Page 35 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

2.5 Verification of the Proposed Progressive Asymptotic Approach 21

Fig. 2.5 Comparison of numerical solution with analytical solution (Dimensionless stream
function)

solution. In this case, the solution for the pore-fluid flow is zero (and, of course, sym-
metric) and the system is in a stable state. However, if the Rayleigh number of the
problem is equal to or greater than the critical Rayleigh number, the solution result-
ing from any small disturbance or perturbation may lead to a non-trivial solution.
In this situation, the solution for the pore-fluid flow is non-zero and the system is in
an unstable state. Since the main purpose of this study is to find out the non-trivial
solution for problems having a high Rayleigh number, Ra ≥ Ra critical ,asmalldis-
turbance or perturbation needs to be applied to the system at the beginning of a
computation. This is why gravity is firstly tilted a small angle away from vertical
and then gradually approaches and is finally restored to vertical in the proposed pro-
gressive asymptotic approach procedure. It is the small perturbation that makes the
non-trivial solution non-symmetric, even though the system considered is symmet-
ric. In addition, as addressed in Sect. 2.3, the solution dependence on the amplitude
of the initially-tilted small angle can be avoided by making this angle approach zero