抛物线形断面渠道共轭水深的直接计算公式

Explicit calculation equations for conjugate depths of channels in parabolic-shaped cross-section

为了给抛物线形断面渠道闸后水跃共轭水深的计算提供显函数计算公式,对抛物线形断面共轭水深函数求一阶导数,并令导数函数为0求出临界水深;利用跃前水深与临界水深之比、临界水深与跃后水深之比,分别作为量纲为一的跃前水深和跃后水深,并引入量纲为一的共轭水深函数,使共轭水跃方程转化为量纲为一的函数方程,在对该方程分离变量后结合跃前、跃后水流能量特征建立跃前水深和跃后水深的迭代计算公式,并证明迭代公式的收敛性;为了进一步提高迭代方程的收敛效率,以量纲为一的跃前水深与跃后水深之积的最大值处,亦即迭代收敛最慢点处的真解的修正值为迭代计算初值,配合迭代方程进行一次迭代得到较为精确的直接计算公式.误差分析表明:在工程常用范围内,提出的跃前水深、跃后水深直接计算式的最大相对误差分别为0.47%和0.55%,公式简捷、准确、适用范围广.

To develop an explicit function for the conjugate depths of hydraulic jumps iu parabolic- shaped channel after sluice gate, the first derivative of the conjugate deplhs was obtained and defined to be equal to zero for calculating the critical depth. A dimensionless conjugate depths formula was proposed by means of two ratios of the depth before water jump and the critical depth, as well as the erilical depth and the depth after water jump. Then the iterative functions of the depths before and after the jump were derived in terms of energy characteristics of upstream and downstream flow. To accele- rate tire convergence speed, applying tire maximum product term of dimensionless depths befoore and al- ter water jump, namely tire true solulion of the slowesl convergenee point, appropriate inilial valves were calculated. As a resuh, an explicit caleulation equation with higher accuracy was obtained with only one iteration. Error analyses indicate that the maximum relative errors of the upstream and down- stream depths, ealeulated by the explicit calculation equations, are 0.47% and 0.55% respectively. The proposed explicit calculation equations are simple, accurate and widely applicable.