Computational simulation of flow over stepped spillwaysAbstractNumerical simulations of water flow over stepped spillways with different step configurations are presented. The finite element computational fluid dynamics module of the ADINA software was used to predict the main characteristics of the flow. This included the determination of the water surface, the development of skimming flow over corner vortices, and the determination of energy dissipation. Since the actual flow is turbulent, the flow model was used. A two-phase solution process was adopted in order to optimize the overall simulation efficiency. In the first phase, a simple yet reasonable water surface consisting of three straight lines was used as an initial guess and was treated as a fixed wall. In the second phase, the results from the first phase were used as initial conditions and the water surface was treated as a free surface that evolved to attain a steady state configuration. For all the cases considered, the predicted water surface profile over the entire length of the spillway was in close agreement with the experimentally measured water surface profile. The predicted energy dissipation was also comparable to the experimentally attained values.Keywords:Stepped spillway; Numerical; Experimental; Modeling; Free surface; Finite element; Turbulent flow1. IntroductionA spillway is a hydraulic structure that is provided at storage and detention dams to release surplus or flood water that cannot be safely stored in the reservoir. When the reservoir s storage capacity is exceeded, water flows over the spillway crest and accelerates down the chute creating high velocities at the toe. This may cause dangerous scour in the natural channel below the hydraulic structure. As a remedy, various forms of energy dissipation mechanisms have been used in practice: simple aprons, stilling basins, straight drops, impact basins, baffled chutes, and plunge pools. Although these methods have proven to be effective in dissipating energy, they incur a substantial increase in the cost of construction of the dam. They may also develop defects that affect the structural integrity of the spillway and may cause failure of the dam.One possible solution is to use a stepped ogee-profile spillway instead of the traditional smooth ogee-profile spillway, where a series of drops are introduced in the invert from the vicinityof the crest to the toe. The stepped spillway is expected to generate substantial energy losses over a wide range of operating flow heads with the steps acting as roughness elements that reduce the flow terminal velocities. In recent years stepped spillways have become popular due to the low-cost and relatively high-speed in construction and rehabilitation of dams using the method of roller compacted concrete.Efficient and safe design procedures for spillways can only be achieved through comprehensive understanding of the intricacies of the flow. Historically, engineers and scientist have resorted to investigating the flow through laboratory experimentation on scaled down models of spillways. This was done in conjunction with simplified flow assumptions based on fluid mechanics concepts. With the advent of high-performance computers and the development of robust computational fluid dynamics (CFD) software, researchers were afforded a complimentary analysis tool. If used judicially, this computational tool is capable of resolving the intricacies of the flow. The long process of validating and establishing credibility of a CFD method starts with demonstrating that the computational results are in close agreement with analytical (closed-form) solution of idealized problems. The next step in the process is the comparisons with well designed experimental results that are believed to closely represent the actual flow. Ultimately, and only after an exhaustive validation process, the design of hydraulic structures can be based on the predictions provided by CFD simulations.The literature surveyed by the authors yielded numerous and wide ranging references on laboratory experimentation using scaled down spillways. In contrast, only a few references on computational modeling were found. The recent and relevant references that dealt with laboratory investigations [1–11] addressed the main characteristics of the flow that influence the design of a stepped spillway. This included scale effects, transition from nappe flow to skimming flow, inception of air entrainment, air concentrations, velocity distributions, pressure distributions, and energy dissipation. As a result, semi-empirical equations have been developed to aid in the design of actual spillways and to lessen the need for individual experimental model studies.The ancestry of modern CFD methods can be traced back to two well established and widely used computational methods: finite difference and finite element. The finite volume method, which has been extensively used to model a wide range of fluid-flow problems, was originally developed as a special finite difference formulation. The volume-of-fluid (VOF) method is an interface tracking scheme that has been used to model free surfaces. All the aforementioned methods have been applied to model flow over a spillway [12–18]. It should be noted,however, that only one relevant reference [16] dealt with flow over a stepped spillway; the others simulated flow over a smooth spillway. Chen et al. [16] used the VOF method in conjunction with the k–e turbulence flow model and reported good agreement with experimental results for the free surface, vortex velocities, and pressure profiles at step surfaces.It is clear that there has been renewed interest in the use of stepped spillways in hydraulic engineering (as evidenced by the numerous experimental works on scale down models). However, there has been little progress in the computational modeling of such structures. Encouraged by the recent work by the authors [18] and recognizing the need for comprehensive computational simulations, the finite element method is used to model the flow over stepped spillways.2. Experimental setupThe experiments were performed in the hydraulics laboratory in a long flume with glass walls. Two pumps supplied the flow into the channel through calibrated orifice meters located in the feed pipes, with two independent valves to control the flow. Discharge was measured using these meters. The channel was kept at approximately zero slope throughout the experiments. Depth of water at any point was measured using a point gauge with accuracy to the nearest millimeter. In the case of fluctuating water surface profile, average values of depths were taken based on several measurements.The profile of the stepped spillway was designed based on the profile of a smooth spillway. Steps were introduced such that the envelope of their tips followed the smooth spillway chute profile. The hydraulic design charts 111-2-/1 of USACE-WES [19] were used for the design of the spillway profile. The upstream portion of the profile before the crest consisted of a vertical face followed by two circular arcs with radii 0.2Hd and 0.5Hd, where Hd is the design head. The downstream portion of the profile after the crest consisted of three segments. T coordinates is located at the crest. Using K =2, n = 1.85, and Hd = 5.08 cm (2 in) gives y = 0.1256x 1.85. The second segment is a straight line with a 60. inclination (slope of 1.73V:1H). The third segment is a circular arc providing a smooth transition between the sloped straight-line segment and the horizontal stilling basin.A total of four stepped spillway configurations were considered operating under a head equal to 1.5Hd. Each spillway was built by assembling two plexiglass parts: an upper part that included the crest region and whose height was 1/3Hspill and a bottom part that includedthe toe region and whose height was 2/3Hspill. The total height of the model spillway from the toe to the crest, Hspill, was equal to 380 mm. In the upper part as well as in the bottom part steps were introduced along the chute such that the envelope of their tips followed the smooth spillway chute profile. In each part the steps were either all large steps of height 1/20Hspill (19 mm), or all small steps of height 1/40Hspill (9.5 mm). The step configurations for all four cases are given in Table 1.While conducting the experiments, some measurements were repeated to ensure that the results are reproducible with the minimum possible errors. The following measurements were recorded: flow rate, water head at the upstream of the spillway, depth at crest, profile of free surface, depth at the bottom or the toe of the model spillway, depth at the downstream end of the hydraulic jump, and the length of the jump.3. Numerical analysisThe computational fluid dynamics module of the ADINA software, ADINA-F [20–23], was utilized to model the flow over the stepped spillway. ADINA-F is a general finite element code that can be used to model a wide range of fluid-flow problems. All four cases outlined in the experimental setup section were modeled. A smooth spillway was also modeled for comparison purposes. A salient feature of the modeling process is the accurate tracking of the entire watersurface: upstream before the crest, over the spillway, and downstream beyond the toe. Since the water surface is not known a priori, a simple yet reasonable surface consisting of three straight lines was used as an initial guess. Although the physical problem being modeled attained a steady state in the hydraulics laboratory, and thus it is reasonable to perform a steady-state analysis, a transient analysis was performed instead. As will be seen later in this paper, the transient analysis did converge to a steady-state solution. The choice of an appropriate flow model (laminar versus turbulent) depends largely on the regime in which the actual flow is most likely to exist. A local Reynolds number criterion [24] was used to confirm that the flow was turbulent in nature. The k–e turbulence flow model was adopted with all the default parameters as provided by ADINA-F.The initial computational domain for case 4 is shown in Fig. 1. A fixed wall boundary condition was imposed on the bottom edges and along the spillway (all lines labeled A). The initial water surface was modeled by the three straight lines labeled B. At the inlet (line labeled C) a uniform velocity equal to 0.1065 m/s was prescribed. This value was calculated by dividing the flow rate by the experimentally measured upstream water depth. The preceding description of the computational domain is applicable to all the other cases with no adjustments. The finite element mesh for case 4 is shown in Fig. 2; it consisted of 5760 triangular three-node element. The mesh resolution is highlighted at three different locations labeled 1, 2 and 3. A mesh with high resolution (small-sized elements relative to the step size) was used along all the steps in order to resolve (or capture) the anticipated vortices. Similar meshes were used in the other three cases. The total number of elements used for each case is given in Table 1.For all the cases considered here the solution process consisted of two phases: (1) Phase-I where the water surface in Fig. 1 was held fixed (by prescribing a fixed wall condition) and a transient solution was sought and (2)Phase-II where the nodal results form Phase-I were used as initial conditions and the water surface is treated as a free surface. This two-phase approach allowed for a faster convergence rate and avoided potential element overlap, particularly for elements whose edges were along the free surface. In ADINA-F a free surface is a moving boundary that is treated as an interface between a liquid and a gas that has negligible mass density and is thus considered as a vacuum. The transient solution in Phase-I was performed using 100 steps with a constant magnitude of 0.01 s for each step, totaling up to 1.0 s of simulation time. During this phase the inlet velocity was applied gradually through a ramp function that attained a unity value at time 1.0 s. The transient analysis in Phase-II was performed using 200 steps with a constant magnitude of 0.005 s for each step totaling up to 1.0 s of simulation time. During this phase the inlet velocity was maintained at 0.1065 m/s.4. Numerical results and discussionThe results of the Phase-II transient analysis were recorded for each of the 200 time steps. The evolution with time of the mesh geometry is shown in Fig. 3 for case 4. The portion of the free surface at the crest and downstream from the crest evolves drastically admitting a wave-like profile for the time span between 0.005 and 0.8 (which constitutes 160 steps). For the time span between 0.8 and 1.0 (which constitutes 40 steps), there are no noticeable changes in the mesh geometry and a steady-state solution is achieved. Similar mesh geometry evolution was observed for all the other cases.In all the ensuing figures the simulation results of Phase-II are reported at time equal to 1.0 s. The free surface profiles are shown in Fig. 4. For all of the four cases the predicted free surface acquires an acceptable and expected shape where: (a) the water surface closely follows the curvature of the crest, the straight line envelope joining the tips of the steps, and the curvature of the toe; and (b) an overall smooth water surface develops at the curvature transition points of the spillway surface. In addition to the acceptable qualitative predictions provided by the numericalresults, a comparison is shown in Fig. 4 to reveal the quantitative agreement with the experimentally measured profiles. Close agreement between the computed and measured profiles is achieved along the entire free surface for all four cases. However, an appreciable discrepancy does exist at a single point at the toe. This could be attributed to the difficulty in measuring the flow depth at the transition to the stilling basin.The velocity vector plots at three different locations downstream from the crest are shown in Fig. 5 for case 4. It is evident that skimming flow develops; where the flow of water skims over the step edges and recirculating zones (or vortices) develop in the triangular recess. A steady state configuration was attained for all time steps from 0.8 to 1.0 (which constitutes 40 steps). formedby the step faces and pseudobottom. This flow. Similar velocity vector plots were obtained for all pseudobottom closely follows the envelope joining the the other three cases.tips of the steps. As can be seen from the velocity legend With an operating head greater than Hd, negative in Fig. 5, the magnitude of the velocity of the recirculat-pressures may develop at certain locations in the spilling water is reduced by about 1/4 that of the skimming way thus increasing the potential for cavitationIn order to investigate the pressure distribution within a step and its variation from one step to another, the pressure profiles at two representative locations are shown in Fig. 6. The first location (which is close to the middle of the upper part of the spillway) cor responds to step number 3, 6, 3 and 6 for cases 1–4 respectively. The second location (which is close to the middle of the bottom part of the spillway) corresponds to step number 12, 24, 18 and 18 for cases 1–4 respectively. Steps are numbered sequentially from crest to toe. The pressure variation over the x and y coordinates of the step surfaces follows a similar pattern for all four cases (irrespective of the step size) and at both locations. First, along the horizontal surface and with increasing distance away from the corner (x-coordinate, Fig. 6and c), the pressure decreases and then increases to reach a maximum just before the tip of the step where it admits a sharp decrease at the tip. This maximum pressure is caused by the impact of the falling water on the step. Second, along the vertical surface Fig. 5. V elocity (m/s) vector plots at three locations downstream from the crest (case 4): (a) just after the crest, (b) at the transition from small to large steps, and (c) just before the toe. increasingdistance upward from the corner (y-coordi-The pressure variation along the straight chute of a nate, Fig. 6b and d), the pressure decreases continuously smooth spillway is also included in Fig. 6 for compariand admits a sharp variation close to the tip. Chen et al. son purposes. The smooth spillway experiences a nega[16] used the VOF method to model flow over a stepped tive or close to zero pressure at the first location andspillway. In their analysis they considered a single spill-small positive pressure at the second location.way with 13 steps and reported pressure patterns that Knowledge of the residual kinetic energy at the toe of are in close agreement with the ones described here. a spillway is crucial in the design of the stilling basin.Fig. 6. Pressure profile plots along the horizontal and vertical surfaces of steps at two.The energy loss over a spillway can be expressed as (H1 . H2)/H1, where H1 and H2 are the respective total heads in the channel upstream and downstream of the stepped spillway (the total energy head is computed as the sum of the height of water surface plus the velocity head). The computed values for this ratio were: 51.6%, 52.5%, 51.9%, and 49.8% for cases 1–4 respectively and 38.1% for the smooth spillway model. An alternative expression for energy loss is (H2S . H2)/H2S, where H2S is the total head at the downstream of a smooth spillway. The computed values for this ratio were: 21.1%, 22.6%, 21.7%, and 18.2% for cases 1–4 respectively. These values are comparable to the energy dissipation ratios reported by Chatila and Jurdi [11] which were close to 20%. This reduction in energy is appreciable specifically at an operating head equal to 1.5Hd. 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