6.6.3 The Density
The corresponding initial normalized density is
e ρ ¼ ψ
λ
λ s
:
ð6:36Þ
6.7 Specification of Initial Conditions
Now, once e p is specified, such as, 1000 atmospheres at the shock front (and γ and B
(γ) are known), then λ s is known since f(η) is a fixed function according to the pointsource solution. Hence, with e p ¼ 1000 then Eq. (6.34) gives
1000 ¼
1
γB γ
ð Þ
λ
À3
s f 1
ð Þ,
as γ ¼ 1.4, B(γ) ¼ 5.31 and f(1) ¼ 1.166, hence,
λ s ¼ 0:054,
ð6:37Þ
so λ s is known once the pressure at the shock front has been specified.
6.8 Results of the Numerical Integration
The discrete from of the equations for the numerical integration have already been
presented in Sect. 6.4. Plots of the pressure, particle velocity and density as a
function of normalized Lagrangian radius are shown in Figs. 6.2, 6.3 and 6.4 for
three different times during the expansion into the surrounding atmosphere. The
plots correspond to times when the peak pressure has dropped to just a few
atmospheres. The pressure at slightly later times is also shown in Fig. 6.5 and one
can observe that the profile develops a negative phase with the pressure dropping
below its ambient value beyond about 3 atmospheres peak pressure.
Let us now investigate in some detail what these plots predict and let us choose
the plots corresponding to τ ¼ 0.1. In the first instance, let us make an estimate of the
velocity of the shock front. In order to do this we will consider plots on either side of
τ ¼ 0.1 and take, for example, plots at τ ¼ 0.09 and at τ ¼ 0.11 and determine the
radial distance the shock front advances in this dimensionless time interval of
δτ ¼ 0.02. This should provide a good estimate of the velocity of the shock front
at τ ¼ 0.1. As before, we can estimate the approximate positions of the shock front in
294
6 Numerical Treatment of Spherical Shock Waves
The corresponding initial normalized density is
e ρ ¼ ψ
λ
λ s
:
ð6:36Þ
6.7 Specification of Initial Conditions
Now, once e p is specified, such as, 1000 atmospheres at the shock front (and γ and B
(γ) are known), then λ s is known since f(η) is a fixed function according to the pointsource solution. Hence, with e p ¼ 1000 then Eq. (6.34) gives
1000 ¼
1
γB γ
ð Þ
λ
À3
s f 1
ð Þ,
as γ ¼ 1.4, B(γ) ¼ 5.31 and f(1) ¼ 1.166, hence,
λ s ¼ 0:054,
ð6:37Þ
so λ s is known once the pressure at the shock front has been specified.
6.8 Results of the Numerical Integration
The discrete from of the equations for the numerical integration have already been
presented in Sect. 6.4. Plots of the pressure, particle velocity and density as a
function of normalized Lagrangian radius are shown in Figs. 6.2, 6.3 and 6.4 for
three different times during the expansion into the surrounding atmosphere. The
plots correspond to times when the peak pressure has dropped to just a few
atmospheres. The pressure at slightly later times is also shown in Fig. 6.5 and one
can observe that the profile develops a negative phase with the pressure dropping
below its ambient value beyond about 3 atmospheres peak pressure.
Let us now investigate in some detail what these plots predict and let us choose
the plots corresponding to τ ¼ 0.1. In the first instance, let us make an estimate of the
velocity of the shock front. In order to do this we will consider plots on either side of
τ ¼ 0.1 and take, for example, plots at τ ¼ 0.09 and at τ ¼ 0.11 and determine the
radial distance the shock front advances in this dimensionless time interval of
δτ ¼ 0.02. This should provide a good estimate of the velocity of the shock front
at τ ¼ 0.1. As before, we can estimate the approximate positions of the shock front in
294
6 Numerical Treatment of Spherical Shock Waves