## Abstract

Numerical models trying to faithfully represent the movement of floating bodies transport in open-channel flow require experimental data for validation. In order to provide an adequate dataset, flume experiments were carried out to analyse the transport of singular and grouped rigid bodies floating on the water surface. Both cylindrical and spherical samples were employed: they were released in a rectangular channel under steady conditions in one-dimensional (plain channel) and two-dimensional (2D) configurations using one rectangular side obstacle, one smooth side obstacle or two rectangular alternate obstacles. The outcomes of the experiments are the planar displacement and the rotation of the samples, which are related to the flow field in the different configurations. The detailed experimental analysis of the floating body motion provides information for the calibration of numerical models simulating floating bodies transport. This dataset is thus employed for the validation of the Eulerian–Lagrangian model ORSA2D_WT, highlighting its strengths and improvable aspects. Similar applications could be carried out with any 2D model which performs the simulation of discrete elements moving on the water surface.

## HIGHLIGHTS

The transport of floating bodies and of large wood is a challenging issue in hydraulic risk modelling.

An experimental dataset is provided, with different open-channel flow regimes, employing floating spherical and cylindrical samples.

Single-body and semi-congested transport of cylindrical samples is tested.

The numerical model ORSA2D_WT is applied to simulate average trajectory, average orientation and final distribution of bodies.

The dataset is available for any model dealing with floating bodies and large wood transport.

### Graphical Abstract

## INTRODUCTION

The numerical modelling of flood flows is a powerful tool for the estimation of the hydraulic risk and nowadays can include several additional effects that are complementary to the inundation. Sediment transport, for example, is generally considered when the main purpose is the evaluation of the morphological evolution or the bridge scouring prediction (e.g. Darby *et al.* 2002; Khosronejad *et al.* 2012; Nones 2019). Another aspect that can be included is the transport of large wood that may interest both the lower part and the upper part of the river basin. In the first case, the interaction with the bridge is the main issue, since wood can accumulate at bridge piers, reducing the free span and leading to a backwater rise which worsens the effects of the flood (e.g. Roca & Davison 2010; Lucía *et al.* 2015). Such interaction is rarely considered, although in recent years, several events involved wood accumulation (Ruiz-Villanueva *et al.* 2014; Silvestro *et al.* 2016; Corsini *et al.* 2017). Past-events reconstruction may also benefit from the inclusion of wood effects (Macchione *et al.* 2019), especially in those catchments where wood transport is known (Comiti *et al.* 2008).

Upper basins are where wooden debris generate since bank erosion, debris flows or landslides that occur in vegetated areas nearby the main channel are the primary sources of floating wood (Comiti *et al.* 2016). Large wood is often collected at check dams, altering their expected functioning (Meninno *et al.* 2020a) and, again, increasing the water level aggravating possible floods. Modelling wood entrainment from such areas requires in-depth knowledge of the vegetated coverings, of the hillslope stability and of the probability of wood uprooting (e.g. Zischg *et al.* 2018). An accurate simulation of the flow through river networks (e.g. Costabile *et al.* 2019) could contribute to properly simulate wood transport from the areas where it originates to the critical locations.

Including wood transport in hydraulic modelling usually requires the coupling between the solution of the shallow water equation (SWE) and the solid phase, i.e. the wood floating on the water surface. Most of the models follow a two-dimensional (2D) approach, with a finite volume method to solve SWE and a discrete element method (DEM) for the transport and rotation of the logs. In some cases, the DEM is fully kinematic (Ruiz-Villanueva *et al.* 2014), with the flow velocity attributed to the logs, while other authors follow a dynamic approach (Alonso 2004; Stockstill *et al.* 2009; Persi *et al.* 2018), selecting the forces and the strategy to calculate the wood linear and angular acceleration with Newton's first law. Kimura & Kitazono (2019) propose also the coupling between the three-dimensional solution of the water flow equations and the 2D transport of floating dowels, while Meninno *et al.* (2020b) followed a different approach, coupling the 2D SWE with the advection–diffusion equation considering wood as a continuous substance transported on the water surface.

Another important step is the computation of the effect of wood on the water, to calculate the backwater effect which becomes important in case of accumulation. This phenomenon is, as said, the main responsible for the aggravated inundations within urban areas and is studied especially with laboratory experiment (e.g. Schalko *et al.* 2020). To take into account the effect of the accumulation, different strategies are available: a statistic approach to evaluate the blockage probability (e.g. Furlan *et al.* 2019), a semi-statistical way which requires the geometrical evaluation approach of the logs to the bridges (e.g. Shrestha *et al.* 2012) or a force-based method, which requires to transfer the drag force acting on the body in the momentum equation (e.g. Ruiz-Villanueva *et al.* 2014).

In the framework of the numerical modelling of wood transport, a dynamic model, named ORSA2D_WT, was developed which includes the entrainment and the computation of the linear and angular acceleration of floating objects (Persi 2018). The forces included in the model are the drag force and the side force (aligned and transversal components of the hydrodynamic force), the added mass and the pressure gradient force. The preliminary calibration was performed on a set of experiments with single release (Persi *et al.* 2019a), and the model was also applied to a real river in uncongested-transport regime (Persi *et al.* 2019b).

The main limits highlighted by these preliminary applications are connected to the high unpredictability of the transport of floating bodies, due to the uncertainty of both the initial conditions and of the real log shape, to the difficulties in replicating exactly the wood–wood or wood–riverbank interaction and to the unexpected effect of the surface turbulence, which is not included in the 2D SWE model used.

Despite trying to simulate exactly the trajectory of single logs, one should try to catch at least the average behaviour (i.e. the average trajectory and orientation, when applicable) together with the deviation from average due to different initial conditions.

To the authors' knowledge, few laboratory experiments about wood transport in clear water are available in the literature, so that a proper validation based on a statistical approach is not possible.

In this paper, we present a series of experiments performed in a laboratory flume with different flow configurations, including the transport of spheres, cylinders and small groups of cylinders. Despite our goal is the numerical modelling of the transport of large wood, which presents an elongated shape, the experimental campaign includes also tests performed by releasing spherical samples. Existing equation of motion (e.g. Maxey & Riley 1983) and study of the hydrodynamic coefficients for spheres (e.g. Batchelor 1967; Truscott & Techet 2009), although always fully submerged, served as an input to derive the model for the floating cylinder transport and are thus fundamental as a first stage for validation. Cylindrical bodies are then considered to approach the large wood transport simulation. These experiments represent a small, although adequate, dataset which can be used as a reference for a semi-statistical calibration of ORSA2D_WT, as well as for any floating transport numerical model.

In the following, we first describe the experimental campaign, providing details about flume configuration, flow regime and sample characteristics. Then, we analyse the spatial and angular average behaviour and distribution of the most significant experiments, finally presenting the results of the application of the code ORSA2D_WT.

## METHODS

### Experimental setup

The experimental campaign was carried out at the University of Zaragoza, in a prismatic horizontal flume, 3.25 m long and 0.24 m wide. The flume had transparent bottom and vertical walls (polymethyl methacrylate, PMMA), while the obstacles were made of PVC with a smooth surface. A constant discharge was supplied under steady-state conditions (15.3 m^{3} h^{−1}, measured by the electromagnetic flow meter COPA-XE DE43F by ABB). Experiments were carried out with four different planar configurations: straight flume, flume with a rectangular obstacle on one side, flume with rectangular obstacles on both sides and flume with a curved obstacle on one side. For conciseness, we focus on the last two configurations (Figure 1(a) and 1(b), with obstacle dimension and positioning), which present non-uniform flow that can be referred to as strong-2D and smooth-2D configuration, respectively. The results for the remaining configurations (straight flume/one-dimensional (1D) flow field and channel with a rectangular obstacle on one side/mid-2D flow field) are shown in the Supplementary Materials.

Four sample types with different shapes and densities were employed, including a wooden sphere, a plastic sphere filled with water and alcohol, wooden cylinders and unbalanced wooden cylinders (Figure 2(a)–2(d), with sample dimension and density). The latter sample had a screw at one end, to simulate the different density distribution of real logs with root wads. The spherical samples were released manually, whereas cylinders were released with a semi-automatic device which let them fall into the water from about 0.02 m above the water surface, reducing the effect of the operator dexterity. The device is described by Persi *et al.* (2019a). In both cases, uncongested transport was tested (Braudrick & Grant 2001) by releasing one sample by one, avoiding any contact between floating elements. In addition, some experiments on semi-congested transport (i.e. limited interactions among transported objects) were also performed by releasing all the cylinders together, both standard and unbalanced. Table 1 summarizes the number of experiments performed for the selected configurations and for each sample.

Flume configuration . | Flow field . | Wooden sphere . | Plastic sphere . | Standard cylinder . | Unbalanced cylinder . | Semi-congested transport . |
---|---|---|---|---|---|---|

1 curved obstacle | Smooth-2D | 9 | 9 | 16 | 20 | – |

2 rect. obstacles | Strong-2D | 6 | 6 | 16 | 20 | 6 |

Flume configuration . | Flow field . | Wooden sphere . | Plastic sphere . | Standard cylinder . | Unbalanced cylinder . | Semi-congested transport . |
---|---|---|---|---|---|---|

1 curved obstacle | Smooth-2D | 9 | 9 | 16 | 20 | – |

2 rect. obstacles | Strong-2D | 6 | 6 | 16 | 20 | 6 |

Water level and velocity measures were performed for each configuration, to provide a useful comparison for the simulated velocities (Persi 2018), in the measurement points shown in Figure 1. Since the flume had horizontal bottom, it is not possible to define uniform flow conditions, so only the velocity and water level ranges are reported, together with the range of the Froude numbers computed where all the data are available (Table 2).

. | u (m s^{−}^{1})
. | v (m s^{−}^{1})
. | h (m)
. | Fr (–) . |
---|---|---|---|---|

Smooth-2D | 0.315–0.81 | −0.3–0.22 | 0.019–0.062 | 0.42–0.59 |

Strong-2D | −0.10–0.88 | −0.34–0.4 | 0.025–0.073 | 0.33–0.85 |

. | u (m s^{−}^{1})
. | v (m s^{−}^{1})
. | h (m)
. | Fr (–) . |
---|---|---|---|---|

Smooth-2D | 0.315–0.81 | −0.3–0.22 | 0.019–0.062 | 0.42–0.59 |

Strong-2D | −0.10–0.88 | −0.34–0.4 | 0.025–0.073 | 0.33–0.85 |

Minimum and maximum values of the average Froude numbers are also reported.

The maximum streamwise velocity *u* is around 0.8–0.9 m s^{−1}, while the minimum value is negative for the strong-2D flow field, due to the strong recirculation observed downstream the abrupt obstacles. The transversal velocity *v* is either positive or negative, due to the obstacle positioning, and covers the wider range for the strong-2D configuration. The water level is strongly variable in all configurations, with the lowest value observed for the smooth-2D configuration, below half of the sphere diameter, not enough for body flotation. In this case, in the downstream part of the flume, the spheres touch the flume bottom and start rotating around their horizontal axis.

### Numerical model highlights

The numerical model named ORSA2D_WT includes a 2D finite volume, Roe-Riemann solver of the SWE for the hydrodynamic part (Petaccia *et al.* 2010, 2016; Petaccia & Natale 2020) coupled with a dynamic DEM that calculates the forces acting on the floating body, its trajectory and orientation (Persi *et al.* 2018, 2019a; Petaccia *et al.* 2018).

*et al.*2019a). Equations (5)–(7) are implemented in the code and solved through a Runge–Kutta 4th order method.where the subscript

*i*refers to each segment of a single body, is the fluid density, is the segment longitudinal area (not projected on the flow direction), and are the flow and body linear velocity at the center of each segment, is the flow angular velocity in the cell corresponding to the segment center, is the body angular velocity, is the body mass,

*I*is the body momentum of inertia, and are the distances between each segment center and the body mass center, is a unit vector for the velocity projection, and the coefficients , , and are the drag, side, added mass and added inertia coefficients, which depend on the body shape and are computed as shown in Table 3. Two types of derivatives appear in the model: the simple time derivative , applied to the body velocity, and the total/Lagrangian derivative for flow variables . The latter is included to consider both the time and the spatial variation of the flow velocity in force computation (Persi 2018; Petaccia

*et al.*2018).

. | Sphere . | Cylinder . | |
---|---|---|---|

C_{D} | |||

C_{S} | |||

C_{AM} | |||

C_{AI} | – | 1.8 |

. | Sphere . | Cylinder . | |
---|---|---|---|

C_{D} | |||

C_{S} | |||

C_{AM} | |||

C_{AI} | – | 1.8 |

For semi-submerged cylinders, the drag and side coefficients where measured and evaluated with CFD simulations (Persi *et al.* 2019c; Alamayreha *et al.* 2020), noticing their variation with the body orientation. For spheres, the drag coefficient depends on the particle Reynolds number , while the side coefficient is variable with the sign of the relative angular velocity, starting from a standard value taken from the literature. The added mass coefficient is computed based on the body density, to consider the effective submergence, while the added inertia coefficient was calibrated for cylinders in Persi *et al.* (2019a). Regarding spheres, since no appreciable planar rotation was observed in the experiments, the rotation formulation cannot be applied, so that the calibration of for spheres is not applicable.

Once that the velocities are computed, kinematic relations are used to calculate the body displacement and angular rotation. Collision between cylinders and with side walls is modelled by implementing Newton's restitution principle adapted from Hecker (1997). The restitution coefficient, calibrated on purpose for cylinders floating on the water surface, is *e* = 0.1 (Persi 2018). Further details about the numerical model can be found in Persi (2018), Petaccia *et al.* (2018), and Persi *et al.* (2019a).

## RESULTS AND DISCUSSION

### Experimental results

The experiments were recorded from top view (Figure 3, strong-2D flow field, after orthorectification) with a Nikon camera (Nikon D810, with a Nikon 24–70 mm f/2.8G lens, set at a focal distance of 24 mm), and the displacement of each object was then acquired through the image analysis software tracker (Brown & Christian 2011). The trajectories of each body center and the planar rotation for the cylinders are evaluated. In this paper, the experimental results for a single-body transport performed in the channel with one side curved obstacle (smooth-2D flow field, Figure 1(a)), which presents the highest number of repetitions for spheres, are shown. The results obtained for the other flume configurations can be found in the Supplementary Material. For semi-congested transport, the flume configuration shown in the paper is the strong-2D one (Figure 1(b)).

#### Single-body transport

The wooden sphere was released manually at about 1.30 m from the inlet and at half flume width. For the first 30–40 cm along the *x* axis, the trajectories diverged transversally, occupying about 1/3 of the flume width. Then, as shown in the boxplots in Figure 4(c), they maintained such transversal dispersion nearly for the entire flume, although they came nearer when approaching the side obstacle, except for one sphere that moved near the left side wall. The trajectories of each repetition show that the sphere, after an initial adaptation due to the release, until about *x* = 1.5 m, followed the measured velocity vectors (Figure 4(a)), avoiding any contact with the side obstacle. In the final part (*x* > 2.75 m), the trajectories return nearly parallel to the walls. The streamwise dispersion (Figure 4(b)) tended to increase in the final part of the flume, due to the different velocities encountered by the spheres which followed different trajectories. Note that the dispersions shown with box plots in Figure 4(b) and 4(c) are calculated at fixed *x* positions for the average trajectory (black line in Figure 4(a)). The box plots report the median (red line), 25th and 75th percentiles (blue box), minimum and maximum values (black whiskers) and outliers values (red crosses, when existing). Such description is valid for all the box plots shown in the paper.

The plastic sphere was released following the same procedure as for the wooden sphere. Also in this case, the trajectories tended to spread transversally until they reached a nearly stable condition which followed for the entire experiment duration (Figure 5(a)). Comparing the trajectories and the velocity vectors, it appears that the sphere followed the flow streamlines, but, with respect to the wooden sphere, the average trajectories were slightly nearer to the left side wall (black line in Figure 5(a)). Both the streamwise (Figure 5(b)) and the transversal dispersion (Figure 5(c)) were slightly larger than those observed for the wooden sphere in the final part of the flume.

The cylinders were released automatically at about (*x* = 1.30 m, *y* = 0.12 m), perpendicularly oriented with respect to the flow main direction. In the first 20–30 cm after the release, they spread transversally and then flowed along the streamline, as shown by the measured velocity vectors displayed in Figure 6(a). The single cylinder trajectories show some differences due to the transversal gradient: the nearer to the side wall, the slower they moved due to the presence of the obstacle. The streamwise dispersion (Figure 6(b)) was extremely reduced and slightly increased in the final part of the flume, reaching a maximum of 0.15 m. The transversal dispersion was maximum when the cylinders went farther than the side obstacle, at *x* = 2.5 m, and then returned to a normal value (Figure 6(c)). With respect to the spheres, the trajectories appeared more grouped. The different shapes of the sample may be one reason for this difference: elongated bodies present a more stable configuration and offer to the flow a smaller frontal area, i.e. a smaller velocity gradient among different repetitions. However, probably the most affecting factor is the release method, which is semi-automatic for cylinders and guarantees a higher repeatability of the experiments.

Cylinders can freely rotate on the planar surface, in the range 0–360° (counter-clockwise direction from vertical axis *y*). To provide a meaningful statistical analysis, the range of variation is limited to 0–90° (*ϑ* = 0°, perfect perpendicularity; *ϑ* = 90°, perfect alignment with the flume walls), which means that angles larger than 90° are manipulated to fit the reduced range although maintaining the correct information about the actual orientation. Note that in the experiments, some cylinders could also make one complete turn in the full range.

The statistical analysis is shown in Figure 6(d). At the beginning, samples were nearly perpendicular to the flow and then tended to rotate reaching a median angle *ϑ* ≈ 68°. Cylinder angles present a large dispersion especially in the second part of the flume, occupying nearly the entire range of variation (standard deviation 80°).

Unbalanced cylinders were released at the same location as standard ones, and the trajectory pattern (Figure 7(a)) was like the one of the above-mentioned samples. At about 1.40 m from the inlet, the trajectories cross each other, and a reduction of the transversal dispersion is observed. Such behaviour, observed for all the experiments, is particularly noticeable for the unbalanced cylinders. This is possibly connected to the displacement of the mass center, which may alter the drop of the cylinders and the initial part of their trajectories. Most of the trajectories are grouped in the left part of the channel (*y* > 0.12 m), while two samples reach a minimum transversal position of 0.075 m, never reached by standard cylinders. However, overall streamwise and transversal dispersions at the points of interest are similar, although at the end of the flume, unbalanced cylinders result in slightly more disperse than standard ones, probably because the higher body inertia affects the trajectory more than for axial-symmetrical bodies. The sample orientation presents a larger dispersion in the central part of the flume, but then, the variation of angles is reduced, the final median angle and standard deviation being, respectively, about 65° and 57°. It seems that the presence of a heavier edge contributes to stabilize the sample rotation.

Overall, the trajectories of single bodies follow quite well the flow streamlines. Despite the samples being released in the same position, they are influenced by the releasing mechanism. The plastic sphere, smallest and lightest, is the mostly affected, with trajectories that rapidly depart from the release position and tend to the left side of the flume (where the operator was placed, probably influencing the release). This reflects in the average trajectory that do not follow a precise streamline but crosses transversally the flume in the upstream part and appears flattened in the second part due to the different curvatures of single paths, especially those very near to the left side wall. On the contrary, the average trajectory for the wooden sphere is nearer to the trend of flow streamlines. This is related to a lower sensitivity to the release conditions that reduce the initial diversion.

If we focus on cylinders, the greater stability of the trajectories is clear, and all the repetitions are well grouped and follow the flow velocity. This is certainly related to the semi-automatic release methodology but may also be due to the elongated shape of cylinders that can rotate to present a more hydrodynamic position. Unbalanced cylinders present a slightly higher variability, both at the beginning and downstream of the obstacle. Such variability, together with the larger angular variation, may depend upon the different mass distribution along the sample.

#### Semi-congested transport

Semi-congested experiments were obtained by releasing all the cylindrical samples together (eight standard cylinders and ten unbalanced cylinders) in the two-side-rectangular obstacle configuration. In the upstream part of the flume, the samples nearly occupied the whole flume width, with random streamwise and transversal dispersion. One exception was video 72, in which the cylinders were aligned in the cross section and parallel to the flow. Interactions between cylinders as well as with the flume walls could occur, while cylinders usually did not hit the two-side obstacles.

Due to the randomness of the initial cylinder configuration, no typical pattern is observed, although it is possible to detect some common characteristics. The analysis focuses on the comparison of the initial and of the final configuration for each video (numbered from 67 to 72), which are shown in Figures 8–10. The initial configuration is taken just after the release, while the final one is the last frame in which all the cylinders appear, in order to have the same number of samples.

Overall, the samples tend to spread longitudinally, while they flow downstream (Figure 8), increasing the streamwise dispersion. On the contrary, the transversal dispersion slightly reduces in the downstream part of the flume (Figure 9), and this is mainly due to the effect of the side obstacles which tend to concentrate the flow, and the floating cylinders in the central part of the flume. In addition, in most videos, the median transversal value is below the flume axis (0.12 m).

Regarding the cylinder orientation (Figure 10), random values are observed at the beginning of the experiments, except for video 72, as anticipated. The final distribution is quite large as well, with an average range of nearly 70°. The median value is on average 61°, confirming the angle observed also for single-body transport.

### Numerical results and comparison

The above-detailed experiments were simulated with ORSA2D_WT under steady-state flow conditions, with constant inlet as the upstream boundary condition and the critical Froude number as the downstream boundary condition. The bed friction is represented through the Manning coefficient, set equal to 0.01 s m^{−1/3}. Side obstacles are simulated as holes in the mesh, a solution that provides a good representation of the flow field as shown in Persi (2018).

For floating bodies, density and dimension are included in the model, together with the initial position, orientation (for cylinders) and initial velocity evaluated after about 1 s after the release (average computed on 0.5 s), in order to discard the effects due to the release of the object. Basically, the model implements an initial distribution of the floating bodies taken from the flume experiments and predicts a possible evolution based on the forces exerted by the flow.

In the following, a qualitative comparison is proposed for the 2D-smooth flow field configuration, while the average simulated trajectory and the final distribution of floating bodies are quantitatively compared for each sample and flume configuration.

#### Flow field simulation

As highlighted in Figures 4(a)–7(a), the samples follow pretty well the flow streamlines, with differences that depend on the body shape and on the release technique (manual or semi-automatic). However, since small differences are observed, it is not straightforward to separate between the two effects, and the flow field is the main factor affecting the path of floating bodies motion. The numerical simulation needs to provide a flow field accurate enough to guarantee that numerical samples are transported with the same velocity as in the experiments. Figure 11 compares the measured and simulated velocity vectors for the considered geometries. In the background, the simulated flow field is shown. In both cases, up to about *x* = 1.80 m, the flow field is uniform, while downstream this section, the deviation due to the side obstacles stands out. Regarding the smooth-2D geometry (Figure 11(a)), simulated results are in good agreement with the measured velocities, with correlation coefficients of 0.989 for the streamwise direction and 0.934 for the transversal component. Local discrepancies are observed, especially in the sections immediately upstream (*x* = 1.98 m) and downstream (*x* = 2.31 m) of the obstacle, affecting mainly the transversal velocity. In the first, the simulation slightly underestimates the flow diversion, while in the latter, the simulated flow presents a higher curvature than in the experiments. Figure 11(b) shows that the measured transversal velocity is larger than the simulated one in most of the flume, except for the sections where the obstacles are located (*x* = 2.00 m and *x* = 2.50 m). The streamwise component appears to be slightly underestimated, too. These differences are reflected in the correlation coefficients, which are 0.663 and 0.636 for the streamwise and transversal velocity, respectively. Two steady recirculation areas are also shown for this geometry, which were observed in the experimental campaign but not measured in detail (only one measure available, as shown). Flow measurements were performed with a digital flowmeter (MiniAri20, with the probe Mini 95.0004 by PCE Instruments) and an Acoustic Doppler Velocimeter (MicroADV 16 MHz, by Sontek). Further details about the comparison of the measured and simulated flow field can be found in Persi (2018).

#### Numerical results for the 2D-smooth flow field

Figures 12–15 show the numerical results for the 2D-smooth flow field (curved side obstacle configuration) and are the numerical analogous of Figures 4–7. The mesh has 16,050 cells, and the code takes about 4 min to simulate 1 s for each single-body transport.

The simulated trajectories of the wooden sphere (Figure 12) interrupt just after the side obstacle because the sphere starts to roll on the flume bottom. Modelling such transport mechanism requires the computation of an anisotropic friction force with a rolling coefficient for movement perpendicular to the axis (applied to large wood, Kang & Kimura 2018; Kang *et al.* 2020) or the inclusion of a ‘transport inhibition parameter’ as proposed by Ruiz-Villanueva *et al.* (2014). In the current version, the numerical model ORSA2D_WT focuses on the transport of floating bodies, and it cannot compute the rolling motion on the flume bottom. Despite the lack of the final part of the trajectories, the numerical and the experimental results have a similar average trajectory, while the single numerical trajectories appear more dispersed if compared with the experimental upper and lower limits (dashed red lines in Figure 12(a)). The boxplots show a similar streamwise distribution (Figures 4(b) and 12(b)) and a larger transversal distribution (Figures 4(c) and 12(c)).

The numerical trajectories of the plastic sphere tend to the left side of the flume, so that the average trajectory presents higher values of the transversal coordinates with respect to the experimental one, while the simulations are included in the experimental limits (Figure 13(a)). From a qualitative standpoint, both the streamwise and the transversal dispersion are like the experimental ones (see Figure 5(b) and 5(c) and Figure 13(b) and 13(c)).

The trajectories computed for the standard cylinder are close to the experimental ones, with nearly overlapped averages. Single numerical trajectories tend to spread a bit more transversally in the final part of the flume, where they exit the upper and lower experimental limits (Figure 14(a)). The streamwise, transversal and angular dispersion are like the experimental ones (see Figures 6(b)–6(d) and 14(b)–14(d)), but the final median angle assumed by the numerical cylinders is lower than the experimental observation (*ϑ* ≈ 48° while in the experiments *ϑ* ≈ 68°).

Numerical and simulated average trajectories are in good agreement also for the unbalanced cylinder, with major differences downstream of *x* = 2.50 m where the experiments strongly tend to the left flume wall (Figure 15(a)). A slightly larger transversal dispersion is observed for the numerical simulation than for the experiments, while the angular dispersions are comparable despite a considerably lower value of the final median angle (*ϑ* ≈ 38° while in the experiments *ϑ* ≈ 65°, see Figures 7(d) and 15(d)).

#### Quantitative comparison

In order to provide a synthetic and quantitative analysis of the comparison between the experimental and the numerical results, the focus is placed on the average trajectories and on the streamwise, transversal and angular dispersion in the final section available for both experimental and numerical data. The coefficient of determination is computed for the average *x* and *y* positions and for the average angle (for cylinders). In order to estimate dispersion, the variances of the streamwise, transversal and angular position are calculated in the final section, and the experimental and numerical values are then compared using Bartlett's test. Tables 4–7 report the coefficient of determination for the average values and the *p*-values resulting from Bartlett's test for final variances, for each sample and for each flume configuration. Note that the results for all the flume configurations are reported (see the experimental setup and the Supplementary Material for the description of the 1D and mid-2D experiments).

. | Average trajectory . | Bartlett's test . | ||
---|---|---|---|---|

Flow regime . | R^{2}
. _{x} | R^{2}
. _{y} | p-value
. _{x} | p-value
. _{y} |

1D | 0.9999 | 0.9938 | 0.4418 | 0.6034 |

mid-2D | 0.9950 | 0.9032 | 4.8478*10 ^{4} | 0.3212 |

strong-2D | 0.9996 | 0.9095 | 0.7373 | 0.2391 |

smooth-2D | 0.9978 | 0.6724 | 0.6221 | 0.1963 |

. | Average trajectory . | Bartlett's test . | ||
---|---|---|---|---|

Flow regime . | R^{2}
. _{x} | R^{2}
. _{y} | p-value
. _{x} | p-value
. _{y} |

1D | 0.9999 | 0.9938 | 0.4418 | 0.6034 |

mid-2D | 0.9950 | 0.9032 | 4.8478*10 ^{4} | 0.3212 |

strong-2D | 0.9996 | 0.9095 | 0.7373 | 0.2391 |

smooth-2D | 0.9978 | 0.6724 | 0.6221 | 0.1963 |

. | Average trajectory . | Bartlett's test . | ||
---|---|---|---|---|

Flow regime . | R^{2}
. _{x} | R^{2}
. _{y} | p-value
. _{x} | p-value
. _{y} |

1D | 0.9999 | 0.9972 | 0.2859 | 0.8421 |

mid-2D | 0.9956 | 0.9669 | 0.0113 | 0.0862 |

strong-2D | 0.9988 | 0.9886 | 0.3242 | 0.8111 |

smooth-2D | 0.9985 | 0.9252 | 0.4489 | 0.9354 |

. | Average trajectory . | Bartlett's test . | ||
---|---|---|---|---|

Flow regime . | R^{2}
. _{x} | R^{2}
. _{y} | p-value
. _{x} | p-value
. _{y} |

1D | 0.9999 | 0.9972 | 0.2859 | 0.8421 |

mid-2D | 0.9956 | 0.9669 | 0.0113 | 0.0862 |

strong-2D | 0.9988 | 0.9886 | 0.3242 | 0.8111 |

smooth-2D | 0.9985 | 0.9252 | 0.4489 | 0.9354 |

. | Average trajectory . | Bartlett's test . | ||||
---|---|---|---|---|---|---|

Flow regime . | R^{2}
. _{x} | R^{2}
. _{y} | R^{2}
. _{θ} | p-value
. _{x} | p-value
. _{y} | p-value
. _{θ} |

1D | 0.9999 | 0.9502 | 0.9898 | 0.9672 | 0.8498 | 0.7443 |

mid-2D | 0.9994 | 0.9248 | 0.9279 | 0.4953 | 0.1007 | 0.3030 |

strong-2D | 0.9877 | 0.7545 | 0.6699 | 0.1523 | 0.0423 | 0.7237 |

smooth-2D | 0.9997 | 0.9358 | 0.8979 | 0.9183 | 0.0409 | 0.5061 |

. | Average trajectory . | Bartlett's test . | ||||
---|---|---|---|---|---|---|

Flow regime . | R^{2}
. _{x} | R^{2}
. _{y} | R^{2}
. _{θ} | p-value
. _{x} | p-value
. _{y} | p-value
. _{θ} |

1D | 0.9999 | 0.9502 | 0.9898 | 0.9672 | 0.8498 | 0.7443 |

mid-2D | 0.9994 | 0.9248 | 0.9279 | 0.4953 | 0.1007 | 0.3030 |

strong-2D | 0.9877 | 0.7545 | 0.6699 | 0.1523 | 0.0423 | 0.7237 |

smooth-2D | 0.9997 | 0.9358 | 0.8979 | 0.9183 | 0.0409 | 0.5061 |

. | Average trajectory . | Bartlett's test . | ||||
---|---|---|---|---|---|---|

Flow regime . | R^{2}
. _{x} | R^{2}
. _{y} | R^{2}
. _{θ} | p-value
. _{x} | p-value
. _{y} | p-value
. _{θ} |

1D | – | – | – | – | – | – |

mid-2D | 0.9993 | 0.9044 | 0.3370 | 0.6281 | 0.4642 | 0.7418 |

strong-2D | 0.9990 | 0.9578 | 0.8676 | 0.3257 | 0.2988 | 0.0034 |

smooth-2D | 0.9997 | 0.9093 | 0.9319 | 0.3239 | 0.0328 | 0.7143 |

. | Average trajectory . | Bartlett's test . | ||||
---|---|---|---|---|---|---|

Flow regime . | R^{2}
. _{x} | R^{2}
. _{y} | R^{2}
. _{θ} | p-value
. _{x} | p-value
. _{y} | p-value
. _{θ} |

1D | – | – | – | – | – | – |

mid-2D | 0.9993 | 0.9044 | 0.3370 | 0.6281 | 0.4642 | 0.7418 |

strong-2D | 0.9990 | 0.9578 | 0.8676 | 0.3257 | 0.2988 | 0.0034 |

smooth-2D | 0.9997 | 0.9093 | 0.9319 | 0.3239 | 0.0328 | 0.7143 |

Overall, a good agreement between the numerical and the experimental average trajectories is observed. The coefficients of determination are generally higher than 0.9 both for the wooden and the plastic spheres, with only one lower value for the wooden sphere in smooth-2D flow regime. Streamwise and transversal averages are alike also for the cylinders, with only one value below 0.8 in Table 6. The angular position shows a slightly lower precision, especially for the standard cylinder in the strong-2D configuration and for the unbalanced cylinder in the mid-2D configuration.

Regarding the distribution, the numerical simulation does not always provide a good prediction of the final possible range. *P*-values below 0.5, in italics in the Tables, are observed for all the samples and for different flume configurations and occur for the 59% of the observations (22 values over the 37 calculated *p*-values).

#### Semi-congested transport

The numerical simulation of semi-congested transport is performed with the same approach as for the single-body transport. The only difference is that, in this case, collisions among cylinders are possible. The initial conditions are taken from the flume experiments, about 1 s after the release in order to discard the effects of the release mechanism. The numerical mesh includes 16,250 cells, and the running time is about 13 min for 1 s of simulation.

The results of the simulations are the planar position and the orientation of all the cylinders employed (18 for each repetition). Figure 16 shows a frame of the experimental video recording together with the numerical results. To highlight the effect of the collision module, both the results with and without collisions are reported (Figure 16(a) and 16(b), respectively). The coordinates of the cylinders do not change much in the two figures, but they arrangement is different, especially near the second obstacle (around *x* = 2.5 m). In Figure 16(b), the logs overlap and are uniformly distributed while in Figure 16(a), they maintain a certain distance among each other and move in small groups as observed in the experiments (see in particular the four logs at the edge of the group, which are similarly arranged as the group of six logs in the experiments). The implementation of the collision results is thus important to obtain the correct arrangement of the transported bodies.

In order to compare the outcome of the simulations with the experimental data, the final distributions of simulated cylinders are shown in Figure 17. Overall, the cylinders appear more dispersed than in the experiments, both in planar and angular configurations. The streamwise position (Figure 17(a)) presents an average median value of 2.40 m, 10% smaller than the experimental value (2.70 m), while the average range, 1.10 m, is larger with respect to the streamwise diffusion observed in the experiments (0.8 m). Figure 17(b) shows that the transversal position is more dispersed than expected and that the cylinders tend to remain nearer to the left wall of the flume (higher median values observed, 0.12 versus 0.10 m).

Final cylinder orientation (Figure 17(c)) presents an average median value of 65°, which is near to the observed value (61°), confirming the good capability of the model to replicate the average orientation of transported cylinders. On the contrary, also in this case, the distribution is larger than expected (80° versus 70°).

## CONCLUSIONS

An experimental dataset about floating body transport is presented, including different flow regimes, from 1D (see Supplementary Material) to strong-2D, and different samples employed, either spheres or cylinders. The analysis of the experimental data in the smooth-2D flow field shows that, despite the samples shape, they tend to disperse transversally and occupy about 1/3 of the flume width. Cylinders change their orientation, from perpendicular to the side walls to nearly parallel to them, although they maintain an angle of about 65–70° from the vertical axis. Semi-congested experiments are also carried out, under strong-2D flow regime. The limited interactions observed between the cylinders do not affect the behaviour of the floating samples, which flow nearly undisturbed and can rotate reaching the typical orientation observed for single bodies, despite the different flume configuration.

This dataset is employed as a reference for the application of ORSA2D_WT, a 2D numerical model for Large Wood transport. The model follows a dynamic approach, computing the forces acting on the floating body, and includes the reciprocal effect of the samples on the flow (two-way coupled). The comparison of the numerical and experimental results of single-body transport shows that the numerical model can replicate quite well the average trajectory and the average angular displacement of the floating samples for any flume configuration and for any sample shape. Regarding the final dispersion in *x*, *y* and *θ*, when applicable, larger errors are observed, with a mismatch in the final variances in nearly 60% of the variables tested. This indicates that the final dispersion, i.e. the final possible range of positions and orientation, is not well caught by the numerical model.

The good replication of the average trajectory is probably connected to the capability of the model to simulate well the deterministic part of the floating body transport that is the effect of the flow velocity on the cylinder or on the sphere. The comparison of the measured and the computed flow velocities shows that the hydraulic simulation reduces its accuracy when the strong-2D flow is considered, but, on average, its effect on the floating bodies is not too noticeable.

The analysis of variances shows that the mismatch among them is not affected by the flow regime nor by the sample typology, so it should not depend on the accuracy of the average flow field simulation (solution of the SWE in ORSA2D_WT) or on the specific formulation of force computation for different body shapes. It may be ascribed to the fact that the model does not include any fluctuation in the flow field and the only variation in the sample trajectory depends on the initial conditions (position and velocities), while in the experiments, additional changes may derive from flow turbulence and water surface fluctuations.

Focussing on the simulation of semi-congested transport, the only well-replicated value is the median angle of the cylinders, while, in general, higher dispersion is observed both for the simulated planar coordinates and orientation than for the experimental ones. Probably, the inaccuracies already observed in the single-body transport simulations can be exacerbated by the interactions between cylinders, which, although small, certainly lead to a variation of the cylinders' trajectories and orientation. Simulation time for semi-congested transport is 3.25 times higher with respect to single-body transport. The increase is not directly proportional to the increase of the number of cylinders because many operations are performed independently on the number of transported logs. However, when considering a real event that may include hundreds of wooden elements, this parameter needs to be taken into consideration. The adoption of the time optimization strategy and the parallelization of the code would be an important step to guarantee the applicability of the model to real scale events.

To sum up, the model makes it possible to predict the average trajectory and orientation of single-body transport, while the estimation of the distribution of these quantities and the simulation of semi-congested transport are not completely satisfactory with the current formulation. Additional efforts are thus required in order to overcome the limits here highlighted, like the evaluation of the effect of the history term (Basset term) which is currently neglected but may contribute to stabilize the numerical results thanks to the inclusion of the viscous-unsteady effects of the flow on the floating body.

Finally, it is the authors' hope that the dataset here proposed could become a reference for the calibration or testing of the numerical models that aim at the simulation of floating body transport. For this reason, the data will be available for anyone interested.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.