Page 229 - Process Modelling and Simulation With Finite Element Methods

216 Process Modelling and Simulation with Finite Element Methods

Re=- PUD
P

This dimensionless parameter rolls the influences of the fluid density p, inlet
velocity U, the diameter D, and viscosity p into one parameter that describes the
dynamic similarity of the flow. Thus changes in the pipe diameter for fully
developed flow are not classed as geometric variation, but rather the more
common parametric variation.
Just as in the last chapter where examples of simulations were given for
stationary models and for transient models, in this chapter we will give examples
of stationary geometric continuation and transient geometric continuation. In the
former, distinct models are solved with slightly different domains and therefore
different meshes. Therefore the solutions are incompatible (different degrees of
freedom) from one geometric parameter value to the next. If the old solution is
to be taken as the initial guess for new geometry, then mapping the old solution
to the new domain in a consistent fashion must be done. In the examples given
here, the system of PDEs for the stationary models are linear, so the solution can
be determined directly in one FEM step. Thus mapping old solutions onto the
new geometry has no additional value. The transient problem, however,
involves a problem in a shrinking domain with a moving front. The domain
changes after each time step, so the mapping of the solution at the old time step
onto the new domain is essential to the model. Consequently, after each time
step, re-meshing must be done as well. One class of problem where this is a
crucial step is the free boundary problem. Film flows and jet flows, for instance,
are cases where the position of the boundary is intimately related to the solution
of the velocity field. The boundaries should be located wherever the stress
balances are satisfied.
The 2-D incompressible, laminar Navier-Stokes equations can be solved by
several standard means (finite difference, finite element, spectral element, lattice
Boltzmann, and multigrid techniques) and have been implemented in standard
simulation engines commercially with fixed boundary conditions and complex
geometries. Standard computational fluid dynamics packages have two standard
engines: (1) the grid generator to cater for complex geometry, and (2) the PDE
engine, which can solve more general systems of transport equations that include
the pressure as in the Navier-Stokes equations as a Lagrange multiplier for the
continuity equation. These two steps are typically conducted separately. The
grid is generated initially, and thereafter many simulations are conducted.
FEMLAB is no different in this respect.
This paradigm for computational fluid dynamics does not deal particularly
well with free boundary problems. An iterative scheme for coupling the flow
solution to grid generation could be envisaged, but automation with standard
packages is difficult to implement. Ruschak [l] described the now standard