平底马蹄形断面的水力计算

Hydraulic calculation of horseshoe cross-section with flat-bottom

平底Ⅱ型马蹄形断面是将标准Ⅱ型马蹄形断面的底拱改成平底而来,是水利水电工程较常用的断面形式之一,但其正常水深和临界水深的水力计算需求解超越方程,无解析解。传统的试算法或查表法计算过程繁琐且计算精度不高。本文通过对平底Ⅱ型马蹄形断面基本方程进行数学变换,并对引入的无量纲参数与无量纲水深的关系进行分析及计算,应用拟合优化原理得到了其水力计算的直接计算公式。该公式不需分段和进行判别选取,直接可得到答案。在工程的适用范围内(即水深与顶拱半径之比:0.05≤h/r≤1.41),计算正常水深的最大误差小于0.41%,计算临界水深的最大误差小于0.20%。其成果对该断面形式的隧洞设计有一定参考价值。

As one of the simplified type of the standard Ⅱ type horseshoe, the Ⅱ type horseshoe cross-section with flat bottom is one of the general used shape types for spillway tunnel and irrigation tunnel. However, the hydraulic calculation for this type tunnels has not been reported. Aiming at this problem, a new hydraulic calculation formula was proposed. The Ⅱ type horseshoe cross-section with flat bottom is consisted of four parts： one flat bottom, two side circular arc arch and one circular arc roof umbrella arch. This simplified horseshoe type is special suitable for the geological conditions with low foundation pressure. The characteristics of hydraulics and mechanics are between horseshoe-shaped and U-shaped arch. Due to the flat bottom, this type can keep a normal water flow regime, and a smoothly connection of the water flow before and behind the tunnel. For obtaining the calculation formula of the normal and critical water depth for Ⅱ type horseshoe tunnel with flat bottom, its geometrical design of hydraulic tunnel were analyzed. In order to guarantee the status without pressure in a free flow tunnel with changing water level, the free space above the water level was not less than 15 % of the whole cross-sectional areas of the tunnel. The upper limit of dimensionless water depth was 1.41. For most of the application engineering, the lower limit of dimensionless water depth was 0.05. After ascertained the utility range of the formula and based on fundamental equations of uniform flow and critical flow, the interrelation between dimensionless water depth and the dimensionless parameter were analyzed. We used the power function as the formula form, and coefficients of the formula were calculated based on the theory of optimization and regression. The calculation formula for normal water depth and critical water depth were obtained. Results showed that the maximum relative error of normal water depth and critical water depth were 0.41% and 0.20%, respectively. This research indicated that the hydraulic calculation formula could be widely used in engineering design and project management with high accurate and simple form.