New dynamic two-layer model for predicting depth-averaged velocity in open channel flows with rigid submerged canopies of different densities

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  • A new dynamic two-layer model, which provides a unified physical basis for vegetated flows with rigid submerged canopies of different densities, is proposed.

  • Two sets of formulae are derived on the basis of genetic programming to predict flow rates for vegetated flows with sparse and dense canopies.

  • On the analogy of rough-wall flows, the two formulae illustrate that relative submergence and canopy density are important in predicting the hydraulic resistance in submerged vegetated flows.

Abstract

The depth-averaged velocity is the commonly used engineering quantity in natural rivers, and it needs to be predicted in advance, especially in flood seasons. A model that can provide a unified physical foundation for open channel flows with different canopy densities remains lacking despite ongoing researches. Here, we use the concept of the auxiliary bed to describe the influence of momentum exchange on rigid canopy elements with varying density and submergence. The auxiliary bed divides the vegetated flow into a basal layer and a suspension layer to predict average velocity in each layer separately. In the basal layer, the velocity profile is assumed to be uniform. In the suspension layer, a parameter called “penetration depth” is applied to present the variations in velocity distribution. We also apply a data-driven method, called genetic programming (GP), to derive Chezy-like predictors for average velocity in the suspension layer. Compared to the hydraulic resistance equation for rough-wall flows, the new formulae calculated by the weighted combination method show sound physical meanings. In addition, comparison with other models shows that the new dynamic two-layer model achieves high accuracy in flow rate estimation, especially for vegetated flow with sparse canopies.

Introduction

The presence of vegetation is of great importance in the overall conveyance capacity of natural channels (Darby, 1999; Stone and Shen, 2002). Numerous studies have been conducted to explore the flow characteristics of vegetated open channels, with many of them focusing on explicit velocity distribution (Huai et al., 2009; Nikora et al., 2013; Tang et al., 2014). However, in natural rivers depth-averaged velocities are used more frequently than distinct velocity profiles in solving practical engineering problems (Li et al., 2015b). Moreover, canopy densities in actual situations are usually variable. This means depth-averaged velocity formulae that have a unified basis for vegetated flow with different canopy densities are still needed. Using rigid cylinders to mimic vegetation, we attempt to develop a new model of depth-averaged velocity in open channel flows with submerged vegetation of different densities. The aim of this model is to explore the physical effects of vegetation stems without the additional biological factors, such as leaves and root systems.

Two major methods, namely, analytical and experimental methods, are currently utilized to address the issue of depth-averaged velocity. The two-layer division approach based on geometric factors is most commonly used (Stone and Shen, 2002; Baptist et al., 2007; Huthoff et al., 2007; Yang and Choi, 2010; Cheng, 2011; Shi et al., 2019). For instance, in the study of Yang and Choi (2010), submerged vegetated flow is divided into two layers, namely, vegetation and surface layers, with the top of the canopy as the boundary, which is shown in Fig. 1(a). In the vegetation layer, the velocity distribution is assumed to be uniform, whereas in the surface layer, a logarithmic velocity profile is adopted. In these cases, the average velocity in the surface layer is calculated by taking the height average scheme. Meanwhile, the depth-averaged velocity in the entire cross section is calculated by using the weighted combination method. Verification by experimental data or simulated data from numerical models shows that these simplified models are generally in good agreement when used to predict depth-averaged velocity in open channel flows with submerged vegetation (Baptist et al., 2007; Huthoff et al., 2007; Yang and Choi, 2010; Cheng, 2011; Li et al., 2015b).

Nonetheless, as stated by Li et al. (2015b), a simple layer division method merely based on geometric factors cannot reflect the turbulence features of vegetated flows, especially when canopy densities change. Luhar et al. (2008) and Nepf (2012) drew attention to the difference in the internal flow mechanisms for vegetated flow with sparse and dense canopies. They demonstrated that the difference depends on the relative importance of bed shear stress and canopy drag force. For sparse canopies, bed shear stress plays a more important role than canopy drag force in the vegetation layer. Thus, the original momentum balance between potential gradients and canopy drag force no longer exists. Hence, the velocity profile in the vegetation layer is close to that shown in open channel flows without vegetation (Fig. 2a and b). For dense canopies, the discontinuity in drag force occurs at the interface of submerged vegetated flow. This phenomenon causes the Kelvin–Helmholtz (K–H) instability, which leads to a region of strong vertical momentum exchange (Fig. 2d). In addition, the transition between sparse and dense canopies takes place at CDahv ≈ 0.1 (Fig. 2c), where CD is the drag coefficient, a is the canopy frontal area per volume, and hv is the height of vegetation (Belcher et al., 2003).

For submerged vegetated flow, the influence of vertical momentum exchange is not limited to the surface layer as it also penetrates into the vegetation layer. This characteristic calls for a layer division method that can show the internal turbulence properties of submerged vegetated flow. Herein, we apply the dynamic boundary based on the concept of auxiliary bed, which is initially proposed by Li et al. (2015b). The boundary divides the flow into an upper layer of vertical turbulent exchange (called the suspension layer) and a lower layer of longitudinal turbulent exchange (called the basal layer), as shown in Fig. 1(b) (Nepf and Vivoni, 2000). For vegetated flow with sparse canopies, this boundary overlaps with the channel bottom because the basal layer of longitudinal momentum exchange vanishes. As for vegetated flow with dense canopies, the dynamic boundary that splits the vertical and longitudinal momentum exchange can easily be observed within the canopies. Therefore, the dynamic division method provides a unified physical basis for vegetated flow with both sparse and dense canopies by effectively representing the turbulence structures.

Applied as a useful formulae derivation tool in previous studies (Baptist et al., 2007; Tinoco et al., 2015; Shi et al., 2019), genetic programming (GP) algorithm is combined with the dynamic boundary to derive the depth-averaged velocity in vegetated flow with rigid submerged canopies. Note that data in this study are collected from flume experiments where canopies are arranged in aligned or staggered forms. The average velocity in each layer is considered separately. In the basal layer, the average velocity is derived from the momentum balance between potential gradients and canopy drag force. In the suspension layer, the average velocity is deduced from a mass of data by using the GP algorithm. Then, data grouping and pre-processing is conducted in terms of the physical basis for the auxiliary bed, which is mainly about the generation and formation of K–H vortices. The maximum dissimilarity algorithm (MDA) is applied to data selection (Camus et al., 2011; Goldstein et al., 2013). The final formulae of depth-averaged velocity for the entire cross section are obtained through weighted combination. The physical meaning and performance of the proposed two-layer model are illustrated in this study.

Section snippets

Genetic programming

GP is a specific evolutionary technique that is frequently used to solve hydraulic engineering problems (Babovic et al., 2001; Huai et al., 2018; Goldstein and Coco, 2014). GP has gradually evolved into a useful and practical method for deriving data-driven predictors since its development by Schmidt and Lipson (2009). In contrast to other techniques, such as linear regression or artificial neural networks, GP is particularly suitable for solving physical problems, since it requires no previous

Results

The depth-averaged velocity in the entire cross section (U) is computed by taking the weighted average of the average velocity in each layer. The equation can be described as follows:U=Us(hs+δe)+Ubv(hvδe)(1λ)H

Then, the formulae for depth-averaged velocity in different situations are obtained.

For condition CDahv > 0.1,U=gHS[2(1λ)3CDvahvδeH3/2+(4.42+0.0105hs+δed)(hs+δeH)3/2]

Meanwhile, for condition CDahv < 0.1,U=Us=gHS(6.26+0.0195H/hvahv)

Nikora et al. (2001) derived the relationship between

Comparison with previous models

The following is a list of previously proposed models for estimating the depth-averaged velocity in the entire cross section.

  • (1)

    Baptist et al. (2007). Baptist et al. obtained the following formula by applying the dimensionally aware GP to the simulated data of the one-dimensional vertical (1-DV) model for submerged vegetation:U=gHS[1g/Cb2+2CDλhv/(πd)+2.5ln(Hhv)]

where Cb is the bed-related Chezy coefficient, which is approximately computed as 60 m0.5s−1 for the smooth bed; and the drag coefficient C

Summary

In this study, we combine the concept of auxiliary bed proposed by Li et al. (2015b) with the GP algorithm to predict the depth-averaged velocity in submerged vegetated flow. We segregate the vegetated flow into the basal and suspension layers while considering the auxiliary bed as the boundary. The average velocity in the basal layer is derived from the momentum balance. For the suspension layer, the GP algorithm is applied to obtain two sets of predictors for open-channel flow with sparse and

CRediT authorship contribution statement

Fan Yang: Conceptualization, Methodology, Software, Validation, Formal analysis, Investigation, Writing - original draft, Writing - review & editing, Visualization. Wen-Xin Huai: Conceptualization, Methodology, Writing - review & editing, Supervision, Project administration, Funding acquisition. Yu-Hong Zeng: Writing - review & editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement

This work is financially supported by the National Natural Science Foundation of China (Nos. 11872285, 11672213, and 51439007). The authors thank V. Pasquino for providing the experimental data to help with the manuscript. Comments and suggestions made by the Editor Dr. D'Odorico and Reviewers have greatly improved the quality of the paper.

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