.2) $$\begin{eqnarray}\frac{U_{G}(r)}{U_{\unicode[STIX]{x1D70F}}}=\frac{U_{WG}}{U_{\unicode[STIX]{x1D70F}}}-\left[\frac{1}{\unicode[STIX]{x1D705}}\ln \left(\frac{rU_{\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D708}}\right)+2\frac{\unicode[STIX]{x1D6F1}}{\unicode[STIX]{x1D705}}\sin ^{2}\left(\frac{\unicode[STIX]{x03C0}r}{2\unicode[STIX]{x1D6FF}}\right)+B\right],\end{eqnarray}$$

where $\unicode[STIX]{x1D708}$ is the kinematic viscosity of the air ( $1.5\times 10^{-5}~\text{m}^{2}~\text{s}^{-1}$ ), $\unicode[STIX]{x1D705}$ the Karman constant (0.41), $\unicode[STIX]{x1D6F1}$ and $B$ are empirical values (0.55 and 5) based on the empirical results provided in Schlichting et al. (Reference Schlichting, Gersten, Krause and Qertel1955), $\unicode[STIX]{x1D6FF}$ the measured thickness of the internal boundary layer, and $U_{\unicode[STIX]{x1D70F}}$ the friction velocity calculated using the interface condition (Spurk Reference Spurk2002):

(3.3) $$\begin{eqnarray}0=\frac{U_{WG}}{U_{\unicode[STIX]{x1D70F}}}-\left[\frac{1}{\unicode[STIX]{x1D705}}\ln \left(\frac{\unicode[STIX]{x1D6FF}U_{\unicode[STIX]{x1D70F}}}{\unicode[STIX]{x1D708}}\right)+2\frac{\unicode[STIX]{x1D6F1}}{\unicode[STIX]{x1D705}}+B\right].\end{eqnarray}$$

In general, the velocity profiles show consistent trends regarding the development of the reverse flow region and the internal boundary layer. Specifically, immediately behind the cavitator (i.e. at $\hat{x}=0.25$ ), the reverse flow covers the smallest vertical span ( $0.5d_{C}$ ) in comparison with other streamwise locations. The primary reason is the effect of the forward flow induced by ventilation jet flow in a relatively narrow cross-section of the cavity. As the cavity increases its diameter, the reverse flow increases its span to approximately $0.8d_{C}$ at $\hat{x}=1.00$ with an appreciable increase of flow speed. Further downstream (for example, at $\hat{x}=1.75$ ), the reverse flow occupies a slightly smaller region (approximately $0.75d_{C}$ ) with a reduced magnitude. For all locations, the forward flow occupies a significantly smaller region compared to the reverse flow.

Figure 7.

Ensemble-averaged (a) internal streamwise velocity contour, (b) streamlines and (c) velocity profile at $\hat{x}=0.25$ , 1.00, 1.75 for the supercavity with TV closure. The symbols follow the same convention as those in the figure 6.

In comparison with figure 6, figure 7 shows the mean internal flow field of the TV supercavity. Compared with the flow conditions of the RJ supercavity, as shown in table 1, the TV supercavity is formed using the same cavitator with the same $Fr$ but increased $Q_{As}$ . Under such conditions, the half-length of the TV supercavity stays unchanged, consistent with a number of prior studies (for example, Logvinovich Reference Logvinovich1972; Epshtein Reference Epshtein1973; Shao et al. Reference Shao, Karn, Ahn, Arndt and Hong2017). Therefore, the streamwise location ( $x$ ) and vertical location ( $y$ ) of mean velocity can still be normalized by the same supercavity half-length ( $L_{Dmax}$ ) and cavitator diameter ( $d_{C}$ ), respectively. As shown in figure 7(a), the contour of the internal streamwise velocity field for the TV supercavity exhibits similar characteristics for the ventilation influence region, the reverse flow region and the internal boundary layer compared to the RJ supercavity. Specifically, the streamwise extent of the ventilation influence region stays approximately the same as that of the RJ supercavity. The trends in development of the reverse flow velocity and the internal boundary layer thickness in the streamwise direction are also similar. However, are still appreciable discrepancies in the flow regions. First, the ventilation influence region has an expanded vertical extent of approximately $0.8d_{C}$ (compared with $0.6d_{C}$ in the RJ case) as estimated from the streamline distribution shown in figure 7(b). The vertical expansion is also illustrated by comparing the velocity profile measured at $\hat{x}=0.25$ in figure 7(c) with that in figure 6(c). The vertical expansion of the ventilation influence region may be attributed to the presence of stronger ventilation jet flow in the TV case. Second, the magnitude of the reverse flow shows an approximately 20 % increase with respect to that in the RJ case, potentially due to a larger adverse pressure gradient. Such an increase is also evidenced more clearly by comparing the velocity profile measured at $\hat{x}=1.00$ in figure 7(c) with that in figure 6(c). Third, the internal boundary layer becomes thinner along the liquid–gas interface in the TV case. Comparing to the RJ case (figure 6 c), the reverse flow also shows a decaying strength towards the centreline of the cavity (figure 7 c) with similar causes as discussed previously (i.e. the asymmetric cavity profile).

Figure 8.

(a) Comparison of measured and modelled internal boundary layer thickness of the RJ and TV supercavities with respect to the streamwise location. (b) Schematic of a boundary layer formed on a moving plate with an external flow moving in the opposite direction.

Furthermore, the variation of the internal boundary layer thickness is quantified and compared between the RJ and TV cases based on the results shown in figures 6 and 7. As mentioned earlier, the boundary layer thickness in our current flow setting is defined as the distance between the liquid–gas interface and the location of zero streamwise velocity at each streamwise location. Figure 8(a) shows the variation of boundary layer thickness along the interface. The thickness $\unicode[STIX]{x1D6FF}$ is normalized by the cavitator diameter $d_{C}$ (i.e. $\hat{\unicode[STIX]{x1D6FF}}=\unicode[STIX]{x1D6FF}/d_{C}$ ). Note that the scaled vertical extent of the forward moving flow in the ventilation influence region (i.e. $\hat{x}\approx 0.8$ ) is defined in the same way as that of the internal boundary layer and plotted in figure 8(a) for comparison. For the RJ supercavity, the experimentally measured thickness ( $\unicode[STIX]{x1D6FF}_{exp,RJ}$ ) shows that the internal boundary layer starts at $\hat{x}\approx 0.8$ with a minimum thickness of ${\sim}0.07d_{C}$ . The boundary layer gradually grows thicker along with the interface. The increasing trend starts to slow down at $\hat{x}\approx 1.7$ due to the increase of adverse pressure gradient in the streamwise direction as the closure is approached, and the boundary layer reaches the maximum thickness of ${\sim}0.15d_{C}$ at $\hat{x}\approx 2$ . Note that the measured thickness shows some jitter along the interface which may be caused by the uncertainties involved in determining the interface and convergence of the mean flow field. In comparison, the internal boundary layer thickness of the TV supercavity ( $\unicode[STIX]{x1D6FF}_{exp,TV}$ ) starts at approximately the same streamwise location (i.e. $\hat{x}\approx 0.8$ ), but with a smaller magnitude of ${\sim}0.04d_{C}$ . Along with the interface $\unicode[STIX]{x1D6FF}_{exp,TV}$ develops in a similar way to $\unicode[STIX]{x1D6FF}_{exp,RJ}$ . Over all measured locations, $\unicode[STIX]{x1D6FF}_{exp,TV}$ is approximately 20 % smaller than $\unicode[STIX]{x1D6FF}_{exp,RJ}$ in an average sense.

To further understand the internal boundary layer growth, a theoretical calculation is proposed to estimate the boundary layer thickness. As illustrated in figure 8(b), after exiting the ventilation influence region (i.e. $\hat{x}>0.8$ ), the internal boundary layer is approximated as a boundary layer developed over a moving plate with an external flow moving in the opposite direction. Under this setting, the shear stresses from the liquid and gas sides at the liquid–gas interface need to be balanced and it should be noted that the dynamic viscosity of the water is significantly higher than that of the air. As a result, the velocity gradient normal to the boundary layer on the water side (referred to as external boundary layer) is negligible compared with that of the internal boundary layer. Under such conditions, according to Spurk (Reference Spurk2002), the liquid–gas interface can be regarded as a plate moving at the same velocity ( $U_{WG}$ ) as the free-stream velocity of water at the corresponding streamwise location. As mentioned above, $U_{WG}$ is approximately constant ( ${U_{WG}\approx U}_{W,\infty }\sqrt{1+\unicode[STIX]{x1D70E}}$ , since $U_{W,\infty }^{2}(1+\unicode[STIX]{x1D70E})\gg 2gy$ ), while the reverse internal flow gradually accelerates upstream from $\hat{x}\approx 2.3$ to $\hat{x}\approx 0.8$ , as shown in the velocity contour in figures 6(a) and 7(a). To our best knowledge, there is no established theoretical model to quantify the boundary layer thickness under such specific conditions. Therefore, we provide a simplified model to estimate the boundary layer thickness at different streamwise locations. According to the numerical simulations of ventilated supercavitation (Kirschner et al. Reference Kirschner, Fine, Uhlman and Kring2001; Cao et al. Reference Cao, Karn, Arndt, Wang and Hong2017; Kinzel et al. Reference Kinzel, Krane, Kirschner and Moeny2017), the adverse pressure gradient in the domain of our PIV measurement of the internal boundary layer thickness (i.e. $0.8\lesssim \hat{x}\lesssim 2.3$ ) is substantially smaller than that near the closure region. Therefore, we do not consider the pressure term in the control equation of the boundary layer. For laminar flow, the control equation is simplified to the well-known Blasius equation:

(3.4) $$\begin{eqnarray}f^{\prime \prime \prime }(\unicode[STIX]{x1D702})+f^{\prime \prime }(\unicode[STIX]{x1D702})f(\unicode[STIX]{x1D702})=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D702}=y\sqrt{U_{WG}/2\unicode[STIX]{x1D708}x}$ , $f^{\prime }(\unicode[STIX]{x1D702})=-U_{G}(\unicode[STIX]{x1D702})/U_{WG}$ . The boundary conditions are specified as follows: $f(0)=0$ due to no penetration from the interface; $f^{\prime }(0)=1$ since the air flow has the same velocity as the interface at $\unicode[STIX]{x1D702}=0$ according to our discussion above; $f^{\prime }(\infty )=-\overline{U_{G,\infty }}/U_{WG}$ , indicating that the flow reaches the velocity of free stream ( $\overline{U_{G,\infty }}$ ) outside the boundary layer. In our calculation, the velocity of free stream $\overline{U_{G,\infty }}$ ( $0.7~\text{m}~\text{s}^{-1}$ for the RJ supercavity and $1.1~\text{m}~\text{s}^{-1}$ for the TV supercavity) is obtained by averaging the streamwise velocity of the reverse flow measured at ${\hat{y}}=0$ and $0.8\leqslant \hat{x}\leqslant 2.3$ . Equation (3.4) can be solved using the shooting method introduced in Cebeci & Bradshaw (Reference Cebeci and Bradshaw2012), and the boundary layer thickness can be calculated using the following equation:

(3.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{cal}=\sqrt{\frac{2\unicode[STIX]{x1D708}x}{U_{WG}}}\unicode[STIX]{x1D702}_{0},\end{eqnarray}$$

where $\unicode[STIX]{x1D702}_{0}$ is similarity variable representing the non-dimensionalized distance from the liquid–gas interface to the location of zero velocity, which is determined from $f^{\prime }(\unicode[STIX]{x1D702}_{0})=0$ by solving (3.4). Due to different values of $f^{\prime }(\infty )$ , $\unicode[STIX]{x1D702}_{0}$ is calculated to be 4.7 for the RJ supercavity and 3.4 for the TV supercavity. Based on (3.5), the modelled boundary layer thicknesses of the RJ and TV cavities are presented in figure 8(a) as solid and dashed curves, respectively. It is clearly observed that the modelled boundary layer thickness of the supercavity TV supercavity is thinner than that of the RJ supercavity at all locations. Compared to the experimental results, the modelled thickness in general shows a good match of the growth trend and magnitude. However, it is noticeable that the measured thickness grows more rapidly compared to the modelled thickness due to the laminar flow assumption used in the model. It is worth noting that for the flat plate the thickness of a turbulent boundary layer $\unicode[STIX]{x1D6FF}$ grows proportional to $x^{0.8}$ in the range of our Reynolds number, which aligns better with the growth rate observed in our measurements. However, no empirical model for the turbulent boundary layer can be derived under our flow settings. In addition, figure 8(a) shows experimental values are first below and then above the modelled values as the boundary layer grows downstream in the range $0.8\leqslant \hat{x}\leqslant 2.3$ . A possible cause of such a difference is the assumption of the constant free-stream velocity of $\overline{U_{G,\infty }}$ , which is below the local $U_{G,\infty }$ in the range $0.8\leqslant \hat{x}\lesssim 1.5$ and above in the range $1.5\lesssim \hat{x}\leqslant 2.3$ . Another cause of the difference between the experiment and the model can come from the fact that 3D effects on the boundary layer thickness are not included in our model. Nevertheless, such effects are estimated to be small considering the significant difference between the local radius of curvature of the interface and the internal boundary layer thickness in our measurement range. In the ventilation influence region (i.e. $\hat{x}<0.8$ ), the vertical extent of the forward moving flow of both the RJ and TV cases measured from the experiment is significantly larger compared with the modelled values, due to the strong influence of the ventilation jets.

Figure 9.

(a) Schematic showing the estimate of gas flow circulation in the reverse flow region at the bottom half of the supercavity based on the axisymmetric assumption. The estimated change of the reverse flow ( $Q_{REV}$ ) compared with the ventilation input ( $Q_{V}$ ) for the supercavity with (b) RJ and (c) TV closure.

Using the internal velocity fields, we further estimate variation of the gas flow rate at each cross-section along the streamwise direction in the reverse flow region to investigate the overall gas circulation in the supercavity. As a first step, the gas flow rates obtained from the 2D plane is employed to approximate the volumetric flow rate based on the axisymmetric assumption. Specifically, as illustrated in figure 9(a), the velocity profile at each streamwise location is extrapolated axisymmetrically in the circumferential direction to cover the entire lower half of the supercavity cross-section (marked as green shaded region). Here the lower half of the cross-sections can be treated as semicircles, since ellipse fitting (introduced in figure 3) shows that the axial ratio of the cross-sections is less than 1.05 for all streamwise locations in the PIV measurement domain. After the extrapolation, the flow rates of the reverse flow (i.e. $Q_{REV}$ ) of the half-supercavity are calculated separately at each streamwise location by integrating the velocity profile at the corresponding range of the vertical span. It is important to note that in the current study we do not estimate the gas flow rate in the internal boundary layer using the same approach above, due to the large uncertainty involved in such an estimate for the boundary layer flow. Such a large uncertainty is associated with the unresolved velocity profile near the interface and the substantial deviation from axisymmetry in the boundary layer caused by the disturbed upper interface. Instead, the gas flow rate moving forward ( $Q_{FWD}$ ) is inferred from the estimate of $Q_{REV}$ based on mass conservation, i.e. $Q_{FWD}=Q_{V}+Q_{REV}$ , where $Q_{V}$ is the ventilation input (estimated to be half of the overall input $Q_{As}$ ). Note that all the volume flow rates (i.e. $Q_{FWD}$ , $Q_{REV}$ and $Q_{V}$ ) are corrected to standard pressure and temperature, i.e. 100 kPa and 273.15 K, and calculated in the units of standard litres per minute (SLPM). Therefore, their values correspond directly to mass flow rates.

For the RJ supercavity, as shown in figure 9(b), it is evident that $Q_{REV}$ is significantly larger than $Q_{V}$ at almost all the streamwise locations. At locations of $\hat{x}>1$ , $Q_{REV}$ increases rapidly as the reverse flow moves upstream, and reaches its maximum near the location of maximum cavity diameter at $\hat{x}=1$ . At upstream locations of $\hat{x}<1$ , the reverse flow rate reduces, with gas partially dispersing into the region near the interface. Correspondingly, $Q_{FWD}$ shows the same trend as that of $Q_{REV}$ . For the TV supercavity, $Q_{FWD}$ and $Q_{REV}$ vary along the streamwise direction in the same fashion as for RJ supercavity, as shown in figure 9(c). However, $Q_{FWD}$ and $Q_{REV}$ of the TV supercavity experience a steeper change at the ventilation influence region (i.e. $\hat{x}<0.8$ ) due to the stronger ventilation jet.

Based on the trends of gas flow rates in the internal boundary layer and reverse flow region presented above, the general picture (common for both RJ and TV supercavities) of the internal gas circulation is depicted in figure 10. Starting from the backside of the cavitator, the ventilation jet impinges directly on the liquid–gas interface as it comes out of the ventilation holes. The jet flow first experiences significant expansion and deceleration as it moves downstream, occupying a large vertical extent between the reverse flow and the liquid–gas interface. Meanwhile, the gas flow that impinges the liquid–gas interface is partially dragged downstream by the moving interface. However, in the ventilation influence region, the amount of ventilation input exceeds the entrainment capability of the moving interface, so the majority of the jet flow is not affected by the interface in this region. As illustrated in the velocity profile of region  $A$ cropped from figure 10, in the majority of the vertical extent of the forward flow, the velocity is very low, and only a small portion of forward flow is influenced by the moving interface. As the interface moves further downstream, more ventilation gas is accelerated by the interface, and at the same time the gas is entrained from the reverse flow region towards the interface. Such an exchange of momentum between the flow near the interface and the reverse flows causes the mass flow rate of the forward flow to increase with a decreasing vertical extent.

Figure 10.

Schematic of gas flows inside the supercavity. The gas exchange between the internal boundary layer and the reverse flow is marked with black arrows.

Outside the ventilation influence region, an internal boundary layer with a velocity profile matching that developed from a moving plate is formed, as illustrated in the region  $B$ of figure 10. In particular, starting at the location of maximum cavity diameter ( $L_{Dmax}$ ), the gas within the internal boundary layer is stripped out and entrained by the reverse flow due to the adverse pressure gradient. Associated with the change of velocity profiles (comparing the velocity profiles between regions  $B$ and  $C$ of figure 10), although the gas entrained inside the internal boundary layer decreases, the boundary layer still thickens according to the trend of a boundary layer on a moving plate. Near the closure region (region  $D$ in figure 10), the internal boundary layer merges from the upper and lower interfaces (more precisely, from all circumferential locations at the 3D interface) according to Savchenko & Savchenko (Reference Savchenko and Savchenko2012). In this region, reverse flow vanishes, and the flow of internal boundary layer increases its velocity as it moves towards the closure due to the decreasing cavity diameter. The supercavity eventually converges and a hydrodynamic wake is formed, in which the gas moves with approximately the same velocity. Due to mass conservation, in this region, the flow rate of the merged boundary layer (i.e. $Q_{FWD}$ ) remains constant and is equal to the ventilation input $Q_{As}$ at all streamwise locations.

Moving upstream from the closure region, as mentioned above, the reverse flow starts at the streamwise location where the internal boundary layer merges. It is driven by the adverse pressure gradient and accelerates as it moves upstream. Besides, the vertical extent of the reverse flow also increases, according to the analysis of Riley & Weidman (Reference Riley and Weidman1989) and Ishak, Nazar & Pop (Reference Ishak, Nazar and Pop2007) for similar flow settings. Specifically, Riley & Weidman (Reference Riley and Weidman1989) and Ishak et al. (Reference Ishak, Nazar and Pop2007) study the boundary layer developed over a stretching plate and a moving wedge, respectively, both with an accelerating external flow in the opposite direction. The flow conditions in their studies share common features as ours (for example, the moving surface, the accelerating reverse flow and the adverse pressure gradient). By performing theoretical calculations based on the Falkner–Skan boundary layer model, they obtained the similarity solution for the flow velocity profile at different streamwise location. The similarity solution shows that the vertical extent of the reverse flow region widens in proportion to its developing length with a power index of $(1-m)/2$ , where $m$ ( ${<}1$ ) is a function of the adverse pressure gradient. From the closure region to the location of maximum cavity diameter ( $L_{Dmax}$ ), the reverse flow continuously entrains gas from the internal boundary layer, and the flow rate increases gradually to its maximum at $x=L_{Dmax}$ . From $x=L_{Dmax}$ to the cavitator, the adverse pressure gradient becomes negligible. Instead, the effect of cavitator blockage and the viscous shear stress associated with a sharp velocity gradient across the reverse flow and ventilation jet regions kick in, reducing the bulk velocity of the reverse flow. In addition, due to the reduction of supercavity diameter and the increase of vertical extent of the forward flow in this region, the reverse flow region narrows down its span and the reverse flow rate drops sharply.

It is worth noting that although the physical picture described above shares some similar insights to those of mechanisms portrayed in prior studies by Spurk (Reference Spurk2002) and Kinzel et al. (Reference Kinzel, Lindau and Kunz2009), some remarkable differences are presented here. Specifically, neither Spurk (Reference Spurk2002) nor Kinzel et al. (Reference Kinzel, Lindau and Kunz2009) consider the air entrainment to be affected by the ventilation jet in the ventilation influence region. Moreover, Spurk (Reference Spurk2002) claims that no reverse flow occurs in the supercavity, and the internal boundary layer keeps entraining gas until the closure region is reached. In contrast, we observe the clear presence of reverse flow and of gas in the boundary layer entrained by reverse flow after the location of maximum diameter. Although the model proposed by Kinzel et al. (Reference Kinzel, Lindau and Kunz2009) takes into account the gas exchange between the internal boundary layer and the reverse flow, they claim that the reverse flow only exists for the TV supercavity and vanishes for the RJ case. However, we observe the presence of reverse flow for both the TV and RJ cavities. In addition, the model from Kinzel et al. does not incorporate the velocity change of the reverse flow along the streamwise location, which is clearly observed from our measurements.

3.3 Gas leakage mechanism and closure modes

In this section, we will investigate the connection of the internal supercavity flow with the gas leakage mechanism and the closure modes of a ventilated supercavity. Using the physical insights gained from the internal flow investigation in §§ 3.1 and 3.2, we will explain the growth of supercavity length and the closure mode transition (i.e. from RJ to TV) as the ventilation input increases. First, we will provide new insights into the physical mechanism that contributes to the formation of RJ closure, in addition to the explanation provided in the prior studies. Then we will discuss in detail the influence of the gas leakage mechanism on the cavity growth and closure transition.

Figure 11.

(a) Time-averaged supercavity profile at the critical point where the visible stable RJ closure is observed with the external boundary layer and the incline angle marked. (b) Schematic showing separation of the external boundary layer on a moving interface (adapted from Sears & Telionis Reference Sears and Telionis1975). (c) Visualization of water flow near the RJ closure. The trajectories of particles in water show they move towards the liquid–gas interface (please refer to the supplementary video S5 for more details).

The mechanism that governs the formation of RJ closure of a ventilated supercavity has been addressed by a number of prior studies (for example, Semenenko Reference Semenenko2001; Spurk Reference Spurk2002; Savchenko & Savchenko Reference Savchenko and Savchenko2012). These studies claim that the RJ closure is formed when the water penetrates the liquid–gas interface and rolls up boundary layer material (i.e. water/air mixture) into a toroidal vortex. Specifically, Semenenko (Reference Semenenko2001) pointed out that the toroidal vortex is formed due to the collisions of fluid particles at closure. As it grows larger, the toroidal vortex is ejected from the supercavity when the friction force on the liquid–gas interface becomes equal to the re-entrant momentum. Spurk (Reference Spurk2002) stated that the small radius of curvature of the liquid–gas interface along with the reversed water flow at the closure region may contribute to the formation of the toroidal vortex, though he did not mention how the reversed water flow is formed. As more water flows into the vortex, the density of the toroidal vortex increases and the local cavitation number becomes negative, causing the vortex to be shed off from the supercavity. Savchenko & Savchenko (Reference Savchenko and Savchenko2012) suggested that Helmholtz–Taylor type mechanisms of instability cause the liquid–gas interface to become unstable, and the gas that fills the defects of the disturbed interface causes splashes at the closure region, which eventually leads to the RJ closure. However, the prior studies provided no detailed explanation of the mechanism that governs the penetration of water flow into the supercavity. Here we use flow visualization, presented in figure 11(a), to offer some further insights into such a mechanism. The supercavity geometry shown in the figure is obtained by averaging the instantaneous supercavity images recorded under 125 f.p.s. and a period of ${\sim}40$  s. From the figure, we notice that the re-entrant jet in this case only appears at the bottom surface of the supercavity, and the inclined angle of the cavity bottom surface ( $\unicode[STIX]{x1D6FC}$ ) is significantly larger than that of the top surface. These observations indicate the water flow close to the interface experiences a stronger adverse pressure gradient at the bottom surface. According to prior theoretical work by Stewartson (Reference Stewartson1954) and Schlichting et al. (Reference Schlichting, Gersten, Krause and Qertel1955), for 2D flow on a fixed wedge which is expanded under an adverse pressure gradient, the expansion angle (corresponding to $\unicode[STIX]{x1D6FC}$ in our study) has a critical value of $17.9^{\circ }$ . Above the critical expansion angle, flow separation occurs at downstream locations. In our experimental results, shown in figure 11(a), the inclined angle of the bottom surface ( $\unicode[STIX]{x1D6FC}$ ) measured from the 2D profile of the RJ supercavity at the critical closure transition point is ${\sim}18^{\circ }$ , very close to the critical angle from the theoretical work on wedge flow. Consequently, though differences between the two flows are recognized, we would like to attribute the water penetration and the formation of the RJ closure to the flow separation of the boundary layer at the water side (i.e. external boundary layer) caused by the strong adverse pressure gradient. The flow separation of a boundary layer on a moving interface has been thoroughly investigated, and a so-called Moore–Rott–Sears (MRS) criterion (co-established by Rott Reference Rott1956; Moore Reference Moore1958; Sears & Telionis Reference Sears and Telionis1975) has been proposed to explain the flow separation. This criterion is illustrated in figure 11(b), a schematic adapted from Sears & Telionis (Reference Sears and Telionis1975), showing the separation of the boundary layer on a downward moving wall (i.e. the interface moves in the same direction as the free stream). The MRS criterion claims that due to the influence of a large adverse pressure gradient, at a certain location downstream, a singular point should exist inside the boundary layer. At the singular point, both the streamwise velocity and the shear stress vanish simultaneously. At downstream locations of the singular point, a reverse flow region appears and the flow inside the region moves towards the singular point. Such reverse flow causes the external boundary layer to separate. As illustrated in figure 11(b), the flow separation leads to an induced velocity of the boundary layer flow towards the moving wall. Using the MRS criterion for our case, we consider the liquid–gas interface of the supercavity as the moving wall. Differing from a solid wall, the liquid–gas interface cannot withstand the momentum of the impinging water. Consequently, the water penetrates the interface and enters the supercavity as a re-entrant jet. As shown in figure 11(c), this phenomenon is clearly observed in a close view of the water flow near the liquid–gas interface at the closure region. By tracking the movement of the tracers in water, the trajectories are plotted and overlaid in the figure to show that the tracers move towards the interface (see supplementary video S5 for more evidence). The water entering the supercavity moves upstream for some distance, and then drops down to the bottom interface due to the gravity. At the closure, the circulating motion of the water forms a toroidal vortex, entraining gas inside the supercavity and exiting the cavity in the form of foams. It is observed in our experiments that foams exit the cavity intermittently through the liquid–gas interface, which is in agreement with Spurk (Reference Spurk2002). The shedding foams also make the liquid–gas interface unstable at the closure.

It is important to note that for the RJ supercavity in our experiment, the twin-vortex tubes still exist in addition to the re-entrant jet at the closure region, as shown in the side view of the supercavity structure in figure 12(a) and bottom view in figure 12(b). In the side view (figure 12 a), such twin-vortex tubes appear as a single rotating bubble column coming out from the closure region, carrying a large amount of gas from the supercavity. The gas flow rate in the twin-vortex tubes is denoted as $Q_{TV}$ . In addition, the foams of the toroidal vortex (associated with re-entrant jet) emerging from the liquid–gas interface carry additional gas (denoted as  $Q_{RJ}$ ). These foams further break into bubbles surrounding the twin-vortex tubes, and move downstream along with the gas flow inside the tubes. From the bottom view shown in figure 12(b), some bubbles from the toroidal vortex blend into twin-vortex tubes, with some gas emerging from the tube and breaking into bubbles.

Figure 12.

Side view (a) and bottom view (b) of the closure region of the RJ supercavity under our experimental conditions. Both the toroidal vortex associated with the re-entrant jet and the twin-vortex tubes are observed.

Since no prior studies on RJ supercavity have mentioned the presence of twin-vortex tubes in the RJ closure, here we provide a mechanism to explain this observation, as illustrated in figure 13. Generally, we think that the mechanism responsible for the formation of the twin-vortex tubes from a supercavity is analogous to the shedding vortices of an aircraft wing. Specifically, as shown in figure 13(a), due to the asymmetric shape of the airfoil of an aircraft wing, the flow distribution at the top surface deviates from that of the bottom surface. Such a deviation results in a net circulation, which induces a lift. According to the fundamentals of aerodynamics (for example, Simons & Simons Reference Simons and Simons1999), due to variations in the cross-section profile along the spanwise direction, the vortex formed by the flow circulation loses its strength along the whole wingspan. Due to Helmholtz’s first theorem, a vortex filament has constant strength along its length. Therefore, the aircraft wing sheds a street of streamwise rotating vortices from the trailing edge. Furthermore, the summation of the strength of the spanwise and streamwise vortices is conservative at all spanwise locations. This phenomenon has been thoroughly explained by the well-known Lanchester–Prandtl lifting line theory. Moreover, the shed streamwise rotating vortex street tends to merge into a pair of counter-rotating vortices while rolling up.

Figure 13.

Schematic showing the formation of streamwise vortices due to the difference of velocity circulation along the spanwise direction for (a) an aircraft wing and (b) a supercavity.

Comparatively, as illustrated in figure 13(b), due to the buoyancy of gas bubbles, the profile of a supercavity results in a flow circulation which induces a downward ‘lift’ (Cox & Clayden Reference Cox and Clayden1955). For a supercavity, the flow circulation associated with the spanwise cross-section is maximum (bound circulation) at the meridian plane, and gradually decreases away from the meridian plane. This trend is similar to that of the aircraft wing shown in figure 13(a), except for the reversed direction. Accordingly, for the supercavity, the streamwise rotating vortex street merges into a pair of counter-rotating vortices at closure, but the rotating direction is reversed for each vortex. In addition, the pair of vortices causes the liquid–gas interface to deform and separate into twin-vortex tubes at closure, known as the TV closure.

In short, the TV closure originates from the 3D asymmetric shape of the supercavity. For the RJ supercavity in our experiment, the 3D asymmetry of the cavity is evident, and thus the twin-vortex tubes are clearly appreciable. The case is different from a number of prior studies on RJ supercavities (for example, Cox & Clayden Reference Cox and Clayden1955; Semenenko Reference Semenenko2001; Spurk Reference Spurk2002), where the coexistence of the RJ and TV closure was not clearly stated. In these prior studies, RJ supercavities were typically investigated under very high $Fr$ . Under such conditions, the gravity effect is negligible and the supercavity has an axisymmetric shape. Therefore, the flow circulation is zero at all spanwise locations and no vortex is shed downstream, resulting in the absence of the twin-vortex tubes. For the same reason, the TV closure does not appear for 2D supercavities which have a uniform cross-section shape along the spanwise direction (for example, the supercavity generated from a cylindrical cavitator).

Figure 14.

(a) Schematic showing the change of the internal flow, the geometry and the closure region of a RJ supercavity with increasing ventilation. (b) Images showing the change of the closure region of a RJ supercavity with increasing ventilation.

For RJ supercavity in our experiments, it is observed that the supercavity length grows and the toroidal vortex size reduces with increasing ventilation, as shown in figure 14(a). Such a trend can be explained based on our insights gained from the internal flow results in § 3.2 and the discussion of RJ closure generation at the beginning of this section. As mentioned earlier (figure 12) and shown in the upper schematic of figure 14(b), the gas leakage for a RJ supercavity is contributed by two mechanisms. One is from the shedding of the toroidal vortex associated with the re-entrant jet (i.e. $Q_{RJ}$ ), and the other from the internal boundary layers that entrain gas and merge near the closure to form twin-vortex tubes ( $Q_{TV}$ ). For a stable RJ supercavity, the ventilation input ( $Q_{As}$ ) should be equal to sum of $Q_{RJ}$ and $Q_{TV}$ according to mass conservation. As the ventilation input increases under fixed $Fr$ , the supercavity changes its geometry, as illustrated in the middle and lower schematics of figure 14(b). In particular, the middle schematic shows a transient state during supercavity growth upon increasing ventilation. As it shows, an increased ventilation input ( $Q_{As}$ ) first causes the vortex tube at the closure to widen its diameter to ensure larger gas leakage, with a corresponding reduction of the incline angle of the bottom interface ( $\unicode[STIX]{x1D6FC}$ ). Such a reduction leads to a weaker external flow separation and a corresponding reduction in the gas leakage from the foamy toroidal vortices (i.e. $Q_{RJ}$ decreases). Meanwhile, due to weakening of flow separation, the pressure at closure (i.e. $P_{out}$ ) recovers, as proved in past studies on hydrofoils (for example, Kubota, Kato & Yamaguchi Reference Kubota, Kato and Yamaguchi1992). Moreover, from our previous experiments (Karn et al. Reference Karn, Arndt and Hong2016a ; Karn, Arndt & Hong Reference Karn, Arndt and Hong2016c ), the pressure at the backside of the cavitator ( $P_{C}$ ) stays approximately the same once the supercavity is formed, regardless of the ventilation input. Consequently, if the cavity length remains unchanged, the adverse pressure gradient within the supercavity increases, stripping a greater amount of gas out of the internal boundary layer as it moves downstream. This phenomenon is observed in the study of adverse pressure gradient boundary layers by Andersen, Kays & Moffat (Reference Andersen, Kays and Moffat1972). Correspondingly, the amount of gas entrained by the internal boundary layer at closure and leaking from the tubes decreases (i.e. $Q_{TV}$ decreases). Since both $Q_{RJ}$ and $Q_{TV}$ decrease upon an increase of ( $Q_{As}$ ) according to our assumption (i.e. supercavity length does not grow), under this condition, mass conservation is not satisfied (i.e. $Q_{As}>Q_{RJ}+Q_{TV}$ ). As a result, more air is stored in the supercavity and the cavity has to elongate itself. After elongation, the cavity is able to entrain more air at closure (i.e. larger $Q_{TV}$ ) to balance the ventilation input. The supercavity stops elongating when the internal boundary layer grows thick enough to fulfil the widened vortex tube at closure, as shown in lower schematic of figure 14(b). In contrast, reduction of $Q_{As}$ causes the RJ supercavity to shorten its length and increase the incline angle  $\unicode[STIX]{x1D6FC}$ . It is worth noting that for $Q_{As}<9$ SLPM in our experiments, the external flow separation becomes so strong that $Q_{RJ}$ exceeds $Q_{As}$ . Mass conservation cannot be satisfied, and the supercavity keeps shortening and eventually collapses. Furthermore, it is also worth explaining why $Q_{As}=10$ SLPM is selected as the ventilation conditions of RJ supercavity used for visualization and PIV measurements in our experiments, but not lower ventilation input. Admittedly, with lower $Q_{As}$ , the toroidal vortex becomes larger (as shown in figure 14 a), and more appreciable differences may be observed between the RJ and TV supercavities. However, larger toroidal vortex rolls more water into the cavity. Driven by the vortex momentum and the adverse pressure gradient, more water droplets are observed intermittently impinging the liquid–gas interface. Therefore, the internal flow field is contaminated and cannot be used for PIV processing under these settings. Specifically, close to 50 % of the PIV images are disturbed with $Q_{As}=9$ SLPM. These droplet impingements are detrimental for achieving high-quality PIV measurements of the internal flows. Therefore, we limited our experiments to a minimum ventilation rate of 10 SLPM.

Figure 15.

(a) Schematic showing the change of the internal flow, the geometry and the closure region of a TV supercavity with increasing ventilation. (b) Images showing the change of the closure region of a TV supercavity with increasing ventilation.

In our experiments, with a further increase of ventilation input, the supercavity closure transitions to TV closure. In contrast to the RJ supercavity, as shown in figure 15(a), we observe that the cavity length does not grow appreciably with increasing ventilation for the TV supercavity. This observation is supported experimentally and numerically by our previous studies (i.e. Cao et al. Reference Cao, Karn, Arndt, Wang and Hong2017; Shao et al. Reference Shao, Karn, Ahn, Arndt and Hong2017), which show the total length increase of a ventilated supercavity to be less than 10 % with doubling of the ventilation input. Compared to the RJ supercavity shown in figure 14(a), the TV supercavity has a longer streamwise extent, a wider vortex tube and a smaller incline angle ( $\unicode[STIX]{x1D6FC}$ ) for the bottom surface near the closure, as shown in the upper schematic of figure 15(b). Additionally, at the closure, the toroidal vortex vanishes and only the twin-vortex tubes remain, indicating all the gas leakage is contributed by internal boundary layer entrainment (i.e. $Q_{As}=Q_{TV}$ ). Accordingly, the external boundary layer does not separate, so the water flow can be well approximated using potential flow theory, as established from many prior works on supercavitation (for example, Cox & Clayden Reference Cox and Clayden1955; Logvinovich Reference Logvinovich1972; Savchenko & Savchenko Reference Savchenko and Savchenko2012, etc.). As the ventilation input ( $Q_{As}$ ) increases, similar to that in the RJ case, the tube becomes wider to allow more air to leak from the tube, and the incline angle of the bottom surface ( $\unicode[STIX]{x1D6FC}$ ) decreases. However, with a decreased  $\unicode[STIX]{x1D6FC}$ , the external water flow near the bottom gas–water interface experiences a weaker expansion as it moves towards the closure. According to potential flow analysis for flow over surfaces with varying curvatures (for example, Jackson & Hunt Reference Jackson and Hunt1975; Rhie & Chow Reference Rhie and Chow1983), the pressure at closure (i.e. $P_{out}$ ) decreases with increasing local water flow velocities. Consequently, the adverse pressure gradient weakens and the internal boundary layer entrains more air at closure (i.e. $Q_{TV}$ increase), which is sufficient to fill the widened tubes. As a result, mass conservation is satisfied for the ventilation input and gas leaking (i.e. $Q_{As}=Q_{TV}$ ), and thus the supercavity length does not increase appreciably.