A.R. Vatankhah; S. Kiani; S. Riahi
Abstract
Introduction: A free overfall offers a simple device for flow discharge measuring by a single measurement of depth at the end of the channel yb which is known as the end depth or brink depth. When the bottom of a channel drops suddenly, the flow separates from sharp edge of the brink and the pressure ...
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Introduction: A free overfall offers a simple device for flow discharge measuring by a single measurement of depth at the end of the channel yb which is known as the end depth or brink depth. When the bottom of a channel drops suddenly, the flow separates from sharp edge of the brink and the pressure distribution is not hydrostatic because of the curvature of the flow. In channels with subcritical flow regime, control section occurs at the upstream with a critical depth (yc). Although pressure distribution at the critical depth is hydrostatic, the location of the critical depth can vary with respect to the discharge value. So, the end depth at brink is offered to estimate the discharge. A unique relationship between the brink depth (yb) and critical depth (yc), known as end-depth ratio (EDR = yb/yc), exist. Since a relationship between the discharge and critical depth exists, the discharge can ultimately be related to yb. However, when the approaching flow is supercritical, critical section does not exist. Therefore, the discharge will be a function of end depth and channel longitudinal slope.
In current study, an analytical model is presented for a circular free overfall with different flat base height in subcritical and supercritical flow regimes. The flow over a drop in a free overfall is simulated by applying the energy to calculate the EDR and end depth-discharge (EDD) relationship.
End-depth-discharge relationship: The flow of a free overfall in a channel can be assumed that is similar to the flow over a sharp-crested weir by taking weir height equal to zero. It is assumed that pressure at the end section is atmospheric, and also streamlines at the end section are parallel. To account for the curvature of streamlines, the deflection of jet due to gravity, the coefficient of contraction, Cc, is considered. At a short distance upstream the end section, the pressure is hydrostatic. By applying the energy equation between end section and control section which is at upstream the end section, the flow depth at the end of the channel yb in terms of depth at the control section can be determined.
Subcritical flow regime: In this case, the approach flow to the brink is subcritical for negative, zero and mild bed slopes with critical depth at the control section. Using the definition of the Froude number at critical depth, the discharge can be determined. As the explicit relationship between discharge and depth at the brink don’t exist, a relationship should be presented through regression analysis between discharge and yb using the different values of yc over the practical range of 0.01 to 0.84. In this study, below explicit equation is presented for computing Q*(dimensionless discharge) in terms of ( ):
Where d is channel diameter, is the ratio of bottom elevation to the channel diameter and . This equation can be used for different values of over the practical range of 0.06–0.6.
Supercritical flow regime: A critical flow occurs upstream of the free overfall under the subcritical approach flow. However, no such critical flow occurs in the vicinity of the overfall under supercritical flow regime. Therefore, the Manning equation for known value of channel bed slope and Manning’s coefficient is exercised to derive the discharge relationship under the supercritical flow regime. Since an explicit equation for discharge in term of yb is impossible, a direct graphical solution for discharge for known end depth, channel bed slope, and ratio of bottom elevation has been provided for supercritical flow regime.
Conclusion: The free overfall in a circular channel with flat base has been simulated by the flow over a sharp-crested weir to calculate the end-depth ratio. This method also eliminates the need of an empirical pressure coefficient. The method estimates the discharge from the known end-depth. In subcritical flows, the EDR has been related to the critical-depth. On the other hand, in supercritical flows, the end-depth has been expressed as a function of the longitudinal slope of the channel using the Manning equation. The mathematical solutions allow estimation of discharge from the known end-depth in subcritical and supercritical flows. The comparisons of the experimental data with this model have been satisfactory for subcritical flows and acceptable for supercritical flows.
S. Riahi; A.R. Vatankhah
Abstract
Introduction: Side weir structures are extensively used in hydraulic engineering, irrigation and environmental engineering, and it usually consists of a main weir and a lateral channel. Side weirs are also used as an emergency structure. This structure is installed on one side or both sides of the main ...
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Introduction: Side weir structures are extensively used in hydraulic engineering, irrigation and environmental engineering, and it usually consists of a main weir and a lateral channel. Side weirs are also used as an emergency structure. This structure is installed on one side or both sides of the main channel to divert the flow from the main channel to the side channel. Lateral outflow takes place when the water surface in the main channel rises above the weir sill. Flow over a side weir is a typical case of spatially varied flow with decreasing discharge. There have been extensive studies on side weir overflows. Most of the previous theoretical analysis and experimental research works are related to the flow over rectangular side weirs in rectangular main channels. In the current study, the flow conditions over a trapezoidal side weir located in a rectangular main channel in subcritical flow regime is considered.
Materials and Methods: The experiments were performed in a rectangular open channel having provisions for a side weir at one side of the channel. The main channel was horizontal with 12 m length, 0.25 m width, and 0.5 m height, and it was installed on a frame; lateral channel that has a length of 6 m, width of 0.25 m, and height of 1 m. It was set up parallel to the main channel; walls and its bed were made up of Plexiglas plates. The side weir was positioned at a distance of 6 m from the channel’s entrance. A total of 121 experiments on trapezoidal side weirs were carried out.
Results and Discussion: For trapezoidal side weir, effective non-dimensionnal parameters were identified using dimensional analysis and Buckingham's Pi-Theorem. Finally, the following non-dimensional parameters were considered as the most effective ones on the discharge coefficient of the trapezoidal side weir flow.
in which Fr1= upstream Froude number, P= hight of the trapezoidal side weir, y1= upstream water depth, z=side slope of the trapezoidal side weir and T=top flow width of the trapezoidal side weir. Water surface profiles were measured along the weir crest, the main channel centerline, and far from the weir section. Different elevations in water surface profile depend on the upstream Froude number in the main channel; depth differences in low Froude numbers are at minimum values, and in high Froude numbers are at maximum amounts. The water surface level along the crest drops at the entrance of the side weir to the first half of the side weir; and it has been attributed to the side weir entrance effect at the upstream. Afterwards, the water level rises towards the downstream of the weir. According to the experimental results, measurements of the water in the centerline of the main channel are reliable and water surface drop is negligible. According to the parameters affecting the discharge coefficient for each value of z, discharge coefficient equations were developed with acceptable accuracy such that the effects of this parameter were shown separately. Finally, the general equation was proposed. The general functional form for discharge coefficient is presented as follows where the effect of the side slope parameter, z, is also considered.
The mean and maximum percentage errors of the discharge coefficient computed using the proposed equation are as 2.6% and 11.5% , respectively.
Conclusion: In this study, the characteristics of trapezoidal side weir overflows in subcritical flow regime were discussed. For this purpose, experimental data related to the water surface profile of the side weir and discharge coefficient were collected and analyzed. The results showed that the most efficient section for measuring water surface profile is located at the center line of the main channel. It was found that for trapezoidal side weir, the discharge coefficient depends on the Froude number, the ratio of crest height to initial depth, the overflow length to initial depth, and the side slope of the weir. In this study, conventional trapezoidal weir theory has been used in order to evaluate the discharge coefficient and provide side weir discharge equation. For this purpose, three reference depths were considered for conventional weir, and for each depth an equation was developed for the discharge coefficient. Comparison between predicted values and experimental data showed that average flow depth results in accurate outcomes for assessing the discharge coefficient. The average value of error for discharge coefficient estimation by the proposed equation is 2.6%. Thus this equation is proposed for use in practice by water engineers.
Keywords: Control structure, Conventional weir, Discharge coefficient, Spatially varied flow, Trapezoidal side weir, Water surface profile
