Page 209 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
l
3
A 4 l 4 201
A 3
Node
A A 5
2
l 5
l 2
A 1

l 1

Figure 7.13 Two-dimensional linear triangular element

Often, it may be necessary to multiply the time-step t by a safety factor due to
different methods of element size calculations. A simple procedure to calculate the element
size in two dimensions is

2Area i
h = min ,i = 1, number of elements connected to the node (7.121)
l i
where Area i are the area of the elements connected to the node and l i are the length of
the opposite sides as shown in Figure 7.13. For the node shown in this figure, the local
element size is calculated as

h = min(A 1 /l 1 ,A 2 /l 2 ,A 3 /l 3 ,A 4 /l 4 ,A 5 /l 5 ) (7.122)
In three dimensions, the term 2Area i is replaced by 3Volume i and l i is replaced by the
area opposite the node in question.

7.6 Characteristic-based Split (CBS) Scheme

It is essential to understand the characteristic Galerkin procedure, discussed in the previous
section for the convection–diffusion equation, in order to apply the concept to solve the real
convection equations. Unlike the convection–diffusion equation, the momentum equation,
which is part of a set of heat convection equations, is a vector equation. A direct extension
of the CG scheme to solve the momentum equation is difficult. In order to apply the
characteristic Galerkin approach to the momentum equations, we have to introduce two
steps. In the first step, the pressure term from the momentum equation will be dropped
and an intermediate velocity field will be calculated. In the second step, the intermediate
velocities will be corrected. This two-step procedure for the treatment of the momentum
equations has two advantages. The first advantage is that without the pressure terms, each
component of the momentum equation is similar to that of a convection–diffusion equation
and the CG procedure can be readily applied. The second advantage is that removing the