Page 191 - Process Modelling and Simulation With Finite Element Methods

178 Process Modelling and Simulation with Finite Element Methods

solutions excited due to numerical noise. Undoubtedly, users of FEMLAB have
already found a straightforward application of nonlinear dynamics theory -
conditioning the computational model on the basis of dimensionless parameters
in the system. Our second case study illustrates the importance of resolving all
scales of the complex system which naively range from the large scale of the
geometric boundaries (dimensionlessly this is termed O( 1) or order unity since
the lengths are usually scaled by a geometric length) down to some small scale
set by nonlinear processes coupled with dissipation. If the parameter that
characterizes nonlinearity is called R, and complex behaviour increases with
increasing R, then one expects creation of complexity down to lengths O(R-’).
Thus, in regions generating complexity, the mesh should be gridded with
resolution O(R-’). Novice modellers routinely fail to recognize that no
satisfactory solution may emerge if all the physics generating complexity are not
resolved. Some physical processes routinely generate large complexity
parameters. Buoyant convection usually has large Rayleigh number Ra.
Pipeline flows are almost always at large Reynolds number Re. Heat transfer
almost always has large Peclet number Pe. Simply, given the small values
naturally occurring for transport coefficients, human scale flows lead to large
complexity parameters. Convergence to a solution does not guarantee that the
dynamics of the model are resolved. Careful modeling requires mesh refinement
studies until a claim that “refining the mesh does not change the result
appreciably” is fully justified. Even experienced modelers can fall into the trap
of unresolved computational models due to the large complexity parameter
problem. For instance, if there are still unresolved motions, but little “sub-grid”
energy transfer, it is convenient to think that laminar solutions to the buoyant
convection problems in double diffusion are, for example, able to ignore small
scale dynamics. Chascheskin et al. [4] argue cogently that there is never a
stationary solution to the double diffusion problem with vertically heated walls.
Internal boundary currents are automatically excited, leading to sharp fine
structure layering the flow. This feature is not captured by high solutal/thermal
Rayleigh number convection since it is not possible even with typical high
performance computing resources. So the “large eddy” simulations with low
subgrid fluxes may still be unresolved, even if there is little change on mesh
refinement - fine structure may influence global dynamics.

5.2 Rayleigh-Benard Convection

Rayleigh-Benard convection is certainly the canonical problem for nonlinear
dynamics and flow stability. You can visualize it by heating vegetable oil in
your kitchen, sprinkling cocoa powder on the surface of a thin layer of oil heated
from below in a frying pan. The hexagonal patterns are clearly visible unless
your cocoa powder has congealed. Still an excellent reference on the history of