Page 205 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
obtain
n+1
φ
∂φ
∂φ
t − φ n =−u 1 ∂x 1 n − u 2 ∂x 2 n 197
∂ ∂φ n ∂ ∂φ n
+ k + k
∂x 1 ∂x 1 ∂x 1 ∂x 2
t ∂ ∂φ ∂φ n
+ u 1 u 1 + u 2
2 ∂x 1 ∂x 1 ∂x 2
n

t ∂ ∂φ ∂φ

+ u 2 u 1 + u 2 (7.106)
2 ∂x 2 ∂x 1 ∂x 2
The standard Galerkin approximation can now be employed for solving the above
equation. Assuming a linear variation of φ within an element as indicated in Figure 7.12,
we can express the variation of φ as
φ = N i φ 1 + N j φ j + N k φ k = [N]{φ} (7.107)
Employing the Galerkin weighting, we obtain
φ n+1 − φ n ∂φ n ∂φ n
T
T
[N] T d
=− [N] u 1 d
− [N] u 2 d

t
∂x 1
∂x 2
∂ ∂φ
n
T
+ [N] k d

∂x 1 ∂x 1
∂ ∂φ n
+ [N] T d

∂x 2 ∂x 2
n

t T ∂ ∂φ ∂φ

+ u 1 [N] u 1 + u 2 d
2
∂x 1 ∂x 1 ∂x 2
n

t T ∂ ∂φ ∂φ

+ u 2 [N] u 1 + u 2 d
(7.108)
2
∂x 2 ∂x 1 ∂x 2
k

i

j
Figure 7.12 Two-dimensional linear triangular element