Figure 1. Flow in a spillway:
Free surface position and streamwise
velocity field at three instants.

Among moving boundary problems, perhaps most challenging is the case of free-surface flows, which involve fluid boundaries whose position is not determined a priori, and has to be obtained as part of the modeling procedure. The space-time finite element formulation of incompressible flows is applicable to free surface flow situations, under the constraints typical of interface tracking methods. These constraints dictate that the deformation of the domain should remain reasonable, and joining and folding of the free surface is avoided. The flows past hydraulic structures such as dams and bridge supports fall into this category. A goal of this research has been to design robust simulation methods for flows which are of importance, e.g., to the U.S. Army Corps of Engineers Waterways Experiment Station.

In addition to the space-time formulation of fluid flow equations in a variable domain, the set of numerical tools we developed includes two other important components. They are a representation of the kinematic conditions which govern the free surface motion, and a mesh update method, which provides interior node displacements for any given deformation of the exterior boundary. The free surface motion in this class of problems may be best characterized by the surface height equation. The hyperbolic nature of this equation requires the use of stabilization techniques such as Streamline-Upwind/Petrov-Galerkin (SUPG). Moreover, steep wave fronts may have to be controlled via a discontinuity capturing operator.

The movement of the interior nodes of the finite element mesh can be obtained by treating the mesh as an elastic solid subjected to prescribed boundary displacements, which are obtained from the discretized kinematic conditions mentioned above. Our research involves two approaches: one approach utilizes mesh movement equations based on linear elasticity; a more robust approach takes into account geometric and constitutive nonlinearities and uses hyperelastic stress-strain relations.

As an application, these methods have been used to compute the flow in a spillway of the Olmsted dam on the Ohio river. The model of the spillway includes underwater energy dissipators and a long stilling basin. In the simulation, the initial condition of flat water surface quickly gives way to a quasi-steady pattern of hydraulic jumps and waves which has been satisfactorily compared with experimental results. The position of the free surface, as well as the color-coded streamwise velocity field, is shown at three equally-spaced instants in Figure 1. An infrequent regeneration of the finite element mesh is sometimes necessitated by the large distortion of the free surface.

Next: Extrusion Dies

References

• Waterways simulation logs: ohio.0418 • cols.0163 • halfpipe.0068