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Simulation Steps

Preliminary Step
The simulation was run on 16 processors on a parallel computer. To distribute the load, a partitioning program (METIS) assigns an equal number of elements to each processor by renumbering them in such a way that all the elements that will be processed by a particular processor form a sequence of consecutive integers. The output file is an integer permutation array.

Three Stages of The Simulation

1

Stokes Stage
The Navier-Stokes equations, in their complete form, take into account inertial effects and unsteadiness. The exact state of flow at t=0 is not crucial to the final solution, but we still need to start with a reasonable approximation. To achieve it, we begin with a simplified version of the equations, assuming no inertia or time-dependence, and we allow no deformations of the mesh for the first 50 iterations. The resulting flow (see figure above) is simplistic, but a good enough starting point.
Notice, for instance, how the water flows fastest (red) and slowest (blue) more or less where you would expect.

2

Navier-Stokes simulation with space-time treatment
Now we may allow surface deformation (see figure) and take inertia and time-dependence into consideration for a much more realistic flow. Instead of four nodes per tetrahedral element, the time-space simulation uses eight, and the number of nodes must be doubled accordingly in the input files. For each time-step, several nonlinear iterations are performed, in the following order:
  1. Solve for position of the surface (advection equation)
  2. Solve for node displacement (linear elasticity)
  3. Solve for flow field (N-S equations)
  4. Return to 1.

3

Reduced Stabilization
After 400 time-steps, much of the initial turbulence has subsided and we can reduce the stabilization parameter from 2 to 1 (see next page).


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