Page 218 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
210
Step 1: Intermediate velocity calculation x 1 component
n
n
n
[M] {˜ u 1 } =−[C]{u 1 } − [K m ]{u 1 } − [K s ]{u 1 } +{f 1 } (7.158)
t
and for the x 2 component
{˜ u 2 } n n n
[M] =−[C]{u 2 } − [K m ]{u 2 } − [K s ]{u 2 } +{f 2 } (7.159)
t
Step 2: Pressure calculation
n 1
[K]{p} =− [G 1 ]{˜ u 1 }+ [G 2 ]{˜ u 2 } +{f 3 } (7.160)
t
Step 3: Velocity correction
n+1 n
[M]{u 1 } = [M]{˜ u 1 }− t[G 1 ]{p}
n+1 n
[M]{u 2 } = [M]{˜ u 2 }− t[G 2 ]{p} (7.161)
Step 4: Temperature calculation
{T} n n n
[M] =−[C]{T} − [K t ]{T} − [K s ]{T} +{f 4 } (7.162)
t
The above four steps are the cornerstone of the CBS scheme for the solution of the
heat convection equations. An extension of the above steps for solving the conservation
form and three-dimensional equations is straightforward. Interested readers should consult
some of the appropriate publications (Nithiarasu 2003; Zienkiewicz et al. 1999).
The mass matrix [M] used in the above steps may be ‘lumped’ to simplify the solution
procedure. This is an approximation, but a worthwhile and time-saving approximation. Mass
lumping will eliminate the need for the matrix solution procedure necessary for consistent
mass matrices. The lumped mass matrix for a linear triangular element is constructed by
summing the rows and placing on the diagonals. The elemental lumped mass matrix of a
linear triangular element is
   
40 0 10 0
A A
[M Le ] =  04 0  =  01 0  (7.163)
12 3
00 4 00 1
If the above mass lumping procedure is introduced into the CBS steps, some small
errors will occur in the transient solution. For steady state solutions, however, no errors
are introduced. However, for transient problems an accurate solution can still be obtained
by appropriate mesh refinement.

7.6.2 Time-step calculation

The time-step restrictions are very similar to the convection–diffusion equation (Equation
7.119). The local time step at each and every node can be computed as follows:

t = min( t c , t d ) (7.164)