1. Introduction
Since the beginning of the plastic industry in the 1950s, it is estimated that roughly ten billion metric tons (
$10^{13}$
kg) of plastics have been produced worldwide, more than half of which have been produced in the last twenty years only (Geyer, Jambeck & Law Reference Geyer, Jambeck and Law2017). Millions of tons of such plastics, the majority of which are positively buoyant, enter the ocean every year in the form of small debris (Geyer et al. Reference Geyer, Jambeck and Law2017). While much attention has been devoted to micro-plastics smaller than 5 mm, recent estimates indicate that 95 % of the mass of buoyant marine plastics is accounted for by macro-plastics larger than 25 mm (Kaandorp et al. Reference Kaandorp, Lobelle, Kehl, Dijkstra and van Sebille2023), which are much greater than the dissipative scales of the turbulence in the upper ocean (Jiménez Reference Jiménez1997). Therefore, to devise predictive dispersion models and sequestration strategies for this harmful pollution, it is imperative to reach a predictive understanding of the transport of relatively large floating objects. This requires merging two challenging branches in the study of turbulence: its behaviour along a free-surface and its ability to transport finite-size particles. Recent studies have posed fundamental related questions in the context of physical oceanography, considering individual and collective properties of floating particles along with the effects of currents, wind, waves and the Earth’s rotation (Beron-Vera Reference Beron-Vera2024; Bonner, Beron-Vera & Olascoaga Reference Bonner, Beron-Vera and Olascoaga2024).
The goal of the present experimental study is to gain insight into the Lagrangian transport of particles of different sizes floating on the quasi-flat free-surface above homogeneous isotropic turbulence, in the absence of surface stresses and significant waves. As the literature on the different involved areas is vast, in the following, we briefly summarise only the background information which is particularly relevant to this work.
1.1. Turbulent dispersion
The modelling of Lagrangian turbulent dispersion originates from the theory of Taylor (Reference Taylor1922) for the evolution in time
$t$
of the position
$x$
of a fluid parcel released at time
$t_0$
from a point source in stationary homogeneous isotropic turbulence. The mean square displacement
$\langle X^2 \rangle (t) = \langle [x(t_0)-x(t)]^2\rangle$
is related to the Lagrangian velocity autocorrelation coefficient
$R_u^L(\tau ) = \langle u(t_0+\tau )u(t_0)\rangle /\langle u^2 \rangle$
via:
\begin{equation} \langle X^2 \rangle (t) = 2\langle u^2 \rangle \int _0^t \int _0^{t^\prime } R_u^L(\tau )\,\mathrm{d}\tau\, \mathrm{d}t^\prime . \end{equation}
Here,
$\langle \boldsymbol{\cdot }\rangle$
denotes ensemble averaging,
$u$
is the fluctuating velocity,
$\tau =t-t_0$
is the time lag and only one component of motion is considered for simplicity of notation. For times much larger than the integral time scale
$T_L = \int _0^\infty R_u^L(\tau )\, \mathrm{d}\tau$
,
$R_u^L (\tau )$
tends to zero and
$\langle X^2 \rangle (t) = 2 \langle u^2 \rangle T_L t$
. This implies that the long-time turbulent dispersion behaves as a Brownian process with diffusivity:
\begin{equation} K_t = \frac {1}{2} \frac {\mathrm{d}}{\mathrm{d}t} \langle X^2 \rangle (t) = \langle u^2 \rangle T_L. \end{equation}
Taylor’s theory finds wide applications in atmospheric sciences and oceanography (Griffa Reference Griffa1996; Wilson & Sawford Reference Wilson and Sawford1996), and it can be extended to inhomogeneous flows upon appropriate stationarisation of the velocity (Batchelor Reference Batchelor1957; Viggiano et al. Reference Viggiano, Basset, Solovitz, Barois, Gibert, Mordant, Chevillard, Volk, Bourgoin and Cal2021). The crux of the problem is evaluating
$R_u^L (\tau )$
: this is challenging to measure, as it requires reconstructing trajectories of duration longer than
$T_L$
while resolving fine temporal fluctuations associated with the highly intermittent Lagrangian acceleration
$a$
(Toschi & Bodenschatz Reference Toschi and Bodenschatz2009). Indeed, significant efforts have been made to model this quantity. Taylor (Reference Taylor1922) assumed a simple exponential,
$R_u^L(\tau ) = \mathrm{e}^{-\tau /T_L }$
, which captures the long-time decay but not the short-time kinematics
$R_u^L(\tau ) = 1-\langle a^2 \rangle \tau ^2 / 2\langle u^2 \rangle$
. This is accounted for by the two-time model proposed by Sawford (Reference Sawford1991) and adopted in several later studies (Mordant et al. Reference Mordant, Lévêque and Pinton2004b
; Jung, Yeo & Lee Reference Jung, Yeo and Lee2008; Huck, Machicoane & Volk Reference Huck, Machicoane and Volk2019; Berk & Coletti Reference Berk and Coletti2021, Reference Berk and Coletti2024; Salmon et al. Reference Sanness Salmon, Baker, Kozarek and Coletti2023):
\begin{equation} R^L_u(\tau ) = \frac {T_1 \mathrm{e}^{-\tau /T_1}-T_2\mathrm{e}^{-\tau /T_2}}{T_1-T_2}. \end{equation}
Here,
$T_1$
and
$T_2$
are associated with the integral and dissipative time scales of the turbulence, respectively. Sawford (Reference Sawford1991) modelled the acceleration of fluid particles with a second-order autoregressive equation in which the random perturbation is time-correlated and therefore differentiable. Thus, the kinematic relationship
$\langle a^2 \rangle R^L_a(\tau ) = - \langle u^2 \rangle \mathrm{d}^2 R^L_u(\tau ) / \mathrm{d}\tau ^2$
(Tennekes & Lumley Reference Tennekes and Lumley1972) can be used to evaluate the Lagrangian acceleration autocorrelation coefficient
$R_a^L(\tau ) = \langle a(t_0+\tau )a(t_0)\rangle / \langle a^2 \rangle$
(Sawford Reference Sawford1991; Huck et al. Reference Huck, Machicoane and Volk2019):
\begin{equation} R^L_a(\tau ) = \frac {T_1 \mathrm{e}^{-\tau /T_2}-T_2\mathrm{e}^{-\tau /T_1}}{T_1-T_2}. \end{equation}
For fluid tracers,
$R_a^L(\tau )$
decays over time scales comparable to the Kolmogorov time
$\tau _\eta$
(Yeung & Pope Reference Yeung and Pope1989; Voth et al. Reference Voth, La, Arthur, Alice, Alexander and Bodenschatz2002).
1.2. Inertial particles in turbulence
Being purely based on kinematics, Taylor’s theory applies to fluid tracers as well as inertial particles, i.e. objects too dense and/or too large to follow the fluid flow (Balachandar & Eaton Reference Balachandar and Eaton2010; Brandt & Coletti Reference Brandt and Coletti2022). The behaviour of small inertial particles (i.e. with diameter
$d_p$
smaller than the Kolmogorov scale
$\eta$
) is usually rationalised in terms of the Stokes number
$St=\tau _p/\tau _\eta$
, where
$\tau _p$
is the particle response time. As long as the particle Reynolds number
${\textit{Re}}_p = \| \boldsymbol{u}_s \| d_p / \nu$
is small, the condition
$d_p \lt \eta$
warrants the approximate validity of Stokes’ drag formulation (Clift, Grace & Weber Reference Clift, Grace and Weber2005). Here,
$\nu$
is the kinematic viscosity of the fluid and
$\boldsymbol{u}_s = \boldsymbol{u}_{\textit{fp}} - \boldsymbol{u}_{\!p}$
is the slip between the fluid velocity at the particle location
$\boldsymbol{u}_{\textit{fp}}$
and the particle velocity
$\boldsymbol{u}_{\!p}$
. Depending on
$St$
, the particle motion has been shown to depart from the fluid flow due to two mechanisms: preferential sampling of regions of high-strain and low-vorticity, prevalent for
$St\lt 1$
(Maxey Reference Maxey1987; Squires & Eaton Reference Squires and Eaton1991); and inertial filtering of small-scale/high-frequency turbulent fluctuations, dominant for
$St\gt 1$
(Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006).
The inertia of small particles stems from their density
$\rho _p$
. When this is much larger than the fluid density
$\rho$
, drag and gravity (of acceleration field
$\boldsymbol{g}$
) dominate over unsteady forces such as added mass, stress gradient and history force (Balachandar Reference Balachandar2009; Ling, Parmar & Balachandar Reference Ling, Parmar and Balachandar2013). The particle is then often conceptualised as a point mass, and its equation of motion reads:
\begin{equation} \frac {\mathrm{d} \boldsymbol{u}_{\!p}}{\mathrm{d}t} = \frac {\boldsymbol{u}_{\textit{fp}}-\boldsymbol{u}_{\!p}}{\tau _p}+\boldsymbol{g}. \end{equation}
Neglecting gravity and assuming an exponential decay of
$R_u^L(\tau )$
, Tchen (Reference Tchen1947) showed how (1.5) implies that
${u}_{\!p}$
is obtained from low-pass filtering
$u_{\textit{fp}}$
with a cutoff frequency of
$\tau _p^{-1}$
. By assuming that
$u_{\textit{fp}}$
is statistically indistinguishable from the unconditional fluid velocity
$u$
, they derived
\begin{equation} \big\langle {u}_{\!p}^2 \big\rangle = \frac {1}{1+\tau _p/T_L} \langle u^2 \rangle . \end{equation}
Csanady (Reference Csanady1963) and Hinze (Reference Hinze1975) used similar but less restrictive assumptions to build a framework later extended by Wang & Stock (Reference Wang and Stock1993), Jung et al. (Reference Jung, Yeo and Lee2008) and Berk & Coletti (Reference Berk and Coletti2021), among others. In particular, assuming the form (1.3) for the Lagrangian autocorrelation coefficient of
$u_{\textit{fp}}$
, closed expressions for the particle fluctuating energy
$\langle {u}_{\!p}^2 \rangle$
, acceleration variance
$\langle a_p^2 \rangle$
and velocity correlation time
$T_p = \int _0^\infty {R^L_{{u}_{\!p}}}(\tau )\, \mathrm{d}\tau$
(where
${R^L_{{u}_{\!p}}}(\tau )$
is the Lagrangian correlation coefficient of
${u}_{\!p}$
) can be derived (see Jung et al. Reference Jung, Yeo and Lee2008; Berk & Coletti Reference Berk and Coletti2021):
\begin{align}& \big\langle {u}_{\!p}^2 \big\rangle = \big\langle u_{\textit{fp}}^2 \big\rangle \bigg [ 1 - \frac {St^2}{(T_1/\tau _\eta +St)(T_2/\tau _\eta +St)} \bigg ], \end{align}
\begin{align}&\qquad \big\langle a_p^2 \big\rangle = \frac {\left\langle u_{\textit{fp}}^2 \right\rangle / \tau _\eta ^2}{(T_1/\tau _\eta +St)(T_2/\tau _\eta +St)}, \end{align}
\begin{align}&\quad T_p = \frac {(T_1/\tau _\eta +St)(T_2/\tau _\eta +St)(T_1+T_2)}{(T_1/\tau _\eta +St)(T_2/\tau _\eta +St)-St^2}, \end{align}
where now
$T_1$
and
$T_2$
are the time scales in the expression of the Lagrangian autocorrelation coefficient of
$u_{\textit{fp}}$
(see Sawford (Reference Sawford1991) and Huck et al. (Reference Huck, Machicoane and Volk2019)). In the following, we will refer to this framework as temporal filtering. It predicts that the variance of the particle fluctuating energy and acceleration,
$\langle {u}_{\!p}^2 \rangle$
and
$\langle a_p^2 \rangle$
, respectively, decrease with increasing
$St$
; while the velocity correlation time
$T_p$
grows and the diffusivity
$K_p = \langle {u}_{\!p}^2 \rangle T_p$
remains essentially constant. Since extreme turbulent fluctuations are short-lived, this view also predicts that acceleration intermittency, quantified by the flatness
$\langle a_p^4 \rangle / \langle a_p^2 \rangle ^2$
, decreases with
$St$
. By modelling
$u_{\textit{fp}}$
, this approach can incorporate the effect of preferential sampling as well as trajectory-crossing, i.e. the particle drift through the turbulence due to body forces such as gravity (Csanady Reference Csanady1963; Wang & Stock Reference Wang and Stock1993); see also Mathai, Lohse & Sun (Reference Mathai, Lohse and Sun2020) for the analogous effect on light particles. While simplified, this framework has been shown to capture important trends of the particle behaviour when
$d_p\lt \eta$
and
$\rho _p/\rho \gg 1$
(see, e.g. Bec et al. (Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006), Jung et al. (Reference Jung, Yeo and Lee2008), Ireland, Bragg & Collins (Reference Ireland, Bragg and Collins2016), Mehrabadi et al. (Reference Mehrabadi, Horwitz, Subramaniam and Mani2018) and Berk & Coletti (Reference Berk and Coletti2021, Reference Berk and Coletti2024)).
Developing a similar framework for particles of finite-size and with a density comparable to (or smaller than) that of the fluid would be desirable, but it poses numerous difficulties. First, for
$d_p \gt \eta$
, Stokes drag cannot be assumed and a closed form for
$\tau _p$
is not available. In particular, the power-law dependence
$\tau _p \sim d_p^2$
(or
$St \sim (d_p /\eta )^2$
) leads to large overestimations, especially when the particle Reynolds number is not small. Moreover, for moderate density ratios, unsteady forces may be comparable to drag, complicating the question of how the particle responds to changes in the surrounding fluid flow. The common proxy is the correlation time scale of
$a_p$
,
$\tau _p \sim \int _0^{T_0} R_{a_p}^L(\tau ) \,\mathrm{d}\tau$
, where
$T_0$
is the zero-crossing time of the Lagrangian autocorrelation coefficient of the particle acceleration
${R^L_{a_p}}(\tau )$
(see, e.g. Calzavarini et al. (Reference Calzavarini, Volk, Bourgoin, Lévêque, Pinton and Toschi2009) and Volk et al. (Reference Volk, Calzavarini, Leveque and Pinton2011)). Additionally, independent of density, for large (finite-size) particles,
$u_{\textit{fp}}$
and
$u_s$
are loosely defined, requiring ad hoc heuristics based on the flow surrounding the particle (Kidanemariam et al. Reference Kidanemariam, Chan-Braun, Doychev and Uhlmann2013; Uhlmann & Doychev Reference Uhlmann and Doychev2014). Moreover, when
$\rho _p/\rho \leqslant 1$
, unsteady forces are significant. The point-particle equation of motion for neutrally buoyant particles then predicts that
$\langle a_p^2 \rangle$
would increase or remain approximately constant with
$d_p/\eta$
(Calzavarini et al. Reference Calzavarini, Volk, Bourgoin, Lévêque, Pinton and Toschi2009; Homann & Bec Reference Homann and Bec2010). This is at odds with experimental measurements and particle-resolved simulations of neutrally buoyant particles, which show how increasing
$d_p/\eta$
leads to decreasing
$\langle a_p^2 \rangle$
, as well as increasing
$T_p$
and
$\tau _p$
(Voth et al. Reference Voth, La, Arthur, Alice, Alexander and Bodenschatz2002; Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007, Reference Qureshi, Arrieta, Baudet, Cartellier, Gagne and Bourgoin2008; Volk et al. Reference Volk, Calzavarini, Verhille, Lohse, Mordant, Pinton and Toschi2008; Brown, Warhaft & Voth Reference Brown, Warhaft and Voth2009; Homann & Bec Reference Homann and Bec2010; Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017; Fan et al. Reference Fan, Wang, Jiang, Sun and Calzavarini2024).
To address the effect of finite-size, Calzavarini et al. (Reference Calzavarini, Volk, Bourgoin, Lévêque, Pinton and Toschi2009) proposed to include Faxén corrections, thus incorporating the non-uniformity of the flow at the particle scale by filtering it over the particle’s surface (to estimate drag) and volume (to estimate added mass and stress gradient forces). This produces qualitatively correct trends for neutrally buoyant particles, though it underpredicts the effects of
$d_p/\eta$
(Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017). The interpretation is that the particle motion is driven by turbulent fluctuations occurring at their scale. In other words, finite-size particles are assumed to apply a spatial filtering (or coarse-graining) of the local turbulence (Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007; Calzavarini et al. Reference Calzavarini, Volk, Bourgoin, Lévêque, Pinton and Toschi2009; Jiang et al. Reference Jiang, Wang, Liu, Sun and Calzavarini2022; Fan et al. Reference Fan, Wang, Jiang, Sun and Calzavarini2024). For particle sizes in the inertial subrange,
$L\gg d_p \gg \eta$
(
$L$
being the integral length scale), velocity fluctuations at the particle scales are thus expected to scale according to the phenomenology put forward by Kolmogorov (Reference Kolmogorov1941),
$\langle [u(x_0+d_p)-u(x_0)]^2 \rangle \sim (\varepsilon d_p)^{2/3}$
(where
$\varepsilon$
is the turbulent dissipation rate), yielding
$(\langle u^2 \rangle - \langle {u}_{\!p}^2 \rangle )/\langle u^2 \rangle \sim (d_p/\eta )^{2/3}$
,
$\langle a_p^2 \rangle / \langle a^2 \rangle \sim (d_p/\eta )^{-2/3}$
and
$\tau _p/\tau _\eta \sim (d_p/\eta )^{2/3}$
. While those are compatible with the observations, the limited available data cannot exclude other scaling behaviours (Voth et al. Reference Voth, La, Arthur, Alice, Alexander and Bodenschatz2002; Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007; Homann & Bec Reference Homann and Bec2010; Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017; Fan et al. Reference Fan, Wang, Jiang, Sun and Calzavarini2024). Moreover, in keeping with the distribution of velocity increments at increasing separations, the spatial filtering framework implies that the particle acceleration (regardless of particle density) becomes less intermittent for larger
$d_p$
(Xu et al. Reference Xu, Ouellette, Vincenzi and Bodenschatz2007; Brown et al. Reference Brown, Warhaft and Voth2009). This is in contrast with experiments and particle-resolved simulations of neutrally buoyant particles, which report high flatness of
$a_p$
even for
$d_p/\eta \gg 1$
(Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007, Reference Qureshi, Arrieta, Baudet, Cartellier, Gagne and Bourgoin2008; Xu & Bodenschatz Reference Xu and Bodenschatz2008; Brown et al. Reference Brown, Warhaft and Voth2009; Homann & Bec Reference Homann and Bec2010; Bellani & Variano Reference Bellani and Variano2012). Even when a reduction of intermittency with particle size was reported, the acceleration flatness was still found to be far higher than the Gaussian limit (Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011).
The above-mentioned works on finite-size neutrally buoyant particles were concerned with three-dimensional (3-D) turbulence. Ouellette, O’malley & Gollub (Reference Ouellette, O’malley and Gollub2008) investigated the motion of spheres in a two-dimensional (2-D) chaotic flow, and found the larger ones had a somewhat larger diffusivity. However, Xia et al. (Reference Xia, Francois, Punzmann and Shats2019) found that the diffusivity of discs floating on wave-driven 2-D turbulence decreased when their diameter increased. Therefore, the behaviour of (quasi-) neutrally buoyant finite-size particles in free-surface turbulence (which, as we describe later, shares similarities with both 2-D and 3-D turbulence) remains an open question.
1.3. Free-surface turbulence
Most previous studies concerned with free-surface turbulence have focused on the effect of the boundary conditions on the sub-surface flow; see seminal studies by Hunt & Graham (Reference Hunt and Graham1978), Perot & Moin (Reference Perot and Moin1995), Shen et al. (Reference Shen, Zhang, Yue and Triantafyllou1999) and Magnaudet (Reference Magnaudet2003), among several others recently reviewed by Ruth & Coletti (Reference Ruth and Coletti2024). In particular, the kinematic boundary condition
$u_z = 0$
at
$z = 0$
influences a so-called ‘blockage layer’ of thickness comparable to the integral scale
$L$
, increasing the surface-parallel velocity fluctuations at the expense of the vertical fluctuations. Here and in the following,
$x$
and
$y$
indicate the surface-parallel directions (
$x$
being streamwise in the presence of a mean flow) and
$z$
, positive upward, is the surface-normal (vertical) direction;
$u_x$
,
$u_y$
and
$u_z$
are the corresponding velocity components. In the limit of large turbulent Reynolds number
${\textit{Re}}$
, the behaviour of the Reynolds stresses in the blockage layer is well predicted by the theory of Hunt & Graham (Reference Hunt and Graham1978) based on rapid-distortion theory (Magnaudet Reference Magnaudet2003; Ruth & Coletti Reference Ruth and Coletti2024).
The dynamic boundary condition
$\partial u_x/\partial z = \partial u_y/\partial z=0$
, however, imposes that vortex lines reorient to be surface-normal within a surface layer of thickness
$L {\textit{Re}}^{-1/2}$
. This has significant consequences for the dynamics immediately below the free-surface (Shen et al. Reference Shen, Zhang, Yue and Triantafyllou1999; Guo & Shen Reference Guo and Shen2010; Aarnes et al. Reference Aarnes, Babiker, Xuan, Shen and Ellingsen2025) but also for the motion along it. Li et al. (Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025) studied the behaviour of millimetric tracers floating along the quasi-flat free-surface above turbulent water past a square-mesh grid. While the statistics of velocity fluctuations, accelerations and dissipation were similar to those in the bulk, and followed the classic phenomenology of Kolmogorov (Reference Kolmogorov1941), the surface-normal vorticity signalled the presence of long-lived vortices (Kumar, Gupta & Banerjee Reference Kumar, Gupta and Banerjee1998; Lovecchio, Zonta & Soldati Reference Lovecchio, Zonta and Soldati2015). This was later confirmed in the detailed measurements of Qi et al. (Reference Qi, Li and Coletti2025a
,
Reference Qi, Xu and Colettib
), who tracked microscopic surface tracers in a homogeneous zero-mean-flow turbulent water tank, highlighting the effect of the dynamic boundary condition on strain-rate and vorticity.
In the absence of significant surfactant effects, the surface velocity field displays a compressibility
$\langle (\partial u_z/\partial z)^2 \rangle$
comparable to the mean square velocity gradients in the bulk, which in turn influences the dispersion of small floating particles (Boffetta et al. Reference Boffetta, Davoudi, Eckhardt and Schumacher2004; Cressman et al. Reference Cressman, Davoudi, Goldburg and Schumacher2004; Lovecchio, Marchioli & Soldati Reference Lovecchio, Marchioli and Soldati2013; Li et al. Reference Li, Wang, Qi and Coletti2024, Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025).
1.4. Focus of the present study
Despite its relevance, the behaviour of finite-size particles floating in free-surface turbulence has rarely been addressed in fundamental fluid mechanics investigations. Valero et al. (Reference Valero, Belay, Moreno-Rodenas, Kramer and Franca2022) studied the behaviour of realistic buoyant litter such as plastic cups, flexible films and face masks in a laboratory flume, highlighting the importance of surface tension and the effect of their size on the transport. In a recent field study, we have reported on the transport of centimetre-sized discs and rods floating in an outdoor meandering stream (Sanness Salmon et al. Reference Sanness Salmon, Baker, Kozarek and Coletti2023). Compared with millimetre-sized floating tracers, the larger particles were observed to have reduced accelerations and more time-correlated motions, which impacted their diffusivity.
In the present study, we use the experimental facilities employed by Li et al. (Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025) to study the transport of tracers on the quasi-flat free-surface above homogeneous isotropic grid turbulence, and focus on the behaviour of marginally buoyant particles in a wide range of sizes,
$d_p/\eta \in [5, 110]$
. We characterise in detail their single-point and two-point/two-time statistics, and discuss to which degree their behaviour is consistent with specific assumptions. We aim to answer the following questions. How does the size of the floating particles affect their motion, in particular, the velocity fluctuations, accelerations and diffusivity? To which extent can the floating particles be modelled as point-particles of a given response time? And to which extent can they be modelled as spatial filters of the underlying turbulent flow?
The rest of the paper is organised as follows: the experimental approach and the parameter space are described in § 2; § 3 presents the results in terms of flow properties below and along the free-surface (§ 3.1) and the particle behaviour (§ 3.2). We then explore to which degree the latter is consistent with the frameworks of temporal filtering (§ 3.3) and spatial filtering (§ 3.4). Section 4 discusses the results, draws conclusions and gives an outlook.
2. Methods
2.1. Experimental apparatus
To cover a wide range of parameters, experiments are performed in two recirculating open-channel flumes, one located at ETH Zürich and the other at the Swiss Federal Laboratories for Materials Science and Technology (Empa). The two facilities are different in scale but identical in architecture, and the same measurement approaches are used in both, as described in detail by Li et al. (Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025). A centrifugal pump drives water through an upstream plenum featuring flow conditioning components (perforated stainless-steel plates, polycarbonate honeycombs and stainless-steel screens) before entering a contraction with a 6–1 area ratio. Turbulence is generated by a stainless-steel grid spanning the entire water cross-section of depth
$H$
and width
$W$
. The grid features squared bars of width
$d$
and mesh size
$M$
, resulting in a grid solidity
$\sigma =({d}/{M})(2-({d}/{M}))=0.31$
. The main hydrodynamic parameters of the two facilities are summarised in table 1. The Reynolds number
${\textit{Re}}_M = U_sM/\nu$
and the Froude number
$\textit{Fr} = U_s/\sqrt {gH}$
are based on the mean surface velocity
$U_s$
. At the start of each experiment, surface residue is removed with a fine net and a standard surface tension
$\gamma = 72\,\mathrm{mN\,m^{-1}}$
is measured via a Du Noüy ring. The values of
$\textit{Fr}$
and of the Weber number
$\textit{We} = \rho \langle u^2 \rangle L/\gamma$
are consistent with the minimal deformation of the water surface, on which only sub-millimetre wave amplitudes are observed. In the Empa facility, two configurations of the flow conditioning components are used, resulting in two different levels of turbulence intensity. Therefore, three cases with different Taylor-scale Reynolds number
${\textit{Re}}_\lambda$
(whose evaluation is detailed in § 3.1) are considered.
Hydrodynamic parameters characterising the free-surface flow in the two used facilities: water depth
$H$
, channel width
$W$
, mean surface velocity
$U_s$
, square mesh size
$M$
, grid Reynolds number
${\textit{Re}}_M$
, Froude number
$\textit{Fr}$
, Weber number
$\textit{We}$
and Taylor-scale Reynolds number of the turbulence
${\textit{Re}}_\lambda$
.
Two types of floating particles are considered (table 2): white polypropylene spheres (RGPBalls Srl,
$\rho _p = 0.87\,\mathrm{g\,cm^{-3}}$
) and polypropylene discs (
$\rho _p = 0.92\,\mathrm{g\,cm^{-3}}$
), photographed in figure 1. Their diameters vary in the range
$d_p \in [1.6, 30]$
mm, with the smallest spheres used as tracers to characterise the free-surface turbulent flow (see Li et al. Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025 for the verification that those particles faithfully follow the flow). In terms of Kolmogorov scales, these tracers correspond to
${\sim} 8\eta$
and
${\sim} 5\eta$
, for the ETH and Empa facilities, respectively. We remind that for 3-D turbulence, neutrally buoyant spherical particles up to
$d_p \sim 5\eta$
behave as flow tracers (Homann & Bec Reference Homann and Bec2010; Fiabane et al. Reference Fiabane, Zimmermann, Volk, Pinton and Bourgoin2012; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017; Berk & Coletti Reference Berk and Coletti2024). For free-surface flows, it has been shown that particles up to
$d_p \sim 10\eta$
capture most of the turbulent kinetic energy (Nikora et al. Reference Nikora, Nokes, Veale, Davidson and Jirka2007; Sanness Salmon et al. Reference Sanness Salmon, Baker, Kozarek and Coletti2023). The discs are produced in-house by laser-cutting 1 mm thick sheets (Vibraplast AG) using a VLS3.50 Desktop Laser (Universal Laser Systems Inc.). The large diameter-to-thickness ratio guarantees that the floating discs remain parallel to the water surface. The particles are mostly submerged, consistent with the weight/buoyancy force balance, which (neglecting surface tension effects) prescribes their fractional submerged volume to be approximately equal to their relative density
$\rho _p/\rho$
.
Physical properties and experimental summary of the floating particles including shape, relative density
$\rho _p/\rho$
and diameter
$d_p$
in dimensional and dimensionless units.
A close-up photograph of the finite-size floating particles; spheres (bottom row) and discs (top and middle row).
2.2. Measurement approach
The sub-surface turbulence properties in the bulk and across the blockage layer are measured by particle image velocimetry (PIV) of
${10}\,{\mathrm{\mu m}}$
hollow glass spheres (LaVision GmbH,
$\rho _p = 1.10\,\mathrm{g\,cm^{-3}}$
). This is performed along the mid-span vertical plane and several surface-parallel planes as close as
${\sim} 1$
mm from the free-surface, as described in detail by Li et al. (Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025).
The properties of the free-surface turbulence and the behaviour of the finite-size particles are obtained by particle tracking velocimetry (PTV) of the floating particles. These are released at the inlet of the test section with a handheld spreader (Gardena GmbH) and collected by a nylon net fixed at the outlet of the test section. Number concentrations that may cause inter-particle interactions are avoided. The particles are imaged by a 12MP CMOS camera (Baumer Ltd, VQXT-120C.HS) mounting Zeiss Milvius lenses. This is suspended above the channel at a downstream location of
$x = 30M$
from the grid, pointing downward at the free-surface flow. The position is chosen such that the field of view (FOV) is within a region where equilibrium conditions have been reached and classic scaling relations for homogeneous isotropic turbulence apply (Mohamed & Larue Reference Mohamed and Larue1990; Lavoie, Djenidi & Antonia Reference Lavoie, Djenidi and Antonia2007; Hearst & Lavoie Reference Hearst and Lavoie2014). Two continuous LEDs (GS Vitec GmbH, MultiLED) illuminate the FOV from both ends of the channel. Diffusers are employed to evenly distribute the light along the water surface, and black background panels are positioned on the transparent walls of the test section to improve image contrast. The imaging parameters for the two facilities are summarised in table 3. The camera calibration is realised by imaging a checkerboard pattern kept at the water surface level, which allows correcting for slight barrel distortion (Zhang Reference Zhang2000). The boundaries of the FOV, which is centred at mid-span and covers approximately half of the channel width, are sufficiently far from the channel sidewalls to ensure that the imaged flow is not influenced by the lateral boundary layers.
Main imaging parameters for the two experimental facilities: frame rate, camera resolution, focal length of the lens, spatial resolution, size of the FOV and non-dimensional distance from the grid spanned by the FOV.
Particles are identified via image segmentation above an intensity threshold, whose exact value is not consequential thanks to the high contrast. The centroids of contiguous groups of pixels exceeding the threshold are identified via a circle-finder algorithm based on the circular Hough transform (alternative algorithms returning the same results within sub-pixel accuracy). Rare occurrences of adjacent particles are discarded in post-processing, imposing a minimum inter-particle gap of twice the capillary length
$\sqrt {\gamma /\rho g} =2.7$
mm or
$d_p$
(whichever is larger). This guarantees that the tracked particles are not significantly influenced by their neighbours.
Particle trajectories are reconstructed using an in-house code implementing a nearest-neighbour PTV algorithm (Baker & Coletti Reference Baker and Coletti2019, Reference Baker and Coletti2021, Reference Baker and Coletti2022; Sanness Salmon et al. Reference Sanness Salmon, Baker, Kozarek and Coletti2023). For the tracer particles, an advective predictor is used which searches in the radius around the centroid shifted downstream by
$\Delta x = \langle U \rangle \Delta t$
(
$\Delta t$
being the inter-frame temporal separation). For particles 5 mm and larger, the advective predictor is unnecessary as the inter-frame displacements are smaller than the particle radius. The particle positions
$\boldsymbol{x}_{\!p}$
, velocities
$\boldsymbol{u}_{\!p}$
and accelerations
$\boldsymbol{a}_{\!p}$
are obtained by convolving the trajectories with a Gaussian kernel and its derivatives, removing most of the high-frequency noise (Voth et al. Reference Voth, La, Arthur, Alice, Alexander and Bodenschatz2002; Mordant et al. Reference Mordant, Crawford and Bodenschatz2004a
). Here and in the following, the bold typeface indicates vectors associated with the two-dimensional motion along the free-surface. Because tracking floating particles along the quasi-flat free-surface is a robust process, most of the reconstructed trajectories have comparable lengths. Still, to avoid possible biases due to varying trajectory length (Mordant et al. Reference Mordant, Crawford and Bodenschatz2004a
; Guala et al. Reference Guala, Liberzon, Tsinober and Kinzelbach2007), we calculate Lagrangian statistics from trajectories of equal length (180 and 140 frames in the ETH and Empa facilities, respectively), trimming longer trajectories.
As tracking errors are negligible, the uncertainties are mostly associated with the finite number of samples. To yield a number of trajectories sufficient for statistical convergence, between twenty and one hundred measurement runs are conducted for each particle-flow condition combination, with 2700–2900 images acquired in each run. For
$d_p \leqslant 7$
mm, more than 10 000 trajectories are obtained for each case. The larger particles have a lower yield due to the constraint of avoiding inter-particle interaction, but the statistics are still based on at least 3700 trajectories. Although statistical convergence of the observables within a few percent is achieved in each experimental run, larger variability is observed between different runs. Therefore, when relevant for the interpretation of the results, the statistical uncertainty is represented with error bars given by the run-to-run standard deviation.
3. Results
3.1. Properties of the turbulence below and along the free-surface
For both facilities, the full characterisation of the sub-surface and free-surface flow is reported by Li et al. (Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025). Here, we provide an account of the main properties, with plots of selected cases to illustrate the flow behaviour.
With the uppermost grid bar located at approximately
$z=-0.86M$
, turbulence is forced over the entire flow volume. Indeed, the behaviour of the sub-surface turbulence measured by PIV is consistent with the predictions of Hunt & Graham (Reference Hunt and Graham1978) for the evolution of homogeneous turbulence below a flat free-surface: the root mean square (r.m.s.) of the vertical velocity fluctuations
$\langle u_z^2 \rangle ^{1/2}$
drops to vanishingly small levels approaching the free-surface, while the horizontal one
$\langle u_x^2 \rangle ^{1/2}$
increases. The dissipation rate
$\varepsilon$
grows in the blockage layer and decreases in the surface layer to approximately recover its bulk value. The compressibility coefficient
$\mathcal{C} = \langle (\partial u_i/\partial x_i)^2 \rangle / \langle (\partial u_i/\partial x_j)^2 \rangle$
(with indices restricted to the surface-parallel components) grows to approach the free-surface condition
$\mathcal{C} = 0.5$
, implying that the in-plane velocity gradients are uncorrelated (Cressman et al. Reference Cressman, Davoudi, Goldburg and Schumacher2004; Boffetta et al. Reference Boffetta, Davoudi, Eckhardt and Schumacher2004). These trends are displayed in figure 2(a) for the representative case
${\textit{Re}}_\lambda = 29$
.
(a) Vertical profiles of the sub-surface properties measured by PIV at
${\textit{Re}}_\lambda = 29$
. (b) Corresponding probability distribution functions (p.d.f.s) of the free-surface turbulence properties measured by PTV. The dashed line indicates a normal distribution.
The lateral velocity fluctuations
$u_y$
and accelerations
$a_y$
of the tracer particles measured by PTV yield the probability distribution functions (p.d.f.s) plotted in figure 2(b), normalised by their respective r.m.s. values. The p.d.f. of surface-normal vorticity
$\omega _z$
is also shown, obtained from Lagrangian tracking of 2 mm long floating rods (see Li et al. Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025). The rods’ r.m.s. rotation rate was found to be an appropriate proxy for the near-surface
$\omega _z$
measured via PIV. While
$u_i$
are normally distributed,
$a_i$
and
$\omega _z$
display levels of intermittency of the small-scale flow features comparable to those in 3-D turbulence (Qi et al. Reference Qi, Li and Coletti2025a
).
The free-surface velocity data are used to obtain Eulerian fields of the mean and r.m.s. fluctuations of the free-surface velocity. The measurements are spatially binned into 5 mm
$\times$
5 mm windows, the size of which is chosen to give at least 100 instantaneous vectors for ensemble-averaging. The fluctuations are obtained by subtracting the local mean velocity, which varies by only a few percent within the FOV. Both r.m.s. components are similar, with the anisotropy ratio
$\langle u_x^2 \rangle / \langle u_y^2 \rangle$
in the range of 0.96–1.18. The streamwise decay of the turbulent kinetic energy of the free-surface flow is well described by a power-law decay:
\begin{equation} \frac {\langle q^2 \rangle }{U_s^2} = A \bigg ( \frac {x}{M}-\frac {x_0}{M} \bigg )^{-m}, \end{equation}
where
$\langle q^2 \rangle = \langle {\boldsymbol{u} \boldsymbol{\cdot } \boldsymbol{u}} \rangle = \langle u_x^2 \rangle + \langle u_y^2 \rangle$
,
$x_0$
is the virtual origin of the grid and the parameter
$A$
is determined via a least-square fit. The decay exponent
$m$
is found to be close to unity, consistent with results in 3-D turbulence (Mohamed & Larue Reference Mohamed and Larue1990; Lavoie et al. Reference Lavoie, Djenidi and Antonia2007; Hearst & Lavoie Reference Hearst and Lavoie2014; Sinhuber, Bodenschatz & Bewley Reference Sinhuber, Bodenschatz and Bewley2015). The mean dissipation rate along the free-surface is evaluated from the spatial decay of
$\langle q^2 \rangle$
, from which we evaluate free-surface values of the Kolmogorov scales, Taylor microscale
$\lambda = \sqrt {15} u_{\textit{rms}} \tau _\eta$
and
${\textit{Re}}_\lambda = u_{\textit{rms}}\lambda /\nu$
, where
$u_{\textit{rms}} = \sqrt {\langle q^2 \rangle /2}$
. To determine the integral length scales
$L$
of the free-surface turbulence, we evaluate the Eulerian velocity correlation coefficient:
\begin{equation} R^E_u(r) = \frac {\langle \boldsymbol{u}(\boldsymbol{r}_0+\boldsymbol{r}) \boldsymbol{\cdot } \boldsymbol{u}(\boldsymbol{r}_0) \rangle }{\langle q^2 \rangle } ,\end{equation}
where the ensemble-averaging is carried out over tracers separated by a distance
$r$
from the reference location
$\boldsymbol{r}_0$
, discretising the separation in bins containing
$\mathcal{O}(10^4)$
data points each. An exponential fit to the form
$R^E_u(r) = \mathrm{e}^{-r/L}$
yields values of the integral scale consistent with the classic relation
$L \sim u_{\textit{rms}}^3/\varepsilon$
(Tennekes & Lumley Reference Tennekes and Lumley1972). Similarly, the integral time scale
$T_L$
is evaluated by fitting an exponential decay to the Lagrangian velocity autocorrelation coefficient,
$R^L_u(\tau )=\mathrm{e}^{-\tau /T_L}$
. The free-surface turbulence properties for the three considered flow conditions are summarised in table 4.
Quantities characterising the free-surface turbulence for the three considered
${\textit{Re}}_\lambda$
: r.m.s. velocity fluctuation
$u_{\textit{rms}}$
, mean dissipation rate of turbulent kinetic energy
$\varepsilon$
, integral length scale
$L$
, integral time scale
$T_L$
, Taylor micro-scale
$\lambda$
, Kolmogorov length scale
$\eta$
and Kolmogorov time scale
$\tau _\eta$
.
3.2. Behaviour of finite-size floating particles
To describe the kinematics of the finite-size particle motion, only selected components of the free-surface velocity and acceleration are illustrated; the behaviour of both components is quantitatively similar, as expected from the quasi-isotropic nature of the free-surface turbulence.
The p.d.f.s of the lateral velocity fluctuations are plotted in figure 3(a), normalised by the respective r.m.s. values, showing a Gaussian distribution for all sizes. Figure 3(b) shows how the particle fluctuating energy
$\langle {u}_{\!p}^2 \rangle = \langle {\boldsymbol{u}_{\!p} \boldsymbol{\cdot } \boldsymbol{u}_{\!p}} \rangle$
remains equal to the one of tracers up to approximately
$d_p/L=0.1$
and drops significantly for larger sizes: increasingly large floating particles are less responsive to the underlying fluid fluctuations. In terms of the Kolmogorov scale, this corresponds to
$d_p/\eta \in [8,20]$
for the different cases. In general, the reduction of fluctuating energy with increasing particle size agrees with findings in 3-D turbulence (Homann & Bec Reference Homann and Bec2010; Calzavarini et al. Reference Calzavarini, Volk, Lévêque, Pinton and Toschi2012; Chouippe & Uhlmann Reference Chouippe and Uhlmann2015; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017). Our results are comparable, for example, with the experiments of Machicoane et al. (Reference Machicoane, Zimmermann, Fiabane, Bourgoin, Pinton and Volk2014) who measured an energy reduction of roughly 40 % for very large (
$d_p/L \sim 0.5$
) neutrally buoyant spherical particles in a von Kármán flow. The fundamental differences between 3-D and free-surface turbulence, however, limit the value of quantitative comparisons. Here and in the following figures, no systematic differences are seen in the behaviour of floating spheres and discs of similar diameter (
${\textit{Re}}_\lambda = 43$
). This suggests that, for the present level of submergence, the motion of floating particles depends on the near-surface flow and how this is modulated by their spatial extension in the surface-parallel direction. This is consistent with the submerged depth of the particles (estimated from a simple force balance and visually verified), which is at most comparable with the surface layer (Hunt & Graham Reference Hunt and Graham1978). Larger particles whose centre of mass resides in the blockage layer may display different dynamics.
(a) P.d.f.s of the lateral velocity fluctuations of the floating particles for
${\textit{Re}}_\lambda =43$
. The dashed line indicates a normal distribution. (b) Particle fluctuating energy versus dimensionless particle size. Error bars represent the run-to-run standard deviation.
The p.d.f.s of the streamwise particle acceleration are reported in figure 4(a). Unlike for the velocity fluctuations, the particle size has a clear influence on the shape of the distributions: the intermittency decreases significantly with increasing size, and the acceleration of the larger particles essentially follows a Gaussian distribution. The total acceleration variance
$\langle a_p^2 \rangle = \langle {\boldsymbol{a}_{\!p} \boldsymbol{\cdot } \boldsymbol{a}_{\!p}} \rangle$
remains approximately constant for
$d_p/L \lt 0.1$
and drops for larger sizes (figure 4
b), and it does so more steeply than the fluctuating kinetic energy. This is likely related to the fact that the fluid accelerations are mostly associated with the fine scales of the turbulence (Toschi & Bodenschatz Reference Toschi and Bodenschatz2009), as discussed in the following sections.
(a) P.d.f.s of streamwise acceleration of the floating particles for
${\textit{Re}}_\lambda =43$
. The dashed line indicates a normal distribution. (b) Particle acceleration variance versus dimensionless particle size. Error bars represent the run-to-run standard deviation.
The degree to which the motion of the finite-size particles is time-correlated is described by the particle velocity autocorrelation coefficient, illustrated in figure 5(a) for
${\textit{Re}}_\lambda =29$
. The decay rate of
${R^L_{{u}_{\!p}}}(\tau )$
decreases with particle size, which is quantified by the increase of the correlation time scale
$T_p$
. This is estimated by fitting an exponential of the form
$\mathrm{e}^{-t/T_p}$
to the measurement of
${R^L_{{u}_{\!p}}}(\tau )$
, reported in figure 5(b) normalised by the fluid integral time scale
$T_L$
. We note that this procedure is associated with significant uncertainty: the accurate measurement of the integral time scales requires integrating over a duration significantly longer than the time scales themselves. This is, however, seldom possible in laboratory experiments due to the limited length of the trajectories (even in the present case, in which most of the trajectories stretch over the entire FOV). Therefore, we rely on the assumption that the autocorrelations decay exponentially (as in Baker & Coletti Reference Baker and Coletti2021 and Sanness Salmon et al. Reference Sanness Salmon, Baker, Kozarek and Coletti2023). The uncertainty associated with possible departures of
${R^L_{{u}_{\!p}}}(\tau )$
from the measured exponential decay at long times also affects the diffusivity (discussed later), but is not expected to overshadow the reported trends. Similar to the velocity and acceleration, the correlation time scale of particles smaller than
$d_p/L = 0.1$
is indistinguishable from the one of the fluid. For larger diameters, an increasing trend of
$T_p/T_L$
is apparent: the motion of larger particles is characterised by more time-correlated velocity fluctuations.
(a) Measured Lagrangian particle velocity autocorrelation coefficients for
${\textit{Re}}_\lambda =29$
. (b) Particle velocity correlation time scale versus dimensionless particle size. Error bars represent the run-to-run standard deviation.
3.3. Comparison with the temporal filtering framework
(a) Particle Stokes number evaluated by integrating their measured acceleration autocorrelation coefficient (filled symbols) and their equivalent analytical form (1.4) inspired by Sawford (Reference Sawford1991) (open symbols). The continuous line is the linear relation (3.3) from Uhlmann & Chouippe (Reference Uhlmann and Chouippe2017) and the dashed line is the
$2/3$
power-law scaling based on Kolmogorov (Reference Kolmogorov1941). (b) Lagrangian particle velocity autocorrelation coefficients for
${\textit{Re}}_\lambda =29$
measured by PTV (filled symbols) and their respective analytical form (1.3) (lines). (c) Lagrangian particle acceleration autocorrelation coefficient for
$d_p/\eta = 21.1$
and
${\textit{Re}}_\lambda =29$
measured by PTV (symbols) and its analytical form (line). The oscillations are due to the small surface waves.
In this section, we evaluate the applicability of the temporal filtering framework to the dynamics of the finite-size floating particles. The first step is to evaluate an effective response time
$\tau _p$
, which can be estimated as the definite integral of
$R_{a_p}^L(\tau )$
up to the zero-crossing time
$T_0$
(Calzavarini et al. Reference Calzavarini, Volk, Bourgoin, Lévêque, Pinton and Toschi2009). Measuring the latter, however, is challenging as it requires high temporal resolution and low noise (Machicoane & Volk Reference Machicoane and Volk2016; Machicoane, Huck & Volk Reference Machicoane, Huck and Volk2017). Inspection of our data indicates that the cases at
${\textit{Re}}_\lambda =29$
provides robust estimates of
$\tau _p$
, which are plotted in figure 6(a) in terms of
$St$
(filled symbols).
An alternative, though related strategy is to use an analytical model for
${R^L_{{u}_{\!p}}}(\tau )$
and differentiate it to obtain
${R^L_{a_p}}(\tau )$
. Following this avenue, we employ the model of Sawford (Reference Sawford1991), with two temporal scales that capture the large-scale and dissipative particle dynamics. In the original model,
$T_1 = 2 \langle u^2 \rangle / C_0 \varepsilon = \mathcal{O}(T_L)$
and
$T_2 = \tau _\eta C_0 / 2a_0 = \mathcal{O}(\tau _\eta )$
, where
$a_0= \langle a^2 \rangle \varepsilon ^{-3/2} \nu ^{1/2}$
and
$C_0$
is the constant in the inertial scaling of the second-order Lagrangian structure function,
$S_2^L(\tau ) = \langle [u(t_0+\tau )-u(\tau )]^2\rangle = C_0 \varepsilon \tau$
(Kolmogorov Reference Kolmogorov1941). By analogy, we define
$T_{1,p} = 2\langle {u}_{\!p}^2 \rangle / C_{0,p} \varepsilon = \mathcal{O}(T_p)$
and
$T_{2,p} = \tau _\eta C_{0,p} / 2 a_{0,p} = \mathcal{O}(\tau _p)$
, where
$a_{0,p} = \langle a_p^2 \rangle \varepsilon ^{-3/2} \nu ^{1/2}$
and
$C_{0,p}$
is found by fitting the measured structure function of the Lagrangian particle velocity as
$S_{2,p}^L(\tau ) = C_{0,p} \varepsilon \tau$
. The resulting forms of
${R^L_{{u}_{\!p}}}(\tau )$
and
${R^L_{a_p}}(\tau )$
(equivalent to (1.3) and (1.4), with
$T_{1,p}$
and
$T_{2,p}$
in place of
$T_1$
and
$T_2$
, respectively) capture reasonably well the behaviour of the measurements, as shown for selected cases in figures 6(b) and 6(c). Because the model parameters are set based on the measurements, the agreement mostly indicates that the forms (1.3) and (1.4) are appropriate to describe the autocorrelation coefficients. Indeed, the values of
$St$
based on integrating the analytical form of
${R^L_{a_p}}(\tau )$
, plotted as open symbols in figure 6(a), are in good agreement with those based on the measurements.
Both these strategies require empirical knowledge of the Lagrangian particle velocities and accelerations, while one would like to estimate
$\tau _p$
based on the turbulence and particle characteristics only. However, as discussed in § 1.2, there is no consensus on the correct expression or even the scaling dependence of
$\tau _p$
with the particle properties. Here, we test the empirical linear relation
\begin{equation} St = 1+0.08 \bigg ( \frac {d_p}{\eta } \bigg ), \end{equation}
which was shown by Uhlmann & Chouippe (Reference Uhlmann and Chouippe2017) to represent well their particle-resolved simulations and the experiments by Volk et al. (Reference Volk, Calzavarini, Leveque and Pinton2011) in 3-D turbulence. Figure 6(a) shows that such a relation is also consistent with the behaviour of our finite-size particles floating in free-surface turbulence. Therefore, for simplicity, we will adopt (3.3) in the following analysis. Two remarks are, however, in order. First, (3.3) is expected to overestimate
$St$
in the range
$d_p/\eta \lesssim 5$
, as neutrally buoyant particles of this size are indistinguishable from tracers both in 3-D and free-surface turbulence (Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007; Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011; Berk & Coletti Reference Berk and Coletti2024; Li et al. Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025). Second, as discussed in § 3.4, the present data are also compatible with the scaling
$St \sim (d_p/\eta )^{2/3}$
predicted by the spatial filtering ansatz (also shown in figure 6
a).
Keeping in mind such caveats, we evaluate
$St$
via (3.3) and test the temporal filtering predictions (1.7) to (1.9). Here, we take
$\boldsymbol{u}_{\textit{fp}} = \boldsymbol{u}$
, i.e. we assume that the particles do not preferentially sample flow regions with specific properties. This is supported by the evidence that finite-size neutrally buoyant particles do not cluster (Fiabane et al. Reference Fiabane, Zimmermann, Volk, Pinton and Bourgoin2012) or do so weakly (Uhlmann & Chouippe Reference Uhlmann and Chouippe2017). In figures 7(a) and 7(b), we then compare the measured fluctuating energy and correlation time scales of the floating particles against the temporal filtering predictions. Overall, the trend of
$\langle {u}_{\!p}^2 \rangle$
is correctly captured. The energy of the small particles is somewhat underestimated, likely because (3.3) overestimates their response time as mentioned previously. Larger discrepancies in
$\langle {u}_{\!p}^2 \rangle$
are found for
${\textit{Re}}_\lambda =29$
, which might be due to the lack of scale separation of the inertial subrange in this case. The correlation time
$T_p$
is somewhat underestimated, although the experimental scatter for the larger particles (due to the limited number of long trajectories recorded) may partly account for the mismatch. Multiplying (1.7) and (1.9) yields
\begin{equation} \frac {K_p}{K_t} = \frac {\big\langle {u}_{\!p}^2 \big \rangle T_p}{\langle u^2 \rangle T_L} = \frac {T_1+T_2}{T_L} \sim 1, \end{equation}
i.e. the diffusivity is expected to be independent of the particle inertia (as originally predicted by Tchen Reference Tchen1947). That is, under the temporal filtering assumption, the increase of
$T_p/T_L$
with particle size balances the decrease of
$\langle {u}_{\!p}^2 \rangle / \langle u^2 \rangle$
. This prediction is consistent with the measurements at
${\textit{Re}}_\lambda =29$
, while it underestimates the diffusivity measured in stronger turbulence, as shown in figure 7(c). There, the diffusivity is evaluated indirectly based on (3.4), i.e. as the product between the velocity variance and the correlation time scale rather than differentiating the mean square displacement, due to the limited length of the trajectories. Still, considering the vast range of particle sizes, the change in diffusivity is relatively modest, implying that (3.4) is a reasonable first-order approximation.
Comparison between measurements (filled symbols) and the temporal filtering framework (lines) for all
${\textit{Re}}_\lambda$
cases; (a) particle fluctuating energy (1.7), (b) velocity correlation time scale (1.9) and (c) particle diffusivity (3.4) versus dimensionless particle size. Error bars represent the run-to-run standard deviation.
Figure 8(a) compares the measured particle acceleration variance and the trend predicted by (1.8). The general trend is captured, though with some quantitative discrepancies. We note that, in the inertial subrange, the temporal filtering framework implies
$\langle a_p^2 \rangle / \langle a^2 \rangle \sim St^{-1}$
(Berk & Coletti Reference Berk and Coletti2024), which according to the assumed linear relation (3.3) is equivalent to
$\langle a_p^2 \rangle / \langle a^2 \rangle \sim (d_p/\eta )^{-1}$
. Recent particle-resolved simulations in 3-D turbulence by Jiang et al. (Reference Jiang, Wang, Liu, Sun and Calzavarini2022) agree with such scaling. The present data, however, suggest an influence of
${\textit{Re}}_\lambda$
on the scaling, which will be discussed in § 3.4.
Finally, we consider the return to Gaussianity of the acceleration p.d.f.s for increasingly inertial particles, which is a hallmark of temporal filtering of small inertial particles in turbulence (Bec et al. Reference Bec, Biferale, Boffetta, Celani, Cencini, Lanotte, Musacchio and Toschi2006). This is quantified by the flatness
$\langle a_p^4 \rangle / \langle a_p^2 \rangle ^2$
plotted in figure 8(b), showing that intermittency in the floating particle acceleration is significant up to approximately
$d_p/\eta = 50$
. According to (3.3), this corresponds to
$St \sim 5$
. The temporal filtering framework does not provide a priori scaling for
$\langle a_p^4 \rangle / \langle a_p^2 \rangle ^2$
, but we can refer to point-particle simulations based on such an assumption. For example, in the homogeneous 3-D turbulence simulations by Ireland et al. (Reference Ireland, Bragg and Collins2016) at
${\textit{Re}}_\lambda = 88$
(comparable to our more turbulent case), the acceleration flatness approximately recovers the Gaussian value of 3 for
$St \sim 10$
(see their figure 11).
(a) Comparison between the measured particle acceleration variance (filled symbols) and the temporal filtering framework (1.8) (lines) for all
${\textit{Re}}_\lambda$
cases. Error bars represent the run-to-run standard deviation and the different power-law scalings are indicated by dashed lines. (b) Particle acceleration flatness as a function of dimensionless particle size. The dashed line indicates a flatness of 3, the value for a Gaussian distribution.
3.4. Comparison with the spatial filtering framework
As discussed in § 1.2, the prevalent view is that neutrally buoyant finite-size particles act as spatial filters of the local turbulent flow. We test such an assumption by considering the amount of energy contained in the flow at scales up to
$d_p$
. This is readily represented by the second-order Eulerian velocity structure function,
$S_2^E (r) = \langle \|\boldsymbol{u}(\boldsymbol{r}_0+\boldsymbol{r})-\boldsymbol{u}(\boldsymbol{r}_0)\|^2\rangle$
, which quantifies the turbulent kinetic energy contained in scales
$r$
and smaller (Davidson Reference Davidson2015). As the particle responds to the remaining turbulent energy
$\langle q^2 \rangle -({1}/{2}) S^E_2(d_p)$
, the kinematic relation
$S^E_2(r) = 2\langle q^2 \rangle [1-R^E_u(r)]$
implies that the fluctuating kinetic energy of a finite-size particle is
\begin{equation} \big\langle {u}_{\!p}^2 \big\rangle (d_p) \sim \langle q^2 \rangle R^E_u(d_p). \end{equation}
Following Kolmogorov (Reference Kolmogorov1941) theory, this argument similarly leads to
$(\langle q^2 \rangle -\langle {u}_{\!p}^2 \rangle )/\langle q^2 \rangle \sim (d_p/\eta )^{2/3}$
in the inertial subrange (Homann & Bec Reference Homann and Bec2010; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017). Alternatively,
$\langle {u}_{\!p}^2 \rangle$
can be estimated directly from (3.5) if a model for
$R^E_u(r)$
is available based on the turbulence properties. Inspired again by Sawford (Reference Sawford1991), we write a two-length exponential:
\begin{equation} R^E_u(r) = \frac {L_1 \mathrm{e}^{-r/L_1}-L_2 \mathrm{e}^{-r/L_2}}{L_1-L_2}, \end{equation}
where
$L_1$
and
$L_2$
are associated with the energy-containing and dissipative scales of the flow, respectively. Because
$R^E_u(r)$
is expected to approximate a simple exponential in the large-scale limit, we take
$L_1 = L$
. Remembering that the Taylor microscale is related to the curvature of
$R^E_u(r)$
at small scales (Pope Reference Pope2000), we take
$L_2 = \lambda$
. This representation proves effective, reproducing well the measured Eulerian velocity autocorrelation coefficient, see figure 9(a). Remarkably, figure 9(b) shows that (3.5) captures the behaviour over the entire range of
$d_p/\eta$
and for all
${\textit{Re}}_\lambda$
considered here.
Let us now consider the other important properties of the particle motion, and their predicted trends based on the spatial filtering ansatz. While the application of Kolmogorov’s theory leads to
$\langle a_p^2 \rangle /\langle a^2 \rangle \sim (d_p/\eta )^{-2/3}$
(Voth et al. Reference Voth, La, Arthur, Alice, Alexander and Bodenschatz2002), the question is complicated by the consideration that the forces acting on finite-size particles are driven by the pressure increments at their scale (Xu et al. Reference Xu, Ouellette, Vincenzi and Bodenschatz2007; Brown et al. Reference Brown, Warhaft and Voth2009). The scaling for the latter is thought to vary with
${\textit{Re}}_\lambda$
, leading to a transition from
$(d_p/\eta )^{-4/3}$
to
$(d_p/\eta )^{-2/3}$
as the turbulence Reynolds number is increased (Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007; Homann & Bec Reference Homann and Bec2010). The floating particle acceleration we measure at the different
${\textit{Re}}_\lambda$
(see figure 8
a) agrees with this picture: the data at
${\textit{Re}}_\lambda = 29$
are consistent with the
$-4/3$
decay, whereas the more turbulent cases are compatible with the
$-2/3$
decay.
Kolmogorov’s inertial subrange theory also predicts the particle response time (taken as the correlation time scale of
$a_p$
) to scale as
$\tau _p \sim \tau _\eta (d_p/\eta )^{2/3}$
. Experiments and particle-resolved simulations of neutrally buoyant finite-size particles in 3-D turbulence, however, find only approximate agreement with such a trend, the comparison being complicated by the limited range of sizes and finite-
${\textit{Re}}_\lambda$
effects (Homann & Bec Reference Homann and Bec2010; Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017; Jiang et al. Reference Jiang, Wang, Liu, Sun and Calzavarini2022; Fan et al. Reference Fan, Wang, Jiang, Sun and Calzavarini2024). Similarly, as shown in figure 6(a), our observations of floating particles are compatible with such inertial scaling, but do not allow us to unambiguously support it with respect to other proposals (such as (3.3)). Like previous studies, we are also limited to a marginal separation of scales, which in this case is inherent to the flow configuration: increasing the intensity of the free-surface turbulence inevitably leads to larger surface deformations, changing the nature of the problem at hand (Brocchini & Peregrine Reference Brocchini and Peregrine2001). Thus, unless fluids of high surface tension are used, turbulent motion along a quasi-flat free-surface can only be obtained at relatively small
${\textit{Re}}_\lambda$
.
The acceleration flatness can also be estimated in the framework of spatial filtering by assuming intermittency corrections for the high-order moments of the velocity increments (Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011; Fan et al. Reference Fan, Wang, Jiang, Sun and Calzavarini2024). For example, using the model by She & Leveque (Reference She and Leveque1994) leads to
$\langle a_p^4 \rangle / \langle a^2 \rangle ^2 \sim (d_p/\eta )^{-0.56}$
. While this is comparable with the trends reported by Volk et al. (Reference Volk, Calzavarini, Leveque and Pinton2011), the acceleration intermittency of finite-size neutrally buoyant particles in 3-D turbulence remains strong for all sizes: the flatness of the acceleration p.d.f. has consistently been reported to be much larger than 3, and often larger than 10, even for
$d_p/\eta \gt 40$
(Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007, Reference Qureshi, Arrieta, Baudet, Cartellier, Gagne and Bourgoin2008; Xu & Bodenschatz Reference Xu and Bodenschatz2008; Brown et al. Reference Brown, Warhaft and Voth2009; Homann & Bec Reference Homann and Bec2010; Volk et al. Reference Volk, Calzavarini, Leveque and Pinton2011; Bellani & Variano Reference Bellani and Variano2012; Uhlmann & Chouippe Reference Uhlmann and Chouippe2017; Jiang et al. Reference Jiang, Wang, Liu, Sun and Calzavarini2022; Fan et al. Reference Fan, Wang, Jiang, Sun and Calzavarini2024). Such a persistent intermittency for particle sizes so deep in the inertial subrange is contrary to a simplistic application of either the temporal or spatial filtering assumption (Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007). Strikingly, unlike in 3-D turbulence, particles floating on the free-surface do display a return to Gaussian acceleration p.d.f. with increasing size (see figures 4
a and 8
b).
The spatial filtering assumption does not offer a specific prediction for the Lagrangian dispersion, in particular, for
$T_p$
and
$K_p$
, nor are we aware of systematic studies of
${R^L_{{u}_{\!p}}}(\tau )$
and its decay for large particles in 3-D turbulence. The exception is represented by the study of Machicoane & Volk (Reference Machicoane and Volk2016), who measured
${R^L_{{u}_{\!p}}}(\tau )$
in a von Kármán flow. They found, however, that the confined nature of the flow crucially influenced
${R^L_{{u}_{\!p}}}(\tau )$
, not allowing one to isolate the effect of particle size.
4. Discussion and conclusions
We have studied experimentally the behaviour of finite-size, marginally buoyant spheres and discs in homogeneous isotropic free-surface turbulence. By using two experimental facilities of identical architecture but different in size, we have spanned a wide range of parameters, with particle sizes up to
$d_p/\eta \sim 100$
and turbulence Reynolds numbers
${\textit{Re}}_\lambda \in [29, 84]$
. The latter is limited by our focus on a regime of quasi-flat free-surfaces, without adding to this already complex system the effect(s) of wind shear and/or surface waves (Falkovich et al. Reference Falkovich, Weinberg, Denissenko and Lukaschuk2005; Farazmand & Sapsis Reference Farazmand and Sapsis2019; Del Grosso et al. Reference Del Grosso, Cappelletti, Sujovolsky, Mininni and Cobelli2019). The motion of the particles is compared with the behaviour of small floating tracers in the same free-surface flows. We find that the behaviour of particle diameters up to approximately
$d_p/L=0.1$
and/or
$d_p/\eta = \mathcal{O}(10)$
is virtually indistinguishable from that of the free-surface flow. For larger sizes, the particle fluctuating energy
$\langle {u}_{\!p}^2 \rangle$
and acceleration variance
$\langle a_p^2 \rangle$
decrease, while their velocity correlation time
$T_p$
and response time
$\tau _p$
(taken as the correlation time scale of the particle acceleration) increase. The opposite and comparable changes in
$\langle {u}_{\!p}^2 \rangle$
and
$T_p$
imply that the long-term diffusivity
$K_p = \langle {u}_{\!p}^2 \rangle T_p$
is weakly dependent on particle size. The accelerations become less intermittent with increasing particle size, displaying a Gaussian distribution above approximately
$d_p/\eta = 50$
. The present data show no systematic differences between spheres and discs when the diameter
$d_p$
(i.e. their maximum extension in the surface-parallel direction) is used to characterise their size. This indicates that the motion is mostly influenced by the near-surface flow, unlike non-spherical particles in 3-D turbulence for which the relevant geometric scale is the volume-equivalent diameter (Jiang et al. Reference Jiang, Wang, Liu, Sun and Calzavarini2022).
We have used our measurements to address the question of whether, and to which degree, the motion of finite-size floating particles can be described by the two fundamentally different approaches commonly used to rationalise the behaviour of inertial particles in turbulence: temporal filtering, which assumes that the particles respond only to fluid fluctuations slower than
$\tau _p$
; and spatial filtering, which assumes that the particles respond only to fluctuations of length scale larger than
$d_p$
. In particular, we have applied the temporal filtering approach in its simplified form that assumes the particles to sample the flow ergodically, i.e. without favouring specific flow regions. This was the original assumption of Tchen (Reference Tchen1947), which fails to capture important trends of small heavy particles that preferentially sample the turbulence (Wang & Stock Reference Wang and Stock1993; Jung et al. Reference Jung, Yeo and Lee2008). In the case of large particles, however, the evidence from 3-D turbulence studies suggests that preferential sampling is weak. Under this assumption, the temporal filtering model provides closed expressions for the particle fluctuating energy, acceleration, velocity correlation time scale and diffusivity, based solely on
$\tau _p$
and the characteristic time scales of the free-surface flow.
Our observations suggest that, in the present range of parameters, the response time (hence, the Stokes number) of the floating particles is reasonably estimated from
$d_p$
via an empirical linear relation derived for finite-size particles in 3-D turbulence. Using that, the temporal filtering approach captures the main observed trends of the transport properties. This is noteworthy, in that a simple particle equation of motion such as (1.5) is very useful in predicting the fate of floating particles in a free-surface flow of known properties. The diffusivity is especially important to parametrise the sub-grid terms in coarse-graining strategies (e.g. in large-eddy simulations and in other large-scale models used to predict the fate of marine pollution) but is usually poorly constrained. For example, in the study of Lagrangian transport of floating plastics in the Mediterranean Sea by Kaandorp et al. (Reference Kaandorp, Dijkstra and van Sebille2020), a range of
$K_p$
between
$1\,\mathrm{m^{2}s^{-1}}$
and
$100\,\mathrm{m^{2}s^{-1}}$
was considered. The prediction that particle diffusivity is, to first order, equal to that of the underlying free-surface turbulence (at least in the considered case of no wind and negligible waves) may prove useful in this sense.
The spatial filtering approach, however, is found to reproduce with quantitative accuracy the reduction of the fluctuating energy of the floating particles with increasing size. In particular, we stress the usefulness of the simple relation
$\langle {u}_{\!p}^2 \rangle \sim \langle q^2 \rangle R^E_u(d_p)$
when the spatial autocorrelation of the free-surface flow is available or can be modelled. Here, we find it to be well represented by a two-length exponential inspired by Sawford’s two-time model for the temporal autocorrelation. This framework predicts power-law scaling relations for the acceleration variance that are compatible with our observations, including the effect of
${\textit{Re}}_\lambda$
.
It is remarkable that the acceleration distributions of the floating particles lose their intermittent character with increasing size. This is consistent with the picture of both temporal and spatial filtering, but in contrast with observations in 3-D turbulence: there, the intermittency remains strong for neutrally buoyant particles of all sizes, indicating that any filtering approach (temporal or spatial) is inadequate or anyway too simplistic (Qureshi et al. Reference Qureshi, Bourgoin, Baudet, Cartellier and Gagne2007). The present finding suggests that finite-size floating particles may be more amenable to such representations.
The different acceleration distributions of large particles in free-surface versus 3-D flows may be interpreted in view of differences in the turbulent dynamics. In 3-D flows, the intense small-scale activity is correlated with large-scale fluctuations of the energy input (Blum et al. Reference Blum, Kunwar, Johnson and Voth2010, Reference Blum, Bewley, Bodenschatz, Gibert, Gylfason, Mydlarski, Voth, Xu and Yeung2011; Carter & Coletti Reference Carter and Coletti2018; Vela-Martín & Avila Reference Vela-Martín and Avila2024) and small vortices are often spatially organised around large-scale shear layers between energetic eddies (Ishihara, Gotoh & Kaneda Reference Ishihara, Gotoh and Kaneda2009; Hunt et al. Reference Hunt, Ishihara, Worth and Kaneda2014). This may partly explain why even particles with inertial subrange sizes exhibit intermittent accelerations. In free-surface turbulence, while large-scale properties such as the turbulent kinetic energy and the integral scales reflect those of the bulk (Li et al. Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025), the velocity gradient tensor of the surface motion follows profoundly different dynamics associated with the specific boundary conditions (Qi et al. Reference Qi, Xu and Coletti2025b
). In particular, the free-surface vorticity and strain rate evolve over time scales comparable to
$T_L$
(rather than
$\tau _\eta$
as in 3-D turbulence) and are related to upwelling/downwelling events moving fluid towards and away from the surface (Kumar et al. Reference Kumar, Gupta and Banerjee1998; Shen et al. Reference Shen, Zhang, Yue and Triantafyllou1999; Lovecchio et al. Reference Lovecchio, Zonta and Soldati2015; Ruth & Coletti Reference Ruth and Coletti2024; Li et al. Reference Li, Salmon Sanness, Hassaini, Chang, Mucignat and Coletti2025; Qi et al. Reference Qi, Li and Coletti2025a
). The equilibrium between upwellings and downwellings implies that stretching and compression of the surface-attached vortices are in balance (unlike in 3-D turbulence; Davidson Reference Davidson2015), impacting the organisation of the intense-fluctuation events and the inter-scale energy transfer along the free-surface (Ruth & Coletti Reference Ruth and Coletti2024; Qi et al. Reference Qi, Li and Coletti2025a
,
Reference Qi, Xu and Colettib
). Further studies that simultaneously capture both the flow and the particle motion shall elucidate how the spatio-temporal structure of the turbulence affects the acceleration of finite-size floating particles.
Taken together, these results demonstrate that both the temporal filtering and the spatial filtering approaches capture important and complementary aspects of the motion of floating particles in free-surface turbulence. Spatial filtering provides a more accurate estimate of the decrease in particle fluctuating energy with size, compared with the prediction based on temporal filtering. The relative success of the latter might actually be rooted in the approximately linear relation between
$d_p$
and
$\tau _p$
, and thus merely reflect spatial filtering. Temporal filtering, however, also yields approximate estimates of the velocity correlation time scale and diffusivity, which are not directly predicted by spatial filtering. As Tchen (Reference Tchen1947) first realised, the concept of response time and temporal filtering are intertwined and, as such, temporal filtering is implicit in point-particle simulations of Lagrangian transport based on (1.5). The present findings suggest that, for finite-size floating particles, alternative approaches fully based on spatial filtering may be desirable, though defining them is outside the scope of the present work. Finally, we remark on an important limitation: for the considered case of negligible waves to be realised,
${\textit{Re}}_\lambda$
(and thus scale separation) cannot be large. This implies that scaling relations such as those predicted by spatial filtering using Kolmogorov’s theory have a limited range of applicability. Such a limitation on the Reynolds number, in turn, constrains the scale separation in the flow. Therefore, one cannot clearly discern whether the relevant dimensionless particle size is
$d_p/L$
,
$d_p/\eta$
or possibly
$d_p/\lambda$
. From basic notions on turbulent kinetic energy and intermittency, the behaviour of the particle velocity and acceleration are expected to depend on the size compared with the integral and Kolmogorov scale, respectively. This may be a simplistic view, considering the relations between distance scales (Blum et al. Reference Blum, Bewley, Bodenschatz, Gibert, Gylfason, Mydlarski, Voth, Xu and Yeung2011) and the peculiar nature of the free-surface (Qi et al. Reference Qi, Li and Coletti2025a
).
Future works shall explore further the influence of floating particle geometry, for example, considering prolate particles whose translation and rotational motions are coupled (Voth & Soldati Reference Voth and Soldati2017) and which represent a large fraction of marine plastics (Kooi & Koelmans Reference Kooi and Koelmans2019). Moreover, research is warranted on how the effect of turbulence combines with that of surface waves, whose impact on the transport of spherical and non-spherical particles has attracted significant interest in recent years (Pizzo, Melville & Deike Reference Pizzo, Melville and Deike2019; Baker & DiBenedetto Reference Baker and DiBenedetto2023; Xiao et al. Reference Xiao, Calvert, Yan, Adcock and Van Den Bremer2024). Finally, in the presence of wind, the submergence will determine the influence of windage, i.e. the drag experienced by objects partly protruding above the free-surface (Beron-Vera, Olascoaga & Miron Reference Beron-Vera, Olascoaga and Miron2019).
Acknowledgements
The authors gratefully acknowledge the support of R. Vonbank for their assistance with the water channel at Empa.
Funding
The present work was supported by the Swiss National Science Foundation (Project No. 207318)
Declaration of interests
The authors report no conflict of interest.
Data availability statement
The corresponding author makes all the data supporting this work available upon reasonable request.
Open access
