Page 210 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
202
pressure term from the momentum equations enhances the pressure stability and allows
the use of arbitrary interpolation functions for both velocity and pressure. In other words,
the well-known Babuska–Brezzi condition is satisfied. Owing to the split introduced in the
equations, the method is referred to as the Characteristic Based Split (CBS) scheme.
The CG procedure may be applied to the individual momentum components without
removing the pressure term, provided the pressure term is treated as a source term. However,
such a procedure will lose the advantages mentioned in the previous paragraph.
For more mathematical details, readers are directed to earlier publications on the method
(Zienkiewicz and Codina 1995; Zienkiewicz and Taylor 2000) and for recent developments,
references (Nithiarasu 2003; Zienkiewicz et al. 1999) are recommended. In order to apply
the CG procedure, the governing equations in two dimensions (note that body forces are
not included for simplicity) may be written as follows:

Continuity equation
∂u 1 ∂u 2
+ = 0 (7.123)
∂x 1 ∂x 2
x 1 momentum equation
!
2
2
∂u 1 ∂u 1 ∂u 1 1 ∂p ∂ u 1 ∂ u 1
+ u 1 + u 2 =− + ν + (7.124)
∂t ∂x 1 ∂x 2 ρ ∂x 1 ∂x 2 ∂x 2
1 2
x 2 momentum equation
2 2 !
∂u 2 ∂u 2 ∂u 2 1 ∂p ∂ u 2 ∂ u 2
+ u 1 + u 2 =− + ν + (7.125)
∂t ∂x 1 ∂x 2 ρ ∂x 2 ∂x 2 ∂x 2
1 2
Energy equation
2 2 !
∂T ∂T ∂T ∂ T ∂ T
+ u 1 + u 2 = α + (7.126)
∂t ∂x 1 ∂x 2 ∂x 2 ∂x 2
1 2
From the governing equations, it is obvious that the application of the CG scheme is
not straightforward. However, by implementing the following steps, it is possible to obtain
a solution to the convection heat transfer equation.
Step 1 Intermediate velocity or momentum field: This step is carried out by removing
the pressure terms from Equations 7.124 and 7.125. The intermediate velocity component
equations, in their semi-discrete form, are
intermediate x 1 momentum equation
! n
n
˜ u 1 − u n n 2 2
1 ∂u 1 ∂u 1 ∂ u 1 ∂ u 1
+ u 1 + u 2 = ν + (7.127)
t ∂x 1 ∂x 2 ∂x 2 ∂x 2
1 2
intermediate x 2 momentum equation
! n
n
˜ u 2 − u n n 2 2
2 ∂u 2 ∂u 2 ∂ u 2 ∂ u 2
+ u 1 + u 2 = ν + (7.128)
t ∂x 1 ∂x 2 ∂x 2 ∂x 2
1 2