2.1. Facility

The experiments were performed in the new multiphase flow tunnel (MPFT) at the Ship Hydromechanics laboratory of Delft University of Technology. The flow is driven by a Voith Inline Thruster (Voith, Heidenheim, Germany) resulting in low noise and vibrations. The operational range of the impeller leads to free-stream velocities $U_{\infty }$ = 2.13–13 m s^–1 and with the addition of an orifice plate downstream of the rim-driven thruster, a minimum $U_{\infty }$ = 0.5 m s^–1 can be reached. To facilitate air lubrication experiments, the MPFT features two kind of degassing mechanisms to prevent air circulation: a de-aeration tower and a large horizontal degasser (figure 1). In the de-aeration tower, the air–water mixture impacts a sloped plate that breaks air patches, allowing air to escape via a separation vessel, while water flows downstream. The horizontal degasser features a specially designed top lid that collects rising bubbles and directs them to the separation vessel. The former degassing mechanism is mainly effective for lower velocities ( ${\lt } 3$ m s^–1) and the latter for higher ones. Before entering the test section the flow is conditioned with a honeycomb and two contractions (figure 1) to ensure a uniform inflow and low turbulence intensity ( ${\lt}0.6\,\%$ as measured by Laser Doppler Anemometry (LDA), see Nikolaidou et al. Reference Nikolaidou, Laskari, van Terwisga and Poelma2026).

Figure 1.

Sketch of the multiphase flow tunnel. Arrows indicate the liquid flow direction. For an indication of the scale: the test section is 2.1 m long.

Figure 1. Long description

A diagram of a multiphase flow tunnel. The diagram includes a de-aeration tower, separation vessel, large contraction, small contraction, honeycomb, test section, differential pressure sensor, temperature sensor, large degasser, and inline thruster housing. Arrows indicate the liquid flow direction through these components. The test section is 2.1 meters long.

The test section is 2.136 m long and the cross-section varies from 0.3 $\times$ 0.3 ${\rm m}^2$ at the inlet to $0.3 \times 0.32$ ${\rm m}^2$ at the outlet due to a sloping bottom. The sloping bottom was designed to compensate for the TBL growth for 2 m s^–1. The test section is tightly sealed with a lid, and it can be pressurised up to 2 bar overpressure. Transparent windows are present on the sides of the tunnel and the bottom, while the top has limited access with 4 viewing windows. The water level in the test section can be monitored with a transparent tube and is kept constant throughout the experiments.

2.2. Test plates

Two different test plates were used during the experiments, one for single-phase flow and one for air lubrication measurements. The former was a simple flat plate without inserts, used to measure the TBL flow for validation of our measurement system (see § 3). The air lubrication plate was more complex and was subsequently used for multiphase flow measurements (see §§ 4 and 5). The test plates were positioned flush with the ceiling of the test section and were submerged in water (depth $d=0.3$ m). Both were made out of transparent polycarbonate, with length $L=1.95$ m and width $B=0.295$ m. The thickness was 11 and 36 mm (in 3 layers of 12 mm) for the single-phase and air lubrication plates, respectively. A zig-zag strip of 0.8 mm thickness was placed 10 cm upstream of the leading edge of the test plates in the tunnel contraction to ensure a TBL.

Figure 2.

Sketch of the air injector. (a) Three-dimensional view from the top and (b) side view. Liquid flow is from right to left.

In the case of the air lubrication plate, air was introduced vertically from the top layer through four manifolds equally spaced along the plate’s width (see figure 2). It was then dispersed in an air box within the second layer, whose downstream end was fitted with a porous material and flow conditioner, ensuring homogeneous distribution. At the last layer, air was injected vertically into the liquid TBL via a $t=4$ mm wide slot-type injector (same as in our previous work Nikolaidou et al. (Reference Nikolaidou, Laskari, van and Poelma2024)), spanning the width of the plate and located 140 mm downstream of its leading edge. The last layer of the plate was also fitted with side fences along its length (2 mm thick and 49 mm in height) to restrict air escaping from the sides (see figure 3). Within the plate’s middle layer an in-house made LED panel was placed downstream of the injection slot (in a watertight housing) to illuminate the entire plate length and width downstream of the injector, as required for the imaging system (see § 2.4).

The air injection was controlled via a mass flow controller (Bronkhorst EL-FLOW Select series) located several metres upstream of the injector location along with an air filter. The flow meter was factory calibrated for a nominal air flow range of 4–500 $\textrm{l}$ min^–1 with an accuracy of $\pm$ 0.5 % error in the reading value and $\pm$ 0.1 % of the full range while the repeatability is 0.2 % of the measured value.

2.3. Drag force measurement system

Both test plates were fitted to a drag force measurement system during their respective uses. This custom-made force balance was designed and built to assess the total friction force of the lower surface of the test plate. The design was based on previously established ones (Zverkhovskyi Reference Zverkhovskyi2014). A load cell (type BM6A by Zemic) with a maximum capacity of 6 kg ( $60\, \textrm{N}$ ) was used. The cell was made from IP68 stainless steel and was suitable for underwater applications both in fresh and salt water.

Figure 3.

Sketch of drag force measurement system attached to the air lubrication plate.

The test plate was suspended by four vertical rods, with their top end fixed to the tunnel sidewalls and their bottom attached to longitudinal bars supporting the plate over its length (see figure 3). This allowed plate motion only in the flow direction with alignment screws enabling precise levelling (angle of attack $\pm 1^\circ$ ). The streamwise displacement of the plate was measured by the load cell, mounted 370 mm from its leading edge. The plate’s width and length were adjusted to maintain a 1.5 mm gap between the plate and the tunnel walls to allow movement and avoid interference. For the air lubrication plate, apart from the side fences, a plastic film at the trailing edge was also used to prevent air from escaping through that gap.

The force balance had to be calibrated in situ. In operating conditions the test section was closed from the top (see figure 1) and filled with water to the brim, resulting in the sensor being fully submerged in water. The calibration was performed without the lid, but with water on top of the plate and the sensor being underwater. Once the lid was closed, the experiments took place and in the end the calibration was repeated to ensure that nothing changed in the system in between. The calibration was performed using standard forces (weights) and a pulley arrangement in accordance to the guidelines given by the ITTC (2017). The calibration involved 16 steps in total of loading and unloading conditions raging from zero to 55 N with a preload as well. A linear relation was found between the output signal in volts and the force in Newtons, with a small standard error ( $\sigma = 6.9 \times 10^{-3}$ $\rm V$ , equivalent to 0.0018 $\rm N$ ).

2.4. Imaging system

For the air lubrication measurements, two different imaging systems were used, to accommodate two different research objectives. First, we wanted to capture the three-dimensional air phase topology synchronously with the measured drag force, in order to gain insights into the different DR mechanisms. This necessitated multiple plane views ( $x{-}y$ , $x{-}z$ ; see figure 4 a for the coordinate system used), high resolution and thus smaller fields of view (FOVs). Second, we wanted to associate the measured drag force with the total air covered plate area, and probe their correlation. This required large FOVs on the $x{-}z$ plane (see figure 4 b).

Table 1.

Camera set-up 1 details.

Table 1. Long description

A table summarizing camera set-up 1 details for air lubrication measurements. The table has four rows and five columns. Column headers are Camera no., FOV (mm x mm), Focal length (mm), f #, and Resolution (px mm^-1). Row 1: Camera no. 1, FOV 740 x 740, Focal length 24, f # 5.6, Resolution 2.75. Row 2: Camera no. 2, FOV 110 x 110, Focal length 105, f # 5.6, Resolution 18.48. Row 3: Camera no. 3, FOV 70 x 70, Focal length 105, f # 5.6, Resolution 30.16.

Figure 4.

Imaging systems used for the multiphase flow campaign. (a) Camera set-up 1. (b) Camera set-up 2.

Figure 4. Long description

Panel A: A diagram of camera set-up 1. The setup includes three cameras labeled Camera 1, Camera 2, and Camera 3. An air-injection system is positioned above a flat plate. The side field of view (FOV) and LED lighting are indicated. The flow direction is shown moving from right to left. The coordinate axes x, y, and z are marked. Panel B: A diagram of camera set-up 2. Similar to set-up 1, it includes three cameras labeled Camera 1, Camera 2, and Camera 3. An air-injection system is positioned above a flat plate. The flow direction is shown moving from right to left. The coordinate axes x, y, and z are marked.

Three Imager Pro X 4M cameras (10-bit, LaVision) with a pixel size of 7.4 $\unicode{x03BC} {\rm m}$ and a resolution of 2048 $\times$ 2048 pixels were used for both set-ups in different configurations and with different lenses, as described below. For wall-parallel imaging planes, illumination was provided by the aforementioned LED panel built within the test plate, while wall-normal imaging was back lit by another LED panel resting on the tunnel sidewall (see figures 2 and 4 a, respectively). Davis 10.2 (LaVision GmbH) was used for data acquisition.

For the first set-up, camera 1 was positioned in a down-up configuration imaging the $x{-}z$ plane downstream of the injection slot. This camera provided information on the global air phase characteristics. Camera 2 captured a smaller FOV of the same plane with an increased spatial resolution. Images from this camera were used for estimating smaller features (e.g. bubble sizes). Lastly, a third camera (camera 3) capturing the $x{-}y$ plane was used to estimate the bubbly/air layer thickness. This camera was placed in the middle of the test section, except for some tests where it was traversed close to the air injector. All three cameras were synchronously capturing independent snapshots with an acquisition frequency of 0.8–7.4 Hz depending on the $U_{\infty }$ . For this campaign $U_{\infty }$ varied from 0.5 to 5.5 m s^–1, and $Q_{\textit{air}}$ from 5 to 400 l min^–1. Details of this camera set-up are given in table 1.

For the second set-up, the same three Imager Pro X 4M cameras were used, but in a different configuration. They were positioned below the test section in a down-up configuration to collectively capture the total $x{-}z$ extent of the test plate downstream of the injection slot. A small overlap between the three FOVs was present to allow stitching. The acquisition frequency varied from 1.11 to 1.94 Hz depending on $U_{\infty }$ , to ensure statistically independent snapshots, whose total number varied from 1000 to 1500. In this campaign $U_{\infty }$ was 2 m s^–1 and $Q_{\textit{air}}$ varied from 5 to 140 l min^–1. The exposure time was set to 200 $\unicode{x03BC} {\rm s}$ to achieve less than one pixel motion blur. Details of this camera set-up are given in table 2.

Table 2.

Camera set-up 2 details.

Table 2. Long description

A table summarizing camera set-up 2 details for three cameras. The table has four columns and three rows. Column headers are Camera no., FOV (mm x mm), Focal length (mm), f#, and Resolution (px mm^-1). Row 1: Camera no. 1, FOV 510 x 510, Focal length 35, f# 4, Resolution 4. Row 2: Camera no. 2, FOV 490 x 490, Focal length 35, f# 4, Resolution 4.22. Row 3: Camera no. 3, FOV 780 x 780, Focal length 24, f# 4, Resolution 2.61.

2.5. Experimental program and procedure

Two experimental campaigns were performed. First, single-phase flow measurements of the drag force acting on the flat plate were performed for a free-stream velocity range of 2–7 m s^–1. These included drag force measurements of the flat plate without imaging. Secondly, single and multiphase flow measurements of the drag force acting on the air lubrication plate were performed with synchronous imaging of the air phase. For this campaign the two imaging systems described above were used. First, a short campaign was performed using camera set-up 2. Secondly, measurements using camera set-up 1 were performed. A summary of the experimental conditions for the latter is given in table 3.

Table 3.

Experimental parameters of the multiphase flow campaign using camera set-up 1.

Table 3. Long description

The table presents experimental parameters for a multiphase flow campaign using camera set-up 1. It has five rows and five columns. The columns are labeled Parameters, Definition, Value, and Units. The parameters listed are Free-stream velocity, Air flow rate, Reynolds number, Froude-depth number, and Non-dimensional air flux. The values and units for each parameter are provided. For example, Free-stream velocity ranges from 0.5 to 7 meters per second, Air flow rate ranges from 5 to 450 liters per minute, Reynolds number ranges from 1 to 14 times 10 to the power of 6, Froude-depth number ranges from 0.29 to 4, and Non-dimensional air flux ranges from 0.012 to 1.4.

For a given free-stream velocity, the mean drag force was recorded at 100 Hz and averaged over 10 minutes, which was sufficient for convergence. Simultaneously, the differential pressure across the contraction and the water temperature (as shown in figure 1) were recorded and averaged. The differential pressure was used to estimate $U_{\infty }$ , with a correction applied, as set by earlier LDA measurements (Nikolaidou et al. Reference Nikolaidou, Laskari, van Terwisga and Poelma2026). The signals were continuously recorded via LabVIEW. For the multiphase cases, measurements were first performed without air followed by those with air, the mass flow rate of which was also recorded and averaged.

3.1. Flat plate TBL characterisation

The flat plate TBL thickness was measured with LDA (Nikolaidou et al. Reference Nikolaidou, Laskari, van Terwisga and Poelma2026) in three streamwise locations along the plate and multiple velocities (see table 4 for one $U_{\infty }$ ). Due to the sloping bottom of the test section, the free-stream velocity was found to slightly decrease at the most downstream position ( $\approx$ 2 %). Due to the position of the flat plate at the tunnel’s ceiling, the TBL was found to originate approximately 30 cm prior to the plate’s leading edge and with a growth that was not strictly canonical.

Table 4.

Experimental parameters of the single-phase flow campaign.

Table 4. Long description

A table with four columns and three rows. The columns are labeled x (m), Re_x x 10^6, delta_99 (mm), U_infinity (m s^-1), and nu/u_tau (micrometers). The rows present the following data: Row 1: x (m) 0.1, Re_x x 10^6 2.4, delta_99 (mm) 16, U_infinity (m s^-1) 5.08, nu/u_tau (micrometers) 4.38. Row 2: x (m) 0.975, Re_x x 10^6 7.6, delta_99 (mm) 51, U_infinity (m s^-1) 5.06, nu/u_tau (micrometers) 4.77. Row 3: x (m) 1.85, Re_x x 10^6 12.8, delta_99 (mm) 77, U_infinity (m s^-1) 4.98, nu/u_tau (micrometers) 5.19.

3.2. Drag force measurement validation

In order to validate the drag force measurement system, measurements of the drag force acting on both the flat plate and the air lubrication plate were performed over a large range of free-stream velocities. The total drag force coefficient was calculated as

(3.1) \begin{equation} C_{D}=\frac {2\textrm{D}_{no air}}{\rho U_{\infty }^2A} ,\end{equation}

where $\rho$ is the calculated water density based on the measured temperature, $D_{no air}$ is the measured drag force acting on the flat plate, $U_{\infty }$ is the free-stream velocity and $A$ is the test plate (bottom) surface area.

Figure 5.

Flat plate total skin friction coefficient of the current measurements along with friction lines from the literature (ITTC 1957; Schlichting Reference Schlichting1979; Grigson Reference Grigson1999; Katsui Reference Katsui2005). Black markers are from measurements with a flat plate. Different marker shapes are from different measurement days. Grey markers are from measurements with the air lubrication plate. Error bars represent the bias and precision errors following ITTC (2002).

Figure 5. Long description

A line graph with the horizontal axis labeled as Reynolds number (R e L) ranging from 3 times 10 to the power of 6 to 14 times 10 to the power of 6 and the vertical axis labeled as drag coefficient (C D) ranging from 2.6 times 10 to the power of negative 3 to 4.0 times 10 to the power of negative 3. The graph includes multiple data lines representing different friction models: ITTC 1957, Prandtl-Schlichting, Katsui, and Grigson. Black markers represent measurements with a flat plate, while grey markers represent measurements with an air lubrication plate. Error bars indicate bias and precision errors. The data points show a general trend of decreasing drag coefficient with increasing Reynolds number.

The flat plate measurements of $C_{D}$ are compared with reference friction lines from the literature (figure 5). More specifically, ITTC’s (1957) and Prandtl-Schlichting’s (Reference Schlichting1979) semi-empirical formulas are plotted. Following Bose et al. (Reference Bose, Insel, Anzbock, Hwangbo, Mewis, Steen, Toki and Zhu2005), Grigson’s (Reference Grigson1999) and Katsui’s (Reference Katsui2005) lines are also obtained through numerical integration of the local friction coefficient in the boundary layer. The Reynolds number is defined as $Re_{L}=U_{\infty }L/\nu$ . Use of the aforementioned TBL origin did not significantly alter the results presented here and as such this correction was not applied. The trend is consistent, and the results show good overall agreement. The experiments appear to be biased towards higher values compared with all the theoretical friction lines. Tests performed without a lid over the test section revealed systematically lower values of the drag coefficient (approximately 5 %–6 %) hinting at possible added resistance from water motion above the plate (supported also by visual observations). Since we are interested in changes in drag rather than absolute values, however, further investigation of such effects was considered out of the scope of the present study. Negligible differences were also found between the drag coefficients of the two test plates (with area differences due to the fences and sealing strip fitted on the air lubrication plate taken into account); thus, any potential form drag effects due to the different geometry of the air lubrication plate were marginal.

An uncertainty analysis of $C_{D}$ was performed following the recommendations by ITTC (2002). For both test plates, the largest source of uncertainty is the bias error from the calibration curve fit, due to the sensor’s large force range. A small precision error is found, as seen in the repeatability of flat plate measurements performed on different days, shown with different (black) markers in figure 5. The total uncertainty ranged approximately from 3 % to 5 %. The air lubrication plate has slightly higher uncertainty due to additional bias from the manufacturer error (material tolerances), caused by the system’s complexity. More details on the experimental uncertainty are given in Appendix A.

Following the aforementioned baseline checks in absence of air (injection), results of multiphase flow conditions are discussed next.

4.1. Baseline drag reduction and air regime topology

We here first introduce the general air phase regime topology, along with the corresponding drag forces. A typical DR curve obtained during the experiments can be seen in figure 6.

Figure 6.

Drag reduction with increasing air flow rate for a nominal free-stream velocity of 2 m s^–1 ( $\textit{Fr}_d=1.24$ ). The vertical dashed line demarcates the transition to the ALR at $Q_{\textit{air}}=Q_{\textit{crit}}$ . Right inset: side view ( $x{-}y$ plane) of the air layer at $Q_{\textit{air}}=140$ l min^–1. Left inset: side view ( $x{-}y$ plane) of the BR at $Q_{\textit{air}}=30$  l min^–1.

In what follows, DR is calculated with

(4.1) \begin{equation} \textrm{DR} = c \times \frac {D_{no air}-D_{\textit{air}}}{D_{no air}} \end{equation}

where $D_{no air}$ in the drag force of the air lubrication plate without air ( $Q_{\textit{air}}=0$ ), $D_{\textit{air}}$ is the drag force of the test plate with air ( $Q=Q_{\textit{air}}$ ) and $c$ is a constant. The constant $c$ is introduced because $D_{\textit{air}}$ is the force experienced by the test plate in its entirety (see figure 3). In reality, only a part of the test plate can be lubricated (downstream of the injector and up to a height equal to the air layer thickness on the side fences). Using the measured maximum air thickness (see § 4.2.2) and assuming a linear increase in $\text{DR}$ with the covered test plate area, we obtain $c=1.34{-}1.35$ . Although introducing $c$ adds an additional bias error to $D_{\textit{air}}$ , this correction is applied since we are interested in DR over the area that can be covered by air: downstream of the air injector and along a part of the side fences area. This also enables us to compare our results with other studies where typically local shear sensors are used to estimate DR. These sensors only consider the local effect of air by definition (Elbing et al. Reference Elbing, Winkel, Lay, Ceccio, Dowling and Perlin2008; Mäkiharju et al. Reference Mäkiharju, Elbing, Wiggins, Schinasi, Vanden-Broeck, Perlin, Dowling and Ceccio2013).

Figure 7.

Characteristic images of the multiphase flow topology: (a) BR ( $Q_{\textit{air}}=10$ l min^–1), (b) TALR ( $Q_{\textit{air}}=70$ l min^–1) and (c) ALR ( $Q_{\textit{air}}=140$ l min^–1), for $U_{\infty }=2$ m s^–1. Air flow rate increases from top to bottom. Black rectangle indicates the injector slot location, while the white region could not be imaged.

For small air flow rates, a BR is observed (figure 6), where dispersed bubbles are present in the flow (figure 7 a). A thin liquid film is evident directly beneath the flat plate (Murai et al. Reference Murai, Fukuda, Oishi, Kodama and Yamamoto2007; Park et al. Reference Park, Tasaka, Oishi and Murai2015; Yoon et al. Reference Yoon, Park, Tasaka and Murai2025), allowing bubbles to slide along it (see left side inset in figure 6). In our configuration, bubbles are not formed due to the injector geometry (e.g. orifice plate or porous layer as used in Madavan et al. Reference Madavan, Deutsch and Merkle1985 and Sanders et al. Reference Sanders, Winkel, Dowling, Perlin and Ceccio2006), but they are formed at the injection slot due to the impact of the air flow with the incoming TBL, which causes the air to break into bubbles. A similar mechanism was presented by Shen et al. (Reference Shen, Ceccio and Perlin2006) and Park et al. (Reference Park, Tasaka, Oishi and Murai2015). Interestingly enough, from $Q_{\textit{air}}=0$ to $Q_{\textit{air}} = 20$ l min^–1 a drag increase ( $\text{DR}\lt 0$ ) is measured, which peaks at $Q_{\textit{air}} = 10$ l min^–1. Drag increase cases are rarely reported in the literature. The drag increase changes to a decrease by further increasing $Q_{\textit{air}}$ . This drag behaviour within the BR will be addressed in detail in § 4.2.1.

By further increasing the air flow rate, the bubbles coalesce and air patches are present in the flow (figure 7 b), as also reported in Elbing et al. (Reference Elbing, Winkel, Lay, Ceccio, Dowling and Perlin2008) and Nikolaidou et al. (Reference Nikolaidou, Laskari, van and Poelma2024). While drag is further decreased in this transitional regime, it happens gradually, unlike the steep change for $\text{DR}\gt 20\,\%$ , previously reported by Elbing et al. (Reference Elbing, Winkel, Lay, Ceccio, Dowling and Perlin2008) for a similar configuration. It is therefore hard to demarcate the bubbly and transitional regimes through the corresponding DR curve. Nikolaidou et al. (Reference Nikolaidou, Laskari, van and Poelma2024) found a steep increase in non-wetted area in this region. The availability of synchronous force measurements and imaging data in the current experiments enables the comparison of the two. This will be further elaborated in § 4.1.2.

For $Q_{\textit{air}}=Q_{\textit{crit}}$ , the individual air patches coalesce and an air layer (figure 7 c) of a certain thickness is formed (see right side inset in figure 6). This jump in morphology is not gradual but sharp (here observed within the 5 l min^–1 resolution of the air flow). It is correlated with ${\approx} 60\,\%$ DR and a clear change in the slope of the $\text{DR}-Q_{\textit{air}}$ curve. On average, the air layer de-wets the top wall across the total length of the test section except for the region close to the air injector, where wetted patches persist. As such, the surface wettability is significantly different from the bubbly/transitional regime where a thin liquid film was always present in between the flat plate and the bubbles/air patches. The air layer extends beyond the FOV of figure 7(c), and the closure can thus not be observed in this image. The integrity of the air layer is disturbed by various factors observed during the experiments. This will be further elaborated in § 4.2.2. Due to the supercriticality of the flow, the air layer is unbounded (Matveev Reference Matveev2003), meaning that the closure of this air layer extends outside of the test section length (until somewhere in the diffusor). In that sense the morphology of the air layer is different from air cavity cases formed behind a backward-facing step or a cavitator (Zverkhovskyi Reference Zverkhovskyi2014) or in the case of subcritical conditions ( $\textit{Fr}_d\lt 1$ ) (Nikolaidou et al. Reference Nikolaidou, Laskari, van and Poelma2024). The latter will be further explored in § 5.

4.1.1. Hysteresis effects and repeatability

For all the experiments performed, the air flow rate was increased from $Q_{\textit{air}}=0$ to $Q_{\textit{air}}=Q_{max}$ and all three regimes (bubbly, transitional and an air layer) were observed. In order to investigate any hysteresis effects the opposite sequence was also tested: starting from the highest air flow rate where an ALR was present, drag was measured as air flow rate was decreased to zero. Both scenarios can be seen in figure 8. The error bars depict the experimental uncertainty due to the bias error. Repeated measurements for each scenario are shown separately.

Figure 8.

Drag reduction measurements in the case of increasing air flow rate from (BR to ALR) and decreasing air flow rate (ALR to BR) for $U_{\infty }=3$ m s^–1 ( $\textit{Fr}_d=1.75$ ). Repeat measurements are also shown (without error bars).

In the BR ( $Q_{air}\lt 60$ l min^–1), differences fall within experimental uncertainty. In the transitional regime (and especially between 100 and 120 l min^–1), larger differences arise, exceeding uncertainty, but repeated measurements also show high variability. This suggests that the nature of the transitional regime itself drives $\text{DR}$ variations rather than hysteresis effects. The importance of repeated measurements is highlighted here. Conversely, the ALR onset is consistent for all measurements, with DR within experimental uncertainty.

Based on the above, we have demonstrated that there is no difference between the air flux required for creation and maintenance of the air layer, which is the desired regime for DR. That means that the supplied air flow rate should always be at least equal to the critical one ( $Q_{\textit{air}}=Q_{\textit{crit}}$ ) to be able to maintain the air layer. Lowering the air flow rate would mean that the air layer would break and drag would be increased. That is one of the main disadvantages of this air injection strategy compared with partial cavities reported by (among others) Arndt et al. (Reference Arndt, Hambleton, Kawakami and Amromin2009), Makiharju et al. (Reference Makiharju, Elbing and Perlin2010), Zverkhovskyi (Reference Zverkhovskyi2014) and Gawandalkar et al. (Reference Gawandalkar, Lucido, Jain, Poelma, Ceccio and Ganesh2025), who measured a difference in formation and maintenance air flow rates.

4.1.2. Non-wetted area and drag reduction correlation

Next, for the same DR results that have been presented in figure 6, the total non-wetted area coverage $A_{nw}$ downstream of the air injector is calculated (see table 2 and § 2 for details on the experimental set-up). To the best of the authors’ knowledge, this correlation has not been explored in this context before, only qualitative discussions have linked the percentage of non-wetted area with DR.

For each $Q_{\textit{air}}$ ranging from 5 to 100 l min^–1, bubbly/air layer coverage was quantified by bubble segmentation using Artificial Intelligence (AI). More specifically, a zero-shot Segment Anything Model (SAM) was implemented as an extension of the flow segmentation framework and modified for bubbly flow images (Khojasteh et al. Reference Khojasteh, van de Water and Westerweel2024). Although SAM can be fine tuned for bubbly flows (Khojasteh et al. Reference Khojasteh, van de Water and Westerweel2024) and related two-phase flow studies have reported retraining the model for complex interfaces (Küçük et al. Reference Küçük, Della Santina and Laskari2025), the zero-shot configuration in the present work provided sufficient performance to proceed. For each image, the automatic mask generator sampled a uniform grid of point prompts to generate candidate bubble masks; candidates were filtered using the model predicted Intersection over Union stability score criterion, and box-based non-maximum suppression, with optional small region cleanup via minimum mask region area (Kirillov et al. Reference Kirillov2023). To isolate dispersed bubbles and ignore the large connected patches that appear at higher air flow rates ( $\geq 70$ l min^–1), masks were post processed by projected pixel area $A$ and only masks within a prescribed range were used. For higher air flow rates ( ${\gt } 80$ l min^–1), where bubble patches became large and image contrast decreased, the automatic segmentation was refined manually using an interactive add/remove step with the SAM predictor: spurious regions were removed by selecting masks, and missing/partial bubbles were added using point prompts and, when needed, bounding box prompts to trim oversized predictions prior to analysis (Kirillov et al. Reference Kirillov2023). Following the calculation of $A_{nw}$ of all the images, an average value is calculated.

Figure 9.

Non-wetted area, calculated from down-up images versus DR measurements for $U_{\infty }=2$ m s^–1 ( $\textit{Fr}_d=1.24$ ). Air flow rate increases from left (5 l min^–1) to right (100 l min^–1). See figure 6 for the air flow rates specification. Dashed line indicates $\textrm{DR} = A_{nw}$ for reference.

Results show that the non-wetted area coverage is not equal to the resultant DR in any of the three air phase regimes. On the contrary, the $\text{DR}$ lags behind the $A_{nw}$ . Overall, the $\text{DR}-A_{nw}$ trend is similar with the $\text{DR}-Q_{\textit{air}}$ one (see figures 9 and 6) for the BR and TALR, while a different trend is seen for the ALR. Although a correlation can be seen between $\text{DR}$ and $A_{nw}$ in each regime separately, no universal correlation exists.

Starting with the BR, it can be seen that, for low $Q_{\textit{air}}$ , increasing $A_{nw}$ (via increasing $Q_{\textit{air}}$ ) results in a drag increase that persists up to a 35 % coverage of the flat plate. Further increasing $A_{nw}$ up to 53 % results in a low DR of around 7 %. It must be noted that, within the BR, the reported $A_{nw}$ does not necessarily represent the true non-wetted wall area. Instead, it is determined from the projected bubble coverage viewed from below. As illustrated in the left inset of figure 6, the wall contact area depends on the bubble shape: more spherical bubbles have a smaller contact area with the wall than flatter bubbles, and some bubbles may not remain in continuous contact with the wall. Consequently, the actual non-wetted area within the BR is expected to be smaller than the values reported here. However, this effect is not expected to alter the observed trend in figure 9.

By further increasing $A_{nw}$ within TALR, $\text{DR}$ increases at a higher rate. As a result, the lag between the parameters is reduced, and a better correlation is seen. In this regime, bubbles/air patches are ‘flatter’ and most likely the calculated $A_{nw}$ is closer to the actual amount of air de-wetting the wall. However, the lag remains considerable: a 75 % non-wetted area corresponds to merely 32 % $\text{DR}$ . Comparing the last two points within the TALR, a big increase in $A_{nw}$ is calculated (17 %). However, this increase is not proportionally translated to a $\text{DR}$ increase (6 %).

A clear jump in $\text{DR}$ is observed when further increasing $A_{nw}$ , which coincides with the onset of ALR ( $Q_{\textit{air}}=Q_{\textit{crit}}$ ): a moderate increase in $A_{nw}$ (4.5 %) results in a large DR increase (22 %) owing to the TALR air patches coalescing into an air layer. At this point, the $A_{nw}$ is already 92 % with only a few wetted patches close to the air injector (see figure 18, discussed in detail later). Interestingly, an even steeper increase of $\text{DR}$ to 69 % is observed when $A_{nw}$ reaches 97 %. Finally, a maximum DR of 77 % (marked by $\times$ in figure 9) is measured for an almost 100 % air coverage. Within the ALR, for a further increase of $Q_{\textit{air}}$ above $Q_{\textit{crit}}$ , the wetted patches near the wall never cease to exist but the air layer gets thicker, which could possibly contribute to the DR increase (figure 17 b). In the next section, the reasons for this DR increase within $\textrm{ALR}$ will be discussed.

From the above, it can be concluded that the non-wetted area coverage alone is not enough to characterise the DR regimes or provide clear insights regarding the governing mechanisms. As a result, drag force measurements and multi-plane images of the air phase are needed for each condition.

4.2. Effect of free-stream velocity on drag reduction curves

Figure 10.

Drag reduction curves, for different $U_{\infty }$ . The dashed lines demarcate the regime transitions.

The effect of $U_{\infty }$ on DR is systematically examined across the different air phase regimes. Figure 10 presents DR as defined in (4.1), plotted against air flow rate $Q_{\textit{air}}$ for varying $U_{\infty }$ . For clarity, repeated measurements are omitted, although good repeatability is achieved (see Appendix B). Across all $U_{\infty }$ , the general trend is consistent: a BR, TALR and ALR emerge for increasing $Q_{\textit{air}}$ with similar morphological features.

In the current experiments, the effect of $U_{\infty }$ is different for low ( $Q_{\textit{air}}\lt 40$ l min^–1) and higher air flow rates: for the former, increasing $U_{\infty }$ results in a monotonic increase of the DR (when keeping $Q_{\textit{air}}$ constant), while the opposite is true for higher air flow rates ( $Q_{\textit{air}}\gt 40$ l min^–1). This enables a distinction between the bubbly and transitional regimes, which is not possible when analysing a single $U_{\infty }$ curve. Further investigation of the physical meaning of this ‘pivot point’ (here at $\text{DR}=8\,\%$ ), potentially via non-dimensionalising $Q_{\textit{air}}$ in figure 10 could provide a more formal distinction between the regimes. The transition from TALR to ALR on the other hand is defined as the first occurrence of $\textrm{DR} = 60\,\%$ , coinciding with a clear reduction in the $\text{DR}-Q_{\textit{air}}$ slope and supported by flow visualisation (for all $U_{\infty }$ ). In Elbing et al. (Reference Elbing, Winkel, Lay, Ceccio, Dowling and Perlin2008), a $\text{DR}$ 20 % and an 80 % classification was made for the BR-to-TALR and TALR-to-ALR limits. However, these limits and the DR levels in general are not directly comparable to those of the present study due to differences in the measurement techniques–specifically, the use of a local versus a global drag measurement. Better DR agreement is expected in the bubbly and transitional regimes, where the air distribution is more homogeneous over the width and length of the domain. In contrast, in the ALR, strong spatial non-uniformity –e.g. wetted patches near the injector and an intact layer downstream –likely lead to lower global DR estimates.

For all $U_{\infty }$ , after transitioning to ALR, further increasing $Q_{\textit{air}}$ has only a small effect on DR. For example, for $U_{\infty }=5$ m s^–1 at $Q_{\textit{air}}=Q_{\textit{crit}}$ , $\text{DR}$ is 63 %. To achieve a 11 % increase in $\text{DR}$ , the air flow rate should be increased by $43\,\%$ . This is especially relevant for real ship applications, where a cost–benefit analysis is used to determine the working conditions of the air lubrication system. The intermediate region ( $40\lt Q_{air}\lt Q_{\textit{crit}}$ ) is characterised by DR lines of various slopes. For increasing $U_{\infty }$ , the slope of the curves becomes lower. That means that, for higher $U_{\infty }$ , relatively more injected air is needed for an increase in $\text{DR}$ .

To further explore these different effects of increasing $U_{\infty }$ across the three regimes, in the next sections the air phase characteristics are quantified using the available imaging data and the results are discussed in tandem with the corresponding DR behaviour.

4.2.1. Characteristics of the bubbly regime

In this section, we will focus on the early stages of the BR ( $Q_{\textit{air}}\lt 40$ l min^–1 in figure 10). For these low air flow rates, the multiphase flow exhibits a different behaviour than in the rest of the regimes: for constant $Q_{\textit{air}}$ , increasing $U_{\infty }$ leads to a decrease in drag, while for all other air flow rates the opposite is true. This points to a potentially different DR (or augmentation) mechanism than that of the TALR and ALR.

Figure 11.

Instantaneous images of the BR in the $x{-}z$ plane for (a) $U_{\infty }=5$ m s^–1 and (b) $U_{\infty }=2$ m s^–1. Here, $Q_{\textit{air}}$ increases from left to right (10, 20 and 40 l min^–1). Flow is from right to left. (c) The corresponding DR curves. The black vertical arrow indicates that DR increases with $U_{\infty }$ for low air injection rates.

Figure 11. Long description

Panel A: Three grayscale images show the bubble regime (BR) in the plane for a flow speed of 5 meters per second. The air flow rate increases from left to right (10, 20, and 40 liters per minute). Flow direction is from right to left. Panel B: Three grayscale images show the bubble regime (BR) in the plane for a flow speed of 2 meters per second. The air flow rate increases from left to right (10, 20, and 40 liters per minute). Flow direction is from right to left. Panel C: A line graph depicts the corresponding drag reduction (DR) curves. The horizontal axis represents the air flow rate (Q_air) in liters per minute, ranging from 0 to 40. The vertical axis represents the drag reduction percentage (DR percent), ranging from -5 to 10. Different colored lines represent different flow speeds (2, 2.5, 3.5, and 5 meters per second). The black vertical arrow indicates that drag reduction increases with velocity for low air injection rates.

To explore this, we zoom in the region of $Q_{\textit{air}}\lt 40$ l min^–1 in the DR profiles and assess it together with corresponding instantaneous bubbly images in the $x{-}y$ and $x{-}z$ planes (figure 11). For the highest liquid velocity ( $U_{\infty }=5$ m s^–1), increasing $Q_{\textit{air}}$ has a negligible effect on bubble size (figure 11 a), while a monotonic increase of DR is measured (figure 11 c). The most notable effect is an apparent increase in the number of bubbles, with a higher concentration along two streaks at the centreline of the plate. Such inhomogeneities can be often traced back to upstream disturbances and small geometry variations yet none could be identified definitively in our facility. For the lowest velocity ( $U_{\infty }=2\,\mathrm{m\,s^{-1}}$ ), increasing $Q_{\textit{air}}$ results in qualitatively larger bubbles (figure 11 b) but the corresponding $\text{DR}$ here does not increase monotonically: it initially decreases from zero to negative values (denoting a drag increase) and reaches a local minimum, before increasing again (figure 11 c). A similar (albeit slightly lower) drag increase was also observed for $U_{\infty }=2.5\,\mathrm{m\,s^{-1}}$ . Further tests with lower $U_{\infty }$ , revealed that the local minimum in $\text{DR}$ increases in magnitude with decreasing free-stream velocity (all the way down to approximately 1.2 m s^–1, see § 5 for more details). Earlier studies on the topic have associated the drag reducing abilities of bubbles to both their size and deformability (Verschoof et al. Reference Verschoof, van der Veen, Sun and Lohse2016), as well as the general topology (closely packed vs dispersed, Biswas & Govardhan (Reference Biswas and Govardhan2024)). In this study we intentionally focused on acquiring synchronous imaging of both the wall-projected bubble area (figure 11) and of the vertical bubble organisation (figure 13) which we will leverage in what follows in order to try and interpret these drag changes.

Figure 12.

Bubble size statistics. (a) Original and segmented images for $U_{\infty }=2$ m s^–1 (left) and $U_{\infty }=3.5$ m s^–1 (right). (b) Probability distribution of the bubble diameter ( $ d_b$ ) for air flow rates ranging from $ Q_{\textit{air}} = 5$ to $ 40$ l min^–1 and $ U_{\infty }=2$ m s^–1. (c) Bubble size range for three $ U_{\infty }$ and various air flow rates. The error bars represent the 2.5th and 97.5th percentiles at the lower and upper ends, respectively. The area-weighted mean bubble size is indicated by ‘x’, while the median is denoted by ‘o’.

In order to identify any other links between bubble topology and $\text{DR}$ , we first quantify the bubble sizes from our imaging data. Towards this end, we detect the bubble boundaries in the $x{-}z$ plane using the AI-based image segmentation introduced in § 4.1.2. An example of the bubble detection can be seen in figure 12(a). For every $U_{\infty }-Q_{\textit{air}}$ pair, 600 images are processed (corresponding to 150–300 bubbles per image for the lowest velocity and more than $1000$ for the highest one). The equivalent bubble diameter $d_b$ is then calculated from the two-dimensional bubble area as $d_b=2\sqrt {A_b/ \pi {}}$ . In figure 12(b), the probability density function (PDF) of all bubble diameters is shown for $U_{\infty }=2$ m s^–1. The bubbles range in size from a few $\mathrm{mm}$ to a maximum of 2 $\mathrm{cm}$ , with a wider $d_b$ distribution and larger values for increasing $Q_{\textit{air}}$ ; for all cases the distribution exhibits a log-normal behaviour, as has been observed for bubbles and droplets in turbulence in a large range of configurations due to the underlying physics of coalescence and breakup (see Mouza et al. Reference Mouza, Dalakoglou and Paras2005; Jacob et al. Reference Jacob, Oliveri, Miozzi, Campana and Piva2010 among many others). For this $U_{\infty }$ , the mode is mostly insensitive to $Q_{\textit{air}}$ ( ${\approx}3{-}4$ mm) and approximately equal to the injector slot width ( $t=4$ mm). For higher $U_{\infty }$ (figure 12 c), the range and mean values of $d_b$ both decrease, indicating that bubble sizes become more uniform and thus their distributions narrower (see also figure 24 in Appendix C); yet, this velocity effect becomes less noticeable for $U_{\infty }\gt 3.5$ m s^–1. Finally, from a quantification perspective, it should also be noted that, for $U_{\infty }=5$ m s^–1 (for which the single-phase $u_{\tau }$ was accurately estimated, see table 4), the extracted $d_b$ range ( $1.86-2.19\,\mathrm{mm}$ ) agrees remarkably well with the maximum bubble diameter predicted by the Kolmogorov–Hinze theory, as adapted for bubbles within the high shear region of wall-bounded flows by Sanders et al. (Reference Sanders, Winkel, Dowling, Perlin and Ceccio2006): $d_{max}=(\sigma /2\rho )^{3/5}(\kappa y/u_{\tau }^3)^{2/5}$ . Here, $y=y_b$ is the outermost bubble wall-normal location (defined in the next paragraphs) and results in a $d_{max}$ between $1.9\ {\rm and}\ 2.2\,\mathrm{mm}$ .

Connecting these quantitative results for $d_b$ with our drag results discussed above and with earlier studies is far from straightforward. Increasing the free-stream velocity (for a constant $Q_{\textit{air}}$ ) results in decreasing bubble size (figures 11 a and 11 b, 12 c) and also DR (vertical line in figure 11 c). This suggests that DR increases with decreasing $d_b$ . This is in line with flat plate results from Henry et al. (Reference Henry, Savio, Koushan, Jenssen, Hanssen and Brink2023), who observed a drag increase in BR for the lowest velocity they tested ( $U_{\infty }=2\,\mathrm{m\,s^{-1}}$ ), a trend that was reversed in higher velocities. Yet, the opposite effect was reported by Biswas & Govardhan (Reference Biswas and Govardhan2024) in a lower Reynolds number experiment, where smaller bubbles were shown to increase drag. This was attributed to smaller bubble deformability, a notion introduced and supported by various other earlier studies (Lu et al. Reference Lu, Fernández and Tryggvason2005; Van Gils et al. Reference Van, Dennis, Guzman, Sun and Lohse2013; Verschoof et al. Reference Verschoof, van der Veen, Sun and Lohse2016). Following the friction velocity Weber number definition by Lu et al. (Reference Lu, Fernández and Tryggvason2005), here, we find $0.5\lt We\lt 15$ , where $We=\rho u_{\tau }^2d_b/\sigma$ . This indicates that, for all our cases, bubbles can be considered deformable. Those present at the lowest velocities are the most deformable ( $We=15$ ) and yet they lead to the highest measured drag, indicating that deformability notions are not sufficient here to predict drag performance. For bubbles of size similar to ours (larger than coherent structures, but smaller than the TBL thickness, $10\times l^{+}\lt d_{b}\lt \delta$ ), a different DR mechanism was proposed by Murai (Reference Murai2014): bubble motion. Due to the various forces acting on the bubbles in such regimes, bubbles move in a cyclic manner similar to ejections and sweeps (Murai et al. Reference Murai, Oishi, Takeda and Yamamoto2006), effectively reducing shear stress. While in our study we did not measure bubble motion, as mentioned before, only our $\text{DR}$ results for $U_{\infty }\geq 3\,\mathrm{m\,s^{-1}}$ (drag decrease) are in line with the regime map from Murai (Reference Murai2014); for lower velocities we measure drag increase instead. What becomes clear from the above is that information from $x{-}z$ planes (either in terms of total non-wetted area as was shown in § 4.1.2, or in terms of $d_b$ and deformability as was shown here), even when it is quantitative, is not sufficient to explain the DR effect of the bubbles.

Figure 13.

Side view images of the BR ( $Q_{\textit{air}}=10$ l min^–1). Here, $U_{\infty }$ increases from top to bottom from 2 to 5 m s^–1. The drag decreases from top to bottom.

Figure 13. Long description

The image contains multiple side view snapshots showing the bubble organisation under different conditions. Panel A: The top photo shows a smooth surface with bubbles residing on a single layer below the wall. Panel B: The second photo from the top shows the same bubble organisation as the previous one. Panel C: The third photo from the top shows a slight dispersion of bubbles below the wall. Panel D: The fourth photo from the top shows a higher dispersion on bubbles below the wall. Panel E: The bottom photo shows the highest dispersion of bubbles below the wall. Each photo is labeled with a scale of 1 centimeter. The drag increases from top to bottom, and the drag reduction (DR) changes from negative to positive.

However, aside from wall-parallel bubble imaging, we also have synchronous imaging in a wall-normal plane ( $x{-}y$ ), which allows us to additionally analyse the vertical organisation of the air phase and further probe its connection to drag characteristics. This is something that, to the best of our knowledge, has not been explored before in the current context. Figure 13 showcases side view snapshots from this vertical plane for $Q_{\textit{air}}=10$ l min^–1 and increasing $U_{\infty }$ (from top to bottom). In the case of the two lower velocities, bubbles are present in a single layer below the wall, and are severely flattened due to shear/buoyancy, while for $U_{\infty }\geq 3\,\mathrm{m\,s^{-1}}$ , smaller, more spherical bubbles are dispersed over multiple wall-normal locations. It is also readily observable that, while the individual bubble size $d_b$ decreases for increasing $U_{\infty }$ (figure 12 c), the overall bubbly layer thickness, $y_b$ increases. More specifically, for the lowest velocity, $y_b\approx 6\,\mathrm{mm}$ , showing a small increase as we increase $Q_{\textit{air}}$ . For $U_{\infty }=3.5 \,\mathrm{m\,s^{-1}}$ , $y_b\approx 6.5\,\mathrm{mm}$ , again showing a slight increase with $Q_{\textit{air}}$ . Lastly, for $U_{\infty }=5 \,\mathrm{m\,s^{-1}}$ , a clear increase in the bubbly thickness is measured: $9\leq y_b\leq 13\,\mathrm{mm}$ . It must be noted that, in the case of the lowest $U_{\infty }$ , $y_b$ is representative of the height of a single bubble, but for higher ones where dispersion is at play, $y_b$ corresponds to multiple bubbles. In all cases, $y_b^{+}$ is much larger than 300 and up to $0.25\delta$ , indicating that bubbles reach the outer layer of the TBL. For all cases it is also interesting to note the slope of the single and multi-bubble layers (see dashed red lines in figure 13–flow is from right to left), reflecting the well-established downstream angle of coherent structures in wall-bounded turbulence. It must be noted that the latter refers to single-phase flow conditions and additional parameters associated with the bubbly dynamics could also affect the slope. Further studies are needed to evaluate the origins of this phenomenon.

Returning to the topological disparity of the bubble layer for different $U_{\infty }$ , it allows us to draw a more straightforward connection with the measured drag. Large bubbles present for $U_{\infty }\lt 3$ m s^–1, sliding over the flat plate in single bubble layers, increase the drag. A possible mechanism could be that this single bubble layer acts similar to roughness and thus initially increases drag. It can be assumed that, as $Q_{\textit{air}}$ increases further, the increase in non-wetted area eventually becomes sufficient for DR, especially when a transitional regime of large air patches is reached. The latter happens for a much lower $Q_{\textit{air}}$ compared with higher free-stream velocities (see also Nikolaidou et al. Reference Nikolaidou, Laskari, van and Poelma2024), and this is why DR for $U_{\infty }\lt 3$ m s^–1 catches up ( $Q_{\textit{air}}=40$ l min^–1, figure 11 c) and eventually outperforms the one for $U_{\infty }\gt 3$ m s^–1 ( $Q_{\textit{air}}\gt 40$ l min^–1, figure 10 and see also next section). On the other hand, for $U_{\infty }\gt 3$ m s^–1, much smaller bubbles are present in the TBL and they are vertically stacked, creating a multi-bubble layer, whose thickness increases with $U_{\infty }$ . Their DR performance is initially far superior to the single bubble layer, providing drag decrease for even the smallest $Q_{\textit{air}}$ tested here. As $Q_{\textit{air}}$ increases further, although the DR keeps increasing, it is at a much slower pace than for lower $U_{\infty }$ (see figure 10), reflecting the much higher needs for $Q_{\textit{air}}$ to reach the TALR and ALR (see §§ 4.2.2 and 6).

From an application perspective, it must be noted that the single layer bubble, present at low $U_{\infty }$ , will not be relevant for full-scale ship applications, since ship service speeds far exceed $3\,\mathrm{m\,s^{-1}}$ , so smaller bubbles dispersed in the vertical would be expected in most conditions.

4.3. Bubbly/transitional regime ( $8\,\%\leq \textrm{DR}\leq 60\,\%$ )

Figure 14.

Slopes of the DR curves within the transitional regime. Horizontal dashed line corresponds to $\text{DR}=40\,\%$ .

Figure 15.

Characteristic transitional regime images ( $x{-}z$ and $x{-}y$ plane) corresponding to $\text{DR}\approx 40\,\%$ for three representative $U_{\infty }$ values. Flow is from right to left in all panels. (a) $U_{\infty }=2$ m s^–1 and $Q_{\textit{air}}=80$ l min^–1, (b) $U_{\infty }=3.5$ m s^–1 and $Q_{\textit{air}}=130$ l min^–1, (c) $U_{\infty }=5$ m s^–1 and $Q_{\textit{air}}=200$ l min^–1.

Figure 15. Long description

Panel A: The top panel is an illustration showing air patches and a liquid film between them. The bottom panel is a side view image showing air bubbles below the flat plate. Panel B: The top panel is an illustration depicting a dense distribution of small bubbles within a liquid. The bottom panel is a side view image showing numerous small bubbles below the flat plate. Panel C: The top panel is an illustration showing a high concentration of bubbles in the liquid. The bottom panel is a side view image displaying a high density of bubbles. Each set of images includes a scale bar indicating one centimeter.

Despite the similar, monotonic increase in DR for all $U_{\infty }$ , there is a distinct decrease in slope, indicating a lower rate of DR increase with $Q_{\textit{air}}$ for higher velocities. For instance, $Q_{\textit{air}}=90$ l min^–1 is needed for $U_{\infty }=2$ m s^–1 to reach a $\text{DR}=40\,\%$ , while for $U_{\infty }=3.5$ m s^–1 and $U_{\infty }=5$ m s^–1 these become 130 l min^–1 and 200 l min^–1, respectively (figure 14). As mentioned above, an increase in $Q_{\textit{air}}$ also leads to a coalescence of bubbles and the appearance of larger air patches, which then contribute to DR by an increase in non-wetted area (figure 9). From the above (and also our observations in Nikolaidou et al. Reference Nikolaidou, Laskari, van and Poelma2024) it is then to be expected that such a transition will happen at a slower rate as $U_{\infty }$ increases. From the available imaging data, both from the wall-parallel as well as the wall-normal imaging planes, this can indeed be seen (figure 15). For the same DR percentage (40 %), a transitional topology is particularly clear for the lowest velocity (see figure 15 a), where air patches, separated by liquid films but forming a single wall-normal layer, are seen very close to the wall. The situation shifts as $U_{\infty }$ increases to 3.5 m s^–1, with the outer boundary of the air phase becoming more deformed, and bubbles appearing around it (figure 15 b). For the highest velocity, only a dense layer of bubbles is present (5 m s^–1, figure 15 c), indicating that a transitional regime might not have been reached despite the very high $Q_{\textit{air}}$ involved. Distinguishing between a bubbly and a transitional regime therefore becomes very challenging. Thus, when also taking into account the topology from figure 13, it becomes clear that for the low velocities, the initially drag-increasing bubbles flatten due to shear, coalesce and lead to a fast rate of DR for $Q_{\textit{air}}\gt 40$ l min^–1, while for the very high velocities, the (slower) monotonic increase in DR is linked instead to an increasingly denser packing and minimal coalescence of much smaller bubbles.

4.4. Air layer regime ( $\text{DR}\geq 60\,\%$ )

As mentioned above, for the last regime of DR, all cases reach a plateau starting at $\text{DR}=60\,\%$ . The main difference across the velocities tested is that, for the highest ones, DR shows a marginal increase with $Q_{\textit{air}}$ (absolute change approximately 5 %), while for low velocities, a much larger increase is observed (approximately 15 %, figure 10). Here, we leverage wall-parallel imaging data downstream of the injector and wall-normal data both in the middle of the plate as well as close to the injector for some more quantitative discussions (figure 16). The goal is twofold: first to characterise the topology of the ALR and how it varies with $U_{\infty }$ , something that is lacking in the current literature. Secondly, to evaluate whether there are any links between that topology and DR. Specifically, we examine the air layer thickness and its coherence. As both affect the non-wetted area, they could lead to changes in DR.

Figure 16.

Instantaneous image of the ALR for $U_{\infty }=2$ m s^–1. The mean air layer thickness $t_{\textit{air}}$ is measured in the middle of the test section (84 cm downstream of the injection location) and initial air layer thickness $t_{\textit{inj}}$ is measured close to the injector (6.2 cm downstream of the injection location).

Figure 17.

(a) Air layer thickness close to the injector $t_{\textit{inj}}$ and exit angle $\theta$ for $U_{\infty }$ = 2 m s^–1 (see also figure 16 for definitions). The vertical dashed line indicates $Q_{\textit{crit}}$ . (b) Mean air layer thickness $t_{\textit{air}}$ development for increasing $Q_{\textit{air}}$ for multiple $U_{\infty }$ . Only $Q_{\textit{air}}\gt Q_{\textit{crit}}$ is shown, meaning that the air layer is formed. Error bars are the standard deviation due to the physical variation in thickness.

Figure 17. Long description

Panel A: A line graph shows the relationship between air flow rate (Q_air) in liters per minute on the horizontal axis and two parameters on the vertical axes: initial air layer thickness (t_inj) in millimeters and exit angle (θ) in degrees. The graph includes two data series: one for initial air layer thickness with bluemarkers and another for exit angle with black markers. The vertical dashed line indicates the critical air flow rate. Panel B: Another line graph illustrates the mean air layer thickness (t_air) in millimeters on the vertical axis against air flow rate (Q_air) in liters per minute on the horizontal axis for different free stream velocities (U_∞). Multiple data series are represented by different colored markers and lines, each corresponding to a specific velocity. Error bars indicate the standard deviation due to physical variation in thickness.

Towards that end, we first employ an edge detection algorithm on the wall-normal imaging planes to identify the outer boundary of the air layer. This allows us to estimate the air layer thickness at the middle of the test section, $t_{\textit{air}}$ (figure 16), and both the initial air layer thickness, $t_{\textit{inj}}$ , and its exit angle, $\theta$ near the injector, where the air jet bends towards the wall due to buoyancy and the liquid momentum (see figure 16).

Figure 18.

Characteristic images of wetted pockets near the injector at $Q_{\textit{air}}=Q_{\textit{crit}}$ and $Q_{\textit{air}}\gt Q_{\textit{crit}}$ for $U_{\infty }$ = 2.5 m s^–1 (a) and (b); and $U_{\infty }$ = 4 m s^–1 (c) and (d). A wetted patch is indicated in (a) for clarity.

Close to the injector, when looking at the evolution of both the thickness and the exit angle of the air phase, minimal variation is observed with increasing $Q_{\textit{air}}$ (while only $U_{\infty }=2$ m s^–1 is shown, this behaviour is similar at all other velocities), especially in the ALR (figure 17 a). A slight decrease in exit angle and increase in $t_{\textit{inj}}$ are seen for $Q_{\textit{air}}\lt 40$ l min^–1, associated with the slight increase in bubble diameter and bubble coalescence behaviour, as discussed in the previous section. Thus, while from a topological perspective the robustness of the exit geometry of the air phase is of note, it cannot be meaningfully linked with the observed increase in DR for the low velocities. Moving to the estimates of the air layer thickness at the middle of the plate, $t_{\textit{air}}$ , the first thing to observe is that $t_{\textit{air}}\approx 1.5t_{\textit{inj}}$ , indicating a growth in air layer thickness along the plate, which is similar for all $U_{\infty }$ (figure 17 b). The second thing to note is that, for lower velocities, there is an increase in $t_{\textit{air}}$ as $Q_{\textit{air}}$ increases (we should repeat here that, since the focus is on the ALR, we only look at $Q_{\textit{air}}\gt Q_{\textit{crit}}$ ). For $U_{\infty }$ = 2 m s^–1 a relative increase of nearly 75 % is calculated, whereas higher velocities yield substantially smaller values. While this trend does mirror the one observed in DR, where an increase of approximately 15 % was observed, and an increase in ALR thickness would be associated with a DR increase (given the increase in non-wetted area in the side fences), the variation observed in $t_{\textit{air}}$ is not significant enough to fully explain it. More specifically, the $t_{\textit{air}}$ growth corresponds to a $3.4\,\%$ increase in terms of non-wetted area, which is minimal considering the DR growth rate with $A_{nw}$ (figure 9). Consequently, the observed increase in DR past $Q_{\textit{air}}=Q_{\textit{crit}}$ for low velocities is associated with an increase in air layer thickness (away from the injector), and can be partly attributed to the resulting increase in non-wetted area, yet there must other factors responsible.

One of these factors is the level of embedment of the air layer within the liquid TBL (the ratio $t_{\textit{air}}/\delta$ ), which has been shown to have a significant effect on the air phase topology (Zverkhovskyi Reference Zverkhovskyi2014; Nikolaidou et al. Reference Nikolaidou, Laskari, van and Poelma2024) and is particularly relevant in full-scale applications where $t_{\textit{air}}/\delta \ll 1$ . For the two lower velocities, this embedment decreases with $Q_{\textit{air}}$ since the air layer thickness increases. Based on our single phase measurements (see § 3), for $Q_{\textit{air}}=90$ l min^–1 we estimate $t_{\textit{inj}}/\delta \approx 0.5$ close to the injector and $t_{\textit{air}}/\delta =0.165$ in the middle of the plate. This is assuming the same TBL growth with and without air, which is unlikely. However, it still provides a rough estimate of embedment. These ratios become 0.47 and 0.27, respectively, at the highest air flow rate. The higher $t_{\textit{air}}/\delta$ ratio, indicates that the air/water interface is further away from the wall and thus encounters a much lower turbulent activity within the TBL. That is expected to act favourably in terms of DR (see also next paragraph), aligning with the $\text{DR}$ increase measured for these velocities and air flow rates. This indicates a compound effect of the $t_{\textit{air}}$ increase: larger non-wetted area and lower turbulent activity within the TBL disturbing the interface. For the higher velocities, the $t_{\textit{air}}/\delta$ ratios are almost constant since the air layer thickness does not vary appreciably with $Q_{\textit{air}}$ . Thus no effect in $\text{DR}$ is expected, aligning again with the relatively unchanged measured $\text{DR}$ with increasing $Q_{\textit{air}}$ (figure 10).

Finally, we turn our attention towards the wall-parallel imaging planes, where clear topological differences are observed between low ( $U_{\infty }=2.5$ m s^–1, figure 18 a,b) and high velocities ( $U_{\infty }=4$ m s^–1, figures 18 c and 18 d). For the latter, the air layer interface is much more deformed, with signs of small-scale features, a clear effect of higher Reynolds number and thus turbulent activity, not seen in the lower velocity cases. Yet, increasing $Q_{\textit{air}}$ past $Q_{\textit{crit}}$ does not lead to any observable change in that topology. On the other hand, for the lower velocities, at $Q_{\textit{air}}=Q_{\textit{crit}}$ , while several small-scale features persist, deforming the air–water interface, these are significantly decreased for $Q_{\textit{air}}\gt Q_{\textit{crit}}$ : as mentioned above, this can be attributed to a larger $t_{\textit{air}}$ , protruding further in outer region of the TBL, such that the interface is exposed to a lower liquid turbulent activity.

In addition, for $Q_{\textit{air}}=Q_{\textit{crit}}$ , several wetted pockets are observed close to the injector, locally breaking the spanwise coherence of the air layer (figure 18 a). These liquid gaps, reflecting high-speed streaks upstream of the injector (see Laskari Reference Laskari2025), also shrink considerably in the lower velocities with increasing $Q_{\textit{air}}$ . This leads to a much smoother and undisturbed interface (figure 18 b). Linking back to the DR behaviour of these cases, a decrease in wetted pockets with increasing $Q_{\textit{air}}$ and a smoother air–liquid interface accompany the increase in DR for the low velocities. This results in a higher maximum DR ( $\text{DR}_{max}\approx 75\,\%$ ) than that achieved for $U_{\infty }=5$ m s^–1 ( $\text{DR}_{max}\approx 65\,\%$ ), despite a similar maximum $t_{\textit{air}}$ (figure 17 b). For the higher velocities, the air–liquid interface is significantly more deformed and at smaller scales than for lower $U_{\infty }$ , owing to the much higher Re. This topology that varies little with increasing $Q_{\textit{air}}$ beyond the critical one, mirroring again the almost constant DR levels observed for that velocity. Thus, expanding on the discussion above regarding the air layer thickness, we can also safely assume here that the observed drag behaviour for $\text{DR}\gt 60\,\%$ as $U_{\infty }$ increases can also be partly attributed to the changes in the air layer spanwise integrity and local breakup.

6.1. Scaling of the critical air flow rate

Figure 22.

Froude-depth number versus non-dimensional air flow rate. The local velocity $U_{\textit{local}}$ at the air–water interface is used for the velocity scale. The dashed vertical line demarcates the shallow/intermediate conditions (to the right) from deep water conditions (to the left).

Finally, the focus is shifted to the critical air flow rate, $q_{\textit{crit}}$ . This air flow rate needed for transition to an air layer is important since the ALR is the most desired regime in terms of DR. Thus, predicting $q_{\textit{crit}}$ is crucial for industrial air lubrication systems and yet, despite a large number of air lubrication studies, there is still no universal scaling available, posing a significant challenge in applying laboratory-scale findings to real life applications. In this section, a scaling is proposed based on the parameters that appear to play a significant role.

For this purpose, the $U_{\infty }-q_{\textit{crit}}$ data points from figure 21, are reused here in a non-dimensional form. For the velocity scale, $\textit{Fr}_d$ is used for normalisation, which was previously shown to dictate the air phase topology for $q_{\textit{air}}\gt q_{\textit{crit}}$ (see § 5). The critical air flow rate per metre width $q_{\textit{crit}}$ , is normalised with the slot width of the air injector, $t$ and the local velocity $U_{\textit{local}}$ . It should be noted that, the resulting coefficient, $C_Q=Q_{\textit{crit}}/(BtU_{\textit{local}})$ , can be interpreted as the ratio of a mean air injection velocity with a local liquid velocity ( $U_{\textit{local}}$ ). Increasing $t$ enhances coalescence of the air phase and favours the transition to the ALR, resulting in a lower $q_{\mathrm{crit}}$ for larger $t$ . This agrees well with visual observations during our experiments but further studies targeting the effect of the slot width are needed. $U_{\textit{local}}$ is the liquid flow velocity at $y=t_{\textit{air}}$ . For small-scale experiments where $t_{\textit{air}} \approx \delta$ it is reasonable to expect $U_{\textit{local}} \approx U_{\infty }$ , but for smaller $t_{\textit{air}}/\delta$ ratios (which is also the case in full-scale ships), $U_{\textit{local}}\lt \lt U_{\infty }$ . Local flow velocities were also used in our previous study to explain incoming TBL differences: a large $U_{\textit{local}}$ (closer to $U_{\infty }$ ) implies a smaller incoming TBL thickness which is shown to promote transition to the ALR (Nikolaidou et al. Reference Nikolaidou, Laskari, van and Poelma2024). As such, $U_{\textit{local}}$ was also chosen here for normalisation, to take into account the effect of local flow conditions rather than free-stream ones. For the results in our previous study (Nikolaidou et al. Reference Nikolaidou, Laskari, van and Poelma2024), $U_{\textit{local}}$ measurements were available, for the current ones, an approximation was made based on separate LDA measurements (Nikolaidou et al. Reference Nikolaidou, Laskari, van Terwisga and Poelma2026) and for Elbing et al. (Reference Elbing, Winkel, Lay, Ceccio, Dowling and Perlin2008) an approximation was made based on the available velocity data in Etter et al. (Reference Etter, Cutbirth, Ceccio, Dowling and Perlin2005).

The resulting non-dimensional plot (figure 22) highlights a vastly different trend between shallow/intermediate and deep water conditions (separated by a dashed vertical line). A steeper slope is shown for the latter, while for shallow water conditions ( $\textit{Fr}_d\gt 0.61$ ) only a very small increase of $C_Q$ with $\textit{Fr}_d$ remains, with the normalisation successfully bounding all data points to within $0.6\lt C_Q\lt 0.9$ . It is good to keep in mind that in absolute terms this corresponds to a very large range of air flow rates (44–15 662 l min^–1) and free-stream velocities (0.57–13.3 m s^–1). While the proposed scaling looks promising, further experiments should be undertaken with different flat plate widths and slot widths to test this further. In addition, the role of other parameters such as roughness and surface wettability should be also examined and potentially added to the scaling as proposed by Peifer et al. (Reference Peifer, Callahan-Dudley and Mäkiharju2020).