Prediction of the flow resistance in non-prismatic compound channels Open Access

The following symbols are used in this paper:

B

total width of compound channel

b

width of the main channel

h

height of the main channel

H

flow depth

α

width ratio (B/b)

β

relative flow depth [(H–h)/H]

δ

aspect ratio (b/h)

θ

converging angle

X_r

relative distance (x/L)

L

converging length

x

distance between two consecutive sections

S_o

longitudinal bed slope

S_e

energy slope

Q_r

discharge ratio (Q/Q_b)

Q

discharge at any depth

Q_b

bankfull discharge

F_r

Froude number

R_e

Reynolds number

A

flow area

P

wetted perimeter

R

hydraulic radius (A/P)

n

Manning's roughness coefficient

C

Chezy's roughness coefficient

f

Darcy–Weisbach resistance coefficient

a

actual values

p

predicted values

ā

mean of actual values

mean of predicted values

N

number of datasets

k

number of variables

y/D

ratio of water depth to culvert diameter

SVM

support vector machine

GEP

gene expression programming

ANN

artificial neural network

ANFIS

adaptive neuro-fuzzy inference system

MARS

multivariate adaptive regression splines

MLP

multilayer perceptron

GMDH

group method of data handling

MLPNN

multilayer perceptron neural networks

NF-GMDH

neuro-fuzzy group method of data handling

CCNPF

compound open channels with non-prismatic floodplains

PS

prismatic section

NPS

non-prismatic section

ADV

acoustic Doppler velocimeter

CM

Cox method

EBM

Einstein and Banks method

LM

Lotter method

SVR

support vector regression

RBF

radial basis function

BPNN

back-propagation neural network

GPR

Gaussian process regression

HDPE

high density polyethylene

R²

coefficient of determination

MSE

mean squared error

RMSE

root mean squared error

MAE

mean absolute error

MAPE

mean absolute percentage error

SI

scatter index

E

coefficient of efficiency

AIC

Akaike information criterion

The consideration of flow resistance is of paramount importance in compound open channels, as it has a direct influence on the hydraulic behavior and overall efficiency of the channel system. Flow resistance is the phenomenon that arises as water moves inside a channel, resulting in the counteracting force against its motion. This resistance induces changes in velocity and causes a dissipation of energy. The comprehension and precise estimation of flow resistance are of utmost importance in a range of engineering applications, including the design and administration of irrigation systems, drainage networks, and flood control structures. The impact of flow resistance on water conveyance capacity is a significant aspect to consider in compound open channels. The velocity of water is substantially influenced by flow resistance, which therefore impacts the total flow rate that can be sustained by the channel. Engineers have the ability to enhance the efficiency of water distribution for irrigation or flood channeling by designing channels that optimize water conveyance via precise quantification of flow resistance. The significance of flow resistance is crucial in the determination of water surface profiles. The presence of flow resistance induces energy losses that have a direct impact on both the water surface slope and the distribution of water depths inside the channel. The precise forecasting of water surface profiles has significant importance in the fields of hydraulic design, floodplain management, and flood control operations. The comprehension of flow resistance is crucial for engineers as it enables them to develop models and simulations to analyze channel behavior. This, in turn, facilitates the accurate prediction of floods and the development of efficient strategies for flood mitigation planning. Several researchers have conducted studies on the flow, velocity, and shear stress distribution on compound sections (Kar 1977; Knight & Demetriou 1983; Myers 1987; Bhattacharya 1995; Patra 1999; Patra & Kar 2000; Patra et al. 2004; Yang & Lim 2005; Khatua et al.2011a, 2011b; Mohanty 2013). Traditional channel division techniques are used for the assessment of discharge capacity in compound channels. It is postulated that the compound section is delineated by hypothetical interface planes located at the confluence of the main channel and the floodplain. The absence of shear stress in these regions suggests a lack of momentum transmission. The momentum transfer phenomena were first examined by Sellin (1964) via laboratory research. Extensive investigations were conducted in laboratory flumes to study both prismatic and meandering compound channel designs. Nevertheless, a comparison between the compound section data of prismatic compound channels and non-prismatic compound channels revealed significant discrepancies in the estimation of shear force carried by floodplains. These errors can be attributed to the omission of additional mass and momentum transfer, as elucidated by Bousmar & Zech (2002), Bousmar et al. (2004), Rezaei (2006), and Rezaei & Knight (2009). The additional momentum exchange associated with this phenomenon must be considered when doing flow modeling for non-prismatic compound channels. Yang et al. (2005) carried out the estimation of the Manning and Darcy–Weisbach resistance coefficients in composite waterway channels. The experimental results suggest that the Darcy–Weisbach resistance coefficient for a compound channel is dependent on the Reynolds number. Bjerklie et al. (2005) developed equations to estimate in-bank river discharge, which are constructed based on theoretical considerations and calibrated and confirmed using a comprehensive experimental database, rely on the Manning's and Chezy equations. The study conducted by Proust et al. (2006) investigated the phenomenon of asymmetric geometry, specifically focusing on its enhanced convergence rate. An augmented convergence angle of 22° leads to heightened levels of mass transfer and head loss. Cao et al. (2006) introduces novel methodologies for the computation of flow resistance and momentum flux in compound open channels. Yang et al. (2007) demonstrated that the methodologies used to investigate the resistance characteristics of inbank and overbank flows in compound channel consisting of a rough main channel and rough floodplains resulted in substantial errors in evaluating the composite roughness. The flow behavior of skewed, two-stage converging and diverging channels was investigated by Chlebek et al. (2010). The researchers noted an elevation in head losses resulting from the transfer of mass and momentum, as well as the homogeneity of velocity on shrinking floodplains. Additionally, they discovered an augmented velocity gradient on the expanding floodplains. The observed variations in flow distribution may be attributed to alterations in the flow force between the main channel and floodplains within different subsections. Rezaei & Knight (2011) conducted a study on compound channels with floodplains that are non-prismatic. The authors examined the depth-averaged velocity, local velocity distributions, and boundary shear stress distributions at different convergence angles. The analysis of depth-averaged velocities provides insights into the impact of contractions on velocity distributions. Notably, these contractions lead to an augmentation of velocity in close proximity to the walls of the main channel. This effect is particularly pronounced in the latter portion of the convergence reach. Additionally, for cases where the relative depth is high, the lateral flow entering the main channel also contributes to these velocity variations. The study conducted by Yonesi et al. (2013) examines the influence of roughness in floodplains on the occurrence of overbank flow in compound channels with non-prismatic floodplains. The rate of change in velocity, as shown by the gradient, exhibited a rise in magnitude as the angle of divergence saw an increase. The magnitude of shear stress increases with the augmentation of surface roughness on the floodplain. Naik et al. (2017) developed novel mathematical equations to predict the spatial variation of boundary shear stress in a composite channel that includes a convergent floodplain. The effective calculation of relationships for stage–discharge with decreased width ratios in compound channels with converging floodplains has been facilitated by the use of developed mathematical expressions. Das & Khatua (2018) conducted experiments on a composite stream that had symmetric geometry. This stream had sections that expanded and narrowed, resulting in zones of flood with varying depths of relative flow. The experiments were conducted to examine the resistance properties of the flow in terms of different roughness coefficients such as C, n, and f.

Creating models to estimate Manning's roughness coefficient using various methods complicates the analysis of interdependencies among components. These models become intricate and time-consuming, leading to a significant reduction in the time required for experiments and labor-intensive calculations. SVM, GEP, ANN, ANFIS, and the M5 decision tree models are commonly used for predicting flow in composite channels (Parsaie et al. 2017). Parsaie & Haghiabi (2017) employed analytical methods, specifically ANN and MARS, to forecast the discharge capacity of compound open channels, revealing that the MARS model outperformed the MLP model in accuracy. Mohanta et al. (2018) used the GMDH-NN technique to estimate Manning's n in meandering compound channels. The evaluation of GMDH-NN performance is conducted using two distinct machine learning approaches, namely SVR and MARS, using a range of statistical methodologies. The results demonstrate that the GMDH-NN model accurately predicts Manning's n in comparison to the MARS and SVR models. Khuntia et al. (2018) developed an ANN model using key characteristics to predict boundary shear stress distribution in straight compound channels, showcasing the effectiveness of BPNN models across diverse parameter spaces. Das et al. (2020) utilized ANN and ANFIS techniques to predict discharge in compound channels with converging and diverging sections, with the ANFIS model demonstrating superior performance over the ANN model. Naik et al. (2022) proposed a unique equation using GEP to estimate water surface profile in converging compound channels using nondimensional variables. Yonesi et al. (2022) utilized soft computation models such as MARS and GMDH to model discharge in compound open channels with non-prismatic floodplains (CCNPF), showing sufficient accuracy in flow discharge estimation. The effective parameters of the Manning's n in HDPE culverts were investigated by Matin (2022) by the utilization of intelligence methodologies, including GPR and SVM. The results of the sensitivity analysis indicate that the parameters y/D and Q play a crucial role in predicting the Manning coefficient for HDPE culvert models. Bijanvand et al. (2023) employed soft computing models like MLPNN, GMDH, NF-GMDH, and SVM to predict water surface elevation in compound channels with converging and diverging floodplains, where SVM model demonstrated superior performance in testing. Kaushik & Kumar (2023a) utilized ANN, SVM, and GEP to forecast water surface profile in compound channels with converging floodplains, with the ANN model displaying high accuracy. Kaushik & Kumar (2023b) formulated a novel equation for compound channels with converging floodplains using GEP, estimating the boundary shear force conveyed by floodplains. Kaushik & Kumar (2023c) utilized SVM to approximate water surface profile in compound channels featuring narrowing floodplains, showing a high degree of concordance with empirical data and previous research findings.

Historically, research has been undertaken on the flow characteristics in compound channels with non-prismatic floodplains. However, there has been little focus on the flow resistance in non-prismatic compound channel with converging floodplains using creative methodologies. The current investigation focused on a compound channel including smooth converging floodplains. To estimate the flow resistance in terms of Manning's roughness coefficient using both nondimensional geometric and hydraulic parameters, a supervised learning approach known as SVM was used. The efficacy of the suggested models was evaluated using theoretical methodologies and established approaches using statistical analysis.

The subcritical flow regime was seen under various circumstances in the two-stage channel, which had a longitudinal bed slope of 0.001. The estimation of Manning's n value was conducted based on data obtained from both in-bank and overbank flows in the floodplains and main channel. The water supply in the experimental channel is sourced from an underground sump and transferred to an above tank by the system. The water originating from the channel is collected inside a volumetric tank that is equipped with a v-notch. The v-notch was appropriately adjusted in order to accurately quantify the discharge emanating from the experimental channel. Subsequently, it facilitates the return of the fluid to the sump situated underneath. Figure 2 depicts the experimental configuration and the assortment of apparatus used in the investigation. A tailgate was constructed at the downstream end of the flume to regulate the water surface profile and enforce a predetermined flow depth inside that section of the flume. A precision point gauge with a precision of 0.1 mm was used to measure the profile of the water surface at distances of 1.0 and 0.3 m in the PS and NPS, respectively. The researchers used an ADV to measure the mean velocity of the cross-section and the three-dimensional velocity distributions around the wetted perimeter. The measurements were taken at vertical intervals of 2.5 cm and horizontal intervals of 10 cm, as seen in the grid format presented in Figure 1. The data obtained via the use of the ADV were processed and refined using the Horizon ADV program. The measurement of lateral distributions of boundary shear stress was conducted using a Preston tube with an outer diameter of 5 mm at the identical portions where the velocity distributions were examined. The pressure differential was measured using a digital manometer. Subsequently, the calibration equations proposed by Patel (1965) were used to compute the shear stress values.

The determination of Manning's n values is subject to various factors, among which are the width ratio, relative depth ratio, aspect ratio, converging angle, relative distance, longitudinal bed slope, energy slope, discharge ratio, Froude's number, and Reynold's number. Several parameters are utilized to facilitate the application of the model equation to various compound channels. The details of the dataset used in the present investigation are given in Table 1.

Table 1

Statistical characteristics of the dataset used

Range .	.	.	.	.	.	.	.	.	.	.	.
Minimum	0.84	0.053	2	4	0	0.001	0.00003	0.98	0.05	76912.63	0.0081
Maximum	2.00	0.600	2	4	1.00	0.001	0.00012	2.43	0.18	229804.35	0.0162
Average	1.48	0.349	2	4	0.50	0.001	0.00008	1.58	0.12	135006.63	0.0116
Median	1.50	0.358	2	4	0.50	0.001	0.00009	1.48	0.12	130987.07	0.0117
Standard deviation	0.34	0.175	0	0	0.31	0	0.00003	0.47	0.04	34764.67	0.0021

Range .	.	.	.	.	.	.	.	.	.	.	.
Minimum	0.84	0.053	2	4	0	0.001	0.00003	0.98	0.05	76912.63	0.0081
Maximum	2.00	0.600	2	4	1.00	0.001	0.00012	2.43	0.18	229804.35	0.0162
Average	1.48	0.349	2	4	0.50	0.001	0.00008	1.58	0.12	135006.63	0.0116
Median	1.50	0.358	2	4	0.50	0.001	0.00009	1.48	0.12	130987.07	0.0117
Standard deviation	0.34	0.175	0	0	0.31	0	0.00003	0.47	0.04	34764.67	0.0021

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Linear kernel:

Polynomial kernel:

RBF kernel:

Sigmoid kernel:

The SVM model employs parameters C, γ, r, and d. The C parameter governs the balance between optimizing the margin and minimizing the classification error on the training dataset. A lesser value of C leads to a broader margin, which may let some instances to be misclassified. Conversely, a bigger value of C aims to minimize misclassification by sacrificing margin width. The parameter γ is unique to the RBF kernel. It regulates the impact of specific training samples. A decrease in the gamma value results in a more refined decision boundary, characterized by a higher degree of smoothness. Conversely, an increase in the gamma value introduces more fluctuations and irregularities to the border, potentially leading to overfitting. The parameter denoted as ‘d’ is exclusive to polynomial kernels. The parameter determines the order of the polynomial function used for the transformation. The attainment of advanced academic degrees has the potential to result in decision boundaries of greater complexity. However, it is important to acknowledge that this pursuit may also elevate the likelihood of overfitting. The efficacy of the SVM model is contingent upon the caliber of the selected settings for said parameters, in addition to the kernel parameters. The selection of values for the parameters C, γ, and r plays a crucial role in regulating the complexity of the regression model. However, the process of determining the optimal parameters is further complicated by the intricate nature of their selection. Kernel functions are utilized to reduce the dimensionality of the input space and facilitate classification. The refinement of the SVM model is achieved through its exposure to data sets, a process that is analogous to the refinement of other neural network models. The flow resistance in terms of Manning's roughness coefficient (n) of compound channel with converging floodplains was evaluated utilizing a reliable and high-quality experimental dataset. Prior to building the SVM model, the data are partitioned into training and testing sets using a 70:30 ratio in MATLAB R (2019). Table 2 illustrates the various models that were developed to predict roughness coefficient, utilizing distinct kernel functions and kernel scales while maintaining a consistent relationship and proportion of training and testing datasets.

In order to assess the effectiveness of the SVM models, statistical metrics including R², MSE, RMSE, MAE, MAPE, SI, E, and AIC were employed, and their corresponding numerical values are displayed in Table 3. The SVM model five has demonstrated superior predictive performance in estimating the Manning's roughness coefficient of composite waterways with converging floodplains, when compared to alternative approaches such as Das & Khatua (2018), as well as established theoretical methods including the Lotter method for composite roughness computation. This is evident from the highest R² value, lowest MSE value, lowest RMSE value, lowest MAE value, and lowest MAPE value. The SI serves as a standardized measure for the quantification of inaccuracy. A lower SI value is indicative of a higher level of model performance. The SVM model five has a significantly reduced SI value of 0.033 in both the training and testing stages, indicating a minimum level of error. The efficiency measure, represented as E, is assigned a value of 1 when there is a complete correspondence between the model and the observed data. The interval of E extends from negative infinity to 1. The SVM model five has an E value of 0.95 for both the training and testing datasets. The AIC is a widely used criterion in statistical model selection. It is used to choose the best suitable model by evaluating the statistical likelihood function. The model that has the lowest AIC value is regarded as the most optimum model. The SVM model five has the lowest AIC score of −510.754 when compared to the other models proposed. The statistical study of several indices indicates that SVM model five exhibits superior performance compared to other models and established methodologies in forecasting the Manning's roughness coefficient in compound channels with narrowing floodplains.

The study employs SVM to estimate the Manning's roughness coefficient (n) in non-prismatic compound channel with smooth narrowing floodplains. The SVM models are developed using high-quality experimental datasets with varying geometrical and flow variables. The study indicates that the proposed model is influenced by a range of features, such as the width ratio, relative depth of flow, aspect ratio, angle of convergence, relative longitudinal distance, longitudinal bed slope, energy slope, discharge ratio, Froude number, and Reynolds number. In the prismatic portion, the Manning roughness coefficient decreases, and in the NPS, it decreases up to the middle. After then, a precipitous increase in the roughness value is seen. The present study aims to examine the correlation between the Manning's roughness coefficient and the nondimensional geometric and flow variables of a composite waterway with narrowing floodplains. It has been discovered that there is a connection that is not linear between each of the parameters. In comparison to previously developed techniques, the newly created model demonstrates superior performance in terms of statistical parameters for a variety of datasets. In line with the evaluation criteria, the SVM model five could fairly forecast the Manning's coefficient in compound channels with converging floodplains. This was shown by the fact that the technique was successful. Since it had the best R² and the lowest MSE, RMSE, MAE, MAPE, SI, and AIC, the SVM model developed using Gaussian kernel function demonstrated superior performance in the prediction of Manning's n in non-prismatic compound channels. The models proposed in this paper possess practical implications for non-prismatic rivers, such as the River Main in Northern Ireland, the Brahmaputra River in India, and other analogous river systems. The results of this research will be valuable in shaping the design of flood control and diversion structures, leading to a reduction in both economic and human losses.

The authors express their gratitude for the help received from the Department of Civil Engineering at Delhi Technological University in Delhi, India.

All relevant data are included in the paper or its Supplementary Information.

The authors declare there is no conflict.