Shock Waves (2001) 11: 157 165
Self-similar implosion of a continuous stratified medium
Nathalie Toqu"
Commissariat ą l Energie Atomique, BP 12, 91680 BruyŁres-le-Chtel, France
Received 8 March 1999 / Accepted 9 April 2001
Abstract. The subject of this paper is the non perturbed self-similar implosion of a continuous stratified
medium in cylindrical and spherical geometry. The main contribution is twofold. The first one is a condition
on the initial density profile which guarantees that the integral of the internal energy over a fixed volume
stays finite at the collapse. The second one is the proof that the self-similar implosion model partly restitutes
the dynamics of an experiment of cylindrical implosion. Numerical values of the self-similar parameter ą
are also given for various kind of stratified media in the two geometries.
Key words: Simulation, Implosion
1 Introduction previous results. More recently, V.D. Sharma and C.H.
Radha (1995) have been interested in the state of the
medium near the collapse such as its high temperature
The self-similar implosion problem is very well known.
and pressure. As the medium could be no more adiabatic,
It was the first studied in the case of an uniform per-
in the energy equation, they assumed that the departure
fect gas medium by G. Guderley (1942), followed by D.S.
from equilibrium was due to vibrational relaxation of a di-
Butler (1954), R.D. Richtmyer and R.B. Lazarus (1977).
atomic gas. It involved a supplementary equation associ-
They proposed a numerical method to calculate the val-
ated to the vibrational energy . They used the Lie group
ues of the self-similar parameter ą for various values of
invariance under infinitesimal point transformations to de-
the polytropic constant ł and from that obtained the so-
termine the complete class of self-similar solutions. In the
lution of the problem. Many other authors have been in-
particular case without vibrational relaxation, they calcu-
terested in this problem. G.B. Whitham (1974) who was
lated ą values which were in agreement with those of G.
attracted by the study of the shock waves in different me-
Guderley, G.B. Whitham, A. Sakurai and P. Hafner.
dia developed an analytic method to evaluate the ą val-
This paper deals with the non perturbed self-similar
ues. His method depends on an approximation which was
implosion of a continuous stratified medium in cylindrical
also made by W. Chester and R.F. Chisnell to study the
and spherical geometry. It follows a thesis (Toqu 1996)
propagation of the shock waves in a non uniform one-
on interface instabilities restricted to the case of an uni-
dimensional medium. Since this period, this approxima-
form medium and introduces a next paper about the linear
tion has been named there after the CCW approximation.
perturbation of the self-similar implosion. It is divided in
In summary, it supposes that a shock front moving down
three parts. The first one is about the classical self-similar
in a channel and perpendicular to its axis may be approxi-
inplosion model applied to media such as gas, liquid or
mated by a characteristic line whose equation is expressed
solid, with a continuous initial density profile. The require-
with the Rankine-Hugoniot conditions. The result is a re-
ment is that the equation of state of this medium could
lation between the Mach number and the cross section of
be approximated by a perfect gas law with an appropri-
the channel.
ate value of the polytropic constant ł. The second part
Later, by the mean of the self-similar implosion model
explains the way to calculate the self-similar parameter ą
of a non uniform gas medium, A. Sakurai (1960) has initi-
according to the numerical method of R.B. Lazarus. It is
ated a model of gas expanding into the vacuum to simulate
demonstrated that the initial density profile depends on a
shock waves in the stars. In the last fifteen years, a mathe-
condition taking into account the geometrical parameter
matical investigation of the self-similar implosion problem
and the polytropic constant ł. A comparison is also made
has been developed. In 1988, P. Hafner (1988) proposed
between the ą values calculated by V.D. Sharma and C.H.
an analytic method to calculate the self-similar param-
Radha (1995) and ours. In the last part, an experiment
eter ą as the limit of a converging sequence ąn with a
of cylindrical implosion and its numerical simulation per-
very high accuracy. His ą values agreed very well with the
formed during the thesis (Toqu 1996) are described. The
"
Present address: Dpartement de Physique de l Universit de
purpose is to proof that the shock and flow trajectories of
Montral, C.P. 6128, Succursale centre-ville, Montral, Qubec
the inner medium are self-similar.
H3C 3J7, Canada (e-mail: toque@astro.umontreal.ca)
158 N. Toqu: Self-similar implosion of a continuous stratified medium
Z
2 Self-similar implosion model of a medium
with an initial density profile
An infinite medium at rest (p = p0 = patm, = 0, u =
u0 a" 0) is processed by a strong convergent shock (M
1 D
+ "). The flow behind the shock is one-dimensional. It is
Q
compressible, unsteady and isentropic ( = 0).
dT
The variables are t and r. The unknown quantities of
the problem are the density , the material velocity u and
1 V
O
the sound speed c. Their relation is approximated by a
łp
perfect gas law: c2 = . ł is the polytropic constant,
Fig. 1. Graph of Z(V )
1 <ł <+ ".
The initial density 0 depends on the variable r in this 1
= . The shock conditions (2) are also developed with
ą
way: 0 = br, with b and e" 0 known constants. b has
self-similar decompositions (3) taken at =1:
the dimension [M][L]-(+3) linked to the characteristic
length and mass of the system.
ł +1 2 2ł(ł - 1)
H(1) = , V (1) = , Z(1) =
The main equations are the conservative ones for the
ł - 1 ł +1
(ł +1)2
flow with the Rankine Hugoniot conditions for the strong
(5)
shock (Zel dovich and Raizer 1967):
At the collapse (t =0, R a" 0), we notice that goes
ńł
1 1 u
ł
to infinite if the value of r is finite and non zero. The self-
ł "t + u "r + "ru +( - 1) =0
ł
ł
r
ł similar decompositions of u and c remain finite under the
ł
ł
ł
conditions:
c2 1
(1)
"tu + u"ru + "r + "rc2 =0
ł
ł
lim+ "V () = 0 and lim+ "Z() =0 ( 6)
ł ł ł
ł
ł
ł
ł
ół
The equations (4) with conditions (5) and (6) are
"t ln c21-ł + u"r ln c21-ł =0
named the self-similar implosion model of a medium with
a continuous initial density profile.
ł +1 2 2ł(ł - 1)
<" , u <" U, c <" U (2)
0 ł - 1 ł +1
(ł +1)2
The value of is 2 in cylindrical geometry and 3 in
3 Numerical calculation
spherical geometry. U is the shock velocity.
of the self-similar parameter ą
The shock wave and the flow are assumed to be
self-similar. Shock position follows the relation: R(t) =
3.1 Theoretical approach
A(-t)ą, with A and ą, unknown positive constants. t =0
is the time of collapse. Before it, values of t are negative.
The unknown quantities of Eqs. (4) are the functions V (),
The unknown quantities , u and c are functions of self-
Z(), H() and the constant ą. In order to calculate the
similar variable (Zel dovich and Raizer 1967):
solution (V (), Z(), H()), we need first to evaluate the
r
value of ą which depends on the geometry of the prob-
= , = brH(), u = X V (), c = X Z()
lem (), the medium (ł) and the initial density (): ą =
R(t)
(3) ą(, ł, ).
The right hand-sides of Eqs. (4) only depend on V (),
H, V and Z are dimensionless. X = U is the shock veloc-
Z(), , ł and . So, the first two equations which are
ity.
d ln
dZ
The conservative equations (1) written with formulas linked as follows: , lead to the differential equa-
d ln dV
(3) are converted into a Cramer system with three un- tion:
dZ dV d ln H
known quantities , and . Its solution is as ńł
d ln d ln d ln Z3 [V ł+2-2-]
2+-
{ }
2
ł dZ ł(V -1)
follows: ł
=
ł
dV
V (V -1)(-V )+Z2 (-2++2) +V
ńł { }
ł
Z3 [V ł+2-2-]
(7)
2+-
ł { }
dZ 2 ł(V -1)
ł 2
Z
ł
ł =
V [ł-3+(1-ł)]+V [(3-ł)-(1-ł)+2]-2
{ }
ł
ł d ln 2
(V -1)2-Z2
ół
ł
+
ł
ł V (V -1)(-V )+Z2 (-2++2) +V
{ }
ł
ł
ł
ł Z 2
V [ł-3+(1-ł)]+V [(3-ł)-(1-ł)+2]-2
ł { }
2
ł
ł +
ł graph Z(V ) are the points (V (1), Z(1))
(V -1)2-Z2
The ends of the
2ł(ł-1)
2
= , and (V (+ "), Z(+ ")) = (0, 0). They
ł
ł+1
(ł+1)2
ł V (V -1)(-V )+Z2 (-2++2) +V
{ }
ł dV ł
ł
=
ł are on each side of the line: Z =1 - V .
d ln
(V -1)2-Z2
ł
ł
ł
The graph Z(V ) cuts the line at a point (Vs, Zs) for
ł
ł
ł
Z2[2-2-+V ł]
ł
which Eqs. (4) are undefinite. The value of the upper parts
ł -[(+)V (V -1)+V (-V )]
d ln H ł(V -1)
ół
=
d ln of the right hand-sides of Eqs. (4) at the point (Vs, Zs) is
(V -1)2-Z2
(4) also zero.
N. Toqu: Self-similar implosion of a continuous stratified medium 159
As Zs = 1 - Vs, the upper parts are multiples of a From the two inequalities (11), only the first one is
quadratic polynomial in Vs: relevant with the limits of parameter ą (Lazarus 1981):
P (Vs) =Vs2ł (1 - ) +Vs
(2 - ł)2
1 >ąe" , (15)
[(2 - ł)( - 1) + ł( - 1) - ]
X
-2( - 1) + (8)
with
roots are:
X =( 2 - ł)2 + ł( - 1)(2 + ł) +(2 - ł)
"
[(2 - ł)( - 1) + ł( - 1) - ] d
Vsą = " (9)
2[ł( - 1)] 2[ł( - 1)] - 4ł2( - 1) [2 - ł( - 2 + 2)].
with
It requires the condition on :
d =( 2 - ł)2( - 1)2
3
0 d" d" - ł - 1 + ł 2( - 1) (16)
+2 [(ł - 2) - ł( - 1)(2 + ł)] ( - 1)
2
+[ł( - 1) + ]2 (10)
We notice that the value ą = 1 corresponds to the
isotherm case ł = 1 which is excluded in this study.
d is positive if ą verifies the inequalities (ł = 2):
For the particular value ł = 2, the inequality (14)
verifies the limits of parameter ą, if:
(2 - ł)2
ą e" "
(2 - ł)2 + ł( - 1)(2 + ł) +(2 - ł) - d
0 d" d"-2( - 1) +2 2( - 1) (17)
or (11)
The condition (16) taken at ł = 2 is the same. So,
it is valid for 1 <ł <+ ".
(2 - ł)2
Taking conditions (13) into account, we have to verify
ą d" "
the inequality:
(2 - ł)2 + ł( - 1)(2 + ł) +(2 - ł) + d
with
3 2ł( - 1)
0 d" d" - ł - 1 + ł 2( - 1) d"
d =4ł2( - 1) [2 - ł( - 2 + 2)] (12)
2 (ł - 2)
d positive results in the conditions:
(18)
If ł <2, then d e" 0 for any = 2 or 3 3
we notice that the part: -ł( -1)+ł 2( - 1), is zero
2
2ł( - 1)
with = 2 and any value of ł. For = 3, it decreases with
If ł >2, then d e" 0 for d" (13)
growing values of ł and equals at ł 83.
(ł - 2)
It is possible to rewrite the condition (16) as follows:
For high ł values (ł + "), we obtain: d" 2( - 1).
If = 2 then d" 2 and if = 3 then d" 4. =2, 0 d" d" 2, for any 1 <ł <+ "
For the specific value ł =2, d positive implies the only
3
=3, 0 d" d" - ł - 1 + ł 2( - 1) < 2.964,
condition:
2
16( - 1)
for any 1 <ł d" 83. (19)
ą e" (14)
16( - 1) + [2( - 1) + ]2
We conclude that conditions (15) and (19) are the only
The parameter ą is restricted to the interval: ones required to verify the limits of the parameter ą. We
2
1 e" ą e" . the upper limit results in the requirement remind that the integral of internal energy e over a fixed
+2
volume about the origin is finite under these two condi-
that the shock velocity is not zero at the collapse (t = 0):
tions.
Later on, the lower limit of inequality (15) will be re-
X = -ąA(-t)ą-1, so ą d" 1.
ferred to as ąinf . It depends on , ł and .
The lower limit results in the requirement that the
The graph Z(V ) can intersect the line: Z =1-V , with
integral of internal energy e over a fixed volume centered
two points, p1 and p2. If the values of and are fixed,
at the origin does not become infinite at the collapse (t =
the first point p1 corresponds to low ł values close to one
0):
and the second one, p2, to ł values higher than two.
At a particular value of ł, named after łc, the solu-
rf f
tion curve Z(V ) changes. For ł = łc, the points p1 and
edv " (ąZ)2[2+(-1)](-t)[ą(+2)-2]d,
p2 are the same (d 0). So, the corresponding value of
0 + "
the parameter ą is ąc ąinf (, łc, ). The value of łc is
2
so, ą e" . calculated numerically.
+2
160 N. Toqu: Self-similar implosion of a continuous stratified medium
Z
ter ą corresponding to the slope L+, for which the function
Z = Z(V ) is continuous at the intersecting points.
D
1
3.2 Numerical results and comparisons
*
P2
*
P1
For low ł values, 1 <ł d" łc, the graph Z(V ) crosses the
*
line Z =1 - V , at the point p1. , ł and fixed, Eq. (7) is
integrated from the point (V (1), Z(1)) to the point p1 =
1 V
O
(Vs-, Zs-) with an arbitrary value of ą verifying condition
(15). The process recurs several times with a new value of
Fig. 2. The two intersections of Z(V ) with the line Z =1 - V
ą and each time under condition (15). The purpose is that
the graph Z(V ) comes the closest to the point p1, with the
At the proximity of the intersecting point, p1 or p2,
slope L+ . Finally, the best convergence gives the closest
the function Z = Z(V ) is assumed linear versus V :
value of ą corresponding to the given ł.
Z - Zs L(V - Vs), (s = s + or s-) (20)
With ł growing and always 1 <ł d" łc, the points p1
and p2 come closer and closer. When they are the same,
So, the differential equation (7) may be written as fol- p1 = p2 (d 0), ł is łc. The value of ąc is calculated by
lows:
the numerical process and we verify it is the same value
as ąinf (, łc, ).
"N
(V - Vs)"N )V +(Z - Zs) )Z
dZ
"V "Z
s s
For high ł values, ł > łc, the numerical method to
L (21)
"D "D
dV
(V - Vs) )V +(Z - Zs) )Z
calculate the parameter ą is unchanged with the exception
"V "Z
s s
of the point of intersection. It is replaced by the point
N(V, Z) et D(V, Z) are the upper and lower parts of p2 =(Vs+, Zs+).
the right hand-side of Eq. (7). Their partial derivatives are
The calculated values of ą are all given in two arrays
given in Annex A.
set in Annex B, one for cylindrical geometry and one for
Equation (21) results in a quadratic polynomial versus
spherical geometry. Several of them are now compared
L: with results published by V.D. Sharma and C.H. Radha
(1995), which are in agreement with those of G. Guderley
"D "D "N "N
(1942), A. Sakurai (1960), G.B. Whitham (1974) and P.
L2 )Z + L )V - )Z - )V = 0 (22)
s s s s
"Z "V "Z "V
Hafner (1988).
Its roots are:
ł calculated ą ą (1995)
"
1.1 0.0 .88524 .8852473632812
"N "D
)Z - )V " "
1.2 0.0 .86116 .8611622802734
"Z "V
s s
L" = (23)
"D
1.4 0.0 .83532 .8353226318359
2 )Z
"Z
s
5/3 0.0 .81563 .8156246582031
2.0 0.0 .80159 .8001124267578
with
3.0 0.0 .77568 .7756673339844
2
6.0 0.0 .75158 .7515630126953
"D "N "N "D
1.2 0.5 .80265 .8026485061645
" = )V - )Z +4 )V )Z (24)
s s s s
"V "Z "V "Z
5/3 0.5 .74540 .7453985717773
2.0 0.5 .72730 .7273047542572
1.2 1.0 .75297 .7529296167195
If " is positive, the function Z = Z(V ) comes closer
5/3 1.0 .68836 .6883719325065
to the intersecting point with the slope L+ or L-.
2.0 1.0 .66898 .6689041662216
The usual solution of Eq. (20) near the intersecting
1.1 2.0 .69697 .7177513122559
point, p1 or p2, is written as follows:
1.2 2.0 .67175 .6717613220215
1.4 2.0 .62833 .6283397583008
+ -
[Z -Zs-L+(V - Vs)]E [Z -Zs-L-(V - Vs)]E (25)
5/3 2.0 .59879 .5994441223144
2.0 2.0 .57900 .5790141997337
3.0 2.0 .55017 .5499990983400
The exponents E+ and E- are linked to the slopes by
6.0 2.0 .52451 .5245207309723
the relations:
"D
)V
"V
s
E" = L" + (26)
"D
This array corresponds to the cylindrical geometry
)Z
"Z
s
( = 2) and the next one, to the spherical geometry
For convenience as in the paper of R.B. Lazarus (1981),
( = 3). In spite of a low numerical accuracy, our results
if we choose |E-| < |E+| then L- will be named after the
are in quite a good concordance with ą values calculated
primary direction and L+ the secondary one. In this pa- by V.D. Sharma and C.H. Radha and are sufficient to eval-
per, we are interested in calculating the values of parame- uate flow and schock trajectories in the next paragraph.
N. Toqu: Self-similar implosion of a continuous stratified medium 161
ł calculated ą ą (1995)
in the following function:
1.1 0.0 .79594 .795968198776
1.2 0.0 .75713 .757140216827
1
V (1)
1.4 0.0 .71717 .717173361778
R1(1) =R0 exp d1 , (29)
1
5/3 0.0 .68841 .688376379013
1(V (1) - 1)
1
2.0 0.0 .66887 .667046318054
3.0 0.0 .63642 .636411895752
with the initial condition: R1(1) = R0 , because the con-
1
6.0 0.0 .61041 .610340917110
tinuity relation of velocity (27) is not verified at the initial
1.2 0.5 .71215 .712164473534
1.4 0.5 .66824 .668258676529 position R0 .
1
5/3 0.5 .63728 .637298974991
After that, the shock trajectory is deduced from the
1.2 1.0 .67297 .672985372543 R1(1)
definition of 1: R(1) = . The dependence on time
1
1.4 1.0 .62616 .626676673889
of the flow and shock trajectories comes from the relation:
5/3 1.0 .59466 .594669113159
R(t) =A(-t)ą, which requires knowledge of the parame-
1.1 2.0 .61200 .658860900879
R0
1
1.2 2.0 .60733 .607337988281
ters ą and A: A = .
(-ti)ą
1.4 2.0 .55882 .558805505371
The following paragraph is about the comparisons be-
5/3 2.0 .52639 .526383563232
tween trajectories calculated according to the way de-
2.0 2.0 .50339 .503395532227
scribed above and those resulting from the bidimensional
3.0 2.0 .47105 .471065814209
hydrodynamical simulation of a cylindrical implosion ( =
6.0 2.0 .44398 .443045935059
2).
4.2 Comparison with direct simulation
4 Flow and shock trajectories
4.1 Calculation method of the self-similar trajectories
The bidimensional simulation is explained in the thesis
(Toqu 1996, Chap. 2) in the case of a constant initial
Once ą is known, the solution (V (), Z(), H()) of
density ( = 0.0). It is the last part of a study which
system (4) can be calculated for fixed values of , ł
begins with several experiments.
and . In the first place, we calculate the entire func-
The experimental device is detailed in Sect. 2.2.1 of the
dZ
tion Z = Z(V ) by integrating the equation: , between
thesis (Toqu 1996). A cylindrical tin shell filled with sili-
dV
the limits (V (1), Z(1)) and (0, 0). After that, we calcu-
con is put in a complex cylindrical wave generator (CWG).
late the functions = (V ) and H = H(). For this,
The CWG is built with different kinds of explosive which
d ln d ln H
Eqs. (4): and , are integrated between the lim- improve the strengh of the shock wave. X radiographs
dV d ln
along the axis of the tin cylinder show the evolution of
its (V (1), 0) and (0, + ") for the first one, (0, ln H(1)) and
its interfaces.
(+ ", ln H(+ ")) for the second one.
In order to restitute the dynamics of the experiment,
The strictly monotonous behaviour of the functions
a particular one is first made with a lead cylinder instead
Z = Z(V ) and = (V ) allows us to deduce the function
of a tin one. Probes are put at different radii in the lead
Z = Z(). We notice that the value of ln H(+ ") is a finite
and the silicon media and at half height of the cylinder
value given by the numerical integration.
to avoid the edge effects. They give the trajectory of the
A trajectory is defined by the lagrangian description.
shock wave. These experimental data help us to built an
We assume that R1 = R1(t, R0 ) is the flow trajectory.
1
one dimensional simulation only taking into account the
It represents a set of points which is at rest at the initial
simple part of the cylindrical wave generator. The 1D nu-
position R0 and at time ti < 0: R0 = R1(ti, R0 ).
1 1 1
merical code is well adapted to the calculation of shock
At later times t >ti, the velocity of these points is not
waves driven by explosive, but not to the study of inter-
zero after the passage of the shock wave. Whatever their
faces instabilities which is the purpose of the thesis (Toqu
positions may be in the flow, the continuity of velocity is
1996). The bidimensional simulations deduced from the
verified:
one dimensional one are lagrangien. They are run with a
X 1 = 1X V (1) (27)
hydrodynamical code named HESIONE that is a 3D finite
1 is the self-similar variable with the definition:
elements code with finite differencies schemes.
R1(t,R0 )
1
1 = . R(t) is the shock trajectory at the position
Before planning simulations with perturbations, the
R(t)
main flow generated inside the silicon by the schock pas-
R0 at time ti. So, 1 is equal to one at this time.
1
1 R1
sage has to be identified. Clues such as the geometry of
The decomposition dR = dR1 - d1, replaced in
1 12
the experimental device, the implosion dynamics and the
the relation (27), involves the equation:
strengh of the schock have involved to the choice of the
self-similar implosion model that is introduced in the pre-
dR1 V (1)
= d1 (28)
vious parts.
R1 1(V (1) - 1)
On the next graphs, the self-similar trajectories in the
It is integrated from one to 1 on the right hand-side silicon medium are calculated with the values = 0.0,
and from R0 to R1(1) on the left hand-side. This results łsilicon =4.4 and ąsilicon = .76038.
1
162 N. Toqu: Self-similar implosion of a continuous stratified medium
1
"
0.9
Self-similar
"
2D simulation "
0.8
"
0.7
0.6
"
"
"
R "
"
"
0.5
"
"
R01
"
"
"
"
"
"
0.4
"
" "
" " "
0.3
0.2
"
0.1
0
0 2 4 6 8 10
1
Fig. 3. Shock trajectory in the silicon medium with a constant initial density ( =0.0)
1
"
Self-similar
2D simulation "
"
0.95
"
0.9
R1
R01
"
"
"
"
"
0.85
"
"
"
"
"
"
"
"
" "
" "
" " "
0.8
"
0.75
0 2 4 6 8 10
1
Fig. 4. Silicon flow trajectory ( =0.0)
2
The parameters ł and ą are obtained by fitting on the r
by the profile: = 0 R10 with 0 = 1.1 g/cm3 and
2D simulation trajectories. In the bidimensional simula-
R0 = 4.6 cm. The next graphs show a good agreement
1
tion, the shock front is quite difficult to localize because it
between the self-similar trajectories and the direct simu-
does not involve a well marked discontinuity of the quan-
lation ones. Consequently, in the silicon medium, the main
tities such as material velocity or density. In spite of it,
flow and the strong shock wave are self-similar.
one couple of parameters (łsilicon, ąsilicon) restitutes the
As noticed that the tin cylinder seems not to affect the
schock and the flow trajectories performed by the direct
dynamics inside the silicon medium. In fact, it behaves like
simulation.
a push which maintains the self-similarity of the silicon
In order to prove that the choice of the self-similar
flow.
implosion model is the right one, further comparisons are
By the way of the Runge-Kutta method, the self-
made in the case of a non uniform initial density profile. By
similar trajectories are rapidly calculated, less than 5 sec-
example, if it is selected as the square of the spatial vari-
onds on HP 5000 computers. With the best lower resolu-
able r ( =2.0), the corresponding value of the parameter
tion, the bidimensional simulation almost needs 4 hours
ą given in Annex B is: ąsilicon = .53366 for łsilicon =4.4.
on the same computers to achieve the collapse of the shock
The comparison is made with a bidimensional simu-
wave.
lation inside of it the initial uniform density is replaced
N. Toqu: Self-similar implosion of a continuous stratified medium 163
1
"
0.9
Self-similar
2D simulation "
0.8
"
0.7
0.6
"
R "
"
0.5
"
R01
"
"
0.4
"
"
0.3 "
"
"
0.2
"
0.1
0
0 2 4 6 8 10
1
Fig. 5. Shock trajectory in the silicon medium with a constant initial density ( =2.0)
1
"
Self-similar
2D simulation "
0.95
"
0.9
R1
R01
"
"
0.85
"
"
"
"
"
"
"
"
0.8
"
"
0.75
0 2 4 6 8 10
1
Fig. 6. Silicon flow trajectory ( =2.0)
For the spherical geometry ( = 3), some comparisons In the case of the cylindrical geometry, the restriction
are introduced in Sect. 4.5.2 of the thesis (Toqu 1996) in of the parameter is not coming from the polytropic con-
the case of an uniform density ( =0.0). The simulations stant ł. But, it depends on it in the spherical case. Under
are run with the 1D numerical code. The self-similar solu- the assumption that the integral of internal energy over a
tion is calculated with the same order of ł value (ł 4.2). fixed volume has to stay finite at the collapse of the shock
A discrepancy is observed for the silicon flow trajectory, wave, the physical meaning of the conditions (19), both
but not for the shock wave. It may be explained by the and separatly, has to be clarified over the only geometri-
push of the tin shell. Too much amplified because of the cal differency.
spherical geometry, it modifies the dynamics of the silicon The second contribution aims at proving that the self-
flow which is no more self-similar. similar implosion model restitutes the main flow of the
silicon medium in the experiment of cylindrical implosion.
The self-similar implosion model only calculates the
evolution of the inner medium, that is the silicon one,
5 Conclusion
without taking into account the interface with the tin
medium. It limits the next studies of instabilities to the
The first contribution of this paper is the mathematical
perturbations growing inside the silicon. The perturbed
demonstration of the conditions (19) on the initial density
simulations are defined with an initial sinusoidal interface
profile.
164 N. Toqu: Self-similar implosion of a continuous stratified medium
ł/ą ą0.0 ą 5 ą0.07 ą0.1 ą0.5 ą1.0 ą2.0
between the tin and the silicon media. They are detailed
1.1 .88524 .88524806 .87775 .87459 .83169 .75931 .69697
in Sect. 2.4.2 of the thesis (Toqu 1996) in the case of an
1.2 .86116 .86116303 .85231 .84859 .80265 .75297 .67175
uniform density. The amplitudes of the instabilities grow-
1.3 .84622 .83807 .83257 .78319 .73048 .64584
ing inside the silicon, close to the interface and to the 1.4 .83532 .83532320 .82521 .82099 .76936 .71490 .62833
1.5 .82675 .82234 .81191 .75881 .70309 .61532
focus, show large oscillations. Without experimental data
1.6 .81970 .81717 .75027 .69368 .60512
restricted to the X radiographs of the tin cylinder, the
5/3 .81563 .81562490 .80476 .80030 .74540 .68836 .59879
linearized form of the implosion model could evaluate the
1.7 .81377 .81109 .79828 .74315 .68592 .59683
amplitudes, not affected by the dynamics of the perturbed
1.8 .80860 .80859994 .79756 .79296 .73713 .67939 .58968
1.9 .80566 .80409908 .79309 .78822 .73188 .67376 .58404
interface and without numerical effects. It could partly ex-
2. .80159 .80011235 .78926 .78407 .72730 .66898 .57900
plain the oscillations and improve the direct simulation.
2.1 .79770 .79799 .78173 .72321 .66462 .57456
2.4 .78777 .78776900 .77613 .77198 .71336 .65419 .56400
3.0 .77568 .77566662 .76381 .75887 .70096 .64034 .55017
3.4 .77001 .77000368 .75807 .75309 .69480 .63406 .54393
A Annex 1
4.0 .76366 .76363465 .75160 .74656 .68700 .62707 .53710
4.4 .76038 .74826 .74325 .68344 .62347 .53366
The details of partial derivatives are:
5.0 .75643 .75640105 .74429 .73923 .67931 .61982 .52949
6.0 .75158 .75156168 .73939 .73430 .67418 .61494 .52451
"N Z3 10. .74188 .74182593 .72955 .72445 .66395 .60372 .51475
= - [-ł - 2 +2+]
50. .73011 .73002154 .71767 .71252 .65168 .59139 .50293
"V
2ł(V - 1)2
100. .72862 .72853594 .71619 .71101 .65013 .58985 .50147
Z
+ {2V [ł - 3+(1 - ł)]
ł/ą ą0.0 ą 5 ą0.07 ą0.1 ą0.5 ą1.0 ą2.0
2
1.1 .79594 .79596980 .79003 .78757 .75589 .71188 .61200
+(3 - ł) - (1 - ł) +2}
1.2 .75713 .75714179 .75042 .74758 .71215 .67297 .60733
1.3 .73378 .73377673 .72655 .72364 .68635 .64566 .57845
1.4 .71717 .71717450 .70977 .70667 .66824 .62616 .55882
1.5 .70440 .70442807 .69685 .69365 .65448 .61238 .54422
"N 3Z2 [Vł+2 - 2 - ]
= 2+ -
1.6 .69419 .69418951 .68644 .68323 .64348 .60104 .53276
"Z 2 ł(V - 1)
5/3 .68841 .68837682 .68060 .67732 .63728 .59466 .52639
1.7 .68572 .68571652 .67795 .67460 .63445 .59175 .52347
1
2
+ V [ł - 3+(1 - ł)]
1.8 .67857 .67855370 .67064 .66737 .62682 .58109 .51572
2
1.9 .67457 .67240014 .66693 .66176 .62030 .57724 .50911
1
2.0 .66887 .66704607 .66114 .65738 .61462 .57148 .50339
+ {V [(3 - ł) - (1 - ł) +2] - 2}
2.1 .66348 .65602 .65276 .60963 .56638 .49838
2
2.2 .65836 .65816533 .65105 .64792 .60521 .56189 .49402
2.4 .65109 .65108461 .64285 .63941 .59961 .55426 .48649
2.6 .64532 .64530018 .63699 .63352 .59315 .54805 .48043
"D
=( - V )(2V - 1) - V (V - 1) + Z2 2.8 .64050 .64048378 .63213 .62863 .58718 .54283 .47535
"V
3.0 .63642 .63641060 .62800 .62448 .58211 .53964 .47105
"D -2 +2+ 3.2 .63295 .63292118 .62446 .62093 .57834 .53586 .46735
=2Z + V (30)
3.4 .62993 .62989873 .62142 .61784 .57516 .53234 .46416
"Z ł
3.6 .62729 .62725578 .61874 .61518 .57235 .52915 .46135
3.8 .62496 .62492541 .61638 .61280 .56985 .52592 .45886
1
We notice that = .
4.0 .62289 .62285554 .61428 .61069 .56763 .52364 .45536
ą
4.5 .61861 .61857036 .60994 .60632 .56304 .51893 .45204
5.0 .61526 .61522398 .60654 .60291 .55944 .51524 .44907
5.5 .61261 .61253956 .60382 .60017 .55654 .51226 .44643
B Annex 2
6.0 .61041 .61033915 .60161 .59795 .55416 .50982 .44398
6.5 .60858 .60850311 .59975 .59607 .55216 .50777 .44170
7.0 .60703 .60694820 .59818 .59449 .55047 .50602 .43958
For =0.0, all the ą values calculated by R.B. Lazarus
8.0 .60454 .60445829 .59565 .59195 .54779 .50322 .43650
and R.D. Richtmyer (1977) are shown next to ours in the
10. .60114 .60104880 .59219 .58847 .54408 .49938 .53267
following arrays. The first one is for the cylindrical ge-
20. .59461 .58555 .58178 .53691 .49190 .42516
ometry ( = 2) and the second one is for the spherical 50. .59086 .59073010 .58555 .57793 .53277 .48758
60. .59045 .58132 .57752 .53235
geometry ( = 3).
80. .58993 .58080 .57700
N. Toqu: Self-similar implosion of a continuous stratified medium 165
References Sakurai A(1960) On the problem of a shock wave arriving at
the edge of a gas, Communications on Pure and Applied
Mathematics, Vol. 13, pp. 353 370
Butler DS (1954) Converging spherical and cylindrical shocks,
Ministry of Supply Fort Halstead Kent UK, A.R.D.E Re- Sharma VD, Radha CH (1995) Similitary solutions for con-
port 54/54 verging shocks in a relaxing gas, International Journal of
Guderley G (1942) Starke Kugelige and Zylindrische Verdich- Engineering Science, Vol. 33, No. 4, pp. 535 553
Toqu N (1996) Contribution to the study of interfaces insta-
tungsste, Luftfahrtsorschung, No. 19, 302 312
Hafner P (1988) Strong convergent shock waves near the cen- bilities in plane, cylindrical and spherical geometry, Thesis
of the University of PARIS VI FRANCE
ter of convergence: a power series solution, SIAM Journal
Applied Mathematics, Vol. 48, No. 6, pp. 1244 1260 Whitham GB (1974) Linear and nonlinear waves, John Wiley
& Sons Inc, pp. 172 177, 196 198, 273 274 and 309 310
Lazarus RB (1981) Self-similar solutions for converging shocks
Zel dovich Ya B, Raizer Yu P (1967) Physics of shock waves
and collapsing cavities, SIAM Journal Numerical Analysis,
and high-temperature hydrodynamic phenomena, Aca-
Vol. 18, No. 2
Lazarus RB, Richtmyer RD (1977) Similitary solutions for con- demic Press Inc., Chap. XII
verging shocks, Los Alamos scientific laboratory of the Uni-
versity of California, Report LA-6823-MS
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