Page 205 - Process Modelling and Simulation With Finite Element Methods

192 Process Modelling and Sinzulation with Finite Element Methods
Ra eigenvalue
1705 0.024
1706 0.01 6
1707 0.009
1708 0.002
1709 -0.006
1710 -0.014
171 1 -0.022
1712 -0.029
1713 -0.037
Table 5.2 Decay rates -hi (largest eigenvalue) with Ra near critical for aspect ratio 1:10.

number leave no doubt that the critical mode oocurs near Rayleigh number 1708.
The triumph of eigenanalysis coupled with FEM is to find numerically the
critical Rayleigh number and approximate length scale associated with the
critical mode. Although the linear stability theory for this problem is not
cumbersome, for many situations with non-trivial and three-dimensional base
states, that cannot always be claimed. FEMLAB, through eigenanalysis,
provides a consistency check on linear stability analysis of stationary states. The
numerical technique is fair more robust, however. Eigenanalysis can be
conducted on any solution, even of transient problems. Recall equation (5.4)
shows that for self-adjoint operators, the eigenanalysis predicts the time
asymptotic dynamics of the linearized system. For non-self adjoint operators, it
has been demonstrated that pseudo-modes that are not eigenmodes can grow
rapidly before the time asymptotic eigenmodes dominate. Trefethen et al. [ 101
identified spiral pseudo-modes as leading to transitions to turbulence in Couette
and Poiseuille flows at much lower Reynolds Numbers than anticipated by linear
theory. This was confirmed experimentally. The extent to which eigenanalysis
of transient flow problems identifies the fastest growing pseudomode in transient
models for instantaneous states is a largely unexplored area, for non-self-adjoint
operators.
Here we have shown that for self-adjoint operators, the FEM model
accurately reproduces the predictions of linear stability theory.

5.3 Viscous Fingering Instabilities

The Benard problem is a paradigm for instabilities of a stationary state, Viscous
fingering is an instability of a non-homogeneous state in motion - a less viscous
fluid displacing a more viscous fluid. Figure 5.7 shows the flow configuration
for miscible viscous fingering, where diffusion tends to spread out viscous
fingers and oppose their formation. Nevertheless, viscous fingering is a long