【文章內(nèi)容簡介】 ble to perform a steadystate analysis, a transient analysis was performed instead. As will be seen later in this paper, the transient analysis did converge to a steadystate 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 ADINAF. The initial putational 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 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 putational 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 threenode element. The mesh resolution is highlighted at three different locations labeled 1, 2 and 3. A mesh with high resolution (smallsized 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) PhaseI where the water surface in Fig. 1 was held fixed (by prescribing a fixed wall condition) and a transient solution was sought and (2) PhaseII where the nodal results form PhaseI were used as initial conditions and the water surface is treated as a free surface. This twophase approach allowed for a faster convergence rate and avoided potential element overlap, particularly for elements whose edges were along the free surface. In ADINAF 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 PhaseI was performed using 100 steps with a constant magnitude of s for each step, totaling up to s of simulation time. During this phase the inlet velocity was applied gradually through a ramp function that attained a unity value at time s. The transient analysis in PhaseII was performed using 200 steps with a constant magnitude of s for each step totaling up to s of simulation time. During this phase the inlet velocity was maintained at m/s. 4. Numerical results and discussion The results of the PhaseII 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 wavelike profile for the time span between and (which constitutes 160 steps). For the time span between and (which constitutes 40 steps), there are no noticeable changes in the mesh geometry and a steadystate solution is achieved. Similar mesh geometry evolution was observed for all the other cases. In all the ensuing figures the simulation results of PhaseII are reported at time equal to 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 numerical results, a parison is shown in Fig. 4 to reveal the quantitative agreement with the experimentally measured profiles. Close agreement between the puted 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 to (which constitutes 40 steps). formed by the step faces and pseudobottom. This fl