5.11.1 The Velocity ϕ
Re-writing Eq. (5.13) again;
f
0
¼ f
À3η þ ϕ 3 þ γ=2
ð
ÞÀ2γϕ
2
=η
Â
Ã
η À ϕ
ð
Þ
2 À f =ψ
h
i
,
and by writing it in the form, we have
f
0
f
η À ϕ
ð
Þ
2 À
f
0
ψ
¼ À3η þ ϕ 3 þ γ=2
ð
ÞÀ2γϕ
2
=η:
ð5:30Þ
In order to eliminate ψ in the latter equation we use Eq. (5.9), namely,
ϕ
0
η À ϕ
ð
Þ¼
f
0
γψ
À
3
2
ϕ,
which gives,
0
1
2
3
4
5
0
0.2
0.4
0.6
0.8
1
Normalized time (see text )
Pressure ratio (see text)
Fig. 5.5 Pressure ratio versus normalized time at a fixed point is shown for γ ¼ 1.4 (see text)
5.11 Taylor’s Analytical Approximations for Velocity, Pressure and Density
235
Re-writing Eq. (5.13) again;
f
0
¼ f
À3η þ ϕ 3 þ γ=2
ð
ÞÀ2γϕ
2
=η
Â
Ã
η À ϕ
ð
Þ
2 À f =ψ
h
i
,
and by writing it in the form, we have
f
0
f
η À ϕ
ð
Þ
2 À
f
0
ψ
¼ À3η þ ϕ 3 þ γ=2
ð
ÞÀ2γϕ
2
=η:
ð5:30Þ
In order to eliminate ψ in the latter equation we use Eq. (5.9), namely,
ϕ
0
η À ϕ
ð
Þ¼
f
0
γψ
À
3
2
ϕ,
which gives,
0
1
2
3
4
5
0
0.2
0.4
0.6
0.8
1
Normalized time (see text )
Pressure ratio (see text)
Fig. 5.5 Pressure ratio versus normalized time at a fixed point is shown for γ ¼ 1.4 (see text)
5.11 Taylor’s Analytical Approximations for Velocity, Pressure and Density
235