Page 197 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

CONVECTION HEAT TRANSFER
Characteristic
n + 1 189
f
x 1
n + 1
∆t

n n n
f f
x 1 − ∆x 1 x 1

x 1 − ∆x 1 x 1
∆x 1

Figure 7.8 Characteristic in a space–time domain

In order to demonstrate the CG method, let us reconsider the simple convection–diffusion
equation in one dimension, namely,

∂φ ∂φ ∂ ∂φ
+ u 1 − k = 0 (7.75)
∂t ∂x 1 ∂x 1 ∂x 1
Let us consider a characteristic of the flow as shown in Figure 7.8 in the time–space
domain. The incremental time period covered by the flow is t from the nth time level to
the n + 1th time level and the incremental distance covered during this time period is x 1 ,
that is, from (x 1 − x 1 )to x 1 . If a moving coordinate is assumed along the path of the
characteristic wave with a speed of u 1 , the convection terms of Equation 7.75 disappear (as
in a Lagrangian fluid dynamics approach). Although this approach eliminates the convection
term responsible for spatial oscillation when discretized in space, the complication of a
moving coordinate system x is introduced, that is, Equation 7.75 becomes

1
∂φ ∂ ∂φ

(x ,t) − k = 0 (7.76)
1
∂t ∂x ∂x
1 1
The semi-discrete form of the above equation can be written as
n
φ n+1 | x 1 − φ | x 1 − x 1 ∂ ∂φ n
− k | x 1 − x 1 = 0 (7.77)
t ∂x ∂x
1 1
Note that the diffusion term is treated explicitly (a definition of explicit schemes has
been given in Chapter 6 and later on in this chapter). It is possible to solve the above
equation by adapting a moving coordinate strategy. However, a simple spatial Taylor series
expansion in space avoids such a moving coordinate approach. With reference to Figure 7.8,