3.1. Morphology of bubbly and air-layer flow

Figure 2 illustrates two distinctly different flow states that emerge under nearly identical free-stream velocity and ventilation gas flux conditions, depending on whether the surface is smooth or equipped with SHPo transverse grooves. As shown in figure 2(a), for the smooth-surface case, the injected gas will go through the following successive stages: bubble growth, deformation, detachment and subsequent convection by the crossflow. This phenomenon occurs under conditions of low ventilation flux, where the injected gas is periodically pinched off by the free stream and forms discrete bubbles, a behaviour that has been reported by Xiong et al. (Reference Xiong, Wang, Dong, Wang and Zhang2024). The pinch-off event is governed by the competition between hydrodynamic forces and surface tension (Oguz & Prosperetti Reference Oguz and Prosperetti1993; Mirsandi et al. Reference Mirsandi, Smit, Kong, Baltussen, Peters and Kuipers2020). At relatively low ventilation flux, the injected gas expands slowly near orifices, leading to an increase in the hydrodynamic force exerted by the liquid crossflow. The hydrodynamic drag causes the air pocket to incline and elongate downstream, accompanied by the development of a local interfacial indentation. As the neck narrows, the increasing local curvature enhances the capillary pressure, eventually triggering neck collapse and pinch-off once it exceeds the local pressure difference across the interface. In contrast, figure 2(b) demonstrates the modulation effect introduced by the SHPo transverse grooves: a continuous air layer develops near the ventilation orifices and remains stable downstream, ultimately enveloping the entire surface of the plate.

Figure 2.

The presence of SHPo transverse grooves leads to a significant transition in the flow regime: (a) bubbly flow and (b) air-layer flow under identical flow conditions, $Re_D\approx 1.8 \times 10^4$ , $Q_j=0.2\,\text{SLPM}$ . The white arrows indicate the flow direction.

Figure 2. Long description

Panel A: A diagram showing bubbly flow with orifices of injector at the top left, labeled Pressure 1, Pressure 2, and Pressure 3 along the flow direction indicated by a white arrow. Panel B: A diagram showing air-layer flow with SHPo transverse grooves at the top, labeled Pressure 1, Pressure 2, and Pressure 3 along the flow direction indicated by a white arrow. The transition from bubbly flow to air-layer flow is highlighted.

3.2. Pressure characteristics of bubbly and air-layer flow

Pressure signals serve as a robust indicator of the underlying flow regime. Figure 3 illustrates the unsteady pressure signatures for the different characteristic flow conditions with and without gas injection, where instantaneous snapshots are synchronised with their respective temporal pressure signals and time–frequency representations. These spectra are generated using the continuous wavelet transform (CWT), which allows for the simultaneous characterisation of the signal in both the time and frequency domains. The wavelet coefficients were computed using a Morlet wavelet, ensuring an optimal balance between time and frequency resolution (Farge Reference Farge1992).

Figure 3.

Temporal evolution and time–frequency spectra of the flow pressure for various cases: (a) non-ventilated free stream ( $Re_D \approx 1.8 \times 10^4$ , $Q_j=0\,\text{SLPM}$ ); (b) bubbly flow over a smooth plate ( $Re_D \approx$ $1.8 \times 10^4$ , $Q_j=0.2\,\text{SLPM}$ ); and (c–e) flows modulated by SHPo transverse grooves in the (c) continuous air layer (CAL) ( $Re_D \approx 1.8 \times 10^4$ , $Q_j=0.2\,\text{SLPM}$ , $\delta ^*\approx0.067$ ), (d) partial air layer (PAL) ( $Re_D \approx 2.5 \times 10^4$ , $Q_j=0.2\,\text{SLPM}$ , $\delta ^*\approx0.067$ ), and (e) bubble shedding (BS) ( $Re_D \approx 3.4 \times 10^4$ , $Q_j=0.2\,\text{SLPM}$ , $\delta ^*\approx0.067$ ) regimes. Here, the horizontal and vertical axes have been normalised by $t^* = tU_\infty /D$ and $p^* = p/\rho _l U_\infty ^2$ . Spectra are spatially averaged, with the non-dimensional frequency given by $f^* = fD/U_\infty$ . The amplitude is derived from the standardised signal $(p^*-\overline {p^*})/\sigma ^*$ , where $\overline {p^*}$ and $\sigma ^*$ represent the mean and standard deviation of the dimensionless pressure.

Figure 3(a–c) presents the temporal pressure pulsations and the corresponding time–frequency spectra under identical free-stream conditions, facilitating a comparison between the non-ventilated and ventilated cases. Comparing figure 3(a–c), pronounced periodic pressure pulsations are observed in the non-ventilated and bubbly flow cases, and the associated pressure magnitudes are of similar order. In contrast, the air-layer regime results in a substantial attenuation of the pressure pulsations. Concurrently, time–frequency analysis reveals that both the amplitude and dominant frequency of the pressure pulsations are markedly suppressed by the injected gas. The emergence of low-amplitude, intermittent pressure pulsation suggests that interactions between discrete bubbles and the near-wall turbulence may disrupt large-scale coherent structures, a mechanism consistent with BDR as discussed by Ceccio (Reference Ceccio2010) and Murai (Reference Murai2014). In addition, a far more pronounced suppression of these pulsations is observed in the air-layer regime. Synthesising these observations of air-layer topology and pressure characteristics, it is evident that the SHPo transverse grooves effectively facilitate the establishment of a continuous and stable air layer.

Figure 4.

Air-layer formation and evolution induced by localised gas injection for the case at $\delta ^* \approx 0.067$ , $Re_D\approx 2 \times 10^4$ , $Q_j=0.2\,\text{SLPM}$ and $\phi _s=50\,\%$ , where $t^*=t_0$ denotes the onset time of gas injection in the Wenzel state.

Figure 4. Long description

Panel A: A diagram showing the Wenzel state where trapped air pockets are lost. Panel B: A diagram showing the Cassie-Baxter state where an air plastron is replenished and the air layer completely covers the groove. Panel C: A diagram showing the initial air-layer state where the air layer is forming. Panel D: A diagram showing the stable air-layer state.

4.1. Dynamic development of an air layer

This subsection presents the dynamic evolution of an air layer, which is essential for understanding the dynamical effects of the SHPo transverse grooves in guiding and promoting air-layer formation and development.

When the specimen was placed in the water tunnel, the air initially trapped within and above the grooves forms a thin air layer, which appeared as a silvery mirror. Subsequently, the free-stream velocity was adjusted to a specified value, and the trapped gas was partially lost when the grooves were subjected to strong external pressure fluctuations (Piao & Park Reference Piao and Park2015; Seo et al. Reference Seo, Garcia-Mayoral and Mani2015) and shear stress (Aljallis et al. Reference Aljallis, Sarshar, Datla, Sikka, Jones and Choi2013). The entire grooves gradually transitioned from Cassie–Baxter state (Cassie & Baxter Reference Cassie and Baxter1944) to Wenzel state (Wenzel Reference Wenzel1936), as illustrated in figure 4(a).

After that, the solenoid valve was opened to inject a predetermined volumetric gas flux into the water through the two ventilation orifices. The injected gas rapidly entered the grooves through the gas wicking effect, which has been demonstrated by Sung et al. (Reference Sung, Choi, Ha, Lee and Park2022). Under the effect of the local gas supply downstream of the orifices, the air layer preferentially advances in the streamwise direction. However, unlike the smooth-wall case, gas pockets farther away from the orifices spread rapidly in the spanwise direction within the transverse grooves but extend downstream slowly. At $t^*=t_0+70$ , the grooves have been completely covered by an air layer and extended outwards to form a partially developed air layer (figure 4 b). Subsequently, the air layer propagates downstream under sustained ventilation (figure 4 c). At the same time, air entrainment occurred continuously in the form of small bubbles at the air-layer closure. After a longer period of gas injection ( $t^*=t_0+2400$ ), portions of the spreading air layer evolved into a continuous and stable state characterised by a time-independent macroscopic interfacial morphology, as shown in figure 4(d). The grooves and the wall were observed to be gas-filled, exhibiting a sharp, transparent gas–liquid interface. The entire process of air-layer development is shown in supplementary movie 1 available at https://doi.org/10.1017/jfm.2026.12219. The air-layer interface was characterised by small-scale capillary waves in both the streamwise and transverse directions, consistent with the findings of Mäkiharju et al. (Reference Mäkiharju, Elbing, Wiggins, Schinasi, Vanden-Broeck, Perlin, Dowling and Ceccio2013). In contrast to the observations by Barbaca et al. (Reference Barbaca, Pearce, Ganesh, Ceccio and Brandner2019), no re-entrant jet was detected at any stage of air-layer development in the present experiments. The formation of this re-entrant jet is fundamentally governed by the dimensionless pressure coefficient difference, ${\Delta } C_{\kern-1.5pt p} = (p_{\mathit{ref}}-p_{\mathit{c}})/(0.5\rho _l U_\infty ^2)$ (Knapp Reference Knapp1955; Callenaere et al. Reference Callenaere, Franc, Michel and Riondet2001), where $p_{\mathit{ref}}$ and $p_{\mathit{c}}$ represent the reference free-stream pressure and the cavity pressure, respectively. While a definitive criterion for the onset of re-entrant jets at the cavity closure remains elusive, previous studies reporting such jets indicate a pressure coefficient difference of order $O(10^{-1})$ or greater (Le et al. Reference Le, Franc and Michel1993; Liu et al. Reference Liu, Wang, Shu, Hong, Fang and Hao2025). However, the time-averaged pressure coefficient difference was estimated to be $\Delta C_{\kern-1.5pt p} \sim O(10^{-2})$ in the present work. At the stable air-layer stage, the grooves remain completely covered by the gas phase. The influence of the localised SHPo grooves on the macroscopic gas dynamics within the air layer is negligible. Thus, $p_c$ is primarily dominated by the gas-phase flow condition. Applying the Bernoulli equation, the pressure coefficient difference is given by

(4.1) \begin{align} \Delta C_{\kern-1.5pt p} \sim \frac {\rho _g}{\rho _l} \left(\frac {U_g}{U_\infty }\right)^{\!2} , \end{align}

where $\rho _g$ is the gas density and $U_g$ is the velocity of the gas phase within the air layer. The minimal-flux ventilation ensures a low $U_g$ , which in turn accounts for the observed small pressure coefficient difference.

Figure 5.

Schematic of the Wenzel–Cassie transition within a groove unit: (a) cross-sectional view of injected gas growing laterally along the space between grooves; and (b) corresponding side view.

4.2. Capillary-induced anisotropic modulation effects of SHPo transverse grooves

According to the observations in § 4.1, the gas motion in the streamwise and spanwise directions is decoupled for analysis to establish an equivalent mechanical model describing the macroscopic plastron evolution within the grooves. The energy-based analysis is conducted to examine the gas spanwise and streamwise transport process, providing both qualitative and quantitative insights into the mechanism by which SHPo transverse grooves influence the evolution of the air layer. It should be noted that the leading edge of the gas layer is generally curved during its dynamic evolution over the grooved surface. To simplify the analysis, the actual spreading process is idealised as a sequence of incremental groove-size filling events in the streamwise direction. Figure 5 illustrates air pocket growth and the capillary force within a single groove unit. The magnitude of the resultant capillary force acting on the gas pocket in the spanwise direction can be expressed as

(4.2) \begin{align} F_{\gamma \parallel } = -3\gamma _{\mathit{LG}}\delta \cos \theta _{\mathit{rec}} -\gamma _{\mathit{LG}} \delta , \end{align}

where $\theta _{\mathit{rec}}$ is the receding contact angle and the subscript $\parallel$ denotes the direction parallel to the transverse grooves. Owing to the fact that surface roughness can trap air pockets on the superhydrophobic surfaces, the receding contact angle is very close to the static contact angle $\theta$ on the surfaces of the SHPo grooves, as shown later in figure 17, which is in good agreement with the measurements reported by Yu et al. (Reference Yu, McClelland, del Campo Melchor, Lee and Lee2023). Considering that the receding contact angle $\theta _{\mathit{rec}}$ is weakly dependent on the velocity of the contact line on the SHPo surfaces (Kim et al. Reference Kim, Kavehpour and Rothstein2015), equation (4.2) can be simplified with the static contact angle $\theta$ as

(4.3) \begin{align} F_{\gamma \parallel } \approx -\gamma _{\mathit{LG}}\delta (1+3\cos \theta ). \end{align}

Considering the pressure difference $\Delta P$ across the gas–liquid interface in the groove induced by the injected gas pocket, the corresponding pressure-induced force acting on the liquid–gas interface can be estimated as

(4.4) \begin{align} F_{P \parallel } \sim \Delta P \delta ^2 \sim \frac {1}{2}\rho _g U_j^2\delta ^2 , \end{align}

which can also motivate the spanwise spreading of the gas pocket. Here $\Delta P$ is assumed to scale with the gas dynamic pressure, i.e. $\Delta P= \rho _g U_j^2/2$ ; and $U_j$ denotes the characteristic gas velocity at the ventilation orifices and is estimated as $U_j=Q_j/2A_{in}$ , where $A_{in}$ is the area of a single orifice. Thus, the competition between the dynamic pressure and capillary forces can be estimated as

(4.5) \begin{align} We_{j} = \frac {F_{P \parallel }}{F_{\gamma \parallel }} \sim \frac {\rho _g U_j^2\delta }{2\gamma _{{LG}}|1+3\cos \theta |} . \end{align}

At $Q_j=0.2\,\text{SLPM}$ , $We_j$ is of the order of $O(10^{-1})$ , indicating that the dynamic pressure force remains smaller than the spanwise capillary force. The spreading of the gas pockets falls strictly within the capillary-dominated regime.

Consequently, the change in interfacial energy may be regarded as the work done by the capillary force in both the spanwise and streamwise directions, i.e.

(4.6) \begin{align} W_\parallel +W_\perp =-\Delta E . \end{align}

According to the spanwise force $F_{\gamma \parallel }$ , the work done by capillary forces in the spanwise direction is $W_\parallel = F_{\gamma \parallel }w$ .

The interfacial-energy change associated with the transition from Wenzel to Cassie–Baxter state of one complete groove unit is now considered to be

(4.7) \begin{align} \Delta E = \gamma _{\mathit{LG}}\Delta A_{\mathit{LG}}+ \gamma _{\mathit{SG}}\Delta A_{\mathit{SG}} +\gamma _{\mathit{SL}}\Delta A_{\mathit{SL}} , \end{align}

where $\gamma _{\mathit{LG}}$ , $\gamma _{\mathit{SG}}$ and $\gamma _{\mathit{SL}}$ are the surface tensions of the liquid–gas, solid–gas and solid–liquid interfaces, respectively.

Considering the solid fraction $\phi _s$ , the streamwise width of the solid ridge of a groove unit can be denoted as $\phi _s\delta [2(1-\phi _s)]^{-1}$ . After the plastron is restored, the curvature of the formed gas–liquid interface is neglected in the present model. According to Young’s equation $\gamma _{\mathit{SG}} - \gamma _{\mathit{SL}} = \gamma _{\mathit{LG}} \cos \theta$ , (4.7) reads

(4.8) \begin{align} {\Delta E = \gamma _{\mathit{LG}}(\Delta A_{\mathit{LG}}+\Delta A_{\mathit{SG}}\cos \theta ) \approx [1+(3-2\phi _s)\cos \theta ] \gamma _{\mathit{LG}}w\Delta x}, \end{align}

where the length of a groove unit in the streamwise direction is $\Delta x$ .

We assume that the local capillary barrier is encountered primarily while the interface traverses the solid ridge to reach the adjacent groove. Consequently, combining equations (4.3), (4.6) and (4.8), the equivalent resistance in the streamwise direction can be expressed as

(4.9) \begin{align} F_{\gamma \perp } = \dfrac {W_\perp }{\phi _s \Delta x} \approx -\gamma _{\mathit{LG}}w(1+\cos \theta ), \end{align}

where the subscript $\perp$ denotes the direction perpendicular to the transverse grooves.

According to the above analysis, the force exerted by the transverse grooves on the development of air pockets is inherently anisotropic, which manifests as a spanwise driving force that facilitates gas expansion along the grooves, while simultaneously providing a streamwise resistance that inhibits the downstream evolution of the air layer. It should be noted that the present analysis should be regarded as a phenomenological, quasi-static description of the air-layer spreading process. Owing to the combined effects of localised gas injection and the crossflow, the dynamic gas-transport process in the region immediately downstream of the injection orifices is difficult to model accurately. In contrast, over most of the grooved region located farther away from the injection orifices in the spanwise direction, the gas-transport process can be reasonably described by the present model. Moreover, the neglect of gas–liquid curvature within individual grooves and the effective contact-angle hysteresis due to contact-line motion may primarily modify the quantitative prefactor of equation (4.9). The effective spanwise and streamwise forces remain of the order of $\gamma _{\mathit{LG}} \delta$ and $\gamma _{\mathit{LG}} w$ , respectively.

The physical mechanisms governing the Wenzel–Cassie transition of grooves and subsequent air-layer formation can be elucidated in detail now. Firstly, the transverse grooves motivate the spanwise transport of injected gas. Considering the pressure outside the grooves, which suppresses the air pockets, it tends to spread more easily rather than escaping outwards from the grooves after the gas is injected into the grooves. Secondly, due to the streamwise resistance $F_{\gamma \perp } \thicksim \gamma _{\mathit{LG}} w$ , capillary force suppresses the shear-induced streamwise transport of the air layer and thereby promotes the accumulation of gas within the groove region. Taken together, these two effects demonstrate the strong modulation exerted by the SHPo transverse grooves on the effect of spreading and the formation of a continuous air layer.

5.1. Regime classification and pressure characteristics

Figure 6 shows the air-layer topologies for $\delta ^* \approx 0.067$ at $\textit{Re}_D \approx 1.8 \times 10^4$ , $2.5 \times 10^4$ and $3.4 \times 10^4$ , revealing three distinctly different regimes. At relatively low $\textit{Re}_D$ (figure 6 a,d), a stable air layer completely covers the surface of the specimen, hereafter referred to as the continuous air-layer (CAL) regime (i.e. $L_a\gt 10D$ , where $L_a$ is the length of the air layer, shown in supplementary movie 2). As $\textit{Re}_D$ increases, the air layer closes on the surface of the specimen and its length decreases dramatically. At the same time, pressure fluctuations at the gas–liquid interface excite more intense interfacial waves, accompanied by gas leakage occurring at the rear of the cavity, and a dynamic balance is reconstructed between the injected and entrained gas, such that the air-layer length remains approximately constant, as shown in figure 6(b,e). This regime is termed the partial air-layer (PAL) regime (i.e. $D\lt L_a\lt 10D$ , shown in supplementary movie 3). With a further increase in $\textit{Re}_D$ , only a localised air layer persists downstream of the orifices within the transverse grooves, while most of the injected gas is stripped off as bubbles and convected downstream, which is referred to as the bubble shedding (BS) regime (i.e. $L_a\lt D$ , shown in figure 6(c,f) and supplementary movie 4). As illustrated by figure 6 and supplementary movies 3 and 4, three-dimensional interfacial instabilities, manifested as capillary-wave disturbances, are observed in both PAL and BS regimes and are particularly pronounced in the PAL regime. Periodic gas entrainment characterises the downstream closure region in both the PAL and BS regimes. A distinguishing feature of the PAL regime is the localised pinch-off of large gas pockets, primarily occurring in the central region of the closure, with a spanwise-uniform air-layer length. In contrast, the air layer in the BS regime exhibits a characteristic V-shape due to the weakened modulating effects of SHPo transverse grooves, reflecting a highly heterogeneous spanwise topology.

Herein, the pressure characteristics of the three distinct air-layer regimes identified in the previous paragraph are compared in figure 3(c–e). It can be observed that the pressure pulsations intensify as the air layer becomes increasingly fragmented/unstable, and a more dispersed distribution of energy occurs in the flow field. At the same time, the pressure signals are significantly modified by the spatial positioning of the transducers relative to the air-layer coverage. Particularly in the PAL and BS regimes, the three transducers are located in fundamentally different zones of the air layer. By correlating the snapshots in figure 3 with the temporal pressure signals, it is evident that, in the PAL regime, the pressure signals of pressure no. 2 is positioned within the highly turbulent closure region, resulting in more intense pressure fluctuations. In contrast, in the BS regime, the presence of grooves enhances the mixing effect of the gas–liquid two phases in the flow field, resulting in a highly chaotic effect, especially in the vicinity of the downstream end of the grooves. Thus, all three transducers are immersed within the bubbly flow, and the amplitude of the pressure pulsations attenuates sharply with increasing downstream distance from the gas leakage source.

5.2. Evolution of air-layer topology versus $\textit{Re}_D$ and $\delta ^*$

During the air-layer regime transition, it is of interest to examine how the air-layer topology evolves with inflow conditions for various groove sizes, as this behaviour is directly relevant to the hydrodynamics of ships or underwater vehicles (Reichardt Reference Reichardt1945). Accordingly, in this subsection, experimental data fitting is employed to establish the relationships of the air-layer length and thickness with the flow conditions and groove sizes.

As $\textit{Re}_D$ gradually increases, the air-layer length and thickness adjust, thereby modifying ${\Delta } C_{\kern-1.5pt p}$ to accommodate variations in the external flow conditions and gradually allow the system to establish a new equilibrium state. The air-layer length and thickness were measured under conditions where the air layer remained in the stable state. Furthermore, the air-layer lengths corresponding to the CAL and PAL regimes at $Q_j=0.2\,\text{SLPM}$ are summarised in figure 7(a). A characteristic L-shaped dependence of the air-layer length on the Reynolds number is observed, which is similar to the variation of cavity length with the cavitation number in natural cavitation (Callenaere et al. Reference Callenaere, Franc, Michel and Riondet2001). The initial air-layer length is approximately 25 times the transverse groove length at low $\textit{Re}_D$ . This length decreases monotonically with increasing $\textit{Re}_D$ , with the most pronounced variation occurring in the CAL regime. As $\textit{Re}_D$ increases further, the rate of reduction in the air-layer length becomes significantly weaker. That is, $|k_1| \gt |k_2|$ , which may result from the closure of the air layer onto the surface of the specimen introducing a reduction of gas leakage rate according to Serebryakov et al. (Reference Serebryakov, Arndt and Dzielski2015). A linear decrease in the time-averaged thickness $th/D$ with $\textit{Re}_D$ is observed across all four cases, as shown in figure 7(b). The procedure for extracting the thickness of an air layer is detailed in Appendix C.

Figure 7.

Dependence of the normalised air-layer (a) length ( $L_a/D$ ) and (b) thickness ( $th/D$ ) on $Re_D$ . Air-layer length data in the BS regime are omitted, as the gas is confined within the grooves. The slopes $k_1$ and $k_2$ from linear fitting are $-3.4\times 10^{-3}$ and $-2.43\times 10^{-4}$ , respectively.

Figure 7. Long description

The image contains two scatter plots labeled as Panel A and Panel B. Panel A: A scatter plot shows the normalized air-layer length (L_a/D) on the vertical axis and the variable Re_D on the horizontal axis. Different symbols represent data for various diameters: 500 micrometers (squares), 1000 micrometers (circles), 1500 micrometers (triangles), and 2000 micrometers (inverted triangles). The plot includes two regimes, CAL and PAL, with dashed lines indicating different relationships. Panel B: Another scatter plot shows the normalized air-layer thickness (th/D) on the vertical axis and the variable Re_D on the horizontal axis. Similar symbols represent data for the same diameters as in Panel A. A dashed line indicates a linear relationship with the slope mentioned in the caption.

6.1. The scaling law on transition from CAL to PAL regime

Following Mäkiharju et al. (Reference Mäkiharju, Elbing, Wiggins, Schinasi, Vanden-Broeck, Perlin, Dowling and Ceccio2013), the intense gas entrainment at the cavity closure is intimately coupled with the interfacial fluctuation dynamics. Through adaptation in the air-layer topology, a new equilibrium between the gas injection and gas leakage rate could be established. As discussed in §§ 4 and 5, three-dimensional capillary waves were observed to travel along the gas–liquid interface. Here, we focus specifically on the streamwise interfacial fluctuations, which are more closely related to gas leakage at the tail. Specifically, these streamwise waves can trigger KH instability as they propagate, which significantly modulates the evolution of the air layer further downstream (Mäkiharju et al. Reference Mäkiharju, Elbing, Wiggins, Dowling, Perlin and Ceccio2010; Podbevšek et al. Reference Podbevšek, Petkovšek, Ohl and Dular2021).

To analyse the effects of interfacial wave characteristics on the CAL–PAL transition, the streamwise instantaneous interface fluctuations $\eta (x,t$ ) were analysed across the relevant velocity range for each groove size. A spatial Fourier transform was applied to these fluctuations to extract the time-averaged dominant wavelength. The detailed methodology and the results are provided in Appendix C. According to the space–time interfacial wave spectrum in figure 19 and quantitative data of the wavelength involved in the CAL and PAL regimes in figure 20, the dominant wavelengths were found to range from approximately 4.5 to 7.5 mm. As demonstrated by Falcon & Mordant (Reference Falcon and Mordant2022), the effect of gravitational stabilisation is rendered physically negligible in the present experiments, and surface tension serves as the dominant stabilising mechanism for the interface.

Clarifying the most relevant physical quantities for the CAL–PAL transition, a dimensional analysis is carried out in the following. The air-layer regimes are characterised by an equivalent air-layer length $L_a$ , the magnitude of which can be written as

(6.1) \begin{align} L_a = F(U_\infty ,Q_j,\delta ,\gamma _{\mathit{LG}},\lambda ,\rho _l,\rho _g,\mu _l,\mu _g), \end{align}

where $\lambda$ is the wavelength of the capillary waves and $\mu _g$ represents the viscosity of the gas.

Non-dimensionalizing equation (6.1), it can be rewritten as

(6.2) \begin{align} L_a/{D} = f(\textit{Re}_D,\overline {Q_j}, \lambda /\delta , We_\delta , \rho _l/\rho _g, \mu _l/\mu _g), \end{align}

where $\overline {Q_j}=Q_j/{U_\infty {D}^2}$ , $\rho _l/\rho _g$ and $\mu _l/\mu _g$ are kept constant in this work. Accordingly, the normalised air-layer length $L_a/{D}$ can be expressed as a function of three dimensionless parameters, namely

(6.3) \begin{align} L_a/{D} = f(\textit{Re}_D,\lambda ^*, We_\delta ) , \end{align}

where $\lambda /\delta$ is denoted by $\lambda ^*$ . Considering that the wavelength of the capillary wave is jointly modulated by the free-stream velocity and the groove sizes, $\lambda ^*$ may thus be regarded as a function of the Weber number ( $We_\delta =\rho _l U_\infty ^2 \delta /\gamma$ ). The quantitative relationship between the dominant wavelength and the Weber number is presented in figure 9, which reveals that $\lambda ^* \thicksim We_\delta ^{-0.9}$ . If we further assess the dependence of the wavelength on the groove size $\delta$ and the free-stream velocity $U_\infty$ , we obtain

(6.4) \begin{align} \lambda \thicksim U_\infty ^{-1.8} \delta ^{0.1}. \end{align}

Subsequently, the relationship between the interfacial wavelength and the free-stream velocity is derived based on the Kelvin–Helmholtz instability analysis. Assuming that the velocity of the injected gas under the air layer ( $U_g$ ) and free stream ( $U_\infty$ ) can be separated with small vorticity thickness, the dispersion relationship under small-amplitude waves, $\eta =|\eta |\,{\mathrm{exp}}(i(kx-\omega t))$ , is

(6.5) \begin{align} \omega = k \dfrac {\rho _l U_\infty + \rho _g U_g}{\rho _l + \rho _g} \pm \mathrm{i} \dfrac {k}{\rho _l + \rho _g} \sqrt {\rho _l \rho _g (U_\infty - U_g)^2 - (\rho _l + \rho _g) \gamma _{\mathit{LG}} k}, \end{align}

where gravity is neglected (Chandrasekhar Reference Chandrasekhar2013). The critical stability condition requires that the imaginary part of the dispersion relation, Im( $\omega$ ), should be negative, i.e. that

(6.6) \begin{align} k\gt k_c = \dfrac {\rho _l \rho _g}{\rho _l + \rho _g} \dfrac {(U_\infty - U_g)^2}{\gamma _{\mathit{LG}}}, \end{align}

where $k_c$ denotes the critical wavenumber. Considering $\rho _l \gg \rho _g$ and $U_\infty \gg U_g$ , equation (6.6) can be simplified to $k_c \thicksim \rho _g U_\infty ^2 /\gamma _{\mathit{LG}}$ , where $k_c=2\pi /\lambda _c$ , i.e. $\lambda _c\thicksim U_\infty ^{-2}$ , which is in close agreement with the experimental result (i.e. $\lambda \thicksim U_\infty ^{-1.8} \delta ^{0.1}$ ). This consistency further indicates that the interfacial wave characteristics of air layers are primarily governed by the external flow conditions, and the influence of the groove size on the interfacial wavelength is physically negligible.

The phenomenon indicates that, once a continuous air layer is established, size effects are effectively mitigated, with the air-layer length depending exclusively on $\textit{Re}_D$ . Thus, the horizontal dashed line in figure 8 provides a clear delineation of the transition from CAL to PAL regime. This stands in marked contrast to conventional SHPo surfaces, where size-dependent stability often poses a major challenge (Yu et al. Reference Yu, McClelland, del Campo Melchor, Lee and Lee2023; Jeon et al. Reference Jeon2025).

Figure 9.

Scaling of interfacial wavelength scaled in $\delta$ versus $We_{\delta }$ . The dashed line indicates a power-law fit with an $R^2$ value of 0.91.

6.2. The scaling law on transition from PAL to BS regime

As discussed in § 4.2, the SHPo transverse grooves play a crucial role in facilitating the formation of a continuous air layer by providing streamwise resistance to the gas phase. In the BS regime, however, the air layer is confined to a localised region downstream of the ventilation orifices. The air pocket releases as discrete bubbles instead of forming a coherent plastron on both sides of the grooves. This phenomenon can be equivalently interpreted as a failure of the streamwise resisting effect working on the air pocket under strong shear conditions. The streamwise force model of a representative air pocket is shown in figure 10, which can be represented as

(6.7) \begin{align} F_{\textit{streamwise}} = F_d-F_{\gamma \perp }, \end{align}

where the driving force $F_d$ and equivalent resistance $F_{\gamma \perp }$ can be defined as

(6.8a) \begin{align} F_d \thicksim \frac {1}{2}C_d \rho _l U_{\kern-1.3pt p} ^2 A_p, \end{align}

(6.8b) \begin{align} F_{\gamma \perp } \thicksim \gamma _{\mathit{LG}} w, \end{align}

where $U_{\kern-1.3pt p}$ denotes the local velocity in the vicinity of the air pocket, which can be defined as $U_{\kern-1.3pt p} = (2/\delta) \textstyle{\int} _{0}^{\delta /2}U(y)\,\text{d}y \approx 0.5U(\delta /2)$ .

Figure 10.

A schematic of the streamwise force balance acting on a representative air pocket within a transverse-groove unit, where the velocity $U(y)$ is a function of the wall-normal distance $y$ .

Assuming that the protruding part of the air pocket has a semicircular cross-section as shown in figure 8, then the projected frontal area facing the flow is $A_p= \delta w/2$ . The drag coefficient $C_d$ , which is a function of the Reynolds number scaled with the air-pocket dimension, is given, according to Clift et al. (Reference Clift, Grace and Weber2005), by

(6.9) \begin{align} {C_d \sim \textit{Re}_p^{-0.625}}, \end{align}

where $\textit{Re}_p = \rho _l U_{\kern-1.3pt p} \delta /2\mu _l \thicksim O(10^2-10^3)$ in our experiments. However, the correlation is generally derived for free-rising spherical bubbles under unbounded conditions. Therefore, the applicability of (6.9) to the present configuration and Reynolds-number range warrants rigorous verification. Additional numerical simulations were performed, and the results have successfully assessed the validity of the adopted drag correlation under the boundary conditions and Reynolds numbers in the present study. The numerical methodology and corresponding results are provided in Appendix D.

When the driving force $F_d$ counterbalances the equivalent resistance $F_{\gamma \perp }$ (i.e. $F_{\textit{streamwise}}=0$ ), the regime transition from PAL to BS regime is achieved. Combining equations (6.7)–(6.9), the critical local velocity $U_{\kern-1.3pt p}$ is found to follow the scaling law

(6.10) \begin{align} U_{\kern-1.3pt p} \thicksim \delta ^{-0.3}. \end{align}

Subsequently, this scaling law is extended to free-stream conditions in the current experimental framework. Considering the air pocket embedded in the turbulent boundary layer, the local velocity $U_{\kern-1.3pt p}$ is assumed to follow the classical $1/7$ power law relative to the free-stream velocity $U_\infty$ (Schlichting & Gersten Reference Schlichting and Gersten2016),

(6.11) \begin{align} U_{\kern-1.3pt p} \thicksim U_\infty \left({\frac {\delta }{\delta _{\textit{BL}}}}\right)^{1/7} , \end{align}

where the thickness of the turbulent boundary layer $\delta _{\textit{BL}}$ scales with the local Reynolds number as $\delta _{\textit{BL}}/x \thicksim \textit{Re}_x^{-1/5}$ , where $\textit{Re}_x=\rho _l U_\infty x/\mu _l$ and $x$ is the distance from the leading edge of the test section in the water tunnel. By substituting this scaling into equation (6.11) and combining it with the scaling law in (6.10), we can obtain

(6.12) \begin{align} \textit{Re}_D\thicksim \delta ^{*-0.4} .\end{align}

If this scaling law between $\textit{Re}_D$ and $\delta ^*$ is plotted on a phase diagram (see figure 8), it is found to be in good agreement with the experimental results, which proves that the PAL–BS transition exhibits a pronounced size effect.

The foregoing analysis in § 6 establishes the scaling laws governing the transitions between different multiphase flow regimes. It should be noted, however, that the intercepts of the dashed lines in figure 8 were determined by fitting the experimental data. Consequently, the critical transition Reynolds numbers in the vicinity of these boundaries may involve a degree of uncertainty due to the finite number of experimental data points.