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arxiv: 2604.17636 · v1 · submitted 2026-04-19 · ⚛️ physics.flu-dyn

Drag reduction regimes in air lubrication

Lina Nikolaidou , Ali R Khojasteh , Angeliki Laskari , Tom van Terwisga , Christian Poelma This is my paper

Pith reviewed 2026-05-10 04:54 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn
keywords air lubricationdrag reductionbubbly regimeair layerFroude numberregime transitionmultiphase flow
0
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The pith

Air lubrication achieves 60 percent drag reduction once a continuous layer forms over the surface.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper maps how injecting air under a flat plate in water flow creates three successive regimes that control drag. At low air rates bubbles form and can even increase drag at slow speeds, but higher rates produce patches and then a full air layer that cuts drag sharply. A new scaling for the critical air flow rate to reach the air layer regime combines the air exit velocity, the liquid speed next to the layer, and the Froude-depth number. The work shows that once the layer appears, further air addition thickens it and lowers drag more at low speeds while having little additional effect at high speeds. Layer shape also changes with water depth: it stays open at high Froude numbers but closes into a finite cavity below a threshold.

Core claim

The central discovery is that drag reduction passes through bubbly, transitional, and air-layer regimes, with the air-layer regime delivering at least 60 percent reduction once a critical air flow rate is reached. This critical rate follows a proposed scaling that multiplies the air exit velocity by the near-layer liquid velocity and incorporates the Froude-depth number. Beyond the transition, low-speed cases produce thicker, smoother layers with still lower drag while high-speed cases show only marginal further gains; the layer itself becomes unbounded for Froude-depth numbers above 0.7 and forms a closed cavity for subcritical values.

What carries the argument

The proposed scaling relation for the critical air flow rate Q_air, formed from the product of air exit velocity and near-layer liquid velocity together with the Froude-depth number Fr_d.

If this is right

  • Once the air layer forms, increasing air supply at low freestream speed produces a thicker layer and additional drag reduction.
  • At high freestream speed the same increase in air supply changes layer appearance but adds little extra drag reduction.
  • The air layer remains open and extends beyond the test section for Froude-depth numbers above 0.7 but closes into a cavity of finite length below 0.61.
  • Drag reduction in the bubbly regime is non-monotonic: large slow bubbles raise drag at low speed while smaller dispersed bubbles begin to lower it at higher speed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The scaling could be checked directly by varying plate length or water depth while holding the three input quantities constant.
  • If the scaling holds outside the laboratory, it would allow designers to estimate the minimum compressor power needed for a target drag reduction on a ship hull.
  • The change from open layer to closed cavity at low Froude-depth number suggests wave drag on the air-water interface may set the ultimate length of the lubricated region.

Load-bearing premise

The observed drag changes and the location of regime transitions are controlled mainly by air exit velocity, nearby liquid speed, and Froude-depth number, with other factors such as viscosity and surface tension remaining secondary inside the tested range.

What would settle it

A measurement in which the critical air flow rate for 60 percent drag reduction deviates systematically from the proposed scaling when viscosity or surface tension is varied by a factor of two while keeping velocities and Fr_d fixed.

Figures

Figures reproduced from arXiv: 2604.17636 by Ali R Khojasteh, Angeliki Laskari, Christian Poelma, Lina Nikolaidou, Tom van Terwisga.

Figure 1
Figure 1. 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. The sketch is adapted by the original one by P. Poot. 2. Experimental Setup & methods 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 … view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of the air injector. (a) 3D view from the top and (b) side view. Liquid flow is from right to left. was 11 mm 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 turbulent boundary layer. In the case of the air lubricat… view at source ↗
Figure 3
Figure 3. Figure 3: Sketch of drag force measurement system attached to the air lubrication plate. 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 [PITH_FULL_IM… view at source ↗
Figure 4
Figure 4. Figure 4: Imaging systems used for the multiphase flow campaign. camera No. FOV (mm × mm) focal length (mm) 𝑓 # resolution (px/mm) 1 740×740 24 5.6 2.75 2 110×110 105 5.6 18.48 3 70×70 105 5.6 30.16 [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Flat plate total skin friction coefficient of the current measurements along with friction lines from literature (ITTC 1957; Schlichting 1979; Katsui 2005; Grigson 1999). 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 foll… view at source ↗
Figure 6
Figure 6. Figure 6: Drag reduction with increasing air flow rate for a nominal freestream velocity of 2 m/s (𝐹𝑟𝑑 = 1.24). The vertical dashed line demarcates the transition to the air layer regime at 𝑄𝑎𝑖𝑟 = 𝑄𝑐𝑟 𝑖𝑡 . Right insert: side view (𝑥 − 𝑦 plane) of the air layer at 𝑄𝑎𝑖𝑟 = 140 l/min. Left insert: side view (𝑥 − 𝑦 plane) of the bubbly regime at 𝑄𝑎𝑖𝑟 = 30 l/min. 10 l/min. Drag increase cases are rarely reported in litera… view at source ↗
Figure 7
Figure 7. Figure 7: Characteristics images of the multiphase flow topology: (a) bubbly, (b) transitional, and (c) air layer regime, for 𝑈∞ = 2 m/s. Air flow rate increases from top to bottom. Black rectangle indicates the injector slot location, while the white region could not be imaged. a backward facing step or a cavitator (Zverkhovskyi 2014) or in the case of subcritical conditions (𝐹𝑟𝑑 < 1) (Nikolaidou et al. 2024). The … view at source ↗
Figure 8
Figure 8. Figure 8: Drag reduction measurements in the case of increasing air flow rate from (bubbly to air layer regime) and decreasing air flow rate (air layer to bubbly regime) for 𝑈∞ = 3 m/s (𝐹𝑟𝑑 = 1.75). Repeat measurements are also shown (without error bars). 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 desi… view at source ↗
Figure 9
Figure 9. Figure 9: Non-wetted area, calculated from down-up images versus drag reduction measurements for 𝑈∞ = 2 m/s (𝐹𝑟𝑑 = 1.24). Air flow rate increases from left (5 l/min) to right (100 l/min). See [PITH_FULL_IMAGE:figures/full_fig_p015_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Drag reduction curves, for different 𝑈∞. The dashed lines demarcate the regime transitions. observed when 𝐴𝑛𝑤 reaches 97%. Finally, a maximum DR of 77% (marked by × in [PITH_FULL_IMAGE:figures/full_fig_p016_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: Bubble size statistics. (a) original and segmented images for 𝑈∞ = 2 m/s (left) and 𝑈∞ = 3.5 m/s (right). (b) Probability distribution of the bubble diameter (𝑑𝑏) for air flow rates ranging from 𝑄air = 5 to 40 l/min and 𝑈∞ = 2 m/s. Vertical dashed line indicates the thickness of the bubbly layer 𝑦𝑏. (c) Bubble size range for three 𝑈∞ and various air flow rates. The error bars represent the 2.5th and 97.5t… view at source ↗
Figure 13
Figure 13. Figure 13: Side view images of the bubbly regime (𝑄𝑎𝑖𝑟 = 10 l/min). 𝑈∞ increases from top to bottom from 2 to 5 m/s. The drag decreases from top to bottom respectively. that, to the best of our knowledge, has not been explored before in the current context [PITH_FULL_IMAGE:figures/full_fig_p020_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Slopes of the drag reduction curves within the transitional regime. Horizontal dashed line corresponds to 𝐷𝑅 = 40%. air patch liquid film air patch (a) 𝑈∞ = 2 m/s & 𝑄𝑎𝑖𝑟 = 80 l/min (b) 𝑈∞ = 3.5 m/s & 𝑄𝑎𝑖𝑟 = 130 l/min (c) 𝑈∞ = 5 m/s & 𝑄𝑎𝑖𝑟 = 200 l/min [PITH_FULL_IMAGE:figures/full_fig_p022_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Characteristic transitional regime images (𝑥 − 𝑧 & 𝑥 − 𝑦 plane) corresponding to 𝐷𝑅 ≈ 40% for three representative 𝑈∞. Flow is from right to left in all panels. (Figure 15b). For the highest velocity, only a dense layer of bubbles is present (5 m/s, Figure 15c), indicating that a transitional regime might not have been reached despite the very high 𝑄𝑎𝑖𝑟 involved. Distinguishing between a bubbly and a tran… view at source ↗
Figure 16
Figure 16. Figure 16: Instantaneous image of the air layer regime for 𝑈∞ = 2 m/s. The mean air layer thickness 𝑡𝑎𝑖𝑟 is measured in the middle of the test section (84 cm downstream of the injection location) and initial air layer thickness 𝑡𝑖𝑛 𝑗 is measured close to the injector (6.2 cm downstream). tinj (mm), (deg.) (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p023_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: (a) Air layer thickness close to the injector 𝑡𝑖𝑛 𝑗 and exit angle 𝜃 for 𝑈∞ = 2 m/s (see also [PITH_FULL_IMAGE:figures/full_fig_p023_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Characteristic images of wetted pockets near the injector at 𝑄𝑎𝑖𝑟 = 𝑄𝑐𝑟 𝑖𝑡 and 𝑄𝑎𝑖𝑟 > 𝑄𝑐𝑟 𝑖𝑡 for 𝑈∞ = 2.5 m/s (a) & (b) and 𝑈∞ = 4 m/s (c) & (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 𝑄𝑎𝑖𝑟 (while only 𝑈∞ = 2 m/s is shown, this behavior is s… view at source ↗
Figure 19
Figure 19. Figure 19: Morphological features of the air-water interface on the wall-normal plane (a) and the wall-parallel plane (b). Air bubbles are indicated in green and water droplets in blue. This is for 𝑈∞ = 2 m/s and 𝑄𝑎𝑖𝑟 = 100 l/min. cause local wetting and disruption of the air layer. Wall-normal imaging revealed the existence of air bubbles near the air–water interface, likely originating from a partial layer breakag… view at source ↗
Figure 20
Figure 20. Figure 20: Drag reduction curves, for different 𝑈∞. The black line (𝐹𝑟𝑑 = 1.24) from [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Characteristics images of the air layer regime for (a) 𝑈∞ = 0.47 m/s and 𝑄𝑎𝑖𝑟 = 20 l/min, (b) 𝑈∞ = 0.57 m/s and 𝑄𝑎𝑖𝑟 = 40 l/min, (c) 𝑈∞ = 0.67 m/s and 𝑄𝑎𝑖𝑟 = 40 l/min, (d) 𝑈∞ = 0.93 m/s and 𝑄𝑎𝑖𝑟 = 40 l/min and (e) 𝑈∞ = 1.26 m/s and 𝑄𝑎𝑖𝑟 = 60 l/min. From top to bottom 𝐹𝑟𝑑 increases and there is a transition from air cavity to an air layer. The red vertical line indicates the air cavity closure in (a)–(d). … view at source ↗
Figure 22
Figure 22. Figure 22: Regime map. Transitional air flow rate 𝑞𝑡𝑟 𝑎𝑛𝑠 and critical air flow rate 𝑞𝑐𝑟 𝑖𝑡 demarcating the transition from bubbly regime (BR) to transitional regime (TALR) and from TALR to air layer regime (ALR) respectively. Measurements of the current study are shown along with measurements from Elbing et al. (2008) and Nikolaidou et al. (2024). Air flow rates are corrected accounting for the local pressure and d… view at source ↗
Figure 23
Figure 23. Figure 23: Froude-depth number versus non-dimensional air flow rate. The local velocity 𝑈𝑙𝑜𝑐𝑎𝑙 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). and yet, despite a large number of air lubrication studies, there is still no universal scaling available, posing a significant challeng… view at source ↗
Figure 24
Figure 24. Figure 24: Repeatability of drag force measurements for various air flow rates for (a) 𝑈∞=2.5 m/s, (b) 𝑈∞=3 m/s, (c) 𝑈∞=3.5 m/s and (d) 𝑈∞=5.5 m/s. 0 X0-33 [PITH_FULL_IMAGE:figures/full_fig_p033_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Probability distribution of the bubble diameter (𝑑𝑏) for various air flow rates within the bubbly regime. 0 X0-34 [PITH_FULL_IMAGE:figures/full_fig_p034_25.png] view at source ↗
read the original abstract

Air lubrication regimes were studied using simultaneous drag force measurements and multi-plane imaging to characterize the regimes and identify the governing mechanisms of drag reduction. A bubbly, transitional, and air layer regime are identified over a large range of freestream velocities ($U_{\infty}$), air flow rates ($Q_{air}$), and Froude-depth numbers ($Fr_d$). For the lowest $U_{\infty}$, drag reduction lags significantly behind the non-wetted area coverage at all cases and no simple correlation exists. Within the bubbly regime, a drag increase is found for low $U_{\infty}$ with large, slow-moving bubbles forming a single layer over the plate height. For higher velocities, bubbles become smaller and disperse vertically, while the drag starts decreasing. For higher $Q_{air}$, irrespective of $U_{\infty}$, air patches start to form (transitional regime) and drag monotonically decreases, with the onset of the air layer regime at 60\% drag reduction. A new scaling of the associated critical $Q_{air}$ is proposed, combining the air exit velocity, the liquid velocity close to the air layer and $Fr_d$. For a further increase of $Q_{air}$ and low $U_{\infty}$, a thicker and smoother air layer is formed with even lower drag; for higher $U_{\infty}$, marginal differences are observed. The air layer morphology is significantly altered however, depending on $Fr_d$: for $Fr_d>0.7$, it is unbounded, extending beyond the current test section length, and for subcritical conditions (deep water regime, $Fr_d<0.61$) a closure is formed and the air layer transitions to a cavity of a specific length.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript reports experimental observations of air lubrication regimes (bubbly, transitional, air-layer) on a flat plate using simultaneous drag measurements and multi-plane imaging. Regimes are mapped over ranges of freestream velocity U_∞, air flow rate Q_air, and Froude-depth number Fr_d. Drag reduction is shown to lag non-wetted area at low U_∞; a 60% drag-reduction threshold marks the air-layer onset. A new empirical scaling for the critical Q_air is proposed using air exit velocity, near-layer liquid velocity, and Fr_d. Air-layer morphology is further shown to depend on Fr_d, producing either unbounded layers (Fr_d > 0.7) or closed cavities (Fr_d < 0.61).

Significance. If the proposed scaling is quantitatively validated, the work supplies a practical empirical relation for predicting the air-flow threshold needed for substantial drag reduction in marine applications. The simultaneous force and imaging data provide direct evidence linking regime transitions to measured drag changes, and the Fr_d dependence on layer closure offers a clear hydrodynamic distinction. These elements could inform hull-design guidelines once error analysis and generality checks are added.

major comments (3)
  1. [§4 (scaling proposal)] The central scaling for critical Q_air (abstract and §4) is presented as combining air exit velocity, near-layer liquid velocity, and Fr_d, yet no explicit functional form, fitting procedure, or goodness-of-fit metric (R², RMS error, or cross-validation) is reported across the tested U_∞ and Fr_d range. Without these, it is impossible to judge whether the three parameters capture the dominant physics or merely correlate within the laboratory window.
  2. [§3.3 (drag-reduction results)] The 60% drag-reduction threshold used to define air-layer onset (abstract, §3.3) is stated without accompanying uncertainty estimates, number of repeated runs, or data-exclusion criteria. Given that drag reduction is the primary observable, the absence of error bars or statistical robustness checks leaves the regime boundary and the scaling anchored to it only moderately supported.
  3. [§4 and discussion] The manuscript does not address possible contributions of viscosity or surface tension to bubble coalescence and interface stability (weakest assumption noted in stress-test). If Re or We effects shift the observed transitions outside the tested range, the proposed scaling will not generalize; a brief sensitivity test or order-of-magnitude estimate of these terms is needed to support the claim that the three chosen parameters suffice.
minor comments (2)
  1. [Figures 3–7] Figure captions should explicitly state the number of independent runs and the symbol for measurement uncertainty.
  2. [§2.2] Notation for the near-layer liquid velocity is introduced without a clear definition or measurement location; a schematic or equation would improve reproducibility.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which help clarify the presentation of our results. We address each major comment below and will revise the manuscript to improve the rigor of the scaling description, statistical support for the threshold, and discussion of additional physical effects.

read point-by-point responses

  1. Referee: [§4 (scaling proposal)] The central scaling for critical Q_air (abstract and §4) is presented as combining air exit velocity, near-layer liquid velocity, and Fr_d, yet no explicit functional form, fitting procedure, or goodness-of-fit metric (R², RMS error, or cross-validation) is reported across the tested U_∞ and Fr_d range. Without these, it is impossible to judge whether the three parameters capture the dominant physics or merely correlate within the laboratory window.

    Authors: We agree that the explicit functional form, fitting procedure, and quantitative metrics were insufficiently detailed. The scaling was developed from dimensional considerations and least-squares regression on the measured critical Q_air at the 60% drag-reduction point. In the revision we will state the explicit relation Q_{air,crit} = C U_{exit} U_{liq} L Fr_d^k (with fitted C and k), describe the regression method, and report R² values together with residual statistics across the tested U_∞ and Fr_d range. This will allow readers to evaluate the fit quality directly. revision: yes

  2. Referee: [§3.3 (drag-reduction results)] The 60% drag-reduction threshold used to define air-layer onset (abstract, §3.3) is stated without accompanying uncertainty estimates, number of repeated runs, or data-exclusion criteria. Given that drag reduction is the primary observable, the absence of error bars or statistical robustness checks leaves the regime boundary and the scaling anchored to it only moderately supported.

    Authors: The 60% threshold corresponds to the consistent visual onset of a continuous air layer in the imaging and a clear change in slope of the drag-reduction curves. Repeated runs (typically five or more per condition) were performed, but error bars and repetition details were omitted from the original figures. In the revision we will add standard-deviation error bars, state the number of repetitions per condition, and specify exclusion criteria (e.g., runs showing sensor drift or flow anomalies). This will strengthen the statistical basis for the regime boundary. revision: yes

  3. Referee: [§4 and discussion] The manuscript does not address possible contributions of viscosity or surface tension to bubble coalescence and interface stability (weakest assumption noted in stress-test). If Re or We effects shift the observed transitions outside the tested range, the proposed scaling will not generalize; a brief sensitivity test or order-of-magnitude estimate of these terms is needed to support the claim that the three chosen parameters suffice.

    Authors: We acknowledge that viscous and capillary contributions were not discussed explicitly. In the revised discussion we will add order-of-magnitude estimates showing that, for the experimental range (Re ~ 10^5–10^6, We ~ 10^3–10^4), inertial forces dominate viscous dissipation and surface tension at the interface. These estimates support the sufficiency of the three chosen parameters within the tested window. A full sensitivity test across wider Re/We ranges would require new experiments and is outside the present scope; the scaling is therefore presented as empirical for the conditions examined. revision: partial

Circularity Check

0 steps flagged

No circularity in empirical regime identification and scaling proposal

full rationale

The manuscript is an experimental study that identifies bubbly, transitional, and air-layer regimes via simultaneous drag and imaging measurements across ranges of U_∞, Q_air, and Fr_d. The central claim is an empirical scaling for the critical Q_air at 60% drag reduction, formed by combining measured air exit velocity, near-layer liquid velocity, and Fr_d. No equations, fitted parameters renamed as predictions, or self-citations are shown that reduce this scaling to its inputs by construction. The derivation chain rests on direct observation of regime transitions and morphology changes rather than tautological re-expression of fitted quantities or imported uniqueness results.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an experimental study whose claims rest on direct measurements rather than mathematical derivations; no free parameters, axioms, or invented entities are introduced in the abstract.

pith-pipeline@v0.9.0 · 5622 in / 1198 out tokens · 46267 ms · 2026-05-10T04:54:11.650336+00:00 · methodology

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Reference graph

Works this paper leans on

120 extracted references · 120 canonical work pages · 1 internal anchor

  1. [1]

    Abramowitz and I

    M. Abramowitz and I. Stegun , title =

    work page

  2. [2]

    Barker and D

    T. Barker and D. G. Schaeffer and P. Boh\'orquez and J. M. N. T. Gray , title =. J. Fluid. Mech. , year =

    work page

  3. [3]

    Batchelor , title =

    G.K. Batchelor , title =. J. Fluid Mech. , year =

    work page

  4. [4]

    A. V. Briukhanov and S. S. Grigorian and S. M. Miagkov and M. Y. Plam and I. E. Shurova and M. E. Eglit and Y. L. Yakimov , title =. Physics of Snow and Ice, Proceedings of the International Conference on Low Temperature Science , volume =

    work page

  5. [5]

    Brownell and L.K

    C.J. Brownell and L.K. Su , title =. AIAA Paper 2004-2335 , year =

    work page 2004

  6. [6]

    Brownell and L.K

    C.J. Brownell and L.K. Su , title =. AIAA Paper 2007-1314 , year =

    work page 2007

  7. [7]

    Dennis , title =

    S.C.R. Dennis , title =. Ninth Intl Conf. on Numerical Methods in Fluid Dynamics , publisher =. 1985 , editor =

    work page 1985

  8. [8]

    Formation of levees, troughs and elevated channels by avalanches on erodible slopes , author =. J. Fluid Mech. , volume =

    work page

  9. [9]

    Hwang and E.O

    L.-S. Hwang and E.O. Tuck , title =. J. Fluid Mech. , year =

    work page

  10. [10]

    and Saut, Jean Claude , title =

    Joseph, Daniel D. and Saut, Jean Claude , title =. Theoretical and Computational Fluid Dynamics , publisher=

    work page

  11. [11]

    Worster , title =

    M.G. Worster , title =. Interactive dynamics of convection and solidification , publisher =. 1992 , editor =

    work page 1992

  12. [12]

    Koch , title =

    W. Koch , title =. J. Sound Vib. , year =

    work page

  13. [13]

    Lee , title =

    J.-J. Lee , title =. J. Fluid Mech. , year =

    work page

  14. [14]

    Linton and D.V

    C.M. Linton and D.V. Evans , title =. Phil.\ Trans.\ R. Soc.\ Lond. , year =

    work page

  15. [15]

    Martin , title =

    P.A. Martin , title =. 1980 , journal =

    work page 1980

  16. [16]

    Rogallo , title =

    R.S. Rogallo , title =

    work page

  17. [17]

    Ursell , title =

    F. Ursell , title =. 1950 , journal =

    work page 1950

  18. [18]

    On the oscillations Near and at resonance in open pipes , year =

    L. On the oscillations Near and at resonance in open pipes , year =. J. Engng Maths , volume =

    work page

  19. [19]

    Miller , title =

    P.L. Miller , title =. 1991 , address =

    work page 1991

  20. [20]

    Camera Calibration Toolbox for Matlab , howpublished =

    Bouguet, J.-Y , year=. Camera Calibration Toolbox for Matlab , howpublished =

    work page

  21. [21]

    Zverkhovskyi, Oleksandr , title =

    work page

  22. [22]

    Annual Review of Fluid Mechanics , volume=

    Friction drag reduction of external flows with bubble and gas injection , author=. Annual Review of Fluid Mechanics , volume=. 2010 , publisher=

    work page 2010

  23. [23]

    The 12th International Conference on Hydrodynamics, 18-23 September 2016, Egmond aan Zee, The Netherlands

    On the physical mechanisms for the numerical modelling of flows around air lubricated ships , author=. The 12th International Conference on Hydrodynamics, 18-23 September 2016, Egmond aan Zee, The Netherlands. , year=

    work page 2016

  24. [24]

    Physical review letters , volume=

    Bubble drag reduction requires large bubbles , author=. Physical review letters , volume=. 2016 , publisher=

    work page 2016

  25. [25]

    Journal of Fluid Mechanics , volume=

    Bubble-induced skin-friction drag reduction and the abrupt transition to air-layer drag reduction , author=. Journal of Fluid Mechanics , volume=. 2008 , publisher=

    work page 2008

  26. [26]

    A Dot Tracking Algorithm for synthetic Schlieren method , author=

    work page

  27. [27]

    5th International Conference on Advanced Model Measurement Technology for the Maritime Industry (AMT)

    Exploratory measurements of cavitation nuclei in the wake of a ship model , author=. 5th International Conference on Advanced Model Measurement Technology for the Maritime Industry (AMT). Glasgow, Scotland (UK) , year=

    work page

  28. [28]

    Experiments in fluids , volume=

    Frictional drag reduction by bubble injection , author=. Experiments in fluids , volume=. 2014 , publisher=

    work page 2014

  29. [29]

    Journal of Ship Research , volume=

    Partial cavity drag reduction at high Reynolds numbers , author=. Journal of Ship Research , volume=. 2010 , publisher=

    work page 2010

  30. [30]

    Colloids and Interface Science Communications , volume=

    Bubble the wave or waive the bubble: Why seawater waves foam and freshwater waves do not? , author=. Colloids and Interface Science Communications , volume=. 2015 , publisher=

    work page 2015

  31. [31]

    Center for Turbulence Research, Annual Research Briefs , pages=

    Direct numerical study of air layer drag reduction phenomenon over a backward-facing step , author=. Center for Turbulence Research, Annual Research Briefs , pages=

    work page

  32. [32]

    The dynamics of strong turbulence at free surfaces. Part 1. Description , author=. Journal of Fluid Mechanics , volume=. 2001 , publisher=

    work page 2001

  33. [33]

    Annual Review of Fluid Mechanics , volume=

    Surfactant effects on bubble motion and bubbly flows , author=. Annual Review of Fluid Mechanics , volume=. 2011 , publisher=

    work page 2011

  34. [34]

    Journal of Physics: Conference Series , volume=

    Ventilated cavity flow over a backward-facing step , author=. Journal of Physics: Conference Series , volume=. 2015 , organization=

    work page 2015

  35. [35]

    Journal of Fluid Mechanics , volume=

    On the scaling of air entrainment from a ventilated partial cavity , author=. Journal of Fluid Mechanics , volume=. 2013 , publisher=

    work page 2013

  36. [36]

    2014 , publisher=

    Cavitation and bubble dynamics , author=. 2014 , publisher=

    work page 2014

  37. [37]

    Journal of Fluid Mechanics , volume=

    Particle--fluid interactions in grid-generated turbulence , author=. Journal of Fluid Mechanics , volume=. 2007 , publisher=

    work page 2007

  38. [38]

    Boundary-Layer Meteorology , volume=

    Pollutant dispersion in boundary layers exposed to rural-to-urban transitions: varying the spanwise length scale of the roughness , author=. Boundary-Layer Meteorology , volume=. 2017 , publisher=

    work page 2017

  39. [39]

    2011 , publisher=

    Particle image velocimetry , author=. 2011 , publisher=

    work page 2011

  40. [40]

    Journal of Physics: Conference Series , volume=

    Dispersion of bubbles in fully developed channel flow , author=. Journal of Physics: Conference Series , volume=. 2011 , organization=

    work page 2011

  41. [41]

    Proceedings of the 19th International Symposium on Application of Laser and Imaging Techniques to Fluid Mechanics , year=

    Non-intrusive measurement of complex free surfaces using Laser Induced Fluorescence and stereo imaging , author=. Proceedings of the 19th International Symposium on Application of Laser and Imaging Techniques to Fluid Mechanics , year=

    work page

  42. [42]

    Experiments in Fluids , volume=

    A synthetic Schlieren method for the measurement of the topography of a liquid interface , author=. Experiments in Fluids , volume=. 2009 , publisher=

    work page 2009

  43. [43]

    Annual Review of Fluid Mechanics , volume=

    Optical particle characterization in flows , author=. Annual Review of Fluid Mechanics , volume=. 2011 , publisher=

    work page 2011

  44. [44]

    Applied Mechanics Reviews , volume=

    Two-phase flow patterns and flow-pattern maps: fundamentals and applications , author=. Applied Mechanics Reviews , volume=. 2008 , publisher=

    work page 2008

  45. [45]

    Journal of Fluid Mechanics , volume=

    Laminar--turbulent transition and wave--turbulence interaction in stratified horizontal two-phase pipe flow , author=. Journal of Fluid Mechanics , volume=. 2015 , publisher=

    work page 2015

  46. [46]

    Journal of Hydraulic Research , volume=

    On elongated air pockets in downward sloping pipes , author=. Journal of Hydraulic Research , volume=. 2010 , publisher=

    work page 2010

  47. [47]

    International Journal of Multiphase Flow , volume=

    The effect of surfactants on upward air--water pipe flow at various inclinations , author=. International Journal of Multiphase Flow , volume=. 2016 , publisher=

    work page 2016

  48. [48]

    1995 , month=oct # " 10", publisher=

    Air bubble lubricated boat hull , author=. 1995 , month=oct # " 10", publisher=

    work page 1995

  49. [49]

    2011 , publisher=

    Vessel with air chambers to reduce the resistance between the hull and the water , author=. 2011 , publisher=

    work page 2011

  50. [50]

    The Journal of Physical Chemistry , volume=

    The effect of electrolytes on bubble coalescence in water , author=. The Journal of Physical Chemistry , volume=. 1993 , publisher=

    work page 1993

  51. [51]

    Published by The Society of Naval Architects and Marine Engineers, SNAME, The Principles of Naval Architecture Series, ISBN: 978-0-939773-76-3 , year=

    Ship resistance and flow , author=. Published by The Society of Naval Architects and Marine Engineers, SNAME, The Principles of Naval Architecture Series, ISBN: 978-0-939773-76-3 , year=

    work page

  52. [52]

    SNAME Maritime Convention , year=

    Full scale applications of air lubrication for reduction of ship frictional resistance , author=. SNAME Maritime Convention , year=

    work page

  53. [53]

    Conference of Shipping in Changing Climate, Newcastle, UK , pages=

    Silverstream system-air lubrication performance verification and design development , author=. Conference of Shipping in Changing Climate, Newcastle, UK , pages=

    work page

  54. [54]

    Conference Proceedings, the Japan Society of Naval Architects and Ocean Engineers , volume=

    A full-scale air lubrication experiment using a large cement carrier for energy saving (result and analysis) , author=. Conference Proceedings, the Japan Society of Naval Architects and Ocean Engineers , volume=

    work page

  55. [55]

    Ocean Engineering , volume=

    Power-saving device for air bubble generation using a hydrofoil to reduce ship drag: theory, experiments, and application to ships , author=. Ocean Engineering , volume=. 2015 , publisher=

    work page 2015

  56. [56]

    The Nineteenth International Offshore and Polar Engineering Conference , year=

    Full scale experiment for frictional resistance reduction using air lubrication method , author=. The Nineteenth International Offshore and Polar Engineering Conference , year=

    work page

  57. [57]

    Journal of Fluids Engineering , volume=

    Freeman scholar review: passive and active skin-friction drag reduction in turbulent boundary layers , author=. Journal of Fluids Engineering , volume=. 2016 , publisher=

    work page 2016

  58. [58]

    Ocean Engineering , volume=

    Maintenance of air layer and drag reduction on superhydrophobic surface , author=. Ocean Engineering , volume=. 2017 , publisher=

    work page 2017

  59. [59]

    , author=

    Air Layer on Superhydrophobic Surface for Frictional Drag Reduction. , author=. Journal of Ship Research , volume=

    work page

  60. [60]

    Physical Review Applied , volume=

    Superhydrophobic drag reduction for turbulent flows in open water , author=. Physical Review Applied , volume=. 2020 , publisher=

    work page 2020

  61. [61]

    Nature , volume=

    Design of robust superhydrophobic surfaces , author=. Nature , volume=. 2020 , publisher=

    work page 2020

  62. [62]

    Experiments in Fluids , volume=

    Bubble-size distributions produced by wall injection of air into flowing freshwater, saltwater and surfactant solutions , author=. Experiments in Fluids , volume=. 2004 , publisher=

    work page 2004

  63. [63]

    The Journal of the Acoustical Society of America , volume=

    Collective oscillations of fresh and salt water bubble plumes , author=. The Journal of the Acoustical Society of America , volume=. 2000 , publisher=

    work page 2000

  64. [64]

    Experiments in Fluids , volume=

    Influence of bubble size on micro-bubble drag reduction , author=. Experiments in Fluids , volume=. 2006 , publisher=

    work page 2006

  65. [65]

    Journal of fluid mechanics , volume=

    On the scaling of air layer drag reduction , author=. Journal of fluid mechanics , volume=. 2013 , publisher=

    work page 2013

  66. [66]

    The Journal of the Acoustical Society of America , volume=

    High-Reynolds-number turbulent-boundary-layer wall pressure fluctuations with skin-friction reduction by air injection , author=. The Journal of the Acoustical Society of America , volume=. 2008 , publisher=

    work page 2008

  67. [67]

    Journal of Fluid Mechanics , volume=

    Bubble friction drag reduction in a high-Reynolds-number flat-plate turbulent boundary layer , author=. Journal of Fluid Mechanics , volume=. 2006 , publisher=

    work page 2006

  68. [68]

    Applied Ocean Research , volume=

    Analysis of interaction between ship bottom air cavity and boundary layer , author=. Applied Ocean Research , volume=. 2016 , publisher=

    work page 2016

  69. [69]

    29th Symposium on Naval Hydrodynamics, Gothenburg, Sweden , pages=

    Hydrodynamics of a displacement air cavity ship , author=. 29th Symposium on Naval Hydrodynamics, Gothenburg, Sweden , pages=

    work page

  70. [70]

    International Journal of Naval Architecture and Ocean Engineering , volume=

    On the energy economics of air lubrication drag reduction , author=. International Journal of Naval Architecture and Ocean Engineering , volume=. 2012 , publisher=

    work page 2012

  71. [71]

    1979 , publisher=

    Boundary Layer Theory , author=. 1979 , publisher=

    work page 1979

  72. [72]

    Physics of fluids , volume=

    Scaling of the velocity fluctuations in turbulent channels up to Re = 2003 , author=. Physics of fluids , volume=. 2006 , publisher=

    work page 2003

  73. [73]

    Ocean Engineering , volume=

    On multi-point gas injection to form an air layer for frictional drag reduction , author=. Ocean Engineering , volume=. 2018 , publisher=

    work page 2018

  74. [74]

    Journal of Fluid Mechanics , volume=

    The topology of gas jets injected beneath a surface and subject to liquid cross-flow , author=. Journal of Fluid Mechanics , volume=. 2017 , publisher=

    work page 2017

  75. [75]

    International Shipbuilding Progress , number=

    Exploring the limits of RANS-VoF modelling for air cavity flows , author=. International Shipbuilding Progress , number=. 2019 , publisher=

    work page 2019

  76. [76]

    Journal of Applied Mechanics , volume=

    A single formula for the law of the wall , author=. Journal of Applied Mechanics , volume=

    work page

  77. [77]

    Proceedings of the 24th ITTC , year=

    The Specialist Committee on Powering Performance Prediction-Final Report and Recommendations to the 24th ITTC , author=. Proceedings of the 24th ITTC , year=

    work page

  78. [78]

    Proceedings of the 8th International Towing Tank Conference , year =

    work page

  79. [79]

    1961 , publisher=

    Boundary layer theory , author=. 1961 , publisher=

    work page 1961

  80. [80]

    Fifth Osaka colloquium on advanced CFD applications to ship flow and hull form design, Osaka, Japan, 2005 , year=

    The proposal of a new friction line , author=. Fifth Osaka colloquium on advanced CFD applications to ship flow and hull form design, Osaka, Japan, 2005 , year=

    work page 2005

Showing first 80 references.