Introduction

The fate of pollutants in stormwater is of interest to protect natural watercourses. To predict how pollutants will disperse in these systems, one-dimensional (1D) modelling based on the advection–dispersion equation (ADE) is commonly used (DHI 2009). The 1D ADE is typically given as:

$$\frac{\partial C}{\partial t} + U\frac{\partial C}{\partial x} = D_{x} \left( {\frac{{\partial^{2} C}}{{\partial x^{2} }}} \right)$$
(1)

where C is cross-sectional mean concentration, t is time, U is mean longitudinal velocity, x is the longitudinal coordinate, and Dx is the longitudinal dispersion coefficient (Fischer et al. 1979). U and Dx are required to use the equation predictively.

Natural watercourses often contain vegetation (O’Hare 2015). Furthermore, prior to entering natural watercourses, stormwater is often treated within vegetated sustainable drainage systems (SuDS) to reduce the quantity of pollutants in surface runoff (Woods-Ballard et al. 2015). Applying the 1D ADE to predict pollutant transport therefore requires estimates of longitudinal dispersion coefficient (Dx) within vegetation. While velocities can be estimated from simple hydraulics or numerical models, reliable estimation of the longitudinal dispersion coefficient based on vegetation characteristics is often problematic (Sonnenwald et al. 2017). This paper investigates the prediction of the longitudinal dispersion coefficient in uniform emergent vegetation at low densities. It presents results from new laboratory measurements of longitudinal dispersion and compares new and existing predictors of longitudinal dispersion coefficient to measured values.

Predicting D x in vegetation

Vegetation can be characterised by stem diameter d, solid volume fraction ϕ, frontal facing area a (the vegetation area perpendicular to the direction of flow per unit volume), and mean stem edge-to-edge spacing s. Assuming vegetation may be represented as an array of vertical cylinders, these parameters are related by ϕ = adπ4−1. Tanino and Nepf (2008) provided:

$$s \approx d\sqrt {\frac{1 + 2\phi }{4\phi } - \sqrt \pi \sqrt {\frac{1 - 2\phi }{4\phi }} \frac{{\left[ {1 - {\text{erf}}\left( {\sqrt {4\phi /1 - 2\phi } } \right)} \right]}}{{\exp \left( { - 4\phi /\left( {1 - 2\phi } \right)} \right)}} - 1 }$$
(2)

to estimate s based on stem diameter and solid volume fraction for a random array of cylinders. As there are multiple stem spacing values for a given vegetation configuration, stem spacing may also be characterised by s50, the median stem spacing. While s ≈ s50 when vegetation stem spacing is uniformly or normally distributed, this is not the case when the distribution of stem spacing is asymmetric (e.g. many small stems but few large ones).

Tanino (2012), in a review of mixing in vegetation, suggested that there are primarily three mixing mechanisms contributing to longitudinal dispersion within emergent vegetation: turbulent diffusion; secondary wake dispersion; and vortex trapping. Turbulent diffusion within vegetation is the result of instantaneous velocity fluctuations caused by eddies generated by stems. Secondary wake dispersion is caused by velocity-field heterogeneity resulting in different travel times for particles (differential advection). Vortex trapping is caused by the temporary entrainment of particles in vortices behind stems.

Of these three processes, only secondary wake dispersion contributes significantly to longitudinal dispersion at low densities (ϕ < 0.1). Turbulent diffusion is the primary transverse mixing mechanism at low densities (Tanino and Nepf 2008). However, as total transverse dispersion is typically an order of magnitude lower than total longitudinal dispersion (Sonnenwald et al. 2017), it does not make a significant contribution to longitudinal dispersion. Similarly, dispersion due to vortex trapping is typically much lower than secondary wake dispersion at low densities (White and Nepf 2003). Therefore, secondary wake dispersion should provide a reasonable approximation of total longitudinal dispersion within vegetation at low densities.

White and Nepf (2003) explained that secondary wake dispersion is the sum of two processes: a wake contribution and a gap contribution. The former is caused by reductions in velocity behind stems and the latter caused by increases in velocity between stems. For low densities, the gap contribution is much lower than the wake contribution, and hence, White and Nepf (2003) suggested that longitudinal dispersion in vegetation may be estimated utilising:

$$D_{x} = 2\sigma_{u}^{*2} s^{*} \sqrt {\frac{{Sc_{t} }}{{Sc_{t} + 1}}} Ud$$
(3)

where σu*2 is the variance of the longitudinal velocity field normalised by mean longitudinal velocity squared, s* = (s + d)d−1, and Sct is the turbulent Schmidt number. The variance term represents the level of velocity perturbation caused by stems throughout the velocity field. These parameters can be non-trivial to estimate for a specific vegetation, and so Lightbody and Nepf (2006) combined several parameter estimates and presented:

$$D_{x} = \frac{1}{2}C_{D}^{3/2} Ud$$
(4)

where CD is the drag coefficient.

Calculating CD in vegetation depends either on having very precise measurements of the energy gradient or on having a direct force sensor (Tinoco and Cowen 2013). Both methods are difficult to apply in practice, and as such, it is often preferable to use an estimate of CD, e.g. CD = 1. CD may also be estimated based on the Ergun (1952) expression for pressure drop in a packed column as:

$$C_{D} = 2\left( {\frac{6475d + 32}{{Re_{d} }} + 17d + 3.2\phi + 0.5} \right)$$
(5)

where Red = Udν−1 is stem Reynolds number and ν is kinematic viscosity (Sonnenwald et al. 2018b).

Comparison of Eq. (4) to experimental data suggested that it provides correct order of magnitude predictions of the longitudinal dispersion coefficient in vegetation when using values of CD estimated with Eq. (5) (Sonnenwald et al. 2018a). However, Eq. (4) depends on several simplifying assumptions for values of σu*2 and Sct and is insensitive to CD. A simplified model has previously been developed by Nepf (2012) for predicting the transverse dispersion coefficient in emergent vegetation based on velocity and stem diameter. This paper aims to explore whether a similar approach could be adopted for predicting the longitudinal dispersion coefficient in emergent vegetation.

Previous studies

Several previous experimental studies have investigated the longitudinal dispersion coefficient in emergent vegetation at low density; these studies are summarised in Table 1. The majority of these studies investigated dispersion in real vegetation. As the natural variability of real vegetation does not lend itself to generalisation, additional experiments investigating longitudinal dispersion in artificial emergent vegetation have been carried out by the authors, as described below.

Table 1 Previous studies investigating longitudinal dispersion in uniform emergent vegetation
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Methodology

A 15-m long, 300-mm-wide recirculating Armfield flume was fitted with uniform artificial emergent vegetation, and dye tracing was conducted. Uniform flow was established at velocities of 7–122 mm s−1 by adjusting flume slope and tailgate, and confirmed using point gauges along the length of the flume. Flow depth was set at 150 mm. A diffuser plate was placed directly after the inlet to straighten the flow.

Three cylinder array configurations representing vegetation were examined. These were a regular periodic configuration, a pseudo-random configuration, and a random configuration. The artificial vegetation was constructed by inserting 4-mm-diameter drinking straws or 8-mm-diameter plastic dowels into plastic base plates laid into the bottom of the flume. Two types of base plates were constructed: one with a regular periodic pattern and one with a random pattern. The regular pattern base plate was drilled such that if every hole were filled with drinking straws, the solid volume fraction would be 0.02. However, stems were placed in only 1 out of every 4 holes in a regular pattern, giving ϕ = 0.005. This allowed for the same number of stems to be repositioned on the regular plates in a new pattern with the stem positions chosen at random, giving the pseudo-random vegetation pattern. The random pattern for the second base plate was chosen so that stems would not touch, but allowed for stems to have a machined face (producing a cylinder segment) that could be placed against the flume walls. These three configurations allow the effects of stem diameter, arrangement, and density to be compared. The vegetation configurations are illustrated in Fig. 1.

Fig. 1
figure 1

Vegetation configuration illustrations a regular periodic vegetation reach layout with 4-mm stems, the vegetation pattern repeats every 0.05 m, b pseudo-random vegetation reach layout with 4-mm stems, the vegetation pattern repeats every 2.5 m, and c random vegetation reach layout with 8-mm stems, the vegetation pattern repeats every 1 m

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The characteristics of the artificial vegetation are given in Table 2. Three different stem spacing measurements are provided: a mean value estimated using Eq. (2), the known mean value, and the median stem spacing. This illustrates how stem spacing measurements can vary depending on configuration. The regular periodic vegetation is an expanded data set from the previous Sonnenwald et al. (2016) study, and hence, comparisons are not made to that previous study.

Table 2 Artificial emergent vegetation characterisation
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Rhodamine WT dye tracing was carried out using four mid-channel mid-depth Turner Designs Cyclops 7 fluorometers to record temporal concentration profiles. The four fluorometers formed three 2.5-m test reaches for the 4-mm stems and three 3-m test reaches for the 8-mm stems, giving total experimental lengths of 7.5 m and 9 m, respectively. The first fluorometer was located 4 m downstream of the inlet diffuser plate. The different reach lengths were to accommodate the difference in vegetation pattern repetition. The 4-mm stems covered the full length of the flume. The 8-mm stems covered between 3 m and 14 m from the inlet diffuser plate, with 1 m before and after the first and last fluorometer.

Three injections were carried out at each velocity, giving nine sets of upstream and downstream temporal concentration profiles recorded at 1 Hz. Manual pulse injections were made directly into the inlet pipe of the flume to ensure that the flow was well mixed. Regular periodic vegetation was investigated at target velocities of 7, 10, 13, 17, 20, 30, 40, and 50 mm s−1, pseudo-random vegetation at 7, 10, 13, 17, 20, 30, 40, 50, 60, 90, and 110 mm s−1, and random vegetation at 10, 30, 55, 80, 100, and 120 mm s−1. Photographs of the experimental setup are shown in Fig. 2.

Fig. 2
figure 2

Photographs of experimental setup, dye is from manual injection to water surface for illustration purpose a regular periodic, showing Armfield flume, b pseudo-random, snapshot of continuous injection at U = 7 mm s−1 and c random vegetation, snapshot of pulse injection at U = 11 mm s−1

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Determining D x from experimental results

Values of Dx were found using the following routing solution to the 1D ADE,

$$C\left( {x_{2} ,t} \right) = \mathop \int \limits_{\tau = - \infty }^{\infty } \frac{{C\left( {x_{1} ,t} \right)U}}{{\sqrt {4\pi D_{x} \bar{t}} }}\exp \left( { - \frac{{U^2\left( {\bar{t} - t + \tau } \right)^{2} }}{{4D_{x} \bar{t}}}} \right){\text{d}}\tau$$
(6)

where C(x1, t) and C(x2, t) are upstream and downstream concentration profiles, respectively, \(\bar{t}\) is travel time, and τ is an integration variable (Fischer et al. 1979). The MATLAB (The MathWorks Inc. 2018) lsqcurvefit function was used to minimise the sum of errors squared between the measured and a predicted downstream temporal concentration profile to produce optimised values of Dx and U.

Results and discussion

Figure 3 presents mean longitudinal dispersion coefficient as a function of velocity, showing the expected linear trend of increasing Dx with U for all configurations. The near identical slope for the pseudo-random and random vegetation was not expected as the vegetation arrays are visually quite different and have different characteristics, notably different stem diameters.

Fig. 3
figure 3

Optimised longitudinal dispersion coefficient plotted against optimised velocity, dashed lines are best-fit linear trend lines with RMSE = 5.073 × 10−5, 3.583 × 10−5, and 1.069 × 10−4 for the regular periodic, pseudo-random, and random vegetation, respectively, vertical and horizontal error bars indicate 95% CIs

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Figure 4 presents non-dimensional longitudinal dispersion coefficient (with respect to stem diameter) as a function of velocity. For velocities greater than approximately 20 mm s−1, the results confirm the linear dependency on U shown in Fig. 3. However, at velocities < 20 mm s−1, the dependency is not linear. The lower velocities correspond roughly to Red < 100. Nepf (1999) suggested that turbulent diffusion could be significantly reduced at low velocities, which by definition corresponds with a greatly increased longitudinal dispersion, as observed here. Notably in Fig. 4, measurements of Dx from all three vegetation configurations do not collapse to a single line, contrary to the relationships suggested by Eqs. (3) and (4), both of which normalise by d. This suggests an alternative scale should be used for normalisation.

Fig. 4
figure 4

Non-dimensional longitudinal dispersion coefficient with respect to stem diameter, Dx(Ud)−1, plotted against velocity, vertical and horizontal error bars indicate 95% CIs

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Figure 5 shows Dx(Us50)−1, the non-dimensional longitudinal dispersion coefficient normalised by median stem spacing, plotted with respect to the stem Reynolds number. Stem spacing is used here given that it changes between the regular periodic and pseudo-random vegetation. The random vegetation has slightly higher values of Dx(Us50)−1 than the regular and pseudo-random vegetation, which behave very similarly. Although there is a significant variation in Dx(Us50)−1 at Red < 100, all three types of vegetation have similarly consistent values of Dx(Us50)−1 at Red ≳ 100. This demonstrates, for the first time, the practicality of normalising longitudinal dispersion coefficient in vegetation by stem spacing. The regular and pseudo-random vegetation have similar non-dimensional dispersion coefficient values at all Red, including Red < 100, as would be expected for two types of vegetation with the same d and ϕ.

Fig. 5
figure 5

Non-dimensional longitudinal dispersion coefficient with respect to stem spacing, Dx(Us50)−1 plotted against stem Reynolds number, the dashed line is Eq. (7), vertical and horizontal error bars indicate 95% CIs

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From the results in Fig. 5, a non-dimensional model for longitudinal dispersion based on stem spacing is proposed:

$$D_{x} = 0.60Us_{50}$$
(7)

where 0.60 is the mean value of Dx(Us50)−1 at Red > 100. Equation (7) is shown as the dashed line in Fig. 5. While Eq. (7) is empirical, it is similar to Eq. (4) by analogy, containing s50 instead of s*, and reflecting velocity-field heterogeneity through U and s50 combined.

Figure 6a shows predictions of Dx made using Eq. (7) compared to measured values of Dx from the present study as well as from other the experimental studies listed in Table 1. Mean s, estimated from d and ϕ, was used instead of s50 for the other studies as s50 was not available. As expected from Fig. 5, Dx is accurately predicted for the regular periodic, pseudo-random, and random vegetation configurations. The data from White and Nepf (2003) and Nepf et al. (1997), collected from cylinder arrays, are reasonably predicted. Notably, the Shucksmith et al. (2010) and Sonnenwald et al. (2017) real vegetation are also predicted well.

Fig. 6
figure 6

Predicted Dx for Red > 100 using a Eq. (7) and b Eqs. (4) and (5) compared to measured Dx, line is line of equality

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Figure 6b shows predictions of Dx made using Eqs. (4) and (5) compared to measured values of Dx. Equations (4) and (5) offer a better prediction of Dx for data from Nepf et al. (1997), White and Nepf (2003), and Sonnenwald et al. (2017) than Eq. (7). The vegetation from this study and the vegetation of Shucksmith et al. (2010) are less well predicted. Neither Eq. (7) nor Eqs. (4) and (5) predict the Huang et al. (2008) real vegetation well. Predictions with Eqs. (4) and (5) are biased towards underestimates of Dx. The scatter of predictions from Eq. (7) may be due to the use of an estimated value of s rather than a measured value.

Figure 7 compares RMSE goodness of fit between measured and predicted Dx made using Eq. (7) and measured and predicted Dx made using Eqs. (4) and (5). It shows the overall quality of predictions to be very similar. Equation (7) predicts the correct order of magnitude dispersion coefficient and is a suitable engineering approximation at ϕ < 0.1. It predicts dispersion coefficient in real vegetation as well or better than Eqs. (4) and (5) in three out of four cases, despite the inherent variability of real vegetation.

Fig. 7
figure 7

Comparison of RMSE between predicted Dx and measured Dx for Red > 100 between Eq. (7) and Eqs. (4) and (5), lower values are better

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Normalisation by median stem spacing, rather than stem diameter, appears to be a useful method of incorporating spatial heterogeneity into non-dimensional longitudinal dispersion coefficient, as it is successful in distinguishing between vegetation arrangements. However, it is worth noting that s50, like d, is still a single length-scale characterisation. Most current theory is based on one such characterisation, while in reality multiple length scales are common, as described in Sonnenwald et al. (2017).

Future work should investigate the suitability of Eq. (7) at higher solid volume fractions and for more types of vegetation. Additional work is also needed to investigate how vegetation is described when calculating dispersion coefficient in complex vegetation arrangements, e.g. with varying stem diameter or stem spacing distributions.

Conclusions

Longitudinal dispersion coefficient (Dx) values obtained from dye tracing in artificial emergent vegetation fitted the expected trend of a linear increase with velocity, but could not be normalised using stem diameter. Stem spacing is suggested here for the first time as the appropriate length-scale normalisation for Dx in vegetation. From this, Dx in vegetation can be modelled by a new simple expression dependent on median stem edge-to-edge spacing. This new model showed reasonable performance when applied to other experimental data, including real vegetation. Although a more complex expression from the literature predicts Dx in vegetation equally well, it has multiple implicit assumptions. The new expression presented here gives robust correct order of magnitude estimates of longitudinal dispersion coefficient in vegetation suitable for engineering purposes.