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gher heads, the discharge is greater than an aerated sharpcrested weir because the negative crest 7 pressure suctions more flow. Although much is understood about the general ogee shape and its flow characteristics, it is also understood that a deviation from the standard design parameters such as a change in upstream flow conditions, modified crest shape, or change in approach channel owing to local geometric properties can change the flow properties. For the analysis of the effects, physical models have been used extensively because a spillway is very important for the safety of dams. The disadvantages with the physical models are high costs and that it can take fairly long time to get the results. Also, errors due to scale effects may increases in severity as the ratio of prototype to model size increases. So, numerical modeling, even if it cannot be used for the final determination of the design, is valuable for obtaining a guide to correct details because putational cost is low relative to physical modeling. In the past few years, several researchers have attempted to solve the flow over spillway with a variety of mathematical models and putational methods. The main difficulty of the problem is the flow transition from subcritical to supercritical flow. In addition, the discharge is unknown and must be solved as part of the solution. This is especially critical when the velocity head upstream from the spillway is a significant part of the total upstream head. An early attempt of modeling spillway flow have used potential flow theory and mapping into the plex potential plane (Cassidy, 1965). A better convergence of Cassidy’s solution was obtained by Ikegawa and Washizu (1973), Betts (1979), and Li et al. (1989) using linear finite elements and the variation principle. They were able to produce answers for the free surface and crest pressures and found agreement with experimental data. Guo et al. (1998) expanded on the potential flow theory by applying the analytical functional boundary value theory with the substitution of variables to derive nonsingular boundary integral equations. This method was applied successfully to spillways with a free drop. Assy (2020) used a stream function to analyze the irrotational flow over spillway crests. The approach is based on the finite difference method with a new representation of Neumann’s problem on boundary points, and it gives positive results. The results are in agreement with those obtained by way of experiments. Unami et al. (1999) developed a numerical model using the finite element and finitevolume methods for the resolution of two dimensional free surface flow equations including air entrainment and applied it to the calculation of the flow in a spillway. The results prove that the model is valid as a primary analysis tool for the hydraulic design of 8 spillways. Song and Zhou (1999) developed a numerical model that may beapplied to analyze the 3D flow pattern of the tunnel or chute spillways, particularly the inlet geometry effect on flow condition. Olsen and Kjellesvig (1988) included viscous effects by numerically solving the Reynoldsaveraged NavierStokes (RANS) equations, using the standardequations to model turbulence. They showed excellent agreement for water surfaces and discharge coefficients. Recently, investigations of flow over ogeespillways were carried out using a mercially available putational fluid dynamics program, FLOW3D, which solves the RANS equations (Ho et al., 2020。在改變表面粗糙度和模型的規(guī)模后,垂直壓力分布幾乎還是一樣上網(wǎng)。 2)建模結(jié)果表明 ,增加的比率引起長(zhǎng)度尺度相似現(xiàn)象,是由于日益增長(zhǎng)的表面粗糙度造成的。在建模過(guò)程中按比例改變后的模型、網(wǎng)格分辨率、表面粗糙度、上游邊界條件和幾何相似度調(diào)整來(lái)排除不同的數(shù)值誤差。彎曲的障礙 、 壁面邊界或其他幾何特征是嵌入在網(wǎng)狀定義分區(qū)和分開(kāi)流動(dòng) 的變 量 。 RNG 模型來(lái)描述更準(zhǔn) 5 確的低強(qiáng)度的湍流流動(dòng)和流具有較強(qiáng)的剪切區(qū)域。 gi是在標(biāo)方向的引力 。A