Diffusion of floating particles in flow through emergent vegetation: Further experimental investigation



[1] In this paper we present the results of a new laboratory investigation aimed at providing a better understanding of the transport and diffusion processes of floating particles (e.g., buoyant seeds) in open channel flow with emergent vegetation. The experiments are designed primarily to study the influence of vegetation density and flow velocity on the relevant interaction mechanisms between particles and vegetation. The aim is also to ascertain the validity of a stochastic model recently proposed by Defina and Peruzzo (2010). We find that (1) the proper definition of plant spacing is given as 1/npdp, with dp being the plant diameter and np being the number of plants per unit area, (2) the particle retention time distribution can be satisfactorily approximated by a weighted combination of two exponential distributions, (3) flow velocity has a significant influence on the retention time and on the efficiency of the different trapping mechanisms, and (4) vegetation pattern and density have a minor influence on the probability of capture and on the retention time of particles. Indeed, the comparison between model predictions and experimental results is satisfactory and suggests that the observed relevant aspects of the particle-vegetation interaction processes are properly described by the model.

1. Introduction

[2] The hydrodynamics of small floating particles in open channel flow through emergent vegetation is a key issue when studying the buoyant seed dispersal by water (nautohydrochory or nautochory) and its role in the structuring of the riparian community. Floating ability of seeds is, in fact, recognized to enhance aquatic seed dispersal [e.g., Nilsson and Danvind, 1997; Van den Broek et al., 2005] and it is thus of great importance in both vegetation dynamics and restoration.

[3] An increasing number of studies and many in situ experimental investigations have focused on this issue: a recent, extensive review is given by Nilsson et al. [2010] which includes a broad discussion on the impact of hydrologic regime, channel and bank morphology, as well as human works on seed dispersal and deposition patterns. Few empirical or semiempirical models have also been proposed mainly aimed at assessing the dispersal distance i.e., the distribution of distances reached by seeds or plant fragments from the point of release. [Riis and Sand-Jensen, 2006; Groves et al., 2009].

[4] All these studies give valuable insight into floating particle dispersal. However, because of the tremendous complexity of the issue, they merely give a qualitative picture of the dynamics that actually govern floating particle transport, diffusion and trapping in the presence of emergent vegetation.

[5] Few studies have addressed floating particle dispersal in the presence of emergent vegetation through controlled laboratory experiments.

[6] Chambert and James [2009] performed laboratory experiments with five different buoyant seeds characterized by markedly different shape, size and density. They used an array of vertical cylinders to mimic emergent plant canopy and found that the main trapping mechanism was through the Cheerios effect whereby floating particles are attracted toward plant leaves by the rising meniscus. They found that trapping frequency increased with increasing stem density and with decreasing particle mass, i.e., particle inertia. They also observed that floating seeds, possibly because of their relatively large size compared to the cylinders diameter, were not often trapped in the wake region behind the cylinders. Since a constant bulk flow velocity (U = 0.16 m s−1) was set for all experiments, the impact of flow velocity, which is known to significantly affect diffusion, was not explored.

[7] Defina and Peruzzo [2010] (hereinafter referred to as Defina and Peruzzo) performed laboratory experiments aimed at providing a better understanding of the relevant particle-vegetation interaction mechanisms responsible for the observed diffusion processes. Two different particles were used in the experiments to mimic buoyant seeds: a wood particle having a diameter of ≈2.5 mm and a relative density of 0.95, and a berry with a diameter of 3.7 mm and a relative density of 0.83; the model plant canopy was the same we use in the present experiments (see section 2). They distinguished different particle-vegetation interaction mechanisms and found that the net trapping mechanism whereby floating particles are trapped by a few leaves of one plant (or, sometimes of two adjacent plants) weaved each other to form a net-like structure, is by far the most effective. They found, in agreement with Chambert and James [2009], that trapping frequency increased with increasing vegetation density whereas the impact of flow velocity was not explored.

[8] Defina and Peruzzo also proposed a stochastic model which includes the relevant, experimentally observed particle-vegetation interaction processes to simulate the transport and diffusion of floating particles in the presence of emergent vegetation.

[9] Since the purpose of the present work is to verify, through specifically designed laboratory experiments, the consistency and validity of most of the assumptions made by Defina and Peruzzo, a brief description of the stochastic model is given below.

[10] The model is one-dimensional and describes particle-vegetation interactions along the curvilinear axis s corresponding to the generic particle trajectory. The particle trajectory is dissected into segments inline image, with inline image being the mean spacing between plants, and the place, along s, where a particle can interact with one leaf (or a few leaves) is referred to as interaction point (Figure 1). The model assumes that the interaction points (one within each segment) are arranged randomly in space with a uniform pdf. In the cases in which a particle interacts with two or more leaves of the same plant, apart from each other, the model assumes one interaction by summing up the single delays.

Figure 1.

Particle trajectory (s) dissected into segments inline image. Open circles denote the interaction points.

[11] When a particle, while being transported by the flow at velocity U0, reaches an interaction point it has the probability Pi of interacting with the vegetation. This interaction leads to either a permanent capture (with probability Pc) or a temporary capture (with probability inline image). Permanent capture occurs primarily through the net trapping mechanism whereby particles are trapped inside the net-like structure formed by few leaves of one plant (or, sometimes of two adjacent plants) weaved each other [Defina and Peruzzo, 2010].

[12] In the case of temporary trapping, the model distinguishes short from long retention time events in order to account for the observed different interaction mechanisms which cause significant particle propagation delay. The main mechanisms responsible for short-time trapping are inertial impaction [Palmer et al., 2004], which occurs when a particle deviates from a streamline because of its inertia and collides with a leaf, and wake trapping which occurs when a particle enters the unsteady recirculation zone behind a plant [e.g., White and Nepf, 2003]. Occasionally, short-time trapping occurs because of the Cheerios effect whereby floating particles are attracted toward leaves by the rising meniscus [e.g., Vella and Mahadeven, 2005].

[13] If the attractive force between a particle and a leaf due to the Cheerios effect overcomes particle inertia, then the particle gets stuck to the leaf. In this case, the particle can escape either because of the leaf vibration induced by the alternate vortex shedding or because the particle is stricken by an energetic turbulent event. This temporary trapping event produces a time delay in the particle propagation of some tens of seconds and it is classified as long retention time event. Accordingly, the authors denote with PL the probability that a particle is trapped for a long retention time (while inline image is the probability that a particle is trapped for a short retention time). The model further assumes that both short and long retention times are random and exponentially distributed with mean retention times TS and TL, respectively. The layout of the model is shown in Figure 2.

Figure 2.

Layout of the model.

[14] On the basis of a few preliminary experiments, which confirmed the reliability of the model, Defina and Peruzzo argued that (1) some features of the diffusion process (e.g., the mean distance a particle travels before being permanently captured by the vegetation) turn out to be independent of the vegetation pattern and density if lengths are properly normalized, (2) the most appropriate length scale is the mean spacing inline image, (3) the different interaction mechanisms, observed in the experiments, are likely to be controlled by local factors including particle characteristics and plant morphology, and (4) turbulent diffusion due to the temporal and spatial heterogeneity of the surface velocity field [e.g., Nepf, 1999; Lightbody and Nepf, 2006], plays a minor role in the longitudinal diffusion process. As in the work by Chambert and James [2009], the impact of flow velocity was not investigated.

[15] The purpose of the present investigation is to gain further insight into some aspects of the floating particle diffusion processes in the presence of emergent vegetation through laboratory experiments. In particular, this work aims to assess the proper length scale by examining the influence exerted by the vegetation density on the observed particle path length distribution.

[16] We also measure retention times actually experienced by particles and verify if the weighted combination of two exponential distributions, as proposed by Defina and Peruzzo, closely fits experimental data.

[17] The impact of flow velocity on the efficiency of the different interaction mechanisms and on the model parameters is also explored.

2. Materials and Methods

[18] We use laboratory experiments to explore the details of the processes that control floating particle transport and diffusion in the presence of emergent vegetation. Observational data are then used not only to strengthen and advance our knowledge, but also to validate and improve the stochastic model proposed by Defina and Peruzzo, which is likely to include all the relevant aspects of floating particle-vegetation interaction processes In particular, we focus on the basic interaction mechanisms between floating particles and vegetation stems and leaves (i.e., inertial impaction, wake and net trapping, and trapping due to the Cheerios effect) which affect particle propagation by promoting propagation delay, permanent captures and diffusion. We also focus on the impact of flow velocity and of vegetation pattern and density on the efficiency of these mechanisms.

[19] The experiments are carried out in a 6 m long, 0.3 m wide tilting flume. Water is recirculated through the channel via a constant head tank that maintains steady flow conditions. A magnetic flowmeter accurately measures the flow rate. Bed slope and a downstream weir are adjusted to achieve uniform flow conditions with a water depth of 0.1 ± 0.002 m.

[20] The model plant canopy consists of plastic plants inserted into a 3.0 m long, perforated Plexiglas board which covers the middle part of the flume (see Figure 3). The plastic plants are 0.15 m high and are composed of approximately nL = 120 leaves. Leaves have an elliptical section with the major diameter d ≈ 2 mm and the ratio of minor to major axes of ≈0.7. Plants are arranged randomly with plant density in the range of inline image plants m−2 to inline image plants m−2; plants are arranged in a staggered pattern for inline image plants m−2. The plant diameter, defined as the diameter of the circle that includes all points where the leaves of one plant pierce the free surface, is inline image m. To mimic buoyant seeds we used small wood cylinders having equal diameter and height of 3 mm and a relative density of ≈0.7.

Figure 3.

Overall view of the flume with plastic plants used in the experiments: np = 55.6 plants m−2, random pattern.

[21] Particles are released all along a cross section approximately 10 cm upstream the vegetated reach. Particles are allowed to dry before being released into the flow and they are released one at a time to avoid the formation of clusters. Indeed, we assume that a particle is permanently captured if retention time is greater than 600 s, consistently with the previous study.

[22] For each vegetation density and bulk flow velocity approximately 400 particles are individually released and monitored. Most of particle runs are recorded with a camera mounted on a moving carriage, supported by a pair of rails along the flume and driven by hand. Recorded frames (frame rate is 12.5 Hz) are then extracted and analyzed to track particle trajectory, to determine particle velocity, and to measure the duration of temporary trapping events (see Table 1). Accuracy in reconstructing instantaneous particle position is sufficiently good (particle position is determined with an error of ±1 mm), however results give a reliable picture of particle path characteristics. The distance traveled by each particle before being permanently captured is also measured. Direct observation and video analysis also allows us to recognize the relevant aspects of the interaction between floating particles and vegetation and the mechanisms responsible for the temporary and the permanent trapping of particles by plants.

Table 1. Summary of Experimental Conditions and Measured Parameters for the Two Sets of Experimentsa
Experiment SetExperimentsU (m s−1)np (m−2)Video RecordingbNumber of RunsP(X > L)Retention TimeTrajectoriesArrival Time
  • a

    Each set is composed of a group of experiments with a unifying physics: in set A, flow velocity is held constant and vegetation density varies in the range of 20–86.7 plants m−2; in set B, flow velocity is varied between 0.033 and 0.167 m s−1, and vegetation density is held constant. X denotes measured data.

  • b

    Only a few runs are video recorded in order to track particle trajectory.

  • c

    Measured by a stopwatch.

AA1 20.00 450X   
 A2 27.78 400X   
 A3 36.67 400X  X
 A40.03346.67 400X   
 A5 55.56X400X X 
 A6 70.00 400X   
 A7 86.67 400X X 
BB10.033 X400XXXX
 B20.050 X340XXXX
 B30.067 X434XXcXX
 B40.08386.67 400X   
 B50.100  400X   
 B60.133 X424XXXX
 B70.167  400X   

[23] Two sets of experiments are performed (see Table 1). Set A is designed to explore the impact of vegetation density on the type and frequency of the different interaction mechanisms. In set B we study the impact of flow velocity on the characteristics of the diffusion process. With data of set B we also explore the adequacy of some model assumptions such as the statistical distribution of residence time.

3. Results and Discussion

3.1. Vegetation Spacing

[24] The proper specification of the (global) length scale inline image is important to assess the model performance. In fact, as argued by Defina and Peruzzo, most of model parameters turn out to be independent of vegetation pattern and density by normalizing lengths to the proper length scale.

[25] The problem has two intrinsic length scales; these are the diameter of the bush dp and the mean center-to-center spacing between adjacent plants inline image, with np being the number of plants per unit area. Other length scales such as the leaf diameter d or the size of the recirculation zone behind leaves are mainly related to turbulent diffusion [White and Nepf, 2003]. Through an analysis of the particle velocity obtained from the reconstructed trajectories, we find that turbulent diffusion coefficient lies in the range between 10−5 and 2 × 10−4 m2 s−1. Since Defina and Peruzzo showed that with such a small diffusion coefficients, turbulent diffusion negligibly contributes to the overall diffusion process, then these length scales are not considered in the present analysis.

[26] Defina and Peruzzo argued that the most appropriate length scale is the spacing inline image. However, this guess was based only on two sets of data with a moderately different vegetation density (i.e., inline image plants m−2 and np = 56 plants m−2). In order to ascertain the most significant length scale we perform suitable experiments (set A in Table 1) in which only vegetation density is changed.

[27] In these experiments we reconstruct and analyze some particle trajectories in order to see if and how they are affected by the presence of the plants. To analyze particle trajectory, we introduce the function r(s) which measures the distance from a point along the trajectory to the nearest vegetation center as shown in Figure 4, and we define the mean distance rm as

display math

where X is the path length.

Figure 4.

Schematic of a particle trajectory with notation defining r(s).

[28] We then introduce the distance R as the average of rm computed over an infinite number of paths and define R0 to be the distance R when paths are straight lines. Interestingly, the same value R0 is found if paths are not straight lines but they are random. Accordingly, if paths are not affected by the presence of plants, i.e., they are random, then their mean distance R should be close to R0. In the case of staggered plants disposition, if a particle follows a path zigzagging its way through the plants as shown in the bottom right corner of Figure 5 (e.g., because of the channeling effect), then a value for R greater than R0 is found. Contrarily, if a particle path joins the centers of adjacent plants as shown in the top left corner of Figure 4, then a value for R smaller than R0 is found. We can then assume that the ratio R/R0 measures the randomness of a particle path. The analysis can be extended to random plants disposition. In this case the limit values of R in Figure 5 are average values.

Figure 5.

Plot of rmr (and R0) versus rms (and R). Points along the line of perfect agreement indicate that the path is random. The theoretical values of R = R0 for inline image plants m−2 and inline image plants m−2 are indicated: the bar length represents one standard deviation ( inline image). Rmax is the value of R when plants are arranged in a staggered pattern and particle trajectory is that shown in the bottom right corner; Rmin is the value of R when particle trajectory is that shown in the top upper corner. Open circles denote (rms, rmr) points for inline image plants m−2 (present experiments). The plot also includes the results of experiments performed by Defina and Peruzzo with different particle types: particle A, inline image plants m−2 (gray circles); particle A, inline image plants m−2 (gray diamond); and particle B, inline image plants m−2 (black triangles).

[29] Since we have a small number of measured paths, a less rigorous procedure is adopted to ascertain whether or not vegetation controls particle trajectory. For each measured path we compute the mean distance rm (rms). We then fit the measured path to a straight line and compute the mean distance of plants from this line (rmr). Figure 5 shows the computed points (rms, rmr) for different vegetation pattern and density and different flow velocity. Since all points cluster about the line R = R0 with a small scatter compared to the theoretical standard deviation we argue that particle paths are random and weakly controlled by plants. This behavior is possibly due to the high porosity that characterize the plants used in the experiments. Plant porosity, here defined as inline image (with nL the number of leaves of one plant), is in fact large and comparable to the overall canopy porosity given as inline image. In addition, random distribution of interaction points (i.e., points where leaves pierce the free surface) is likely to prevent the formation of preferential flow paths and the occurrence of channeling effect.

[30] For a straight path (or, equivalently, for a random path) the mean spacing is given by inline image [White and Nepf, 2003]. Therefore, we can safely assume that this spacing is the proper length scale, at least when vegetation is highly porous.

[31] Some ambiguity however arises in defining the spacing this way since the plant diameter cannot be precisely defined or easily determined. However, as we show below, possible small errors in estimating dp can be compensated by suitable tuning of the interaction probability Pi.

[32] In this series of experiments (set A in Table 1), in which only vegetation density is changed, we also measure the distance X traveled by particles before permanent capture, whose distribution is given as

display math

where inline image is the particle mean path length before permanent capture, while inline image is commonly referred to as retention coefficient [e.g., Riis and Sand-Jensen, 2006].

[33] By fitting experimental data to equation (2) we determine the mean path length which is plotted against vegetation density in Figure 6. The results suggest that mean path length goes as the inverse of vegetation density (solid line in Figure 6).

Figure 6.

Mean path length as a function of vegetation density (set A). Distance λ and vegetation density np are normalized by the plant diameter dp. Since 1/npdp is the mean spacing between two plants, a particle travels, on average, a distance of 3.6 times the mean plant spacing before a permanent capture occurs.

[34] It is interesting to observe that, when the probability Pc is moderately small, the mean path length can be approximated as

display math

[35] Moreover, we can safely assume that the probability Pc depends on local factors (i.e., on particle and vegetation characteristics, and flow velocity) and it is weakly affected by the vegetation pattern and density. Accordingly, and from equation (3), the ratio inline image, which is the mean spacing between two successive interactions, has the same trend as that of inline image. Since both inline image and inline image go with the inverse of np we argue that the probability Pi is likely to be independent of vegetation pattern and density, in agreement with Defina and Peruzzo.

[36] It is worth noting that any error in evaluating inline image (e.g., because of an incorrect evaluation of the plant diameter) can be compensated through the tuning of Pi.

3.2. Retention Time Distribution

[37] Defina and Peruzzo assumed in their model that retention time is distributed according to the sum of two exponential distributions weighted by the probability PL that a particle experiences a long retention time event:

display math

where TS and TL are the short and long mean retention times, respectively. The reason for assuming two different time scales stems from the different interaction mechanisms responsible for particle propagation delay observed in the experiments. The reliability of the assumption was indirectly verified through the comparison of measured and modeled arrival time distribution. However, retention time distribution was not directly measured.

[38] In this work we use video-recorded paths of experimental set B to quantitatively evaluate retention times. Frame by frame analysis of the recorded paths allows us to measure the time that each particle spends at rest because of a temporary trapping event.

[39] The cumulative probability P(T > t) is then estimated on the basis of the relative cumulative frequency as

display math

where i is the number of measured data T with a value smaller than the reference value t, Ni is the total number of measured data, and N0 is the guessed number of temporary trapping events that, for being too short, cannot be measured. In fact, we cannot accurately measure retention times shorter than about 0.25 s, the frame rate being 12.5 s−1.

[40] Parameters TL, TS, PL, and N0 are then determined by fitting measured retention times to equation (4) according to the procedure outlined in Appendix A (see Table 2).

Table 2. Summary of Experimental Conditions and Results for Video-Recorded Experiments of Set B
 U (m s−1)
Nip10221175 6849

[41] The results are plotted in Figure 7 which also includes the data measured with a stopwatch while visually observing traveling particle behavior. In this case, retention times shorter than 2 s were not considered in the analysis because as a result of the measuring technique, they are likely affected by large uncertainty. Since shorter retention time events are lacking in this case, the chosen value for N0 is rather uncertain because of the long extrapolation involved.

Figure 7.

Experimental probability distribution inline image (symbols) compared to equation (4) (solid lines). The inset shows the linear relationship between bulk flow velocity and the probability PL.

[42] The experimental data are fitted reasonably well by equation (4). We also explore the possibility of using different probability distribution functions to fit the measured data. For example, we find that the Weibull distribution performs very well when bulk flow velocity is large (e.g., U = 0.133 m s−1); however, it largely fails in fitting slow flow data.

[43] It is worth noting that mean retention time Trest increases with increasing bulk flow velocity (see Table 2). This experimental result is rather surprising and far from being intuitive since one would expect the opposite. In fact, interaction mechanisms like the net trapping and the Cheerios effect do not depend on flow velocity whereas particle inertia increases with flow velocity. A possible explanation of this behavior is that at slow flow velocity only weak and short-lasting interaction events produce a temporary trapping, whereas stronger interactions produce permanent capture (i.e., events which are not considered in the analysis of retention time distribution). This is the reason why the probability PL decreases with decreasing flow velocity. The behavior is also consistent with the proposed model in which the probability Pi filters out all “interaction events” characterized by a retention time so short that it can be regarded as zero. In fact, at each interaction point, particles have the probability 1 − Pi of experiencing a trapping event characterized by a negligibly short retention time. The proposed model is thus formally equivalent to a model in which particles actually interact with vegetation at each interaction point and the retention time distribution is given as

display math

[44] Equation (6) suggests that the overall retention time (i.e., the total time a particle spends at rest during one run) decreases with increasing flow velocity provided that the probability Pi decreases rapidly enough with increasing flow velocity.

[45] We use experimental data of set B to estimate how Pi and Pc, separately, vary with flow velocity. Using the measured path length Xi of the ith experimental run we compute the total number of interaction points Nip as

display math

where int( ) denotes the integer part of a real number and Nruns is the number of runs (see Table 1). Given the number of permanent captures (Nc) and the number of measured (Ni) and guessed (N0) temporary interaction events (see Table 2), the probability Pi is estimated to be

display math

[46] The probability Pc is then computed from inline image. The behaviors of probabilities inline image, Pi, and Pc as a function of bulk flow velocity, as well as that of inline image are plotted in Figure 8; the numerical values are given in Table 3.

Figure 8.

Mean path length and probabilities Pi, Pc, and PiPc as a function of bulk flow velocity.

Table 3. Summary of Experimental Conditions and Results for Experiments of Set B
 U (m s−1)
λ (m)
Pi0.7980.5690.32a  0.106 
Pc0.4990.504   0.494 
U0 (m s−1)0.0460.0600.072  0.133 

[47] Figure 8 shows that the probability Pi actually decreases with increasing flow velocity. We speculate that, when flow velocity is (relatively) high then the Cheerios effect, alone or in combination with inertial impaction or wake trapping, is not able to determine a “significant” interaction event. Accordingly, the distance inline image between two successive interactions increases with increasing velocity. On the contrary, permanent trapping events, which are mainly due to the net trapping mechanism, are moderately affected by flow velocity, and the probability Pc remains fairly constant. However, at very low flow velocity, the Cheerios effect is sufficient to overcome friction drag and particle inertia, and both Pi and Pc are expected to approach unity as U goes to zero (Figure 8).

[48] Inspection of recorded paths, as well as visual observation, allows us to evaluate the frequency of occurrence of the different mechanisms responsible for permanent capture events, at different flow velocity and vegetation density. Figure 9a shows that the Cheerios effect is effective only at slow flow velocity whereas the net trapping mechanism, enhanced by the Cheerios effect, is by far the most frequent permanent capture mechanism also at moderately slow flow velocity. We also observe that, at slow flow velocity, a small fraction of permanent captures (≈5%) occurs because particles get stuck to the channel wall through the Cheerios effect. In fact, near the wall the velocity is very small and the drag force is not sufficient to reentrain particles into the main flow. These events are not included in the present analysis. Figure 9b compares the efficiency of the different permanent capture mechanisms at different vegetation densities. No evident trend can be observed further confirming that particle-vegetation interaction process is a local one, marginally affected by vegetation pattern and density.

Figure 9.

Efficiency of the observed permanent capture mechanisms as a function of (a) flow velocity ( inline image plants m−2) and (b) vegetation density (U = 0.033 m s−1). Net trapping (black) and Cheerios effect (gray).

[49] For each particle run we also compute the mean particle velocity inline image, tX being the time required for a particle to travel the whole path length, X. The mean particle velocity is distributed according to a bimodal probability density function (Figure 10) thus reinforcing the idea that the diffusion process is governed by two, dramatically different time scales. Note that the number of temporary trapping events inline image is, on average, less than two in the present experiments. Moreover, since inline image, most of the events are short-time captures, whereas in the few remaining events particles experience (at least) one long-time trapping event, characterized by a mean retention time inline image s which is much longer than the time a particle takes to travel its trajectory when only short-time trapping events occur. This is the reason why the mean particle velocity uX is extremely slow when one (or, less frequently, more than one) long-time interaction event occurs, whereas in the absence of long retention time events the mean velocity uX is just smaller than the bulk flow velocity.

Figure 10.

Probability density function of mean particle velocity uX for inline image plants m−2 and different bulk flow velocities. The inset shows the plot of average mean particle velocity UX versus bulk flow velocity U.

[50] Figure 10 also shows that the second peak is skewed to the left indicating that the diffusion process is not Gaussian even excluding long-time trapping events.

3.3. Arrival Time Distribution

[51] Using video-recorded paths (experiments A3 and experiments of set B), we also determine the time Ta spent by particles to reach some fixed cross sections and we use these data to construct the cumulative arrival time distributions. The experimental results are compared to the model predictions in Figure 11. Importantly, the values for the model parameters are not determined through a calibration procedure. In fact, probability PL, and mean retention times TS and TL, whose values are given in Table 2, are determined through a best fitting regression procedure on the measured retention times; probabilities Pi and Pc have been determined as discussed in sections 3.1 and 3.2 (see equations (8) and (3)).

Figure 11.

Cumulative arrival time distributions: comparison between experimental results (symbols) and model predictions (solid lines).

[52] The velocity U0 is the only parameter we adjust in order to fit the experimental data. We observe that in the present experiments (as well as in the experiments discussed by Defina and Peruzzo) the assessed velocity U0 is, on average, just slightly greater than the bulk flow velocity, and that the difference inline image decreases with increasing of U (see Table 3). We speculate that the relatively large surface velocity U0 at slow bulk flow velocity is mainly determined by the experimental arrangement. In fact, at the downstream end of the flume is a weir that concentrates the flow at the free surface; this high-velocity layer slowly grows upstream and expands over the whole flow depth. However the distance upstream from the weir required for a significant vertical mixing is relatively large and increases with decreasing bulk flow velocity.

[53] Indeed, the presence of this high-velocity surface layer is possibly the reason why, at low bulk flow velocity, particle velocity ux in Figure 10 extends well beyond U.

[54] Figure 12 compares the model predictions to the experimental results for experiments A3 and B1 characterized by the same bulk flow velocity of 0.033 m s−1 and different vegetation density. The cumulative arrival time distributions are computed and measured at the same non dimensional positions inline image so that, according to the model, the shape of the curves should be exactly the same. However, since the absolute position x of the monitored cross sections are different (see Figure 12), data of experiments B1 are accordingly delayed in time. Experimental data of the two series actually do not overlap perfectly: the sharply rising limb of the curves for experiments A3 have a smaller inclination. This is possibly due to turbulent diffusion, which acts over the greater absolute distances traveled by particles in this case.

Figure 12.

Cumulative arrival time distributions: comparison between model predictions (solid lines) and experimental results for experiments B1 (open symbols) and A3 (solid symbols).

[55] The comparison shown in Figures 11 and 12 indeed serves to provide an indication of whether or not the model includes all the relevant mechanisms responsible for the observed particle dispersion. Therefore, in judging the results one should keep in mind that the computed straight rising limb of the cumulative arrival time distributions, particularly evident in Figures 11a and 11c, is determined by particles that reach the monitored cross sections without interacting at all with the vegetation. The comparison could be largely improved by introducing in the model a weak turbulent diffusion, with a diffusion coefficient of the order of 10−4 m2 s−1. Moreover, the model retention time should include the delay due to particle acceleration after the sharp slowdown determined by an interaction event (see Defina and Peruzzo) whereas TS and TL given in Table 1 are the mean time intervals a particle actually spends at rest. Finally, the average number of interaction events experienced by particles in each run is small and the experimental paths cannot be safely regarded as realizations of a purely random process. At low flow velocity (U = 0.033 m s−1 and U = 0.050 m s−1) and np = 86.67 plants m−2, the mean path length inline image = 0.22 – 0.28 m is comparably small to the mean center-to-center spacing between plants of 0.11 m. Therefore, the experimental arrival time data are not statistically suitable enough to unambiguously serve as a basis for a definitive comparison with the model predictions. Indeed, the discrepancies shown in (Figure 11) can be partly ascribed to the irregular morphology of the plants (see Figure 3).

[56] In view of these considerations, we can say that model predictions compare favorably with experimental observations, and that the model actually includes, and correctly describes, all the relevant aspects of the investigated diffusion process.

4. Conclusions

[57] In this work the focus was on the effects of vegetation density and flow velocity on the interaction processes between small floating particles and emergent vegetation in an open channel flow, through laboratory experiments.

[58] We showed that the characteristic length scale of the problem is inline image. This is actually true when plants are characterized by a high porosity, and channeling effect is thus negligible. To further expand our knowledge, it would be interesting to investigate the case of small or even zero porosity, as in the case of rigid cylinders used to mimic, e.g., Phragmites).

[59] The experimental results confirmed that the model parameters Pi, Pc, PL, TS, and TL do not actually depend on the vegetation configuration and density, whose impact is controlled by the average spacing inline image. We also showed that the retention time distribution can be approximated by a weighted combination of two exponential distributions.

[60] Flow velocity, on the other hand, has a significant influence on model parameters whose behavior reflects the relative importance of the Cheerios effect (which does not depend on flow velocity) to drag and inertia. Surprisingly, the experiments showed that mean retention time (for each temporary interaction event) increases with increasing bulk flow velocity. This is because the Cheerios effect becomes less and less effective as the flow velocity increases, and most of temporary trapping events are by a weak net trapping mechanisms characterized by longer retention times. However, because of the weakness of the Cheerios effect, the number of interaction events reduces with increasing flow velocity and, on the whole, the time spent by a particle to travel a given distance decreases, as expected, at higher flow velocities.

[61] This study has confirmed that the proposed model is able to describe floating particle dispersal in the presence of emergent vegetation. However the evaluation of model parameters still requires a well designed calibration procedure.

[62] A next step of the research should address the issue of finding relationships between model parameters, flow velocity, and particle and vegetation characteristics. One way could be through the use of mathematical models able to solve the detailed near-free-surface velocity field in the presence of stems or leaves piercing the free surface, and the dynamics of floating particles including particle-leaf interaction processes.

Appendix A:: Model Parameters Estimation

[63] The procedure adopted to estimate the number of lacking data N0 in equation (5) and to assess model parameters PL, TS, TL is here detailed. To this aim we use measured retention times ti, which are sorted in increasing order to form a set of pairs inline image, i = 1 to Ni.

[64] As a preliminary step, we compute a first guess for N0 using a subset of pairs inline image containing only the shortest retention times ti ( inline image 0.8 s). Combining equations (4) and (5), we obtain

display math

[65] The exponential functions in the above equation are approximated with a first-order Taylor expansion about t = 0; moreover, since TL is much greater than TS, we neglect inline image compared to inline image and write

display math

which is rearranged to read

display math

[66] N0 is then given as the intercept of the above straight line (linear regression).

[67] The second step then finds a first approximation for TL. We use a subset of pairs inline image containing only the longest retention times ( inline image 10 s). Note that, since TS is expected to be of the order of 1–2 s then inline image turns out to be negligibly small when inline image 10 s. We then write

display math

[68] We fit experimental data to the above equation and find TL and PL.

[69] The last step considers the full set of data. Equation (A1) is rewritten as

display math

and TS is computed as

display math

[70] We then compute the determination coefficient R2 using the full set of data and use a trial and error approach to fine tune model parameter in order to maximize R2.


[71] We wish to acknowledge Paolo Rachello, Pierfrancesco Da Ronco, and Fabrizio Benetton for their contribution to the experimental investigations. We also thank Chris James for his comments on the present work.