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Calculation of a hydrodynamic cumulation
251
Short communications
Here M,GO, K,Go, K’=K(O, 0). It follows from the expansions max[M(A), K(A)]=
(10) that as A-O
K+KiA+o(A),
if
M
or
M=KandK,BM,,
M+M,A+o (A),
if
M>K
or
M=KandK,
(11)
MsK’,
if
if
M>K’ an,dK”
if
M>K’ andK”>O.
(12)
It follows from (1 l), (12) that the behaviour of the function R”(A)=max(M(A),K(l),
is described in a linear approximation
W(I) =
sup min[K(e,A),M(e)]} OSzed5
for sufficiently
small A>0 as follows:
M+o(A),
if
K
M+M,A+o (A),
if
M>mas(K,
K’)
K+K,A+o(A),
if
,M
M=K>K’mdK,>M,.
or
or
M=K>K’andK,&If,,
The author thanks Yu. B. Germeier for his comments. Translated by J. Berry REFERENCES 1.
GERMEIER, Yu. B., On two-person games with a fixed sequence of moves. Dokl. Akad. Nauk SSSR, 198, S, lOOl-1004,197l.
2.
KUKUSHKIN, N. S., A game with incomplete information. Zh. vjchisl. Mat. mot. Fiz., 13, 1, 210-216, 1973.
3.
GERMEIER, Yu. B., Games with non-antagonisticinterests (Igry s neprotivopolozhnymi MGU, Moscow, 1972.
interesami), izd-vo
CALCULATION OF A HYDRODYNAMIC CUMULATION* G. A. ATANOV Donetsk (Received 25 September 1973) A SOLUTION for the case when a shock wave is incident described. The computational and the integration
on a concave cavity in a liquid is
method used is the method of “distitegration
of the discontinuity”,
is carried out in moving nets. Some results for water in the case of a conical
cavity are presented.
A description of a hydrodynamic cumulation, consisting of the interaction of a shock wave with the concave free stirface of a liquid, is given in [l] . A self-similar solution for the case when a weak shock wave is incident
on an infinite plane wedge-shaped cavity with small opening angle is
given in [2]. In this paper the solution for the Asymmetric *Zh. vychisl. Mat. mat. Fiz., lS, 3,800-802,197s.
case without
these restrictions
is
252
G. A.
Atanov
described. The calculation is performed by the method of “disintegration of the discontinuity” in a moving net [3]. 1. We consider the following problem. In a cylindrical tube (Fig. 1) there is a certain volume of liquid. On the left of it the surface is flat, on the right it is concave. A piston strikes the flat face, generating a shock wave in the liquid. Reaching the free surface on the right, the shock wave deforms it, forming an ultrajet.
.4 t
FIG. 2.1 is for a! = 30”, 2 is for or = 45*, 3 is for tll = 6C”
FIG. 1
We will consider not very strong shock waves (with a pressure at the front of up to 30 kbar), then the calculation can be carried out in the quasi-acoustic approximation, regarding them as isentropic. Using as the equation of state of the liquid Tait’s equation
where P is the pressure, p is the density, B = const, and n = const, we eliminate the pressure from the equations of gas dynamics. Choosing as scales: of velocity - the initial velocity of sound a*==r’[nB/po],of density - the initial density po, of length - the radius R of the tube, of time - the ratio R/so, we write in divergent form the dimensionless system of equations aPr -iic-spur -
i?pZT
apur +-+-=o, ax
at
i?r
d
r
-c x;
dpuwr
(p*+pn+-I)+
8puur 8 dpur ++ z,
7
(1)
&P-l
r (pQ+pnLP-1)s
-*
Pk
u(xp,
8X
at
= 0,
n
The initial conditions are discontinuous: ~~(0) =3ssc u(x,
fr, 01,
r, O)=O,
IGKO,
v (2, r, 0)
=o,
r, 0)-l,
r, O)=uy.
(2)
The condition at the piston d’L (qj, 1)
dt
PPOR *
= -
s MP 0
(I-p”)tdr,
(3)
253
Short communications
on the free surface p(xc. r)==l,
(4)
on the body of the tube u(x,
where U, v are the velocity components, coordinate
~p=~p(t)
1, f)=O,
(5)
is the coordinate
of the free surface, F is the cross-sectional
of the piston, xC=xC (r, t) is the
area of the tube, Mp is the mass of the piston,
and u,, is the speed of the shock. 2. The system (1) with conditions
(2)-(5)
was solved by the method of “disintegration
of
the shock” in a moving net. At the initial instant the net is shown in Fig. 1. Along the coordinate r the domain of integration
Cas subdivided into M parts with constant
step h,= I/M. Along the
coordinate x on each time layer the net was constructed anew. Its step h, was found from the condition of subdividing the line r=mh, into equal parts between the piston and the free surface: h,,
Xcm-X P = ~
K
where the upper suffix corresponds
h,”
xcn’--Z P. = ___
K
’
’
to values at the instant t + 7, the lower suffix to the instant t, T
is the time step, K is the number of subdivisions
along the x-axis. The position of the free surface
is defined by the formula
where the half-integral
subscripts correspond
is defined by the formula
xp(t+t)
to parameters in the cells, the position of the piston
=xp(t)+up(t)q
The difference formulas for determining
and its speed by condition
(3).
the flow parameters at the instant r t T are
constructed just as in [3]. The quantities on the surfaces bounding the cells are found from the calculation of the disintegration of the discontinuity. As the conditions of dynamic compatibility for waves admitting of linearization, the conditions on the characteristics [ul* +J were taken, the “plus” sign corresponding In the case of waves not admitting
Wn-I)]=
0,
to the family I, the sign “minus” of linearization,
to the family II.
it is necessary to solve the exact
equations c[~l+[P”l=o, for waves propagating
c[l;p]--n[u]=O
to the left or downwards from a boundary b[u]-[p”]=O,
line, and
O[l/p] +n[ul=O
for waves propagating to the right or upwards. The quantities b and c are determined like b and a in [3] . The disintegration occurs for velocity components normal to the boundary. Disintegrations on the boundaries
of the domain are represented
by only one wave.
3. As an example we present below the results of the calculation
of the flow of water (B
n=7.15) for a conical cavity with various semivertical angles CLFigure 2 shows the speeds of the head of the jet obtained when a shock wave with a pressure of 600 bar in incident on the
=3045 bar,
2.54
G. A. Atanov
front. The lower dashed line gives the value of the speed at the front of the shock wave, the upper one gives the value of the speed for the shock wave reflected from the flat free surface. As we see, the concave nature of the free surface leads to a considerable increase of the speed. Figure 3 shows the pressure profiles in the section x = 0 for the instants indicated there (a=&“).
FIG. 4
FIG. 3
The calculation was carried out on the “Dnepr-21” computer, the computing time for one version boated to about 6 hours. C~culations were carried out for M = 10, Ic = 12. The step along the t-axis was chosen as in [3], the safety factor was 0.4. It was shown by a check that in the case of a heavy piston the initial conditions can be specified as shown in Fig. 4 (the water compressed by the shock wave is crosshatched). Translatedby J. Berry ~FERENCES 1.
POPOV, S. G., Some problems and methods of experimental aerodynamics (Nekotorye zadachii metody eksperimental’noi aerodinamiki), Gostekhizdat, Moscow, 1952.
2.
TULIN, M., The formation of ultrajets. Mechanics. Periodical collection of ~ans~tio~s of foreign papers (Mekhanika. Period. sb. perev. in. statei), No. 6,65-82, 1969.
3.
GODYNOV, S. K., ZABRODIN, A. V. and PROKOPOV, G. P., A computational scheme for twodimensional non-stationary problems of gas dynamics and calculation of the flow from a shock wave approaching a stationary state. Zh. vjkhisl. Mat. mat. Fiz., 1,6,102~1050,1961.
Short communications
Here M,GO, K,Go, K’=K(O, 0). It follows from the expansions max[M(A), K(A)]=
(10) that as A-O
K+KiA+o(A),
if
M
or
M=KandK,BM,,
M+M,A+o (A),
if
M>K
or
M=KandK,
(11)
MsK’,
if
if
M>K’ an,dK”
if
M>K’ andK”>O.
(12)
It follows from (1 l), (12) that the behaviour of the function R”(A)=max(M(A),K(l),
is described in a linear approximation
W(I) =
sup min[K(e,A),M(e)]} OSzed5
for sufficiently
small A>0 as follows:
M+o(A),
if
K
M+M,A+o (A),
if
M>mas(K,
K’)
K+K,A+o(A),
if
,M
M=K>K’mdK,>M,.
or
or
M=K>K’andK,&If,,
The author thanks Yu. B. Germeier for his comments. Translated by J. Berry REFERENCES 1.
GERMEIER, Yu. B., On two-person games with a fixed sequence of moves. Dokl. Akad. Nauk SSSR, 198, S, lOOl-1004,197l.
2.
KUKUSHKIN, N. S., A game with incomplete information. Zh. vjchisl. Mat. mot. Fiz., 13, 1, 210-216, 1973.
3.
GERMEIER, Yu. B., Games with non-antagonisticinterests (Igry s neprotivopolozhnymi MGU, Moscow, 1972.
interesami), izd-vo
CALCULATION OF A HYDRODYNAMIC CUMULATION* G. A. ATANOV Donetsk (Received 25 September 1973) A SOLUTION for the case when a shock wave is incident described. The computational and the integration
on a concave cavity in a liquid is
method used is the method of “distitegration
of the discontinuity”,
is carried out in moving nets. Some results for water in the case of a conical
cavity are presented.
A description of a hydrodynamic cumulation, consisting of the interaction of a shock wave with the concave free stirface of a liquid, is given in [l] . A self-similar solution for the case when a weak shock wave is incident
on an infinite plane wedge-shaped cavity with small opening angle is
given in [2]. In this paper the solution for the Asymmetric *Zh. vychisl. Mat. mat. Fiz., lS, 3,800-802,197s.
case without
these restrictions
is
252
G. A.
Atanov
described. The calculation is performed by the method of “disintegration of the discontinuity” in a moving net [3]. 1. We consider the following problem. In a cylindrical tube (Fig. 1) there is a certain volume of liquid. On the left of it the surface is flat, on the right it is concave. A piston strikes the flat face, generating a shock wave in the liquid. Reaching the free surface on the right, the shock wave deforms it, forming an ultrajet.
.4 t
FIG. 2.1 is for a! = 30”, 2 is for or = 45*, 3 is for tll = 6C”
FIG. 1
We will consider not very strong shock waves (with a pressure at the front of up to 30 kbar), then the calculation can be carried out in the quasi-acoustic approximation, regarding them as isentropic. Using as the equation of state of the liquid Tait’s equation
where P is the pressure, p is the density, B = const, and n = const, we eliminate the pressure from the equations of gas dynamics. Choosing as scales: of velocity - the initial velocity of sound a*==r’[nB/po],of density - the initial density po, of length - the radius R of the tube, of time - the ratio R/so, we write in divergent form the dimensionless system of equations aPr -iic-spur -
i?pZT
apur +-+-=o, ax
at
i?r
d
r
-c x;
dpuwr
(p*+pn+-I)+
8puur 8 dpur ++ z,
7
(1)
&P-l
r (pQ+pnLP-1)s
-*
Pk
u(xp,
8X
at
= 0,
n
The initial conditions are discontinuous: ~~(0) =3ssc u(x,
fr, 01,
r, O)=O,
IGKO,
v (2, r, 0)
=o,
r, 0)-l,
r, O)=uy.
(2)
The condition at the piston d’L (qj, 1)
dt
PPOR *
= -
s MP 0
(I-p”)tdr,
(3)
253
Short communications
on the free surface p(xc. r)==l,
(4)
on the body of the tube u(x,
where U, v are the velocity components, coordinate
~p=~p(t)
1, f)=O,
(5)
is the coordinate
of the free surface, F is the cross-sectional
of the piston, xC=xC (r, t) is the
area of the tube, Mp is the mass of the piston,
and u,, is the speed of the shock. 2. The system (1) with conditions
(2)-(5)
was solved by the method of “disintegration
of
the shock” in a moving net. At the initial instant the net is shown in Fig. 1. Along the coordinate r the domain of integration
Cas subdivided into M parts with constant
step h,= I/M. Along the
coordinate x on each time layer the net was constructed anew. Its step h, was found from the condition of subdividing the line r=mh, into equal parts between the piston and the free surface: h,,
Xcm-X P = ~
K
where the upper suffix corresponds
h,”
xcn’--Z P. = ___
K
’
’
to values at the instant t + 7, the lower suffix to the instant t, T
is the time step, K is the number of subdivisions
along the x-axis. The position of the free surface
is defined by the formula
where the half-integral
subscripts correspond
is defined by the formula
xp(t+t)
to parameters in the cells, the position of the piston
=xp(t)+up(t)q
The difference formulas for determining
and its speed by condition
(3).
the flow parameters at the instant r t T are
constructed just as in [3]. The quantities on the surfaces bounding the cells are found from the calculation of the disintegration of the discontinuity. As the conditions of dynamic compatibility for waves admitting of linearization, the conditions on the characteristics [ul* +J were taken, the “plus” sign corresponding In the case of waves not admitting
Wn-I)]=
0,
to the family I, the sign “minus” of linearization,
to the family II.
it is necessary to solve the exact
equations c[~l+[P”l=o, for waves propagating
c[l;p]--n[u]=O
to the left or downwards from a boundary b[u]-[p”]=O,
line, and
O[l/p] +n[ul=O
for waves propagating to the right or upwards. The quantities b and c are determined like b and a in [3] . The disintegration occurs for velocity components normal to the boundary. Disintegrations on the boundaries
of the domain are represented
by only one wave.
3. As an example we present below the results of the calculation
of the flow of water (B
n=7.15) for a conical cavity with various semivertical angles CLFigure 2 shows the speeds of the head of the jet obtained when a shock wave with a pressure of 600 bar in incident on the
=3045 bar,
2.54
G. A. Atanov
front. The lower dashed line gives the value of the speed at the front of the shock wave, the upper one gives the value of the speed for the shock wave reflected from the flat free surface. As we see, the concave nature of the free surface leads to a considerable increase of the speed. Figure 3 shows the pressure profiles in the section x = 0 for the instants indicated there (a=&“).
FIG. 4
FIG. 3
The calculation was carried out on the “Dnepr-21” computer, the computing time for one version boated to about 6 hours. C~culations were carried out for M = 10, Ic = 12. The step along the t-axis was chosen as in [3], the safety factor was 0.4. It was shown by a check that in the case of a heavy piston the initial conditions can be specified as shown in Fig. 4 (the water compressed by the shock wave is crosshatched). Translatedby J. Berry ~FERENCES 1.
POPOV, S. G., Some problems and methods of experimental aerodynamics (Nekotorye zadachii metody eksperimental’noi aerodinamiki), Gostekhizdat, Moscow, 1952.
2.
TULIN, M., The formation of ultrajets. Mechanics. Periodical collection of ~ans~tio~s of foreign papers (Mekhanika. Period. sb. perev. in. statei), No. 6,65-82, 1969.
3.
GODYNOV, S. K., ZABRODIN, A. V. and PROKOPOV, G. P., A computational scheme for twodimensional non-stationary problems of gas dynamics and calculation of the flow from a shock wave approaching a stationary state. Zh. vjkhisl. Mat. mat. Fiz., 1,6,102~1050,1961.