Page 29 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

2.3 Solving Steady-State Natural Convection Problems in Fluid-Saturated Porous Media 15

a zero pore-fluid flow solution for natural convection of pore-fluid, even though
the Rayleigh number is high enough to drive the occurrence of natural convection
in a fluid-saturated porous medium. In order to overcome this difficulty, a modi-
fied Horton-Rogers-Lapwood problem, in which the gravity acceleration is assumed
to tilt a small angle α, needs to be solved. Supposing the original Horton-Rogers-
Lapwood problem has a Rayleigh number (Ra) and that the non-zero solution for the
modified Horton-Rogers-Lapwood problem is S(Ra,α), it is possible to find a non-
zero solution for the original Horton-Rogers-Lapwood problem by taking a limit of
S(Ra,α) when α approaches zero. This process can be mathematically expressed
as follows:

lim S(Ra,α) = S(Ra, 0), (2.42)
α→0
where S(Ra, 0) is a solution for the original Horton-Rogers-Lapwood problem;
S(Ra,α) is the solution for the modified Horton-Rogers-Lapwood problem; S is
any variable to be solved in the original Horton-Rogers-Lapwood problem.
It is noted that in theory, if S(Ra,α) could be expressed as a function of α
explicitly, S(Ra, 0) would follow immediately. However, in practice, it is necessary
to find out S(Ra, 0) numerically since it is very difficult and often impossible to
express S(Ra,α) in an explicit manner. Thus, the question which must be answered
is how to choose α so as to obtain an accurate non-zero solution, S(Ra, 0). From
the theoretical point of view, it is desirable to choose α as small as possible. The
reason for this is that the smaller the value of α, the closer the characteristic of
S(Ra,α) to that of S(Ra, 0). This enables a more accurate solution S(Ra, 0) to
be obtained in the computation. From the finite element analysis point of view, α
cannot be chosen too small because the smaller the value of α, the more sensitive
the solution S(Ra,α) to the initial velocity field of pore-fluid. As a result, a very
small α usually leads to a zero velocity field due to any inappropriate choice for the
initial velocity field of pore-fluid. To avoid this phenomenon, α should be chosen
big enough to eliminate the strong dependence of S(Ra,α) on the initial velocity
field of pore-fluid. For the purpose of using a big value of α and keeping the final
solution S(Ra, 0) of good accuracy in the finite element analysis, S(Ra,α) needs to
approach S(Ra, 0) in a progressive asymptotic manner, as clearly shown in Fig. 2.1.
This leads to the following processes mathematically:

lim S(Ra,α i ) = S(Ra,α i+1 ) (i = 1, 2, ......, n − 1), (2.43)
α i →α i+1
lim S(Ra,α n ) = S(Ra, 0), (2.44)
α n →0
1
α 1 = α, α i+1 = α i , (2.45)
R
where n is the total step number for α approaching zero; R is the rate of α i approach-
ing α i+1 . Generally, the values of α,n and R are dependent on the nature of a problem
to be analysed.