Drag reduction of a miniature boat with superhydrophobic grille bottom

Friction drag, resulted from the surface fluidic shear stress, occurs in all the cases where a relative motion exists between solid surface and fluid. Drag reduction techniques have received more and more attention in recent years due to their promising applications with great benefits in many areas. For example, any significant amount of drag reduction in ground, marine and aeronautic transportations can bring us a remarkable saving of fossil fuel. In the airplane design, only a few percent of friction drag reduction may improve its performance to a high level. In the micro- and nano-fluidics, the drag reduction plays an important role in controlling the delivery speed and accuracy of a picoliter of fluid. Several techniques have been proposed so far to reduce the friction drag: water-repellent wall,^1,2 microbubbles,^3,4 surface microstructure,^5,6 turbulent flow,^7,8 surfactant additives,^9,10 and so on.

Over the past decades, scientists constantly get inspiration from nature, such as lotus leaf,¹¹ water strider,¹² mosquito legs,¹³ etc., to study and mimic superhydrophobic surfaces. Experiments indicated that superhydrophobic surfaces have considerable friction drag reduction effect.^1,14 It is commonly explained that the superhydrophobic surface reduces friction drag by supporting a shear-free air-water interface over which flow slip occurs. According to the slip law, the friction drag reduction is b/h for the parallel plates with relative motion as shown in Fig. 1, where b is the slip length and h is the gap. The slip length usually ranges from nanometer scale to micrometer scale depending on the surface property.¹⁵ This indicates that only when the gap h is on the same order as the slip length, can the friction drag be reduced dramatically. One interesting question is: what kind of superhydrophobic surface or structure can achieve considerable drag reduction? Recently, the movement of a superhydrophobic sphere falling down in fluid was investigated by McHale et al.¹⁶ and Su et al.¹⁷ respectively. However, the two research groups reported opposite conclusions. In the former group, an increase in the terminal velocity of the sphere with superhydrophobic surface (contact angle 150°) was observed. They believed that when the superhydrophobic surface was submerged into water it could retain an air film (a plastron), which was responsible for the drag reduction. In the latter group, however, the experiments showed that the superhydrophobic sphere (contact angle 167°) fell more slowly than the normal one. They found that the dense microbubbles trapped at the solid-water interface acted not as a reducer, but as an enhancer for the drag force. One of the possible reasons for the opposite results may be the difference of surface roughness structure (the two groups used microscale and nanoscale particles for surface coating, respectively). However, a reasonable explanation for the difference of the two experiments as mentioned above is still not available now.

As we know, water strider can stand and slide fast and effortlessly on water surface due to its long and water repellent legs. The surface tension around one of its superhydrophobic legs can provide a load support as much as 15 times the total body weight of the insect.¹² Inspired by this phenomenon, we design here a miniature boat fabricated with a superhydrophobic grille bottom. The grille consists of a number of parallel superhydrophobic copper wires, just like parallel “legs” arrayed with the same spacing. The miniature boat covered with this structure at the bottom surface was used to study the drag reduction effect, as shown in Fig. 2. We measured the drag force of the grille boat and compared it with that measured from a normal flat bottom boat. In addition, the maximum loading capacity of the grille boat was also studied. The influences of grille spacing on drag force and loading capacity were further analyzed to provide a better understanding of this design.

In our experiments, all the hulls of miniature boats were made of polyester polyurethane foam (density of ∼0.04 g/cm³) with the outline size of 160×90×10 mm (length × width × height). Two types of miniature boats were fabricated: the boat with normal flat bottom and the boat covered with a superhydrophobic grille bottom. In order to compare the drag force, the gross weight of all miniature boats used in the experiments was unified as 22 g.

(a) Flat plate boat. This type of boat was made of the foam as mentioned above with a thin hydrophilic copper film wrapped around. Therefore, the boat bottom could be considered as a smooth flat plate (see Fig. 3(a)).

(b) Superhydrophobic grille bottomed boat (grille boat for short). First, copper wires with radius R = 0.3 mm were made into a uniform shape (see Fig. 2). The effective length of the wires, L_e, ranges from L to L+2r, where L (= 152 mm) is the length of the wires in the straight portion and r (= 4 mm) is the curve radius of the wires at the two ends. In theoretical prediction we selected the effective length L_e = L+r = 156 mm. Then these copper wires were arranged with the same spacing, S, at the bottom of flat foam to form a grille structure (Fig. 3(b)). Because the hydrophilic copper density (8.9 g/cm³) is greater than water density, this structure is not able to give a considerable loading capacity to support the boat to float on the surface. Therefore, the copper wires were treated chemically in ethanol solution of tetradecanoic acid for a few hours at room temperature.¹⁸ After such a treatment, the surfaces of copper wires showed superhydrophobic property (contact angle 150°, sliding angle 2°, Fig. 3(c)), and the boat could easily stand on the water surface (Fig. 3(d)). In this work, three grille boats were fabricated with spacing S = 6.5, 5.0 and 3.0 mm, respectively.

Figure 4(a) shows the open water circulation experimental system for the measurement of drag force. The size of water channel was 160×30×30 cm. Water was circulated using a centrifugal pump and sent to the water channel through pipes. The flow rate was controlled by a flow adjusting valve. Subsequently, the water was forced to flow through the left baffle with a row of small holes (as shown in Fig. 4(b)) for the purpose of getting a steady flow. When the water level was higher than the height of the right baffle (5 cm), the water would overflow back into the tank. The typical moving velocities of water, v, were 4–12 cm/s at the surface, hence giving a Reynolds number (Re = ρvL_e/μ) in the range of ∼6×10³–20×10³ and a Froude number (Fr = v/(gL_e)^0.5) in the range of ∼0.03–0.1, where ρ and μ are density and dynamic viscosity of water, and g is the acceleration of gravity. Therefore, all the experiments were carried out under laminar and tranquil flow and thus the wavemaking drag was negligible.

The miniature boat was put on the water surface and was drawn by a soft thread (cotton, diameter of 0.1 mm) as shown in Fig. 4(c). The thread then went around a steel bar (diameter 0.4 mm) and was finally connected to the digital balance (model BT25S, Sartorius AG, Germany, accuracy 0.01 mg). The friction coefficient between the thread and the steel bar is less than 0.03. Therefore, the measuring error of the drag force induced from the surface friction is less than 3%.

The miniature boat was put on the water surface when the water reached steady flow. With the flow of water, the drag force of miniature boat could be measured by the force sensor (digital balance) directly (see Fig. 4(a)). The maximum loading capacity of grille boat was investigated by increasing a load carefully until the grille structure was immersed into the water.

In the present experiment, the values measured by the force sensor are considered as the total drag forces of miniature boats. Figure 5 shows the total drag forces of the flat plate boat (square) and the grille boat with spacing S = 6.5 mm (circle) as a function of the velocity. It shows that the superhydrophobic grille has a significant drag reduction effect in the range of the reported experimental velocity. We interpreted the drag reduction effect as the following two mechanisms: (a) grille structure reduces the contact area between the boat and the water, and (b) grille structure allows the water to flow through between the spacing of the grilles. Figure 6(a) shows the relationship between the drag reduction ratio (DR) and Reynolds number. The drag reduction ratio is defined as:

where D_t-plate and D_t-grille are the total drag force of the flat plate boat and the grille boat, respectively. The dotted line indicates the general trend of the data. The figure reveals that the drag reduction effect is affected by the Reynolds number. When the Reynolds number is 6.6×10³ (v = 4.12 cm/s), a high drag reduction ratio of 78.9% is obtained. Although with the increase in Reynolds number the drag reduction ratio is decreased, a considerable drag reduction ratio of 58.1% is still obtained at the maximum experimental Reynolds number 19.8×10³ (v = 12.37 cm/s).

Furthermore, the friction drag, one important component of the total drag force, is also analyzed. The friction drag of the smooth flat plate boat for laminar flow can be calculated by Blasius formula:¹⁹ 

where A is the contact area between the flat plate boat and water. The friction drag of a single cylinder moving in fluid can be calculated by formula:²⁰ 

Here, we assume that the friction drag of a copper wire in the grille structure is not influenced by its adjacent wires, and then the friction drag of the grille structure is roughly calculated by linear accumulation:

where n is the number of copper wires, and β is the ratio of submerged area (2φRL_e) of the cylinder to its total surface area (2πRL_e), i.e., β = φ/π (see the definition in Fig. 8) ranging from 0 to 1. In Fig. 6(b), the friction drag reduction ratio as a function of Reynolds number is shown. Here the drag reduction ratio is defined similarly as in Eq. (1), but subscript t is replaced by f. As shown in the figure, the grille plays an important role in the friction drag reduction. With the increase of Reynolds number, the reduction remains at over 80% and show a growing trend.

For the purpose of a better understanding of the drag reduction mechanism of this grille structure, the influence of the grille spacing on the drag force was studied experimentally. We also measured the drag force of the other two miniature boats with S = 3.0 mm and S = 5.0 mm. As mentioned above, the gross weight of boats was also unified as 22 g. Figure 7 shows the experimental values of the drag force of three types of grille structures as a function of Reynolds number. Although more copper wires were arranged under the boat with the decrease of spacing, the drag force only increased slightly. Even at the maximum Reynolds number, the drag force of the grille boat with S = 3.0 mm was only about 125 dynes, much less than the value of flat plate boat moving with the same Reynolds number. Although the number of the copper wires contacting with the water surface, n, was increased, the load shared by each copper wire was reduced, leading to the decrease in the ratio β, i.e., the decrease in contact area between a single copper wire and the water surface. As a result, in the case of a constant load, the friction drag would not change significantly according to Eq. (4).

For practical applications, it is necessary to know the maximum loading capacity of the grille boat. As shown in Fig. 8, assuming that the edge effect of the grille structure could be neglected, due to the periodicity of this structure, the analysis model is simplified as parallel thin long rigid cylinders. In the two dimensional plane, the meniscus profile between two cylinders is expressed by a function y = y(x), and obeys the Laplace-Young equation:

where γ is the surface tension of water, and g is the acceleration of gravity. If the submerge angle φ is given, Eq. (5) could be solved. Then the buoyancy force could be obtained by means of the total volume of the extruded liquid, which consists of the volume of the wetted segment of the cylinders and the volume of the water dimple displaced by air. A more detailed study on the equilibrium conditions for the floating objects could be found in Ref 21.

The loading capacity, F, of a long cylinder with small radius is mainly provided by the water surface tension and influenced by the length of cylinder, i.e., F = 2γL_esin(φ+θ-π).²² The cylinder acquires its maximum loading capacity when the submerge angle φ reaches the critical value φ_c = 270°-θ, while the vertical force that the meniscus exerts on the cylinder almost reaches the maximum value.²³ Thus, we estimate approximately the maximum loading capacity of the grille structure with equation F_max = 2nγL_e. Table I shows the experimental and calculated values of the maximum loading capacity of the three grille boats, as well as the corresponding drag reduction ratios with velocity v = 4.12 cm/s. As listed in the table, the maximum loading capacity increases with the increase in the number of copper wires, and generally agrees with the prediction value. However, the drag reduction ratio reduces slightly with the decrease in the grille spacing. Therefore, we need to keep a balance between the loading capacity and the drag reduction effect in practical applications.

According to the estimation equation of the loading capacity, two effective methods can be used to enhance the loading capacity. One is to increase the length of copper wires, and the other is to increase the number of copper wires. However, in practice we can not arbitrarily increase the length or the number of the copper wires without any limit.

On one hand, the hypothesis that the copper wire was a rigid cylinder is limited by the wire length. Considering the constraint condition at both ends of the copper wire, if the copper wire is simplified into a fixed end beam subjected to a uniform load q, the maximum deflection occurring at the central portion is δ = qL_e⁴/(96EπR⁴) (E is the modulus of elasticity of copper).²⁴ In the case of the grille boat with spacing S = 6.5 mm under a total load of 22 g, the load exerted on a copper wire is the combination of the gravity (downwards) and the water surface tension (upwards). If the water tension exerted on the wire were roughly considered as a uniform load, the maximum deflection of the wire occurring at the central point should be ∼0.17 mm. However, based on the rigid beam assumption as mentioned above, the maximum dimple depth of the superhydrophobic copper wire is ∼2.2 mm. This indicates that the rigid beam assumption used in section III may give a less than 8% of estimation error for the dimple depth for the present geometry design. The theoretical prediction error for the maximum loading capacity (Table I) resulting from the rigid assumption should be less than 1.7%. Therefore, the length of the copper wire designed here is reasonable. However, if a boat has to be designed long enough, the hypothesis will be no longer valid and the following two methods can be used to increase the loading capacity: (a) to design the boat bottom with several short grille structures linked in succession, (b) to replace the copper wires with narrow thin sheets, such as superhydrophobic blades that will not give the vertical bending.

On the other hand, although the loading capacity increases with the number of the copper wires, the deviation of the experiment from the prediction also increases, as listed in Table I. In reality, as the spacing decreases (n increases) to near the capillary length, the water dimples among the copper wires of the grille start to overlap together.^21,22 Therefore, the two copper wires located at the two sides of the grille structure will reach the maximum vertical load and then break the water surface prior to the wires between them.²¹ This may be one of the reasons why the measured loading support differs from the predicted value. As a result, in order to prevent water from entering the interspaces between the side wire and the boat bottom, one can either seal the interspaces with a hydrophobic thin sheet or insert other superhydrophobic copper wires in the interspaces.

The miniature boat designed in this paper can easily stand on the water surface by using the grille structure combined with superhydrophobic surface. Compared with superhydrophobic surface itself, the grille can provide more air-water interface for drag reduction. The miniature boat needs only a little power to overcome the drag force. Because of the considerable loading capacity, this type of miniature boat can also carry mini-equipments and devices, offering potential applications in cleaning, monitoring analysis on dams, lakes and rivers, which is similar to the artificial water strider reported by Song et al.²³ We believe that through rational optimization of the grille length and spacing, the drag reduction structure proposed here can be used in the design of microfluidic channel, lab on a chip, delivery of a droplet on a grille structure surface, and so on.

In the present paper, we propose a novel design of drag reduction structure inspired by the water strider legs, i.e., superhydrophobic grille fabricated with parallel wires. A miniature boat fabricated with this type of structure on the bottom can stand on the water surface. The drag force was measured using the designed water circulation experimental system. It is found that the boat covered with such superhydrophobic grille has a significant drag reduction effect compared with the same size of flat plate boat. For the grille with 6.5 mm spacing, the drag reduction reaches 78.9% when the moving velocity is 4.12 cm/s. The dramatic drag reduction of the superhydrophobic grille results from two possible reasons: (a) the real contact area between the grille and water surface is very small, and (b) the grille structure allows free passage of water through the spacing. Meanwhile, the maximum loading capacity of the grille bottomed boat is also investigated. It is found that an effective way to reduce the fluid drag and keep the loading capacity as high as possible is to appropriately lengthen the grille and widen the spacing among the wires. Our experiments demonstrate the feasibility of drag reduction of superhydrophobic grille structure.

This work was supported by the National Natural Science Foundation of China (90816025, 10925209, and 10972050).