Document Type : Research Paper
In addition to rivers and canals, which always have a free water surface, in some cases, closed conduits such as partially-filled pipes and culverts also are gravity-fed. The flow in these sections is very different from the hydraulics of flow in closed or under pressure pipes. One of the distinguishing aspects of these two types of flow is the higher probability of sedimentation occurring in the free flow conditions. Sedimentation in these sections may occur naturally or artificially. In sewers, this sedimentation generally occurs in non-self-cleaning pipes due to a reduction in flow velocity (formation of a backwater profile, M1, caused by downstream obstacles) or the entry of sediment particles larger than the particle mobility threshold at the time of pipe design. In road culverts, a sediment bed may be used at the bottom of the culvert to create a suitable environment and substrate for fish to live and passage. Over time, the sediment layer deposited at the bottom of the section becomes stable and is considered a new geometry for the culvert. So far, the hydraulic design and analysis of flow in these channels has been mainly one-dimensional, using empirical and semi-analytical relations or equations, and two or three-dimensional simulations in this field are limited. In this study, a quasi-two-dimensional model was used to analyze the flow in partially-filled deposited pipes, which has a lower run time than two- and three-dimensional models and greater accuracy than one-dimensional models.
Nowadays, there are many mathematical models that, while having the accuracy of two-dimensional and three-dimensional models, are similar in simplicity to one-dimensional models. These models are called depth-averaged models or lateral distribution model (LDM) and calculate hydraulic parameters (such as flow velocity, discharge, bed shear stress, etc.) across the channel. Among the important and well-known models in this field are Wark et al (1990), Shiono and Knight (1991), and Ervine et al. (2000). Due to the widespread applications of the Shiono and Knight model (SKM) by researchers, this model was used in this study. SKM is a quasi-two-dimensional mathematical model based on the Navier-Stokes equations which is generally used to calculate the transverse distribution of flow velocity and boundary shear stress in open channels. In this study, this model is used for flow analysis of a partially-filled pipe with flat bed. For calibration of friction coefficient (f), the Rameshwaran and Shiono (2007) equation was used. Also for calibration of two coefficients of eddy viscosity (λ) and the secondary flow (β), a numerical optimization process was used. The optimization of these two coefficients is based on the simultaneous solution of transverse distribution of flow velocity and boundary shear stress with a minimum error in a deposited pipe with three different flow depths. Afterward, this process is validated based on another flow depth. SKM has been numerically solved using finite difference methods. All steps of solving this model, as well as optimization of eddy viscosity and secondary flow coefficients, have been carried out in Excel.
By comparing the SKM results with Sterling's experimental data, it was found that this model behaves differently in estimating the transverse distribution of flow velocity and boundary shear stress in deposited pipes. The results of the transverse velocity distributions at all flow depths have very good accuracy, with a mean error of about 6 percent and a maximum of nearly 7.1 percent. The average flow velocity obtained from the Manning equation is higher than both the calculated and observed velocities. The highest error of the Manning formula (about 37%) occurred at the lowest flow depth (40.7 mm). After calculating the transverse velocity distribution, the flow discharge is obtained by lateral integration of this distribution. The stage-discharge curves obtained from the SKM are in much better agreement with the experimental data compared to the Manning formula results.
However, the lateral distribution of shear stresses is not very satisfactory. The mean and maximum of shear stress prediction errors are 18.5 and 22 percent, respectively. Although this behavior is not unusual and has been reported in studies by various researchers, it could also have another reason. One of the reasons for this issue is the complexity of the geometry of the partially filled pipes with flat bed.
In this study, a simple and practical quasi-two-dimensional model based on flow hydrodynamics was used to simulate the lateral distribution of flow velocity and bed shear stress, as well as to extract the stage-discharge curve in partially-filled deposited pipes. The results showed that:
1. The results obtained have appropriate accuracy and the use of the SKM proposed in this study can reduce the error in flow rate calculations by up to 30% compared to the widely used Manning formula.
2. For the lateral distribution of local bed shear stress, several simulations were performed using the SKM at different flow depths. The results of this simulations showed that the estimation of bed shear stress values in partially-filled deposited pipes is less accurate than the flow velocity values. The error of the traditional formula (τ=γRS0), which is widely used in estimating shear stress in rivers and open channels, is more than 72%. Therefore, the results of the SKM, as initial and preliminary solutions, well meet the needs of researchers in the design and management of hydraulic structures.
3. The quasi-two-dimensional Shiono and Knight model, using the equations considered in this study, has high efficiency and accuracy in estimating the distribution of flow velocity and shear stress across partially filled and deposited pipes and can be a very good alternative to the conventional one-dimensional models used in this field.
This work was funded by Gorgan University of Agricultural Sciences and Natural Resources (GUSNR) under grant no. 04-525-69.
All authors contributed equally to the conceptualization of the article and writing of the original and subsequent drafts.
The authors didn’t use any generative AI and AI-assisted technologies in the writing process of this manuscript.
The data that support the findings of this study are available from the corresponding author upon reasonable request.
The authors would like to thank financial supports of Gorgan University of Agricultural Sciences and Natural Resources in the present study.
The authors avoided data fabrication, falsification, and plagiarism, and any form of misconduct.
The authors declare no conflict of interest.
