Research papersPredicting the bulk average velocity of open-channel flow with submerged rigid vegetation
Introduction
An open-channel flow with submerged vegetation is a common flow condition in natural rivers and artificial channels, especially during flood seasons. The interplay between a flow and vegetation results in a specific and complicated flow structure. It has also aroused the interest of hydraulic and hydrology researchers. Accordingly, several researchers (Huai et al., 2009, Nikora et al., 2013, Tang et al., 2014) have conducted detailed experiments and established models to describe the vertical distribution of velocity in an open-channel flow with submerged vegetation. Nikora et al. (2013) stated that flows can be divided into five sub-layers: bed-boundary, uniform, mixing, logarithmic and wake layers (Fig. 1). The bed-boundary layer is usually thin. The velocity in this layer rapidly increases with the distance from the riverbed. In the uniform layer, the drag force balances with the sliding force, causing a relatively uniform distribution of velocity. The mixing layer is the most complicated layer and can be described by a hyperbolic tangent profile (Raupach et al., 1996, Nepf, 2012b, Nikora et al., 2013). In the logarithmic layer, the velocity follows the logarithmic profile. In the wake layer, this profile should be revised with a wake term. However, this kind of velocity distribution is complex. In river engineering, the flood discharge capacity can be evaluated only with bulk cross-sectional average velocity (Ub). As such, people sometimes only care about Ub rather than the detailed velocity distribution. Nonetheless, even only for Ub, predicting is still difficult because the submerged vegetation in an open-channel flow exerts forces and remarkably changes the hydrodynamics of the flow (Su and Li, 2010, Okamoto and Nezu, 2010, Okamoto et al., 2012, Li and Li, 2015, Etminan et al., 2017). Numerous models have been established to estimate the average velocity in open-channel flows with submerged vegetation. Examples of these models are the single-layer (Cheng, 2015, Tinoco et al., 2015, Gualtieri et al., 2018) and two-layer approaches (Stone and Shen, 2002, Baptist et al., 2007, Huthoff et al., 2007, Yang and Choi, 2010, Cheng, 2011, Li et al., 2015b; a review is provided by Pasquino & Gualtieri, 2017).
The single-layer model proposed by Cheng (2015) is derived from the conventional Darcy–Weisbach formula with a revision in the hydraulic radius, which considered the obstruction from vegetation. A relationship between the Darcy–Weisbach coefficient and other parameters, such as the energy slope, submergence, and vegetation density, was established by analyzing the existing data bank. Tinoco et al. (2015) applied genetic programming (GP) and used the Froude number as the target dimensionless parameter in formula searching. Among the alternative predictors obtained by GP, a Chezy-like equation was selected by the authors as the final formula. Gualtieri et al. (2018) compared the performance of different conventional resistance equations when used in vegetated flow with high submergence and argued that the Keulegan equation performs best. These single-layer models are simple but ignore the different physical mechanisms of vegetation-induced drag force and bed-roughness-induced resistance. The conventional flow resistance equations used (Cheng, 2015, Tinoco et al., 2015, Gualtieri et al., 2018) were originally established in flumes without vegetation, in which the friction resistance arises from the bed roughness. However, in vegetated flow, the drag force is often the dominant source of resistance, making the bed resistance negligible (Huthoff et al., 2007). Moreover, there is a huge difference between the physical processes of vegetation-induced drag force and bed-roughness-induced resistance. Thus, directly applying these conventional flow resistance equations and searching for corresponding scales, such as roughness height and friction factors, may need more consideration.
Most of the two-layer models separate the flow with submerged vegetation into the vegetation layer (also called as the resistance layer) and the surface layer by using the top of the vegetation as the boundary (Fig. 1) (Stone and Shen, 2002, Baptist et al., 2007, Huthoff et al., 2007, Yang and Choi, 2010, Cheng, 2011). The average velocities of each layer (Uv and Us) were estimated individually, and the bulk average velocity (Ub) was derived by weighted combination. The mean velocity in the vegetation layer was calculated based on the force balance between vegetation-reduced drag force and the sliding force (Stone and Shen, 2002, Baptist et al., 2007, Huthoff et al., 2007, Yang and Choi, 2010, Cheng, 2011), whereas the average velocity in the surface layer can be estimated with the logarithmic velocity distribution assumption (Baptist et al., 2007, Yang and Choi, 2010), based on similarity considerations (Huthoff et al., 2007) or an equation similar to the Darcy–Weisbach equation (Cheng, 2011).
In this study, Uv and Us are separately estimated. Uv is estimated by analysing the momentum equations (force balance), and Us’s predictor is based on GP. If good predictors for Uv and Us could be obtained, then Ub could be estimated by weighted combination method. Different from the single-layer GP-based model proposed by Tinoco et al. (2015), the model presented in this study separates the flow into the vegetation and surface layers. Then, the drag force is disassociated with bed roughness. Previous studies (Azamathulla and Ahmad, 2012, Tinoco et al., 2015) have shown that the GP algorithm provides accurate predictors without additional assumptions. The performance of this model in terms of average velocity in submerged vegetated flow will be evaluated in this study.
Section snippets
Geometric scales and two-layer approach
The vegetation geography can be described via individual vegetation scales and their density (Nepf, 2012a). Vegetation is categorised in accordance with shape: a circular cylinder and a strip-like cylinder (Nepf, 2012a). For rigid vegetation, circular cylinder arrays have been mostly used to simulate the vegetation for experiments and theoretical analysis (Shimizu et al., 1991, Dunn et al., 1996, Meijer and van Velzen, 1999, Stone and Shen, 2002, Poggi et al., 2004, Murphy et al., 2007, Liu et
Results
After 18 h of evaluating formulae in generations, a series of solutions is obtained by Eureqa and shown in Table 2. A total of 14 solutions are found with a minimum complexity of 1 and maximum complexity of 33. The MAE and mean squared error (MSE) of these solutions could also be found in Table 2:
The non-continuous relationship between the complexity and MAE is shown in Fig. 2. Thus, a stair-like figure (frontier) called Pareto front is drawn in Fig. 2, where
Rationality and stability
To test the performance of Eq. (31) for the training, validation, and test groups, MAE and MSE are calculated and shown in Table 3. The discrepancy ratio (DR) shows the discrepancy between the measured and predicted values (Aghababaei et al., 2017), which is also calculated. The percentage of DR between −0.3 and 0.3 is often used to evaluate the performance of models (Zeng and Huai, 2014, Aghababaei et al., 2017, Huai et al., 2018), which could also be found in Table 3. Among the three data
Summary
Previous studies have applied various formulae for bulk average flow velocity in open channels with submerged vegetation that are deduced with different theoretical considerations. In this study, we employ GP, a data-driven method, in the conventional two-layer approach to craft the mathematical correlations between the f-like coefficient fs and the relevant dimensionless variables. fs is then applied to calculate the average velocity of the surface layer and finally the bulk average velocity
Acknowledgments
This work is financially supported by the Natural Science Foundation of China (Nos. 51439007, 11672213, and 11872285). The authors are very grateful to the anonymous referees for their inspiring comments and suggestions.
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