معادله صريح براي برآورد ارتفاع آبپايه انتهايي لبه‌پهن براي كنترل جهش هيدروليكي در مقاطع ذوزنقه‌اي

An explicit equation for calculating board crested end sill height to control hydraulic jump in trapezoidal sections

يكي از مسايل مهم در كنترل جهش تعيين حداقل ارتفاع آبپايه انتهايي است. اگرچه پژوهش‌هاي زيادي در مورد تاثير آبپايه انتهايي روي مشخصات جهش هيدروليكي در حوضچه‌هاي آرامش با مقاطع مستطيلي انجام شده است، ولي در مورد حوضچه‌هاي آرامش با مقطع ذوزنقه‌اي پژوهش‌هاي زيادي گزارش نشده است. در اين پژوهش علاوه بر مقايسه نتايج آزمايشگاهي با روش‌هاي انرژي و اندازه حركت كه با روش‌هاي عددي حل شده‌اند، يك معادله صريح براي برآورد ارتفاع آبپايه انتهايي با روش انرژي ارايه شده است. ارزيابي نتايج بيانگر هماهنگي مناسب بين داده‌هاي آزمايشگاهي و محاسباتي است و فرمول صريح ارايه شده با تقريب مناسب ارتفاع آبپايه انتهايي را برآورد مي‌كند.

In this study, an explicit equation is presented for estimating the height of the end sill in the trapezoidal stilling basin based on the energy equation. In addition to the experimental results were compared with energy and momentum methods which have been solved by numerical methods. Evaluation results indicate proper coordination between experimental and computational data and confirmed that the presented explicit equation can compute end sill height with suitable accuracy. In recent decades, several studies aimed at understanding the function of hydraulic jumps in trapezoidal sections have been made Forrester and Skrinde (1950) to find an analytical equation based on the Froude number, initial depth, sequent depth and sill height for controlling jump in rectangular sections which used rectangular broad-crested weir equation for determining discharge which was created by Barker and Doeringsfeld (1941). Achour and Debabeche (2003) investigated the effects of broad-crested end sill on the U-shaped sections as well as the effect of the sharp-crested end sill on the triangular sections with the apex angle of 90°, by comparing the theoretical and experimental equations in U-shaped sections. They also found that the required sharp-crested end sill for controlling the hydraulic jump with the same conditions should be slightly taller than broad-crested end sill in U-shaped cross sections. Achour and Debabeche (2003) presented an explicit relationship in rectangular sections to find the minimum broad-crested end sill height necessary to control the hydraulic jump which indicated good agreement with the results of Forrester and Skrinde (1950). If a broad-crested end sill placed on the supercritical flow path in the stilling basin, hydraulic jump is formed. The minimum height necessary for the sill according to the specific energy concept can be calculated. In this case, due to the existence of critical depth on the sill, this section will act as a control section. To evaluate the results of the proposed explicit equation to determine the end sill height, a number of tests were carried out. Experiments were done in a horizontal channel with symmetrical trapezoidal cross-section, and with side slopes z=0.5, 1, 1.5, Length of 3 m and a width of 5.0 meters to measure the characteristics of flow jumps in the discharges range of 19.3 < Q < 79.4 liters per second in the water research laboratory of the Department of Irrigation and Reclamation Engineering of Tehran University. To slow down and measure the input flow in laboratory model, upstream primary reservoir, with dimensions of 1.25 m wide, 1 m long and 1 m high were designed and built. Inlet water flow to model supply through the circulated flow system with the closed circuit pipes in the laboratory, and through the branch of pipe near the model was connected to the first tank. To calm the approaching flow to the weir, a mesh with 1.25 m width and 1 meter height was used. Discharges measured with a calibrated weir in upstream tank. With the crossing of water over the weir, the flow was entered to supply-height reservoir. The purpose of constructing this reservoir was providing the required energy for the formation of the hydraulic jump with desired Froude number. Initial and sequent depth was measured with an accuracy of 1.0 mm by a point gage. In order to create jump without the end sills, flow was controlled by a terminal movable valve as a tailgate. For each of the experimental data obtained from the Fr1, in the range of 3 < Fr < 10, the end sill height was chosen in such a way that ?x, the distance between the gate and the initial depth, to be less than 10 cm. Horizontal distances were measured with an accuracy of 1 mm with a graduated tape. Due to fluctuations, sequent depth was measured with ?5 mm accuracy. To reduce the effect of fluctuations on the accuracy of measurement, data reading was done in a transparent sink that there were a few small holes on one side of it. To control the hydraulic jump, a Plexiglas broad- crested with 40 cm width and with a height at the range of cm. Given that the critical depth was created on the broad-creased sill, by writing momentum equation at the section of the sequent depth and sections with critical depth on the sill, the sill height can be calculated. As well as the sill height can be calculated by Bernoulli equation without taking into account the energy loss. Using any of these two methods is needed to solve nonlinear equations implicitly. With the proposed explicit equation, estimation of height end sills is provided easily in terms of the flow characteristics and canal cross sections with high accuracy. The values obtained from the proposed explicit equation for the height of broad-creased end sills with predicted values of energy and momentum equations are approximately the same, and all three methods have little difference with the experimental results. Also, it can be said that this equation can be predicted experimental data with high accuracy and indicates good performance as terms of practicality. In this study, the control of hydraulic jump with broad-creased end sills in trapezoidal sections with three side slope of 1: 0.5, 1:1 and 1: 1.5 was investigated. Flow analysis over the broad-creased sill allowed that the minimum sill height necessary for controlling the hydraulic jumps to be estimated. The results of the comparison of the proposed equation with momentum and energy equations and experimental results showed good accuracy in estimating the required broad-creased sill height to control the jumps in trapezoidal sections. Because there are no more experimental results, presented equation can be used as a guide in designing of energy dissipators to determine the broad-creased sill in trapezoidal sections with the explicit equation. It is noteworthy that the broad-crested sill is superior in comparison with other dissipators hydraulic jump structures, due to its structural stability and lower costs.