S. Khodadoust Siuki; M. Nemati; R. Estakhr

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**Abstract **

Introduction: For a velocity profile in turbulent flows, the flow conditions in the vicinity of the wall are described by logarithmic law of the wall. However, it has been extensively verified that the log-law does not apply in the outer region of the boundary layer. For example, in free surface flows, ...
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Introduction: For a velocity profile in turbulent flows, the flow conditions in the vicinity of the wall are described by logarithmic law of the wall. However, it has been extensively verified that the log-law does not apply in the outer region of the boundary layer. For example, in free surface flows, the law of the wall holds only for 20 percent of the flow depth from the wall. Coles (1956) conducted an important advancement and argued that away from the wall, the deviations of the profiles of measured velocity from those obtained from the law of the wall could be explained by another universal law, called the wake-law. Combining both laws (wall and wake), a complete approximation to the time-averaged velocity profile in turbulent flows is then feasible (White, 1991). On the other hand, the fundamental problem of characterizing the mean velocity profile in sediment-laden flows remains unresolved. While existence models have been developed to estimate velocity profile, but there is a lack of generalization in the sediment-laden flows. For several decades, it has been controversial about the effects of suspended sediment on hydraulic characteristics of the flow, including flow resistance and velocity distribution. Fig. 1 shows the variations of velocity distribution due to introduction of the suspended sediment. As it is seen in this Figure, the suspended sediment moves faster than the water in the inner layer; on the other hands, there is a velocity-lag due to the introduction of sediment into the outer layer. Accurate estimate of the rate of sediment loads is important in sediment-laden flow. Because velocity distribution is one of the required parameters to estimate the sediment discharge. Until now, many equations have been introduced by many researchers for estimating the velocity distribution in open channels. Generally, there are two different views about the velocity distribution in sediment-laden flows. The first view suggests that the log-law is also applied in the sediment-laden flows and von Karman constantly decreases with increasing sediment concentration. Such researchers as Vanoni (1946), Einstein and Chen (1955), Elata and Ippen (1961) supported this idea. Another view is that von Karman constantly does not decrease with increasing sediment concentration and velocity distribution deviates from the main region of the flow. Because of these contradictions about the effects of suspended sediments on characteristics of the flow and given the existence of several developed models , this question may be raised whether which one is markedly superior to the others or what model gives accurate results in the sediment-laden flow. No attempt was made to make an exhaustive comparison of the models with available experimental data. The present study evaluates and discusses the performance of seven models, by comparing these with experimental data selected from four sources. Then these equations will be assessed using the experimental data, and the best model will be introduced by means of statistical analysis.
Materials and Methods: In this paper, the velocity distribution of sediment-laden flow has been investigated. Such equations as Log-law, modified log-law, wake-law, modified log-wake, log-modified wake, and parabolic law have been studied. The accuracy of each equation has been assessed by using statistical analysis. The mean average error (MAE) is a quantity used to measure how close predictions are to the eventual outcomes. The root-mean-square error (RMSE) is a frequently used measure of the differences between value predicted by a model or an estimator and the values actually observed. Determination coefficient (R2) is a number that indicates how well data fit a statistical model. Experiment data related to Wang and Qian (1989), Vanoni (1946), and Coleman (1981) have been used to test the proposed models. In most data sets, the width-depth ratios are less than 5, i.e., the maximum velocity occurs below the water surface. Thus, the boundary layer thickness is defined as the distance from the bed to the maximum velocity position, where the velocity gradient is zero.
Results and Discussion: The accuracy of each equation has been assessed using some statistical indices. The results showed that the log-wake velocity distribution in both the inner and outer regions estimated the velocity values with reasonable accuracy (with a relative error of 5%). It is recommended that this equation is used to calculate the suspended sediment discharge. On the other hand, parabolic-law doesn’t have a good accuracy and it will cause large errors (with a relative error up to about 15%). In addition logarithmic distributionsare only able to estimate accurately the velocity in the inner region. It was also found that in sediment-laden flows, in the region where y/h ≥ 0.2, the effect of sediment concentration can be neglected as the sediment concentration becomes very low. Therefore, it is more reasonable to look for an equation having acceptable accuracy in the inner layer.