4.1 Operating point

The agreement with the experimental operating point is assessed by evaluating the temporal mean, denoted by an overbar, $\overline{\bullet }$ , of quantities recorded at probing locations $1$ and $2$ , and by comparison of the velocity profile upstream of the wedge.

Table 6 shows the obtained mean velocity $\overline{u}_{1}$ and pressures $\overline{p}_{1}$ , $\overline{p}_{2}$ . Additionally, the mean cavitation numbers $\overline{\unicode[STIX]{x1D70E}}_{1}$ and $\overline{\unicode[STIX]{x1D70E}}_{2}$ are given. The quantities $\overline{u}_{1}$ and $\overline{p}_{2}$ , controlled with the boundary conditions, match the experimental references and vary only little for the different grid levels. Correspondingly, an almost exact agreement is obtained for the cavitation number $\overline{\unicode[STIX]{x1D70E}}_{2}$ . In contrast, the mean upstream pressure $\overline{p}_{1}$ is consistently higher in the simulations for all three grid levels. As already discussed, the upstream pressure, and thus also the upstream cavitation number, is part of the solution. Compared with the experiments, $\overline{p}_{1}$ and $\overline{\unicode[STIX]{x1D70E}}_{1}$ are larger by approximately 10 %, indicating that the pressure loss $\unicode[STIX]{x0394}p=p_{1}-p_{2}$ in the test section is higher in the simulation.

Two factors contribute to the higher pressure drop in the simulation. First, vapour structures, causing a blockage effect, induce additional pressure losses. For the present case and operating point, the extent of vapour structures intermittently reaches almost the size of the channel cross-section. The pressure losses associated with cavitation thus exceed the losses due to viscous effects within the test section. As discussed in § 4.6, the mean amount of vapour produced is larger in the simulations compared with the experiments, thus causing a larger contribution to the pressure drop in the simulation. Second, the pressure on the upstream shoulder of the wedge has a large influence on the overall pressure drop as well. See appendix C for an analytical model that can be used to explain the observed differences between the pressure drop in the simulations and the experimental references. These are the primary reasons for the larger pressure drop $\unicode[STIX]{x0394}p$ and thus the deviations found in $\overline{p}_{1}$ and $\overline{\unicode[STIX]{x1D70E}}_{1}$ . In view of the uncertainty in the measurements, the agreement with the experimental operating point can still be considered to be satisfactory. The analytical model derived in appendix C shows that the present configuration generates a pressure loss, even under inviscid assumptions. It thus is analogous to the generation of wave drag in inviscid supersonic flow. Here, it is caused solely by the presence of two-phase flow, which prevents a pressure recovery on the back side of the wedge.

Due to the inviscid assumption, viscous effects, such as boundary layers or corner vortices within the channel, are not captured by the numerical simulation. This could potentially alter the effective inflow conditions to the cavitating region. The velocity profile upstream of the wedge is analysed in appendix D. It is found that upstream of the wedge, the near-wall viscous layer spans less than 2 % of the channel height. Due to the subsequent acceleration of the flow on the convergent part of the wedge, the boundary layer is expected to be even thinner when reaching the sharp wedge apex. From this investigation, it is concluded that the incoming boundary layer does not have a significant influence on the sheet cavity, and that the assumptions of inviscid flow and slip boundary conditions in the two-phase computations are sound.

With only small changes between the quantities $\overline{u}_{1}$ , $\overline{p}_{1}$ and $\overline{p}_{2}$ obtained on grids $\mathit{lvl}_{1}$ and $\mathit{lvl}_{2}$ , table 6 already gives an indication that the numerical results on the finest level can be considered as grid-converged. This is confirmed by a detailed analysis of grid convergence, provided in appendix E. The following discussion hence focuses on results obtained on the finest grid.

4.2 Instantaneous flow topology

Figure 5 compares instantaneous simulation results with snapshots from experimental high-speed videos in top view. For a visualisation of the numerical results, isosurfaces of the 10 % void fraction are chosen. Furthermore, instantaneous vortical structures are shown with isosurfaces of $\unicode[STIX]{x1D706}_{2}=-2\times 10^{6}~\text{s}^{-2}$ , coloured by the axial vorticity $\unicode[STIX]{x1D714}_{x}$ . Here, $\unicode[STIX]{x1D706}_{2}$ denotes the second eigenvalue of $\unicode[STIX]{x1D64E}^{2}+\unicode[STIX]{x1D734}^{2}$ , where $\unicode[STIX]{x1D64E}$ and $\unicode[STIX]{x1D734}$ are the symmetric and asymmetric parts of the velocity gradient tensor $\boldsymbol{\unicode[STIX]{x1D6FB}}\unicode[STIX]{x1D66A}$ respectively. Dash-dotted lines indicate the location of the apex $(x=0)$ and rear edge ( $x=x_{w}$ ) of the wedge. For illustration of the shedding process, five representative time instants are selected in figure 5(a–e), exhibiting typical topological features and coherent flow structures. It is unknown which precise value of $\unicode[STIX]{x1D6FC}$ corresponds best to a visual examination of cavitation from experiments. Furthermore, it is unclear whether a single value actually exists, considering the broad range of cavitation topologies (bubble cavitation, sheet cavitation, cloud cavitation) that can be covered with the homogeneous mixture approach. We choose the value of $\unicode[STIX]{x1D6FC}=0.1$ for visualising the numerical results. A further aspect for such a juxtaposition is the highly transient nature of the flow and the lack of repeatability of the shedding process. Due to the inherent chaotic nature of cavitating flow, this equally affects the simulation and the experiments. We thus do not expect to find identical flow structures for any two arbitrarily selected time instants. However, for a qualitative assessment of flow structures, the presented analysis is suitable.

Figure 5. Illustration of a typical shedding cycle (top view); comparison between experiment (left, from direct communication with H. Ganesh) and simulation on the $\mathit{lvl}_{2}$ grid (right). The numerical results show vapour structures with the isosurface of $\unicode[STIX]{x1D6FC}=0.1$ (grey) and vortical structures with the isosurface of $\unicode[STIX]{x1D706}_{2}=-2\times 10^{6}~\text{s}^{-2}$ , coloured by the axial vorticity $\unicode[STIX]{x1D714}_{x}$ . The white boxes indicate coherent flow structures, and the dash-dotted lines denote the wedge apex ( $x=0$ ) and the wedge rear point ( $x=x_{w}$ ).

Figure 5(a) shows the time instant of maximum attached sheet cavity length. The sheet, curved towards both sidewalls, attains its largest streamwise extent at mid-span. The experimental picture shows small-scale streamwise-oriented cavitating vortices (‘streamers’; see Laberteaux & Ceccio Reference Laberteaux and Ceccio2001) trailing from the attached sheet at mid-span. Furthermore, cavitating vortices are situated along both sidewalls. A streamwise-oriented vortex is located close to the left sidewall and is connected to the sheet. Another vortex, oriented normal to the bottom wall and propagating downstream, is located close to the right sidewall. The simulations equally exhibit these flow structures. Side views provided by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) show that the streamwise-oriented vortices in the vicinity of the lateral walls are typically detached from the bottom wall. They are different from corner vortices found in turbulent channel flow. Rather, they are characteristic features of the cavitating flow, similar to ‘streamers’, resulting from the non-uniform distribution of circulation in the spanwise direction. The latter originates from the non-planar and variable-strength condensation shock front, causing the downstream advection of vortex lines. Furthermore, the isosurfaces show perturbations of the attached sheet along the sidewalls, developing downstream of the apex, which are also found in the experimental picture.

Figure 5(b) depicts the situation of a condensation shock propagating through the attached sheet towards the wedge apex. As indicated by the dashed line, the progression of the shock is faster at mid-span, while it is slightly slower close to the sidewalls. Simultaneously, a large cavitating horseshoe vortex develops at the downstream part of the vapour structure. Just upstream of the horseshoe vortex, a so-called crescent-shaped region is found, a typical flow phenomenon of cavitating flow; see, e.g., Reisman et al. (Reference Reisman, Wang and Brennen1998). The simulations also predict the occurrence of the condensation shock phenomenon, with a non-uniform rate of progression across the spanwise direction. Likewise, large cavitating horseshoe vortices, as well as crescent-shaped regions, can be found.

Figure 5(c) depicts the instant when the condensation front touches the wedge apex. For most shedding cycles, this does not occur simultaneously across the complete span. Instead, the shock reaches the apex first close to mid-span, while it lags slightly behind near the lateral walls. A similar behaviour is found in the simulation. When the shock reaches the apex, a detached cloud is generated, concentrating strong vorticity along the spanwise direction. Typically, ‘streamers’, as shown by the experimental picture, or horseshoe vortices, as depicted in the numerical visualisation, develop at the trailing part of the cloud.

Figure 5(d) shows the situation just after the detachment along the entire spanwise direction, and development of a fully separated cloud further downstream. The region just downstream of the apex, denoted as ‘leading-edge’ separation in the figure, is characterised by complex flow patterns. Cavitating vortices are mainly oriented along the spanwise direction, partially forming horseshoe-type structures. Streamwise-oriented vortices connect these patches of cavitation. The flow topology predicted by the simulation resembles the experimental picture. The formation of separated flow at the apex differs from viscous flow separation. Comparable to the detachment caused by a re-entrant jet, it is induced by the reverse flow following behind the upstream-propagating condensation front, which reaches the apex.

Finally, at the instant shown in figure 5(e), the separated cloud is convected further downstream, while a new cavity sheet develops. Typical for this situation is the observation of large cavitating vortices wrapping around the cloud, oriented primarily in the streamwise direction. The simulation exhibits comparable structures.

On comparing experimental and numerical panels (a–e), it can be observed that the simulation exhibits slightly less small-scale cavitation. Yet, the $\unicode[STIX]{x1D706}_{2}$ -criterion illustrates the existence of complex vortical structures. For example, smaller streamwise-oriented vortices can be found at the downstream part of the cloud as well as concentrated close to the tunnel sidewalls. This indicates that the pressure drop in these vortex cores is not sufficient for cavitation to occur. Resolution of the vortex cavitation within these small-scale structures would require a much finer numerical grid. However, the larger cavitating structures (e.g. horseshoe cavitation on a variety of length scales, crescent-shaped regions, streamers, vortices along the sidewalls in the streamwise direction as well as normal to the bottom wall) are predicted in close agreement with the experiments. From this qualitative comparison, it can be concluded that the selected numerical model is able to capture the primary features present in the cavitating flow of sheet-to-cloud transition.

The shedding process can also be visualised with the spanwise average, denoted by $\langle \,\bullet \,\rangle _{z}$ in the following. For this purpose, figure 6 shows a series of six consecutive time instants in the form of the instantaneous spanwise-averaged void fraction $\langle \unicode[STIX]{x1D6FC}\rangle _{z}$ , the streamwise velocity $\langle u\rangle _{z}$ and the pressure $\langle p\rangle _{z}$ in the vicinity of the wedge. The plots of velocity and pressure include isocontours of $\langle \unicode[STIX]{x1D6FC}\rangle _{z}$ to relate these fields to the occurrence of cavitation.

Figure 6. The spanwise-averaged instantaneous flow field during a shock-dominated shedding cycle for six consecutive time instants; numerical prediction on the $\mathit{lvl}_{2}$ grid. Comparison between (a) the void fraction $\langle \unicode[STIX]{x1D6FC}\rangle _{z}$ , (b) the streamwise velocity $\langle u\rangle _{z}$ and (c) the pressure $\langle p\rangle _{z}$ . Isocontours of $\langle \unicode[STIX]{x1D6FC}\rangle _{z}$ are superimposed on $\langle u\rangle _{z}$ and $\langle p\rangle _{z}$ .

The first instant, $t=t_{0}$ , depicts the instant when the attached sheet reaches its maximum length in the streamwise direction; here, $s_{L}\approx 100~\text{mm}$ . The spanwise-averaged void fraction attains values close to $\langle \unicode[STIX]{x1D6FC}\rangle _{z}\approx 0.8$ . Within the sheet, the local flow velocity is small and is directed downstream, while the spanwise-averaged pressure attains values close to the vapour pressure. Shortly thereafter, a condensation shock forms at the rear part of the attached sheet. In the subsequent time instants, $t=t_{0}+6~\text{ms}$ and $t=t_{0}+12~\text{ms}$ , the condensation shock propagates upstream through the sheet. Simultaneously, the spanwise-averaged void fraction in the attached part of the sheet increases, reaching values of $\langle \unicode[STIX]{x1D6FC}\rangle _{z}\approx 0.9$ . The condensation front spans almost the complete height of the attached sheet and causes condensation. Across the shock, the direction of flow is reversed and the static pressure increases. A shear layer forms between the free-stream and the upstream-directed flow behind the shock, exhibiting classical Kelvin–Helmholtz instabilities. Within the low-pressure vortex cores, evaporation takes place. This leads to the formation of new cavity structures behind the shock, subsequently rolling up into a cloud. Across the condensation front, the more dense liquid displaces the lighter mixture region within the sheet. As a consequence, Rayleigh–Taylor instabilities develop along the interface. In addition to the non-uniform rate of progression noted above, the resulting perturbations of the interface contribute to the fact that it does not appear as a sharp front in the spanwise average. At $t=t_{0}+17~\text{ms}$ , the shock touches the wedge apex. Again, caused by variations in the spanwise direction, this does not happen uniformly across the wedge, and the spanwise-averaged void fraction is non-zero close to the apex for this instant. Simultaneously, the downstream cloud is continuously fed by the shear layer. Due to the upstream-directed flow following behind the shock front, the cloud is not yet detached. Detachment across the complete spanwise direction is shown at the next instant, $t=t_{0}+21~\text{ms}$ . The upstream-directed flow breaks down, and the cloud separates and starts to propagate downstream. With the cloud being convected downstream, the growth of a new sheet is initiated, as shown at the last instant, $t=t_{0}+27~\text{ms}$ .

For a total of 27 shedding cycles, the maximum attached sheet length $s_{L}$ is determined from the spanwise-averaged void fraction. The average yields a mean attached sheet length of $\overline{s}_{L}=108.4~\text{mm}$ . A measure for the cycle-to-cycle variation is given by the standard deviation of $\overline{s}_{L}$ , which amounts to $\pm 13~\text{mm}$ .

4.3 Temporal evolution of the shedding process

The shedding process across multiple cycles can be further analysed by recording the temporal evolution of the spanwise-averaged flow on a plane at a normal distance $n=n_{p}$ parallel to the wedge surface. Choosing $n_{p}=5.2~\text{mm}$ , figure 7 shows the thus obtained variation in time for the void fraction $\left.\langle \unicode[STIX]{x1D6FC}\rangle _{z}\right|_{n_{p}}$ and axial velocity $\left.\langle u\rangle _{z}\right|_{n_{p}}$ , plotted along the $s$ -direction for a time span of $1~\text{s}$ .

Figure 7. The time evolution of the shedding process over a period of $1~\text{s}$ ; numerical prediction on the $\mathit{lvl}_{2}$ grid. Spanwise-averaged quantities are extracted from a wedge-parallel plane at a normal distance of $n=n_{p}=5.2~\text{mm}$ and plotted along $s$ . Comparison of (a) the void fraction $\left.\langle \unicode[STIX]{x1D6FC}\rangle _{z}\right|_{n_{p}}$ and (b) the axial velocity component $\left.\langle u\rangle _{z}\right|_{n_{p}}$ . The slopes indicated by black lines denote processes of cavity growth ( $\overline{u}_{growth}=5.5~\text{m}~\text{s}^{-1}$ ) and cavity collapse ( $\overline{u}_{shock}=-4.5~\text{m}~\text{s}^{-1}$ ).

Individual shedding cycles can be identified by the triangular shape in the plot of $\left.\langle \unicode[STIX]{x1D6FC}\rangle _{z}\right|_{n_{p}}$ . The black solid lines exemplarily highlight several cycles. Positive slopes denote processes of cavity growth, i.e. the formation of a new attached sheet. For most cycles, typical void fractions in the attached sheet exceed 70 %, with a tendency to increase during the growth process and towards the wedge apex, as already noted above. In general, the flow is directed downstream within the sheets and the velocity magnitude is small. After reaching a maximum sheet length, a new process of cavity collapse is initiated, indicated by negative slopes in figure 7. Due to the restriction to the plane $n=n_{p}$ , the structures observed here are slightly shorter than the maximum attached sheet length $\overline{s}_{L}$ computed above. The majority of collapse processes are associated with the occurrence of a condensation shock. Behind the shock, the flow is directed upstream and the magnitude of the flow velocity is significantly increased. In most cycles, the void fraction drops across the shock to values close to zero, as expected. For some cycles, however, non-zero vapour content is present just downstream of the condensation shock. On one hand, this is associated with the cavitating shear layer following immediately behind the shock. On the other hand, the front propagation velocity may be inhomogeneous along the spanwise direction, as mentioned above. During the collapse process, the cloud, still being connected to the sheet, moves downstream only moderately. For most cycles it does not pass beyond $s\approx 150~\text{mm}$ . At the time instant when the shock touches the apex, the cloud completely separates and is convected further downstream.

The slopes in figure 7 can be used to estimate the characteristic velocities for the cavity growth and collapse processes. The cavity growth velocity, on average $\overline{u}_{growth}=5.5~\text{m}~\text{s}^{-1}$ , is approximately constant during a given shedding cycle. The velocity of cavity collapse, corresponding to the propagation velocity of the condensation shock front $\overline{u}_{shock}$ , is found to be $\overline{u}_{shock}\approx -4.5~\text{m}~\text{s}^{-1}$ on average. The front undergoes a slight acceleration when it approaches the wedge apex. Figure 8 compares the value of $\overline{u}_{shock}$ obtained numerically with experimental measurements for upstream cavitation numbers in the range $1.88\leqslant \unicode[STIX]{x1D70E}_{1}\leqslant 2.19$ . The computed value for the front propagation velocity is in close agreement with the experimentally reported values.

Figure 8. Average propagation velocity of condensation shocks $\overline{u}_{shock}$ . Comparison between simulation (

, red) and experiments of Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) (

, grey). The error bars indicate cycle-to-cycle variation.

The acceleration of the condensation shock towards the wedge apex is caused by the fact that the void fraction in the sheet close to the apex increases during the process of cavity collapse, as seen from figures 6 and 7. This is in agreement with the findings of Brennen (Reference Brennen1995), who, neglecting bubble dynamics, surface tension and phase change, derived the following equation for the front propagation velocity relative to the upstream fluid $u_{shock,rel}$ in a bubbly flow:

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle u_{shock,rel}^{2}=\frac{p_{2}-p_{1}}{\unicode[STIX]{x1D70C}_{L}}\cdot \frac{1-\unicode[STIX]{x1D6FC}_{2}}{(1-\unicode[STIX]{x1D6FC}_{1})(\unicode[STIX]{x1D6FC}_{1}-\unicode[STIX]{x1D6FC}_{2})}. & \displaystyle\end{eqnarray}$$

States upstream and downstream of the shock are denoted by subscripts ‘1’ and ‘2’ respectively. Assuming the pressure drop $p_{2}-p_{1}$ to be almost constant across the front, $u_{shock,rel}$ increases when $\unicode[STIX]{x1D6FC}_{1}\rightarrow 1$ . As seen from figure 7, the velocity of the flow upstream of the front also remains approximately constant. Thus, the absolute velocity of the condensation front, $u_{shock}$ , increases as well.

Figure 7 shows that the predicted shedding process does not undergo perfect repeatability. This is documented for the current configuration by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016). It is also a general feature of sheet-to-cloud shedding, as, e.g., described by Reisman et al. (Reference Reisman, Wang and Brennen1998). For the present case, irregular processes of cavity growth in between two main sheddings can be found in figure 7, e.g. around $t=0.08~\text{s}$ , $t=0.61~\text{s}$ , $t=0.79~\text{s}$ and $t=0.82~\text{s}$ . These are associated with the occurrence of a classical re-entrant jet instability. According to Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016), this is more frequent at higher cavitation numbers. As reported by these authors, however, it can also appear for the chosen operating point. Furthermore, it is observed from the simulations that a large coherent collapse of a cloudy structure can prematurely stop the sheet before it reaches its ‘natural’ maximum sheet length, as dictated by the adverse pressure gradient. Thereby initiating a condensation front, this downstream cloud collapse depends, e.g., on the previous cycle, neighbouring cavity structures, etc. Furthermore, a cloud collapse does not necessarily occur at mid-span. This can initiate sheet retraction primarily at one side, while the opposite side lags behind. This in turn affects the next cycle and may promote variations in the spanwise direction. The entire shedding process thus is chaotic. This contributes to the cycle-to-cycle variations in figure 7, e.g. regarding the maximum void fraction reached in the sheets, the internal structures in the sheets and the attached sheet length. The visualisation in figure 7 exhibits a subharmonic behaviour at a frequency lower than the shedding frequency, which can, e.g., be seen in the maximum vapour content in the sheets and the sheet length, which might also be a result of the abovementioned mechanism of premature sheet collapse. This is compared with the experiments in § 4.6.4.

4.4 Mean flow topology

Figure 9. The time- and spanwise-averaged flow field in the test section; numerical prediction on the $\mathit{lvl}_{2}$ grid. Comparison of (a) the pressure $\langle \overline{p}\rangle _{z}$ , (b) the axial velocity $\langle \overline{u}\rangle _{z}$ and (c) the void fraction $\langle \overline{\unicode[STIX]{x1D6FC}}\rangle _{z}$ . The direction of flow is indicated by streamlines; the red solid line indicates a region of reversed flow $\langle \overline{u}\rangle _{z}=0$ ; the white dashed frame highlights the field of view for the experimental x-ray densitometry of Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016).

In order to analyse the time-averaged flow field in this section, the results obtained on the $\mathit{lvl}_{2}$ grid are sampled for a time span of $1.66~\text{s}$ , comprising approximately 30 shedding cycles. Figure 9 depicts the two-dimensional flow field obtained by averaging in time and the spanwise direction. The contour plots show the mean pressure $\langle \overline{p}\rangle _{z}$ , axial velocity $\langle \overline{u}\rangle _{z}$ and void fraction $\langle \overline{\unicode[STIX]{x1D6FC}}\rangle _{z}$ in the vicinity of the wedge. The direction of the mean flow is indicated by streamlines; a contour line of $\langle \overline{u}\rangle _{z}=0~\text{m}~\text{s}^{-1}$ highlights a region of reversed mean flow. Due to the convergent shape of the channel, the mean flow is accelerated above the wedge. Correspondingly, the local static pressure drops, reaching values close to the vapour pressure on the back side of the wedge. Accordingly, a contiguous region of cavitation is present in the temporal mean. Located above the back side of the wedge, it originates from the sharp wedge apex. A narrow region of reversed flow exists just above the wedge in the mean flow, spanning less than half of the cavity height.

The mean flow is not homogeneous in the spanwise direction. This is illustrated by figure 10, showing the time-averaged three-dimensional flow field. The mean cavitation pattern is visualised by an isosurface of the 20 % void fraction. Slices perpendicular to the spanwise direction show the local time-averaged void fraction $\overline{\unicode[STIX]{x1D6FC}}$ above the wedge, with a cutoff at $\overline{\unicode[STIX]{x1D6FC}}<0.1$ . On the bottom and left sidewalls of the channel, the contour of the mean velocity in the streamwise direction, $\overline{u}$ , is given. An isocontour of $\overline{u}=0~\text{m}~\text{s}^{-1}$ denotes a region of reversed flow present in the time-averaged flow field. Vortical structures are shown by isosurfaces of $\overline{Q}=1\times 10^{5}~\text{s}^{-2}$ , with $\overline{Q}$ denoting the time average of the instantaneous $Q$ -criterion, $Q=1/2(\Vert \unicode[STIX]{x1D64E}\Vert ^{2}-\Vert \unicode[STIX]{x1D734}\Vert ^{2})$ . The orientation of the vortices is given by colouring the isosurface by the local mean axial vorticity $\overline{\unicode[STIX]{x1D714}}_{x}$ . Despite the fact that the long time span of the temporal average contains many shedding cycles, fluctuations in the isosurface of the vapour volume fraction can be observed. However, these deviations can be regarded as small, and the statistics are considered as sufficiently converged.

Figure 10. The time-averaged flow field in the vicinity of the wedge; numerical prediction on the $\mathit{lvl}_{2}$ grid. Vapour structures are shown by isosurfaces of $\overline{\unicode[STIX]{x1D6FC}}=0.2$ and vortical structures by isosurfaces of $\overline{Q}=1\times 10^{5}~\text{s}^{-2}$ , coloured by the axial vorticity $\overline{\unicode[STIX]{x1D714}}_{x}$ . Contours of $\overline{\unicode[STIX]{x1D6FC}}$ are given in three slices perpendicular to the spanwise direction, with a cutoff at $\overline{\unicode[STIX]{x1D6FC}}<0.1$ . The tunnel walls show the contours of the axial velocity $\overline{u}$ . The red solid line ( $\overline{u}=0$ ) indicates a region of reversed flow.

The streamwise extent of the time-averaged vapour structure is largest at mid-span. Simultaneously, the region of reversed flow extends further downstream close to the sidewalls. Two counter-rotating vortices along the streamwise direction are present in the temporal mean. These vortices originate from the corner between the sidewalls and the bottom wall of the test section, just downstream of the contiguous mean vapour structure. They are inclined slightly away from the bottom wall and oriented such that fluid is transported along the lateral walls in the positive $y$ -direction, converging towards mid-span. This spanwise inhomogeneity of the time-averaged flow is caused by the presence of the sidewalls.

A direct verification of the three-dimensional mean flow structure with the experiments is not possible, as no corresponding data are available. However, it is in alignment with the observations made in the context of figure 5 above. All presented instantaneous images exhibit noticeable variations along the spanwise direction, e.g. the development of horseshoe vortices in the centre of the channel and vortices along the sidewalls, and show that the extent of the attached sheets is largest at mid-span. Thus, the presence of the sidewalls cannot be neglected in the simulations. We conclude that the global flow topology is correctly captured, even though slip boundary conditions due to the inviscid modelling are used in the present study.

4.5 Spectral analysis

In order to identify the dominant frequencies and their spatial distribution in the system, a spectral analysis for signals recorded within the entire test section is carried out. For this purpose, the local axial velocity component $u$ and pressure $p$ are recorded along the bottom wall of the test section by a total of 55 numerical probes. The probes are located at mid-span and distributed evenly in the streamwise direction with a spacing of $\unicode[STIX]{x0394}x\approx 11~\text{mm}$ between two consecutive probes. For the spectral analysis, Welch’s method is applied. The signals are sampled with a time resolution as given in table 5 and then linearly interpolated on a constant time step of $1\times 10^{-6}~\text{s}$ . The spectra are estimated using Hanning window segments with equal window length in the time domain of $0.2~\text{s}$ and $50\,\%$ overlap between subsequent segments.

Figure 11. Frequency spectra for the signals of (a) velocity $u$ and (b) pressure $p$ recorded by probes distributed within the test section along the bottom wall at mid-span. The dashed lines (– –) indicate the locations of the wedge apex ( $x=0$ ) and the rear point ( $x=x_{w}$ ), and the dash-dotted lines (— - —) indicate the mean of the maximum attached sheet length ( $x=\overline{x}_{L}$ ).

Figure 12. The time-averaged flow quantities used for non-dimensionalisation of the frequency spectra plotted in figure 11. Juxtaposition of (a) mean velocity $\overline{u}_{bottom}$ and (b) pressure $\overline{p}_{bottom}$ recorded at mid-span along the bottom wall. The dashed lines (– –) indicate the locations of the wedge apex ( $x=0$ ) and the rear point ( $x=x_{w}$ ), and the dash-dotted lines (— - —) indicate the mean of the maximum attached sheet length ( $x=\overline{x}_{L}$ ).

The frequency content is displayed in figure 11. The plots show the square root of the premultiplied power spectral density, $\sqrt{f\,PSD(\bullet )}$ , as a function of the streamwise position. The computed spectra for the axial velocity $u$ , figure 11(a), and the pressure $p$ , figure 11(b), can thus be interpreted as the RMS amplitude at a given frequency. A non-dimensional representation of the spectra is obtained using the local mean quantities for normalisation, $\sqrt{f\,PSD(\bullet )/\overline{\hspace{2.22198pt}\bullet \hspace{2.22198pt}}^{2}}$ . Figure 12 shows the time-averaged axial velocity, figure 12(a), and pressure, figure 12(b), at mid-span along the bottom wall, which are used for the non-dimensionalisation. The resulting dimensionless frequency spectra are given in figure 13 for the velocity signals, figure 13(a), and pressure signals, figure 13(b). The locations of the wedge apex, $x=0$ , and rear edge, $x_{w}=177~\text{mm}$ , are indicated by dashed lines, while the projection of the maximum attached sheet length on the $x$ -axis, $\overline{x}_{L}=\cos (\unicode[STIX]{x1D711}_{2})\overline{s}_{L}=107.3~\text{mm}$ , is shown with a dash-dotted line in the figures.

Figure 13. Normalised frequency spectra of (a) velocity $u$ and (b) pressure $p$ recorded by probes distributed within the test section along the bottom wall at mid-span. The spectra are normalised by the local mean of the velocity and pressure, as shown in figure 12(a,b) respectively. The dashed lines (– –) indicate the locations of the wedge apex ( $x=0$ ) and the rear point ( $x=x_{w}$ ), and the dash-dotted lines (— - —) indicate the mean of the maximum attached sheet length ( $x=\overline{x}_{L}$ ).

The spectrum for the axial velocity, figure 11(a), shows a dominant peak (global maximum) in the region $\overline{x}_{L}\lesssim x\lesssim x_{w}$ . The associated frequency $f_{1}\approx 19.1~\text{Hz}$ corresponds to the frequency of cloud shedding. It is also found in the spectra of discrete probes, as discussed in appendix E. With an RMS amplitude of approximately $6.4~\text{m}~\text{s}^{-1}$ , substantial variations are found for the axial velocity component. These are caused by the flow reversal following behind the propagating condensation front and are comparable to the upstream velocity $u_{1}$ . The normalised spectrum, figure 13(a), shows that the fluctuations exceed the local mean velocity significantly. The amplification locally exceeds a factor of ${>}15$ and remains high within the entire region of attached sheets.

Due to the chaotic nature of the shedding process, the spectra do not show a sharp peak at a single frequency. Instead, the fluctuations appear as smeared over a range of frequencies: taking 80 % of the peak RMS amplitude, the shedding occurs in a frequency band of approximately $f_{1}\pm 2~\text{Hz}$ . The spectrum further shows the existence of harmonics of the shedding frequency.

The spatial region with noticeable variations in the velocity signals extends downstream of the attached sheet and behind the wedge. This is caused by downstream-propagating clouds that lead to a perturbation of the near-wall velocity, recurring coherently with the shedding frequency. Furthermore, RMS amplitudes exceeding approximately $2~\text{m}~\text{s}^{-1}$ can be found in the frequency band $10\leqslant f\leqslant 100~\text{Hz}$ throughout the test section, but are concentrated around the maximum attached sheet length and the wedge apex.

As expected, the pressure spectrum, figure 11(b), equally exhibits the strongest fluctuations at the shedding frequency $f_{1}$ and harmonics thereof. In contrast to the velocity, where the largest RMS amplitudes are found within a single spatial region, two distinctive peaks can be identified in the pressure spectrum at $f_{1}$ . At the first location, just upstream of the maximum attached sheet length $x\lesssim \overline{x}_{L}$ , the RMS amplitude amounts to approximately $15~\text{kPa}$ . This corresponds to the region where the condensation shock is formed, connected to a pressure rise, within each cycle. In addition, a second maximum can be found centred around $x\approx x_{w}$ . Here, the majority of shed clouds collapse coherently. Intense pressure peaks emitted by these collapses cause RMS amplitudes that are approximately 40 % higher than for the first peak. In between these two regions, a contribution at a lower frequency is visible, which might be caused by an interference between these two mechanisms. The normalised pressure spectrum, figure 13(b), shows that the variations caused by the condensation shock are of the same order as the local mean pressure, which is essentially equal to the vapour pressure. The relative pressure fluctuations hence are smaller by approximately one order of magnitude than variations in the velocity, as seen above.

While no significant velocity fluctuations are present upstream of the wedge ( $x<0$ ), noticeable levels of pressure fluctuations are found in this region. These are caused by shocks, originating from coherent cloud collapses, that propagate upstream through the liquid medium. Downstream, in the region $300\leqslant x\leqslant 500~\text{mm}$ , the pressure spectrum shows significant contributions at frequencies $f\geqslant 1~\text{kHz}$ . These are generated by high-frequency incoherent collapse events created during the disintegration of downstream-propagating clouds into smaller vapour structures.

Two artefacts can be found in the pressure spectrum. First, in the vicinity of the wedge, the amplitude rises when approaching the upper bound of the spectrum, i.e. at frequencies $f>10~\text{kHz}$ . Due to the intermittent presence of vapour directly adjacent to the wedge surface, the pressure signals recorded in this area are clipped at the vapour pressure. This leads to the observed ringing artefacts. Second, the spectrum gives a sharp peak at $f\approx 300~\text{Hz}$ when approaching the exit of the test section. This is caused by interactions between cavity collapse acoustics and the free-stream jet exiting the test section into the larger downstream tubing. The normalised pressure spectrum, however, shows that the relative amplitude of this phenomenon is small.

The shedding process can be characterised by a non-dimensional Strouhal number. In agreement with Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016), it is computed here as $St=f_{1}\overline{s}_{L}/u_{1}$ using the shedding frequency $f_{1}$ , the mean attached sheet length $\overline{s}_{L}$ and the upstream velocity $u_{1}$ . With $f_{1}=19.1\pm 2~\text{Hz}$ and $\overline{s}_{L}=108.4\pm 13~\text{mm}$ , it amounts to $St=0.262\pm 0.03$ . The frequency $f_{1}$ is obtained equally by the presented spectral analyses of the pressure and velocity signal record along the bottom wall, as well as by the pressure and void fraction signals recorded with discrete numerical probes at position 1 and $P_{D}$ , as discussed in appendix E. On the other hand, in order to determine the shedding frequency experimentally, Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) used pressure probes as well as transient spanwise-averaged void fraction measurements obtained by x-ray densitometry. Figure 14 shows the comparison with the experimental references for upstream cavitation numbers in the range $1.88\leqslant \unicode[STIX]{x1D70E}_{1}\leqslant 2.19$ . The computed Strouhal number is in close agreement with the experimentally reported value for periodic shedding cavities, which is given by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) as $St=0.26\pm 0.02$ .

Figure 14. The Strouhal number $St$ for cloud shedding. Comparison between simulation (

, red) and experiments of Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) (frequency obtained from pressure transducer signals

, black, and void fraction measurements

, grey). The error bars indicate cycle-to-cycle variation.

Figure 15. The time- and spanwise-averaged void fraction field $\langle \overline{\unicode[STIX]{x1D6FC}}\rangle _{z}$ . (a) Comparison of contour plots obtained numerically (left) and with x-ray densitometry by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) (right; data from direct communication with H. Ganesh). (b) Void fraction profiles $\langle \overline{\unicode[STIX]{x1D6FC}}\rangle _{z}(n)$ extracted along the dashed lines indicated in (a) (simulation, red; experiment, black).

4.6 Comparison with x-ray densitometry

4.6.1 Time-averaged void fraction

The dashed rectangle drawn in figure 9(c) indicates the field of view for the x-ray densitometry conducted by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016). A zoom in to this view is given in figure 15(a), juxtaposing the temporal mean of the experimentally obtained void fraction with the corresponding numerical result of $\langle \overline{\unicode[STIX]{x1D6FC}}\rangle _{z}$ . Figure 15(b) compares profiles of the experimentally and numerically obtained void fraction, plotted as functions of the normal coordinate $n$ . These profiles are extracted along the $y$ -direction, at locations spaced evenly in the spanwise direction for $10~\text{mm}\leqslant x\leqslant 120~\text{mm}$ , with $\unicode[STIX]{x0394}x=10~\text{mm}$ , as indicated by dashed lines in figure 15(a).

The time-averaged void fraction field is only a simplified representation of this highly transient system. While being a straightforward measure of comparison with the experiments, its significance remains limited as it does not correspond to a physical realisation of the flow. Due to the transient shedding process, cloudy structures and sheet cavitation intermittently occur for most of the measurement stations. Furthermore, the statistical sampling has to be taken into account when considering these quantities, because of the large cycle-to-cycle variations obtained equally in the experiments as well as in the computations. The experimental references for the mean and RMS were obtained from 788 samples, covering a total time span of $0.79~\text{s}$ , corresponding to approximately 15 shedding cycles. On the other hand, the numerical data comprise $1.66~\text{s}$ or 30 cycles, sampled at $0.4\times 10^{-6}~\text{s}$ , containing more than $4\times 10^{6}$ samples. Due to strong cycle-to-cycle variations, including the subharmonic behaviour noted in §§ 4.3 and 4.6.4, we believe that both the numerical result and the experiment require $O(100)$ cycles in order to fully ensure statistical convergence.

The simulation exhibits larger vapour production in the temporal and spanwise mean: the extent of the mean vapour structure in the streamwise direction, as, e.g., measured by the 20 % void fraction, is larger than for the experiments. The local maximum average void fraction in the simulation is up to 25 % higher compared with the experiments. The agreement in the front part of the wedge, $0\lesssim x\lesssim 50~\text{mm}$ , corresponding to the location of the sheet cavity, is better than in the region further downstream. As shown by the vapour profiles, the local mean void fraction attains an almost constant value of 50 % across the height of the sheet, being consistently higher in the simulation. The void fraction then abruptly drops to $\unicode[STIX]{x1D6FC}\approx 0$ , with good agreement for the cavity thickness and the sharpness of the liquid–vapour interface. Further downstream, $x\gtrsim 50~\text{mm}$ , the agreement deteriorates. In this region, the maximum amount of vapour volume is overestimated by the simulation. However, the wall-normal extent of the time-averaged vapour fraction, i.e. the $n$ -position where $\langle \overline{\unicode[STIX]{x1D6FC}}\rangle _{z}\approx 0$ , agrees well with the experimental references for all investigated stations. Furthermore, the computation gives excellent agreement of the mean vapour content near the bottom wall.

Four factors may contribute to the larger mean amount of vapour in the simulation. First, the shape of the wedge apex influences the local vapour production. As no information about the actual manufactured apex geometry is available to us, it is modelled as a sharp corner in the simulation. A finite apex radius would mitigate the suction peak in the pressure, which would contribute to an attenuation of cavitation. Second, the equilibrium speed of sound $c_{eq}$ is used for obtaining the barotropic equation of state. As discussed in appendix A, this is a lower estimate of the true speed of sound in mixture regions. With lower values of $c$ being connected to stronger evaporation, this may cause a higher vapour content. Third, neglect of the effect of gas content in the simulations may also contribute to a higher mean amount of vapour, although an exact quantification is difficult. Qualitatively, the presence of free gas increases the compressibility of the bulk. For the present configuration, it is expected that the suction peak at the wedge apex is mitigated, thereby reducing the local rate of vapour production. Furthermore, the acoustics in the bulk are damped, which also leads to an attenuation of expansion waves. As these in turn may induce cavitation, this may additionally contribute to a lower vapour content. Finally, in the region $x\gtrsim 50~\text{mm}$ , the main contribution to the mean void fraction originates from cloudy structures, periodically detaching from the sheet cavity. It is believed that the spatial resolution in this region is insufficient to capture the complete three-dimensionality and granularity of the vapour structures. As mentioned initially, Kato et al. (Reference Kato, Yamaguchi, Maeda, Kawanami and Nakasumi1999) found that the bubble number density in clouds can be of the order of $10^{3}~\text{bubbles}~\text{cm}^{-3}$ or higher. The necessary spatial resolution is beyond what is currently computationally feasible for us when considering a large-scale system such as the one under investigation.

The mean amount of vapour produced may further be affected by viscous lateral boundary layers present in the experiments, by introducing additional three-dimensionality. As demonstrated in the context of figure 10, the presence of the tunnel sidewalls causes spanwise inhomogeneities in the simulation as well. Since the walls are modelled with slip boundary conditions, this is, however, purely attributed to the effect of flow restriction. As discussed initially, it is believed that this effect has negligible influence on the temporal mean for the current configuration.

Subsequently, the fluctuations of the spanwise-averaged void field $\langle \unicode[STIX]{x1D6FC}\rangle _{z}$ are analysed in terms of their RMS level and compared with the experiments. For this purpose, figure 16(a) shows a contour plot of $(\overline{\langle \unicode[STIX]{x1D6FC}\rangle _{z}^{\prime }\langle \unicode[STIX]{x1D6FC}\rangle _{z}^{\prime }})^{1/2}$ and figure 16(b) shows the corresponding profiles along the $y$ -direction, evaluated at identical locations to those used before. It should be noted that the definition of the RMS used here differs from that utilised in the context of the grid convergence study in appendix E, $\langle \overline{\unicode[STIX]{x1D6FC}^{\prime }\unicode[STIX]{x1D6FC}^{\prime }}\rangle _{z}^{1/2}$ , as it is not possible to measure the true fluctuations of the three-dimensional void fraction field with the available experimental measurement techniques. As such, the peak RMS fluctuations documented here are smaller by a factor of approximately 1.6 compared with the actual fluctuations of the void fraction given in figure 34 in appendix E.

Figure 16. The RMS of the time- and spanwise-averaged void fraction field $(\overline{\langle \unicode[STIX]{x1D6FC}\rangle _{z}^{\prime }\langle \unicode[STIX]{x1D6FC}\rangle _{z}^{\prime }})^{1/2}$ . (a) Comparison of contour plots obtained numerically (left) and with x-ray densitometry by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) (right; data from direct communication with H. Ganesh). (b) Profiles of $(\overline{\langle \unicode[STIX]{x1D6FC}\rangle _{z}^{\prime }\langle \unicode[STIX]{x1D6FC}\rangle _{z}^{\prime }})^{1/2}$ extracted along the dashed lines indicated in (a) (simulation, red; experiment, black).

Similar conclusions to those from figure 15 can be drawn. The simulation and experiment agree best in the upstream part, $0\lesssim x\lesssim 50~\text{mm}$ . Here, the RMS level matches with an almost constant plateau in the centre of the attached sheet. Across the liquid–vapour interface, the RMS level drops abruptly. At slightly larger wall-normal distances, the computed fluctuations follow a nearly linear trend before they vanish in the free stream. This can be ascribed to periodic perturbations of the sheet interface. A similar trend is also observed for the experiments, albeit weaker. The fluctuations recorded in the experiments do not vanish in the free stream, presumably due to noise in the measurements. In the downstream region, $x\gtrsim 50~\text{mm}$ , dominated by the existence of detached clouds, the characteristics of the RMS fluctuations agree qualitatively between computation and experiment. As already discussed, the utilised grid is insufficient for resolving the full fine-scale complexity of the detached clouds, which may affect the predicted fluctuations.

4.6.2 Instantaneous void fraction

In the previous section, it was conjectured that the agreement between the computed and experimental void fractions depends on whether cavitation occurs as attached sheets or as cloudy structures. As the two mechanisms may alternate in time at a specific location, a clear differentiation is difficult when considering time-averaged data only. In order to better assess the representation of different regimes of cavitating flow with the computations, the instantaneous void fraction is analysed in the following.

For this purpose, two consecutive time instants are selected. Situation $A$ , $t=t_{0}$ , corresponds to the presence of an attached sheet reaching its maximum streamwise extent. For situation $B$ , the instant $t=t_{0}+12~\text{ms}$ is chosen, with a condensation shock being propagated midway through the sheet. Instants $A$ and $B$ are depicted in figure 17, comparing the instantaneous spanwise-averaged void fraction field $\langle \unicode[STIX]{x1D6FC}\rangle _{z}$ from the simulation with the experimental result obtained by x-ray densitometry. In addition, the comparison is carried out in terms of instantaneous spanwise-averaged void profiles along the $y$ -direction, extracted at identical locations to those above.

Figure 17. The instantaneous spanwise-averaged void fraction field $\langle \unicode[STIX]{x1D6FC}\rangle _{z}$ at the instant of maximum attached sheet length (panels (a,b), $t=t_{0}$ ) and with the condensation shock at $x\approx 40~\text{mm}$ (panels (c,d), $t=t_{0}+12~\text{ms}$ ). (a,c) Comparison of contour plots obtained numerically (left) and with x-ray densitometry by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) (right; data from direct communication with H. Ganesh). (b,d) Void fraction profiles $\langle \unicode[STIX]{x1D6FC}\rangle _{z}(n)$ extracted along the dashed lines indicated in (a,c) (simulation, red; experiment, black).

At instant $A$ , the simulation and experiment show an attached sheet attaining an almost identical length $s_{L}\approx 90~\text{mm}$ . The profiles show that the sheet thickness predicted by the simulation matches the experiment up to $x\approx 40~\text{mm}$ . In the rear part, $x\gtrsim 40~\text{mm}$ , the sheet is slightly thicker in the simulation, due to a disturbance of the liquid–vapour interface propagating through the cavity for the selected cycle. Despite this difference, excellent agreement is obtained for the void fraction within the attached sheet, reaching values of ${\gtrsim}80\,\%$ . In addition, the downstream profiles within the sheet correctly capture the trend of increasing void fraction at larger normal distances from the wedge.

At instant $B$ , the condensation shock reaches the position $x\approx 40~\text{mm}$ . The numerical void fraction profiles extracted upstream of the shock position, i.e. within the attached sheet, again match excellently with the experimental references. In close agreement, the void fraction decreases for both the simulation and the experiment from $\langle \unicode[STIX]{x1D6FC}\rangle _{z}\gtrsim 0.85$ just upstream to $\langle \unicode[STIX]{x1D6FC}\rangle _{z}\approx 0.4$ across the shock. Compared with the simulation, the decrease of the void fraction in the streamwise direction appears more gradually in the experiments. It is believed that this is caused by a larger variation of the condensation shock front in the spanwise direction for the selected time instant. Larger deviations in the void fraction can be found downstream of the condensation shock. In this region, cloudy vapour structures develop. These originate from the shear layer located at a distance of $n\approx 10~\text{mm}$ above the wedge, i.e. between the free-stream and the upstream-directed flow following behind the shock. The amount of vapour produced in the shear layer is higher in the simulation than in the experiment and, correspondingly, also in the cloud downstream.

From this analysis, it can be concluded that the agreement in the local instantaneous void fraction depends on the topology of the cavitating flow. In a sheet cavity, the model closely agrees with the experimental references, yielding equivalent values of $0.80\lesssim \langle \unicode[STIX]{x1D6FC}\rangle _{z}\lesssim 0.95$ . The agreement is less good when cloudy or foamy cavitation structures exist. In this case, the amount of vapour tends to be larger in the simulation than in the experiment. This leads us to the conclusion that the spatial resolution is adequate for the case of a sheet cavity, but may be insufficient for representing detached and highly fragmented cloudy structures. It is beyond the computational resources available to us to resolve this regime better. We believe that the spatial resolution is the main reason for the observed deviations, in conjunction with the other aspects discussed in the previous section (shape of wedge apex, equilibrium speed of sound and neglect of free gas content). Due to the transient shedding process, cloudy structures and sheet cavitation intermittently occur for most of the measurement stations analysed with figures 15 and 16. In consequence, a larger vapour content for the shed clouds causes the higher mean and RMS of the vapour content observed previously, even if sheet cavities are well reproduced.

4.6.3 Individual shedding cycle

Over the period of a shedding cycle, figure 18 juxtaposes the vapour volume obtained from x-ray densitometry and the computed instantaneous spanwise-averaged void fraction field $\langle \unicode[STIX]{x1D6FC}\rangle _{z}$ . In addition, the void fraction extracted at mid-span, $\left.\unicode[STIX]{x1D6FC}\right|_{z=0~\text{mm}}$ , is provided. The shedding process is exemplified by six consecutive time instants, beginning with the instant $t=t_{0}$ , when an attached sheet reaches its largest streamwise extent. The time intervals for the experiment and numerics are identical.

Figure 18. Instantaneous vapour void fraction during a shedding cycle for six consecutive time instants. Comparison between (a) x-ray densitometry by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) (data from direct communication with H. Ganesh) and (b) numerical void fraction spanwise averaged, as well as (c) in the mid-plane slice.

The spanwise-averaged field compares well with the experimental references. During the existence of an attached sheet, the volume fraction within the attached part of the cavity matches well. Comparing $t=t_{0}$ , $t=t_{0}+6~\text{ms}$ and $t=t_{0}+12~\text{ms}$ , the spanwise-averaged void fraction increases, as noted before, exhibiting values of $\langle \unicode[STIX]{x1D6FC}\rangle _{z}\gtrsim 0.8$ . With good agreement of the propagation velocity $u_{shock}$ , as discussed above, the position of the condensation shock is almost identical for the simulation and experiment for the presented frames. The void fraction in the downstream developing cloud differs, as discussed. Nevertheless, the location and spatial extent of downstream cloudy structures, e.g. cavitation in the shear layer and subsequent roll-up, match reasonably well with the experiment.

Further insight can be gained from the void fraction field extracted in the mid-span slice. At the beginning of the shedding cycle, $t=t_{0}$ and $t=t_{0}+6~\text{ms}$ , the sheet is not a contiguous region of constant vapour volume fraction. Rather, its internal structure exhibits local patches of low and high void fractions. In addition, the void fraction in slices parallel to the mid-span shows noticeable variations in the spanwise direction, especially at the beginning of a shedding cycle. Correspondingly, the condensation front is difficult to identify during the early stages of a shedding cycle in individual slices. This finding cannot be compared directly with the x-ray densitometry as it is an integral measurement in the spanwise direction. However, experimental high-speed videos (see figure 5) do not show a clear sheet either, but inhomogeneous cavity structures. Furthermore, the propagation velocity of the condensation front, being a function of the local void fraction, as shown by (4.1), is inhomogeneous as well.

When the shock progresses towards the wedge apex, i.e. at $t=t_{0}+12~\text{ms}$ , the void fraction in the mid-span slice attains an almost constant value throughout the remaining part of the sheet, accompanied by the occurrence of the condensation shock. This observation agrees with the fact that the spanwise average of the void fraction increases over the period of a shedding cycle, leading to an acceleration of the condensation front. As already noted above, this observation can also be made from the experimental measurements. Void fractions $\langle \unicode[STIX]{x1D6FC}\rangle _{z}\gtrsim 0.9$ imply that for these time instants the flow transitions to a contiguous vapour sheet.

4.6.4 Shedding process

The temporal evolution of the shedding process, in conjunction with the cycle-to-cycle variation of the void fraction and velocity in the simulation, has been discussed in figure 7 in § 4.3. In order to further assess the numerical results, an equivalent analysis is repeated here also for the experimental data. Figure 19 compares the variation in time of $\left.\langle \unicode[STIX]{x1D6FC}\rangle _{z}\right|_{n=n_{p}}$ in simulation and experiment over a period of $1~\text{s}$ .

Figure 19. The time evolution of the shedding process over a period of $1~\text{s}$ . Spanwise-averaged void fraction $\left.\langle \unicode[STIX]{x1D6FC}\rangle _{z}\right|_{n_{p}}$ extracted from a wedge-parallel plane at a normal distance $n=n_{p}=5.2~\text{mm}$ and plotted along $s$ . Comparison of (a) numerical prediction on the $\mathit{lvl}_{2}$ grid, and (b) experimental x-ray densitometry of Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016) (data from direct communication with H. Ganesh).

Similar features are observed in the time evolution of the void fraction, for both the experiment and the simulation. Growth and collapse processes are clearly present. Experiment and simulation equally show a noticeable level of cycle-to-cycle variation, e.g. regarding the maximum attained cavity sheet lengths. The maximum level of vapour volume fraction within the sheets also varies. For the experiment, the void fraction varies from $\langle \unicode[STIX]{x1D6FC}\rangle _{z}\approx 0.55$ at $t\approx 0.68~\text{s}$ to $\langle \unicode[STIX]{x1D6FC}\rangle _{z}\approx 0.95$ at $t\approx 0.4~\text{s}$ . Furthermore, irregular processes of cavity growth, presumably due to the occurrence of re-entrant jets, are observed experimentally as well, e.g. at $t=0.08~\text{s}$ , $t=0.47~\text{s}$ and $t=0.71~\text{s}$ . The subharmonic behaviour of the maximum void fraction within the sheets and the attached sheet length is also found in the experimental data.

Overall, the growth and collapse behaviour predicted by the simulation tends to be slightly more irregular. Furthermore, the internal structures in the sheets appear to be more scattered in the simulation. As mentioned in § 4.6.1, free gas content, which is neglected in the computation, may have a damping effect on the acoustics in the bulk. As these in turn may induce cavitation, this damping thus could cause a reduction of such scatter.

4.7 Condensation shock phenomenon

In the following, the propagating condensation front is analysed with the help of Rankine–Hugoniot jump conditions. These relations hold for any hyperbolic conservation law. For one-dimensional flow, this can be written in the generic form

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\,t}\boldsymbol{U}+\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}\,x}\boldsymbol{F}(\boldsymbol{U})=0, & \displaystyle\end{eqnarray}$$

with the state vector $\boldsymbol{U}$ and flux $\boldsymbol{F}$ . The solution supports the existence of discontinuities, i.e. shocks. These must satisfy the Rankine–Hugoniot relations

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle \left[\boldsymbol{F}(\boldsymbol{U})\right]_{L,R}=s\boldsymbol{\cdot }\left[\boldsymbol{U}\right]_{L,R}, & \displaystyle\end{eqnarray}$$

where $\left[\,\bullet \,\right]_{L,R}=\left(\,\bullet \,\right)_{L}-\left(\,\bullet \,\right)_{R}$ , the subscripts $L$ and $R$ denoting the left- and right flow states, and $s$ is the constant propagation velocity of the discontinuity. Equations (4.3) can be utilised (i) for assessing whether the observed condensation fronts fulfil the Rankine–Hugoniot conditions and thus represent a compressible shock-wave phenomenon and (ii) to compute the shock strength or propagation velocity $s$ . It should be noted that this is a simplified analysis, assuming a planar one-dimensional front propagating with constant velocity within a homogeneous medium.

Neglecting bubble dynamics, surface tension and viscosity, the system can be modelled by the compressible Euler equations, with the vectors of conserved quantities and flux given by $\boldsymbol{U}=[\unicode[STIX]{x1D70C},\unicode[STIX]{x1D70C}u,\unicode[STIX]{x1D70C}E]^{\text{T}}$ and $\boldsymbol{F}(\boldsymbol{U})=[\unicode[STIX]{x1D70C}u,\unicode[STIX]{x1D70C}u^{2}+p,\unicode[STIX]{x1D70C}u\left(e+u^{2}/2\right)+p]^{\text{T}}$ respectively. The Rankine–Hugoniot conditions then read as

(4.4a-c ) $$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D70C}u\\ \unicode[STIX]{x1D70C}u^{2}+p\\ \unicode[STIX]{x1D70C}u\left(e+\frac{1}{2}u^{2}+p/\unicode[STIX]{x1D70C}\right)\end{array}\right]_{L,R}=s\cdot \left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D70C}\\ \unicode[STIX]{x1D70C}u\\ \unicode[STIX]{x1D70C}E\end{array}\right]_{L,R}. & & \displaystyle\end{eqnarray}$$

For this analysis, pre- and postshock states are extracted from the simulations at several stages during the shedding cycles. We discuss a suitable time instant, depicted in figure 20. The visualisation shows the void fraction and streamwise velocity at mid-span ( $z=0$ ), figure 20(a,b), and the respective spanwise average, figure 20(c,d). The pre- and postshock states used for the evaluation are extracted at the locations indicated in the figure. Table 7 summarises the obtained velocity perpendicular to the front $u_{\bot }$ , density $\unicode[STIX]{x1D70C}$ , void fraction $\unicode[STIX]{x1D6FC}$ and pressure $p$ . For reference, the pre- and postshock speeds of sound, under the assumption of frozen and equilibrium conditions, (A 1) and (A 2), are also provided in table 7. Due to the nonlinearity of these relations, application of (A 1) and (A 2) is not commutative with the spanwise-averaging operation. The speed of sound thus is first computed locally within each computational cell, and then spatially averaged.

Figure 20. Instantaneous flow field exhibiting a condensation shock at $x\approx 75~\text{mm}$ . Comparison of the flow field $(a,b)$ in the mid-span slice and (c,d) as a spanwise average, showing (a,c) the vapour volume fraction and (b,d) the axial velocity, with isocontours of the void fraction superimposed. The indicated locations are used for extraction of the pre- and postshock quantities shown in table 7.

Upstream of the discontinuity, the local flow velocity is small and mainly oriented downstream. The sheet is characterised by a void fraction exceeding 95 % in the slice and a spanwise average of 84 %. Across the shock, the direction of flow is reversed, with a significant increase in magnitude. Complete condensation to $\unicode[STIX]{x1D6FC}=0$ occurs behind the front in the slice. In contrast, the shock appears to be smeared for the spanwise average, and the postshock void fraction is non-zero. The pressure rise across the front for the chosen instant is $1.7~\text{kPa}$ on the spanwise average, and only slightly larger in the slice. In general, values between $1$ and $5~\text{kPa}$ are observed. The compression is thus very weak, which is in close agreement with the values reported by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016).

Table 7. Representatively chosen pre- and postshock flow states for Rankine–Hugoniot analysis, extracted from the time instant depicted in figure 20. Comparison between quantities in the mid-span slice and the spanwise average.

Utilising the quantities extracted at discrete locations for various time instants in (4.4a,b ), it is found that the discontinuities indeed satisfy the Rankine–Hugoniot conditions, with an error of ${\lesssim}1\,\%$ . In contrast, errors of $10$ – $15\,\%$ are obtained with the spanwise-averaged values. The reason lies in the fact that the jump conditions are strictly applicable only at discrete locations, while not being necessarily valid for spanwise-averaged flow states. Interestingly, this approach still yields a good approximation of the bulk front propagation velocity. With the values from table 7 for the chosen example, a value of $\langle s\rangle _{z}=-5.43~\text{m}~\text{s}^{-1}$ can be computed for the spanwise average. This is in reasonable agreement with the average value of $\overline{s}\approx -4.5\pm 0.5~\text{m}~\text{s}^{-1}$ obtained from the $s$ – $t$ diagram, figure 7. The propagation velocity computed in the slice for the chosen example is $s=-5.75~\text{m}~\text{s}^{-1}$ . Ranging, in general, between $-4.5$ and $-8~\text{m}~\text{s}^{-1}$ , it tends to be higher than the bulk propagation velocity. This can be explained with the assumption of one-dimensional flow, thereby disregarding any existing perturbations of the interface and inhomogeneities in the pre- and postshock states. As discussed, the actual front experiences Kelvin–Helmholtz and Rayleigh–Taylor instabilities. It thus is strongly non-planar and has a non-uniform propagation velocity.

It should be noted that the Rankine–Hugoniot condition for the conservation of energy, (4.4c ), can be utilised for computing the temperature variation associated with the condensation shock. In the present example, assuming that the liquid behind the shock is at a reference temperature $T_{ref}$ , this variation can be evaluated as $\unicode[STIX]{x0394}T\approx 0.3~\text{K}$ . Although this is larger in magnitude than for the example chosen in appendix B, the temperature variation across the shock is still negligibly small.

Using the quantities in the slice, we compute a shock Mach number $\mathit{Ma}_{s}$ , relating the front propagation velocity, relative to the preshock flow, to the preshock speed of sound. As already mentioned, estimates for the upper and lower bounds of the thermodynamic speed of sound are given by (A 1) and (A 2). By excluding any phase transfer from the estimation, i.e. using the frozen speed of sound $c_{fr}$ , the Mach number is evaluated as $\mathit{Ma}_{s}^{fr}=0.63$ for the current example, so that a condensation shock could not exist. In contrast, with the equilibrium speed of sound $c_{eq}$ , $\mathit{Ma}_{s}^{eq}=5.2$ for the chosen instant. In the simulation, $\mathit{Ma}_{s}=\mathit{Ma}_{s}^{eq}>1$ , because the equilibrium speed of sound is utilised in the thermodynamic model. The local speed of sound in the experiments is unknown. However, since a propagating shock wave was observed by Ganesh et al. (Reference Ganesh, Mäkiharju and Ceccio2016), we conclude that in the experiments $\mathit{Ma}_{s}>1$ , too.

With only a few kPa, the pressure rise $\left[\,p\,\right]_{L,R}$ across the condensation shock is small. In order to assess the sensitivity of the computed shock propagation velocity to the pressure rise, the time instant discussed above is taken as a reference. The postshock pressure $p_{R}$ is then systematically varied between $-40\,\%$ and $160\,\%$ around the probed reference value of $p_{R}^{ref}=4.4~\text{kPa}$ . This gives a pressure difference across the shock of $0.3~\text{kPa}\leqslant \left[\,p\,\right]_{L,R}\leqslant 4.7~\text{kPa}$ , corresponding to a range between 15 % and 230 % of $[\,p^{ref}\,]_{L,R}$ .

The propagation velocity of the condensation shock can be computed with the formula of Brennen (Reference Brennen1995) (equation (4.1), denoted as variant $A$ in the following) or, alternatively, can be obtained by repeating the above analysis and solving (4.4a,b ) for $s$ (variant $B$ ). Additionally, (4.4b ) can be utilised for determining $s$ directly (variant  $C$ ).

With the reference conditions from table 7, Brennen’s formula gives $s^{ref,A}=-7.45~\text{m}~\text{s}^{-1}$ , which is larger than both the numerical results and the experimental references. This is expected, as phase change is neglected in (4.1), thereby implicitly assuming a higher value for the speed of sound, i.e. the frozen speed of sound. The computed reference velocities of variants $B$ and $C$ with $s^{ref,B}=-5.75~\text{m}~\text{s}^{-1}$ and $s^{ref,C}=-5.83~\text{m}~\text{s}^{-1}$ are very similar, and agree well with the cavity dynamics actually observed in the experiments and the simulation.

Figure 21. Sensitivity of the condensation shock propagation velocity. Relative change $s/s^{ref}$ as a function of $\left[\,p\,\right]_{L,R}/\left[\,p^{ref}\,\right]_{L,R}$ , computed with (4.1) by Brennen (Reference Brennen1995) (variant $A$ , — - —) and by Rankine–Hugoniot analysis (equation (4.4a,b ), variant $B$ , ——; equation (4.4b ), variant $C$ , – –).

The sensitivity of the computed velocities is summarised in figure 21, plotting the normalised velocities $s/s^{ref}$ versus $\left[\,p\,\right]_{L,R}/\left[\,p^{ref}\,\right]_{L,R}$ . The Rankine–Hugoniot analysis is insensitive to the pressure rise: a $\pm 10\,\%$ difference in $\left[\,p\,\right]_{L,R}$ leads to a change in the predicted propagation velocity of at most $\pm 0.7\,\%$ . In contrast, Brennen’s formula shows a considerably larger sensitivity of $\pm 8\,\%$ for the same variation in the pressure rise.

While (4.1) gives reasonable results for the values from the mid-span slice, a considerably lower shock speed of $\langle s\rangle _{z}^{A}=-3.04~\text{m}~\text{s}^{-1}$ is obtained by using the spanwise-averaged values. The reason is the observed strong sensitivity of Brennen’s formula to the pressure difference in combination with the fact that the pressure difference in the spanwise average is lower than for the discrete slice, $\left[\,\langle p\rangle _{z}\,\right]_{L,R}<\left[\,p\,\right]_{L,R}$ . Hence, the shock speed computed with (4.1) is considerably affected by the degree of variation in the spanwise direction. This is not the case for the Rankine–Hugoniot analysis, which gives comparable values for the spanwise averages and the discrete slices, and is less sensitive to the pressure rise.

With the simulation, flow field data can be extracted at any position in the computational domain, including taking spanwise-averaged data. Thus, a clear distinction between the two approaches is possible. For the experimental data, on the other hand, some care has to be taken. Void fraction measurements as obtained by the x-ray densitometry are only available as spanwise averages, while the pressure is measured by pressure probes at discrete positions in the field.