Page 259 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
Here, i, j and k represent the three nodes of a linear triangular element. Refer to
Chapter 7 for the definitions of b i , b j , b k , c i , c j and c k . The momentum diffusion matrix is
 2   2  251
b b i b j b i b k c c i c j c i c k
1 i 2 1 i 2
[K me ] =  b j b i b j b j b k  +  c j c i c j c j c k  (8.49)
4ARe 2 4ARe 2
b k b i b k b j b c k c i c k c j c
k k
The characteristic stabilization matrices have been ignored, but can be included for the
purpose of oscillations at very high Reynolds and Rayleigh numbers (see Chapter 7). At
lower Reynolds and Rayleigh numbers, however, these terms may be neglected in order to
save computational time.
The matrix form of the discretized second-order term for Step 2 is
b b i b j b i b k c c i c j c i c k
 2   2 
1 i 2 1 i 2
[K p1e ] =  b j b i b j b j b k  +  c j c i c j c j c k  (8.50)
4A 2 4A 2
b k b i b k b j b c k c i c k c j c
k k
The first-gradient matrix in the x 1 direction is
 
b i b j b k
1
[G p1e ] =  b i b j b k  (8.51)
6
b i b j b k
and the second-gradient matrix in the x 2 direction is

 
1 c i c j c k
[G p2e ] =  c i c j c k  (8.52)
6
c i c j c k
The matrices due to the fluid drag on the solid are

1
[M p1e ] = [M pe ]
Re
C |V|
[M p2e ] = √ [M pe ] (8.53)
Da 3/2
The forcing vectors (boundary terms) are, for the x 1 momentum component,

 n

1 b i u 1i + b j u 1j + b k u 1k
{f 1 }=  b i u 1i + b j u 1j + b k u 1k  n 1
4A Re
0
 n

c i u 1i + c j u 1j + c k u 1k
1
+  c i u 1i + c j u 1j + c k u 1k  n 2 (8.54)
4A Re
0