Figure 4a. Flow in a spillway: boundaries of the computational domain.
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Figure 4b. Flow in a spillway: mesh deformation mechanism.
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Figure 4c. Flow in a spillway: streamwise velocity and free surface position.
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The DSD/SST formulation has been used to simulate a flow in a spillway of a dam. The computational domain, shown in Figure 4a, is 500 ft long and 32 ft wide. The upstream and downstream water levels are at a reference height of 301.5 ft and 280 ft, respectively. The spillway is modeled after the existing Olmsted dam on the Ohio river, which is being studied by the U.S. Army Corps of Engineers Engineering, Research and Development Center for erosion abatement. The spillway bed includes four obstacles designed to disspate the energy of the flow. Periodic boundary conditions are applied at the lateral boundaries, in order to simulate a wider section of the dam.

The mesh consists of 147,256 space-time nodes and 418,249 tetrahedral elements. To allow for periodicity, the surface meshes at both lateral boundaries are identical. For the purpose of mesh motion, the mesh is assumed to act as a linearly elastic solid, with the displacements of the interior nodes determined by the displacement of the boundaries. The response of the interior mesh to the motion of the free surface is illustrated in Figure 4b. The free surface elevation is governed by a kinematic condition expressed in a variational form.

The fluid enters the upstream boundary with a free-stream velocity of (7.42, 0.0, 0.0) ft/s. The Reynolds number based on the upstream velocity and the upstream water depth is approximately 1x10⁷. In the Smagorinsky turbulence model we use C=0.15. Slip is allowed at all spillway surfaces, and a normal stress corresponding to the target downstream depth is imposed at the outflow. The steady-state Stokes flow solution serves as the initial condition, and the unsteady flow is computed for 1000 time steps with a time step size of 0.05 s. Figure 4c illustrates the evolution of the free surface from initial fictitious state, and the color-coded streamwise velocity distribution at selected domain surfaces.

This computation has been carried out on the CRAY T3E-1200. At every time step the coupled, nonlinear equations are solved with 5 Newton-Raphson iterations. The coupled, linear equations that need to be solved at each Newton-Raphson step are solved also iteratively, with the GMRES update techniques with a Krylov space size of 50. More information on this approach and on this simulation can be found in Güler et al. (1999) and Behr et al. (1999).