Document Type : Original Article

Authors

• English Darvishi ¹
• Salah Kouchakzadeh ²

¹ Assistant Professor, Department of Water Engineering, College of Agriculture, Razi University, Kermanshah, Iran

² Department of Irrigation and Reclamation Engineering, University of Tehran, Karaj, Iran

Abstract

Introduction

In steady flow, the minimum specific energy and the specific force occur at critical depths. But will it be the same in the unsteady flow? In this research, using laboratory data of water surface profile, the correctness of these equations in unsteady flow and the location of critical flow, minimum specific energy and specific force will be investigated.

Methodology

Laboratory equipment

The experiments were performed in a channel with a width of 0.37 m, a height of 0.6 m and a length of 3 m made of Plexiglas with a thickness of 2 cm located in the hydraulic laboratory of the Technical University of Vienna.

Due to the importance of bed slope, especially in different water measurement structures, Darvishi et al. (2017) attempted to correct the Boussinesq equations Eq (6).

Numerical model

The numerical scheme used by Darvishi et al. (2015) to separate the derivative term of the variable f at point n from the distance of the four-point finite difference. The upstream boundary condition was considered as the flow discharge changes with time. This hydrograph was used as an upstream boundary condition in the unsteady flow.

Results and Discussion

In Figure 5, the position of the minimum specific force, initially has a displacement in the positive direction x, but over time, its position moves upward. While the position of the minimum specific energy and critical flow is constantly moving downstream to stabilize its position after reaching a steady flow. Therefore, determining the critical flow position to measure the flow in unsteady flow using the critical Froude number and minimum energy has a higher confidence than the minimum specific force. In order to investigate Equation 2 in the unsteady flow, the diagram of the changes of specific energy changes in the critical flow relative to the unit discharge is plotted in Figure 7. These two graphs are exactly the same, which means that Equation 2 in the unsteady flow also calculates the unit discharge correctly. These results are consistent with the results of Chanson and Wang 2013. A comparison of this chart with the chart provided by Castro-Orgaz and Chanson 2016 shows a significant difference. In their diagram, the unsteady flow line does not correspond to Equation 2 and for qc greater than 0.04 it deviates from Equation 2. The reason for this discrepancy can be related to the Saint-Vanant equations used by them to simulate unsteady flows.

Figure 8 shows the changes in the depth slope at the critical depth (-hx)c relative to the product of the critical depth multiplied by the curvature of the bed hczbxx. Equation 4 is also plotted in the diagram. According to Castro-Orgaz and Chanson 2016 the relative error of estimation (-hx) c using Equation 4 varies from 8.5% to 17.5%. For unit discharge of 0.03559 m2/s in paper by Sivakumaran et al. 1983 is also plotted on the chart. As can be seen, at values -hczbxx less than 0.05, the data correspond to the graph of Equation 4, but with increasing this value, it moves away from this curve. In this case, the relative error is about 17%. So that the laboratory data does not match this curve. As Fenton and Darvishi (2016) have stated, Equation 4 often does not provide the right results. While Castro-Orgaz and Chanson 2016 have considered this Equation valid for values -hczbxx less than 0.15.

Conclusions

The position of the critical Froude number, the minimum specific energy, and the specific force on the trapezoidal broad-crest for the inlet incremental hydrograph to channel were investigated using the numerical solution of the modified Boussinesq equation and laboratory data. The results showed that the position of the minimum specific energy and the critical Froude number move in a short distance from each other and continuously in the direction of flow. While the position of the specific force, first moves in the forward direction and after a while in the opposite direction of the flow. The unit discharge is very close to each other in all three positions, and in the minimum specific energy and the critical Froude number are exactly the same. Specific energy was also plotted in critical flow versus unit discharge. This diagram is completely consistent with the specific energy relationship in the steady critical flow. In order to judge this relationship more accurately, it is necessary to examine hydrographs with different shapes. The singular point relationship for different beds in the steady flow was also investigated. This relationship has a good accuracy for values less than 0.05 times the product of the second order differential of the bed at critical depth on the studied beds. At values higher than this, there is a relative error of up to 18%.



Keywords: Froude number, singular point, curved bed, minimum specific energy, minimum specific force.

Keywords

• Froude number
• singular point
• curved bed
• minimum specific energy
• minimum specific force

Main Subjects

• Hydraulic