Hydrodynamic regimes and drag on horizontally pulled floating spheres

Hydrodynamic drag on a sphere is a classical fluid dynamics problem that is related to a wide range of industrial, military, and sport applications.^1–4 Major features of this drag are captured by the drag curve, which gives the dependence of the drag coefficient, C_D = 2F_D/(πR²ρU²) on the Reynolds number, Re = 2ρRU/μ. Here F_D is the drag force on the sphere, ρ the fluid density, R the sphere radius, μ the dynamic shear viscosity of the fluid, and U the sphere velocity. When the sphere is moving in the bulk the hydrodynamic drag has two major components, the skin friction and the pressure-induced form drag. The viscous skin friction is dominant only at small Reynolds numbers in the Stokes-flow regime. With the increase in the Reynolds number, the form drag becomes dominant due to flow separation and formation of a low-pressure wake at the back of sphere.¹ A major feature of the drag curve is the drag crisis transition happening at Re ∼ 2.5 × 10⁵, which is characterized by a sudden drop in C_D due to the transition to a turbulent boundary layer that moves the flow separation to the rear of the sphere.^1–5 

The drag on a macroscopic floating sphere moving along a liquid interface has been far less studied compared to the drag on a sphere moving in the bulk fluid. For the case of a floating sphere in addition to the skin friction and form drag, a third major drag component is the wave drag due to energy carried away by the free-surface waves produced by the sphere movement. For smaller floating objects the wave drag force starts to be pronounced at a velocity close to the capillary-gravity wave minimum force.⁶ Merrer et al.⁷ have studied the wave drag by measuring the deceleration of a liquid nitrogen droplet surfing over a water interface due to the Leidenfrost effect. Following this, Benusiglio et al.⁸ have investigated the wave drag on a fully submerged sphere near the interface, using an apparatus that can differentiate the wave drag on the sphere from the total drag. Of some relevance to the drag on a sphere moving along a liquid interface are studies of the drag on spheres moving horizontally through a density-stratified fluid.⁹ 

Although the shape of a ship hull is clearly different from that of a sphere, many aspects of drag on a floating sphere have parallels with the drag on marine vessels. The drag on boats and ships is also composed of skin friction, form drag, and wave drag.^10–12 Drag on a ship is usually evaluated as a function of the Froude number, $Frhull=Uhull/gLhull$⁠, where L_hull is the length of the hull waterline, U_hull is the ship speed and g is the gravitational constant. Due to the more slender shape of a ship, at low Fr_hull the skin friction is a far more dominant component than in the case of a sphere. However, with the increase in velocity the wave drag rapidly increases, giving rise to the natural-hull-speed limit at velocities corresponding to Fr_hull ≈ 0.45.¹² For a water-displacement ship hull that is supported mostly by buoyancy this speed is difficult to exceed by simply increasing the ship engine power. Another ship hydrodynamics phenomenon that is relevant to the present study of floating spheres is the water-displacement-hulls “sinkage and trimming” problem which emerges with the increase in the Froude number due to buildup of hydrodynamic pressure over the wetted surface of the ship hull.^12–14 

In the present work, we investigate the drag on a floating sphere pulled along the water interface using a counterweight pulley system. Pulling the sphere along the water interface with a fine line instead of towing it at a fixed position enables us to investigate the unrestricted movement of the floating sphere in response to the hydrodynamic forces. By conducting simultaneous side- and top-views high-speed camera imaging of the sphere motion, we observe the characteristic hydrodynamic regimes when the pulling force is gradually increased to cover an extended range of subcritical Reynolds numbers. Using a calibrated counterweight pulley allows us to evaluate the hydrodynamic drag on the floating sphere in a similar manner as in the classical free-falling sphere experiments.^5,15,16

The experimental setup is shown in the Fig. 1 schematic and the supplementary material Fig. S1 photograph. Experiments were conducted in the KAUST water channel, which is 10 m long and has a cross section of 1 × 1 m². The channel's transparent clear acrylic walls allow side-view observation of the floating sphere movement. In the present experiments, the channel was filled with water to about 0.6 m depth.

We use a hollow alumina sphere of 10 cm diameter, and the effective density is half that of water. The sphere density is changed only by adjusting the thickness of the alumina shell. This guarantees that the sphere center of the mass is the same as the sphere geometrical center. The sphere surface is smooth due to a fine polish finish. For consistency, the same alumina sphere was used in all experiments presented. The sphere was cleaned with ethanol and pure water before each set of measurements. The dried clean sphere surface is hydrophilic with a static water contact angle of about 30°.

The sphere is dragged along the water interface with the aid of a pulley system shown schematically in Fig. 1. A small hole is drilled through the sphere surface and used to thread the sphere to a fine fishing line (0.3 mm). The line is passed through the pulley and is fixed to a counterweight whose mass was adjusted to vary the pulling force. The force with which the sphere was pulled was determined by attaching a Hooke's law spring in the front of the sphere in calibration runs as seen in the supplementary material, Fig. S2. The springs were not used in the actual data runs.

The sphere was release at about 3.5 m from the end of the tank. Side-view and top-view high-speed cameras were positioned at about 1.5 m from the tank end (see the supplementary material, Fig. S1), allowing the sphere to accelerate for about 1.5 m before coming into the field of view of the cameras. For the range of the pulling force used the sphere average velocity has reached a terminal value when the sphere enters the camera view, as confirmed by the measured velocity profiles.

The side-view high-speed video camera was a Photron-FASTCAM-SA5, and the top-view high-speed camera was a Photron-FASTCAM-SA3. The high-speed videos contain 1024 × 1024 pixels and were taken using a typical rate of 250 frames per second (fps) and setting the shutter speed at up to 1/8000 s for sharper imaging when needed. We use a 55 mm lens giving a typical resolution of about 0.45 mm/pixel, or a wider viewing angle 20 mm lens with a typical resolution of about 1.0 mm/pixel.

As usual, we characterize the dependence of drag coefficient on the Reynolds number. Even though in our experiment the sphere is not fully submerged in water, we calculate the drag coefficient, C_D = 2F_D/(πR²ρU²) and Reynolds number, Re = 2ρRU/μ, using the conventional fully submerged definitions, with R the sphere radius, U the sphere velocity, and the water density ρ, and dynamic viscosity μ. That way it is convenient to compare the drag on the floating sphere to the drag on a sphere moving in bulk water. One might expect the lower limit for the drag on a half-submerged floating sphere, moving slowly along the water interface, to be close to half the drag on a fully submerged sphere. Another important dimensionless number in the case of a floating sphere is the sphere Froude number, $Fr=U/gR$ reflecting the balance between the inertia and gravity forces acting on the sphere. We also notice that our sphere size is much larger than the water capillary length, $λc=γ/(ρg)$ ∼ 3 mm, where γ is the water surface tension.

To study the floating sphere movement when pulled along the water interfaces, we conducted a series of experiments, using the hollow alumina sphere of diameter, 2R =10 cm and density equal half of the water density, ρ_S = 0.5 g/cm³. The static sphere is floating on water half-submerged with the waterline passing along the sphere equator. In our experiments, we use pulling forces in the range of F ≈ 0.07 to 4.0 N, which resulted in sphere average horizontal velocity along the water interface of U ≈ 0.2 to 2.0 m/s. Figure 2 summarizes the U vs F dependence for the applied forces range used. The corresponding Reynolds numbers range was Re ≈ 2 × 10⁴ to 2 × 10⁵, and the Froude number range, Fr ≈ 0.25 to 2.3.

In a series of experiments, the sphere movement was tracked for about 0.5 meters using side-view and top-view high-speed video imaging. Figure 3 (Multimedia view) shows snapshots from the representative combined side- and top-view videos. In addition, we also conducted a series of experiments using large viewing angle objective to track the sphere movement for a longer distance of about 1 meter which is exemplified in Fig. 4 (Multimedia view). Representative sphere trajectories extracted from such videos are shown in Fig. 5(a) for the sphere movement in the water interface plane (X–Y plane in Fig. 1) using the top-view observation and Fig. 5(b) for sphere movement in the vertical plane (X–Z plane in Fig. 1) using the side-view observation. Figure 5(c) gives the corresponding horizontal velocity of the sphere. In all our considerations of the horizontal velocity of the sphere, U is measured only accounting for the X axis component of sphere velocity and does not include the Y and Z axis sideways components.

Figure 2 shows the experimental results of the average horizontal sphere-velocity U vs the pulling force, F. It also includes the corresponding Froude numbers. With the increase in the pulling force, we observe three characteristic regimes of the sphere motion. Formally we will refer to these three regimes as: low Froude number regime, Fr < 0.6, intermediate Froude number regime, 0.6 < Fr < 1.2, and high Froude number regime, Fr > 1.2. The transition between these regimes is gradual.

Figure 3(a) exemplifies the sphere movement in the low Fr regime (F = 0.07 N, Fr = 0.28). The side-view observation shows that the half-submerged sphere moves along the water surface with the water line close to the sphere center. The top-view observation, however, shows that the sphere movement in the horizontal plane deviates significantly from rectilinear, with prominent periodic transverse oscillations. The extended trajectory example for this case can be seen in the lower magnification top-view Fig. 4(a). One can also notice the propagation of asymmetric features in the sphere wake. Figure 5(a) trajectory 1 shows an example of the low Fr regime characteristic sinusoidal trajectory in the horizontal plane. The corresponding side-view trajectory in Fig. 5(b) is straight, indicating that the sphere remains at the same depth, while oscillating horizontally.

The intermediate Fr regime is illustrated in Fig. 3(b) (F = 0.48 N, Fr = 0.71), Fig. 3(c) (F = 1.07 N, Fr = 0.97), and Fig. 3(d) (F = 1.70 N, Fr = 1.11) snapshots and videos. As seen in Figs. 3(b)–3(d), side-views, with the increase in the pulling force and sphere velocity, we observe the development of a pronounced front wave and a matching wake depression trailing the sphere. At the same time with the sphere velocity increase, the sphere center is gradually dipping below the water interface. The top-views in Figs. 3(b)–3(d) show the front wave and the wake pattern. The trajectories 2, 3, and 4 in Figs. 5(a) and 5(b) show that in the intermediate Fr regime the sphere motion becomes stabilized in both observation planes, with the sphere center gradually dipping below the water interface.

The high Fr regime is exemplified by Fig. 3(e) (F = 4.15 N, Fr = 2.30). The side-view video in this regime shows the horizontal movement of the sphere switching to a periodic vertical dipping down and surfacing above the water interface. Figure 3(e) snapshot shows the fully submerged sphere during the dipping phase and Fig. 3(f) snapshot shows the following surfacing phase with the sphere center rising above the original water surface. The entire cycle of this vertical motion can be seen in the lower magnification Fig. 4(b). The top view indicated that the sphere moves straight in the horizontal plane. The top-views in Figs. 3(e) and 3(f) show a surface wave pattern that is different from the intermediate Fr regime, with a less pronounced front wave and a more intense but narrower wake. The characteristic vertical trajectory of the sphere is seen in trajectory 5 of Fig. 5(b).

Figure 6 shows the drag coefficient C_D vs the Reynolds number Re, for the half-water density floating sphere pulled along the water interface. In the same graph for reference, we plot the drag coefficient of fully submerged sphere using experimental data obtained for free-falling spheres in water.¹⁶ The Reynolds numbers of span values, Re = 2 × 10⁴ to 2.0 × 10⁵, corresponding to the pre-drag-crisis range. This regime is characterized by flow separation close to the sphere equator resulting in a steady drag coefficient, C_D ≈ 0.50 ± 0.05. The predominant drag in this regime is the form drag with the skin-friction drag less than 5% of the total.¹ For the floating sphere pulled along the water interface, we expect that additional drag due to wave motion will have a major contribution.^6–8 

The data in Fig. 6 confirm that for the lower Fr = 0.28, corresponding to Re = 2.4 × 10⁴, the drag on the half-submerged sphere, C_D = 0.28 ± 0.02, which is close to half of the drag on a sphere moving in the bulk. This agrees with Fig. 3(a) showing that the water line is close to the sphere center and the surface waves are hard to notice. However, with an increase in the pulling force, the drag coefficient begins to steadily increase until it reaches a peak value of C_D = 0.70 ± 0.02 at Re = 7.8 × 10⁴. The increase in the drag is related to the dipping of sphere with the velocity increase as well as with the increase in energy dissipation from the generation of surface waves. At the peak value, the drag coefficient on the floating sphere exceeds the drag on a sphere moving in bulk liquid. This effect is demonstrated in Fig. 7 (Multimedia view) video comparing the movement of the floating sphere with a neutral buoyancy sphere being pulled underwater with an identical force F = 1.70 N. If we assume that at the peak C_D the form drag on the floating sphere is close to that on a fully submerged sphere, the excess drag can be interpreted as arising from the wave drag. This suggests that about 20 to 30% of the total drag is due to wave drag.

For the high Fr range, the picture is complicated because the sphere translational velocity and related drag force fluctuate significantly during the dipping and surfacing of the sphere. Figure 5(c) shows that the velocity fluctuates between 1.8 m/s corresponding to C_D = 0.30 and 1.2 m/s corresponding to C_D = 0.7. The drag coefficients present in Fig. 6 are calculated using the average horizontal velocity of the sphere over the complete dipping and surfacing cycle. Because of the nonequilibrium nature of the sphere movement in this regime, these drag coefficients should be considered only as an average resistance measure. We notice, however, that the general trend in the high Fr regime is for gradual lowering of the average drag with increase in the Reynolds number past the peak value in the intermediate Fr regime. As the average dipping of the sphere in this regime remains approximately the same as the case with the peak value of the drag [compare Fig. 5(b) trajectories 4 and 5], the drag-lowering trend could be related to reduction in the wave drag. This hypothesis is supported by the Fig. 3 videos showing a seemingly less intense wave pattern produced by the sphere in the high Fr regime compared to the intermediate Fr regime.

We observe two separate transverse oscillatory instabilities. At low Fr, we see horizontal oscillations [Fig. 4(a)] while at high Fr, the oscillations are in the vertical direction [Fig. 4(b)]. On the other hand, at intermediate Fr, the trajectories are straight without any oscillation.

Deviation from rectilinear motion is characteristically observed in free-falling sphere experiments conducted in the precrisis range of Reynolds numbers.^17–19 These deviations from the rectilinear motion are explained by the appearance of nonaxisymmetric vortical structures in the wake that perturb its trajectory.^17–24 We assume that the horizontal oscillations for the low Fr regime in our experiment are of the same nature. In the literature, there are different modes of oscillatory movement of free-falling spheres. In a detailed study, Horowitz and Williamson¹⁷ map the oscillations of free-falling and free-rising spheres as a function of the Reynolds number and the mass ratio of the sphere to fluid density. They identify several regimes consistent with single-sided and double-sided vortex rings shedding, as well as a new regime of four vortex-ring sheddings. A general feature for a sphere of Re = 2 × 10⁴ to 2 × 10⁵ is that the low-frequency disturbances in the flow structure f are found to be independent of the Reynolds number, with a Strouhal number value close to 0.19, i.e., $St=Df/U$ ≈ 0.19.^20–23 This means that the streamwise distance at which the vortex structure are shed, λ = U/f = 2R/St, should be independent of the sphere velocity, and scale with the sphere radius as, λ ≈ 10.5R.

In Fig. 8(a), we plot the top-view trajectories for the sphere movement at low Fr regime, for a range of sphere velocity U = 0.18 to 0.45 m/s. It is seen that for this range the sphere trajectory is independent of the sphere velocity and the oscillatory movement has half-wavelength, λ_S/2 ≈ 0.53 m, or λ_S ≈ 21R. This is consistent with alteration of the sphere movement at each two cycles of vortex shedding at St ≈ 0.19. To confirm this scaling, we also conducted some measurements using a smaller 3D-printed sphere of 5 cm in diameter and of the same density as the 10 cm sphere. Representative trajectories for the 5 cm sphere are shown in Fig. 8(b). Once again, the wavelength of the oscillatory movement is independent of the sphere velocity and consistent with the alteration of the sphere trajectory at each two cycles of vortex shedding at frequency corresponding to St ≈ 0.19. We also note that Yun et al.²³ have observed a similar scaling of about 10.5R for the vortical structures behind a sphere at subcritical Reynolds numbers.

In the intermediate force regime, the sphere trajectory is stabilized both in the vertical and the horizontal planes. The stabilization is probably due to the front wave and wake depression dominating the nonaxisymmetric vortex shedding.

In the high Fr regime, the sphere movement in the vertical plain transits to the dipping and resurfacing mode of movement. It seems that this instability is triggered once the entire sphere is covered by water and the downforce on the partially submerged sphere, discussed in Sec. III E, is no longer compensating the buoyancy force on the fully submerged sphere. However, this instability is far more complex to analyze than the sideways swinging of the sphere in the horizontal plane, in the low Fr regime. The analysis is further complicated by the constantly changing drag during the dipping and surfacing of the sphere. A more quantitative treatment of the dipping and surfacing mode of the pulled sphere movement will be subject of future investigations.

One major feature of the sphere movement in the intermediate Fr regime is that with the increase in the pulling force and velocity the sphere center gradually dips below the water interface. We speculate that this sphere dipping effect is due to the dynamic pressure increase on the lower half of the sphere. The resulting increase in the downforce is balanced by the increase in the buoyancy force.

Here we derive a semiempirical expression to estimate the dipping dependence on the Froude number. First, we consider the idealized case of a half-submerged sphere moving horizontal in inviscid fluid, e.g., potential flow approximation. The pressure variation around the lower part of the sphere surface in potential flow is of the form^1,15

where φ is the angle relative to the X axis along which the sphere is moving. The  Appendix details the calculation of the resulting downforce, F_d, reaching the expression,

In a realistic situation due to flow separation and formation of lower-pressure, wake at the sphere back the pressure profile deviates from the potential flow form.¹ For the floating sphere in addition to the sphere dipping the situation is further complicated by the development of the front wave and the wake depression. Nevertheless, it is reasonable to assume that similar to the potential flow approximation Eq. (2), the downward force will depend on the characteristic dynamic pressure $12ρU2$ and the area of the sphere cross section with the water interface, pr². We suggest a semiempirical downforce equation in the form,

Here r is the radius of the sphere cross section with the water interface, characterizing the vertical asymmetry in the pressure (see Fig. 9 inset schematic) and c is a proportionally coefficient to be determined from the best fit of the experimental data. The dynamic pressure downforce is balanced by the increase in the buoyancy force. Ignoring the changes in the wave profile, as schematized in the Fig. 9 inset, the increase in the buoyancy force is given by

From the force balance, we obtain

For each c, Eq. (5) can be solved as a parametric equation to obtain the h/R vs Fr dependence. Keep in mind that this formulation applies only for a sphere of half the liquid density, i.e., which statically is half submerged. Figure 9 compares h/R vs Fr dependence given by Eq. (5), with the corresponding experiments. The experimental data used are limited to the low and intermediate Fr regimes. Here the experimental h values are taken as the position of the sphere center regarding the undisturbed water interface. Best agreement with the experiential values was found using c ≈ 1. For comparison, we also show Eq. (5) using the c = 11/16 coefficient for the potential flow approximation from Eq. (2). We note that the best fit value of c accounts for the approximation used in both, the downforce Eq. (4) and the buoyancy force Eq. (5). These considerations are relevant only for sphere movement in the low and intermediate Fr regimes. In the high Fr regime, h/R varies significantly with time and the sphere gets completely covered with water, a situation for which the simplified model we use is no longer adequate.

In the first instance, the sphere dipping with the increase in the pulling force might seem to be contraintuitive compared with the behavior of a high-speed boat that tends to be lifted instead of dipping with increasing speed. Typically, a high-speed boat will switch to planning mode at which the hull is supported mostly by hydrodynamic forces instead of buoyancy, at Fr_hull > 1.0.¹² However, as mentioned in the Introduction, in contrast to the planning hull of a high-speed boat, a water-displacement-hull ship moving at lower Froude numbers is known to experience “sinkage.”^12–14 The physics underlying sinkage is the same as that for a dipping sphere, i.e., due to buildup of hydrodynamic pressure below the ship hull with the increase in the velocity. For a water-displacement-hull ship, the sinkage increases with the Froude number following the same type of dependence as the one shown in Fig. 9 for the half-water-density floating sphere.^12–14 

We note that although that the ship-hull sinkage is a known phenomenon and is sometimes the reason for limiting the ship's velocity in shallow waters, the amount of sinkage for ships seems to be less dramatic than the one observed here for the floating sphere.^12–14 One reason for the less dramatic effect in the case of ships could be the natural-hull speed limit, which restricts the water-displacement-hull ships to operation at relevantly lower Froude numbers, Fr_hull < 0.45.¹² Heuristically, it is clear that the hydrodynamic forces grow with area, or the square of the size, while buoyancy scales with the volume or cube of the size. The ratio of the two forces for similar velocity is thereby inversely proportional to the object size, as is expressed by Fr². The Fr values for ships are therefore much smaller than our orange-sized spheres and the sinkage of common-sized ships is dominated by buoyancy and much less affected by the hydrodynamic pressures.

Using high-speed imaging, we investigate the movement of a half-submerged floating sphere when it is pulled horizontally along the water surface in a test channel. Depending on Froude number we observe three characteristic hydrodynamic regimes for the precrisis range of Reynolds numbers. In the lower Froude number regime, Fr < 0.6 the sphere moves horizontally without noticeable disturbance of the water interface but has an oscillatory trajectory in the horizontal plane. The wavelength of the oscillatory trajectories is consistent with the frequency of vortical structure shedding in the sphere wake. In the intermediate Froude number regime, 0.6 < Fr < 1.2 a pronounced front wave and trailing wake depression develops together with the sphere center gradually dipping below the water interface. The dipping effect is due to the increased dynamic suction pressure on the submerged part of the sphere resulting in a downforce that is compensated by the increase in buoyancy. The sphere drag reaches a peak value at the intermediate Fr regime that exceeds by about 30% the drag on the fully submerged sphere. In the high Froude number regime, Fr > 1.2, the sphere periodically dips below and then resurfaces above the water interface. This mode of the sphere movement seems to lower the average drag on the sphere compared to the partially submerged sphere and will be the subject of further investigation. Other future directions of investigation might include studying the behavior of floating spheres of various densities and the effect of superhydrophobic coatings^16,25,26 on the drag on floating spheres.

See the supplementary material for a Word file containing supplementary Fig. S1: photograph of the laboratory experimental setup; supplementary Fig. S2: example of sphere with attached “Hooke's law” spring used for pulling force calibration.

We acknowledge an anonymous referee for suggesting the sphere dipping analogy with ships sinkage. The work was supported by the King Abdullah University of Science and Technology (KAUST) under Grant No. URF/1/3723-01-01.

It is convenient to calculate the downforce using spherical coordinates (r, θ, φ) as schematized in Fig. 10. Because of symmetry, the pressure distribution on the lower half of a half-submerged sphere in potential flow approximation will be identical to that of a fully submerged sphere which is of the form,¹ 

To calculate the total lift or downforce, one takes the Z component of the pressure force acting on the sphere surface,

The downforce is obtained as integral of the pressure force Z component over the surface of the hemisphere,

Completing the integration over θ,

Completing the integration using the table integrals for sin²φ and sin⁴φ, we obtain Eq. (2) in the manuscript,

The data that support the findings of this study are available from the corresponding author upon reasonable request.