Page 28 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

14 2 Simulating Steady-State Natural Convective Problems

e T
M = ΨΨ dA. (2.40)
p
A
It needs to be pointed out that ε is a penalty parameter in Eq. (2.37). For the pur-
pose of obtaining an accurate solution, this parameter must be chosen small enough
to approximate fluid incompressibility well, but large enough to prevent the result-
ing matrix problem from becoming too ill-conditioned to solve.
By assembling all elements in a system, the finite element equation of the system
can be expressed in a matrix form as

Q −B U F F
= , (2.41)
0 E(U) T G
where Q, B and E are global property matrices of the system; U F and T are global
nodal velocity and temperature vectors of the system; F and G are global nodal load
vectors of the system. Since Equation (2.41) is nonlinear, either the successive sub-
stitution method or the Newton-Raphson method can be used to solve this equation.

2.3 The Progressive Asymptotic Approach Procedure
for Solving Steady-State Natural Convection Problems
in Fluid-Saturated Porous Media

To solve the steady-state Horton-Rogers-Lapwood problem with the full nonlinear
term of the energy equation included in the finite element analysis, the asymptotic
approach concept (Cook et al. 1989, Zhao and Steven 1996a, b, c) needs to be used in
a progressive fashion (Zhao et al. 1997a). If the gravity acceleration is assumed to tilt
at a small angle, α, in the Horton -Rogers-Lapwood problem, then a non-zero veloc-
ity field in a fluid-saturated porous medium may be found using the finite element
method. The resulting non-zero velocity field can be used as the initial velocity field
of the pore-fluid to solve the original Horton-Rogers-Lapwood problem with the
tilted small angle being zero. Thus, two kinds of problems need to be progressively
solved in the finite element analysis. One is the modified Horton-Rogers-Lapwood
problem, in which the gravity acceleration is tilted a small angle, and another is
the original Horton-Rogers-Lapwood problem. This forms two basic steps of the
progressive asymptotic approach procedure. Clearly, the basic idea behind the pro-
gressive asymptotic approach procedure is that when the small angle tilted by the
gravity acceleration approaches zero, the modified Horton-Rogers-Lapwood prob-
lem asymptotically approaches the original one and as a result, a solution to the
original Horton-Rogers-Lapwood problem can be obtained.
Based on the basic idea behind the progressive asymptotic approach procedure,
the key issue of obtaining a non-zero pore-fluid flow solution for the Horton-Rogers-
Lapwood problem is to choose the initial velocity field of pore-fluid correctly. If the
initial velocity field is not correctly chosen, the finite element method will lead to