Page 22 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

8 2 Simulating Steady-State Natural Convective Problems

different materials, numerical methods are always needed to solve the aforemen-
tioned problems.
From the mathematical point of view, the Horton-Rogers-Lapwood problem pos-
sesses a bifurcation. The linear stability theory based on the first-order perturbation
is commonly used to solve this problem analytically and numerically (Nield 1968,
Palm et al. 1972, Caltagirone 1975, 1976, Combarnous and Bories 1975, Buretta
and Berman 1976, McKibbin and O’Sullivan 1980, Kaviany 1984, Lebon and Cloot
1986, Pillatsis et al. 1987, Riley and Winters 1989, Islam and Nandakumar 1990,
Phillips 1991, Nield and Bejan 1992, Chevalier et al. 1999). However, Joly et al.
(1996) pointed out that: “The linear stability theory, in which the nonlinear term
of the heat disturbance equation has been neglected, does not describe the ampli-
tude of the resulting convection motion. The computed disturbances are correct
only for infinitesimal amplitudes. Indeed, even if the form of convective motion
obtained for low supercritical conditions is often quite similar to the critical distur-
bance, the nonlinear term may produce manifest differences, especially when strong
constraints, such as impervious or adiabatic boundaries, are considered.” Since it is
the amplitude and the form of natural convective motion that significantly affects or
dominates the contaminant transport and mineralization in a fluid-saturated porous
medium, there is a definite need for including the full nonlinear term of the energy
equation in the finite element analysis.
From the finite element analysis point of view, the direct inclusion of the full
nonlinear term of the energy equation in the steady-state Horton-Rogers-Lapwood
problem would result in a formidable difficulty. The finite element method needs
to deal with a highly nonlinear problem and often suffers difficulties in establish-
ing the true non-zero velocity field in a fluid-saturated porous medium because the
Horton-Rogers-Lapwood problem always has a zero solution as one possible solu-
tion for the velocity field of the pore-fluid. If the velocity field of the pore-fluid
used at the beginning of an iteration method is not chosen appropriately, then the
resulting finite element solution always tends to zero for the velocity field in a
fluid-saturated porous medium. Although this difficulty can be circumvented by
turning a steady-state problem into a transient one (Trevisan and Bejan 1987), it
is often unnecessary and computationally inefficient to obtain a steady-state solu-
tion from solving a transient problem. Therefore, it is highly desirable to develop
a numerical procedure to directly solve the steady-state Horton-Rogers-Lapwood
problem. For this reason, a progressive asymptotic approach procedure has been
developed in recent years (Zhao et al. 1997a, 1998a). The developed progressive
asymptotic approach procedure is based on the concept of an asymptotic approach,
which was previously and successfully applied to some other fields of the finite ele-
ment method. For instance, the h-adaptive mesh refinement (Cook et al. 1989) is
based on the asymptotic approach concept and can produce a satisfactory solution
with the progressive reduction in the size of finite elements used in the analysis.
The same asymptotic approach concept was also employed to obtain asymptotic
solutions for natural frequencies of vibrating structures in a finite element analysis
(Zhao and Steven 1996a, b, c). To solve the steady-state Horton-Rogers-Lapwood
problem with the full nonlinear term of the energy equation included in the finite