Page 199 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
The above equation is equal to zero only if all the element contributions are assembled.
For a domain with only one element, we can substitute
191
T N i
[N] = (7.84)
N j
On substituting a linear spatial approximation for the variable φ, over elements as
typified in Figure 7.9, into Equation 7.83, we get
{φ − φ } T ∂ n
n+1 n
T
[N] [N] d
=−u 1 [N] ([N]{φ}) d

t
∂x 1
t 2 T ∂ 2 n
+ u 1 [N] ([N]{φ}) d
2
∂x 1 2
∂ 2
n
+ [N] T 2 ([N]{φ}) d
(7.85)

∂x 1
Before utilizing the linear integration formulae, we apply Green’s lemma to some of
the integrals in the above equation. Green’s lemma is given as follows:
∂β ∂α

α d
=− β d
+ αβn 1 d

∂x 1
∂x 1
∂β ∂α

α d
=− β d
+ αβn 2 d (7.86)

∂x 2
∂x 2
where n 1 and n 2 are the direction cosines of the outward normal n,
is the domain and
is the domain boundary. The second-order derivatives can also be similarly expressed
(see Appendix A). Applying Green’s lemma to the second-order terms of Equation 7.85,
we obtain
{φ }− {φ } T ∂ n
n+1 n
T
[N] [N] d
=−u 1 [N] ([N]{φ}) d

t
∂x 1
T
t 2 ∂[N] ∂[N]
− u 1 {φ} d
2
∂x 1 ∂x 1
t 2 T ∂[N]
+ u 1 [N] {φ}n 1 d
2 ∂x 1
T
∂[N] ∂[N]
− k {φ} d

∂x 1 ∂x 1
∂[N]
T
+ [N] k {φ}n 1 d (7.87)
∂x 1
The first-order convection term can be integrated either directly or via Green’s lemma. In
this section, the convection term is integrated directly without applying Green’s lemma. How-
ever, integration of the first derivatives by parts is useful for the solution of Navier–Stokes
equations, as demonstrated in Section 7.6. It is now possible to apply a shortcut for the