dυ
dω
¼ Æ
1
2ω 0
υ i À υ f
À
Á
Cos
ω
ω 0
:
ð4:38Þ
Since we are considering the case where the shock wave moves to the right, then
υ i > υ f and as a result of the assumption that (dυ/dω) ! 0 in the shock region, the
positive sign in Eq. (4.38) is taken, hence,
dυ
dω
¼
1
2ω 0
υ i À υ f
À
Á
Cos
ω
ω 0
! 0,
so that ω/ω 0 is confined to the range,
À
π
2
ω
ω 0
π
2
:
Accordingly, Eq. (4.37) can be written as
υ ¼
υ i þ υ f
2
þ
υ i À υ f
2
Sin
ω
ω 0
,
ð4:39Þ
which has a continuous solution within the shock region; À(π/2) (ω/ω 0 ) (π/2).
For ω ¼ À (π/2)ω 0 we have υ ¼ υ f and for ω ¼ (π/2)ω 0 we have υ ¼ υ i . When
Eq. (4.39) is pieced together with the particular solutions; υ ¼ υ i and υ ¼ υ f , as stated
by Von Neumann and Richtmyer, one obtains the composite continuous solution as
shown in Fig. 4.2. The specific volume within the shock region is shown in Fig. 4.2
as a solid line while the particular solutions outside the shock region are shown as
broken lines. Hence, requirement 1 is satisfied.
We have already seen that the shock region extends over an interval of width πω 0 ,
where ω 0 is given by Eq. (4.36), hence, the width of the shock region is
π
2
γ þ 1
1=2
κΔx,
which is the order of Δx if κ is close to unity and this satisfies requirement 2.
Outside the shock region ∂υ/∂t is, in general, very small and in the case of a
steady-state shock it is zero, so in this region q is negligible in comparison to the
pressure p due to the factor (Δx)
2 in the equation for q; this satisfies requirement 3.
Consequently, Von Neumann and Richtmyer have demonstrated that their
expression for q meets all requirements: the equations describing the flow have
continuous solutions and these equations can be used for the entire calculation as if
no shocks were present at all. Instead, the shocks automatically appear as regions
where there are rapid but continuous changes in the velocity, density etc. and have
jumps in value that matches the conditions supplied by the Rankine-Hugoniot
equations.
144
4 Numerical Treatment of Plane Shocks
dω
¼ Æ
1
2ω 0
υ i À υ f
À
Á
Cos
ω
ω 0
:
ð4:38Þ
Since we are considering the case where the shock wave moves to the right, then
υ i > υ f and as a result of the assumption that (dυ/dω) ! 0 in the shock region, the
positive sign in Eq. (4.38) is taken, hence,
dυ
dω
¼
1
2ω 0
υ i À υ f
À
Á
Cos
ω
ω 0
! 0,
so that ω/ω 0 is confined to the range,
À
π
2
ω
ω 0
π
2
:
Accordingly, Eq. (4.37) can be written as
υ ¼
υ i þ υ f
2
þ
υ i À υ f
2
Sin
ω
ω 0
,
ð4:39Þ
which has a continuous solution within the shock region; À(π/2) (ω/ω 0 ) (π/2).
For ω ¼ À (π/2)ω 0 we have υ ¼ υ f and for ω ¼ (π/2)ω 0 we have υ ¼ υ i . When
Eq. (4.39) is pieced together with the particular solutions; υ ¼ υ i and υ ¼ υ f , as stated
by Von Neumann and Richtmyer, one obtains the composite continuous solution as
shown in Fig. 4.2. The specific volume within the shock region is shown in Fig. 4.2
as a solid line while the particular solutions outside the shock region are shown as
broken lines. Hence, requirement 1 is satisfied.
We have already seen that the shock region extends over an interval of width πω 0 ,
where ω 0 is given by Eq. (4.36), hence, the width of the shock region is
π
2
γ þ 1
1=2
κΔx,
which is the order of Δx if κ is close to unity and this satisfies requirement 2.
Outside the shock region ∂υ/∂t is, in general, very small and in the case of a
steady-state shock it is zero, so in this region q is negligible in comparison to the
pressure p due to the factor (Δx)
2 in the equation for q; this satisfies requirement 3.
Consequently, Von Neumann and Richtmyer have demonstrated that their
expression for q meets all requirements: the equations describing the flow have
continuous solutions and these equations can be used for the entire calculation as if
no shocks were present at all. Instead, the shocks automatically appear as regions
where there are rapid but continuous changes in the velocity, density etc. and have
jumps in value that matches the conditions supplied by the Rankine-Hugoniot
equations.
144
4 Numerical Treatment of Plane Shocks