3.1. Baseline turbulent boundary layer characteristics

The incoming liquid TBL developing over the wall is first characterised. This is done in the absence of air injection. The presence of the latter was found to impose an upstream APG on the incoming TBL (Anand Reference Anand2021). Therefore, to avoid any complications while comparing the behaviour of the TBL over the cavity with respect to the baseline case (over the wall), we characterise the incoming TBL without the cavity. The inner-normalised mean streamwise velocity and root-mean-square (r.m.s.) streamwise turbulent velocity fluctuations of the baseline TBL are shown in figure 2. In comparison with laser Doppler anemometry (LDA) measurements of DeGraaff & Eaton (Reference DeGraaff and Eaton2000) at comparable Reynolds number, the logarithmic behaviour in the present incoming TBL is not strictly canonical. The absence of a well-formed wake can be explained as a result of a free stream that is more noisy than expected. The underestimation in the r.m.s. streamwise velocity fluctuations observed here compared with the LDA measurements can be explained due to the relatively large interrogation areas employed. The incoming TBL in the present study is not a strictly canonical ZPG TBL as seen in the shortened log region and a mild FPG present due to TBL development ( $K \approx 3\times 10^{-9}$ ), the latter, however, much lower compared with the local FPG imposed on the windward side of the cavity (see discussion in section 3.3). Although sufficiently different global incoming TBL conditions such as $\delta$ and $u_{\infty }$ are shown to affect the air cavity characteristics (Nikolaidou et al. Reference Nikolaidou, Laskari, van Terwisga and Poelma2024), here, we focus on the relative TBL changes. Thus, some deviations from canonicality are not expected to affect the general trends discussed here. This is further supported by the $\delta /t_{max} = 3$ case added for comparison in what follows (see section 3.3).

Figure 2. Inner-normalised (a) mean streamwise velocity and (b) streamwise velocity intensities for the baseline TBL. Also shown is reference data using LDA at comparable friction Reynolds number ( ${Re}_{\tau}$ ) by DeGraaff & Eaton (Reference DeGraaff and Eaton2000).

3.2. Air cavity identification

As discussed in section 1, in TBL flows over solid bumps, the shape of the bump, and in particular its height and length, have been observed to influence the development of the TBL (Baskaran et al. Reference Baskaran, Smits and Joubert1987; Webster et al. Reference Webster, DeGraff and Eaton1996). Given the similarity in geometry of the spanwise-uniform region of the cavity with solid bumps in the literature, the shape of the cavity is then also expected to influence the development of the incoming TBL. Therefore, before studying the behaviour of the TBL over the cavity and comparing the flow development with previous solid bump studies, the time-averaged geometry of the cavity needs to be identified and characterised. To this end, an identification technique based on the thresholding of correlation values obtained after image post-processing was employed.

Figure 3. Top: contour of mean correlation map $\overline {R}$ along with the mean cavity identified (in red). Here, $\overline {c}$ is the chord length, $t_{max}$ is the maximum thickness of the mean cavity and $x_{o}$ is the leading edge of the mean cavity. Inset: variation of $\overline {R}$ with wall-normal distance at the streamwise location of $t_{max}$ ( $\overline {R}_{th}$ : correlation threshold chosen to identify the cavity). Bottom: mean cavity interface (solid black line) with r.m.s. fluctuations of the cavity thickness along the length of the cavity (shaded grey region). Error bars depict the uncertainties in the identification technique. Inset: example of an instantaneous cavity interface detected (in red). See also supplementary movie 1.

The instantaneous correlation values are obtained through a PIV routine following a multi-pass approach, with a final window size of $24\times 24\;\mathrm{pix}$ and an overlap of $50\;\%$ (see figure 3). A smaller final window size (higher spatial resolution) than the one applied to acquire the velocity fields allowed a more accurate estimate of the air–water interface, and therefore the cavity thickness and length through the correlation values. It was, however, found to be too noisy for velocity estimates; hence, a lower resolution final interrogation window was used there (see section 2). A threshold to identify instantaneous geometries of the cavity is obtained using the time-averaged correlation map (figure 3 top). We exploit the fact that correlation values within the cavity should be inherently low compared with the surrounding liquid as there are no tracer particles to correlate with, while patterns within the cavity region (e.g. reflections) also varied significantly between frames and thus did not correlate. This leads to a sharp gradient in correlation between cavity and flow, clearly delineated in the time-averaged map ( $\overline {R}$ , see inset of figure 3 top). Given the fact that the cavity interface can lie anywhere within this gradient (in the range of $0.15\lt \overline {R}\lt 0.40$ : see shaded red area in inset of figure 3 top), an average of $\overline {R}$ in this region is identified as a representative threshold ( $\overline {R}_{th}=0.27$ ). This results in an uncertainty (error bars in figure 3 bottom) in the location of the cavity interface of approximately $\pm 2\;\mathrm{mm}$ along its length and dominates any spatial resolution effects. Regardless, the chosen threshold does not affect the analysis in the trends discussed later. This threshold is then applied to every image to detect the instantaneous cavity shape, which is found to match relatively well with visual inspection of the raw images (see supplementary movie 1). The mean air–water interface is then obtained by averaging over these instantaneous realisations; comparison with the mean directly identified from the time-averaged correlation map (Nikolaidou et al. Reference Nikolaidou, Laskari, van Terwisga and Poelma2024), yields similar results ( $\lt 1\;\mathrm{mm}$ difference). The discontinuities observed in the detected interface (figure 3, e.g. near the $\overline {c}$ label) are due to light path obstructions in the raw particle images. The smoothness of the rest of the field, however, allowed interpolation for the mean shape in these locations.

The mean cavity shape reveals a well-defined asymmetric bump-like geometry at the spanwise-uniform part of the cavity upstream (also supported by visual observations). The highly dynamic nature of the shedding region of the cavity downstream, its limited thickness and significantly lower correlation gradients, did not allow for a valid identification of the same. The maximum thickness and chord length of the mean cavity were found to be $t_{max} = 13.2\pm 2\;\mathrm{mm}$ and $\overline {c} = 143\pm 2\;\mathrm{mm}$ , respectively. The ratio of the incoming boundary layer thickness to the maximum thickness of the cavity was found to be $\delta /t_{max} = 12$ , which is considerably higher than the ratios encountered in the solid bump studies of Baskaran et al. (Reference Baskaran, Smits and Joubert1987) and Webster et al. (Reference Webster, DeGraff and Eaton1996) ( $\delta /h = 0.25$ and $\delta /h = 1.5$ , respectively).

Unlike solid bumps for which length and thickness are constant, these parameters vary in space and time for the air cavity studied here. However, when looking at the instantaneous cavity shapes (see supplementary movie 1) and r.m.s. fluctuations of the cavity thickness across its length (grey shaded region around the mean cavity in figure 3 bottom), there are no significant deviations of the cavity from a bump-like geometry. The r.m.s. deviation of the maximum thickness and chord length of the cavity detected (approximately $5.8\;\mathrm{\%}$ and $5.2\;\mathrm{\%}$ compared with the mean, respectively), however, are lower than the estimated experimental uncertainty (approximately $7\;\mathrm{\%}$ of $t_{max}$ ) (see inset of figure 3 bottom and figure 12 in the Appendix), and as such no conclusions can be drawn on it. As detailed analysis of small-scale features (capillary waves or ripples) of the gas–liquid interface lies beyond this study’s scope, we will use the average cavity shape in the analysis that follows. Measurement techniques tailored to interfacial physics would better capture these instantaneous features (for example Gomit et al. (Reference Gomit, Chatellier and David2022); van Meerkerk et al. (Reference van Meerkerk, Poelma and Westerweel2020)).

With the mean cavity geometry defined, the flow behaviour around it can then be analysed. In what follows, a modified coordinate system based on the mean cavity geometry will be employed. The normalised streamwise coordinate is defined as $x' = (x-x_{o})/\overline {c}$ , with $x_{o}$ being the leading edge of the cavity. Using the instantaneous cavity lengths instead of $\overline {c}$ did not alter any of the presented results, corroborated by no significant deviations of the instantaneous cavity lengths from the mean (approximately $5.2\;\mathrm{\%}$ ). Thus, normalisation based on $\overline {c}$ was chosen for the analysis. The wall-normal coordinate, $y'$ is normalised with $\delta$ , and is defined with its zero location based on the mean interface height, and maintained normal to the flat plate at the interface positions, the latter employed owing to a relatively large $\delta /t_{max}$ (and minimal streamline curvature) (Webster et al. Reference Webster, DeGraff and Eaton1996).

3.3. Turbulence statistics

The mean streamwise velocity field clearly illustrates that the incoming TBL experiences an alternating streamwise pressure gradient over the length of the cavity (figure 4). The TBL first experiences an APG, as indicated by the moderate elevation of the contour lines at the leading edge of the cavity ( $x'=0$ ). The flow then accelerates around the apex of the cavity, indicating a FPG, while it decelerates past the apex due to an APG, with no indication of incipient separation. To get some further insight on the influence of the cavity on the TBL, the streamwise variation of the local boundary layer thickness $\delta$ is also computed and compared with the solid bump study by Baskaran et al. (Reference Baskaran, Smits and Joubert1987). Estimation of $\delta$ in non-equilibrium TBLs (for example subjected to pressure gradients) is typically difficult as the mean velocities and wall-normal gradients beyond the boundary layer edge do not exactly go to zero (see also the discussion in Vinuesa et al. (Reference Vinuesa, Bobke, Örlü and Schlatter2016)). Here, the local boundary layer thickness is defined as the average wall-normal location where the streamwise, wall-normal and Reynolds shear stresses approach an approximately constant value in the free stream. A similar technique using turbulent quantities to identify the boundary layer thickness in open channel flows was also used by Das et al. (Reference Das, Balachander and Barron2022).

Figure 4. Mean streamwise velocity contours with some streamlines (solid black lines); $x'=0$ is the leading edge of the cavity where $x' = (x-x_{o})/\overline {c}$ . The local $\delta$ from the present study (circles; $\delta /t_{max}=12$ ) is compared with that over a thicker air cavity (Anand (Reference Anand2021): upward-pointing triangles; $\delta /t_{max}=7$ ), both normalised by the maximum cavity thickness $t_{max}$ , and that over a solid bump (Baskaran et al. (Reference Baskaran, Smits and Joubert1987): downward-pointing triangles; $\delta /h=0.25$ ), normalised by the maximum bump height $h$ . Flow is from left to right.

The streamwise variation of $\delta$ in the present study (see figure 4) follows the trend set by alternating pressure gradients discussed previously: a relatively small growth around the leading edge of the cavity due to an APG, followed by a thinning over the windward side of the cavity until the apex due to a FPG and finally a thickening over the leeward side of the cavity due to an APG. This is also in line with trends observed in the studies of Anand (Reference Anand2021) and Baskaran et al. (Reference Baskaran, Smits and Joubert1987) (air cavity and solid bump, respectively). The TBL here (and in Anand (Reference Anand2021)), was found to not separate at the leeward side of the cavity under APG conditions, whereas incipient separation was observed in Baskaran et al. (Reference Baskaran, Smits and Joubert1987) at the leeward side of the solid bump. This is due to the marked difference of the $\delta /t_{max}$ ( $\delta /h$ ) ratio between the three studies, with low values of this ratio typically leading to separation under APG conditions. We note that the presence of a free-slip boundary condition (at the cavity) can delay flow separation, as has been observed in the numerical study of flow over a Gaussian bump (Ceccacci et al. Reference Ceccacci, Calabretto, Thomas and Denier2022). However, the $\delta /t_{max}$ ratio in the current study is higher than the $\delta /h = 1.5$ used by Webster et al. (Reference Webster, DeGraff and Eaton1996) for a solid bump, where no TBL separation occurred. Thus, the present ratio is expected to be sufficiently high to prevent flow separation irrespective of whether the boundary condition at the bump is no slip (solid bump) or free slip (air cavity).

In order to more closely investigate these alternating pressure gradient effects on the TBL across the cavity, the mean velocity and turbulent stresses profiles are evaluated next, at selected streamwise locations. The profiles have been normalised by local outer units, that is, using the local $\delta$ and $U_{\infty }$ at each streamwise location. As mentioned previously, all profiles are evaluated over the cavity, thus using the local coordinate system $(x', y')$ .

Figure 5. Variation of mean streamwise velocity (markers) for $\delta /t_{max}\approx 12$ (left) and $\delta /t_{max}\approx 3$ (right) across different streamwise locations over the cavity, normalised with local outer units. Colours represent different streamwise locations along the cavity, as shown in the inset. The profile at $x'_{1}$ without a cavity is denoted with a black line. Flow is from left to right.

The mean velocity profiles at different streamwise locations exhibit deviations in the inner and logarithmic regions compared with the baseline case (figure 5, left). An upstream influence imposed by the leading edge of the cavity on the incoming boundary layer can be observed at streamwise locations $x_{1}'$ and $x'_{2}$ , where an APG causes a deficit in the mean velocity compared with when no cavity is present. This influence has been estimated to last until approximately $0.9\delta$ upstream of the leading edge of the cavity, beyond which the TBL relaxed to its baseline state (Anand Reference Anand2021; Nikolaidou et al. Reference Nikolaidou, Laskari, van Terwisga and Poelma2024). Over the windward side of the cavity, a switch in the pressure gradient to a favourable one causes the TBL to accelerate, as highlighted by the increase in mean streamwise velocity until the apex (approximately $x'_{4}$ ). The profiles display a systematic deviation from log-law behaviour, which was also reported by Narasimha & Sreenivasan (Reference Narasimha and Sreenivasan1979) in strongly accelerating flows as due to the stabilising effects of FPGs. At the leeward side of the cavity ( $x'_{5}$ ), the boundary layer decelerates compared with the mean velocity at $x_{4}'$ due to an APG, however, it is still faster than the incoming flow due to the upstream FPG. Overall, the mean velocity profile variation over the cavity follows the trend set by the alternating streamwise pressure gradients, in line with the boundary layer thickness variations discussed previously (see figure 4). In an independent experimental campaign in the same set-up, but with the air injector located further upstream such that a relatively low $\delta /t_{max} \approx 3$ was achieved (Nikolaidou et al. Reference Nikolaidou, Laskari, van Terwisga and Poelma2024), similar trends in the mean streamwise velocity over the cavity were observed (figure 5 right), again with no TBL separation. The recovery of the TBL at $x'_{6}$ back to its state upstream of the cavity (at $x'_{1}$ ), occurs within one $\delta$ from the trailing edge of the cavity. Further downstream, the return of the TBL to its baseline state (at $x'_{1}$ without a cavity) occurs within a cavity length ( $\approx 3\delta$ ) from the trailing edge of the cavity (figure 5, right).

Figure 6. Variation of the streamwise normal stress $\overline {u'u'}$ across different streamwise locations over the cavity (markers) normalised with local outer units. Colours represent different streamwise locations along the cavity, as shown in the inset. The profile at $x'_{1}$ , when no cavity was present, is denoted with a black line. Flow is from left to right.

Figure 7. Variation of the wall-normal stress $\overline {v'v'}$ across different streamwise positions of the cavity (markers) normalised in local outer units. Colours represent different streamwise locations along the cavity, as shown in the inset. The profile at $x'_{1}$ , when no cavity was present, is denoted with a black line. Flow is from left to right.

Figure 8. Variation of the Reynolds shear stress $-\overline {u'v'}$ across different streamwise positions of the cavity (lines with markers) normalised in local outer units. Colours represent different streamwise locations along the cavity, as shown in the inset. The profile at $x'_{1}$ , when no cavity was present, is denoted with a black line. Flow is from left to right.

The variation of the streamwise normal ( $\overline {u'u'}$ , figure 6), wall-normal ( $\overline {v'v'}$ , figure 7) and Reynolds shear stress profiles (- $\overline {u'v'}$ , figure 8) across the length of the cavity are investigated next. Higher than anticipated reflections from the cavity did not allow for in-depth analysis of the stresses for the prior independent experimental campaign where $\delta /t_{max}\approx 3$ (although qualitatively similar behaviour was observed overall) and as such will not be included in the forthcoming discussion. The strongest variations as compared with the no cavity case are observed in the $\overline {v'v'}$ and $-\overline {u'v'}$ profiles, and they extend throughout the majority of the TBL height. Variations in $\overline {u'u'}$ , on the other hand, are less pronounced (figure 6) and are restricted to $y' \lt 0.4\delta$ , with the rest of the outer region showing a near collapse under outer scaling. Furthermore, the stress profiles do not exhibit “knee” points, which are typical markers for an internal layer triggered within the TBL (Balin & Jansen Reference Balin and Jansen2021; Parthasarathy & Saxton-Fox Reference Parthasarathy and Saxton-Fox2023; Webster et al. Reference Webster, DeGraff and Eaton1996) (see section 1 ). Internal layers have been reported to be formed in TBLs subjected to relatively strong favourable–adverse pressure gradients (quantified by $K\gtrsim 2\times 10^{-6}$ ) typically imposed at low $\delta /h$ ratios (Parthasarathy & Saxton-Fox Reference Parthasarathy and Saxton-Fox2023); however, internal layers are not expected to be triggered in the TBL here ( $K\approx 0.3\times 10^{-6}$ , $\delta /t_{max}= 12$ ).

As the TBL moves over the windward side of the cavity where it is subjected to a FPG, a suppression in $\overline {u'u'}$ and $-\overline {u'v'}$ profiles throughout the boundary layer is observed as expected (Volino Reference Volino2020). However, $\overline {v'v'}$ exhibits a marked increase below $y\approx 0.3\delta$ throughout the length of the cavity (figure 7), even in the region where an attenuation of stresses is expected due to a FPG (station $x'_{3}$ ). In addition, the general shape of the $\overline {v'v'}$ profiles over the length of the cavity appears to be largely different in character compared with its state at $x'_1$ and $x'_{2}$ . The amplification of $\overline {v'v'}$ at $x'_{3}$ and $x'_{4}$ relative to $x'_{2}$ , does not follow the expected trend based on the influence of a FPG, the latter which is observed in $\overline {u'u'}$ and $-\overline {u'v'}$ . Instead, the vertical momentum induced by the air injection seems to dominate the $\overline {v'v'}$ profiles overshadowing any potential pressure gradient effects. This behaviour of $\overline {v'v'}$ in addition to a free-slip boundary condition is what sets it apart from solid bump studies. Variations in $-\overline {u'v'}$ follow the alternating streamwise pressure gradient trend (as observed in $\overline {u'u'}$ in figure 6), with a notable increase below $y\approx 0.3\delta$ at $x'_5$ compared with $x'_3$ and $x'_4$ . This increase can be explained due to the combined effects of the APG imposed by the cavity’s leeward side and the momentum induced by the air injection (reflected in $\overline {v'v'}$ ). A weakening of $\overline {v'v'}$ and $-\overline {u'v'}$ above $y\approx 0.3\delta$ at $x'_{4}$ and $x'_{5}$ is observed relative to the stations over the windward side of the cavity. Weakening of stresses in the outer region has been attributed to effects of convex streamline curvature (Balin & Jansen Reference Balin and Jansen2021; Baskaran et al. Reference Baskaran, Smits and Joubert1987) or a combination of convex streamline curvature and FPG effects (Uzun & Malik Reference Uzun and Malik2021). However, the weakening in $\overline {v'v'}$ and $-\overline {u'v'}$ observed here is not expected to be due to streamline curvature, but is instead solely driven by the effects of FPG. In the solid bump study of Baskaran et al. (Reference Baskaran, Smits and Joubert1987) where $\delta /h = 0.25$ , they noticed a decline in the turbulent stresses in the outer region which was attributed to a prolonged streamline curvature. While in the solid bump study of Webster et al. (Reference Webster, DeGraff and Eaton1996) with $\delta /h = 1.5$ , only mild variations were observed in the outer region of the stresses, which was attributed to a minimal impact of streamline curvature. In addition, Anand (Reference Anand2021) reported a near collapse above $y\approx 0.4\delta$ in the $\overline {u'u'}$ and $-\overline {u'v'}$ profiles across different streamwise positions along the cavity for $\delta /t_{max} = 7$ . Therefore, if streamline curvature induced by the shape of cavity was indeed important in the current study, we would expect $\overline {u'u'}$ to show a decrease (similar to $\overline {v'v'}$ and $-\overline {u'v'}$ ) at higher distances from the wall ( $y' \gt 0.4\delta$ ). Moreover, at higher $\delta /t_{max}$ ratios (compared with $\delta /h = 1.5$ (Webster et al. Reference Webster, DeGraff and Eaton1996) and $\delta /t_{max} = 7$ (Anand Reference Anand2021); in the current study, $\delta /t_{max} = 12$ ), we would expect streamline curvature effects induced by the shape of the cavity (or a solid bump) to be even less important. Alternating streamwise pressure gradients and air injection are therefore expected to be the dominant perturbations to the incoming TBL in the current study, the latter being more prominent in the windward side of the cavity.

Up until this point, only single-point statistics of the TBL have been considered, focusing on magnitude variations across $\delta$ due to the presence of the air cavity; $\overline {v'v'}$ profiles were those impacted the most in both magnitude and overall shape. To further investigate how these variations translate to a change in spatial coherence, two-point correlations of the wall-normal fluctuations ( $R_{v'v'}$ ) were computed (see figure 9 and table 2), at two reference wall-normal locations ( $y_{ref}=0.1\delta$ and $0.4\delta$ ). The extent of $R_{v'v'}$ in the streamwise ( $L_x$ ) and wall-normal direction ( $L_y$ ), was estimated based on an elliptical fit to $R_{v'v'}$ at a correlation level of $0.3$ (Wu & Christensen Reference Wu and Christensen2010). Qualitatively, with no cavity present, $R_{v'v'}$ was found to be narrow and elongated in the wall-normal direction (red line contour in figure 9) consistent with previous observations of $R_{v'v'}$ over smooth walls (Krogstad & Antonia Reference Krogstad and Antonia1994; Wu & Christensen Reference Wu and Christensen2010); the spatial extent of $R_{v'v'}$ evaluated at $y_{ref}= 0.15\delta$ and $R_{v'v'} = 0.3$ was found to agree reasonably well with similar estimates in canonical TBLs in the literature (current study: $L_x=0.11\delta$ and $L_y= 0.14\delta$ ; Wu & Christensen (Reference Wu and Christensen2010): $L_x= 0.16\delta$ and $L_y= 0.18\delta$ ).

Table 2. Orientation ( $\theta$ ) and extent ( $L_x$ and $L_y$ ) of $R_{v'v'}$ (at a level of $0.3$ , based on figure 9). Note that the streamwise positions chosen (A to D) match those in figure 9 for clarity, while the no cavity case at the most upstream position is also added for comparison (denoted with A^′ )

In the presence of the cavity, the spatial structure of $R_{v'v'}$ undergoes marked changes as the TBL travels over the cavity. This is more prominently observed at $y_{ref}=0.1\delta$ compared with $y_{ref}=0.4\delta$ (see figure 9 and bottom row of table 2). The coherence in $R_{v'v'}$ at $y_{ref}=0.4\delta$ remains relatively similar at all streamwise regions examined with no significant changes in the spatial extent of coherence compared with the baseline case (no cavity present). It should be noted here that this similarity in spatial coherence of $R_{v'v'}$ at $y\gt 0.4\delta$ is, however, coupled with marked changes in magnitude, as shown in the intensity profile evolution of $v'v'$ over the cavity (see figure 7).

Figure 9. Two-point correlations of wall-normal velocity fluctuations $R_{v'v'}$ at four different streamwise regions ( $A$ to $D$ ) and at two wall-normal locations ( $y= 0.1\delta$ and $y= 0.4\delta$ ), for the cases with (black lines) and without (red lines) the cavity present. Vertical dashed lines demarcate the different streamwise regions where $R_{v'v'}$ is computed. Note that the local $\delta$ is used for normalisation. Contour levels used: [ $0.3\;0.4\;0.6\;0.8\;1$ ]. Inset: illustration of the elliptical fit (blue dash-dotted line) used for the extent ( $L_x$ and $L_y$ ) and orientation ( $\theta$ ) estimates of $R_{v'v'}$ at a $0.3$ correlation level (see table 2). Approximate inclinations of the cavity: windward side $\theta _W \approx 14.3^\circ$ and leeward side $\theta _L \approx -7.7^\circ$ . Flow is from left to right.

At $y_{ref}=0.1\delta$ , changes to both the aspect ratio and the orientation of $R_{v'v'}$ are observed over the cavity (regions $B$ and $C$ in figure 9), while just upstream of the leading edge (region $A$ ) they are both relatively similar compared with the baseline case. On average, the spatial structures in regions $B$ and $C$ are found to approximately follow the inclinations imposed by the cavity. In terms of aspect ratio, at region $B$ , contours of $R_{v'v'}$ exhibit an increase in the streamwise spatial extent increasing the aspect ratio beyond $1$ . This can be attributed to the acceleration imposed by the FPG at the windward side of the cavity: a similar increase in the spatial extent of $R_{v'v'}$ under a FPG was observed in the study of Volino (Reference Volino2020), where the TBL was subjected to changing pressure gradients in a variable-height water tunnel. This anisotropy is further amplified in region $C$ , with aspect ratios reaching almost twice those at region $A$ , followed by a recovery of $R_{v'v'}$ at region $D$ . Considering that the leeward side of the cavity experiences an APG, it is interesting to observe this increased spatial extent of $R_{v'v'}$ at region $C$ . This is in contrast to previous studies where a TBL influenced by an APG alone was reported to undergo little to no change in the spatial extent of $R_{v'v'}$ at similar reference wall-normal locations (Krogstad & Skåre Reference Krogstad and Skåre1995; Gungor et al. Reference Gungor, Maciel, Simens and Soria2014). Volino (Reference Volino2020) also reported almost no change in $R_{v'v'}$ in an APG region which was preceded by a FPG and a ZPG region.

In an attempt to explain the increased extent of coherence of $R_{v'v'}$ observed at region $C$ , we compare the local mean advection ( $\mathcal{T}_{adv} \sim \overline {c}/\overline {u}$ ) and eddy turnover time ( $\mathcal{T}_{eddy} \sim L/u'_{rms}$ ) scales, $L$ being the spatial extent of $R_{v'v'}$ discussed previously (for comparison, the large eddy turnover time $= \delta /u_{\tau } \sim 5\mathcal{T}_{eddy}$ ). The form and response of such structures is expected to be influenced by the competition between $\mathcal{T}_{adv}$ and $\mathcal{T}_{eddy}$ . It is expected that for $\mathcal{T}_{adv}\gt \gt \mathcal{T}_{eddy}$ , structures in the TBL have sufficient time to reach local equilibrium with the local conditions of the cavity interface. As a result, the influence of upstream (history) effects are then expected to be relatively small. On the other hand, at relatively lower $\mathcal{T}_{adv}$ , eddies in the TBL do not have sufficient time to reach local equilibrium with the local conditions of the cavity interface. As a result of a lag in their response, upstream influences (and their cumulative effect) are expected to be more important. In the current study we find $\mathcal{T}_{eddy} \sim 2\mathcal{T}_{adv}$ . Therefore qualitatively speaking, we expect the structural response of the TBL to be relatively slow to adjust to the local conditions of the cavity interface, bringing possible upstream effects into play. The study of Parthasarathy & Saxton-Fox (Reference Parthasarathy and Saxton-Fox2023) involving the application of a sequence of streamwise pressure gradients on a TBL highlighted the importance of the history of pressure gradients influencing the response of the TBL (also see Vinuesa et al. (Reference Vinuesa, Örlü, Vila, Ianiro, Discetti and Schlatter2017)). In the APG studies of (Krogstad & Skåre Reference Krogstad and Skåre1995) and (Volino Reference Volino2020) where they featured an APG region that was preceded by a ZPG development segment, either directly (Krogstad & Skåre Reference Krogstad and Skåre1995) or following a ZPG recovery phase (Volino Reference Volino2020), little to no change in the spatial extent of $R_{v'v'}$ in the APG region was reported. Furthermore, in the APG study of Gungor et al. (Reference Gungor, Maciel, Simens and Soria2014), where the APG region was preceded by a laminar separation bubble, a similar conclusion was drawn. Thus, in the current study, the fact that an increased spatial extent of $R_{v'v'}$ is observed at region $C$ (where an APG acts), the former typically observed in TBLs under a FPG (Volino Reference Volino2020), is expected to be a consequence of a cumulative effect of previous pressure gradients imposed. It is important to note that two-point correlations are a statistical measure of the average spatial structure; instantaneously, the spatial coherence is expected to exceed (or decrease from) the estimated extents. Additionally, $R_{v'v'}$ is estimated here in streamwise regions chosen in accordance with the alternating pressure gradients expected around the mean cavity shape (as reflected in the mean and turbulence intensity profiles). The latter, although not significantly changing in geometry, can still impose a pressure gradient switch at streamwise locations which vary instantaneously from the average. It is therefore possible that large-scale structures of the order of half the cavity length or longer (for example, $R_{v'v'}$ at region $C$ ), are at times subjected to an inhomogeneity in pressure gradients along their spatial extent, thereby making it difficult to distinguish the exact source of the observed deviation from the baseline case. Overall, situations involving a continuously applied APG or FPG or a different sequence of streamwise pressure gradients or even the way the pressure gradients are imposed (for example variable tunnel heights, ramps, bumps etc.), can alter the statistics and structure of the TBL differently, an area that still requires further exploration.

Figure 10. Left: Probability density function (pdf) of the wall-normal velocity fluctuations $v'$ (normalised by the r.m.s. $v'_{rms}$ ) at the windward side of the cavity (region $B$ in figure 9). The pdf of $v'$ at the leeward side (region $C$ ) not shown here for clarity. Differently coloured areas of the pdf correspond to different conditional averages of $R_{v'v'}$ : $R|_{v'\lt v'_{rms}}$ (mustard) and $R|_{v'\gt -v'_{rms}}$ (light blue); $R|_{v'\gt v'_{rms}}$ (red) and $R|_{v'\lt -v'_{rms}}$ (dark blue). Right panel: contours of the conditioned two-point correlations (at $y_{ref} = 0.1\delta$ ) at the two streamwise positions and coloured according to the area of the pdf considered (top: $v'\lt v'_{rms}$ (of either sign); bottom: $v'\gt v'_{rms}$ (of either sign)). Black lines in the contour represents the unconditioned correlation $R_{v'v'}$ (also shown in the pdf). Contour levels used: [0.4 0.6 0.8 1]. Flow is from left to right.

In order to further examine the anisotropy observed in the wall-normal velocity coherence around the cavity, $R_{v'v'}$ is conditioned based on the standard deviation of $v'$ ( $v'_{rms}$ ) and the sign of $v'$ ( $v'\gt 0$ : $R|_{v'\lt v'_{rms}}$ and $R|_{v'\gt v'_{rms}}$ ; $v'\lt 0$ : $R|_{v'\gt -v'_{rms}}$ and $R|_{v'\lt -v'_{rms}}$ , see figure 10). The conditional correlations (coloured lines in figure 10) are compared with the unconditional $R_{v'v'}$ (black lines in figure 9 and figure 10). The spatial coherence and anisotropy observed in the unconditional $R_{v'v'}$ are clearly due to high intensity events. For fluctuations with intensities $\lt v'_{rms}$ (of either sign), spatial coherence is limited and more isotropic compared with the global behaviour, while those with intensities $\gt v'_{rms}$ (of either sign), a significant elongation in $x$ is observed, leading to the observed anisotropy in the unconditional correlations. At the leeward side of the cavity, this high intensity, anisotropic behaviour is also seen to be prominent for $v'\lt 0$ , thus for structures moving towards the cavity interface (see table 3 for an estimate). These relatively large coherent structures (with $v'\lt 0$ ) near the trailing edge of the cavity (where the cavity is observed to breakup and shed), can play an important role in the cavity’s form and stability: regions of downwards moving liquid (directed towards the cavity) can lead to pinch-offs at the cavity’s trailing edge, promoting its breakup and resulting in a shortened length. Along with this mechanism, other factors like the presence of re-entrant jets and small-scale wave instabilities at the cavity interface proposed in the literature (see for example Mäkiharju et al. (Reference Mäkiharju, Elbing, Wiggins, Schinasi, Vanden-Broeck, Perlin, Dowling and Ceccio2013); Zverkhovskyi (Reference Zverkhovskyi2014)) can shorten the cavity length. This shortening can have negative implications in ship air lubrication applications, where a minimal contact area between water and solid surface is sought, that is, achieving and maintaining the maximum possible cavity lengths. To establish the extent of such interactions, however, simultaneous time-resolved measurements of the instantaneous cavity shape and stability and TBL structuring would be required, which is out of the scope of the present study.

Table 3. Length scale ( $L_x$ and $L_y$ ) estimates of conditioned $R_{v'v'}$ : $R|_{v'\gt v'_{rms}}$ (red) and $R|_{v'\lt -v'_{rms}}$ ) (dark blue) contours in figure 10 (right panel bottom) measured at $0.4$ correlation level using an elliptical fit (see figure 9). $L_x$ and $L_y$ are lengths along the streamwise and wall-normal directions respectively