Assessing a Smart Technology versus a Mathematical model to Estimate the Transverse Shear Stress Distribution in a Rectan- gular Channel

One of the most important subjects of hydraulic engineering is the reliable estimation of the transverse distribution in rectangular channel of bed and wall shear stresses. This study makes use of the Tsallis entropy, Genetic Programming (GP) and (ANFIS) methods to assess the shear stress distribution (SSD) in rectangular channel. To evaluate the results of the Tsallis entropy, GP and ANFIS models, laboratory observations were used in which shear stress was measured using an optimized Preston tube. This is then used to measure the SSD in various aspect ratios in the rectangular channel. To investigate the shear stress percentage, 10 data series with a total of 112 different data for were used. The results of the sensitivity analysis show that the most influential parameter for the SSD in smooth rectangular channel is the dimensionless parameter B/H, Where the transverse co-ordinate is B, and the flow depth is H. With the parameters (b/B), (B/H) for the bed and (z/H), (B/H) for the wall as inputs, the modeling of the GP was better than the other one. Based on the analysis, it can be concluded that the use of GP and ANFIS algorithms is more effective in estimating shear stress in smooth rectangular channels than the Tsallis entropy-based equations.


Introduction
Knowledge of boundary shear stress is necessary when studying sediment transport, flow pattern around structures, estimation of scour depth and channel migration. The determination of boundary shear stress, i.e. at the wall and bed depends on the channel geometry and its associated roughness. Various direct and indirect methods have been extensively discussed in experimentally measure the wall and bed shear stresses in channels with different cross sections [1][2][3][4][5]. Bed shear stress can be estimated based on four techniques (1) bed slope product nents, respectively and 20 . 0 1  C methods of genetic programming (GP) and (ANFIS) were examined to determine the precision of these models in estimating bed and wall shear stress. These three methods are compared with each other and with the experimental results of Lashkar-Ara and Fatahi [9]. This study aimed at using Tsallis entropy method to predict the SSD in smooth rectangular channel. The results of the Tsallis entropy, GP and ANFIS methods compared with experimental results of Lashkar-Ara and Fatahi [9]. Although this analysis was performed in parallel with Sheikh Khozani and Bonakdari [17] research, it can be said in a practical contrast that the data used in this study is based on the measurement of shear stress using the optimal diameter of the Preston tube, which was designed by Lashkar-Ara and Fatahi [9], so the comparison of findings is more precise and less uncertain.

Data collection
Information on the SSD has been collected in the Lashkar-Ara and Fatahi [9] experiments of a smooth rectangular channel, performed in a flume 10-meter long, 60 cm wide and 70 cm high. All measurements were performed in the range of 11.06 to 102.38 liter per second flow rate. Flow rate variations led to observable changes in water depth ranging from 4.3 cm to 21 cm and the aspect ratio of 2.86 to 13.95. The values of static and total pressure difference in various aspect ratios of B/H were measured and reported using pressure transducer apparatus with a capacity of 200 mill bar and 50 Hz measuring frequency. In order to create uniform flow condition and to match the hydraulic gradient with the flume bed slope a weir at the end of the flume was installed. The notation used for a smooth rectangular channel conduit is illustrated in Figure 1. Figure 2 shows the experimental setup.
where is the average wall shear stress, is the average bed shear stress, ? ? is the density, is the kinematic viscosity, g is the gravity acceleration, V is the flow velocity, H is the flow depth, B is the Flume floor width, Sw is the Water surface slope, is the roughness height, (Re) is the Reynolds number and (Fr) is the Froude number. The theorem of Buckingham-π is used to obtain the following five independent dimensional parameters: For GP simulation, 160 data of bed shear stress (τb) and 100 data of wall shear stress (τw) were collected in a smooth rectangular channel with different flow depths. Approximately 70 percent of the total data were chosen for training and the remaining 30 percent for testing. If a random variable (τ) in a cross section of a channel is assumed to be a consistent shear stress, then, according to Tsallis entropy of [25] the SSD or shear stress probability density function f(τ), can be identified as [20]: where τ is the shear stress, q is a true number, and Tsallis 's entropy function is H(τ). The τ value varies from 0 to τmax, and with these restrictions, the integral value of H(τ) will be 1.
Using the maximum entropy theorem, the f(τ) can be calculated to maximize the entropy function subject to specified constraints like Eqs. (7 ) and ( 8) respectively [26].
where the minimum and maximum shear stress values are τmean and τmax, respectively.
At this stage, using maximization of Lagrange coefficients by equations 6 to 8, the Lagrange function L can be written down as Eq. (9): where λ0 and λ1 are the Lagrange multipliers. By ∂L/∂(τ) = 0 to maximize entropy, the f(τ) yields as: which λ′ = 1 q−1 + λ0. f(τ) as Eq. (10) represents the shear stress probability distribution function (PDF). The SSD's Cumulative Distribution Function (CDF) is introduced as Eq. (11): where y is the direction of the channel wall, which varies from 0 at the free surface to L, and L is the entire wetted perimeter. f(τ) is the derivative of F(τ), so the following equation is done by a partial derivation of F(τ) with respect to y: The shear stress function is represented as Eq. (13) by substituting Eq. (10) into Eq. (11) and Eq. (12) and solving the integral and simplifying it.
where k = q/ q−1 and q is a real parameter in the Tsallis relationship and has an actual value of 3/4 [11,27]. λ1 and λ′ are Lagrange multipliers that can be derived by trial and error from two implicit equations that follow. Indeed, by inserting and integrating Eq (10) into two constraints (Eqs. (7) and (8)), two Eqs. (14) and (15) are returned as: (14) and (15) solve to obtain two undefined Lagrange multipliers (λ1 and λ′). To estimate the SSD, a pair of mean and maximum shear stresses is required. It is also possible to use Knight et al. [2] analytical relationships to approximate the values of τmax and τmean [20,28,29]. In a laboratory plexiglass flume, [9] conducted her experiments. The flume was 10 meters in length, 0.6 meters in width, 0.7 meters in depth and its slope was set at where τmean (w) and τmean (b) the mean and τmax (w) and τmax (b) are the maximal shear stress on the wall and bed of the channel, each of the respective. The transverse SSD can therefore be calculated for any given channel, dependent upon the depth of the water and the slope of the bed.

Genetic Programming (GP)
In the second part of this analysis, the GP model is applied as one of the evolutionary algorithms (EA) to improve the accuracy of the given relations. The GP is an automated programming method to solve problems by designing computer programs GP is widely used for modeling structure recognition technology applications concerns. For this aim the GP technique is used to understand the basic structure of a natural or experimental process. In the GP method a population is slowly improved by selectively omitting the population who is not so fit and selecting from better species. EAs perform defining a target in the form of a quality criterion and then using that objective to evaluate candidates for a solution in a step-by-step improvement of data set frameworks and returning within a few iterations an optimum solution. The GP can optimize both the structure of the model and its parameters. One of the advantages of the GP algorithm is that it can extract an equation based the input and output parameters and it is more effective than other ANN models [30]. The outcomes of the GP model were analyzed by using the statistical indexes and compared with the experimental results.

ANFIS
ANFIS is designed to provide the requisite inputs and outputs for adaptive networks to build fuzzy rules with acceptable membership functions. FIS is a common and cardinal programming method that uses fuzzy theory to write fuzzy if-then rules and fuzzy logic bases that map from a given input information to the desired output. An adaptive network is a multi-layer feed-forward Artificial Neural Network (ANN) with; partially or entirely adaptive nodes in which the outputs are predicted on adaptive node parameters and the parameter adjustment is specified by the learning rules due to the error term. In adaptive ANFIS, hybrid learning is generally a learning form [31].

Criteria for Statistical Assessment
Maximum Error (ME), Key Absolute Error (MAE), Root Mean Square Error (RMSE) and Efficiency Coefficient (E) are the five statistical evaluation parameters used to determine the Tsallis entropy, GP model and ANFIS model performance, which are measured as follows: where Oi is the observed parameter value, Pi Predicted parameter value, is the mean value observed parameter value and n number of samples.

Modeling of GP
In this section, Sensitivity of the GP model for any input parameter is evaluated by adding all four inputs to the models first. Each parameter is then omitted and a total of three separate versions are checked. The GP models used for data on the bed and wall are described as: For the bed  (1). By sensitivity analysis performed, since the flow situation is fully developed, can be ignored the Reynolds number, the parameter has been eliminated in model 2. As shown in Table 4, by omitting Reynolds number (Re) in the input parameters, there is no significant difference. On the other hand, because all the experiments have examined the subcritical flow conditions, can be ignored the effect Froude number, the parameter has been eliminated in model 3. By eliminating the Reynolds number and Froude number parameters, the GP model performance does not change much, and the GP model can be deduced to be insensitive to the B/H parameter. The B/H ratio is obviously important in the estimation of shear stress, as this parameter plays a significant role in the equations stated. Therefore, the model 3 for the bed and wall is chosen as the most suitable model. The results of the most accurate GP model and experimental bed and wall data are shown in the form of the scatter plots in Figures 3 and  4. As seen in statistical analysis, the GP model outcomes are very similar to the bed and wall shear stress line fitted. Dimensionless bed shear stress modeling with GP was superior to dimensionless wall shear stress modeling with average R 2 of 0.945 and 0.8266, respectively, and both models were superior to the other GP models in this study. In order to decide the best answer, the best feedback should be treated as a pattern. Different important parameters in modeling, such as population members, number of generations, tree structures size etc. should be carefully determined in the first step with regard to the consumer of the data examined. The scale of each configuration of the tree will play a major role in the final model's accuracy. Determining the greater numbers than optimal value reduces the precision of the test results prevented from displaying the models are not presented largely because the models generated by genetic programming were very long-scale in order to measure the shear stress. The method of fitting models resulting from genetic programming against experimental results of parameters 2.86, 4.51, 7.14 and 13.95 are shown in Figure 4. The analysis of tables 3, 4 and 5 shows that the GP model performed higher than the Tsallis entropy and ANFIS methods in terms of comparison. The statistical analysis results of GP model predictions tabulated in Table 3.

ANFIS modeling
For this purpose, 70% of the experimental data is used for network training and the remaining 30% is used for testing results. As input parameters to the model, the parameters b/B & B/H for bed and z/H & B/H for wall were presented. Figure 5 shows the performance of the ANFIS model to estimate the bed SSD (τb) and Figure 6 shows the performance of the ANFIS model to estimate the wall SSD (τw), 30% of the data which were not used in the training stage would be used to evaluate the performance of the model. The results of statistical indexes for modeling shear stress with ANFIS are summarized in Table 4. As well, the estimating bands of the four above parameters using to determine the shear stress are shown in Figure 5. Skewness results obtained from statistical prediction dimensionless parameters.

Comparison of the GP model, Tsallis entropy and ANFIS
The best GP models and Tsallis entropy are compared to the equations proposed by Lashkar-Ara and Fatahi [9] in this section. Table 5 shows the statistical parameter results of the estimation of shear stress by the Tsallis entropy model. Figures 7 and 8 shows the SSD in a smooth rectangular channel for B/H is 2.85, 4.51, 7.14,13.95. As shown in these statistics, all of the test evidence used to model the SSD using the GP model is realized. For training stage for modeling SSD in rectangular channel using GP model, 70 percent of all data were used, and 30 percent data were used for testing process. As shown in Figure 5, for B/H= 2.86, 4.51, 7.14, 13.95, the GP model predicts the bed shear stress better than the Tsallis entropy model. In Figures 6c and 6d, for B/H = 4.51, 7.14, the GP model predicts wall shear stress better than the Tsallis entropy model, but in Figures 6a and 6d, the Tsallis entropy is more accurately models to predict wall shear stress than the GP model. Additionally, the GP model estimates bed and wall shear stress better than the Tsallis entropy-based model at rising flow depth. It is understandable that the channel architecture is challenging when a model expects higher shear stress values. It is therefore not cost-effective to use the Tsallis entropy method. When the GP model's observations are more accurate, it can be used to design stable channels more consistently. The GP model estimates the bed shear better than the ANFIS model for B/H= 2.86, 4.51, 7.14, , the ANFIS model estimates the shear stress better than the GP model, but the GP model estimates the wall shear stress better than the ANFIS model in B/H = 4.51,7.14,13.95. The GP model demonstrates superior efficiency to the Tsallis entropy-based model, while both models neglect the influence of secondary flows. It can be inferred that the GP model of bed and wall shear stress estimation is more sensitive than the Tsallis entropy method overestimated the values of bed shear stress and the GP model's outcomes are greater. The bed shear stress values decrease at the middle of the channel (Figure 5), which varies from other situations. From Figures 7 and 8, it can be shown that the GP model's fit line is similar to the 45-degree line than the other ones, and with a higher R 2 value, its predictions are more reliable. In predicting the position of maximal shear stress, both the GP and Tsallis-entropy based models display the same pattern as the centerline of the channel, which is consistent with the experimental outputs.

Conclusion
An important problem that engineers have been trying to solve is evaluating the SSD on open channels. This study makes use of the Tsallis entropy, model ANFIS and GP method to predict the SSD in rectangular channel. Sensitivity analysis was introduced to define the efficient parameters in SSD modeling with Tsallis, and 3 separate GP models were used. The findings show that in SSD estimation, B/H is a sensitive parameter. With increasing the flow depth, both the GP model and the Tsallis entropy-based equation performed better than other cases. The GP model, with an average RMSE of 0.0893 and R 2 of 0.9450 in bed and RMSE of 0.0904, R 2 of 0.8266 in wall for aspect ratios (B/H) 2.86, 4.51, 7.14 and 13.95, approximately performance over the Tsallis entropy-based equation, with RMSE of 0.0714, R 2 of 0.9109 in bed and RMSE of 0.0880, R 2 of 0.8006 in wall and ANFIS model, with RMSE of 0.1380, R 2 of 0.9344 in bed and RMSE of 0.0846, R 2 of 0.8470 in wall for all flow depths. The GP model predicts wall shear stress better than the ANFIS and Tsallis entropy-based models to compare the efficiency of the models as the flow depth increases. As a practical achievement of the results of this study, it can be said that if the model overestimates the shear stress, this unrealistic estimate will cost a lot to design the channel. Based on the analysis, it can be deducted that the use of the Tsallis entropy methodology in the shear stress evaluation is not economical.