Plastron trapping trenches and uses thereof

Abstract

The invention, according to one embodiment, discloses the geometric (i.e., shape) and dimensional (i.e., size) details and other properties of trenches that can retain trapped gas under a fluid such as water (called "plastron") while subjected to the flow conditions common for traveling watercraft. A solid surface of numerous trenches filled with air allows the surface to experience a smaller friction drag in flowing water. Retainment of the plastron under high-speed conditions of regular watercraft has proved elusive. Here, the geometric and dimensional details of the trenches are disclosed that can retain the plastron under the flow conditions common for watercraft -- under the shear rate of water flow that a boat traveling even over 10 knots would experience. The trenches may also be applied to other objects or surface that come into contact with flowing fluid.

Description

2023-204-2 PLASTRON TRAPPING TRENCHES AND USES THEREOF Related Application

[0001] This Application claims priority to U.S. Provisional Patent Application No. 63 / 458,068 filed on April 7, 2023 which is hereby incorporated by reference. Priority is claimed pursuant to 35 U.S.C. § 119 and any other applicable statute. Technical Field

[0002] The technical field generally relates to trenches or surfaces containing trenches that are used with liquid flows of a relatively high shear rate. Under a liquid, the trenches or surfaces containing trenches are able to retain the plastron in a pinned state even when the liquid flows with a relatively high shear rate. Statement Regarding Federally Sponsored Research and Development

[0003] This invention was made with government support under 1336966 awarded by the National Science Foundation. The government has certain rights in the invention. Background

[0004] Superhydrophobic (SHPo) surfaces have been one of the most popular topics in science and engineering over the last two decades because of their unique potentials, such as hydrodynamic drag reduction, self-cleaning, anti-icing, anti-biofouling, and anti-corrosion. Among them, drag reduction of watercraft has been cited as a motivating factor in nearly every publication on SHPo surfaces for its global-scale impact on energy saving and environmental protection. When a SHPo surface is completely immersed in water, a substantially continuous layer of air, commonly called a plastron, may be formed on the surface and produce a slip boundary that reduces skin friction drag. While all numerical studies over the years and many experimental studies in the 2010s have reported a significant drag reduction, successful drag-reduction experiment in fully turbulent flows in open water, which represents the field condition of watercraft, has not been reported until 2020. These most recent successes have eased the skepticism that had grown against the SHPo drag reduction, after two decades of research without any successful field experiments. Importantly, the successful reports strongly suggested that most of the inconsistent2023-204-2 experimental results in the past may have been simply due to the loss or deterioration of the plastron. In other words, the original notion of SHPo drag reduction is valid as far as the plastron remains in a good shape. The tortuous path to the current state of knowledge also indicates how difficult yet important it is to accurately monitor the state of the plastron during experimental studies of SHPo drag reduction, leading to the two-camera observation technique. Focusing on longitudinal trench SHPo surfaces, which have been the most effective for drag reduction and to help the design of SHPo surfaces capable of reducing the drag for watercraft, this work teaches the range of trench geometries that can maintain a pinned or slightly degraded plastron, which has its water-air interfaces pinned or slightly depinned at the top edges of the trench and trench tops over the entire or nearly entire trench length so that much of the pristine slip capability is preserved, even in high-speed open-water flows.

[0005] Over the years, hydrostatic pressure and air diffusion have been found to affect the plastron stability in stationary and flowing water. More recently, the shear-driven drainage, which was developed to understand the loss of infused oil on liquid infused surfaces (LIS) caused by the shear in flowing water, has been borrowed to explain the plastron loss on trench SHPo surface in very high shear flows. However, while the diffusional loss of the infused oil on LIS in water is negligible and was justifiably ignored in the shear drainage model, the diffusional loss of the trapped air in water cannot be ignored and would require a revised theory applicable to SHPo surfaces. Summary

[0006] Focusing on longitudinal trench SHPo surfaces, which have been the most effective for drag reduction, and to help design SHPo surfaces capable of reducing the drag for watercraft, a range of simple trench geometries are investiaged that can maintain a pinned or slightly degraded plastron, which has its water-air interfaces pinned or slightly depinned at the top edges of trench tops over the entire or nearly entire trench length so that much of the pristine slip capability is preserved, even in high-speed open-water flows. To establish a theoretical model that can describe the plastron morphology on the longitudinal trench SHPo surfaces in high-speed flows under water, two theories are used: (i) the plastron stability theory based on the hydrostatic pressure and air diffusion and (ii) the shear-drainage theory modified for plastron (i.e., not oil) stability. Since the diffusional loss of the trapped air in water (unlike the trapped oil for the case of LIS) cannot be ignored, in this disclosure the2023-204-2 diffusional loss of air is analyzed and the loss was found to be still small under the common flow conditions of watercraft in practice. This finding, as well as a few other considerations, lead to the discovery that the above two theories can be used together and the "combined theory" can predict the fate of plastron (i.e., would it stay pinned or not) so that one can design the geometric details of trench SHPo surfaces that can retain the plastron. Importantly, this discovery also implies a realization of the most important parameters, i.e., the parameters of the first order importance: immersion depth (which is intertwined with air saturation level), shear rate, and trench length. In addition, as the parameters of the second order importance, the water pressure on the plastron is expected to deviate from the theory at the front (leading) and rear (trailing) end of the trench by the dynamic effect of flows, leading to a negative effect when compounded with the interfacial contaminations such as surfactants that accumulate at the rear end. While based mostly on the combined theory and the parameters of the first order importance, this disclosure still addresses the parameters of the second order importance in designing the geometric details of trenches or surfaces containing trenches.

[0007] The combined theory leads to the design of simple trenches for an experiment validation, and a successful result confirm the validity of the combined theory and other considerations as explained in the bulk of this specification.

[0008] To evaluate the combined model experimentally, a combinatorial series of SHPo surfaces were prepared and their plastron status monitored in fully turbulent flows under a motorboat at various flow speeds on seawater. For drag reduction applications, ideally one would like to have a pinned plastron, where the trapped air fills the entire depth and length of trench. To enable differentiating pinned and slightly degraded plastron, which can retain an acceptably substantial amount (e.g., > 30%) of the drag reduction induced by pinned plastron, from degraded and no plastron, which is left with an unacceptably small amount (e.g., < 30%) of the drag reduction by pinned plastron, a new observation scheme was devised using two underwater cameras by applying the approach of to the current goal. Since the plastron was observed to be intact at all speeds (i.e., 2.3 m / s < flow seed < 5.1 m / s or 10000 s-1< shear rate < 62000 s-1) in the previous boat experiments but found depleted at the higher speeds (i.e., 6.1 m / s < flow speed < 10.1 m / s, or 55000 s-1< shear rate < 140000 s-1) in the towing tank experiments while using similar trench SHPo surfaces, the boat was modified to increase its top speed to 7.2 m / s (or shear rate up to ~83000 s-1) so that the plastron can be depleted by shear-driven drainage in the current boat study.2023-204-2

[0009] The invention, according to one embodiment, discloses the geometric (i.e., shape) and dimensional (i.e., size) details of longitudinal trenches that can retain trapped gas under water (called "plastron") while subjected to the shearing flows common for traveling watercraft. A well-known utility is a solid surface of numerous trenches filled with air that allows the surface to experience a smaller friction drag in flowing water. Retainment of the plastron under the realistic conditions of regular watercraft (i.e., not small toy boat) has not been shown possible before the UCLA lab led by C.-J. Kim demonstrated the first successful drag reduction under a motorboat on seawater (up to 10 knots) using trenches with pitch of 50-200 microns, length of 7 cm, and gas fraction of 90%. See M. Xu et al., “Superhydrophobic Drag Reduction for Turbulent Flows in Open Water,” Physical Review Applied, Vol.13, 2020, 034056. However, when subjected to a more challenging condition (e.g., up to 20 knots in a towing tank), the same surface was found to lose the plastron at high speeds. See M. Xu et al. “Superhydrophobic drag reduction in high-speed towing tank”, Journal of Fluid Mechanics, Vol.908, 2021, A6. Further research identified the key parameters of the trenches that are important for the retainment of plastron and how they affect the retainment. Here, the geometric and dimensional details of the trenches are disclosed that can retain the plastron under the flow conditions common for watercraft – under the shear rate of water flow that a boat traveling over 10 knots (or 5 m / s, which creates over 60000 / s of shear rate on the traveling solid surface) would experience.

[0010] While the above embodiment was experimentally confirmed with SHPo surfaces made of simple trenches designed by the combined theory, the analyses leading to the combined theory and considerations of the second-order effects further allow for the design of a solid surface made of more generalized trench shapes as far as they are closed in the flow direction (and not necessarily closed in the transverse direction). A generalization leads to the design a SHPo solid surface consisting of trenches that are closed in the flow direction (and not necessarily closed in the transverse direction) with each trench covered with trench tops (or gratings) aligned to the flow direction.

[0011] In one embodiment, the key parameters of the trenches are as follows: ^ A solid surface with a plurality of trenches. ^ Each trench has a bottom surface and is closed in the fluid flow direction by a leading end and a trailing end. ^ Each trench is covered with a plurality of trench tops, which are in contact with the liquid.2023-204-2 ^ All of the top surfaces, including the top surfaces of trench tops, the leading end and the trailing end, are substantially flush and hydrodynamically smooth to the flow. ^ The trench tops are substantially straight and aligned nominally parallel to the flow direction of a fluid over the solid surface. ^ The width (w) of the opening between trench tops on the trench is satisfied as follows: 10 µm ≤ w ≤ 100 µm. ^ The pitch (p) of trench tops relates to the width of trench tops as (p–w). ^ Gas fraction of trench top (∅ = w / p) is equal to or larger than 70%: ∅ ≥ 0.7. ^ The depth of the trench is larger than 1 / 2 of the trench opening width (w). ^ The thickness of trench top (t) is larger than the width of the trench top (p–w) and smaller than the trench depth (d): p–w ≤ t ≤ d. ^ Trenches are closed in the flow direction, which is length direction of the trench tops. Specifically, the trenches have a leading or front end that is closed and a trailing or back end that is closed in the flow direction. As the distance between the leading and trailing end, trench length (L) should be between 1 and 20 cm. If longer, the plastron is easily lost by the hydrodynamic effect of common watercraft. If shorter, the drag reduction is not large enough for practical utility. ^ Below the trench top, trenches do not necessarily need to be closed in the width direction: t ≤ d. See t in FIGS.2B-2C. For example, the trench top may not come into contact with the bottom of the trench in some embodiments (e.g., FIGS.2B-2C and 3A-3C). ^ Trench tops may have discontinuities in the length direction and form a gap (g) so long as the discontinued gap is smaller than the width of a trench top, i.e., g < p–w. ^ The top edge of trench tops (see FIG.2B) should be sharp with the radius of curvature (r) smaller than 10 microns, i.e., r < 10 microns. If too round, the solid- liquid contact area becomes large enough to compromise the slip effect. ^ Substantial portions of the inner surfaces of the trenches, including all the side surfaces of the trench top, the bottom surfaces of the trenches, and the side surfaces of the leading and trailing ends, should be hydrophobic with the local (intrinsic) contact angle of water ( ^w) larger than 100 degrees ( ^w > 100º).2023-204-2 ^ In some embodiments, the width (w) of the trench opening is tapered along the flow direction near the leading end and / or front end of the openings as illustrated in FIG. 5. Preferably, the taper is less than 2 mm in length. ^ The trench top may be structurally supported by one or more columns that connect the trench top to the trench bottom. ^ One or more of the inner surfaces of the trench may be covered with nano- roughness. Nano-roughness encompasses submicron-sized features that impart roughness to the surface. ^ Top edges of the trench tops and leading and trailing ends may have a re-entrant shape.

[0012] In another embodiment, a solid surface having a plurality of trenches for maintaining a plastron on the solid surface during flow of a fluid over the solid surface, wherein each trench is defined by a leading end and a trailing end, a bottom surface and trench tops, wherein each trench satisfies the following conditions:

[0013] (a) a width (w) of an opening between the trench tops arranged with a pitch (p) is within the range of 10 µm and 100 µm;

[0014] (b) a pitch (p) and a width (p–w) of the trench tops are arranged so that a gas fraction (∅ = w / p) of the trench tops is equal to or larger than 70% (∅ ^ 0.7);

[0015] (c) a depth (d) of the trench is greater than w / 2;

[0016] (d) a thickness (t) of the trench tops is equal to or larger than a width of the trench tops (p–w) and equal to or smaller than the trench depth (d) (p–w ^ t ^ d);

[0017] (e) a length (L) of each trench is between 1 cm and 20 cm;

[0018] (f) the trench tops comprise respective top edges having a radius of curvature (r) smaller than 10 microns;

[0019] (g) wherein the trench tops comprise side surfaces and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º);

[0020] (h) wherein the leading ends and trailing ends comprise side surfaces and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º);

[0021] (i) wherein the trench comprises a bottom surface and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º); and2023-204-2

[0022] (j) wherein the trench tops are aligned nominally parallel to a flow direction of the fluid over the solid surface and the top surfaces of the trench tops are substantially flush relative to one another and to top surfaces of the leading end and the trailing end.

[0023] In another embodiment, a solid surface having a plurality of trenches for maintaining a plastron on the solid surface during flow of a fluid over the solid surface, wherein the trenches are defined by a leading end and a trailing end, a bottom surface and trench tops, wherein each trench of the plurality of trenches satisfies the following conditions:

[0024] (a) a width (w) of an opening between the trench tops arranged with a pitch (p) is within the range of 10 µm and 100 µm;

[0025] (b) a pitch (p) and a width (p–w) of the trench tops are arranged so that a gas fraction (∅ = w / p) of the trench tops is equal to or larger than 70% (∅ ^ 0.7);

[0026] (c) a depth (d) of the trench is greater than w / 2;

[0027] (d) a thickness (t) of the trench tops is equal to or larger than a width of the trench tops (p–w) and equal to or smaller than the trench depth (d) (p–w ^ t ^ d);

[0028] (e) a length (L) of each trench is between 1 cm and 20 cm;

[0029] (f) the trench tops comprise respective top edges having a radius of curvature (r) smaller than 10 microns;

[0030] (g) wherein the trench tops comprise side surfaces and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º);

[0031] (h) wherein the leading ends and trailing ends comprise side surfaces and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º);

[0032] (i) wherein the trench comprises a bottom surface and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º);

[0033] (j) wherein the trench tops are aligned nominally parallel to a flow direction of the fluid over the solid surface and the top surfaces of the trench tops are substantially flush relative to one another and to top surfaces of the leading end and the trailing end;

[0034] (k) wherein the trench tops are continuous between the leading end and the trailing end; and

[0035] (l) wherein all of the trench tops contact the bottom surface.2023-204-2

[0036] In some embodiments, all of the trench tops of the solid surface contact the bottom surface. Brief Description of the Drawings

[0037] FIG.1 illustrates a watercraft or boat at least partially covered with the disclosed solid surface having a plurality of trenches.

[0038] FIG.2A illustrates a perspective view of one embodiment of a portion of two trenches.

[0039] FIG.2B illustrates a partial cross-sectional view of one trench illustrated in FIG. 2A taken along the line A-A’.

[0040] FIG.2C illustrates a partial side view of the trenches illustrated in FIG.2A seen from line named Side view.

[0041] FIG.3A illustrates a perspective view of another embodiment of a portion of two trenches.

[0042] FIG.3B illustrates a partial cross-sectional view of one trench illustrated in FIG. 3A taken along the line A-A’.

[0043] FIG.3C illustrates a partial side view of the trenches illustrated in FIG.3A seen from line named Side view.

[0044] FIG.4A illustrates a perspective view of another embodiment of a portion of two trenches.

[0045] FIG.4B illustrates a partial cross-sectional view of one trench illustrated in FIG. 4A taken along the line A-A’.

[0046] FIG.4C illustrates a partial side view of the trenches illustrated in FIG.4A seen from line named Side view.

[0047] FIG.5 illustrates another embodiment of a portion of three trenches. In this embodiment, the trench opening is tapered in its width at the leading (or front) end and / or the trailing (or back) end of the trench. This creates a zig-zagged configuration of the leading (or front) end and / or the trailing (or back) end along the direction transverse to the flow.

[0048] FIG.6A illustrates a magnified view of the top edge of trench top according to one embodiment. The top edge is formed where the top surface of the trench top meets the side surface of the trench top. The radius of curvature (r) is shown for the edge in detail A.

[0049] FIG.6B illustrates a magnified view of the top edge of trench top according to another embodiment. The top edge is formed where the top surface of the trench top meets2023-204-2 the side surface of the trench top. The edge in this embodiment has a re-entrant configuration as seen in detail B.

[0050] FIGS.7A-7F is a schematic illustration of plastron being compromised on a SHPo surface with vertical sidewalls of t = d, as embodied in FIGS.4A-4C. Because the water pressure is usually higher than the trapped air pressure, the air-water interface is concave when pinned as seen in FIG.7A. If the water pressure is large enough to make the local (intrinsic) contact angle of water on the trench sidewall exceed the advancing contact angle θa, the contact line is depinned from the top edges and slides into the trench (FIG.7B) until the trench is fully wetted (FIG.7C). Although not common, if the water pressure is lower than the trapped air pressure, the meniscus is convex when pinned (FIG.7D). If the local contact angle of water on the trench top decreases below the receding contact angle θr, the contact line is depinned from the top edges and lets the neighboring air pockets merge (FIG. 7E). The merged air may form isolated bubbles off the surface, shrinking the plastron (FIG. 7F), which grows back to the pinned state (FIG.7D).

[0051] FIGS.8A-8D illustrate pressure distributions along the trench top in the flow direction. For the air-water interface to stay pinned on the top edge of the trench top at the length position x, the pressure difference between the water and the plastron, ∆P(x) = Pwater(x) – Pair(x) (blue vertical arrows), should be sustainable by the Laplace pressure of meniscus ∆P^or ^P^,min< ∆P(x) < ^ ^P^,max. FIG.8A illustrates the effect of immersion depth H and air saturation level s. In static water, the water pressure on the trench surface (thick green line) is Pwater,st = Patm + PH. The partial pressure of air dissolved in water Pair,st is sPatm (thick red line), which equals Patmif the water at the free surface (in contact with ambient air) is saturated with the atmospheric air. FIG.8B illustrates the effect of shear stress by water τw. In flowing water, the shear stress τwmakes the air pressure in the plastron Pair(x) (thick red line) increase linearly with x, decreasing the pressure difference ∆P(x) along the trench. FIG.8C illustrates that when immersed in flowing water, the two trends of (a) and (b) are combined to suggest a more general trend. FIG.8D illustrates that the above trend may be deviated by the dynamic effects of water flow near the front and rear (back) end of trench.

[0052] FIGS.9A-9B schematically illustrate an opening between two neighboring trench tops on a SHPo surface embodied by FIGS.4A-4C and submerged in longitudinally flowing water (i.e., water that flows in parallel to the trench tops) with the contact lines pinned on top edges of the trench tops. FIG.9A illustrates an exemplary illustration of plastron morphology. The curved arrow in the trench shows the air circulation inside the plastron.2023-204-2 FIG.9B has side-view profiles of the x-direction air flow inside plastron. The net air flow profile consists of three different flow profiles: shear-driven, Laplace pressure-driven, and air diffusion-driven. The first two profiles follow those of Wexler, J. S. et al., Shear-driven failure of liquid-infused surfaces, Physical review letters 114, pp.168301 (2015), which is incorporated herein by reference, developed for liquid-infused surface (LIS), and the third profile is newly introduced to account for the air diffusion across the air-water interface, which varies along the x-direction. The air flux (in the x-direction) induced by the air diffusion varying along x turns out to be small for the flow conditions of this study.

[0053] FIGS.10A-10D illustrate the experimental setup used to test the SHPo surfaces. FIG.10A is a schematic cross-section view of boat setup. FIG.10B is a schematic cross- section view of the testing unit, including shear sensor and camera setup. FIG.10C is a picture of the boat. FIG.10D is a picture of the bottom of testing well, taken by looking up from below the boat in air.

[0054] FIGS.11A-11B illustrate the SHPo samples embodied by FIGS.4A-4C and prepared for the experimental verification. FIG.11A illustrates three (3) different trench types depending on the top-edge shape of trench tops and the local roughness of the trench inner surfaces. The SEM pictures reveal a top edge of a cross-cleaved trench top as well as the local roughness (nano-grass in this experimental study) around the top edge. FIG.11B illustrates how each sample carries ten (10) trenches with each trench containing around 30 or 42 trench tops. All trenches in this study have a gas fraction w / p = 0.9. The 40 mm × 70 mm sample of a solid surface has a 30 mm × 60 mm micromachined surface surrounded by a smooth surface. For this test sample, the micromachined region has repeated trenches with side ends, which close the trench in the transverse direction and are not explicitly shown in FIGS.2-4. Divided by leading / trailing ends or a side end between two neighboring trenches, the repeated trenches are to test varying geometries, which combine longitudinal trench tops with p = 75 µm and p = 100 µm with trench length L = 2.5, 5, 10, 30, 60 mm. The inset SEM picture shows a divider (which functions as a leading end for one trench as we a trailing end for the next trench) between trenches repeated in the flow direction. The same arrangement was used for all the 12 samples (3 roughness types × 4 trench depths), providing 120 different trench and trench-top geometries of t = d with one photomask. The SEM pictures of cleaved samples show two different trench-top pitches and one trench depth d = 67.5 µm.

[0055] FIGS.12A-12E illustrate sample images for key trends. FIG.12A shows the effect of trench opening w shown by the side camera. Narrower trench openings w maintained the2023-204-2 plastron better. FIG.12B shows the effect of shear stress ^τw shown by the side camera. Slower flows maintained the plastron better. FIG.12C shows the effect of trench depth d shown by the side camera. Deeper trenches maintained the plastron better. FIG.12D shows the effect of nano-grass shown by the two cameras. For each pair of images, the top image was taken by the side camera, and the bottom image was taken by the rear camera. While the plastron was lost significantly on RE at this high flow speed (U = 6.4–6.7 m / s), a pinned or slightly degraded plastron was found for all trenches on NG and RE+NG, demonstrating the effectiveness of adding nano-grass. FIG.12E shows the effects of dynamic water pressure and interfacial contamination shown by the rear camera. Regions with trench length L = 2.5 mm, 10 mm, and 60 mm are outlined. The inset picture shows the pinned or slightly depinned interfaces at the front end of the 60 mm trenches.

[0056] FIG.13 shows the wall shear stress on a smooth surface τw0at different boat speeds. The experimental data fit the power regression line. Detailed Description of Illustrated Embodiments

[0057] With reference to FIG.1, a surface 10 includes a plurality of trenches 12 formed therein or disposed thereon. The surface 10 is configured to be exposed to a fluid environment. The fluid may include aqueous fluids such as water or other types of fluids including non-polar fluids. This includes, for example, fluids made from a polymer (e.g., heated polymers in the form of a fluid may also be used). In some embodiments, the surface 10 is exposed to a water environment. For example, the surface 10 may include all or portions of the hull of a watercraft or boat. The surface 10 may also include a surface of a pipe or a channel (e.g., inner surface) that is exposed to flow. In other embodiments, the surface 10 may include contact surfaces of a mold that contacts a heated fluid used to mold solid objects. Other objects or devices that are exposed to the flow of fluid such as water are contemplated. Regardless of the type of object, one or more surfaces 10 contains the plurality of trenches 12. The plurality of trenches 12 are covered with a plurality of trench tops 16 that are generally straight and aligned nominally parallel to the flow direction of the fluid (e.g., water) over the surface 10 as the object bearing the surface 10 travels through the fluid or as fluid travels over a stationary object as seen by the arrow in FIG.1. With reference to FIGS.1, 2A-2C, 3A-3C, 4A-4C, 5, 6A, and 6B each trench 12 is defined by a bottom surface 14 (which may be the surface of the object or a separate surface located on top of surface of the object), trench tops 16, a leading end 22 and a trailing end 24. In this2023-204-2 regard, the trenches 12 are closed in the fluid flow direction by the leading end 22 and a trailing end 24 of the trenches 12. The leading end 22 and trailing end 24 extend downward to the bottom surface 14 with fluid-contacting side surfaces.

[0058] The surface 10 itself may have the plurality of trenches 12 formed therein or thereon. For example, the plurality of trenches 12 may be formed directly on a native surface 10 of an object or article of manufacture (e.g., boat, watercraft, mold). Alternatively, the plurality of trenches 12 may be applied to a surface 10 as a cover, coating, or overlay. In some embodiments, the plurality of trenches 12 may be applied in smaller sheets, modules, or tiles that are assembled together to cover a larger area of the surface 10. As one example, a polymer sheet containing the plurality of trenches 12 may be applied to a surface 10 to create the plastron-retaining surface. Alternatively, the trenches 12 may be manufactured or otherwise directly formed in the native surface 10.

[0059] The trench tops 16 include a top surface 18 that defines the top of the trench tops 16. The trench tops 16 further include sidewalls 20 that define the side surfaces of the trench tops 16. The trench tops 16 have top edges formed at the transition from the top surface 18 to the sidewalls 20 (FIGS.2B, 3B, 4B, 6A, and 6B). These edges are preferably sharp with the radius of curvature (r) smaller than 10 microns as illustrated in FIG.6A. The radius of curvature may be considered to be negative if the top edges have a re-entrant shape FIG.6B. Note the trenches 12 are closed in the flow (i.e., length) direction. Specifically, each trench 12 has a leading end 22 that is closed in the flow direction and a trailing end 24 that is closed in the flow direction but does not necessarily have a closed end in the transverse direction. Multiple trenches 12 may be placed one after the other with the trench tops 16 aligned substantially in the flow direction. In this configuration, the leading end 22 and trailing end 24 together act as a common divider between two trenches 12 adjacent in the flow direction. This enables the trenches 12 to cover a larger area of the surface 10.

[0060] As illustrated in FIGS.2A-2C, a trench 12 is covered with trench tops 16, which define trench top opening width (w), trench top pitch (p), trench top width (p–w), and trench top thickness (t). Trench top 16 may be interrupted by gaps (g) located along their length, and the gap (g) should be less than the trench top width (p–w) such that (g < p–w). Also, trench top thickness (t) may equal the trench depth (d) at one or more selected locations to form post(s) or column(s) that supports the trench tops 16. One such post or column is seen in FIG.2C which is formed by sidewall 20 extending all the way from the top surface 18 to the bottom surface 14. All of the top surfaces 18, including the top surfaces of trench tops2023-204-2 16, the leading end 22 and the trailing end 24, are substantially flush and hydrodynamically smooth to the fluid flow.

[0061] In one embodiment, such as the embodiment illustrated in FIGS.3A-3C, the trench tops 16 may not contact the bottom surface 14 but are largely suspended over the bottom surface 14. In this embodiment, the trench tops 16 resemble a grill used on a barbeque. For example, the trench tops 16 may extend laterally between the leading end 22 and the trailing end 24. In this embodiment, the trenches 12 are fully open in the width direction below the trench tops 16. In some instances, a small portion of the trench tops 16 may contact the bottom surface 14 via a post or column, but substantially all of the trench(es) 12 is / are still open in the width direction even with the presence of these posts or columns.

[0062] In another embodiment, such as the embodiment illustrated in FIGS.4A-4C, the trench tops 16 contact the bottom surface 14 along substantially their entire length. In this embodiment, the trench 12 is closed in the width or transverse direction. In some embodiments, the trench tops 16 may be interrupted by gaps (g) located along their length. The gap (g) should be smaller than the trench top width (p–w) such that (g < p–w). However, even with the presence of these gaps (g) substantially all of the trench(es) 12 is / are still closed in the width direction.

[0063] In another embodiment, as illustrated in FIG.5, the width of the trench opening (w) near the leading end 22 and / or the trailing end 24 is tapered along the flow direction. In addition, the tapered opening may have a tapered length of less than 2 mm. The smaller trench top opening width (w) retains the plastron better in the front (i.e., toward leading end 22) and the rear (i.e., toward trailing end 24) of the trench 12, where the plastron is more vulnerable. The vulnerability is increased by the hydrodynamic discontinuity at the leading end 22 and trailing end 24, where the flow is interrupted, and the surfactant concentration (if present) at the trailing end 24, where the surfactant (which are commonly present in bodies of water) is accumulated by advection of the flow. The plastron is more robust if the trench top opening has a smaller width.

[0064] In one embodiment, the parameters of the plurality of trenches 12 satisfy the following restrictions found in Table 1 below:2023-204-2 Table 1 Parameter Restriction Pitch of trench top: p Restricted via w and ∅. g e trenches 12 each covered with a plurality of trench tops 16 over bottom surface 14. FIGS.2A and 2C illustrates two trenches 12 arranged end-to-end in the fluid flow direction. Portions or segments of the trench tops 16 may contact the bottom surface 14 via posts or columns that support these segment(s) of the trench tops 16 from the bottom surface 14. The parameter restrictions associated with this embodiment are contained in Table 1.

[0066] FIGS.3A-3C illustrates another embodiment of a solid 10 surface having one or more trenches 12 each covered with a plurality of trench tops 16 that substantially do not contact the bottom surface 14. FIGS.3A and 3C illustrates two trenches 12 arranged end-to- end in the fluid flow direction. This embodiment resembles a barbeque grill as the trench tops 16 are mostly suspended between the leading end 22 and the trailing end 24. In other words, trench 12 is mostly open in the transverse direction of the flow.

[0067] FIGS.4A-4C illustrates another embodiment of a solid surface 10 having one or more trenches 12 each covered with a plurality of trench tops 16 that extend downward and contact the bottom surface 14. The sidewalls 20 of the trench tops 16 extend downward fully to the bottom surface 14. FIGS.4A and 4C illustrates two trenches 12 arranged end-to-end in the fluid flow direction. In this particular implementation, t = d everywhere or nearly everywhere. In other words, trench 12 is mostly closed in the transverse direction of the flow. However, is should be appreciated that some gaps (g) may still be present along the2023-204-2 length of one or more of the trench tops 16. In such a configuration, the trench 12 is still substantially closed in the traverse direction even with the presence of these gaps (g).

[0068] FIG.5 illustrates another embodiment of a solid surface 10 having a plurality of trenches 12 arranged in an end-to-end configuration each covered with a plurality of trench tops 16. In this embodiment, the trench top opening is tapered in width (w) along the flow direction near the leading end 22 and / or the trailing end 24. This configuration creates a zig- zagged configuration of the leading end 22 and / or the trailing end 24 in the width direction (i.e., transverse to the flow). In one embodiment, the length of the taper Ltaperis less than 2 mm.

[0069] It should be appreciated that the surfaces of the trench 12 may be roughened in one or more ways. In one embodiment, nano-grass (NG), such as that illustrated in FIG.11A, is formed on one or more of the surfaces of the solid surface 10 and or the trench 12 structures. This may include the trench tops 16, top surfaces 18, sidewalls 20, and bottom 14. In another embodiment, the trench 12 and trench top 16 may be formed with its top edges in a re-entrant (RE) shape as illustrated in FIG.6B. Another embodiment combines both NG and RE edge like that of FIG.6B into the trenches 12 and trench tops 16.

[0070] Note that the plastron that is generated on the surface 10 with the trenches 12 may be initially generated by dissolved gases contained in the fluid which flows over the trenches 12. The dissolved gases naturally create the plastron on surface 10 with the trenches 12 and remains in place for an extended period. This plastron may also be continuously replenished or regenerated with gas contained in the fluid. Alternatively, the plastron may initially be seeded onto the surface 10 with the tranches 12 from an outside source. This may include, for example, a source of gas the leads to the delivery of gas withing the trenches 12. Chemical or electrochemical reactions may also be used to generate the gas that forms the plastron on the surface 10 of the trenches 12.

[0071] Theory and Experimental Results

[0072] Theories and Deviations from the Model

[0073] Acceptable and unacceptable plastron for drag reduction

[0074] Considering a solid surface embodied in FIGS.4A-4C immersed in water, as illustrated in FIGS.7A-7F, which also defines the trench top pitch p, trench opening or trench top opening width w, and trench depth d. The gas fraction of the surface is defined as w / p. Although most numerical studies of SHPo drag reduction assume air-water interfaces (or menisci) to be flat and pinned on the trench top edges, in reality menisci are rarely flat and2023-204-2 may not be pinned on the top. For watercraft applications, which usually involve open water in nature (whose air saturation level hovers around 100%) and hydrostatic pressure, menisci would either be pinned and concave, as shown in FIGS.7A-7C, or depinned-in and concave, as shown in FIGS.7D-7F. Note that the amount of depinning is expressed as the water intrusion depth h, which is the distance between the trench top 16 and the meniscus contact line. Compared with the pinned-and-flat meniscus, the pinned-and-concave meniscus would degrade the slip only slightly, but the depinned-in meniscus (even if flat) would degrade the slip significantly. Both numerical and analytical studies have predicted that the slip length, which determines the drag-reducing ability of a SHPo surface, on longitudinal trenches 12 would decrease by ~50% when the contact line slides down from the top surface 18 by merely 10% of the trench top opening width, i.e., h / w = 0.1, and by nearly 70% when h / w = 0.2, for the trench 12 with 0.9 of gas fraction (w / p = 0.9). Such a small amount of depinning has been unnoticeable in previous high-speed flow experiments, where the only practical way to confirm the plastron was by observing its silvery sheen, which indicates its existence but not thickness. Note even a significantly depinned interface, e.g., FIGS.7D-7F, may still appear bright on trench SHPo surfaces 10. Compounded by the fact that even a marginal depinning would lead to a substantial decrease in slip length, the common practice of confirming the existence of plastron only with its brightness helps explain the frustratingly inconsistent experimental results even with trench SHPo surfaces 10 that have been hampering the progress of SHPo drag reduction research. Considering the stringent condition that little depinning is allowable for a successful drag reduction, an acceptable plastron was defined as one with contact lines pinned or slightly depinned on the edge of the trench top 18 (e.g., h / w < 0.1, or h / w < 0.2). Note this new definition of acceptable plastron, which is useful for drag reduction, differs from the common definition of plastron lifetime that includes all shades of plastron until the meniscus hits the trench bottom 14 and instead resembles the stringent definition of plastron lifetime that includes only pinned interfaces. Assuming the depinning-caused loss of drag reduction by up to 70% (which means down to 15% of drag reduction if the pinned plastron was to provide 50% of drag reduction) is acceptable (somewhat arbitrarily), a new observation scheme was devised and implemented that can differentiate h / w ≤ 0.17 (acceptable plastron) from h / w > 0.17 (unacceptable plastron), as explained in the experimental sections.

[0075] For the contact line to stay pinned as in FIGS.7A and 7D, the pressure difference between the water above and the air inside the plastron, ∆P = ^Pwater – Pair, should be balanced2023-204-2 by the Laplace pressure of the air-water interface ^P^at the trench top 16, ∆P = ^P^. Since the trench geometry determines the minimum and maximum value of ^P^possible at the trench top 16, the range of pressure difference allowable for pinning can be expressed as: ∆^^ఙ,^^^^ ^^ ^^ ൌ ^^ ^^ఙ^ ^^ ^^ఙ,^^௫ (1.1a)^^ ^^ ^^ ^^^where σ is the surface pressure by more thancontact line will be depinned in and slide into the trench 12, as illustrated in FIG.7B. Note that the above ranges of Laplace pressure were based on the simple trench geometry with vertical sidewalls. If one adds re-entrant edges to the trenches 12, the maximum Laplace pressure increases to ∆Pσ,max = 2σ / w, expanding the pinned state, as introduced in the previous open-water drag reduction experiments. On the other hand, if the water pressure is lower than the plastron pressure by more than the minimum Laplace pressure, ∆P < ∆Pσ,min = –2σcos(θr– 90°) / w, the contact lines will be depinned out and let neighboring air pockets merge, as illustrated in FIG.7E. The latter case, i.e., FIGS.7D-7F, may occur when a SHPo surface is placed shallow in supersaturated water. While the merged air pockets may grow large and leave by buoyancy in static water as shown in FIG.7F, in fast flowing water, the overgrown plastron is mostly prevented by the shear.

[0076] The effect of hydrostatic pressure and air diffusion on plastron morphology

[0077] Diffusion of air between the plastron and surrounding water on a hydrophobic trench 12 in stationary water has been well studied using a 2D model and experimentally verified. Based on Henry’s law, the partial pressure of dissolved air in water is p = kHc, where kHis Henry’s constant and c is the concentration of dissolved air. The partial pressure of dissolved air in water can also be expressed as p = sPatm, where s is the pressure ratio of the dissolved air in the water to the atmospheric air above the water or the percentage saturation of air in water, also simply called the air saturation level. The volumetric diffusion rate of air into the plastron can be approximated by Fick’s law as ^^ ^^^^^^^ ^^^^^ ^^ ^^^^ ^^ ^^^^ ^^ ^^^ ^^^where V is thethe air-water interface, Pairis the air pressure in the plastron, A is the air-water interfacial2023-204-2 area, x is the position along the trench 12, and t is time. In static water, where the condition is uniform along the trench 12 so that Pair(x,t) = Pair(t) and A(x,t) = A(t), the above diffusion rate can be simplified as ^^ ^^^ ^^^ ^^ ^^ൌ ^^^^^^^ ^^^^ ^^ ^^^^ ^^ ^^െ ^^^^ ^^ ^^^ ^^^൧(1.3)

[0078] If the pressure in plastron^^^^ ^^ ^^, ^^ ^^ൌ ^^ ^^^^ ^^ ^^ (1.4)where the subscript st indicates static water as opposed to the dynamic water, which flows and imposes a shear stress on the plastron. Since in static water, the water pressure on the plastron is Pwater,st = PH + Patm, where PH is the hydrostatic pressure at immersion depth H, (1.4) allows the pressure difference between two sides of the meniscus to be expressed in the following way: ∆^^^௧ൌ ^^௪^௧^^,^௧െ ^^^^^,^௧ൌ ^^ு^ ^1 െ ^^^ ^^^௧^ (1.5)

[0079] To trench SHPovisualizes how the pressure difference ΔP(x) (vertical arrows in the FIG.) is determined by the hydrostatic pressure PHand air saturation level s. If ΔP(x) is larger than the largest sustainable Laplace pressure, i.e., ^P^,max, or smaller than the smallest sustainable Laplace pressure, i.e., ^P^,min, by the air-water interface, depinning would occur at location ^^.

[0080] The effect of shear by water flow on plastron morphology

[0081] SHPo surface vs. LIS: If water is not static, the flowing water will drag the trapped air with it, causing a shear-driven flow of air inside the plastron, hence increasing the air pressure toward the rear (trailing) end 24 of the trench 12. The increased air pressure at the rear end, in turn, will cause a pressure-driven flow of air in the opposite direction to the water flow. In accordance with the pressure distribution, the plastron morphology can be depicted as shown in FIG.9A, which is drawn for a simple trench 12 with length L, opening width w, and depth d and assuming L » w ~ p. The analysis for the loss of air on SHPo surfaces starts by noting there exists air diffusion across the air-water interface. By combining the three types of air flows (i.e., shear-driven, Laplace pressure-driven, and air diffusion-driven), as indicated in FIG.9B, the air pressure in the plastron along the trench 12 is obtained and the resulting meniscus morphology.2023-204-2

[0082] Water shear-driven flux: To analyze the air flux driven by the flowing water, it was assumed: (i) the air flow inside the plastron is laminar, and (ii) the meniscus is flat. Based on Liu, Y. et al., Effect of viscosity ratio on the shear-driven failure of liquid-infused surfaces. Physical Review Fluids 1, pp.074003 (2016), which is incorporated by reference herein, shear-driven flux qs can be expressed as ^^ ^^ ^^3^^ 2 ^^^^ ^^ ^^^^ൌ1 ^ 2 ^^ ^^ ^^(1.6)^^ ^^ ^^where D is the normalized the plastron that is determined by trench aspect ratio d / w andif d / w = 1 and w / p = 0.9, which are the typical parameters used for the experiments in this study, then D = 0.201. N is viscosity ratio, which is N = µwater / µair= 55 for SHPo surfaces, where µwaterand µairare dynamic viscosities of water and air, respectively. csl is a factor determined by d / w; if d / w = 1, csl= 0.108. τwis the shear stress of flowing water applied on the SHPo surface.

[0083] Laplace pressure-driven flux: In addition to the above assumptions, for simplicity, it is further assumed: (iii) the air pressure in the plastron changes linearly with x. The Laplace pressure-driven flux inside the trench 12 is ^^ ^^^^^ ^^3^^ ^^^^ ^^^ ^^^ ^^^^^ൌ െ(1.7)where c is a factor

[0084] Air diffusion-infused oil was considered to not diffuse into the surrounding water. While such an assumption was reasonable due to the insolubility of silicone oil in water, the same assumption is not reasonable for SHPo surfaces, for which the solubility of air in water is appreciable, e.g., ~0.8 mM, compelling the analysis of how the air diffusion between the plastron and flowing water would affect the plastron morphology. Note the diffusion rate across the meniscus varies along the trench 12 because the pressure of trapped air varies along the trench 12, as indicated with "air diffusion" in FIG.9B. At a steady state, for example, air would diffuse into the plastron on the leading half of the trench 12 and diffuse out from the plastron on the trailing half of the trench 12, inducing a new air flux qd that is referred to as the air diffusion-driven flux, as shown in FIG.9B. The varying air pressure along the trench 12 would also change the meniscus curvature, and thus, the meniscus area, which would affect the air diffusion rate across the meniscus. However, for simplicity here, the curvature effect is ignored leaving it for a future study. In any case, interestingly and2023-204-2 somewhat surprisingly, it was found that the air diffusion-driven flux, although clearly relevant to SHPo surfaces, is negligibly small when compared with the shear-driven and pressure-driven flow for typical flow conditions of watercraft, as analyzed in Appendix B. ^^^^≪ ^^^^; ^^^^≪ ^^^^ (1.8)

[0085] The net flux: the above the net flux of the air in the trench 12 is practically zero.^^^^^ ^^^^^ ^^^^^ ^^^^^ ^^^^ൌ 0(1.9)which means the shear- SHPo surfaces as well.

[0086] By integrating (1.6) and (1.7) into (1.9), one can get the gradient of air pressure along the trench 12 as ^^ ^^^^ ^^ ^^^ ^^^ 2 ^^ ^^2^^ ^^ൌ^^ ^^^^ 1 ^ 2 ^^ ^^ ^^3^^ ^^(1.10)

[0087] The shear as shown in FIG.8B, for a given trench geometry(i.e., D and N). As the flow speed increases (along with the shear stress), depinning would occur at the front end of trench 12 when ∆P(0) > ^P^,max (FIG.7B) or at the rear end when ∆P(L) < ^P^,min (FIG.7E) depending on which one would occur first. Based on (1.10), decreasing the trench depth d, increasing the trench opening width w, or increasing the shear stress of water τwon the trench 12 would lead to a larger pressure gradient of air, which promotes depinning on the leading or the trailing end of the trench 12, as shown in FIG.8B.

[0088] General: To understand the state of plastron on a trench SHPo surface 10 covering the hull of a traveling watercraft, one should consider all three – the flow speed, the immersion depth of the position of interest on the hull, and the air saturation level of the water. For this more general situation of interest, FIG.8C, which combines FIGS.8A and 8B, is presented to help one understand the trends of how the three main factors affect the plastron stability.

[0089] Deviations by water dynamic pressure, interfacial contamination, and turbulent fluctuation

[0090] The above subsections focused on the effects of water pressure and shear stress and ignored the effects of trench boundaries. The water pressure was assumed to be uniform on the trench 12, i.e., Pwater(x) = Pwater, ignoring the effect of solid surfaces before (x < 0) and after (x > L) the trench 12 for simplicity. However, the water pressure would decrease and2023-204-2 increase momentarily as water flows past the front (leading end 22) and rear (trailing end 24) end of the trench 12, where the boundary condition changes from no-slip to slip and from slip to no-slip, respectively. Such dynamic pressure effect has been studied on SHPo surfaces with posts but not on longitudinal trenches, which are typically modeled to be infinitely long. A qualitative analysis is presented of the dynamic pressures to understand their effects on plastron morphology, as illustrated in FIG.8D. The pressure difference between the water and the plastron at the front end would be smaller than the expected, suppressing the depinning-in at the front end. In other words, as the shear stress of water flow increases, depinning-in would start to occur slightly away from the front end. On the other hand, the pressure difference at the rear end would be larger than the expected, promoting the depinning-in at the rear end. Dedicated investigations would be needed in the future to quantitatively assess how the dynamic pressure affects the plastron morphology on longitudinal trench SHPo surfaces.

[0091] In the above subsections, the surface tension of the air-water interface was assumed to be constant, ignoring the effects of potential contaminants inevitable in the environmental water. Surfactants in water can adsorb onto the air-water interfaces, where they can be advected by the shear and accumulate at the rear end of trench 12. The accumulation may lead to the formation of a stagnant-cap region, where the surfactant reaches its maximum interfacial concentration and reduces the surface tension by ~50% for a typical surfactant such as SDS. Although the surfactant effect may dominate and practically eliminate the drag reduction for some cases, the detrimental effect by the stagnant cap is confined to a relatively short range (e.g., ~1 mm) at the rear end of trench 12 for typical flow conditions. Accordingly, the surfactant effect is relatively small for the long (> 10 mm) trenches 12 used for drag reduction in turbulent flows. Nevertheless, the surfactant may induce a premature depinning at the rear end when the lowered surface tension is compounded by the dynamic pressure. Lastly, it should be noted that numerous other effects, such as the small particles and micro-organisms that may accumulate on the meniscus and decrease surface tension, the impact of solid particles onto the meniscus, and the influence of salinity level of seawater. These and other unforeseeable environmental effects are important motivations behind performing flow experiments in a field condition, such as a passenger motorboat on natural seawater as was done here.2023-204-2

[0092] Furthermore, the above subsections considered steady-state flows with time- averaged values. For the typical flow conditions of watercraft, however, the turbulent pressure fluctuations are significant. The water pressure in turbulent flow is ^^^^ ^^ ^^ ^^ൌ ഥ ′^^ ^^ ^^ ^^ ^^,^^^^ ^^ ^^ ^^ ^^േ ^^^^ ^^ ^^ ^^ ^^ (1.11)where P̅ is the time- P' In circulating air inside thefluctuation. Since the air is confined in the trench 12, the fluctuation in water would compress and decompress the trapped air, inducing a reactive fluctuation in air that opposes the fluctuation of water.

[0093] Accordingly, the pressure difference across the water-air interface for turbulent water flow over the air trapped in trench 12 may be expressed as ∆^^^ ^^^ ൌ ^^௪^௧^^,௧௨^^െ ^^^^^^ ^^^ ൌ ^ത^௪^௧^^െ ^ത^^^^^ ^^^ ^ ^^௪ᇱ^௧^^െ ^^^ᇱ^^ (1.12)describe the plastron with the above equation to account for the turbulent fluctuations, at this point one can point to reports in the literature. For example, Rastegari, A. et al., On drag reduction scaling and sustainability bounds of superhydrophobic surfaces in high Reynolds number turbulent flows, Journal of Fluid Mechanics 864, pp.327-347(2019) studied the probability density function (p.d.f) of the wall pressure fluctuations in turbulent channel flows of water on longitudinal trench SHPo surfaces 10, assuming a shear-free interface (τw,air= 0) and infinitely deep and long trench (i.e., d ^ ^, L ^ ^), and showed an estimate of the upper limit for P'waterto be ′^ ^ 3^^^^ ^^ ^^ ^^ ^^ൌ 4^ ^^′^^ ^^ ^^ ^^ ^^^^^ ^^ ^^ൌ 4 ^2.32 ln ^^ ^^^^4^ ^ 2.31 ln^ ^^ ^^^^^ െ 14^(1.13)for w+≳ 5 with 99.75% confidence.

[0094] The pressure fluctuation was normalized by the wall shear stress of the SHPo surface τw, i.e., P'water+= P'water / τw. The trench opening width was nomalized by the wall unit of the turbulent bounday layer, i.e., w+= w / δv, where the wall unit is defined as δv = ν(τw / ρ)-1 / 2, ν is kinematic viscosity of water, and ρ is the density of water. The friction Reynolds number is defined as Reτ = δ / δν, where δ is the boundary layer thickness. Because the infinitely deep trench would have no air compression, making P'air= 0 in (1.12), (1.13) may be viewed as an extreme case of (1.12). On the other hand, Piao, L. et al., Two-dimensional analysis of air–water interface on superhydrophobic grooves under fluctuating water2023-204-2 pressure, Langmuir 31, pp.8022-8032 (2015) studied how the pressure fluctuation in water affect the lifetime of plastron on a longitudinal trench SHPo surface 10, which have a finitely deep (limited d) and infinite length (L ^ ^) trench, by considering the gas compression (i.e., P'air ്0 in (1.12)) and viscous dissipation induced by the fluctuation. Using the fluctuation data reported by Tsuji, Y. et al., Pressure statistics and their scaling in high-Reynolds-number turbulent boundary layers. Journal of Fluid Mechanics 585, pp.1-40 (2007), which is incorporated herein by reference, for common high Reynolds number flows and assuming the trench geometry similar to that investigated herein, it was found the fluctuation would not affect the plastron stability in the shallow water used for the current flow experiments.

[0095] Experiments and Methods

[0096] The boat and underwater cameras

[0097] The motorboat (13-foot Boston Whaler) retrofitted for the drag reduction research used in Xu, M. et al., Superhydrophobic drag reduction for turbulent flows in open water. Physical Review Applied 13, pp.034056 (2020) (incorporated by reference herein) was used for experiments. Since shear-induced wetting, which was not observed in the boat test of Xu et al. (2020), was found during the high-speed tow tank test by Xu, M. et al., Superhydrophobic drag reduction in high-speed towing tank. Journal of Fluid Mechanics 908, pp. A6 (2021) (incorporated by reference herein) for similar SHPo surfaces, the boat was revamped to increase its top speed. By adding a hydrofoil stabilizer (Doel-Fin Hydrofoil, Davis Instruments) to the outboard motor, the boat top speed was increased from 10 knots to 14 knots, increasing the maximum shear rate attainable on the sample surface from ~5500 s-1to ~8300 s-1. A test well, which replaces a portion of the boat hull with a testing unit including sample surfaces, was installed on the boat as shown in FIG.10A. A custom- developed shear stress sensor was used, as shown in FIG.10B, to measure the shear stress on the SHPo surface during the boat test with uncertainties of 0.1τw0, where τw0 is the measured shear stress on a smooth surface. An overall picture of the retrofitted boat is shown in FIG. 10C.

[0098] Two miniature underwater cameras with waterproof rating IP67 (TODSKOP 5.5 mm WiFi Borescope) were used to monitor the plastron status on the SHPo surface during the boat test. Each camera was held in its own 3D-printed housing with a streamlined profile and installed as shown in FIG.10D (one black and one white) to observe the sample from a specific distance and direction, so that together, the two cameras can accurately monitor the plastron states over the entire sample surface. The side camera observed the SHPo surface in2023-204-2 the spanwise direction of the trench 12 with an elevation angle ^ = 10±2°, which is the angle between the sample surface and the camera central axis. Here, the trenches 12 used resembled the configuration of FIGS.4A-4C. When the sample is observed from this specific elevation angle ^ = 10±2°, the regions with 0 ≤ h / w ≤ 0.17±0.04 (i.e., pinned and slightly depinned interface) appeared bright with the well-known silvery sheen, while the regions with h / w > 0.17±0.04 (i.e., depinned and no interface) appeared dark. The smallest detectible depinning is determined by the minimum elevation angle, which is limited by the camera's depth of focus and the size of the surface to observe. On the other hand, from the rear camera, which observed the surface in the parallel direction of the trench 12, the regions with h / w < d / w (i.e., any plastron) appeared bright, while the regions with h / w = d / w (i.e., no plastron) appeared dark. For the experiments in this study, if a type of trenches 12 appears bright from the side camera, it has a pinned or slightly degraded plastron (i.e., deemed acceptable for drag reduction). If a type appears dark from the side camera, it has a degraded or no plastron (i.e., deemed unacceptable). Although not used to determine the acceptable and unacceptable plastron, the rear camera helped to understand how the plastron is morphed inside the trench 12 by differentiating the depinned interface from no interface along the trench length.

[0099] Preparation of SHPo surface samples

[0100] A series of SHPo surface samples with trenches 12 were prepared, as shown in FIG.11A. To test different Laplace pressure limitations, 3 different roughness types of longitudinal trenches 12 shown in FIG.11A were prepared. The first roughness type was trenches 12 with a re-entrant shape at the top edge of the trench 12 (named RE). The second roughness type was trenches 12 without a re-entrant edge but covered with nano-grass (named NG). The third roughness type had both the re-entrance and nano-grass (named RE+NG). For each roughness type, 4 different trench depths were prepared, making a total of 12 samples (40 mm × 70 mm in size) each diced out from a 4-inch silicon wafer. Since there are 10 different combinations of trench opening widths and lengths on each sample, a descriptive name was used for each trench geometry. For example, NG_d90-p75L30 points to the section filled with trenches 12 of 75 µm pitch and 30 mm length on the sample of the nano-grass (but no re-entrance) type and 90 µm trench depth.

[0101] The trenches 12 were made on silicon wafer by developing 3 different fabrication processes of micro electro-mechanical systems (MEMS) based on photolithography, deep reactive ion etching (DRIE), and atomic layer deposition (ALD). For the 3 roughness types shown in FIG.11A, the first type (RE) was trenches 12 with re-entrance at the top edge of the2023-204-2 trench 12. This type was used for the boat study by Xu et al. (2020) and tested for a comparison in this study. The DRIE recipe was modified to create a ~250 nm of undercut below the ~500 nm thick silicon dioxide layer on top of trenches 12, thus creating the re- entrance, which is shown in the top SEM of FIG.11A. The sawtooth-like sidewall below the re-entrance is by how DRIE works and should be considered smooth in nanometer scale. The second type (NG) was removed of the re-entrant edge by adding hydrofluoric wet etching after the DRIE. Following the wafer dicing, the surface was conformally coated with a ~55 nm thick Al2O3layer by ALD and then immersed in a 60 °C deionized water bath for 10 minutes to roughen the Al2O3 into a nano-grass. The middle SEM picture of FIG.11A shows the top edge with no re-entrance and the entire surfaces uniformly covered with nano-grass with ~100 nm of roughness. The third type (RE+NG) had both the re-entrance and nano- grass, as shown in the bottom SEM picture of FIG.11A, by omitting the hydrofluoric wet etching in the processing steps of the second type. For each of the three roughness types, 4 samples with increasing trench depths (i.e., d = 50.6, 67.5, 90, 153 µm) were prepared by increasing the etching time of DRIE. Hence, each of the 12 samples has a unique roughness type and trench depth. Once the trenches 12 were formed, all the samples were cleaned by O2 plasma and then coated uniformly with the self-assembled monolayer (SAM) of 1H,1H,2H,2H-perfluorodecyltrichlorosilane (FDTS) in a custom-made vapor-based coater to achieve superhydrophobicity. The contact angles of water on FDTS-coated smooth silicon and Al2O3 nano-grass were measured with an in-house contact angle measurement apparatus and summarized in Table 2. Table 2 Type of surface θ (°) θa(°) θr(°)

[0102] On each sample, trenches 12 with a combination of 2 different pitches (p = 75, 100 µm) and 5 different lengths (L = 2.5, 5, 10, 30, 60 mm) were fabricated, as schematically shown in FIG.11B. A sample was cleaved along the vertical broken line drawn on the schematic to obtain the two SEM pictures (p75 and p100), which show the two different pitches. The gas fraction of all trenches 12 was kept at 90%, i.e., w / p = 0.9. The 30 mm × 60 mm micromachined area in the middle was divided into 10 parallel sections each ~3 mm wide and containing 42 or 30 parallel trenches 12 of p = 75 µm or 100 µm, respectively. The2023-204-2 section width was, in part, designed based on the resolution of the side camera. To provide the 5 different trench lengths, 8 of the 10 parallel sections were further divided into multiple (2, 6, 12, or 24) shorter trenches 12, the top SEM showing one such partition. The smooth area outside the micromachined area was to prevent the flow disturbances by the gap between the sample and the surrounding plate. Since 12 different samples were fabricated to provide combinations of 3 roughness types (i.e., RE, NG, and RE+NG) and 4 trench depths (i.e., d = 50.6, 67.5, 90, 153 µm), a total of 120 different trench geometries have been prepared for flow experiments.

[0103] The flow experiments

[0104] To comprehensively compare the effects of hydrostatic pressure, air diffusion, and shear stress on different SHPo samples, all the flow tests were performed in brackish water with air saturation level at 100–101% in the mouth of a creek (Ballona Creek, Los Angeles, California, U.S.A.) meeting the Pacific Ocean. The air saturation level was monitored regularly by a total gas sensor (Point FourTMtracker, PENTAIR), and the specific testing area was determined for each test based on the air saturation level within the 2-mile range inside the creek. One end of the range was the creek’s entry point into the ocean, where the air saturation level tended to be 104–106% due to the wind and waves on the ocean, while the other end was the farthest upstream point allowed by the transportation rules, where the air saturation level was measured to be constantly below 99%. The air saturation level gradually decreased away from the ocean but varied significantly by the tide and wind conditions, requiring the measurement of the air saturation level regularly and often. At high tide, the ocean water would enter the creek, increasing the air saturation in the upstream end to as high as 100–101%, while at low tide the ocean water would retreat from the creek, decreasing the air saturation in the downstream end (i.e., the entrance point) to as low as 99–100%.

[0105] Each sample was tested with boat speeds varying from 2 m / s to 7.2 m / s with ~0.5 m / s intervals. For each test, the boat remained stationary at first, then accelerated to the target speed in ~5 seconds and maintained the target speed for ~40 seconds for observation. The sample was kept under water during the entire test trial (typically 30–40 min), and its immersion depth was measured to be 0.15±0.03 m for all tests. The boat was carefully trimmed (i.e., weight distributed carefully) to maintain a ~3º running (tilting) angle, measured by an inclinometer (H4A1-45 Inclinometer, RIEKER), and a constant waterline at all the target speeds. To estimate the shear stress on the SHPo surface for a given boat speed, a smooth 40 mm × 70 mm silicon sample, diced from a 4-inch bare silicon water, was attached2023-204-2 to the shear stress sensor and its shear stress τw0 was measured at different speeds multiple times. Based on the shear stress versus speed data, the relation between smooth surface shear stress τw0 and boat speed U, was derived using the power regression method. The shear stresses on the SHPo surface were, then, estimated from that on the smooth surface from τw~ 0.7τw0, which was found in the previous research using similar surfaces and the same boat (Xu et al.2020). After the flow experiments, the samples were cleaned, dried, and examined under SEM to confirm their integrity including the nano-grass structures.

[0106] Results and Discussions

[0107] Image pairs collected, plastron length measured, and key trends confirmed

[0108] The images from the side and rear cameras were analyzed as pairs to determine the state of plastron along the trench 12: (i) pinned or slightly depinned interface (i.e., h / w ^ 0.17 in this study, limited by the underwater cameras availability), (ii) depinned interface (i.e., 0.17 < h / w < d / w), and (iii) no interface (i.e., h / w = d / w). The plastron length Lp was obtained by measuring the length of plastron in the first state. In other words, the depinned interface is excluded when defining Lp in this study. If a trench 12 is filled with the plastron of the first state of interface (i.e., h / w ^ 0.17) over the entire length (i.e., Lp= L < Lss), the trench 12 is deemed to have a pinned or slightly degraded plastron, which is acceptable drag reduction. For all other cases (i.e., Lp = Lss < L), the trench 12 is deemed to have degraded or no plastron, which is unacceptable. All the sample images obtained from the boat tests – a pair of images at each of ~10 different boat speeds for each of the 12 samples, i.e., a total of ~120 image pairs with each covering 10 different trench types were analyzed, producing ~1200 data points of Lp.

[0109] Throughout the collected data, the plastron length Lp increased with trench depth d and decreased with trench top opening width w (or pitch p) and boat speed U, as expected from the theory. Several sample images were selected in FIGS.12A-12E to reveal key trends. The selected ones were more often RE samples because the loss of plastron was rare (i.e., difficult to spot trends) on NG and RE+NG samples. FIG.12A shows a rear-view picture of an RE sample with d = 67.5 µm (i.e., RE_d67.5) at U = 5.5 m / s. The image revealed trenches 12 with p = 75 µm had longer plastron than those with p = 100 µm on 60 mm long trenches, indicating a stronger plastron stability on narrower trenches 12, as expected. FIG.12B presents 4 side-view pictures of 4 RE samples with d = 153 µm (i.e., RE_d153) taken at 4 different flows speeds (U = 3.8, 4.6, 5.5, 6.7 m / s). The images of 60 mm long trenches 12 revealed pinned or slightly degraded plastron (i.e., Lp = L < Lss) at speeds up to U = 5.5 m / s2023-204-2 but degraded plastron (i.e., Lp = Lss < L) at U = 6.7 m / s, indicating weakened plastron stability at higher flow speeds, as expected. Incidentally, most of the 60 mm long trenches 12 on RE sample (i.e., RE_d153-p75L60 and RE_d153-p100L60) were found maintaining a pinned or slightly degraded plastron up to U = 5.5 m / s, corroborating the existence of plastron reported in Xu et al. (2020). FIG.12C presents 4 side-view pictures of 4 RE+NG samples with 4 different trench depths (i.e., RE+NG_d50.6, RE+NG_d67.5, RE+NG_d90, and RE+NG_d153) at a high speed (U = 6.3–6.7 m / s). While the depinning of interfaces by high shear stress was apparent on shallow trenches 12 (RE+NG_d50.6), the degraded plastron on the front region of the trench 12 was shortened and disappeared with increasing trench depth, as predicted by the theory.

[0110] FIG.12D presents 3 pairs of pictures taken from 3 samples of different roughness types with d = 90 µm at a high speed (U = 6.4–6.7 m / s). On the sample with re-entrance but without nano-grass (e.g., RE_d90), most trenches 12 had regions of no plastron. In comparison, on the samples with nano-grass regardless of re-entrance (e.g., NG_d90 and RE+NG_d90), nearly all trenches 12 were found to have a pinned or slightly degraded plastron, demonstrating the effectiveness of adding nano-grass to the trench 12. Incidentally, note the RE sample was populated with no interface and pinned or slightly depinned interface but no depinned interface. The lack of the depinned interface on RE was likely because once the meniscus is depinned from the top edge, where the re-entrance (on which θa~ 180°, effectively) maximizes the Laplace pressure, the smooth sidewalls (on which θa ~ 116°) could not provide the same level of Laplace pressure, letting the contact line slide down quickly to the fully wetted state (i.e., no interface). On the other hand, while the region of no interface was negligible on the NG and RE+NG samples, depinned interface were found on shallow trenches 12 at high speeds. The depinned interface was likely because the rough sidewalls (on which θa~ 166°) provided a similarly large Laplace pressure as the top edge. In other words, the nano-grass, while increasing the plastron stability, especially calls for an appropriate observation method, such as the two-camera system used in this study, to detect the degraded plastron, which may otherwise be interpreted as a pinned or no plastron.

[0111] Deviations from the linear increase of air pressure along a trench

[0112] As noted herein, additional effects may cause the plastron morphology to deviate from the trend of linearly decreasing pressure difference along the trench 12. The magnitude of pressure difference is expected to be smaller at the rear end than at the front end, as depicted in FIG.8C, because Pwater> Pairin the current experimental conditions. First, for an2023-204-2 example, as shown on the 60 mm length trenches 12 in FIG.12E (i.e., p75L60, p100L60), while a significant portion of the front region had no plastron, the very front end was found to have a plastron. This is a deviation from the linear theory, which predicts the pressure difference increasing toward the front of trench 12. This small but interesting deviation from the front wetting can be explained by the pressure of the flowing water decreasing right past the front end, as depicted in FIG.8D. Second, throughout the collected images, including FIG.12E, the plastron was frequently found to be lost at the rear end. This is a deviation from the linear theory, which predicts the pressure difference decreasing toward the rear of trench 12. This deviation, which is termed "rear wetting", may be partially explained by the water pressure increasing near the rear end, as explained with FIG.8D. However, the deviation at the rear end was found to be more common and more pronounced than the deviation at the front end. For example, rear-wetting was observed on all trenches 12 of all RE samples at U > 4.6 m / s and some trenches 12 on NG and RE+NG samples. The stronger deviation at the rear end may be explained by the pressure increase by the dynamic flow exasperated by the negative effects of interfacial contaminants, as explained herein. Also, the rear wetting was not directly affected by the trench length, making its wetting effect more significant on shorter trenches 12. For an example, on the RE_d67.5 sample shown in FIG. 12E, the rear wetting had a relatively small effect (< 5%) on p75L60, but a large effect (~50%) on p75L2.5. In addition, the rear wetting tended to be more significant on deeper trenches 12, possibly because the trapped air there was more compressible and provided less dynamic resistance against depinning. In any case, the rear wetting was found to be ~4 times shorter on NG and RE+NG than on RE, manifesting another significant benefit of nano-grass for future applications.

[0113] The mechanism of rear wetting calls for a significant investigation in the future, as it seems inevitable for drag-reducing SHPo surfaces. As discussed herein, the rear wetting may arise from the increased local water pressure when the boundary condition changes from slip to no-slip at the trench end 24, combined with the stagnant cap formed by the surfactant (or particles) advected to the rear end. While the former would require numerical and experimental studies of hydrodynamic issues involving free surfaces, the latter would further involve diffusion and interfacial phenomena.

[0114] Comparisons with the theoretically estimated steady-state plastron length

[0115] While further advanced analysis is necessary for unifying all the factors for the prediction of the plastron length in turbulent flow, for convenience here a preliminary2023-204-2 estimation of the steady-state plastron length is prepared to compare with the experimental conditions, where the leading end 22 of trench 12 tends to be depinned first. Based on the equilibrium state of the air pressure in static water, (1.4), and linear gradient due to the shear, (1.10), the air pressure is estimated to be in the scale of ^^ ^^ ^^ ^^ ^^ ^^2^^ ^^2^ഥ^ ^ ^^^ ~ ^^ ^^ ^ ^^^^ ^^1 ^ 2 ^^ ^^^^^^^ െ ^^^(2.1)^^3^^^^2which shows the

[0116] Thethe plastron along the trench 12 can be expressed based on (1.12) with the following trend. ^^ ^^ଶ∆ ^^^ ^^^ ~ ^ത^௪^௧^^^ ^^௪ᇱ 2 ^^^^^^ ^௧^^െ ^^ ^^^௧^െ ᇱ^^^ ^^ ^^ ^^^^ ൬ ^^ െ ^ െ ^^ (2.2)expected to reach the Laplace pressure limitation prior to the trailing end 24 of trenches 12, leading to ΔP(0) = ΔPσ,max, which leads to an estimated trend of the steady-state plastron length as ^^^^3′ ′1 ^ 2 ^^^^^^ ^^ ^^ ~ ^∆ ^^ ^^ ^^ ^^ െ ^ഥ^ ^^ ^^ ^^ ^^ ^^ ^ ^^ ^^ ^^ ^^ ^^ െ ^ ^^ ^^ ^^ ^^ ^^ ^^ െ ^^ ^^ ^^ ^^ ^^(2.3) where,. of the pressure fluctuation terms remains unclear at this point, the nano-grass coverage would certainly make the plastron more stable on NG and RE+NG, compared with RE used in the previous open-water studies (Xu et al.2020; Xu et al.2021).

[0118] To qualitatively show the effects of nano-grass, trench dimensions, and flow conditions (i.e., wall shear stress), the actual plastron lengths Lpon 60 mm long trenches 12 were measured from all the images using ImageJ and the estimated theoretical plastron lengths Lssfrom (2.3) were compared and accordingly showing similar trends. If Lss< Lmax= 60 mm, the interface at the front of the 60 mm long trench 12 should be depinned, and the plastron length could be observed as Lp= Lss. For the calculation of the theoretical estimation, the flow conditions of the experiments were used: air saturation level within s = 100–101%, average water pressure as ^ത^௪^௧^^~ 1500 Pa, and the wall shear stress on the SHPo surface τw estimated from the boat speeds U measured using the regression equation in Appendix D. Besides, the pressure fluctuation term, i.e., P'water – P'air, was intentionally ignored in the2023-204-2 estimation range to allow the comparison. By increasing the boat speed beyond the speeds used by Xu et al. (2020), which did not observe any shear-driven wetting, severely degraded plastron was observed on the same RE sample. In comparison, the NG and RE+NG samples were confirmed to have a clearly improved plastron stability and showed a better matching between the estimated ranges and experimental results. Although the theoretically estimated range of plastron length on RE was similar to those on NG and RE+NG, the metastable state of the re-entrant edge on RE was vulnerable to the many fluctuations in the environmental water and the pressure fluctuation of the highly turbulent flows under the boat.

[0119] The rear wetting made the plastron shorter than the estimation by (2.3) on NG and RE+NG especially for d = 90, 153 µm, but the effect was small (< ~8%). There was no significant difference between NG and RE+NG, as expected from the theory. It should be noted that, for simplicity, the theoretical wall shear stress on the SHPo surface τwwas estimated from the wall shear stress on a smooth surface τw0 by assuming 30% drag reduction for all the speeds, which was the typical drag reduction value from the previous works for p100 (Xu et al.2020; Xu et al.2021) with gas fraction w / p = 0.9 in turbulent boundary flows for U > ~5 m / s. Although the theoretical wall shear stress should be 10–20% larger than the estimated ones for U < ~5 m / s, this effect was expected to be ignorable because a larger Lss would not change the fact that all the surfaces should have a pinned plastron (i.e., Lp = L < Lss) at low speeds (i.e., U < ~5 m / s) anyway due to the small wall shear stress. Besides, the theoretical shear stress for p75 should be 5–10% larger than that on p100 in the same water flow due to the smaller pitch (Xu et al.2021), increasing Lssvalues for p75 surfaces by 5– 10%, while the experimental values will still fit the estimated values reasonably well. It is noted that the theoretically estimated steady-state length Lssof (2.3) was for pinned plastron, while the experimentally measured Lp was for both pinned and slightly degraded plastron due to the finite resolution of observation. Unfortunately, the current shear-driven drainage models do not allow one to quantitatively estimate how a slight degradation of plastron would affect its steady-state length. However, it is believed that the effect was minor because a slightly degraded plastron is unstable with a very short lifetime in the current experiment, making its population small in the measured data. Most importantly, NG and RE+NG have been demonstrated to maintain a pinned (including slightly degraded) plastron in the 60 mm long trench 12 in turbulent boundary layer flows up to 7.2 m / s in accordance with the theoretical estimation, suggesting a direction toward high-performance SHPo surfaces for drag reduction.2023-204-2

[0120] To evaluate longitudinal trench SHPo surfaces 10 in high-speed flows of open water, which represent the operating conditions of common watercraft, the sustainability of pinned plastron was studied how the pinned plastron is affected by the pressure, air saturation level, and wall shear of the water, and how the trends may be distorted by other factors, such as trench boundaries, surfactant, and turbulent fluctuation. To model the effect of water pressure, an existing theory was used. To model the effect of wall shear stress of flowing water, another existing theory was used after a scale analysis revealed the diffusion of trapped air by the wall shear is small for the tested flow conditions. Distortions by the dynamic effect of flows were anticipated at the front and rear ends of the trench 12 and corroborated by a numerical simulation. To evaluate the theoretical models and the distorting effects, trench SHPo surfaces 10 with combinatorial variations of trench opening width, trench depth, trench length, and nano-roughness have been prepared and tested underneath a 13-foot motorboat in brackish water at a sea mouth. A unique observation technique using two underwater cameras was employed to differentiate pinned (and slightly degraded) plastrons from degraded (and no) plastrons rather than the common practice of determining whether the plastron is present or depleted. The experimental results corroborated the theoretical estimations reasonably well, considering the many assumptions in the models and the uncertainties inevitable in the field tests. When the trench surfaces were coated with nano-grass, nearly all the trenches 12 tested were confirmed to have a pinned (or slightly degraded) plastron. This work contributed to designing SHPo surfaces geared toward field conditions for drag reductions, anti-biofouling, anti-corrosion, etc.

[0121] Appendix A. Plastron loss by shear drainage at high-speed flows

[0122] Underwater videos of the longitudinal trench SHPo surface used by Xu et al. (2020) and tested at two different maximum speeds of boat. The sample was filled with 7 cm long trenches 12 made of re-entrant edges and smooth sidewalls 20 (i.e., type RE by the designation of this report). For these close-up videos, two side cameras were used simultaneously (differently from the use of one side camera in the main study) to cover an entire sample. The videos confirmed a pinned plastron being maintained at speeds up to 8 knots (4.1 m / s), which was near the maximum boat speed tested by Xu et al. (2020), and the videos showed the plastron being drained out by the shear stress at 10 knots (5.1 m / s) and completely lost at 13 knots (6.7 m / s), motivating the current study of developing the nano- grass-covered SHPo surfaces 10. Following the test procedures by Xu et al. (2020) and unlike the current disclosure, the air saturation level was not measured for this visualization.2023-204-2

[0123] Appendix B. Scaling comparison of the three air fluxes in a trench

[0124] The air diffusion across the air-water interface (meniscus) will lead to a diffusion- driven air flow inside the plastron. Since air diffusion rate across the meniscus varies with the Laplace pressure and meniscus area, the diffusion-driven flow flux scales as qd~ kpσL, where kp is the interfacial mass transfer coefficient, σ is the air-water interfacial tension, and L is the trench length. In turbulent boundary layer flows, kpis defined by “film theory” (Cussler & Cussler 2009) as ^^ ^^^^ൌ^^ ^^ ^^ ^^^^ ^^ ^^ ^^ ^^ ^^ ^^ ^^(B1)

[0125] where Dairis the molecular weight of air (i.e., 29diffusion length, which depends on the flow condition. Henry’s constant kH(i.e., 1.21–1.34 atm / mM) is irrelevant to the hydrostatic pressure unless the immersion depth is very large (e.g., kHincreases ~14% at immersion depth ~1000 m. The effect of Reynolds number on diffusion length in turbulent boundary layer flow over a SHPo surface has been previously studies and the relation between the Sherwood number SH^0and friction Reynolds number Reτ0, where subscript 0 indicates a smooth surface, was found as SH^0 = 0.34Reτ00.913. Here, SH^0= ^0 / δc, where ^0is the momentum boundary layer thickness on a smooth surface and approximated to be ^0 / x̂ = 0.01277Rex-0.1341for the purpose of scaling estimation, with the Reynolds number definedx̂ is the streamwise distance from the leading edge of immersed boat hull to the sample surface (x̂ is labeled as ~2 m in FIG.10A). Since the boat is tilted and elevated (by planing) when it speeds up, i.e., x̂ decreases slightly with U, the tilting angle and waterline of the boat were measured for all the individual experiments to estimate x̂ for each test run. For the purpose of estimating friction Reynolds number Re^ ^ ^ ^^ ^δ0(τw0 / ρ)0.5 / ν, the boundary layer thickness on the smooth surface δ0can be approximated to be δ0 / x̂ = 0.16Rex-1 / 7for the purpose of scaling estimation, and the shear stresses on smooth surface τw0 were established. In the turbulent boundary layer flow under the boat setup, using the relations above, one can estimate δcat the minimum and maximum Reynolds number (Rex = 5.17×106and 1.10×107, corresponding to the minimum and maximum boat speed U = 2.0 m / s and 7.2 m / s) to be 6.75 and 1.93 µm, respectively. The parameters estimated using the relations above are listed in Table 3, where U and x were measured from experiments and other parameters were estimated from theoretical equations. By inputting these values in (B1), one can obtain kp = 0.5×10-10–1.6×10-10m / (s‧Pa).2023-204-2 Table 3 U (m / s) x̂ (m) Rex^0(10-3m) Reτ0δc(10-6m) 2.0 2.3 5.17×1063.70 3255 6.75 7.2 1.5 1.10×1072.21 6549 1.93

[0126] By considering trenches 12 with aspect ratio d / w = 1 and gas fraction w / p = 0.9, based on (1.6)–(1.7), the shear-driven flow and the pressure-driven flow scale as qsl ~ 10-3τw3µair-1and qp~ 10-2h3σµair-1L-1, respectively. Then, one can estimate the magnitude of each flux inside the 12 the exemplary values into the scaling equations. By assuming (i) thetrench top opening width and depth of w = h = 90 µm and length of L = 60 mm and (ii) the shear stress on the SHPo surface is τw ~ 50 Pa, the scaling equations lead to qd / qsl~ O(10-4) and qd / qp~ O(10-4). Therefore, it was concluded that, for the trench geometries and flow conditions relevant to the current study, the diffusion-driven flow is negligibly small compared with the shear-driven flow and pressure-driven flow.

[0127] Appendix D. Experiments for shear stress vs. boat speed

[0128] The shears stress on a smooth surface τw0underneath the boat has been measured with the custom shear sensor over the range of boat speeds U used in the current study and plotted in FIG.13. The power regression line, which fits the experimental data reasonably well (especially considering the varying environmental conditions the field tests are subjected to), is used to estimate the wall shear stress for the plastron observation runs. The power regression also ensures the shear stress is zero when the boat speed is zero. The shear stress on the SHPo surface, τw, is estimated as 0.7τw0, which assumes 30% drag reduction. The estimation for 30% drag reduction of SHPo surface was confirmed by previous work on the same boat (Xu et al.2020) and in a high-speed towing tank under similar flow condition (Xu et al.2021). The ~30% drag reduction has also been proven to be consistent by the authors' recent experiments (unpublished), which used the same flow conditions as this study.

[0129] While embodiments of the present invention have been shown and described, various modifications may be made without departing from the scope of the present invention. It should be appreciated that reference to the “top” or “bottom” surfaces does not necessarily refer to the orientation of the respective surfaces in the final state. For example, consider trenches 12 formed on the hull of a ship. The trenches 12 have their respective top surfaces 18 of the trench tops 16 that are actually oriented below the bottom surface 14 given the orientation of the trenches 12 on the underside of the surface 10 of the ship hull. The2023-204-2 invention, therefore, should not be limited, except to the following claims, and their equivalents.

Claims

2023-204-2 What is claimed is:

1. A solid surface having a plurality of trenches for maintaining a plastron on the solid surface during flow of a fluid over the solid surface, wherein the trenches are defined by a leading end and a trailing end, a bottom surface and trench tops, wherein each trench of the plurality of trenches satisfies the following conditions: (a) a width (w) of an opening between the trench tops arranged with a pitch (p) is within the range of 10 µm and 100 µm; (b) a pitch (p) and a width (p–w) of the trench tops are arranged so that a gas fraction (∅ = w / p) of the trench tops is equal to or larger than 70% (∅ ^ 0.7); (c) a depth (d) of the trench is greater than w / 2; (d) a thickness (t) of the trench tops is equal to or larger than a width of the trench tops (p–w) and equal to or smaller than the trench depth (d) (p–w ^ t ^ d); (e) a length (L) of each trench is between 1 cm and 20 cm; (f) the trench tops comprise respective top edges having a radius of curvature (r) smaller than 10 microns; (g) wherein the trench tops comprise side surfaces and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw> 100º); (h) wherein the leading ends and trailing ends comprise side surfaces and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º); (i) wherein the trench comprises a bottom surface and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw> 100º); and (j) wherein the trench tops are aligned nominally parallel to a flow direction of the fluid over the solid surface and the top surfaces of the trench tops are substantially flush relative to one another and to top surfaces of the leading end and the trailing end.

2. A solid surface of claim 1, wherein substantially all of the trench tops do not contact the bottom surface.2023-204-2 3. A solid surface of claim 1, wherein substantially all of the trench tops contact the bottom surface.

4. A solid surface of claim 1, wherein at least some of the trench tops or portions thereof contact the bottom surface.

5. The solid surface of claim 3, wherein the trench tops contact the bottom surface via a post or column.

6. The solid surface of claim 4, wherein the trench tops contact the bottom surface via a post or column.

7. The solid surface of any of claims 1-6, wherein the trench tops are continuous between the leading end and the trailing end.

8. The solid surface of any of claims 1-6, wherein the trench tops have one or more gaps (g) between the leading end and the trailing end and wherein the gap (g) is smaller than the width (p–w) of a trench top (g < p–w).

9. The solid surface of claim 1, wherein the trench tops are continuous between the leading end and the trailing end and wherein substantially all of the trench tops do not contact the bottom surface.

10. The solid surface of claim 9, wherein posts or columns connect at least some of the trench tops to the bottom surface.

11. The solid surface of claim 10, wherein one or more gaps (g) are located in the trench tops and wherein the gap (g) is smaller than the width (p-w) of a trench top (g < p-w).

12. The solid surface of claim 1, wherein the width (w) of opening between the trench tops is tapered along the flow direction near the leading end and / or the trailing end of the trench.2023-204-2 13. The solid surface of claim 12, wherein the tapering comprises a tapered length of less than 2 mm in the flow direction.

14. The solid surface of claim 1, wherein the top edges of the trench tops comprise a re-entrant edge.

15. The solid surface of claim 1, wherein the trench comprises nano-roughness disposed substantially over one or more of the bottom surface, leading end surface, trailing end surface, and a surface of the trench tops.

16. A solid surface having a plurality of trenches for maintaining a plastron on the solid surface during flow of a fluid over the solid surface, wherein the trenches are defined by a leading end and a trailing end, a bottom surface and trench tops, wherein each trench of the plurality of trenches satisfies the following conditions: (a) a width (w) of an opening between the trench tops arranged with a pitch (p) is within the range of 10 µm and 100 µm; (b) a pitch (p) and a width (p–w) of the trench tops are arranged so that a gas fraction (∅ = w / p) of the trench tops is equal to or larger than 70% (∅ ^ 0.7); (c) a depth (d) of the trench is greater than w / 2; (d) a thickness (t) of the trench tops is equal to or larger than a width of the trench tops (p–w) and equal to or smaller than the trench depth (d) (p–w ^ t ^ d); (e) a length (L) of each trench is between 1 cm and 20 cm; (f) the trench tops comprise respective top edges having a radius of curvature (r) smaller than 10 microns; (g) wherein the trench tops comprise side surfaces and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º); (h) wherein the leading ends and trailing ends comprise side surfaces and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw> 100º); (i) wherein the trench comprises a bottom surface and substantial portions thereof are hydrophobic with a local (intrinsic) contact angle of water (θw) larger than 100 degrees (θw > 100º);2023-204-2 (j) wherein the trench tops are aligned nominally parallel to a flow direction of the fluid over the solid surface and the top surfaces of the trench tops are substantially flush relative to one another and to top surfaces of the leading end and the trailing end; (k) wherein the trench tops are continuous between the leading end and the trailing end; and (l) wherein all of the trench tops contact the bottom surface.

Images