Discussion of 'Hydraulic Characteristics of Flow over Sinusoidal Sharp-Crested Weirs' by Zahra Oreizi, Manouchehr Heidarpour, and Sara Bagheri

摘要

The discusser would like to thank the authors for introducing a sharp-crested weir with sinusoidal walls. The discusser, however, would like to add a few points. The proposed model by the authors produces the following equation for computing theoretical discharge, Q_t: Q_t = (2g)~(1/2) ∫_0~(h_1) (h_1-y)~(1/2)[b_1 + 2b_2 sin(2πny/D)]dy (1) where h_1 = upstream head over the weir crest; b_1 = crest width; b_2 = amplitude of the sine function (0 ≤ b_2/b_1 ≤ 0.5); g = gravitational acceleration; n = cycle number of the sine function; and D = weir-notch height. Assuming y = h_1t, Eq. (1) takes the form Q_t = h_1(2gh_1)~(1/2) ∫_0~1 (1-t)~(1/2)[b_1 + 2b_2 sin(2πnh_1t/D)]dt (2) where t = nondimensional lump variable. Because b_1 ≠ 0, Q_t=2b_1h_1(2gh_1)~(1/2)[1/3+k∫_0~1(1-t)~(1/2)sin(ηt)dt](3) where k = b_2/b_1; and η = 2πnh_1/D. Eq. (3) can be expressed in terms of Fresnel integrals as Q_t = 2b_1h_1(2gh_1)~(1/2)｛1/3+k/η-k/(η~(3/2))(π/2)~(1/2)[cos(η)FresnelC((2η/π)~(1/2)) + sin (η)FresnelS((2η/π)~(1/2))(4) Eliminating Fresnel integrals yields Q_t = 2b_1h_1(2gh_1)~(1/2)(1/3+k/η)(5) The percentage deviation of Eq. (5) compared with that of Eq. (3) is PD = 100 ×[ 1-(1/3+k/η)/(1/3+k∫_0~1(1-t)~(1/2)sin(ηt)dt)](6) For the weir under investigation, the cycle number, n, is equal to one; thus, 0 ≤ η ≤ 2tt. Eq. (6) is depicted in Fig. 1 for different values of k. For k = 0 (rectangular weir), there is no deviation between Eqs. (5) and (3), but for other k values the deviation is considerable, especially for η ≤ 2.297. Thus, in general Fresnel integrals cannot be eliminated from the theoretical discharge equation [Eq. (4)]. Based on Fig. 1, the generality of Eq. (5) is questionable.