Page 218 - Process Modelling and Simulation With Finite Element Methods

Simulation and Nonlinear Dynamics 205

Now pull down the Solve menu and select the Parameters option. This pops
up the Solver Parameters dialog window.

Solver Parameters
8 Select time stepping tab, set output times 0:0.01:0.5
Set time stepping algorithm odel5s
Select general tab, set solution form to weak, solver type time
dependent
Solve
Cancel
OK
You should animate the solution. The final time should look similar to Figure
5.14. Compare with Figure 5.15 which shows the initial condition. The red band
is the initial slug of unit concentration. Clearly, the nonuniform streamlines are
due to the vorticity generated by the concentration noise. The effect of
channeling of the less viscous fluid in the slug eventually becomes so
pronounced that the upstream and downstream more viscous fluid connects
through the slug, isolating islands of the less viscous fluid. In an immiscible
fluid, a topological change forming droplets would have had to have occurred.
Here, the interface remains smooth due to diffusive mixing, so channeling has
occurred. The longer the slug, the less likely the complete channeling through
the slug is to occur in a fixed time. The trailing front of the slug is stable since it
has less viscous fluid displaced by more viscous fluid. As the animation shows,
it is the instability of the leading front that becomes so pronounced, it eventually
breaks through the slug. The success of this simulation in capturing unstable
frontal dynamics , in comparison with the first model in the Darcy’s Law mode,
is predicted solely on the use of periodic horizontal boundary conditions.
Otherwise, the model equations are dynamically equivalent to the built-in modes
that were used before.
Figure 5.16 shows the formation of viscous fingers just on the leading
edge with accompanying vorticity generation. Otherwise, the trailing edge
simply has diffused out somewhat with still uniformity in the cross-stream
direction.
Figure 5.17 shows the leading edge gouging out large, broad fingers into the
slug. Viscous fingering is not a symmetric process - back-fingering of the more
viscous fluid into the less viscous displacing fluid does not necessary mirror the
dynamics of forward fingering. Here, the slug is poor in less viscous fluid.
Figure 5.14 shows that we eventually use up the supply of finger-forming less
viscous fluid if the slug length is too short in comparison to its breadth. Figure
5.17 has much shorter fingers than seen in [ 141 with a long slug.