CP620, Shock Compression of Condensed Matter - 2001
edited by M. D. Furnish, N. N. Thadhani, and Y. Horie
© 2002 American Institute of Physics 0-7354-0068-7/02/$ 19.00
INVARIANT FUNCTIONAL FORMS FOR K(p,P) TYPE
EQUATIONS OF STATE FOR HYDRODYNAMICALLY DRIVEN
FLOW
George M. Hrbek
Los Alamos National Laboratory, Los Alamos, NM 87545
A complete Lie group analysis of the general ID hydrodynamic flow problem for a general K(p,P)
type equation of state is presented. Since many equations of state (e.g., Vinet, Birch-Murnaghan,
Shaker, and Tait) are of this form, one can use the techniques presented here to study their behavior.
Making use of the full transformation group, the hyperbolic conservation laws for mass, momentum,
and energy reduce to non-linear, ordinary differential equations. These equations describe the density, particle velocity, and pressure behind the shock as functions of the shock front Mach number.
Note that unlike the classical result that defines the flow only at the strong shock limit, in this analysis the Mach number is allowed to be one or greater. This allows the behavior of a shocked material
to be described down to the acoustic limit. This technique is illustrated using the Tait EOS for a
shock moving through NaCl. Finally, the ratios of the group parameters are shown to have definite
physical meanings defined in terms of the equation of state and the physical conditions occurring
during the event that has produced the shock wave.
INTRODUCTION
plicable for ID hydrodynamic flow with a K(p,P)
closure/EOS relation.
Group analysis of the general ID hydrodynamic
flow problem has only been superficially interpreted and applied. Traditional work has centered
on the classical self-similar solution of the first
kind. The typical application being the generation
of analytic test problems of a physically degenerate, asymptotic flow useful in benchmarking general computer codes.
However, this represents a small and limited class
of solutions. A much richer interpretation exists if
one considers the full group transformation.
The goal of this paper is to provide a general technique that allows the researcher to accurately describe the hydrodynamic flow behind the shock
front for Mach numbers at the shock front greater
than or equal to one. This is done via a similarity
transformation using the full Lie group and is ap-
GROUP STRUCTURE
The 1-D flow equations for this problem are as
follows;
pt + pur + upr jupr~l=Q
(1)
ln
~~ P
"
r =0
(2)
0
(3)
Where t and r are the temporal and spatial variables, p is the density, u is the particle velocity, P
is the pressure, K(p,P) is the EOS/closure relation,
and j=0, 1, or 2 for rectangular, cylindrical, and
spherical geometry respectively.
139
Application of the Lie algorithm leads to the following five-parameter group generator;(3)
REDUCTION OF THE FLOW EQUATIONS
The conservation equations (1-3)
can be reduced
by the substitution of the similarity variable and
the transformed dependent variables (eqns. 7 - 10).
The continuity equation is then
c5pd
U =
(4)
Dividing by c2, these five group parameters can be
reduced to the four values, K= Ci/c2 the expansion
rate of the shock wave, a= c3/c2 the time delay till
the onset of shock formation,\|/=c4/c2 the material
resistance ratio, and \i= c5/c2 the material uniformity in front of the shock. The problem is further restricted by the invariance condition that dictates the
structure of the bulk modulus K.
^
and the conservation of energy is written
WhereK = K/P.
Reduction of this coupled set via Cramer's rule
yields f£]=A£]/0£], g'£]=0£]/0£], and
From the group generator a set of characteristic
equations can be written that can be used to generate various coordinate transformations of the flow
equations
dr _ dt _ dp _
du
(ID
The conservation of linear momentum equation
becomes
(5)
fr~t + <r~/ip~
~
], where
(14)
(15)
(6)
dP
(16)
Given this set of equations, it is then possible to
construct the following general class of coordinate
transformations. Starting with the similarity variable.
and
(17)
(7)
Note that %= -1+A, +j A, and i3= -2 +2 A+ jx.
and the new dependent variables for the density
p($,t) = (t+aY
(8)
and the velocity
u(s,t)=(t+<
INVARIANCE CONDITIONS FOR
THE BULK MODULUS
(9)
and finally the pressure
The invariance condition on the bulk modulus (5)
allows the following four mathematical forms;
• The full form with material resistance and
non-uniformity (vj^O and (
¥
(P+ (-2+2A+/Q
K(p,P)
]
(18)
0\-^-r^/VT^,/^
•Q\.
(10)
Note that these variables are expressed in terms of
the independent similarity variable £, time, the
group parameter ratios as described above, and the
reduced functions f(£), g(^), and ]
•
no material resistance \|/^0, but non-uniform
material [i^O
K(p,P)
_
p<-2+2
140
= c[p-
• • uniform
material
µ≡0,
but
resistance
uniform
material
|0^0,
but
resistanceψ≠0
(20)
K K(p,P)
(ρ ,P) = =
( 2(2P(A-l)
P ( λ − 1) + +
ψ y))CC[[ρp ]]
(20)
• • uniform
material
µ≡0,
and
nonoresistance
uniform
material
ji^O,
and
resistanceψ≡0
\|/=0
(21)
KK
( ρ( ,pP, P) =
]
) =PCPC[ ρ[p]
(21)
These
forms
can
be
used
to
represent
the
bulk
These forms can be used to represent the bulk
modulus
in invarious
modulus
variousways
waysininthe
thereduced
reducedequations
equations
(11-13).
Note
that
the
material
resistance
(11-13). Note that the material resistanceψ,\|/,does
does
notnot
appear
in in
thethesolution
appear
solutionofofthe
thereduced
reducedfunctions
functions
f,g,f,g,
and
h, h,butbutrather
and
ratheris isseen
seenininthe
theexpression
expressionfor
for
thethe
pressure
(10)
pressure
(10)
The
Theexpression
expressionforforthethe pressure
pressure without
without this
this
material
resistance
turn
material
resistance
turnout
outtotobebedifferent
differentfrom
fromthe
the
oneonewhere
wherethethematerial
materialresistance
resistanceisisincorporated
incorporated
into
intothetheanalysis.
analysis. Estimates
Estimates ofof the
the effect
effect ofof
including
will
consideredinina afuture
futurepaper.
paper.
including
ψ \|/
will
bebeconsidered
(5)
The
is shown
shown in
in Figure
Figure 1.
1.
Thevariation
variationof
of ψ
\|/ for
for NaCl
NaCl(5) is
Note
that
µ=2
(3+j)λ
in
order
to
maintain
invariNote that |i=2 - (3+j)X in order to maintain invariance
anceofofthe
theenergy
energy integral.
integral.
The
TheRankine-Hugoniot
Rankine-Hugoniot relations
relations
ρp1luu1l == ρ 0 u 0
(27)
(27)
Compression Ratio (β)
Compression Ratio (p)
Material Resistance (ψ)
0
1
1.25
1.5
1.75
2
-100
-100
Planar (j=0)
Cylidrical (j=1)
« -200
-200 - -
Spherical (j=2)
-300
-300
Figure
for NaCI
NaCl Ko=23.81,
Ko=23.81,Ko'=5.68
Ko'=5.68
ψ for
Figure 11 Tait
Tait y
ILLUSTRATIVE
EXAMPLE
ILLUSTRATIVE
EXAMPLE
PP
ρ
−1+ exp([ 1− o ][1+ K o' ])
ρ
e =e = •
(
1
+
ρ
ρ o [1+ K o' ]
K o ( −1+ exp([1−
P p==
K (ρ , P) =
×-x
ρ
exp([ 1− o ][1+ K o' ]
ρ
e1 − e 0 =
)
(23)
(23)
1+ K o'
ρo
ρ
exp([ 1 −
ρo
][1 + K o' ])
ρ
ψ
(−2 + 2λ + µ )
(24)
exp([ 1 −
(24)
1
][1 + K o' ])( β − 1) ×
β
2 β 2 [ 2 + K o' ] exp( −
M2[l+K'0]
M 2 [1+ K o' ]
(25)
1+ K o'
β
(30)
(30)
) -X
×
(exp[ 1 + K o' - exp[ 1 + K' J) = 0
(exp[
] − exp[ 1 + K o ]) = 0
β
where we have made use of the fact that K=c22b and
where
we have
usewave
of the
fact that
K=cc is
b and
the speed
of themade
shock
R=cM,
where
the
the
speed
of
the
shock
wave
R=cM,
where
c
the
speed of sound through the medium and M isis the
( −2 + 2λ + µ ) K o
(26)
×
(26)
β (1 + K o' )
(ft + exp([ 1 ( β + exp([ 1 − 1 ][1 + K o' (1 − β + K o' ]))
β
Where P is the compression
ratio and K and K '
0
(29)
(29)
( − 1 − β + K o' ( β − 1)) −
(25)
This yields the material resistance function \|/
This yields the material resistance function ψ
ψ =
P1 1
1
(
−
)
2
2 ρ 0 ρ1
(1 − β 2 )(1 + K o' ) +
where p0 is the initial density.
where
is the
initial
If weρostart
with
the density.
general form (18) we can write
If the
we Tait
start bulk
withmodulus
the general
form (18) we can write
as follows
the Tait bulk modulus as follows
K (ρ , P) = P +
(28)
(28)
areused
usedtotogenerate
generate the
the boundary
boundary conditions
conditions at
are
at the
the
shock
front
(^=1).
From
the
conservation
shock front (ξ=1). From the conservation of
of
energy relation
relation (29),
(29), an
an estimate
estimate of
energy
of the
the
compression ratio
ratio for
for the
the particular
particular equation
compression
equation of
of
state
(i.e.,
Tait)
can
be
defined
in
terms
state (i.e., Tait) can be defined in terms of
of the
the
Mach Number
Number and
and the
the first
first derivative
derivative of
Mach
of the
the bulk
bulk
modulus
K
'.
0
modulus K0’.
(22)
(22)
ρo
][1+ K o' ])
ρ
ρ0
)
ρPi1
P1 = u 02 ρ 0 (1 −
example,
considerthe
theTait
TaitEquation
Equationofof
AsAs
anan
example,
consider
(4) (4>
State
State
speed
sound through
the medium
and M is and
the
Mach ofNumber.
For NaCI
(Ko=23.81GPa
Mach
Number.
For
NaCl
(Ko=23.81GPa
and
Ko'=5.68) the compression ratio asymptotically
Ko’=5.68)
compression
approaches the
1.6823
(see Tableratio
1). asymptotically
Note that this
approaches 1.6823 (see Table 1). Note that this
0
Where
β is
themodulus
compression
ratio
Ko and Ko’
are the
bulk
and its
firstand
derivative.
are the bulk modulus and its first derivative.
141
Mach No. (M)*
Compression Ratio (β)
Mach No. (M)*
1.1
1.1 1.2
1.1
1.2 1.5
1.2
1.5 2.0
1.5
2.0 5,0
5,0 10.0
10.0 20.0
10.0
20.0 50.0
580.0 (→∞)
50.0
(->°o)
580.0 (→∞)
Ratio ((3)
Compression1.0584
(β)
1.0584
1.1135
1.1135
1.2547
1.2547
1.4101
1.4101
1.6329
1.6697
1.6329
1.6791
1.6697
1.6818
1.6791
1.6823
1.6818
1.6823
Note that in figures 2-4, the lowest curves corre-
Note
thattoinM=1.1
figures
correfigures
2-4,
the
lowest
curves
correspond
and2-4,
thatthe
thelowest
curves curves
for M ≥5
are
spond
to M=l.l
and that
curves
for
M
spond
M=1.1
that the
the
curves
forlimit.
M >5
≥5 are
are
merging
at the asymptotic,
strong
shock
merging
merging at
at the
the asymptotic,
asymptotic, strong
strong shock
shock limit.
limit.
*Note that the high Mach flow (M>5) are only pre-
The profiles vary with Mach number and should
1+ K
=
illustrate how the density, particle velocity, and
(14-17) yield solutions to (11-13). Figures 2 - 4 ilpressure profiles vary from the shock front (ξ=1)
lustrate how the density, particle velocity, and
back towards the center of initiation (ξ=0) for a
pressure
profiles vary from the shock front
(^=1)
(ξ=1)
spherically expanding shock wave front
in NaCl.
back towards the center of initiation (ξ=0)
(§=0) for a
Dimensionless
Position (ξ)
spherically expanding
shock wave
in NaCl.
Reduced Density f( )
Reduced Density f( )
1.7
1.6
1.5
1.4
M=1.2
M=1.5
M=1.1
1.4
1.2
1.1
M=2.0 M=20.0
M=5.0 M=50.0
1
1.3
M=2.0
M=1.2 M=5.0
M=1.5 M=10.0
1.3
0.5 M=10.0
0.55 M=580.0
0.6 0.65
1.2
1.1
0.7
0.75
0.8
0.85
0.9
M=20.0
Figure 2 NaCl Variations by Mach No. (j=2,
0.95
0.55 M=580.0
0.6 0.65
0.5
0.7
0.75
0.8
0.85
0.9
REFERENCES
1
0.95
(1) Hrbek, G. M., in Proceedings Shock Compression of
REFERENCES
REFERENCES
1
Condensed Matter -1999 (CP505), 169-172 (1999).
Dimensionless
Position
Figure 22NaCl
NaCl Variations
Variations
by Mach
Mach
No.(ξ)
(j=2, µ=0.0)
Figure
by
No.
(j=2,
(2)
Hrbek, G.
M., inProceedings
Proceedings ShockCompression
Compression of
(1)
Hrbek,
(1)Condensed
Hrbek, G.
G. M.,
M., in
in Proceedings Shock
Shock Compression of
of
Matter
-2001
(O4.003),
(2001).
Condensed
Matter -1999
(CP505),
169-172 (1999).
Condensed
(CP505),
(3) Hrbek,Matter
G. M.,-1999
Group
Analysis169-172
of Shock(1999).
Wave Phe(2)
Hrbek,
G.
M.,
Proceedings
Shock
Compression
of
(2)nomena
Hrbek,in
G.Solids,
M., in
in Ph.D.
Proceedings
ShockUniv.
Compression
of
Dissertation,
of IL, UrCondensed
Matter
-2001
(O4.003),
(2001).
Condensed
Matter -2001 (O4.003), (2001).
bana, IL, 1992.
(3)
Hrbek,
G.
Group
Phe(3)(4)
Hrbek,
G. M.,
M.,
Group
Analysis
of Shock
ShockBWave
Wave
PheKushwah,
S.S.,
and J.Analysis
Shanker, of
Physica
253, 90-95
nomena
nomena
in Solids,
Solids, Ph.D.
Ph.D. Dissertation,
Dissertation, Univ.
Univ. of
of IL,
IL, UrUr(1998).in
bana,
bana,
IL, 1992.
1992. L. C., and A.L. Ruoff, J. of Appl. Phys.
(5) IL,
Chbabildas,
(4)
Kushwah,
S.S.,
J.J. Shanker,
(B13), 4182-4187
(4)47
Kushwah,
S.S., and
and(1976).
Shanker, Pkysica
PhysicaBB 253,
253, 90-95
90-95
(1998).
(1998).
(5)
(5) Chbabildas,
Chbabildas, L.
L. C.,
C., and
and A.L.
A.L. Ruoff,
Ruoff, /.J. ofAppl.
of Appl. Phys.
Phys.
47
47 (B13),
(B13), 4182-4187
4182-4187 (1976).
(1976).
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
Reduced Velocity g(ξ)
Reduced Velocity g(ξ)
M=1.1
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
M=1.2
M=1.5
Dimensionless Position
Position(ξ)
(c)
Dimensionless
M=1.1 M=2.0
M=1.2 M=5.0
M=1.5 M=10.0
M=2.0 M=20.0
M=5.0
M=50.0
0.5
0.55M=580.0
0.6 0.65
M=10.0
0.7
0.75
0.8
0.85
0.9
0.95
1
Figure 3 NaCl Variations by Mach No. (j=2, µ=0.0)
M=20.0
M=50.0
0.5
0.55M=580.0
0.6 0.65
0.7
0.75
0.8
0.85
CONCLUSION
CONCLUSION
Group theoretical methods have been shown to be
µ=0.0)
M=50.0
1
CONCLUSION
a powerful method in the evaluation of flow behav-
Dimensionless Position
Position (ξ)
(|)
Dimensionless
1.6
1.5
approach
acoustic
wave
M→1.
As
The
profiles
with
number
and
should
profilesanvary
with Mach
Machsolution
numberas
and
should
M→∞, the
profiles asymptotically
a
approach
an acoustic
wave
as
wave solution
solutionconverge
as M—>1.
M→1.asAs
As
consequence
of theasymptotically
conservation ofconverge
energy reM—»oo
the
profiles
as
a
M→∞,? the profiles asymptotically converge as a
quirement atofthe
shock front (30)ofand
generate the
consequence
of the
the conservation
conservation of energy
energy rerestrong shock, self-similar solution.
quirement
quirement at
at the
the shock
shock front
front (30)
(30) and
and generate
generate the
the
strong
strong shock,
shock, self-similar
self-similar solution.
solution.
Group
theoretical
methods
have
shown
to
methods
have been
been
shown
to be
be
ior behind
a shock
front. Using
the full
group
a powerful
method
in
the
evaluation
of
flow
behavpowerful
method
the
evaluation
of
flow
behavtransformation and the Tait EOS, the passage of a
iorshock
behind
a shock
front.
the
full
front.ofUsing
Using
the been
full group
group
through
a block
NaCl has
simulated
transformation
and
the
EOS,
passage
of
as a function of
Mach
number
at the
the
front.
transformation
and
the Tait
Tait
EOS,
the shock
passage
of aa
shock
shock through
through a block
block of
of NaCl
NaCl has
has been
been simulated
simulated
as a function
function of
of Mach
Mach number
number at
at the
the shock
shock front.
front.
M=1.1
1.7
DISCUSSION
DISCUSSION
DISCUSSION
+ K o' ) exp[o 1]−+exp[
K o' ])1+ K o' ]) β
(1(exp[
(31)
' β
1+ K o
'
+ K(11-13).
(exp[
]− exp[ 1to
(14-17)
yieldβsolutions
Figures 2 - 4
o ]) β
K
P
M=1.2
M=1.2
0.45
M=1.5
0.45
0.40
M=1.5
0.40
M=2.0
0.35
M=2.0
0.35
0.30
M=5.0
0.30
0.25
M=5.0
M=10.0
0.25
0.20
M=10.0
0.20
0.15
M=20.0
0.15
M=20.0
0.10
M=50.0
0.05
0.10
M=50.0
M=580.0
0.00
0.05
M=580.0
0.00 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9
0.95 1
0.5 Figure
0.55 4
0.6
0.65
0.7 0.75
0.75
0.8 0.85
0.85
0.9 µ=0.0)
0.95 11
0.5
0.55
0.6
0.65
0.7
0.8
0.9
0.95
NaCl
Variations
by Mach
No. (j=2,
Figure 44 NaCl
NaCl Variations
Variations by
by Mach
Mach No.
No. (j=2,
(j=2, µ=0.0)
jj,=0.0)
Figure
to show
the onset
of (M>5)
asymptotic
behavior.
*Notesented
that the
high Mach
flow
are only
preflow
the Mach-Compression
relationship
(30),
the onset of asymptotic
behavior.
sentedUsing
to show
integratingrelationship
(11-13) for (30),
the Tait κ
Usingand
thenumerically
Mach-Compression
equation integrating (11-13) for the
and numerically
the Tait K
κ
(1+ K o' ) exp[ 1+ K o' ])
K
equation
κ = P =
(31)
'
κ =
/ ——M=1.1
^M=1.1
Reduced Pressure h(ξ)
Table 1. Energy Conservation Across
Table 1. Energy Conservation Across
The Shock Front for NaCl
The Shock Front for NaCl
Dimensionless Position (ξ)
Dimensionless
Dimensionless Position
Position (ξ)
(£,)
Reduced Pressure h(ξ)
shows
the speed
of the
shock
only
table table
shows
how how
the speed
of the
shock
cancan
only
shows
only
produce
a well
defined
eq. 30)
compression
produce
a well
defined
(via (via
eq. 30)
compression
or or
willconserve
not conserve
energy
across
shock
front.
it willitnot
energy
front.
across
the the
shock
front.
0.9 0.95
0.95
0.9
11
Figure33NaCl
NaCl Variations
Variations by
by Mach
Mach No.
No. (j=2,
(j=2, µ=0.0)
^=0.0)
Figure
142