Page 34 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

20 2 Simulating Steady-State Natural Convective Problems

864 nine-node quadrilateral elements of 3577 nodes in total. The mesh gradation
technique, which enables the region in the vicinity of problem boundaries to be
modelled using finite elements of small sizes, has been employed to increase the
solution accuracy in this region. The following parameters associated with the pro-
o
gressive asymptotic approach procedure are used in the calculation: α = 5 , n = 2
and R = 5.
Figures 2.4, 2.5, 2.6 and 2.7 show the comparison of numerical solutions with
analytical ones for dimensionless velocity, stream function, temperature and pres-
sure modes respectively. In these figures, the plots above are analytical solutions,
whereas the plots below are numerical solutions for the problem. It is observed from
these results that the numerical solutions from the progressive asymptotic approach
procedure associated with the finite element method are in good agreement with the
analytical solutions. Compared with the analytical solutions, the maximum error in
the numerical solutions is less than 2%. This demonstrates the usefulness of the
present progressive asymptotic approach procedure when it is used to solve the
steady-state Horton-Rogers-Lapwood problems.
At this point, there is a need to explain why both the analytical and the numer-
ical solutions for the pore-fluid flow are non-symmetric, although the geometry
and boundary conditions for the problem are symmetric. As stated previously, the
Horton-Rogers-Lapwood problem belongs mathematically to a bifurcation problem.
The trivial solution for the pore-fluid flow of the problem is zero. That is to say, if
the Rayleigh number of the problem is less than the critical Rayleigh number, the
solution resulting from any small disturbance or perturbation converges to the trivial

Fig. 2.4 Comparison of numerical solution with analytical solution (Dimensionless velocity)