A stem spacing-based non-dimensional model for predicting longitudinal dispersion in low-density emergent vegetation
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Abstract
Predicting how pollutants disperse in vegetation is necessary to protect natural watercourses. This can be done using the one-dimensional advection dispersion equation, which requires estimates of longitudinal dispersion coefficients in vegetation. Dye tracing was used to obtain longitudinal dispersion coefficients in emergent artificial vegetation of different densities and stem diameters. Based on these results, a simple non-dimensional model, depending on velocity and stem spacing, was developed to predict the longitudinal dispersion coefficient in uniform emergent vegetation at low densities (solid volume fractions < 0.1). Predictions of the longitudinal dispersion coefficient from this simple model were compared with predictions from a more complex expression for a range of experimental data, including real vegetation. The simple model was found to predict correct order of magnitude dispersion coefficients and to perform as well as the more complex expression. The simple model requires fewer parameters and provides a robust engineering approximation.
Keywords
Stem spacing Longitudinal dispersion Solute transport 1D modelling Vegetated flows Cylinder arraysIntroduction
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 (D_{x}) 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
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.
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 C_{D} estimated with Eq. (5) (Sonnenwald et al. 2018a). However, Eq. (4) depends on several simplifying assumptions for values of σ_{u}*^{2} and Sc_{t} and is insensitive to C_{D}. 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
Previous studies investigating longitudinal dispersion in uniform emergent vegetation
Description | d (mm) | ϕ | s (mm)^{a} | U (mm s^{−1}) | References |
---|---|---|---|---|---|
Mixed real species | 1–10 | 0.002–0.022 | 7–26 | 14–38 | Huang et al. (2008) |
Artificial random | 6 | 0.010–0.055 | 8–25 | 29–74 | Nepf et al. (1997) |
Carex | 10–55 | 0.002–0.059 | 70–104 | 99–232 | Shucksmith et al. (2010) |
Phragmites australis | 3 | 0.002 | 28–30 | 171–242 | Shucksmith et al. (2010) |
Regular periodic | 4 | 0.005 | 51.9^{b} | 7–50 | Sonnenwald et al. (2016) |
Typha latifolia | 10–19 | 0.013–0.047 | 29–36 | 9–29 | Sonnenwald et al. (2017) |
Artificial random | 6 | 0.010–0.064 | 7–25 | 12–97 | White and Nepf (2003) |
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.
Artificial emergent vegetation characterisation
Description | d (mm) | ϕ | Estimated s (mm)^{a} | s (mm)^{b} | s_{50} (mm) |
---|---|---|---|---|---|
Regular periodic^{c} | 4 | 0.005 | 24.9 | 51.9 | 51.9 |
Pseudo-random | 4 | 0.005 | 24.9 | 29.4 | 23.9 |
Random | 8 | 0.027 | 18.0 | 22.7 | 22.3 |
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.
Determining D _{x} from experimental results
Results and discussion
Figure 6b shows predictions of D_{x} made using Eqs. (4) and (5) compared to measured values of D_{x}. Equations (4) and (5) offer a better prediction of D_{x} 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 D_{x}. The scatter of predictions from Eq. (7) may be due to the use of an estimated value of s rather than a measured value.
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 s_{50}, 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 (D_{x}) 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 D_{x} in vegetation. From this, D_{x} 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 D_{x} 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.
Notes
Acknowledgements
The authors thank Ayuk Merchant, Nathan Wilson, Alexandre Delalande, and Zoe Ball who conducted the laboratory work, and Ian Baylis for his technical support at the University of Warwick. This work was supported by the Engineering and Physical Sciences Research Council (EPSRC Grants EP/K024442/1, EP/K025589/1).
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