Page 44 - Introduction to Computational Fluid Dynamics

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2.4 DISCRETISATION
From the point of view of obtaining stable and convergent numerical solutions, 10:49 23
observations 2 and 3 are significant. The associated matter will become clear in a
later section.
2.4.2 IOCV Method
In this method, the RHS and LHS of Equation 2.5 are integrated over a control
volume x and over a time step t. Thus,

∂ ∂T
t e t e

Int(LHS) = kA dx d t + q Ad x d t, (2.19)
t w ∂x ∂x t w

where t = t + t.Itisnow assumed that the integrands are constant over the time
interval t. Further, q is assumed constant over the control volume and since the

second-order derivative is evaluated at a fixed time, we may write

∂T ∂T
Int(LHS) = kA − kA t + q A x t. (2.20)
P
∂x ∂x
e w
It is further assumed that T varies linearly with x between adjacent nodes. Then

∂T T E − T P ∂T T P − T W
= , = . (2.21)
∂x x e ∂x x w
e w
Note that when the cell faces are midway between the nodes, these represen-
tations of the derivatives are second-order accurate (see Equation 2.11). Using
Equation 2.21 therefore gives

kA kA
Int(LHS) = (T E − T P ) + (T W − T P ) t
x e x w

+ q A x t. (2.22)
P
Similarly,

t e ∂(CT )
Int(RHS) = ρ A dx dt
∂t
t w
n
o
= (ρ A x) P [(CT ) − (CT ) ] . (2.23)
P
Substituting Equations 2.22 and 2.23 into the integrated version of Equation 2.6,
therefore, we can show that

n
ρ VC n n n
+ ψ (AE + AW) T = ψ AE T + AW T + S, (2.24)
P E W
t P