λ nþ1,j ¼ λ n,j þ Δτ
e u nþ1,j þ e u n,j
2
ð6:24Þ
e υ nþ1,j ¼ e υ n,j
þ e υ 0,j λ nþ1,j Δτ 2e u nþ1,j
λ nþ1,j À λ nþ1,jÀ1
Δχ 0,j
þ λ nþ1,j
e u nþ1,j À e u nþ1,jÀ1
Δχ 0,j
!
:
ð6:25Þ
e q nþ1,j ¼ κΔχ 0,j
À
Á 2
2
e υ nþ1,j þ e υ n,j
e u nþ1,j À e u nþ1,jÀ1
Δχ 0,j
Â
e u nþ1,j À e u nþ1,jÀ1
Δχ 0,j
À
e u nþ1,j À e u nþ1,jÀ1
Δχ 0,j
!
ð6:26Þ
e p nþ1,j ¼
γþ1
γÀ1 e υ n,j À e υ nþ1,j
e p n,j þ 2e q nþ1,j e υ n,j À e υ nþ1,j
À
Á
γþ1
γÀ1 e υ nþ1,j À e υ n,j
:
ð6:27Þ
where λ 0, j ¼ jΔλ and Δχ 0,j ¼ λ
2
0,j Δλ. An alternative form of the equation for the
pressure when written in terms of the density rather than the specific volume is
e p nþ1,j ¼
γþ1
γÀ1
e ρ nþ1,j À e ρ n,j
h
i
e p n,j þ 2e q nþ1,j e ρ nþ1,j À e ρ n,j
À
Á
γþ1
γÀ1
e ρ n,j À e ρ nþ1,j
:
ð6:28Þ
Brode [3] has also approximated the differential equations for the air motion by a
set of finite difference equations and these are presented below. These equations are
correct to second order for small quantities Δχ and Δτ. The momentum equation, that
is, Eq. (6.12), is written in the following difference form (with the normalized
density replacing the normalized specific volume);
e u
nþ
1
2
j
À e u
nÀ
1
2
j
Δτ
¼ À
λ
n
j
2
γΔχ
0
j e ρ
0
j
e p
n
jþ
1
2
À e p
n
jÀ
1
2
þ e q
nÀ
1
2
jþ
1
2
À e q
nÀ
1
2
jÀ
1
2
h
i
,
and where, for convenience, the time-stepping index n is written here as a superscript, hence,
6.4 Difference Equations
289