1. Introduction
The assessment of mass transport in the ocean plays a crucial role in determining the migration of sediments, pollutants (Iwasaki et al. Reference Iwasaki, Isobe, Kako, Uchida and Tokai2017; Dobler et al. Reference Dobler, Huck, Maes, Grima, Blanke, Martinez and Ardhuin2019), tracking buoys (Nath Reference Nath1978; DeBok & Roehrig Reference DeBok and Roehrig1981; Wilson Reference Wilson1982; Tanizawa, Minami & Imoto Reference Tanizawa, Minami and Imoto2001), ice floes (Arikainen Reference Arikainen1972; Perrie & Hu Reference Perrie and Hu1997; Meylan et al. Reference Meylan, Yiew, Bennetts, French and Thomas2015), as well as trajectory forecasting for search and rescue operations (Röhrs et al. Reference Röhrs, Christensen, Hole, Gö. Broström and Sundby2012; Breivik et al. Reference Breivik, Allen, Maisondieu and Olagnon2013). The recent surge in interest in this topic is driven by the alarming increase in plastic pollution, which requires effective management and clean-up, beginning with an improved understanding of their global transport patterns (Law et al. Reference Law, Morét-Ferguson, Maximenko, Proskurowski, Peacock, Hafner and Reddy2010; Eriksen et al. Reference Eriksen, Lebreton, Carson, Thiel, Moore, Borerro, Galgani, Ryan and Reisser2014; Van Sebille et al. Reference Van Sebille, Wilcox, Lebreton, Maximenko, Hardesty, Van Franeker, Eriksen, Siegel, Galgani and Law2015; Law Reference Law2017; Van Sebille et al. Reference Van Sebille2020). In this study, we investigate the drift motion of large floating objects induced by surface gravity waves.
1.1. Reasons for enhanced drift
The orbit of a fluid particle in waves is not fully closed over a wave cycle, resulting in a net drift in the direction of wave propagation known as Stokes drift (Stokes Reference Stokes1847). To study the transport of floating objects, several studies have superimposed the Stokes drift onto the Eulerian current field obtained from ocean general circulation models or observations (see Van Sebille et al. (Reference Van Sebille2020) for a review). There are three reasons why this approach may not result in a correct prediction of the drift of floating objects in the real ocean. First, this approach ignores the modification of the Eulerian-mean flow by the surface waves: on the rotating Earth, the Stokes–Coriolis force drives an Eulerian mean. This so-called Ekman–Stokes flow needs to be added to the Stokes drift to properly estimate the wave-induced Lagrangian-mean flow, which is the flow responsible for transporting objects floating on the ocean surface (Higgins, Vanneste & van den Bremer Reference Higgins, Vanneste and van den Bremer2020b ). On the time and spatial scales associated with typical laboratory experiments, this Ekman–Stokes flow does, of course, not arise. Second, it has been shown that the drift at the surface of a breaking wave is much larger than the prediction of classical Stokes drift for non-breaking waves (Deike, Pizzo & Melville Reference Deike, Pizzo and Melville2017), as confirmed in experiments (Lenain, Pizzo & Melville Reference Lenain, Pizzo and Melville2019; Sinnis et al. Reference Sinnis, Grare, Lenain and Pizzo2021), and breaking may make a significant contribution to the Lagrangian drift of realistic wave fields (Pizzo, Melville & Deike Reference Pizzo, Melville and Deike2019). In the context of irregular waves, Eeltink et al. (Reference Eeltink, Calvert, Swagemakers, Xiao and van den Bremer2023) have recently proposed a stochastic differential equation that uses a Brownian motion to capture the non-breaking drift-diffusion effect associated with a random non-breaking sea state and a compound-Poisson process to capture the rapid jumps in particle position that take place during wave breaking. Third, unlike an idealised fluid particle (that is, a perfect Lagrangian tracer), floating marine objects exhibit significant variations in size, which can lead to non-Lagrangian drift behaviour (Nath Reference Nath1978; Santamaria et al. Reference Santamaria, Boffetta, Afonso, Mazzino, Onorato and Pugliese2013; Ren et al. Reference Ren, He, Dong and Wen2015; Calvert et al. Reference Calvert, McAllister, Whittaker, Raby, Borthwick and van den Bremer2021; DiBenedetto, Clark & Pujara Reference DiBenedetto, Clark and Pujara2022).
This non-Lagrangian drift behaviour is the subject of the present paper. Studies that have examined the drift of non-Lagrangian particles or ‘objects’ can be classified into those that ignore the disturbance of the wave field by the object and those that take this disturbance into account. Typically, the effect of the object on the wave field can be neglected when the relative size of the object (characterised by
$l/\lambda$
, the ratio of the object’s characteristic dimension
$l$
to the incident wavelength
$\lambda$
) is smaller than 0.2 (Isaacson Reference Isaacson and Shaw1979).
1.2. Drift of objects that do not disturb the wavefield
Beginning with studies that have ignored the disturbance of the wave field by the object and focusing on fully submerged particles, Eames (Reference Eames2008) and Santamaria et al. (Reference Santamaria, Boffetta, Afonso, Mazzino, Onorato and Pugliese2013) used the Maxey–Riley equations (Maxey & Riley Reference Maxey and Riley1983) to examine how far slightly positively or negatively buoyant objects are transported by regular waves as they either rise to the free surface or sink. Santamaria et al. (Reference Santamaria, Boffetta, Afonso, Mazzino, Onorato and Pugliese2013) predicted that positively buoyant objects in deep-water waves experience an increase in drift owing to their inertia (see DiBenedetto et al. (Reference DiBenedetto, Clark and Pujara2022) for an extension to finite depth). Also considering fully submerged objects, DiBenedetto & Ouellette (Reference DiBenedetto and Ouellette2018) showed that non-spherical objects have a preferential orientation under waves, confirming this result numerically (DiBenedetto & Ouellette Reference DiBenedetto and Ouellette2018) and experimentally (DiBenedetto, Koseff & Ouellette Reference DiBenedetto, Koseff and Ouellette2019), but not examining the effect of the object’s inertia or its modified drift. Even without direct modification of the drift, the orientation changes the drag on slightly negatively buoyant objects, which results in objects of different shapes being transported different distances before ‘raining out’ (DiBenedetto, Ouellette & Koseff Reference DiBenedetto, Ouellette and Koseff2018). Taking a similar approach, Pujara & Thiffeault (Reference Pujara and Thiffeault2023) have shown that wave action reduces the vertical dispersion of particles but increases horizontal dispersion.
Turning to floating objects, but still ignoring the disturbance of the wave field they cause, theoretical models that apply forces derived from the wave field onto a moving object, assuming the wave field is unaffected by the presence of the object itself (e.g. through Morison’s equation (Morison, Johnson & Schaaf Reference Morison, Johnson and Schaaf1950)) are thus commonly used to study the drift motion of small objects (Rumer, Crissman & Wake Reference Rumer, Crissman and Wake1979; Shen & Ackley Reference Shen and Ackley1991; Shen & Zhong Reference Shen and Zhong2001; Grotmaack & Meylan Reference Grotmaack and Meylan2006; Huang, Huang & Law Reference Huang, Huang and Law2016). Calvert et al. (Reference Calvert, McAllister, Whittaker, Raby, Borthwick and van den Bremer2021) used a perturbation expansion in wave steepness to derive a closed-form solution for the enhanced drift of a floating sphere, extending the models of Rumer et al. (Reference Rumer, Crissman and Wake1979) and Huang et al. (Reference Huang, Huang and Law2016). Calvert et al. (Reference Calvert, McAllister, Whittaker, Raby, Borthwick and van den Bremer2021) attributed the enhanced drift to two physical mechanisms: the increased vertical oscillatory (linear) motion of the floating sphere and the (viscous) drag-induced phase difference between the motion of the inertial sphere and that of an idealised Lagrangian particle, giving rise to a net horizontal component of the buoyancy force when averaged over a wave cycle.
1.3. Drift of objects that do disturb the wavefield
In those models that do take the disturbance to the wave field by the (floating) object into account, the object’s surfaces are often treated as part of the boundary of the fluid domain, and the problem is approached based on potential-flow theory (Haskind Reference Haskind1946; Faltinsen Reference Faltinsen1993; Newman Reference Newman2018). Apart from the (linear) oscillatory (unsteady) exciting forces leading to (linear) oscillatory motion, objects exposed to waves also experience steady (i.e. wave-averaged) nonlinear forces, giving rise to a slow drift motion in the direction of wave propagation. This steady second-order force is often referred to as the (mean) drift force and affects the trajectories of objects in waves (Suyehiro Reference Suyehiro1924; Watanabe Reference Watanabe1938; Havelock Reference Havelock1942; Skejic & Faltinsen Reference Skejic and Faltinsen2008). There are three main approaches to calculating the drift force: the near-field formulation (Pinkster & Hooft Reference Pinkster and Hooft1976; Pinkster & Van Oortmerssen Reference Pinkster and Van Oortmerssen1977), which solves for the second-order disturbance in the near field by direct integration of pressure on the surface of the wetted body; the far-field formulation (Newman Reference Newman1967), which is derived from conservation of momentum and thus sets the change in momentum (flux) of the fluid equal to the mean force acting on the object; and the middle-field formulation (Chen Reference Chen2007), which applies momentum conservation to a fluid control volume and introduces control surfaces around the body.
All three approaches focus on the calculation of a force when the object’s drift speed is zero and/or on the calculation of motion when an object is moored. They do not (directly) predict the drift of a freely floating object. Typically, the objects considered are three-dimensional with relatively large sizes, often comparable to the wavelength. Moreover, few studies (e.g. Newman Reference Newman1967) have examined cases in following waves where objects are travelling in the direction of wave propagation (cf. ships), characterised by heading angles of
$0^{\circ }$
(most of the cases are in oblique waves or head waves). A floating object travelling with a constant velocity on the surface of otherwise calm water experiences a non-viscous (i.e. potential-flow) drag, reflecting the energy required to push the fluid out of the way of the object, known as the wave-making resistance (Michell Reference Michell1898; Havelock Reference Havelock1909). In principle and analogously to this wave-making resistance, the so-called wave drift damping force (Nossen, Grue & Palm Reference Nossen, Grue and Palm1991; Aranha Reference Aranha1994), which acts on a slowly drifting object, could balance the drift force discussed above, perhaps aided by a further viscous or turbulent drag term in realistic flows, leading to force equilibrium and resulting in steady drift motion. However, the authors are unaware of existing work that uses such an equilibrium of drift and drift damping forces to obtain estimates of the (enhanced) drift of floating objects, and note that the required analysis would not be straightforward based solely on existing work (e.g. Nossen et al. Reference Nossen, Grue and Palm1991; Aranha Reference Aranha1994).
To consider the effect of diffraction on the wave-induced drift of freely floating objects Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) used a two-dimensional (2-D) hybrid numerical solver, which deals with both viscosity and wave–body interaction, to predict significant drift enhancement of large (5–10 % of the wavelength) floating rectangular boxes. The drift enhancement was explained in small part by the effect of viscosity (as by Calvert et al. Reference Calvert, McAllister, Whittaker, Raby, Borthwick and van den Bremer2021), but in large part by the standing wave pattern generated by the presence of the (oscillating and drifting) object in the wave field. To explain their numerical results, Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) derived a ‘diffraction-modified Stokes drift’ akin to Stokes (Reference Stokes1847), but based on the combination of incident, diffracted and radiated wave fields, which are based on potential-flow theory and obtained using the boundary element method (BEM). This ‘diffraction-modified Stokes drift’ could explain both qualitatively and quantitatively the increase in drift predicted by the hybrid numerical model.
1.4. Relative object size and steepness: previous experimental studies
Experimental investigations of the wave-induced motion and drift of floating objects have been conducted for nearly half a century. Table 1 provides a summary of relevant experimental studies, including the present paper. This table shows that the drift of an object depends on its relative size. The drift of very small objects is equal to the theoretical Stokes drift. However, as the objects become larger, enhanced drift is observed (we refer to drift as ‘enhanced’ in this paper when it is larger than the theoretical Stokes drift). Nath (Reference Nath1978) reported Lagrangian drift behaviour for small discs and spheres on regular waves but found enhanced drift for larger objects with deep drafts (such as spar-type buoy drifters) in low wave steepness. Measurements by Murray, Guy & Muggeridge (Reference Murray, Guy and Muggeridge1983) suggested that the maximum horizontal drift attained by floating spheres differs substantially from that of a water particle only when
$l/\lambda \gt\,\rm 12.5$
$\%$
. Similar conclusions were drawn by Alsina, Jongedijk & van Sebille (Reference Alsina, Jongedijk and van Sebille2020) regarding the drift of floating spherical particles with
$l/\lambda$
ranging from 0.1
$\%$
to 1.4
$\%$
, which were not enhanced. Calvert et al. (Reference Calvert, McAllister, Whittaker, Raby, Borthwick and van den Bremer2021) suggested that the drift of a sphere increases with its relative size, and enhanced drift was found for objects as small as
$l/\lambda =6\%$
. Le Boulluec, Forest & Mansuy (Reference Boulluec, Forest and Mansuy2008) found that, regardless of shape, the drift velocity increases with relative size of the object
$l/\lambda$
, considering a range of relative size from 6
$\%$
to 50
$\%$
.
Table 1. Summary of previous laboratory studies of wave-induced drift of freely floating objects on waves. Some values not explicitly provided in the cited papers were obtained from digitised figures or estimated using the linear dispersion relationship. In 2-D experiments, the ‘third dimension’ shows the approach used to ensure the experiment’s 2-D nature.
$l$
represents the length of the object in the direction of wave propagation,
$\lambda$
the wavelength,
$h$
the height of the object,
$\rho$
the density of the object,
$k$
the wavenumber,
$d$
the water depth, and
$a_w$
the wave amplitude. The wave types R, IR, and WG correspond to regular waves, irregular waves, and wave groups, respectively.
Drift behaviour is also influenced by the incident wave steepness. Tanizawa et al. (Reference Tanizawa, Minami and Imoto2001) reported Lagrangian drift behaviour for a spherical buoy with a relative size of
$l/\lambda=\,\rm 6$
$\%$
at low wave steepness, while enhanced drift was observed for all objects with
$l/\lambda \gt\,\rm 10$
$\%$
across various wave steepnesses. They proposed that, for small objects, drift is proportional to the square of the wave steepness, and as the object size increases, drift gradually becomes linearly proportional to the wave steepness. Huang, Law & Huang (Reference Huang, Law and Huang2011) also found enhanced drift behaviour for objects with
$l/\lambda$
ranging from 12.8
$\%$
to 15.8
$\%$
, and the magnitude of this drift enhancement increased with wave steepness.
1.5. 3-D versus 2-D objects
All the experimental studies mentioned above are three-dimensional (3-D), meaning that while the wave conditions may be unidirectional, the objects do not span the entire wave crest. The scarcity of 2-D studies can be attributed to the greater experimental difficulty in designing objects that are truly 2-D and capable of moving freely in a wave tank, especially for very small objects. Additionally, lateral variations in surface elevation for unidirectional waves pose challenges. Previous 2-D studies have typically used models that occupy the full width of the flume, with a small gap between the walls and models to ensure free movement.
Harms (Reference Harms1987) conducted systematic 2-D laboratory measurements on the drift of ice floes, which exhibit significant variations in size. He proposed an empirical formula for wave-induced drift based on the ratio of wave period to the object’s natural roll period, wave steepness, and the length of the floe. His formula suggests that the object’s drift changes with its relative size, and there is a critical size, the value of which depends on the object’s aspect ratio (typically approximately
$l/\lambda =50\%$
for an object with
$h/l=0.08$
, where
$h$
is its height and
$l$
its length, and density
$\rho =920\,\rm kg\,m^-{^3}$
). When the object is smaller than the critical size, the drift increases approximately linearly with its relative size. However, when the object is larger than the critical size, the drift decreases substantially and nonlinearly with further increases in relative size. The value of this critical size does not vary significantly with wave steepness. For objects with different aspect ratios (but the same density), when the relative size is smaller than the critical size for all objects, the drift increases as the submergence increases with increasing
$h/l$
. This finding is consistent with the measurements of Huang & Law (Reference Huang and Law2013), which focused on relatively large objects with
$l/\lambda \gt 60\%$
. Tanizawa et al. (Reference Tanizawa, Minami and Imoto2001) and He, Ren & Qiu (Reference He, Ren and Qiu2016) also observed enhanced drift for intermediate sizes (
$l/\lambda$
ranging from 10
$\%$
to 60
$\%$
), and the relationship between drift velocity and wave steepness shifted from a quadratic to a linear relationship. Specifically, He et al. (Reference He, Ren and Qiu2016) found that the ratio of the magnitudes of the quadratic components to the linear components shifted from 10 to 0 for objects ranging from
$l/\lambda =9 \%$
to
$30\%$
.
1.6. Present paper
This paper reports the results from laboratory experiments examining the enhanced wave-induced drift, large, 2-D, rectangular floating objects. We set out to make two contributions.
First, this paper aims to observe and study experimentally the mechanism for enhanced drift described by the diffraction-modified Stokes drift theory proposed by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024). To do so, we perform a comparison with the numerical results of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024). While documenting the dependence of enhanced drift on different parameters, the experimental literature surveyed above has not examined the mechanism(s) explaining the drift enhancement or the relationships between drift velocity and the oscillatory motion of objects, nor has it examined the surface elevation in the vicinity of the object. We aim to do this here.
Second, we explore a gap in the parameter space that the existing literature has left open. To the best of the authors’ knowledge, the drift behaviour of objects with relative sizes ranging from
$l/\lambda \lt 6\%$
to
$l/\lambda =27\%$
has not been examined in 2-D experiments previously. In an effort to address this gap and explore object sizes ranging from very small to intermediate size, near or crossing the critical size reported by Harms (Reference Harms1987), we conduct 2-D laboratory experiments investigating the drift behaviour of objects with relative sizes in the range
$l/\lambda =2.6$
–
$27\%$
in regular deep-water waves. In the ocean, relative size depends on both the size of the litter and the wavelength. On the one hand, pollutants in the ocean greatly vary in size. While most of the litter is small, such as microplastics (
${\lt } 5\,\rm mm$
) (Setälä et al. Reference Setälä, Fleming-Lehtinen and Lehtiniemi2014; Cózar et al. Reference Cózar2014), there is also large litter, such as wreckage of vessels and ships, which can be tens of meters in length (Carlton et al. Reference Carlton2018). On the other hand, typical ocean wavelength consists of scales from 10 cm to 1 km (Lefevre & Cotton Reference Lefevre and Cotton2001; Toffoli & Bitner-Gregersen Reference Toffoli and Bitner-Gregersen2017). As a result, our experiments cover the range from tiny pieces of litter in long waves to large debris in shorter waves.
In this paper, we have chosen to perform experiments with 2-D objects (i.e., objects spanning the entire regular wave crest) for two reasons. Most importantly, 2-D experiments allow a mechanism that probably also occurs in 3-D, but is strongest in 2-D to be isolated, namely the diffraction-modified Stokes drift of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024), which is proposed in a 2-D context therein. Secondly, while most real-world applications are 3-D, specific applications, such as ice floes (Arikainen Reference Arikainen1972; EI-Tahan et al. Reference EI-Tahan, EI-Tahan and Venkatesh1983; Wadhams Reference Wadhams1983; Harms Reference Harms1987), are distinctly 2-D. Ice floes can take the form of so-called ice edge bands, which are 2-D-like (typically, 10 km in length and 1 km in width (Wadhams Reference Wadhams1983)). To ensure two-dimensionality in our experiments, we utilise carefully designed objects equipped with low-friction bearings.
The outline of the paper is as follows. In § 2 we present our experimental method, followed by a brief introduction to the theoretical model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) and the empirical parametrisation of Harms (Reference Harms1987) in § 3. We present our experimental results in § 4, where we also propose a new empirical parametrisation, and compare our experimental results to the theoretical model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) and the empirical parametrisation of Harms (Reference Harms1987). Finally, we draw conclusions in § 5.
Figure 1. Schematic layout of the wave flume and experimental set-up, indicating the positions of the two cameras, the coordinate system, and the dimensions of the floating object.
2. Experimental method
2.1. Wave flume and camera set-up
Experiments were conducted in the Wave and Current Flume at the University of Oxford, UK. The flume is 10 m long, 1.1 m wide and 1 m deep with working water depth of 0.85 m. Figure 1 provides a diagram of the experimental set-up. Waves were generated using three hinged flap-type wavemakers, located at one end of the flume. At the other end, a wave-absorbing beach was positioned. It is important to note that the flume is not fully closed, and includes pumps and a pipe system for fluid recirculation underneath the flume that can be used to create a current. Although a current was not actively generated in the present experiments, the presence of the pipe system allows fluid recirculation during wave generation, which may influence the Eulerian current in the flume (it does not provide the no-flow boundary condition a closed tank would provide, see § 2.2). To measure wave properties, seven resistance-type wave gauges were distributed along the flume, sampling the surface elevation at a rate of 128 Hz. The gauges were mounted on a steel frame in five rows. The distances of the rows from the wavemaker were as follows: 2.14 m, 4.59 m, 6.02 m, 6.32 m, and 6.59 m. Within the rows located at distances of 6.02 m and 6.32 m, two pairs of gauges were placed to assess any lateral variation in the wave field. For each pair of gauges, one was positioned at
$y=0.0\,\rm m$
(the middle of the flume), while the other was located at
$y=0.275\,\rm m$
, away from the middle position.
Two E 2–4K Ultra-HD Cinematic Cameras (Z-CAM-E1503) equipped with a Panasonic 12 mm–60 mm digital interchangeable lens were used to measure the movement of the floating objects at 50 frames per second (fps), as depicted in figure 1. Side Camera 1 was positioned at the same height as the still-water level, located 12 m away from the flume. In this configuration, the camera’s field of view encompassed the entire length of the flume, and it was horizontally orientated normally to the region of interest including the water surface. Its calibrated camera images were used to measure the drift and oscillatory motion of the floating objects (introduced below) in the
$(x,z)$
-plane. Top-side Camera 2 was mounted 0.65 m above the still-water level, facing down the wave flume towards the wavemakers. In this set-up, the camera’s field of view covered approximately 4 m of the flume length. Camera 2 was specifically used to measure the drift of small yellow balls in the
$(x,y)$
-plane, as these balls were not visible from the side of the flume due to their size. To ensure proper illumination and consistent lighting conditions for all experiments, three LG-600SC daylight dimmable LED lights (not shown in figure 1) were strategically placed along the length of the flume. These lights illuminated the ends of the objects, which were painted yellow. This illumination scheme also facilitated consistent lighting conditions for subsequent image processing. Optical distortion and pixel scale quantification were achieved by capturing calibration images of a 13 by 10 chequerboard with 50 mm square size using the cameras.
The parameters describing the waves used in our experiments are presented in table 2. Initially, we conducted experiments with wave gauges installed in the flume, without any objects, to measure the undisturbed wave field. These experiments also allowed us to quantify any unabsorbed reflections and lateral variation of the wave field within the flume. The lateral variation for a single frequency was calculated by determining the relative difference in amplitude measured by the gauges. A reflection analysis based on the methods described by Goda & Suzuki (Reference Goda and Suzuki1976) and Mansard (Reference Mansard1980) was performed. As shown in table 2, the lateral variation was found to be low and the wave reflection coefficients were predominantly below
$5\%$
, with a maximum value of
$10\%$
. For experiments involving the measurement of object motion, the wave gauges were removed. At the beginning of each experiment, the objects were placed in the quiescent flume 1.5 m away from the wavemakers. Waves were then continuously generated until the objects reached a position 1 m from the beach, when the wavemakers were turned off. Recording of the camera and the wavemaker started and ended simultaneously. In cases where large drift speeds were expected (i.e. for large objects in steep waves), the experiments were repeated 2–3 times to ensure a minimum duration of 35 wave periods of data after the drift has reached a quasi-steady state. Following each experiment, a 10-min wait time was observed to allow the flume to settle. To assess the repeatability of our experiments, we conducted additional experiments for four out of the twelve values of input wave steepness (
$k{a_w}=0.06$
, 0.10, 0.20 and 0.32) for each frequency and each object. For these experiments, we performed five repeats, which are analysed in Appendix B.
Table 2. Experimental matrix showing the different wave parameters. We have examined a total of 48 wave conditions, combining four different frequencies, each with 12 different wave steepness values ranging from 0.04 to 0.32. The wave steepness interval was 0.02 for
$ka$
values between 0.04 and 0.2, and 0.04 for
$ka$
values between 0.2 and 0.31. The table includes the following parameters:
$f$
(wave frequency),
$a_w$
(measured wave amplitude),
$k$
(wavenumber),
$\epsilon$
(wave steepness),
$d$
(water depth), and
$\lambda$
(wavelength). The term
$\Delta a_w$
represents the lateral variation of the wave amplitude calculated by measuring the wave amplitude using two pairs of gauges positioned at
$x=6.02\,\rm m$
,
$y=0.0\,\rm m$
;
$x=6.02\,\rm m$
,
$y=0.275\,\rm m$
;
$x=6.32\,\rm m$
,
$y=0.0\,\rm m$
; and
$x=6.32\,\rm m$
,
$y=0.275\,\rm m$
. The average value across all wave steepness values is reported.
$R=a_I/a_R$
refers to the reflection coefficient, which corresponds to the ratio of the incident wave amplitude
$a_I$
to the reflected wave amplitude
$a_R$
.
Figure 2. Shapes, bearing layouts, and dimensions of the objects used: (a) Object 0 – small yellow polypropylene balls with a diameter of 2 cm and a density of
$\rho =920\,\rm kg\,m^-{^3}$
), serving as Lagrangian tracers; (b) Object 1– including flange; (c) Object 2 – including flange; (d) Object 3 – structure with six bearings; (e) Object 4 – structure with six bearings; (f) Object 5 – structure with nine bearings. (g) Top view of the objects with flanges (Objects 1 and
$ \&$
2). The flanges are made of perspex sheet and have a width of 0.15 m, a height of 0.1 m, and thickness of 0.006 m.
2.2. Floating objects
To ensure that our experiments were as close to 2-D as possible, we carefully designed the width
$b$
of our floating objects to match that of the flume. To minimise friction between objects and the walls, and to reduce the potential for rotation about the
$z$
-axis, we installed bearings (Xiros polymer ball transfer unit BB-515-B180-POM) at each end of the objects. Without a certain amount of restraint, smaller objects are prone to twisting caused by small lateral perturbations in the wave-induced forces. The bearings were specifically chosen for their low friction properties. The three dimensions of the objects are denoted as the length
$l$
parallel to the
$x$
-axis, the width
$b$
parallel to the
$y$
-axis and the height
$h$
parallel to the
$z$
-axis, and diagrams showing these dimensions are given in figures 1 and 2, which also depict the layout of the bearings that are fixed to the ends of each object (see Appendix A for additional details). The draft (submergence depth) of an object
$h_d$
is the vertical distance from its bottom to the free surface when the object is afloat in still water. The values of the dimensions of the objects are provided in table 3. The smallest objects with length
$l$
less than 0.16 m (objects 1 and 2), were fitted with 6-mm-thick perspex flanges on their ends to provide enough surface area to mount four bearings and to prevent the objects from rotating about the
$z$
-axis. To increase the range of relative object sizes
$l/\lambda$
, alongside varying the size of objects themselves, we also varied the wavelength of the incident waves (see table 4). Figure 3 provides a side and a top view of the smallest and largest objects in the flume.
Table 3. Object dimensions. Object 0 spherical ball, functioning as a current drogue. The number of bearings indicates the number installed at each end of the object. Objects 1–5 are solid and were constructed from teak wood. The density of these objects is
$\rho =6.36\times 10^2\,\rm kg\,m^-{^3}$
.
Table 4. Matrix of relative object size
$l/\lambda$
. For each relative object size, experiments with 12 different values of incident wave steepness were performed.
To determine the strength of Eulerian-mean currents at the surface, we also carried out reference experiments, in which we used small floating balls with a relative size of
$l/\lambda =1.6\%$
, as shown in figure 2(a). Previous studies have shown that objects that are this small (
$l/\lambda =1.6\%$
) behave as Lagrangian tracers and are thus transported at the speed of the Stokes drift in the absence of Eulerian-mean currents (Harms Reference Harms1987; Alsina et al. Reference Alsina, Jongedijk and van Sebille2020; Calvert et al. Reference Calvert, McAllister, Whittaker, Raby, Borthwick and van den Bremer2021). Thus, their drift corresponds to that of a Lagrangian particle and may be used to determine the strength of Eulerian-mean currents at the surface. In doing so, we further avoid the complexity of directly measuring the Eulerian-mean current (e.g. through particle image velocimetry) as these are very challenging to measure near the surface and around a moving object.
Figure 3. Side and top view of two objects in the wave flume. (a) On the left is the small object with
$l=0.05\,\rm m$
, and on the right is the large object with
$l=0.35\,\rm m$
. (b) On the left is the small object with
$l=0.05\,\rm m$
and on the right is the large object with
$l=0.35\,\rm m$
.
2.3. Data acquisition
Our goal is to examine the wave-induced drift of floating objects, investigate the effects of relative size and wave steepness, and understand the underlying mechanisms of drift enhancement by analysing the objects’ oscillatory motions and local surface elevation amplitude distribution in their vicinity. To achieve this, we first measured the wave amplitude for each wave condition in the flume using the mean value obtained from all seven wave gauges. This measured wave amplitude is used to calculate wave steepness and, subsequently, the theoretical Stokes drift. Next, we extracted the objects’ trajectories and the local surface elevation from the calibrated camera images. Image-processing techniques were used to calibrate the images, perform object tracking, and obtain the horizontal and vertical trajectories of the objects (only the horizontal trajectory for the small yellow balls). We used edge detection and curve line extraction methods to obtain the free surface elevation.
2.3.1. Image processing and object tracking
The small yellow balls were tracked using the CSRT algorithm in OpenCV in Python using images from Camera 2, allowing us to obtain their
$(x,y)$
-locations in pixel space. Care was taken to identify potential issues such as sphere collision and loss of tracking. The 2-D objects were tracked from the side of the flume using images from Camera 1. As the objects passed the vertical upright members of the wave flume structure, the camera’s view of the objects was partially or entirely obscured depending on object size. This loss of sight meant that it was not possible to use the CSRT tracker for tracking objects from the side of the flume. Instead, the objects, which were painted yellow, were identified using an HSV mask in the HSV Colour space with OpenCV for each frame, followed by temporal advancement of the frame to get the time history of object motion. To remove background noise, unwanted stationary yellow elements were removed from the frame in a preliminary step. After applying the HSV mask, we extracted the contours of rectangular boxes with fixed sizes for each object. From these contours, we identified the position of the object’s centre and calculated its roll angle in pixel space (we define the rotation of our objects around the
$y$
-axis as its roll motion). This method, in which we enforced that the tracked objects have a constant shape and size, meant it was possible to maintain tracking of the objects when they were partially obscured. Once the object trajectories were identified in pixel space, they were undistorted and transformed into physical space. The calibrated Camera 1 had an overall mean reprojection error of approximately 0.09 pixels (0.2 mm), while Camera 2 had a mean reprojection error of 0.46 pixels (0.6 mm). Examples of the horizontal trajectories are provided in figure 4(a).
2.3.2. Edge detection to measure the free surface elevation
It was not possible to operate wave gauges while the objects were in the flume, as they would obstruct the floating objects. To estimate the instantaneous free surface elevation while objects were in the flume, we used the images recorded by Camera 1. Surface elevation was extracted using edge-detection techniques. We set up virtual wave gauges that travelled with the objects, to obtain measurements of surface elevation a fixed positions relative to the objects. Specifically, we first determined the location of the objects for each frame and then established a group of virtual gauges with pre-set
$x$
-coordinates. The distance between the virtual gauges and the objects was kept constant. We set up 60 gauges with their relative position,
$x-{x_c}$
, ranging from –1.1
$\lambda$
to 1.1
$\lambda$
, in which
$x$
represents the location of the wave gauges and
$x_c$
the location of the centre of the object. To perform edge detection, we found that standard algorithms such as Canny edge detection and threshold approaches did not yield satisfactory results for our study. Therefore, we used the Holistically Nested Edge Detection (HED) algorithm, which employs a deep neural network capable of learning rich hierarchical edge maps (see Xie & Tu Reference Xie and Tu2015 for more details). We used a pre-trained HED Caffe model to leverage the benefits of the HED algorithm (see Xie Reference Xie2015). It is worth noting that while HED preserves boundaries in the image better than other methods, it produces more blurry boundaries than Canny edge detection. To enhance the sharpness of the edges, we applied the Zhang–Suen thinning algorithm using the ‘ximgproc.thinning’ function in OpenCV. By skeletonising the image, we obtained the surface elevation at the pre-set locations of the virtual wave gauges. Throughout the experiment, the positions of the objects and virtual wave gauges were updated with each frame as time progressed. Figure 4(b) provides examples of the time history of several virtual gauges, illustrating the variations in surface elevation.
Figure 4. Example time history of objects’ horizontal trajectories and surface elevation in regular waves with
$f=0.9\,\rm Hz$
. (a) Horizontal trajectories of objects with different relative sizes for
$k{a_w}=0.10$
, where
$x_c$
is the horizontal position of the objects’ centre, and
$T_w$
denotes the wave period. (b) Local surface elevation of virtual gauges at various locations for
$k{a_w}=0.20$
, where
$x$
is the horizontal position of the virtual gauges, and
$\eta$
is the local wave elevation. During the extraction process of the surface elevation for virtual wave gauges, the data are removed when the gauges are obscured by the flume’s structural members or leave the pixel space (that is, when the position of the wave gauges is beyond the scope of the camera). Extraction begins when the object reaches
${x_c}=2.3\,\rm m$
and ends when it reaches
${x_c}=8.2\,\rm m$
, away from the wave maker.
2.3.3. Data post-processing
We aim to analyse the drift velocity, oscillatory linear motion in both the horizontal and vertical directions, as well as the magnitudes of local free surface elevations using the data obtained in §§ 2.3.1 and 2.3.2. The time-average drift and horizontal oscillatory motions are derived from the horizontal trajectories of the objects. Since it takes time for the object to accelerate and for its drift to reach a quasi-steady state, in which the drift speed is constant, we eliminate the initial transient from the data. The time required for an object to reach a quasi-steady state depends on its size and wave conditions. Generally, larger objects in larger waves take longer to reach a quasi-steady state due to their greater inertia and larger motion amplitudes. However, smaller objects move more slowly and are more sensitive to environmental disturbances such as lateral wave variations and friction. Consequently, there are some instances where we must wait longer for smaller objects to achieve a quasi-steady state before collecting data. Typically, the initial transient ranges from approximately 15
$T_w$
to 30
$T_w$
in duration, although in some cases, we wait up to 100
$T_w$
before commencing data collection. For small objects in low wave steepness, it may take over an hour for the object to reach the end of the flume, whereas for large objects in high wave steepness, this process can take as little as 35 s (in such cases, we conduct 2–3 repeats to ensure data over a sufficient number of wave periods). We analyse trajectories over durations ranging from approximately 450
$T_w$
to 1000
$T_w$
for smaller objects and 35
$T_w$
to 160
$T_w$
for large objects. Once the trajectory data are obtained, the horizontal motion is filtered by using low- and high-pass frequency filters to separate the sub-harmonic (
$f/f_p\leq 0.4$
) motion and linear motions (
$0.4 \lt f/f_p\leq 1.7$
), thus removing higher-order motion. The drift velocity is examined by performing a linear regression of the sub-harmonic motion with time. The amplitude of horizontal oscillatory motion (corresponding to the
$x$
-direction) is determined by performing zero-up-crossing analysis on the signal and calculating the mean magnitude over the full time duration. The analysis of vertical oscillatory motions and local wave amplitude distribution of the surface elevation follows a procedure similar to that of horizontal motions. Each trajectory and time history of the surface elevation undergoes a manual inspection as part of the quality-control process.
3. Existing theoretical and empirical models used for comparison
In this section, we outline the two models for drift enhancement (compared with the Stokes drift) that we will use to compare with our experimental results in § 4. The Stokes drift in finite water depth can be calculated by
\begin{equation} {u_S} = c{({a_w}k)^2}\frac {{\cosh \left ( {2k\left ( {d + z} \right )} \right )}}{{2{{\sinh }^2}\left ( {kd} \right )}}, \end{equation}
where
$c=\omega /k$
is the wave celerity,
$\omega =2\pi f$
the angular frequency and
$z$
the vertical coordinate of the fluid particle. For deep water (
$kd\gg 1$
), (3.1) becomes
${u_S} = c{({a_w}k)^2}{e^{2kz}}$
. For the floating objects that we study, we evaluate the Stokes drift at
$z=0$
. The only two models that we believe are suitable for comparison with our experimental results for large objects are the diffraction-modified Stokes drift model recently developed by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) and the empirical formulas for drift enhancement proposed by Harms (Reference Harms1987). We will summarise the key equations of both models in turn.
3.1. Diffraction-modified Stokes drift of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024)
Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) showed that diffraction of the wave field affects wave-induced drift and cannot be neglected even for objects of
$l/\lambda \lt\,\rm 10$
$\%$
. They proposed a diffraction-modified Stokes drift akin to Stokes (Reference Stokes1847) but considering the combination of the incident, diffracted and radiated wave fields. Here, we follow Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) and use a simplified boundary element method (BEM) model based on potential-flow theory to solve for the linear wave field. The total linear wave field is decomposed into its incident, diffracted, and radiated parts. If we describe the flow by a linear velocity potential
$\Phi$
, it can be written as
\begin{equation} \Phi (x,z,t) = {\Phi _I}(x,z,t) + {\Phi _D}(x,z,t) + {\Phi _R}(x,z,t), \end{equation}
where
$\Phi _I$
,
$\Phi _D$
and
$\Phi _R$
are incident, diffraction and radiation potentials, respectively, then the diffraction-modified Stokes drift
$u_{S,O}$
can be calculated by a second-order accurate expansion that is analogous to Stokes (Reference Stokes1847), as shown by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024)
\begin{align} u_{S,O} &= \overline {{\xi _x}\frac {{{\partial ^2}\Phi }}{{\partial {x^2}}} + {\xi _z}\frac {{{\partial ^2}\Phi }}{{\partial x\partial z}}}\nonumber\\& =\overline { {\xi _x}\left ( {\frac {{{\partial ^2}{\Phi _I}}}{{\partial {x^2}}} + \frac {{{\partial ^2}{\Phi _R}}}{{\partial {x^2}}} + \frac {{{\partial ^2}{\Phi _D}}}{{\partial {x^2}}}} \right ) + {\xi _z}\left ( {\frac {{{\partial ^2}{\Phi _I}}}{{\partial x\partial z}} + \frac {{{\partial ^2}{\Phi _R}}}{{\partial x\partial z}} + \frac {{{\partial ^2}{\Phi _D}}}{{\partial x\partial z}}} \right )}, \end{align}
where
${\xi _x}={\mathop {\textrm {Re}}\nolimits } \{ {A_x}{e^{ - \iota \omega t}}\}$
and
${\xi _z}={\mathop {\textrm {Re}}\nolimits } \{ {A_z}{e^{ - \iota \omega t}}\}$
are the linear horizontal and vertical harmonic oscillatory motions of the object, and the overline denotes averaging over a wave period. To evaluate (3.3), the linear oscillatory motions, the three potentials, and their derivatives are all calculated using a standard BEM model, as described in detail by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024).
Unlike the perturbation expansion in position applied by Stokes (Reference Stokes1847) to an (infinitesimally small) Lagrangian particle to derive the Stokes drift, (3.3) is applied to an object of finite size. Equation (3.3) requires the evaluation of second derivatives of the velocity potential at a location where the velocity potential is not defined (i.e. at
$\xi _x=\xi _x=0$
). Ideally, these second derivatives should be evaluated at the object’s centre (of mass), which is outside of the domain where the potential-flow solutions apply (i.e. outside of the fluid domain). In practice, however, the second derivatives in (3.3) are determined based on the velocity potential evaluated at the object’s boundaries, as described in Appendix A.5 of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024). For (3.3) to be a good approximation, the horizontal and vertical distances between the object’s boundaries and its centre of mass need to be small. Formally, this requires the ratio of length of the object
$l$
and the wavelength
$\lambda$
to be small for the diffraction-modified Stokes drift model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) to apply, but not so small that the effects of diffraction themselves become insignificant.
3.2. Empirical model of Harms (Reference Harms1987)
In addition to the theoretical model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024), we also compare the data with the empirical formulae proposed by Harms (Reference Harms1987) based on their extensive set of laboratory experiments for 2-D objects. According to Harms (Reference Harms1987), object drift can be estimated by
\begin{equation} u_{S,O} = \left \{ {\begin{array}{*{20}{c}} {0.87\sqrt {gl} \frac {{2{a_w}}}{\lambda }{{\left ( {\frac {{{T_w}}}{{{T_{O,R}}}}} \right )}^{2.81 + \log \frac {{2{a_w}}}{\lambda }}}}&{\frac {{{T_w}}}{{{T_{O,R}}}} \le 1.15,}\\ {1.15\sqrt {gl} \frac {{2{a_w}}}{\lambda }{{\left ( {\frac {{{T_w}}}{{{T_{O,R}}}}} \right )}^{0.87 + \log \frac {{2{a_w}}}{\lambda }}}}&{\frac {{{T_w}}}{{{T_{O,R}}}} \gt 1.15,} \end{array}} \right . \end{equation}
where
$T_w$
is the wave period and
$T_{O,R}$
is the natural period of the roll motion of the object. Harms (Reference Harms1987) evaluated the natural period of the roll motion of the object
$T_{O,R}$
in the slender-object limit (
$l/h\gg 1$
) and using empirical values of two coefficients obtained from the literature. This result is valid for the slender objects (
$l/h\gt 3$
) that Harms (Reference Harms1987) considered, whereas our objects all had an aspect ratio of
$l/h=1.5$
. To estimate the natural roll period for each object, we conducted numerical simulations using the BEM model and identified the local peak value of the roll-mode response in waves of different frequencies. These responses were calculated for a range of frequencies from
$f=0.01$
to
$4\,\rm Hz$
(corresponding to relative size
$l/\lambda$
ranging from nearly 0 to 0.5, which covers the range of relative sizes considered in our study) with
$0.01$
-Hz increments. We also identified the frequency corresponding to the peak value of the heave motion within this frequency range. For standard rectangular floating objects, the natural roll period can be estimated using (e.g. Baniela Reference Baniela2008)
\begin{equation} T_{O,R}=2\pi \sqrt {\frac {\rho h}{\rho _w g}\frac {(l^2+h^2)}{[l^2+6\rho /\rho _w(\rho /\rho _w-1)h^2]}} \end{equation}
where
$\rho$
and
$\rho _w$
denote the densities of the object and of water, respectively, and
$g$
is the gravitational acceleration. We estimate the values of the natural roll and heave periods directly using the BEM. These periods,
$T_{O,R,BEM}$
and
$T_{O,H,BEM}$
, and the roll period obtained from (3.5) are listed in table 5 for each object. From the table, it can be seen that the values of the roll period obtained from (3.5) (
$T_{O,R}$
) and from the BEM (
$T_{O,R,BEM}$
) agree closely.
Table 5. Estimated natural periods of object heave and roll motion. Object numbers 1–5 correspond to the objects specified in table 3. The periods
$T_{O,R,BEM}$
and
$T_{O,H,BEM}$
are the estimated natural periods of roll and heave motions for the objects calculated from the BEM simulations. The natural roll period
$T_{O,R}$
can be obtained directly using (3.5).
To further validate our approach to calculating the roll period, we compared our estimate with the value obtained for a floating cuboid in the study by He et al. (Reference He, Ren and Qiu2016). The cuboid in their study had dimensions
$b=0.42\,\rm m$
,
$l=0.3\,\rm m$
,
$h=0.2\,\rm m$
, and
$h_d=0.1\,\rm m$
, with a density of
$\rho =500\,\rm kg\,m^-{^3}$
. For this geometry, our estimate of the roll period is
${T_{O,R,BEM}}=1.85\,\rm s$
and the period derived from the experiments of He et al. (Reference He, Ren and Qiu2016) is 1.6 s. We note that the free-damped oscillation decay test used by He et al. (Reference He, Ren and Qiu2016) will be affected by viscosity, whereas our (potential-flow) estimate only accounts for radiation damping. Nevertheless, we can conclude that our BEM-based estimates provide reasonable estimates of the natural roll period. We note that the empirical parametrisation (3.4) suggests that the drift peaks when
${T_w}/{T_{O,R}}=1.15$
. Values of
${T_w}/{T_{O,R}}$
we use for all of the objects considered in our study, alongside values of
${T_w}/{T_{O,H}}$
, are listed in table 6.
4. Results and analysis
In this section, we present the results of our laboratory experiments, focusing on the linear oscillatory motion, drift velocities, and the local wave amplitude distributions of the wave field surrounding the objects. We compare the results of our laboratory experiment with the models in § 3 and analyse the effects of object size and wave steepness in turn. Before analysing our results for drift, we must take into account the Eulerian-mean background current. Appendix B provides a comprehensive assessment of the repeatability of our experiments.
Figure 5. Lagrangian drift of small objects and the Eulerian-mean velocity on the surface as a function of wave steepness in waves with frequencies of
$f=0.9\,\rm Hz$
and
$f=1.0\,\rm Hz$
. (a) Lagrangian drift, normalised by wave celerity
$c$
where
$c=\omega /k$
is the ratio of angular frequency to wave number, of the 3-D small yellow balls (
$l=D=0.02\,\rm m$
) with
$l/\lambda =\,\rm 1.6$
$\%$
and the smallest 2-D objects with
$l/\lambda =\,\rm 2.6$
$\%$
and
$l/\lambda =\,\rm 3.2$
$\%$
. (b) Comparison of wave-celerity normalised Eulerian-mean velocity derived from experiments and theory (4.1). The values of
$\overline {u}_E$
for
$l/\lambda =2.6\%$
and
$l/\lambda =1.6\%$
are obtained by
$\overline {u}_E=\overline {u}_{L,O}-{u_{O,S}}=\overline {u}_{L,O}-{u_{S}}$
, in which
$\overline {u}_{L,O}$
is the Lagrangian drift of the corresponding objects that is directly measured in the wave flume and
$u_{S}$
is the theoretical Stokes drift (3.1), which is equal to the Stokes object drift for such small objects.
4.1. Eulerian-mean background current
After estimating the Lagrangian-mean drift of objects
$\overline {u}_{L,O}$
based on the data acquired in § 2.3, we can calculate what we define as the ‘Stokes object drift’
$u_{S,O}$
. The Stokes object drift is defined analogously to the Stokes drift (cf. Bühler Reference Bühler2014) as
$u_{S,O}\equiv \overline {u}_{L,O}-\overline {u}_{E}$
, in which
$\overline {u}_{E}$
is the Eulerian-mean velocity. In flumes,
$\overline {u}_{E}$
usually takes the form of a return current. We note that the appearance of the Eulerian-mean current (at the front of the wave train, i.e. from the beginning of wave generation) does not entail a temporal delay. The first arrival of waves from the wave maker is like the first half of a wave group, and a group-like structure immediately experiences divergent Stokes transport and induces the Eulerian-mean return flow (e.g. Van Den Bremer & Taylor Reference Van Den Bremer and Taylor2015; Van den Bremer & Taylor Reference Van den Bremer and Taylor2016). The factors dependent on time (that are thus not instantaneous) are the development, transport and diffusion of vorticity, which subsequently alter the Eulerian current in a complex way (Longuet-Higgins Reference Longuet-Higgins1953; Van den Bremer & Breivik Reference Van den Bremer and Breivik2018). To examine the Eulerian-mean velocity in our flume, we use small (
$D=0.02\,\rm m$
), yellow, spherical 3-D balls with a density close to that of water (
$\rho _p=920\,\rm kg\,m^-{^3}$
) as current drogues, which are expected to behave as accurate Lagrangian tracers based on their size and density (cf. Nath Reference Nath1978; Harms Reference Harms1987; Alsina et al. Reference Alsina, Jongedijk and van Sebille2020; Eeltink et al. Reference Eeltink, Calvert, Swagemakers, Xiao and van den Bremer2023). As the drift behaviour of 3-D balls is assumed to be purely Lagrangian, they are transported at the speed of the Stokes drift when the Eulerian-mean velocity is removed (i.e.
$u_{S,O} \equiv \overline {u}_{L,O}-\overline {u}_E=u_S$
when
$\overline {u}_E=0$
). For equivalent wave conditions and similar time durations in the same tank, we assume the Eulerian-mean flows at the surface for all objects are equal. Figure 5(a) shows the Lagrangian drift
$u_{L,O}$
of the small yellow balls and of the smallest 2-D object (
$l=0.05\,\rm m$
) as a function of wave steepness. The Lagrangian drifts of the balls, which are 3-D, and the small 2-D objects are found to be similar, indicating that the 2-D objects with a relative size of
$l/\lambda =\,\rm 2.6$
$\%$
exhibit predominantly Lagrangian drift behaviour and are transported at a speed equal to the Stokes drift when the Eulerian-mean velocity is removed. We also compare the measured Eulerian-mean drift obtained from the experiments with theoretical values in figure 5(b). The volume flux induced by the Stokes drift is equal to
$Q_S=\int _{-d}^{0} u_S(z)\textrm {d}z$
; the volume flux associated with the Eulerian-mean return current is
$Q_E=\int _{-d}^{0} u_E(z)\textrm {d}z$
. By assuming the Eulerian-mean return current
$\overline {u}_E$
is depth uniform (i.e., it has zero vorticity) and by balancing the volume fluxes (i.e.
$Q_S+Q_E=0$
), we obtain
$\overline {u}_E=-Q_S/d$
and thus
\begin{equation} \overline {u}_E=- \frac {{\int _{ - d}^0 {{u_S}(z)\textrm {d}z} }}{d} =- \frac {1}{d}\int _{ - d}^0 {\frac {{c{{(a_wk)}^2}\cosh 2k(d + z)}}{{2{{\sinh }^2}(kd)}}\textrm {d}z}, \end{equation}
where
$d$
is the water depth,
$u_S(z)$
is the vertical distribution of Stokes drift,
$c$
the wave celerity,
$k$
the wavenumber and
$a_w$
the wave amplitude. Surprisingly (see Van den Bremer & Breivik (Reference Van den Bremer and Breivik2018) and Monismith (Reference Monismith2020) for context), the Eulerian-mean velocity in the flume appears to be predicted very well by (4.1) (for an incident wave frequency with
$f=0.9\,\rm Hz$
). It is well known that analytical estimates and experimental measurements of the return current for regular waves are sensitive to the exact assumptions used and the boundary conditions within a given flume and that predicted and measured values often differ (Van den Bremer & Breivik Reference Van den Bremer and Breivik2018). Nevertheless, our observations indicate that the theoretically estimated Eulerian-mean flow agrees with calculations derived from the reference experiments with yellow balls.
Given this good agreement, we use the theoretical values of Eulerian-mean velocity (4.1) to evaluate the Stokes object drift
$u_{S,O}$
from the measured Lagrangian-mean object drift
$\overline {u}_{L,O}$
for all objects. The expression for the Stokes object drift is then given by
\begin{equation} u_{S,O}=\overline {u}_{L,O}-\overline {u}_{E}=\overline {u}_{L,O}+\frac {1}{d}\int _{ - d}^0 {\frac {{c{{(a_wk)}^2}\cosh 2k(d + z)}}{{2{{\sinh }^2}(kd)}}\textrm {d}z}. \end{equation}
4.2. Effect of object size and wave steepness
To examine the effects of relative object size and wave steepness, we will first look at the oscillatory motion and then the Stokes object drift followed by the local wave amplitude distribution. The linear oscillatory motions in both the horizontal and vertical directions are given in figure 6 as a function of wave steepness, and in figure 7 as a function of relative size. The motions are normalised by the corresponding wave amplitudes
$a_w$
measured using wave gauges in the wave flume without objects. The Stokes object drift obtained using (4.2) is reported in figure 8 as a function of wave steepness and relative size. For clarity, we do not plot all the measured data in each figure (that is, we do not present the data for
$l/\lambda =12.5\%$
,
$15.6\%$
,
$22.4\%$
and
$24.8\%$
in figure 8(a), but include these data in figure 8(b). By time averaging the time history of the local surface elevations of all moving virtual wave gauges (as described in § 2.3), we obtain the local wave amplitude distribution
$a(x)$
as a function of the relative distance between the gauges and the object’s centre. Figure 9 plots the normalised local wave amplitude distribution
$a(x)/{a_w}$
for objects of different relative sizes when
$k{a_w}=\,\rm 0.21$
and figure 10 plots this distribution for an object of
$l/\lambda =18.3\%$
for different values of wave steepness.
4.2.1. Oscillatory motion
Figure 6. Linear oscillatory motion amplitudes as a function of wave steepness for objects of different relative size. (a) Magnitude of horizontal motion normalised by wave amplitude. (b) Magnitude of vertical motion normalised by wave amplitude.
Figure 7. Linear oscillatory motion amplitudes as a function of the relative size of the objects for different wave steepness. (a) Magnitude of horizontal motion normalised by wave amplitude. (b) Magnitude of vertical motion normalised by wave amplitude.
Figure 8. Celerity-normalised Stokes object drift velocity (a) as a function of wave steepness for different relative object sizes and (b) as a function of relative size for different wave steepness.
We start by looking at the horizontal oscillatory motion. As shown in figure 6(a) and figure 7(a), when the objects are small, with a relative size of
$l/\lambda \lesssim\,\rm 5$
$\%$
, the magnitude of the horizontal motions is very close to the incident wave amplitude
$a_w$
, implying behaviour is essentially Lagrangian. The amplitude of horizontal motion decreases at an approximately linear rate as the relative object size increases. This trend applies to all objects in the size range we considered and is independent of wave steepness. For fixed relative size, the non-dimensional motion does not vary significantly with wave steepness (
$k{a_w} \lesssim 0.25$
). For the highest wave steepness,
$k{a_w}=0.31$
, the non-dimensional horizontal motion is slightly larger than for lower values of wave steepness, and this difference is greatest for smaller objects
$l/\lambda \lesssim\,\rm 15.6$
$\%$
. This could be a nonlinear phenomenon for which we cannot account for, but may also arise because smaller objects are more sensitive to imperfections of the wave field in the flume due to their lower inertia. As waves become very steep (near the limits of the wavemaker), both wave reflection and lateral variations in the wave field deteriorate, affecting both the value of the incident wave amplitude
$a_w$
(used for normalisation) and the motion response of the objects.
The vertical motion of the objects exhibits a more complex behaviour, as shown in figures 6(b) and 7(b). When the objects are very small, they behave like Lagrangian particles. However, as the object size increases, the non-dimensional vertical motion becomes dependent on both wave steepness and relative object size. Considering the dependence on relative size first, at low wave steepness,
$k{a_w} \lesssim 0.15$
, the vertical motion increases with relative object size when
$l/\lambda \lesssim\,\rm 15.6$
$\%$
, but decreases with size when
$l/\lambda \gtrsim$
18.3
$\%$
. At high wave steepness,
$k{a_w} \gtrsim\,\rm 0.20$
, the vertical motion always decreases with relative size, with a more pronounced decrease for larger objects,
$l/\lambda \gtrsim\,\rm 22.4$
$\%$
. The behaviour of vertical motion as a function of wave steepness also varies depending on the object size. When the object is either very small,
$l/\lambda \lesssim\,\rm 5.2$
$\%$
or very large,
$l/\lambda =\,\rm 27.2$
$\%$
, its non-dimensional vertical motion remains independent of wave steepness. However, for objects with intermediate sizes with
$8.4 \% \lesssim l/\lambda \lesssim 24.8 \%$
, their non-dimensional vertical motion decreases with increasing wave steepness. This decreasing trend is most pronounced for objects with a relative size of
$l/\lambda =15.6\%$
, 18.3
$\%$
and 22.4
$\%$
. We believe that this behaviour may be attributed to the effect of a resonance in the vertical direction, although we do not fully explain it here.
Figure 9. Non-dimensional local surface elevation amplitude distribution,
$a(x)/{a_w}$
, as a function of horizontal distance (scaled by wavenumber) from the centre of mass of the object,
$k(x-x_c)$
, for objects of different relative sizes with incident wave steepness
$k{a_w}=0.21$
, and frequency
$f=0.9\,\rm Hz$
. The black dashed line denote the results from the numerical model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) for
$l/\lambda =10\%$
, where the incident wave steepness is
$k{a_w}=0.034$
and the wave frequency is
$f=0.65\,\rm Hz$
. The red lines represent the left and right sides of the largest object with a relative size of
$l/\lambda =18.3\%$
.
Figure 10. Local surface elevation amplitude distribution,
$a(x)$
, as a function of horizontal distance (scaled by wavenumber) from the centre of mass,
$k(x-x_c$
), for the largest object with a relative size of
$l/\lambda =18.3\%$
at a frequency of
$f=0.9\,\rm Hz$
. (a) Difference between the local surface elevation amplitude and the incident wave amplitude,
$a(x)-a_w$
, for three different wave steepnesses. (b) Ratio of local surface elevation and incident wave amplitudes,
$a(x)/a_w$
, for three different wave steepnesses. The black dashed lines denote the results from the numerical model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) for
$l/\lambda =10\%$
with incident wave steepness
$k{a_w}=0.09$
and wave frequency
$f=0.65\,\rm Hz$
. The red lines represent the left and right sides of the object.
4.2.2. Stokes object drift
The Stokes object drift, normalised by wave celerity, shown in figures 8(a) and 8(b), exhibits a clear dependence on both object size and wave steepness. In general, for low wave steepness,
$k{a_w} \lesssim 0.15$
, the Stokes object drift increases with relative size for all the object sizes covered here (i.e.,
$2.6\% \leq l/\lambda \leq 27.2\%$
). However, for relatively high wave steepness,
$k{a_w} \gtrsim\,\rm 0.17$
, the drift shows a different trend. It increases with object size for
$l/\lambda \lesssim 15.6$
but exhibits a decreasing trend for
$l/\lambda \gtrsim 15.6$
. Changes in the functional dependence of the Stokes object drift on the wave steepness are also evident. Drift always increases with increasing wave steepness. For small objects (i.e.
$l/\lambda =\,\rm 2.6$
$\%$
), this increase appears to be quadratic, whereas for larger objects, the relationship between drift and steepness appears to be closer to linear – we will consider this in more detail in § 4.3 (see also Tanizawa et al. (Reference Tanizawa, Minami and Imoto2001) and He et al. (Reference He, Ren and Qiu2016) for similar observations). The relative size at which this transition in both the vertical motion and the drift behaviour occurs lies in the range
${T_w}/{T_{O,H}}=[1.07,1.31]$
(or
${T_w}/{T_{O,R}}=[0.6,0.84]$
). In contrast to the suggestion by Harms (Reference Harms1987) that the drift reaches a maximum at
${T_w}/{T_{O,R}}=1.15$
(cf. (3.4)), our observations suggest that for objects with small aspect ratio, for which relative size the maximum (peak) values in vertical motions and drift rates occur, depends on
${T_w}/{T_{O,H}}$
(e.g. the peak value of the drift occurs when the ratio is equal to 1.15).
4.2.3. Local wave amplitude distribution
Next, we examine the wave field in the vicinity of the object. Specifically, we examine the wave amplitude distribution at virtual probes that are stationary in the reference frame of the object (the reference frame drifts and oscillates horizontally with the object). In this reference frame, the amplitude field has a spatial distribution (consisting of nodes and antinodes), which is obtained by time averaging the amplitudes at each virtual probe location (in the reference frame of the object) over time after reaching a quasi-steady state. The local wave amplitude distribution in the vicinity of the moving object thus obtained is shown in figure 9. Note that we have corrected the wave elevation obtained from edge detection to address potential measurement errors in the camera data. This correction is achieved by using reference data of wave elevation in the absence of the object and ensuring consistency of this data with the wave gauge data collected in the tank (i.e.
$a_w$
), which is used for normalisation.
Figure 9 also shows the results from the numerical simulations of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024). There are two important differences between the surface amplitude distribution obtained from experiments in the present paper and from numerical simulations of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) that only make a qualitative comparison between the two possible. First, the results from Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) are for lower wave steepness (
$ka_w=0.034$
). When the wave steepness becomes too high, the large drift of the object causes large mesh deformation, making the dynamic mesh-based numerical model less accurate. This challenge does not arise in our experiments, where it is, in fact, easier to measure the surface amplitude distribution for higher steepness. The second difference is the reference frame in which the surface amplitude distributions are obtained. Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) used ‘gauges’ to output the surface elevation that were fixed in the stationary reference frame. In Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024), implementing moving gauges was shown to be prohibitively computationally expensive, as the gauges move with the object horizontally every time step. The reason Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) could still observe a standing wave pattern in the stationary reference frame is probably due to the relatively small magnitude of the drift therein.
It is clear from figure 9 that a standing wave pattern forms that is clearest for relatively large objects,
$l/\lambda =\,\rm 10.5$
$\%$
and 18.3
$\%$
. The local amplitude of the surface elevation is normalised by the wave amplitude in the absence of the object,
$a_w$
. Similar to observations by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024), for small objects,
$l/\lambda =\,\rm 2.6$
$\%$
, the local time-averaged wave amplitude does not vary significantly with distance to the object’s centre. It remains relatively constant at a value of
$a(x)/{a_w}=1.0$
, suggesting negligible disturbance to the wave field and no standing waves. As the object size increases, a standing wave pattern becomes apparent. On the upstream side, the time-averaged wave amplitudes show a pattern of smaller and larger amplitudes compared with the incident wave field (or compared with the case of a very small object) at different distances from the object’s centre. The standing wave pattern thus identified is more evident as the object becomes larger. On the downstream side, (somewhat) smaller amplitudes are observed for all locations, and there is a discernible difference in the magnitude of the surface elevation between the two sides of the object for large enough objects. The formation of the standing wave pattern is explained by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) using their numerical model. As the object is large enough to diffract the wave field, it ‘disrupts’ the wave field and ‘impedes’ the passage of waves, leading to local blocking. This disturbance results in a difference between wave amplitudes on the upstream and downstream sides of the object. We note that, as described in § 2.3.2, the locations of the virtual wave probes are continuously updated in time to maintain a constant horizontal distance relative to the object’s centre. Consequently, the standing wave pattern observed only occurs in the reference frame of (and thus stationary relative to) the object.
In figure 10, we show how the standing wave pattern varies with steepness for relatively large objects (large enough to generate a clear standing wave pattern). The standing wave pattern is plotted as
$a(x)-{a_w}$
in figure 10(a) and as
$a(x)/{a_w}$
in figure 10(b), for an object of size
$l/\lambda =\,\rm 18.3$
$\%$
. The pattern is clearer when plotted as an absolute value (that is, as
$a(x)-{a_w}$
) as incident wave steepness increases from
$k{a_w}=0.10$
to
$k{a_w}=\,\rm 0.21$
, while the pattern is less evident when plotted as a normalised value (that is, as
$a(x)/{a_w}$
) as steepness increases. This observation aligns with previous findings of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024), as increasing wave steepness increases the wave-celerity normalised Stokes object drift, while decreasing the Stokes-drift-normalised Stokes object drift. If we consider two identical waves with an angular frequency of
$\omega$
, wavenumber
$k$
and wave amplitude
$a_w$
, propagating in opposite directions (i.e. 100 % reflection by the object), the resultant wave elevation
$\eta (x,t)$
would take the form
$\eta (x,t)=2a_w \sin (kx) \cos (\omega t)$
. By taking the modulus of the wave amplitude distribution (
$|2\sin (kx)|$
), we obtain the standing wavelength, which is half that of the incident waves. Indeed, the wavelength of the standing wave pattern depicted in figures 9 and 10 is
$\lambda _{sw}=1.05\,\rm m$
, approximately half of the incident wavelength
$\lambda = 1.91\,\rm m$
. We note that the effect of evanescent waves may be included in the estimate of
$\lambda _{sw}$
, as well as the effects of phase differences that result from the distance to the wave probe being measured relative to the object’s centre of mass, rather than its boundaries (the horizontal positions of the boundaries relative to the (virtual) probe change constantly as the object moves and rotates). Moreover, the (virtual) wave probes are set with a certain spatial resolution, which may introduce small errors in determining maxima and minima and their exact locations, from which
$\lambda _{sw}$
is estimated.
Despite obvious general qualitative agreement, there are two ways the standing wave pattern found here is different from that identified by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024), as shown in figures 9 and 10: (1) the phase difference between the standing wave patterns formed by objects of different sizes and (2) the sign of the difference in amplitude between the two sides of the object. First, Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) found that the local maxima and minima in the amplitude distribution occurred at similar relative distances for objects of different sizes, which is not the case here (cf. Figure 9). Second, in Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024), the local surface elevation amplitude on the upstream side of the object (i.e., the locations just to the left of the leftmost vertical red line in figure 9) is smaller than on the downstream side (i.e. the location just to the right of the rightmost vertical red line in figure 9), while our experiment shows the opposite. The discrepancies are likely due to our (virtual) gauges moving with the objects (i.e. the distance between the gauges and the object is constant), while the gauges in Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) were fixed in a stationary reference frame, causing a phase shift and a non-stationary standing wave pattern.
Table 7. Coefficients of the regression of wave-celerity normalised Stokes object drift
$u_{S,O}/c$
on wave steepness
$\epsilon =k{a_w}$
given by (4.3). The error bounds represent
$\pm$
one standard deviation of the coefficients, and
$R^2$
is the coefficient of determination.
4.3. A new empirical parametrisation for enhanced Stokes object drift
To quantitatively describe the effects of relative size
$l/\lambda$
and wave steepness
$\epsilon =k{a_w}$
in our experimental observations, we perform the following polynomial regression for the wave celerity-normalised Stokes object drift
$u_{S,O}/c$
\begin{equation} u_{S,O}/c=p_{00}+p_{10}\left (\frac {l}{\lambda }\right )+p_{01}\epsilon +p_{20}\left (\frac {l}{\lambda }\right )^2+p_{11}\left (\frac {l}{\lambda }\right )\epsilon +p_{02}\epsilon ^2, \end{equation}
where
$p_{00}$
,
$p_{10}$
,
$p_{01}$
,
$p_{11}$
and
$p_{02}$
are the coefficients to be determined. We have chosen a second-order bi-variate polynomial functional form in (4.3) for its simplicity and because previous experimental studies have revealed a mixture of linear and quadratic correlations between drift, object size and steepness (Harms Reference Harms1987; Tanizawa et al. Reference Tanizawa, Minami and Imoto2001; Huang & Law Reference Huang and Law2013) (as discussed in § 1.4). We note that idealised Lagrangian particles have a (celerity-normalised) drift, the Stokes drift (3.1), that is (only) quadratic in steepness. We consider cases with all terms retained as well as those with several terms set to zero. Table 7 shows the coefficient values obtained from the regression with 95
$\%$
confidence bounds. To ensure independence between the covariates, which were varied independently in our experimental matrix, we have also estimated the population Pearson correlation between relative size
$l/\lambda$
and wave steepness
$\epsilon$
, finding a very small sample correlation coefficient
$O(10^{-18})$
. There is clear evidence that the Stokes object drift is dependent on both linear and quadratic terms in wave steepness (
$\epsilon$
and
$\epsilon ^2$
); the coefficient on the quadratic term is the most significant followed by the linear term. Ideally, if both the wave steepness and relative size of the object are 0, the drift should be zero, so we set
$p_{00}=0$
. The fact that this (setting
$p_{00}=0$
) significantly changes the regression coefficients suggests that the simplified combination of linear and a quadratic terms has limitations. The quadratic term in
$l/\lambda$
is insignificant, and we set it to zero (
$p_{02}=0$
). We thus propose the simplified relationship
\begin{equation} u_{S,O}/c=0.0008\left (\frac {l}{\lambda }\right ) + 0.03 \epsilon +0.0060 \left (\frac {l}{\lambda }\right ) \epsilon +1.0\epsilon ^2. \end{equation}
This simplified model provides a reasonably accurate representation of the wave celerity-normalised Stokes object drift in terms of relative size and wave steepness, with the most significant contributions coming from the linear and quadratic terms of wave steepness. However, it is important to note that this simplified model is specific to the experimental data and may not capture all possible complexities of the drift behaviour. We have also conducted a similar polynomial regression of drift normalised by Stokes drift as a function of relative size and wave steepness. However, the results suggest that higher-order terms (at least a third order) of wave steepness need to be considered to make
$R^2$
larger than 0.9, indicating potentially complex behaviour that goes beyond a simple polynomial model.
4.4. Comparison with existing theoretical and empirical models
In this section, we set out to assess the range, in terms of relative object size and wave steepness, over which the classical Stokes drift (given by (3.1)), the diffraction-modified Stokes drift of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) (given by (3.3)) and the empirical formulae of Harms (Reference Harms1987) (given by (3.4)) are valid. This has been done by comparing the drifts from experimental measurement to those predicted by three different models and by examining their effectiveness in a parameter space made up from two parameters: relative size and wave steepness. To facilitate the comparison, we use two different normalisation techniques for the drift: celerity normalisation and Stokes-drift normalisation, and then examine the normalised drift as a function of relative object size and wave steepness. We first compare the experimental measurements to the theoretical Stokes drift and the diffraction-modified Stokes drift model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) by normalising the drift by the theoretical Stokes drift and showing this as a function of relative object size and wave steepness, respectively. Next, a comparison is drawn between the experimental results and the empirical formula by (3.4), along with the diffraction-modified Stokes drift, as a function of relative size for different values of wave steepness. To help explain any discrepancies between measurements and model predictions, the amplitude of the linear oscillatory motion values derived from experiments are compared with those predicted by the BEM model, which is used to calculate the diffraction-modified Stokes drift in (3.3).
Figure 11. Stokes drift-normalised Stokes object drift (a) as a function of wave steepness for different relative object sizes and (b) as a function of relative size for different values of wave steepness. An idealised Lagrangian particle with zero enhanced drift would have
$u_{S,O}/u_S=1$
. The red dashed line with red square labelled DMS represents the diffraction-modified Stokes drift given by (3.3). As the diffraction-modified Stokes drift for a certain object size does not change with wave steepness (we use a linear ‘BEM’ model to solve (3.3), and (3.3) is a second-order accurate quantity), it is shown as a single line in panel (b).
The experimentally measured Stokes object drift scaled by the theoretical Stokes drift as a function of wave steepness and relative object size are presented in figures 11(a) and 11(b), respectively. As shown in panel (a), small objects
$l/\lambda \leq 2.6\%$
follow the theoretical Stokes drift. As the object size increases, the amplification factor
${u_{S,O}}/{u_S}$
increases, suggesting a larger drift enhancement. This is true for all values of wave steepness but is especially evident for smaller values of wave steepness. For objects of a certain size, the amplification factor decreases drastically as wave steepness increases in the low to intermediate wave steepness range, reaching its peak value at the lowest wave steepness. As the wave steepness increases further, the amplification factor either becomes constant or decreases slowly with steepness. A decreasing trend with wave steepness is more distinct for larger objects and was also observed in measurements by Huang et al. (Reference Huang, Law and Huang2011), although they do not explicitly discuss this.
Comparison between the experimentally measured Stokes object drift and the diffraction-modified Stokes drift normalised by theoretical Stokes drift is made in figure 11(b). In addition, a comparison between the measured Stokes object drift, the diffraction-modified Stokes drift proposed by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) and the empirical formula of Harms (Reference Harms1987) is made in figure 12 by examining the drift normalised by wave celerity
$c$
as a function of relative size for different values of wave steepness. The figures show that for small objects, the diffraction-modified Stokes drift accurately predicts Stokes object drift for various values of steepness, but this agreement breaks down for relatively large objects, for which the expansion that underlies (3.3) is no longer valid. As shown in both figures 11(b) and 12, the diffraction-modified Stokes drift is capable of giving a good prediction beyond the range of the classical Stokes drift.
Figure 12. Comparison of the Stokes object drift from experiments with the diffraction-modified Stokes drift (3.3) proposed by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) and the empirical formula (3.4) of Harms (Reference Harms1987) as a function of relative object size for different values of wave steepness. The drift is normalised by wave celerity. Markers connected by continuous lines correspond to the results from experiments, dashed lines (of the same colour, corresponding to the colour in figure 11(b) to the diffraction-modified Stokes drift, and dash-dotted lines (of the same colour) to the empirical formula of Harms (Reference Harms1987).
Figure 13. Comparison of the amplitudes of oscillatory motion for objects of different sizes and in different wave steepness from experiments to results predicted by a linear BEM model. (a) Horizontal oscillatory motion. (b) Vertical oscillatory motion. The motion amplitudes in panels (a) and (b) are normalised by the incident wave amplitude
$a_w$
. Markers connected by continuous lines correspond to the results from experiments, while dashed lines of the same colour are those from BEM simulations.
As the diffraction-modified Stokes drift (3.3) is calculated based on the linear oscillatory motion of the object, figure 13 compares the amplitudes of the horizontal and vertical components of motion between experiments and the linear BEM that underlies (3.3) (see Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) for details) as a function of relative size for different values of wave steepness. The linear BEM model performs well in predicting the horizontal oscillatory motions for all objects and wave steepnesses, but fails to accurately predict the vertical motions of larger objects where the assumption of linearity underlying the BEM model probably fails.
Figure 14. Domains of validity of the three different models for Stokes object drift within the parameter space covered by experiments in this paper: the classical Stokes drift for idealised Lagrangian particles (3.1), the diffraction-modified Stokes drift (3.3), and the empirical formula of Harms (Reference Harms1987) (3.4). The black dot represents the experimental data points.
Figure 15. Domain of validity of the three models compared with the experiments in this paper. Each circle represents an experiment, with the colour of the filled circle denoting the model(s) that has an error of less than
$20\%$
compared with the experiment. The black circle corresponds to an experiment for which none of the three models is able to predict the drift observed in experiments with an error of less than
$20\%$
. Blue and green denote the classical Stokes drift and the diffraction-modified Stokes drift, respectively. Purple denotes the empirical formula of Harms (Reference Harms1987) (3.4). A combination of colours within a circle suggests multiple models have an error of less than
$20\%$
compared with experiments.
Despite the difference in density between the objects used here (
$\rho =636.2\,\rm kg\,m^-{^3}$
) and by Harms (Reference Harms1987) (
$\rho =0.92\,\rm kg\,m^-{^3}$
), the empirical formula by Harms (Reference Harms1987) still works well for small wave steepness and all object sizes considered here. However, the formula by Harms (Reference Harms1987) underestimates the drift significantly for all sizes when the wave steepness is relatively large (
$ka_w\geq 0.16$
).
The empirical formula of Harms (Reference Harms1987) is a discontinuous function of
$T_w/{T_{O,R}}$
, with one parametric fit for short-period waves,
$T_w/{T_{O,R}}\le 1.15$
(corresponding to objects of large relative size) and another for long-period waves,
$T_w/{T_{O,R}}\gt 1.15$
(corresponding to objects of small relative size). The transition from short-wave to long-wave drift behaviour occurs when
$T_w/{T_{O,R}}=\,\rm 1.15$
. According to (3.4), Stokes object drift is fundamentally different in long and short waves. When the object is small, that is, in the long-wave regime, the drift increases monotonically with relative size, while when the object is large, that is, in the short-wave regime (i.e. when
$l/\lambda \ge\,\rm 50$
$\%$
) drift decreases continuously with relative size. The drift peaks at the transition point between the two regimes.
Finally, in figures 14 and 15, we present the domains over which the different models we have discussed agree with the experimental data in this paper. When an object is very small
$l/\lambda \lesssim\,\rm 3.2$
$\%$
, it is transported at a speed equal to the theoretical Stokes drift for the full range of wave steepness considered. As object size increases, up to
$l/\lambda =\,\rm 8.4$
–
$23.6\%$
depending on steepness, drift behaviour is enhanced compared with the theoretical Stokes drift, and the diffraction-modified Stokes drift gives a relative error of less than 20
$\%$
when compared with experimental data. The diffraction-modified Stokes drift predicts drift enhancement as a result of increased vertical motion (relative to the incident wave field) and the difference between the diffracted and radiated wave fields. The range of object sizes for which the diffraction-modified Stokes drift result of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) is valid depends on the wave steepness (see figure 11). As the wave steepness increases, its drift enhancement amplification (relative to the theoretical Stokes drift) decreases to a smaller number compared with the same ratio in low wave steepness. As the relative object size is increased further, the assumptions underlying the diffraction-modified Stokes drift derivation by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) break down, and the theory becomes ineffective. The empirical formula by Harms (Reference Harms1987) for large object sizes is then more accurate provided the steepness is low. To investigate the correlation between object drift and both relative size and wave steepness, we perform the polynomials regression analysis based on our experimental data. The detailed analysis including our own parametrisation is given in Appendix 4.3.
5. Conclusion
This paper has presented an experimental study of the drift of large rectangular 2-D objects under the action of waves and, in doing so, evaluated the effectiveness of the diffraction-modified Stokes drift model proposed by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024). It is experimentally difficult to examine the drift of 2-D objects (as objects may spin under the influence of lateral variation and become jammed). To overcome these difficulties, we have designed objects that float freely on the water’s surface, while occupying the full width of the wave flume used, meaning our experiments are as close to 2-D as possible. To achieve this, the objects are equipped with low-friction bearings and ‘flange’ structures to support these bearings for small objects.
We have considered rectangular objects with sizes up to 27 % of the length of the incident waves. Very small objects (less than 3 % of the wavelength) essentially behave as Lagrangian tracers, and their drift is well captured by the classical Stokes drift of Stokes (Reference Stokes1847). However, for larger objects, we found drift much greater than that predicted by the classical Stokes drift; we observed drift velocities up to 17 times larger than the Stokes drift in the extreme case (for the largest object in the smallest steepness waves considered).
In trying to understand the behaviour of larger objects, an important transition occurs when the ratio of the incident wave period to the natural frequency of heave is of order 1 (
${T_w}/{T_{O,H}}=\mathcal {O}(1)$
), as previously observed by Harms (Reference Harms1987). Around this point, the vertical oscillatory motion peaks and the drift behaviour changes from increasing with relative size to decreasing with relative size. For objects that are large enough to diffract the wave field and whose natural period of oscillation is significantly smaller than that of the incident waves, the enhanced drift appears to be primarily caused by the disturbance to the wave field the object causes. Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) proposed a diffraction-modified Stokes drift, which appears to capture the drift behaviour well in this regime. Once the object becomes too large (
$\sim l/\lambda \gt 10$
%), the small-parameter assumption underlying the diffraction-modified Stokes drift starts to break down. The domain of validity of the diffraction-modified Stokes drift extends to larger objects for steeper waves, which we attribute to the fact that the drift enhancement amplification
$u_{O,S}/u_S$
is greatly reduced (more linear) for high wave steepness, which, somewhat counter-intuitively, extends the range the underlying linear BEM model is valid.
The transition behaviour is related to the behaviour of the objects’ natural heave motion. For objects whose natural period of motion is much smaller than the period of the incoming waves, the oscillatory motion of the object change monotonically (vertically increased and horizontally decreased). The drift increases with object size. The drift enhancement for such objects is attributed by Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) to increases in their heave motion and a positive imbalance between the effects of the diffracted and radiated wave fields. For objects whose periods are much larger than those of the incoming waves, the oscillatory motions are greatly reduced (e.g.
$l/\lambda =\,\rm 27.1$
$\%$
), and their drift increases with relative size at a slow rate or becomes constant.
The dependence of drift behaviour on wave steepness is found not to be purely quadratic (as predicted by Stokes (Reference Stokes1847)) but a mixture of linear and quadratic, and changing with relative size. In addition, if the waves are too steep, the linearised BEM model we use to predict diffraction, and thus the diffraction-modified Stokes drift of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024), breaks down, and a higher-order model would be needed to predict the diffracted and radiated wave field. The empirical formula of Harms (Reference Harms1987) does well for larger object sizes and small steepness, but breaks down for larger steepness. Based on our experimental data, we proposed a new parameterisation that predicts the wave-induced drift as a function of relative size (the ratio of object size to the wavelength) and wave steepness, including linear and quadratic terms.
We make the following four recommendations for future work. First, the diffraction-modified Stokes drift model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) should be made suitable for more nonlinear circumstances by expanding (3.3), which is based on small-parameter assumptions and only retains first-order terms, to include higher-order terms. Due to the linearity of the BEM model used to calculate the radiated and diffracted wave fields and the approximate nature of the model, the diffraction-modified Stokes drift theory based on (3.3) from Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024) is ineffective in predicting the drift for large objects in high wave steepness.
Second, while (second-order) drift forces are routinely calculated (e.g. Newman (Reference Newman1967); Faltinsen (Reference Faltinsen1993)), it remains unclear how a force balance (between the drift force and the wave drift damping force (Nossen et al. Reference Nossen, Grue and Palm1991; Aranha Reference Aranha1994)) can be used to predict an enhanced steady drift of large objects analogous to the diffraction-modified Stokes drift of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024). This should be explored in future work.
Third, for ocean waves, including both internal and surface waves, the Lagrangian-mean object drift is given by
$\overline {u}_{L,O}=u_{S,O}+\overline {u}_{E,O}$
(e.g., Haney & Young Reference Haney and Young2017; Van den Bremer & Breivik Reference Van den Bremer and Breivik2018; Thomas, Bühler & Smith Reference Thomas, Bühler and Smith2018; Higgins, van Den Bremer & Vanneste Reference Higgins, van Den Bremer and Vanneste2020a
). To examine the extent to which the Lagrangian-mean object drift is affected by the introduction of finite object size, we have exclusively examined how the Stokes object drift is affected by the finite size of the object, leaving the Eulerian-mean flow unaffected (as in the deep-water diffraction-modified Stokes drift model of Xiao et al. (Reference Xiao, Calvert, Yan, Adcock and van den Bremer2024)). Under certain circumstances, it is not inconceivable that the Eulerian-mean flow may also be affected by the presence of the object, when this becomes very large and prohibits fluid passage. This could result in flow blockage; the Eulerian-mean flow would exhibit a distinctly different structure compared with that assumed in the current study.
We hypothesise that the effect of finite object size on the Eulerian-mean flow field is more evident in shallow water depths, where the contribution of the second-order velocity potential to drift cannot be neglected (Pessoa & Fonseca Reference Pessoa and Fonseca2015). Therefore, we recommend the effects of finite depth to be examined in future work.
Fourth, although the focus of this paper has been on two-dimensional geometries, we acknowledge that such a configuration is a very strong simplification of wave-induced drift in many real-world environments. Future work is thus recommended to explore how the presence of a third dimension and the influence of object width affect the diffraction-modified Stokes drift mechanism. As diffraction of the wave field is the main cause of enhanced drift, we expect drift enhancement to be reduced in 3-D compared with 2-D for a unidirectional incident wave field, as waves can diffract around the object in 3-D instead of just being either reflected or transmitted (2-D). Finally, it would be of interest to explore the effects of directional spreading.
Acknowledgements.
Q.X. is supported by the China Scholarship Council-PAG Oxford Scholarship. T.S.vdB was supported by a Royal Academy of Engineering Research Fellowship.
Declaration of interests.
The authors report no conflict of interest.
CRediT author statement.
Qian Xiao: Conceptualisation, experimental design and execution, data collection and analysis, writing-original draft preparation. Mark McAllister: Experimental design, supervision, writing-reviewing and editing. Tom Adcock: conceptualisation, supervision, writing-reviewing and editing. Ton van den Bremer: conceptualisation, supervision, writing-reviewing and editing.
Appendix A. Photographs of objects with bearings
Figure 16 provides side and front views of the objects outside of the water. It shows the yellow-painted ends of the objects, where the bearings are installed.
Figure 16. Two-dimensional-type objects. (a) Side view of five objects (painted yellow) installed with bearings. Objects 1, 2, 3, 4 and 5 are shown from left to right, top to bottom. Objects 1 and 2 are designed with ‘flange’ structure shown in figure 2. (b) Front view of objects.
Figure 17. Uncertainty envelopes as a function of relative size for different wave steepness: (a) Stokes object drift, these data are shown without normalisation, and we only plot data for objects with relative sizes of
$l/\lambda =2.6\%$
,
$5.2\%$
,
$8.4\%$
,
$10.5\%$
,
$18.3\%$
and
$27.1\%$
; (b) celerity-normalised Stokes object drift velocity; (c) wave-amplitude-normalised magnitude of horizontal motion; (d) wave-amplitude-normalised magnitude of vertical motion. Mean values, denoted by lines, are obtained using five repeat experiments. The error bars, shown as shaded areas, correspond to two standard deviations either side of the mean. The transparency of panel (c) has been adjusted to make it clearer for overlapping parts.
Appendix B. Measurement error and repeatability
To assess the repeatability of the experiments we present and the precision of the approaches we use to acquire measurements, we performed repeated experiments for four wave steepnesses (
$k{a_w}=0.06$
, 0.10, 0.20 and 0.32) for each object size
$l/\lambda$
. We conducted five repeats for these experiments and present the results in figure 17, where we show normalised linear oscillatory motions and drift as a function of relative size for different wave steepness. For drift, our experiments demonstrate very small variability for low wave steepness and small objects; variability is larger for high wave steepness and larger objects. We attribute this to two factors. Firstly, the way we conducted the experiment plays a role. For very small objects in low wave steepness, we limited the object to drift a shorter section of the flume (only the middle two sections, 4 m) instead of allowing it to drift along the entire length. This is because these objects travel very slowly. The objects are thus not subject to any potential variation in conditions along the length of the flume, leading to less variability. However, for larger objects in high wave steepness, which travel more quickly, for one experiment, we needed to perform 2–3 runs to ensure that we acquired sufficient data for at least 35 wave periods (after reaching a quasi-steady state). This introduces more variability in the measurements. The second factor that contributes to the variability is the repeatability of the waves in the flume, as shown in figure 18. The variability of the oscillatory motion appears larger due to the normalisation process, where the standard deviation is amplified. Unlike the drift, the relatively large variability of the oscillatory motions occurs primarily in cases with small objects and intermediate wave steepness. Apart from the wave repeatability and ways to conduct the experiment, this is also due to the fact that the larger the object is, the easier it is to capture its bobbing motion using the camera.
Figure 18. Uncertainty envelopes of wave amplitude as a function of wave steepness for different wave frequencies. Mean values, denoted by lines, are obtained using five repeated experiments. The error bars, shown as shaded areas, correspond to two standard deviations either side of the mean.
Table 8. Measurement error of the Lagrangian object drift
$\overline {u}_{L,O}$
during fitting of the horizontal trajectory for objects with different sizes and in varying wave steepness. We chose three different sizes of objects to report here:
$l/\lambda =2.6 \%$
for Object 1 in wave
$f=0.9\,\rm Hz$
;
$l/\lambda =12.5 \%$
for Object 3 in wave
$f=1.1\,\rm Hz$
;
$l/\lambda =22.4\%$
for Object 5 in wave
$f=1.0\,\rm Hz$
.
$1-R^2$
refers to the goodness of fit with
$R^2$
the uncertainty in the prediction of the fitted polynomial.
$\sigma$
refers to the standard deviation (uncertainty of the coefficients) of the linear coefficient determination in the fit, and thus
$\sigma /\overline {u}_{L,O}$
represents the standard deviation of the fitted linear coefficient.
We assess the repeatability of the waves in the absence of the object here, to better understand the nature of the uncertainty we observe above. The wave gauges set up are the same as in § 2.1. We chose four wave steepnesses (
$k{a_w}=0.06$
, 0.10, 0.20 and 0.32) for each frequency in table 2. We conducted five repeats for each experiment. The mean values of wave amplitude for each wave gauge are calculated, and we use the average value of all seven gauges as our wave amplitude for one repeat. The measured data are given in figure 18. The main variability occurs at high wave steepness, and at intermediate wave steepness for a frequency of
$f=1.1\,\rm Hz$
. This variability is most likely caused by reflections that are not absorbed by the beach. We conclude wave gauge measurements are a small cause of error.
We calculate drift velocity by performing a linear regression on the measured trajectories once a steady state is reached. To evaluate how well this works for our measured object trajectories and estimate the uncertainty in the gradient (drift velocity,
$\overline {u}_{L,O}$
), we report the measurement errors, including the coefficients of determination in table 8. The quantity
$1-R^2$
increases monotonically with wave steepness. This is expected, as larger wave steepness leads to larger horizontal oscillatory motions of the object, causing deviations from a constant gradient and increasing the uncertainty in the fit. The relative uncertainties for the parameters are all very small with a maximum value of 1.5 %.
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