Potential flow solution for a free surface flow past a sudden slope change

Abstract

An analytical model was developed to calculate the pressure distribution in a free surface flow past a sudden change in channel slope. A conformal transformation technique was used to solve the problem analytically in a way that there is no need for trial and error to find the location of the free surface. Two methods were employed for this simulation: flow at a corner and free streamline theory. It was shown that free streamline theory is more accurate. Experiments were conducted to verify the ability of the analytical model to calculate the pressure distribution in a channel with a sudden change in slope. Slope changes of 6.22°, 10°, and 15° were tested with various flow discharges. The analytical model predictions of pressure distribution along the channel bed and with depth agreed well with the experimental measurements. Pressures up to 25 times the hydrostatic pressure were experimentally measured near the point of sudden change in slope. These pressures were reproduced by the model. The analytical model predictions of the water surface profile over a ramp in a prototype spillway were compared with those of a numerical model. The comparison showed a good agreement.Key words: pressure distribution, free surface flow, analytical model, chute spillway, aerator ramp, potential flow.

Bottom-Pressure Development Due to an Abrupt Slope Reduction at Stepped Spillways

“…Immediately downstream of the slope reduction, larger values of the mean as well as of the 5 th and 95 th percentiles were recorded (steps +1 and +2, Figure 4b). A similar trend was observed on an abrupt slope reduction on stepped [15] and on smooth chutes [14]. From the model studies of the Upper Stillwater The above findings are consistent with those of others (e.g., [2][3][4][5][6][7][8][9][11][12][13]22,24]).…”

“…By using the ratio p/(γ d up cosθ), values increasing from 6 (step −2) to 21 (step +1 or +2) were observed in the vicinity of the slope reduction, for 2.6 ≤ d c /h ≤ 4.6 (0.16 m ≤ d c ≤ 0.28 m) (not shown herein). A similar trend was reported by [14] on an abrupt slope reduction at an aerator-deflector, namely for ∆θ = 15 • . For such slope reduction, they reported pressure heads ranging between 3 and 23 times the upstream uninfluenced hydrostatic pressure, at a distance to the slope reduction of less than twice the upstream flow depth, for 0.14 m ≤ d c ≤ 0.22 m.…”

“…Immediately downstream of the slope reduction, larger values of the mean as well as of the 5th and 95th percentiles were recorded (steps +1 and +2, Figure 4b). A similar trend was observed on an abrupt slope reduction on stepped [15] and on smooth chutes [14]. From the model studies of the Upper Stillwater dam stepped spillway, with a slope reduction from 72 • to 59 • (∆θ = 13 • ), mean pressures obtained with piezometers located near the step edge were shown to increase shortly upstream of the slope reduction, with highest values recorded at this location, regardless of the tested reservoir heads [15].…”

“…In the latter study, 95th pressure percentiles of p/(γh) ≈ 6.5 were obtained in the vicinity of the step edge (x/l ≈ 0.06), for d c /h = 2. dam stepped spillway, with a slope reduction from 72° to 59° (∆θ = 13°), mean pressures obtained with piezometers located near the step edge were shown to increase shortly upstream of the slope reduction, with highest values recorded at this location, regardless of the tested reservoir heads [15]. Similarly, [14] observed that pressures were influenced by the slope reduction slightly upstream of the latter, despite supercritical flow. Therein, three slope reductions were tested, ∆θ = 6.22°, 10° and 15°.…”

“…As for smooth chutes, the effect of slope reductions on the bottom-pressure development was addressed by few experimental and numerical studies. A study [14], among others, observed a sudden change in pressure and velocity distributions as the flow passed over an abrupt slope reduction at an aerator-deflector. They reported that a slope reduction of ∆θ = 15 • lead to a pressure increase up to 25 times the hydrostatic pressure near the slope reduction.…”