Page 43 - Fundamentals of Computational Geoscience Numerical Methods and Algorithms

2.6 Application of the Progressive Asymptotic Approach Procedure 29

of point A in the ξ, η and ζ directions of a local coordinate system; x i , y i and z i are
the coordinate components of nodal point i in the x, y and z directions of the global
coordinate system; n is the total nodal number of the element containing point A; φ i
is the interpolation function of node i in the element containing point A.
In Eqs. (2.70), (2.71) and (2.72), the coordinate components of point A in the
global coordinate system are known, so that the coordinate components of this point
in the local system can be determined using any inverse mapping technique (Zhao
et al. 1999f). Once the coordinate components of point A in the global coordinate
system are determined, the velocity components of point A in the global system can
be straightforwardly calculated as follows:

n

u A = φ i (ξ A ,η A ,ζ A )u i (2.73)
i=1
n

v A = φ i (ξ A ,η A ,ζ A )v i (2.74)
i=1
n

w A = φ i (ξ A ,η A ,ζ A )w i , (2.75)
i=1
where u A , v A and w A are the velocity components of point A in the x, y and z direc-
tions of the global coordinate system; u i , v i and w i are the velocity components of
nodal point i in the x, y and z directions of the global coordinate system, respectively.
The above-mentioned process indicates that in the finite element analysis of fluid
flow problems, the trajectory of any given particle can be calculated using the nodal
coordinate and velocity components, which are fundamental quantities and therefore
available in the finite element analysis.
To demonstrate the applicability of the progressive asymptotic approach proce-
dure for simulating convective pore-fluid flow in three dimensional situations, the
3
example considered in this section is a cubic box of 10 × 10 × 10 km in size. This
box is filled with pore-fluid saturated porous rock, which is a part of the upper crust
of the Earth. In order to simulate geothermal conditions in geology, the bottom of the
box is assumed to be hotter than the top of the box. This means that the pore-fluid
saturated porous rock is uniformly heated from below. For the system considered
here, the classical analysis (Phillips 1991, Nield and Bejan 1992, Zhao et al. 1997a)
indicates that the convective flow is possible when the Rayleigh number of the sys-
tem is either critical or supercritical. For this reason, the parameters and properties
of the system are deliberately selected in such a way that the Rayleigh number of
the system is supercritical.
Figure 2.12 shows the finite element mesh of 8000 cubic elements for the three
dimensional convective flow problem. For the purpose of investigating the pertur-
bation direction on the pattern of convective flow, two cases are considered in the
following computations. In the first case, the perturbation of gravity is applied in the
x-z plane only. This means that the problem is axisymmetrical about the y axis so that