Page 198 - Process Modelling and Simulation With Finite Element Methods

Simulation and Nonlinear Dynamics 185

all the eigenvalues are real. It follows that the unstable mode is not propagating,
but stationary and growing in strength until it saturates.
To permit the possibility that waves might propagate, however, we need to
change the horizontal boundary conditions from the earlier simulation which had
no flux and slip boundary conditions on the horizontal bounding planes. Gravity
waves cannot propagate through such planes, since they are transverse and
require up-and-down motion. Furthermore, the model was stationary, so
although complex eigenvalues are possible, transient motion was prohibited. To
implement periodic boundary conditions, a minor change is necessary.

Recipe for Periodic Boundary Conditions

Pull down the Mesh menu and select the Parameters option.

Mesh Parameters
Set symmetry boundaries: 1 4
Apply
OK

These boundaries are now equivalent, but not necessarily periodic. To make that
constraint, we need to require that all variables are identical on the boundary
nodes. Pull down the Boundary menu and select View as Coefficients, then
select Boundary Settings. All four equations are displayed simultaneously in
Coefficient View

Click on the h-tab, and set the main diagonal to 1 and the off
diagonal elements to zero. Make sure the r-tab has u,v,p, and T
for the four equations.
Select domain 4
Click on the h-tab, and set the main diagonal to -1 and the off
diagonal elements to zero. Make sure the r-tab has -u,-v,-p, and
-T for the four equations.
Apply
OK
Since the boundaries 1 and 4 are equivalent, these two conditions are
sequentially added to the boundary constraints. So the condition on u is that u
on boundary I and -u on boundary 4 sum to zero. The same holds for v, p, and
T, ensuring horizontal periodicity of all variables.
Because the domain is about as long as the most dangerous mode
(wavenumber 3.117 implies that we have nearly a period of 2d3.117 = 2, i.e.