Published by SLIT,
Shanghai, China
Applied Mathematics and Mechanics
(English Edition, Vol.9, No.2, Feb 1988)
ON G E N E R A L F O R M OF N A V I E R - S T O K E S E Q U A T I O N S AND
IMPLICIT FACTORED SCHEME
Wang Bao-guo ( ~ E ~ )
(Institute of Engineering Therraophysics, Academia Sinica, Beoing )
(Received Aug. 1, 1986; Communicated by Bian Yin-gui)
Abstract
A general weak conserval'iveform of Navier-Stokes equations expressed with respect
to non-orthogonat curviiinear coordinm'es and with primitive variables was obtained by
Abstract
using tensor analysis technique, where the comravariant and covariant velociiy components
were employed. Compared
current
method, the
The one-dimensional
problem ofwith
the the
motion
of acoordinate
rigid flyingtransformm'ion
plate under explosive
attack has
an analyticestablished
solution equations
only when
polytropic
index and
of detonation
equals
three. In
are the
concise
and forlhright,
they are moreproducts
convenient
to be to
used
general, a for
numerical
analysis in
is body-fitted
required. In
this paper,
however,
by utilizing
thefaciored
"weak" shock
solving problems
curviiinear
coordinate
syslem.
An implicit
behavior ofscheme
the reflection
shock
in
the
explosive
products,
and
applying
the
small
parameter
purfor solving the equations is presented with r
discussions in this paper. For nterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
dimensional flow the algorithm requires n-steps and for each step only a block tridiagonal
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
matrix
needs obtained
tobe~o&ed,
it avoids
inverling
the matrix for
large
of Thus
Final velocities ofequation
flying plate
agree
very well
with numerical
results
bysystems
computers.
enhances
the speed ofof
arithmetic.
In this stud.v,
the Beam- Warming's
implicit
an analyticequations
formulaand
with
two parameters
high explosive
(i.e. detonation
velocity and
polytropic
index) for factored
estimation
of the is
velocity
of flying
plate is established.
sJhceme
extended
and developed
in noh-orlhogonai curviiinear coordinate
system.
I.
Introduction
1.
Introduction
Explosive
technique
ffmds its important
in the
study
of behavior
of
In
order to driven
obtain flying-plate
physical solutions
withsuflicient
accuracy,use
there
arises
the need
for accurate
materials
under
intense
impulsive
loading,
shock
synthesis
of
diamonds,
and
explosive
welding
and
numerical representation of boundary conditions. Such representation is best accompiished when
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
the coordinate lines coincide with the curved body contours. Therefore, general non-orthogonal
of common interest.
curvilinear
are of
applied
widetyO-31.plane
At present,
a variety
of flying
formsplate,
of Na~Jier-Stokes
Under coordinates
the assumptions
one-dimensional
detonation
and rigid
the normal
equations
were
given
re!several
types
ofcooralnatesystems
14~J,~ut
the
coordinate
approach of solving the problem of motion of flyor is to solve the following system transformation
of equations
method isthestill
widely
usedtTI
and equations
so obtained
are(Fig.
complicated.
In this paper, tensor
governing
flow
field of
detonation
products behind
the flyor
I):
analysis technique is directly adopted and Navier-Stokes equations in conservative form can be
directly derived in generalized co6rdinates.
The equations
--ff
=o, are umversal and the form is neater. In
ap +u_~_xp+ au
addition, the paper has a detail discussion.of
Navier-Stokes equations solved by using implicit
au
au
approximate factorization algorithms.
y1
=0,
If.
The Gradient, Curl and D iaS
v e r g e nacse of Higher-Order Tensor Fields
a--T
(i.0
=o,
The gradient, curl and divergence of higher-order tensor fields are equivalent to operators
p =p(p,
s), on the tensor. Consider, for exaniple, the arbitrary
9 ~(O/Ox ~), e' x ( O / O x ' ) , e ' . (O/Ox
~) acting
scalar field f, vector field a and second-order tensor fields eJ0"r~,, when they are acted onwith
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
the previouswith
operators,
we findRthat
respectively,
the trajectory
of reflected shock of detonation wave D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state para179 rarefaction wave behind the detonation wave
meters on it are governed by the flow field I of central
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
180
Wang Bao-guo
v f = e ' v , f =e'Of./Ox'
(2.1)
va=e~e~v~a ~
(2.2)
V (e~e~r~) : e~e~e~v~v~
(2.3)
el
1
V • a ---- ~'~e~v~aa-"
- - - M,- ff
82
e3
O/ax ~ o/ax ~ alax ~
gz
as
V X (e~e~r~) =~'~e~:e~
1 O(~"~a'.)
V" a = ~--~
Ox i
V" (e~e~r~~)
=
(2.4)
(2.5)
v~a ~
e~v ~r~
(2.6)
(2.7)
where fa iJ,
~ {a'} are covariant and contravariant components of a , respectively; { r ~ } , ,~r~}
Abstract
are covariant and contravariant components of a second-order tensor, respectively; { o~ }, { e ~} are
correspondingly
the local
base of
vectors
and of
reciprocal
base plate
vectors
general attack
curvilinear
The one-dimensional
problem
the motion
a rigid flying
underinexplosive
has
V~ is covariant
differentiation
of a tensor
metric tensor
of any
coordinates
ancoordinates;
analytic solution
only when
the polytropic
index ;g.~-is
of detonation
products
equals
to three.and
In its
general,
a numerical
analysis
is required.
In this tensor.
paper, For
however,
by utilizing
the "weak"
shock
determinant
is denoted
by g;{~Jk}
is Eddington
an exhausti.ve
account
o f the previous
behavior
of the
shock references
in the explosive
applying
the small
parameter
purexpressions,
onereflection
should consult
[8] andproducts,
[1 I]. In aand
similar
manner,
we find
that Laplace's
terbation
method,
an
analytic,
first-order
approximate
solution
is
obtained
for
the
problem
of
flying
operator acts on f, a and second-order tensor e~e*r~.k, respectively,i.e.
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
Final velocities of flying plate obtained agree very
well with numerical results by computers. Thus
V. v / = e ' .--6W-x~~, e
(2.8)
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
index) for estimation of the velocity of flying plate is established.
V" v e ~ e k r ~ =1. g ~ Introduction
e ~ e " v ~ v j.r~o
(2.10)
whereExplosive
{ gO'}are driven
second-order
contravariant
tensor.
flying-plate
technique metric
ffmds its
important use in the study of behavior of
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
III. V i s c o u s - S t r e s s T e n s o r in G e n e r a l Curvilinear C o o r d i n a t e s
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
of common
interest.that V is velocity of fluid flow, then the strain-rate tensor becomesml
I f w e specify
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
~={ V
+ ( v Vis) oto}/2
(3.1)
approach of solving the problem of motion
ofv flyor
solve the following system of equations
governing
the
flow
field
of
detonation
products
behind
the
flyor
(Fig.
I):
In non-orthogonal curvilinear coordinates it can be written
e = e~e ~'(V ,vj Jr V.~v, ) / 2 = e,e~ (V*V~H- V~v ~) / 2 = e r eJei.~ = e, e~e =-~
--ff
ap +u_~_xp+
au
=o,
( 3.2 )
where {v~}, {v~}, are covariant and contravariant components of V , respectively;
au
au
{ e~ }, { e ~ } are covariant and contravariant components
of strain-rate tensor, respectively; As is
y1
=0,
(i.0
well - known, e=j and
e *j are both symmetric tensors; We make the usual hydrodynamical
aS
as
assumption that the viscous-stress tensor
a--T is a linear
=o, function of the strain-rate tensorU0k it can be
written[~.~l:
p
H = e , e j ~ 'y-- [ - - (
=p(p, s),
P'~--~l.Z~iv'l~'J'~-2#
8i'~]e,ey=(v"J--,g"P)e,e~
(3.3)
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
where {v=~} is contravariant components of the stress tensor; {r,=~} is contravariant
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state paracomponents
viscous-stress
p, the
pressure;/~,
the coefficient
of viscosity;
meters on it of
arethe
governed
by the tensor;
flow field
I of hydrostatic
central rarefaction
wave behind
the detonation
wave
rD~ and
andby initial
r ' ~ stage
are symmetric
Substituting
Eq.of (3.2)
Eq. (3.3),
the expression
of motion tensors.
of flyor also;
the position
F andinto
the state
parameters
of products o f
viscous-stress tensor becomes
293
General Form of Navier-Stokes Equations
181
v'~ =/z (V ~v~+ V~Vi) - - ( 2 / 3 )/.z,.q{ J v / r il
(3.4)
=tz(gl"g ~p+ g~#g~"-- (2/3)g~g~'#)V,,v#
where V ~ and V~ are contravariant and covariant derivatives, respectively.
IV.
Navier-Stokes
Equations
of W e a k
Conservative
Form
in Generalized
Curvilinear Coordinates
The continuity equation:
ap
at
Op
1 o(~/ gpv ~) : 0
~-v'(P~")~-Ti-+-E-/-Eo
where p is density, t is time, and
The momentum equation"
(4.1)
ax'
v ~ is contravariant comporrents of velocity V 9
Abstract
Or, q vSvjv/~__e~
p ]
~v~
+v,-~t I
e p( at - problem of theL motion
at +pu'v.~v,+v,V.,"
(pv')
The one-dimensional
of a rigid flying plate
under explosive attack has
an analytic solution only when the polytropic index of detonation products equals to three. In
general, a numerical analysis is required.
this paper,
however,]by
utilizing the "weak" shock
= e ~ [ InO(pvD#t
~-V~(pv~v~)
=e~V~rj'~
behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
is a mixed second-order stress tensor: The dot product of the preceding equation
wheredriven by various
plate
high explosives with polytropic indices other than but nearly equal to three.
el;
is
taken
and the
results
are agree very well with numerical results by computers. Thus
with velocities of flying
Final
plate
obtained
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
O(pvk ) /OI"F V ~ (pv:vk--'c~.~ ) =0
index) for estimation of the velocity of flying plate is established.
Multiplying both sides of this equation byg ~. one gets
1. /Ot-F
Introduction
O(pv~)
v ~(pv~v~--'r ~) ----0
(.t 2)
The
energydriven
equation:
Explosive
flying-plate technique ffmds its important use in the study of behavior of
~"
be respectively
the total shock
energysynthesis
per unit of
volume
and per
mass, welding
where the
Let under
e and intense
materials
impulsive loading,
diamonds,
and unit
explosive
andtotal
cladding
of metals.
of estimation
flyor
velocity
way of raising
it are
energy will
includeThe
notmethod
only internal
energyofbut
also
kineticand
andthepotential
energ.y
u.tmquestions
Thus, the
of
common
interest.
energy equation is
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
approach of
solving the(/-/.V)
problem
of flyor is to solve~'+~"
the following+V~(pv
system of
p---di-=V.
+Vof9motion
()LvT)=p---#u-+pv~v~
~) equations
governing the flow field of detonation products behind the flyor (Fig. I):
=Oe/Ot + V ,( ev ~)
(4.3)
where
2
--ff
=o,with
is thermal conductivity;
temperature;
ap Tis
+u_~_xp
+ au
au
au
1
y
=0,
V" (/7. V) =V~(r~JvD
V"
~)
aS(2vT)=V,(Zg~JOT'/ax
as
equation (4.3) becomes,
(4.4)
(4.5)
(i.0
a--T
=o,
p =p(p, s),
e/Ot + V~
(ev' --entropy
ruv~ -- and
2gUaT/Ox
= 0 of detonation products(4.6)
where p, p, S, u are pressure,adensity,
specific
particle j)
velocity
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
Combining
Eqs.
(4.1), (4.2)
and (4.6)
Navier-Stokes
equations
of weak
conservative
trajectory
F of flyor
as another
boundary.
Bothyields
are unknown;
the position
of R and
the state
parameters
it are governed
by the flow field I of central rarefaction wave behind the detonation wave
form on
in ~generalized
coordinates
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
182
Wang Bao-guo
pv'
fLI
]
pv,v, + g,,e/
0 I v.l+
0
0--7-1; 1 -0~
pyre,' + az,P|=
"~v3v' + g"'PI
( e + P)v ~ J
I
v,I
I
t.-c"Jv~+,~o'~aT/Ox~ .I t. --F~,(e+P)vJ
I
(4.7)
_J
Abstract
here v is the pressure; v ~,g~rand r '~j are accordingly the contravariant components of velocity,
The one-dimensional
the motion
of a rigid flying
plate under
attacktensor;
has
contravariant
componentsproblem
of metricoftensor
and contravariant
components
of explosive
viscous-stress
an/ ' ~analytic
solution
only
when
the
polytropic
index
of
detonation
products
equals
to
three.
In
'is the Christoffel symbol of the second kind; i,j, k= 1 to 3;
general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock
behavior
the reflection shock Sinc hthe
V. A n ofLrnp!.~citFactored
e m eexplosive
f o r t h eproducts,
N a v i e r -and
S t o kapplying
e s E q uthe
a t i small
o n s parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
Introducing
the fc.llowing
symbols:
plate driven
by various
high explosives
with polytropic indices other than but nearly equal to three.
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
PV1detonation
]
p with two parameters of high explosive (i.e.
an analytic formula
velocity and polytropic
index) for estimation
p v 1 of the velocity of flying plate is established.
pvlv I + gllp
U~
pv 2
pv 3
,
1.
pv2v z+'9~lp
E(U)~
Introduction
pvSv I + gs~p
( e + P)v 1use in the study of behavior of
Explosive driven flying-plate technique ffmds its important
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding(5ol)
and
pvlv s +and
918p
q.-g12p
cladding of metals. ThepvlvZ
method
of estimation of flyor velocity
the way of raising it are questions
of common
interest.
F(U
) ~ [ pv2v~+ g2=_P , G(U)~
ovZv3+ 9zsP
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
pvsv z + 9sLP
pvSv 3+ .qssp
approach of solving the problem of motion of flyor is to solve the following system of equations
e + P)v
s I):
( e detonation
+ P)v ~ products behind the(flyor
governing the flow field of
(Fig.
m ~ 0~] 1
n~pv ~, b_~pv 8, ( V ) ~ . . V . V
g
--ff
=o,
ap +u_~_xp+ au
where P is the hydrostatic pressure and its expression is
au
au
y1
=0,
-P= (y-- 1) r e - - ( 1 / 2 0 ) (g:lmZ+gzznZ+.q~bZ+2g,~mn-V2g,smb+2g.3nb) ]
aS
as
a--T
(i.0
(5.2)
=o,
where y is ratio of specific heats. Because the flux vectors E,Fand G are homogeneous functions
p =p(p, s),
of degree one in U, so that
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
E =ofAreflected
U , F =shock
B U , ofGdetonation
=CU
5.3)
respectively, with the trajectory R
wave D as a boundary and (the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parawhere on
.A,Bit and.
C are Jacobian
matrixes,
meters
are governed
by the flow
field Ii.e.
of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
General Form of Navier-Stokes Equations
183
f
.-.
I
I
I
v
I
~:~
I
~
v
v
I
~
~
I
~
v
~
v
I
.~.-
~.~
%
v
v
%
%
e.=4
I
I
v
I
v
I
-~
I
~
~"
I
~"
I
I
%
-
Abstract
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has
an analytic solution only when the polytropic index of detonation products equals to three. In
general, a numerical analysis
by utilizing
I is Irequired.
~
~ In this I paper,"~however,
~
.-"~the "weak" shock
~.
~I
~.
I
=
,-~
I
behavior ofv the reflection
shock
in the explosive
products,
and applying
the small
parameter
v
I
I purI
I
v
I
"
"
~"
"
I
terbation method,
anI analytic, first-order approximate
solution
is
obtained
for
the
problem
of
flying
v
v
I
%
I
~
I
plate driven by various high explosives with polytropic indices other than but nearly equal
I to three.
I
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
o)
index) for estimation
of the velocity of flying plate is established.
v
I
~-
v
1. ~ Introduction
I
I
I
~"
I
I
v
I
v
I
I of
Explosive driven flying-plate
technique
ffmds
use in% the study
of behavior
%
%
I its important
i
I
I
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
c,l
cladding of metals. The method of estimation of flyor velocity
and the way
it are questions
%
% of raising
%
%
%
of common interest.
V
v
Under I the assumptions
plane
detonation
and rigid
I normal
I
I flyingI plate, the
I
I
I
I of one-dimensional
approach of solving the problem of motion of flyor is to solve the following system of equations
v
V
V
governing
behind
thev flyor~(Fig.~ I): v
o the
v flow vfield of detonation products
,~..O~
--ff
ap +u_~_xp+
C~
au
au
Ul
III
~
9
II
C~I
aS
as
au
y1
a--T
=o,
II
p =p(p, s),
=o,
C~
I
=0,
ILl
"r
(i.0
lad
II
r,.)
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
}84
Wang Bao-guo
where
glg 3ilvl I
az
t a~
ga, ga~ gaa
vz
gal ga~ g~a
t'~
~l'e
~ 7--I ( V ) ' - - ( V - - 1 )
ye
P
"p--1
--(V)~--(v_l)v~a~
2
a,=--- (1,-- 1) ( U ) ~ t, ~
~"ev~ '
a~
a =_(p_l)(U),_v ~
~,ev z
P '
a,=--
"~ev ~
?e
a~--------p
a =_(p_l)(V)Zv
s
--.p
,
?--1
2
via,
(V)Z--(l'--l)v~a~
In order to study covariant derivative of viscous-stress tensor, equation (3.4) can be rearranged
as tbllows:
Abstract
~ ' " = b ~ ' d ~ o W / o x o - W"~~ ~,,~ F ~ , ~
(5.4)
The one-dimensional problem of the motion of a rigid flying plate
under
explosive
attack
has
(ij)
~I"~I'= 1~(9,~gt# -F 9~#9~ - - ( 2 / 3 ) 9'19" # ),
:' =9,,e a,~products equals to three. In
an analytic
solution only when the polytropic index ofb~d
detonation
general,
numerical analysis
is required.
however,
by up
utilizing
It isa emphasized
that indices
k, a, Infi this
andpaper,
o- are
summed
from the
I to "weak"
3. This shock
clearly
behavior
of
the
reflection
shock
in
the
explosive
products,
and
applying
the
small
parameter
purshows
terbation method, an analytic, first-order approximate solution is obtained for the problem of flying
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
Therefore
equation
can be written
in aexplosive
more compact
way as velocity
follows, and polytropic
an analytic
formula
with (4.7)
two parameters
of high
(i.e. detonation
index) for estimation of the velocity ofPv
flying
~ plate is established.
o[
ot [
pv'
P v~
+
O..__.~_
Ox ~
l..
PV'V~ + g ~ P
J
OH(U,U,~,)
pv~vi +1.g a PIntroduction
I =
Ox ~
pv~v , + g a s p tl
OM(U,Ux,)
Ox a
PZ"~ ]
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
(e+P)v'
J
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
O N ( UThe
, U,,,)
o s lestimation
( u , u , ~ , , u xof
, ) flyor_~velocity
o s 2 ( u , and
u , , , ,the
u , ~way
o of raising it are questions
cladding of metals.
method of
Ox I
Ox ~
-+
of common interest.0 x z
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
OS~(U,U.v,U,,,) + OQ,(U)
R ( U ,is Utox 'solve
, U x ,the
, U ~following
)
(5.6)
approach of solving the problem of motion of +flyor
system of equations
Ox a
Ox ~
governing the flow field of detonation products behind the flyor (Fig. I):
where
H(U,Ux,)=[o,b,~
--ff
=o,
Jr
/Ox 1 , + b~2>av~/Ox
L,
apav +u_~_xp
au
(II)
b ~ O v k / O x ', f l ? 9
au / ~ au, b;'-~)Ov*/Ox
M(U,U,~,)-[O,,.,~ar~'~-*"~:~,u
y1
=0, "-, b~y~ov*/Ox', f~j*
(i.0
N ( U , Ux~) -= [ 0, b~';Ov*/Ox
aS
a 3,
s b~2~Ov*/Ox s,
Q,(U)-=-[O
-,sg(33~a"*/a~'~,/v.~
~
, .(3] ~
a--T
=o,
p =p(p, s),
~ aa#
~1~ ~ a*# v J , ~~ a a#
( m F . p~v , ~ w~ a ('s),~ F~pv~ i,f7. ]~
, wsk
0
(11 ~
}
2
|
where p, p, S, u are pressure, density, specific
entropy
particle
of detonation products
bt, x'Ov /Ox + and
b(ll~av
v /,.,_ velocity
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
(12)
J are 2unknown;
h(12)a~,k /l,q.~
~ position of R and the state paratrajectory F ofS1
flyor
another
( U , asU,~,,
U~..,)boundary.
=bj, a Both
Ov /Ox
+ ~ j 8 ~c~ the
I
meters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
!
f,
l
293
General Form of Navier-Stokes Equations
-
185
0
b,,, Ov / o x + b~V~ov'/ox 3
(2,)
S z ( U , Ux', UxO =
I~1
~
'-" ~
l
l ~" -'~ +
It3
/
0
b~J,,,ow/ox, + b Z ' o W / O x 2
S3(U,
U~,, U.'.) =
b~"Ov*/Ox ' + b ~ ' O v * / O x ~
-
f.
Abstract
The one-dimensional problem of thepI'~j
motion
rigidv~v
flying
plater '~'
under
v Jr 'of
+ apFJj
~-l-'~J
-- Fexplosive
l j r' ij attack has
an analytic solution only when the polytropic index of detonation products equals to three. In
pF~j
v~v 2paper,
+ pF#~however,
v ' v J - F J by
a r,~z
F~jthe
r' 'j"weak" shock
, U , , , , U.~,,U~,)
-- required.
-general,R a( Unumerical
analysis is
In this
utilizing
behavior of the reflection shock in the explosive products, and applying the small parameter purF ~ vJv 3+ pF#~
v ' Wis- -obtained
FI~ r '~3b problem
r' '~
terbation method, an analytic, first-order papproximate
solution
for Fthe
of flying
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
( e + P ) F ~ v ~ - F ~ r ' ~ % ~ - - 2 F ~ 9~'OT/Ox ~
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
an
analytic
formula
two parameters
of highupexplosive
(i.e.3; detonation
and the
polytropic
where
indicesi,
j, k,with
a and
fl are summed
from I to
superscriptvelocity
T denotes
transpose
index) for estimation of the velocity of flying plate is established.
of an arbitrary matrix; notationsf~ to f7 are defined as follows:
f t ~ gj~b~ ~ vPOv~/O xl 1.+ 2 gIntroduction
"OT/Ox 1
-(2J)
fz=g~pb~z
v # 3v tf /Ox z +2gZZOT/Ox z
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
materials under f3
intense
impulsive loading,
shock synthesis
=- gJpb(kZsJ)vaOv'/Ox
3+ 2gSSOT
/Ox 3 of diamonds, and explosive welding and
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
(,j)
Or*
~r
. Ovk
. ,z aT
, ,3 a T
of common interest._
, = g j a b , , of
va--ff~x~
+ g~a~,~,3 plane
v.--g-Zx3
+ /tg --6-U-x~
aa flying plate, the normal
Under the [assumptions
one-dimensional
detonation
and +rigid
approach of solving the problem of motion of flyor is to solve the following system of equations
f 6 ~field
. -...~.# l j of
b(ZJ).,a
Ovk _L.._t4zJ~.,a
Ov~
. z,(Fig.
OTI):+ Ag
: 28 OT
governing the flow
the -}flyor
, detonation
t
v "-~'~xt products
. ~ , , j p ~ , 3 behind
,., --~xs
Ag
--~-f-fl
_
~
b(3J~.pOvk --ff
•
au
f
__a(fj)
t~3J;v~ Ovk =o,
ap +u_~_xp+
k
0.
au
au
y1
OT
2 32 OT
=0,
(i.0
In the preceding expressions, itaS
is emphasized
that index./is summed up from 1 to 3; Ux,,
as
a--T with respect
=o, to x ~, x-"and x3-coordinates, respectively. A
Ux,and Uxs denote the partial derivative
single-step temporal scheme is givenp bylSl
=p(p, s),
OAt
O
At
O
- - U "entropy
+ 1and
- ~ Aparticle
U"-'+O
[ ( O - - 1of- -detonation
6 ) A t Z + A t products
3
]
A Up," =p,. S, u are pressure,
. A U "density,
4
where
specific
velocity
1 + ~e - a t
1 + ~e
at
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
(5.7)
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters
it are governed
by the
I of central
wave behind for
the detonation
wave
where on
superscript
n denotes
theflow
timefield
level;.
0 andrarefaction
$ are parameters
specified schemes.
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293
186
Wang Bao-guo
When 0=0 and ~ = - - 1 / 2 , for example, then equation (5.7) becomes the explicit-leapfrog
scheme. Note that in equation (5.7) and the remainder of this paper, notation A does not denote.
Laplace operator. A is defined as the increment, for example AU"=---U"+ I U" . If H, M, Nand
Q; can be expanded in Taylor's series, then the following relationship are obtained:
AH'=( D:--~) AU"+a-~r(P~AU)"
+0(At ~)
A M " = ( D ; - - ~ / O P ; ~ AU. + o_~ (P2AU). +O( AtZ)
AN"=(D; --~],-,,-,OP" ~^rr.+ O__~3.(P3AU).+O(Att
AQ~=W~AU'+O(At ~)
D~=-SH/OU, D2-OM/OU
, Ds--ON/OU
Abstract
P,~aH/OUx,,
The one-dimensional
problemP2--aM/OUx~,
of the motion of P3-ON/OUx,
a rigid flying plate under explosive attack has
an analytic solution
only
when
the
polytropic
index
of detonation products equals to three. In
W~=-oQ~/oU ( i = 1 , 2 , 3 )
general,
a
numerical
analysis
is
required.
In
this
paper,
however, by utilizing the "weak" shock
Substituting (5.6) into (5.7) and rearranging, we obtain
behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
I+0AtF O iA D
OPI\"
0=
+ O
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
an analytic
explosive
(i.e.
__ formula
O" p , with two
O parameters of highOP3
"
Oz detonation velocity and polytropic
C--D3--W3
+-~s
)
(ex3)2PI ]}AU"=/~
( O x ' ) ' - ' of+the
~ ( velocity
(5.8)
index) for estimation
of flying plate
is established.
where Iis the identity matrix; superscript n denotes the time level; symbol ~
1.
,~-=+---r
~"'~ r~,,,~,,.-,
L]
o
Introduction
is defined as follows:
+ ~. <,-~,.-,
,
+,o-> <,,,~,--' + <,,,>.-'
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
" E +ofHestimation
+ s, ).+ofa -flyor
~ - (velocity
. F + M +and
S :the
) " +wayoof- ~raising
( - C +it Nare+ questions
S: >
cladding of metals. The method
of common interest.
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
approach of solving the problem of motion of flyor is to solve the following system of equations
governing the flow field of detonation products behind the flyor (Fig. I):
+r~L.~_,_<:_A,,o
+~+,. ]+~<,<,.-,+o[(o-{.-~),,,~+,,,~ ]
Note that in equation (5.8) we adopt the same notation convention as in equation (6) of
=o,
reference [5] (for example, notation --ff
of the form [O- ~(
A- - Di) .lAU, denotes 0 - ~ [ (A-ap +u_~_xp+
au
au [5],au
1
DI )"AU"] , etc.). According to reference
an approximate
factorization of equation (5.8) can be
y
obtained
=0,
(i.0
aS
a--T
as
=o,
OAt
O
OP, \ "
O= p ,
{,+--r-+v
[~(~-*:',-,',",
+p. ~=p(p,
, J s), <o~,,,.-,
})
[ o ( density, specific OP2
" and particle velocity of detonation products
where p, f,I+
p, S, uOAt
are pressure,
entropy
(Ox~)= P; ]}
-Tg-~-k-6-~-t
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
trajectory F of OAt
flyor as another
are\"unknown; the position of R and the state paraO C boundary. BothOP3
meters on it are governed by the flow field I of central rarefaction
(0xS)= P , wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
"t
B-D~-W~ +"ff'Tx )
,__:_' .1>,.
=/~ + O ( A t s)
(5.9)
293
General Form of Navier-Stokes Equations
187
Implementation of equation (5.9) is achieved by writing it in a four-step t'otm given by:
step 1 (along xl-direction):
OAt
O
0Pl
"
02
step2 (alongx2-direction):
OAtr
02
OPz ~"
O IB
-
,-
0:s
<o,:> ,P:
step 3 (along x3-direction):
OAt
0
{I+ l---~[O-~-(
C
-
step 4
-
D z--,,rx:3---6-~- OPs ~"
]
-O'
--9 ]}AU=AO
(Oxs)2P,
(5.10c)
Abstract
U " +motion
I = U ' +of
A Ua" rigid flying plate under explosive attack
(5.10d)
The one-dimensional problem of the
has
an analytic
solution
polytropiconindex
of detonation
equals
to require
three. Inthe
In general,
the xonly
I-, x 2-when
and xthe
3- operators
the left
side of Eqs.products
(5.10a,b,c)
each
general, aofnumerical
analysis is required.
this paper,with
however,
by utilizing
"weak" shock
solution
a block-tridiagonal
system ofInequations
each block
havingthedimensions
5 x 5.
behavior of the reflection shock in the explosive products, and applying the small parameter purSolving block-tridiagonal matrix is main computation for solving Navier-Stokes equation by means
terbation method, an analytic, first-order approximate solution is obtained for the problem of flying
of
implicit
scheme.
This decomposition
avoids
to invert
the matrix
for large
plate
drivenfactored
by various
high explosives
with polytropic
indices
other than
but nearly
equal systems
to three. of
equations
and
computational
time
(CPU)
is
significantly
reduced.
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
It is important
notetwo
thatparameters
equation of
(5.10)
here
is more
general
than reference
[5]. The
an analytic
formula to
with
highderived
explosive
(i.e.
detonation
velocity
and polytropic
index)
for
estimation
of
the
velocity
of
flying
plate
is
established.
equation obtained in generalized curvilinear coordinates reduces to equation (6) of reference [5] for
Cartesian coordinates in the ordinary two-dimensional space. Finally, it is remarked that the
numerical dissipation terms have been added
to the right side of equation (5.9) to damp wiggles or
1. Introduction
recouple solutions that have become decoupled. We choose fourth-brder terms recommended in
Explosive
ffmds
its important
in the
study
of behaviorchange
of
reference
[5] ordriven
other flying-plate
appropriatetechnique
dissipat.i%e
terms021
when use
solving
flow
parameters
materials
under
intense
impulsive
loading,
shock
synthesis
of
diamonds,
and
explosive
welding
and
severely.
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
of common interest.
The
work
hereof
is one-dimensional
performed underplane
the direct
supervision
offlying
Prof. plate,
Wu Chung-hua
Under
thereported
assumptions
detonation
and rigid
the normal of
Institute
Thermophysics
Yin-gei of
approach of
of Engineering
solving the problem
of motion(Chinese
of flyor Academy
is to solveoftheSciences).
following Prof.
systemBian
of equations
governing othe
flow field (Chinese
of detonation
products
behind theread
flyorthe
(Fig.
I): manuscript and gave helpful
Institute
f Mechanics
Academy
of Sciences)
entire
criticisms. The author is greatly indebted to them for their valuable advice and encouragement.
References.
--ff
ap +u_~_xp+
au
=o,
au
au
[ 1 ] Wu Chung-hua, Basic Aerothermodynamic
Equations
y1
=0,Governing FluidFlow in Turbomachines
Expressed in Terms of Non-Orthogonal Curvilinear Coordinates, Lecture Notes,(i.0
China
aS
as
University of Science and Technology
(1975).
a--T
=o, (in Chinege)
[ 2 ] Thompson, J.F., Grid generation techniques in computational fluid dynamics, J. AIAA, .22
p =p(p, s),
(1984), 1505- 1523.
[where
3 ] Wang
Strongly
implicit
procedure
applied
to transonic
equations
and new
p, p, S,Bao-guo,
u are pressure,
density,
specific
entropy
and particle
velocitystream
of detonation
products
approach
determination
of Computational
4 (1985),
respectively,
withtothe
trajectory R of
of density-field,
reflected shockJournal
of detonation
wave D as a Physics,
boundary2,and
the
trajectory
of flyor
as another boundary. Both are unknown; the position of R and the state para4 7 4 -F 481.
(in Chinese)
on it are
governed by
the flow field
I of centrala nrarefaction
the detonation
wave in
[meters
4 ] Chien
Wei-zang,
.Variational
principles
d g e n e r a l iwave
z e d behind
variational
principles
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
hydrodynamics of viscous fluids, Applied Mathematics and Mechanics, 5, 3 (198.4),
293
1282 - 1296.
188
Wang Bao-guo
[ 5 ] Beam, R.M. and R.F. Warming, An implicit factored scheme for the compressible NavierStokes equations J. AIAA, 16 (1978), 393-402.
[ 6 ] MacCormack, R.W., A numerical method for solving theequations of compressible viscous
flow, AIAA paper 81-0110 (1981).
[ 7 ] Steger, J.L., Implicit finite-difference simulation of flow about a~itrary two-dimensional
geometries, J. AIAA, 16 (1978), 679-686.
[ 8 ] Wang Bao-guo, Matrix solution of compressible flow on S~ surface through a turbomachine
blade row with splitter vanes or tandem blades, dissertation, Institute of Engineering
Thermophysics, Chinese Academy of Sciences (1981). (in Chinese)
[ 9 ] Chaprnan, S. and J.G. Cowling, The Mathematical Tlleorv of Non-Uniform Gases. Cambridge
(1970).
[I0] Bian Yin-gui, Boundar)' Layer Theoo,, Vol. I and II, China University, of Science and
TechnoJogy (1979). (in Chinese)
Abstract
[1 I] Kelichefskii, N.A.. Elements of Tensor
Calculation ~md Applications in Mechanics, GETL
(1954). (in Russian)
The one-dimensional problem of the motion of a rigid flying plate under explosive attack has
[12]
Zhang
Han-xin,only
Thewhen
Exploration
of the spatial
oscillations
finite difference
an analytic solution
the polytropic
index of
detonationin products
equals tosolutions
three. Infor
shocks, Acta
Aerodynamica
SiHica.however,
1 (1984),by12-19.
(inthe
Chinese)
general,Navicr-Stokes
a numerical analysis
is required.
In this paper,
utilizing
"weak" shock
behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying
plate driven by various high explosives with polytropic indices other than but nearly equal to three.
Final velocities of flying plate obtained agree very well with numerical results by computers. Thus
an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic
index) for estimation of the velocity of flying plate is established.
1.
Introduction
Explosive driven flying-plate technique ffmds its important use in the study of behavior of
materials under intense impulsive loading, shock synthesis of diamonds, and explosive welding and
cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions
of common interest.
Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal
approach of solving the problem of motion of flyor is to solve the following system of equations
governing the flow field of detonation products behind the flyor (Fig. I):
--ff
ap +u_~_xp+
au
au
aS
as
au
y1
=o,
=0,
(i.0
a--T
=o,
p =p(p, s),
where p, p, S, u are pressure, density, specific entropy and particle velocity of detonation products
respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the
trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it are governed by the flow field I of central rarefaction wave behind the detonation wave
D and by initial stage of motion of flyor also; the position of F and the state parameters of products
293