Page 261 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION IN POROUS MEDIA
The Step 2 pressure calculation becomes
1 ∂ 2 n+θ 1 1 C |V| ∂ ˜u i 253
(p ) = + + √ (8.59)
∂x 2 t ReDa Da 3/2 ∂x i
i
Step 3 is also different and is given as

1 1 C |V| n+1
+ + √ u i =
t ReDa Da 3/2
1 1 C |V| 1 (∂p )
n+θ
+ + √ ˜ u i − (8.60)
t ReDa Da 3/2 ∂x i
Although extra complications were introduced in the semi-implicit form at Step 1 for steady
state solutions, we can avoid simultaneous solution of the algebraic equations by taking
the coefficient

1 1 C |V|
CO = + + √ 3/2 (8.61)
t ReDa Da
on to the RHS. Thus, the system can be enabled for the mass lumping procedure (Nithiarasu
and Ravindran 1998) when discretized in space. The final matrix form of the three steps are

Step 1: Intermediate velocity calculation
x 1 momentum component

{u 1 } −1 n
[M p ]{˜ u 1 }= [M p ] + CO −[C p ]{u 1 }− [K p ]{u 1 }+ {f 1 } (8.62)
t
x 2 momentum component

{u 2 } −1 n
[M p ]{˜ u 2 }= [M p ] + CO −[C p ]{u 2 }− [K p ]{u 2 }+ {f 2 } (8.63)
t
Step 2: Pressure field

n+1 CO n
[K p1 ]{p} =− [G p1 ]{˜ u 1 }+ [G p2 ]{˜ u 2 }− {f 3 } (8.64)
t
Step 3: Momentum correction
n+1 −1 n+1
[M p ]{u 1 } = [M p ]{˜ u 1 }− CO [G p1 ]{p}
n+1 −1 n+1
[M p ]{u 2 } = [M p ]{˜ u 2 }− CO [G p2 ]{p} (8.65)
The quasi-implicit (QI) form is very similar to that of the above scheme but now the
viscous, second-order terms are also treated implicitly (θ 2 = 1) (Nithiarasu et al. 1997).
The important difference, however, is that the quasi-implicit scheme does not benefit from
mass lumping when solving for the intermediate velocity values. A simultaneous solution
of the LHS matrices is essential here. It has been proven that both the QI and SI schemes
generally perform well (Nithiarasu 2001).