Page 258 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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Finally, Step 3 can be written in a weak form as
n+1

∂p
T
T T CONVECTION IN POROUS MEDIA
[N] u i d
= [N] ˜u i d
− t [N] d
(8.41)

∂x i
Other field variables, such as temperature and concentration, can be established in a
similar fashion via Step 1 and will be discussed later.
The final matrix form of the assembled equations is obtained by introducing
Equation 8.38 into Equations 8.39 to 8.41 and are written in a matrix form, as follows:
Step 1: Intermediate velocity calculation
x 1 momentum component
n
[M p ]{ ˜ u 1 }= t −[C p ]{u 1 }− [K p ]{u 1 }− [M p1 ]{u 1 }− [M p2 ]{u 1 } +{f 1 } (8.42)
x 2 momentum component
n

[M p ]{ ˜ u 2 }= t −[C p ]{u 2 }− [K p ]{u 2 }− [M p1 ]{u 2 }− [M p2 ]{u 2 } +{f 2 } (8.43)
Step 2: Pressure field
n+1 1 n
[K p1 ]{p} =− [G p1 ]{˜ u 1 }+ [G p2 ]{˜ u 2 } −{f 3 } (8.44)
t
Step 3: Momentum correction
n+1
[M p ]{ u 1 }= [M p ]{ ˜ u 1 }− t[G p1 ]{p}
n+1
[M p ]{ u 2 }= [M p ]{ ˜ u 2 }− t[G p2 ]{p} (8.45)
The matrices in the above equations are the assembled global matrices. The elemental
matrices of the porous medium equations, for linear triangular elements, are (similar to the
ones reported in Chapter 7)
Elemental mass matrix
 
21 1
A
[M pe ] =  12 1  (8.46)
12
11 2
Elemental convection matrix
 
(usu + u 1i )b i (usu + u 1i )b j (usu + u 1i )b k
1
[C pe ] =  (usu + u 1j )b i (usu + u 1j )b j (usu + u 1j )b k 
24 2
(usu + u 1k )b i (usu + u 1k )b j (usu + u 1k )b k
 
(vsu + u 2i )c i (vsu + u 2i )c j (vsu + u 2i )c k
1
+  (vsu + u 2j )c i (vsu + u 2j )c j (vsu + u 2j )c k  (8.47)
24 2
(vsu + u 2k )c i (vsu + u 2k )c j (vsu + u 2k )c k
where
usu = u 1i + u 1j + u 1k
vsu = u 2i + u 2j + u 2k (8.48)