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Hypervelocity fragment formation technology for ground-based laboratory tests
Acta Astronautica 104 (2014) 77–83
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Hypervelocity fragment formation technology for ground-based laboratory tests V.F. Minin, I.V. Minin n, O.V. Minin Siberian State Geodesic Academy, Novosibirsk, Russia
a r t i c l e i n f o
abstract
Article history: Received 11 July 2014 Received in revised form 19 July 2014 Accepted 21 July 2014 Available online 30 July 2014
Cumulative jet formation was regarded aimed at the formation of hypervelocity fragments up to 14 km/s for the investigation of space debris effects on shielding screens. The basic physical problems of jet formation process are analyzed. It is shown that in process of realization of hyper-cumulation conditions for jet formation without complete stagnation point involving formation of the inner zone of constant pressure (dead zone), the flow mass is always greater than slug mass. That opens wide the possibilities for increasing fragment mass in laboratory tests. & 2014 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Hypervelocity impact Explosion Shaped charge Cumulative jet Hyper-cumulative jet
1. Introduction Space debris environment is a serious hazard to space missions [1–3]. Thus effective protection measures should be undertaken to increase survivability of space structures. That turns to be of a tremendous importance for developing the systems containing constellations of low Earth orbiting satellites as a space segment [4,5]. Developing the concepts of such systems it is necessary to take into consideration the space debris environment the systems will operate in. The speed of the strike of an object placed on LEO and element of space debris depending on the collision angle might be as high as 15 km/s [4,5]. The problem of space debris evolution is governed by processes of debris reproduction as a result of different break-ups and high-speed collisions on LEO [6–8]. Secondly, the problem of spacecrafts shielding from small particles could be solved protecting them by fluid-filled containments, which consume the impact energy and redistribute momentum [7]. n
Corresponding author. Tel.: þ7 983 3032239; fax: þ7 383 3433937. E-mail address: [email protected] (I.V. Minin).
http://dx.doi.org/10.1016/j.actaastro.2014.07.027 0094-5765/& 2014 IAA. Published by Elsevier Ltd. All rights reserved.
Evaluation of the results of hypervelocity impacts of space debris fragments on space vehicles of different structures were performed in many papers [8–10]. Those research are of great importance for developing strategies for shielding space structures, or recommending avoidance maneuvers. However, all those complex models of hypervelocity impacts need verification in experiments. Experimental data for hypervelocity impacts for velocities higher than 10 km/s and fragment size larger than 10 mm is hardly available in laboratories. The available experimental accelerators (light gas guns, explosive shaped charges, rail guns, electrostatic guns, etc.) could accelerate up to 14 km/s only very small fragments (less than 1 mm). The present paper investigates basic physical principles of cumulative jet formations and suggests the technology making it possible essentially increasing the mass of accelerated fragment by making use of a principle of hyper-cumulation. 2. Physics of cumulative jet formations Cumulative jets may be obtained in process of inertial collapse toward the symmetry axis of the conical metal
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liner. Conical liner collapse may result either from the explosion product effect of the explosive charge, with the explosive adjacent to the liner from outside, or from ablative pressure due to the high-power pulse of laser emitter affecting the liner outer surface. The first experimental theoretical research of cumulation effects [11–19] showed that the model problem whose solution is to contribute to understanding of the cumulative charge principles, is a problem of incompressible liquid jets collision. On the basis of this model the first methods were developed, which allowed for making sufficiently correct calculation of cumulative charges characteristics. Basic results of non-stationary theory of jet-formation are achieved by applying the law of conservation of mass, pulse and energy to some liner elements. Each element is considered independently, in the corresponding coordinate system. In [14], the experimental check of the calculation on such non-stationary cumulation theory was conducted. A good agreement between the theory and the experiment was stated. There are several hydrodynamic effects and properties of the materials, which may cause departure from the simple pattern of quasi-stationary jet formation: the effect of gradients caused by the cone convergence, time changes of the incoming flow velocity, phase velocity and pressure at stagnation points, shearing tension effect on the throwing process, and the effect of strength and viscosity on cumulative jet-formations. Strength and viscosity cause hysteresis of the process of compression-unloading in the zone of the flow turn. Therefore jet velocities experimentally fixed in works [16,20] are lower than theoretical ones, with this difference growing with reduction of the throwing angle. The model, taking into account viscosity [20] effect, predicts analogous dependencies, that is the main argument in favor of the determinative contribution of viscosity to the observed departure. Strength forces also prevent conical surface from deformation along its axis. In [21] some results of comparison of calculation and experimental data on the cylinder liner expansion in process of explosive charge detonation are presented. Analogous deformation model was used in calculation. Viscosity dissipation turned out to be able to efficiently impede the liner at its initial velocity of 110 m/s, max, with final deformation being no more than 15–20%. High velocities caused liner deformation, with characteristic deformations in jet-forming process being 7–10 times as large. Experiments [20] state metal viscosity reduction by a factor of 104 with deformation rate growing up to the values characteristic for shock wave stressing (108 s 1). It is noted that viscosity effect on conical liner collapse is by a factor of 102 or 103 less and can be neglected everywhere except the area adjoining the cone base, where the explosive layer is thinner. The formulae of the cumulation theory (for example, [13]) do not have any limits for maximum velocity achievable for cumulative jets. It rises limitlessly with approaching collapse angle of conical liner, but in the corresponding experiments cumulative jet was not revealed. Moreover, it was found out that the jet is not formed at the angles of collapse less than some critical one, whose values depend on the materials of cumulative liner. Relying on the result of works [16,22] it was established that this limitation is a matter of principle, being related to the
compressibility of the conical liner material (the problem of the compressed liquid plane jets collision was considered). It is well known in gas dynamics that for the flow with the velocity of the free line of flow there is a maximal rotation angle at which the attached shock wave may exist. It is also true for the substance, which is not an ideal gas. For axisymmetric flows, it is much more difficult to theoretically estimate compressibility effect. It was mentioned in [23] that jetless flow is impossible in case of cone collapse. But as it was noted in [24], by more detailed analysis it was found out that this conclusion refers to the conical flow, that is to the flow whose properties are constant along the rays originating from the cone vertex. There is no reason to think the flows are conical in case of cumulative charge liner collapse. In works [24,25] mathematical modeling techniques were used to reveal jet formation criteria. The results [25] for supersonic conditions agree with papers [16,22]. Numerical investigations of axisymmetric flows in inert and chemically reacting mixtures showed, that cumulative jet formations due to geometrical factors plays an important role in detonation onset [26–29]. More detailed research, using mathematical modeling made it possible for the authors [24] to formulate the following specified criteria of jet formation:
Jet formation is to take place at the flow velocities (in
the reference system related to the critical point) which are less than the sound velocity; jet formation takes place also at supersonic velocities of the flow, if the angle of collision is large enough for the incoming flux jump not to contact with the collision point; and the jets, forming at supersonic velocities of collision, atomize in radial direction; at subsonic velocity jets remain continuous.
Qualitative results in revealing jet-formation criteria are the same for both plane and axisymmetric cases. However, it is important that in axial symmetry for jetless configuration, there appeared a jet in the calculation, whose mass was max. 2% greater than that of the analogous jet in case of incompressible liquid. The authors of the work of reference made an assumption, that when revealing jet-formation criteria, such jet formation may be neglected. The authors have not come to the certain conclusion concerning the appearance of jet in axisymmetric case. It may be explained either by the numerical effect or physical reality reflection. The effect of jet appearance in axisymmetric case for “jetless” configuration was explained in works [35,36]. Using strict mathematical methods the authors generalized Walsh criterion for axisymmetric case and formulated criteria of jet-formation in this geometry. It was noted by the authors [24] that numerical experiments did not model the desired conditions for the flow in all cases. The offered criteria of jet formation relate to the steady flow. In case of the plane flow, the steadiness was achieved. But in configuration used in calculations for axisymmetric case stationary conditions are unachievable, as the mass of the inward flow is continually growing. The
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authors [24] consider, that for the exact assessment of the criteria, in case of axial symmetry, some additional calculations with specified initial statement are required. It was considered in [35,36]. Preliminary numerical research on the jet-formation beginning process conducted by the authors showed [37] that in the turn zone the flow character is considerably nonstationary, wave and quasiperiodic. Such behavior is characteristic for the viscous medium. At the initial time, when the gradients are large, the substance heating is maximal and there is an increase of the characteristic disturbance propagation velocities. At the stationary stage, the heating of the substance due to viscosity is lower, the sound speed is also lower, and, consequently, the oscillation period is to be somewhat longer. In other words, this feature of the flow near the certain point is caused by dissipative processes, with oscillation amplitude decreasing with the increase of viscosity. The period slightly grows in spite of the fact that with growing viscosity the substance heating also grows, resulting in the sound speed increase. After the shock wave departure from the symmetry axis, pulsations (whose period is equal to that of the pressure oscillations at the critical point (Fig. 1)) start propagating both in the direction of the slug and in that of the cumulative jet.
3. Peculiarities of converging flows In preliminary numerical research of the flow it was detected that after the shock wave reflection from the symmetry axis two additional pressure maxima, diverging
Fig. 1. Successive steps of cumulative jet formation. Iso-levels of pressure are shown. The jet pressure pulsations are clearly seen.
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across the plane, are formed. At some time the distance between the two maxima reaches its highest point, and then the maxima begin converging and join at the time corresponding to the first peak of the pressure pulsation. After that, the pressure pulsations begin propagating up the flow. Their oscillation period is the same as that at the critical point. This pattern of the flow contradicts the generally accepted Lavrentiev theory, according to which maximal pressure is to be observed at the flow turning point on the symmetry axis. Pressure variations in the region of the flow turn results in the velocity value fluctuation along the jet, as well as fluctuation of its surface. Thus, we can come to the conclusion, that the models using stationary flow cannot be applied for calculation. Since the time of classical works of Lavrentyev and Taylor the problem on symmetric collision of jets, or problems of conical liner collapsing have been taken as a model problem whose solution allowed for understanding basic laws of cumulative charges. With some modifications due to the necessity of taking into account the rate of liner elements collapse along the generatrix, this model is used as a basis for most of the engineering approaches for calculating cumulative charges characteristics. It is logical to assume that some other classical schemes of jets collision studied in ideal fluid dynamics may serve as a proper basis for describing the corresponding physical process, i.e. the opposite direction of research is offered to be chosen: from mathematical abstraction to the real phenomenon. An interesting problem of the classical ideal fluid dynamics is that of a glancing collision of different thickness jets [30]. The point is, the conservation laws may work at different angles of outgoing jets, i.e. there is no unique solution. There are two view points concerning this problem: (1) there exists a unique solution of this problem, and different variants of correlation, closing the system of conservation laws are offered; (2) there is no unique solution, and this non-uniqueness testifies either to the dependence of the stationary flow on the solutions on the initial conditions, or to the solutions instability. Experimental research of water jets collisions at the velocities of several m/s [31,32] confirms existence of the unique solution. It is revealed in the papers [31,32] that the stationary flow configurations are non-symmetric (relative to the velocity sign change), whereas equations of motion do have this symmetry. It may be concluded that in these experiments, the stationary flow configuration is considerably affected by dissipative processes in the flow turn zone, i.e. to describe these experiments some other equations are required, those taking into account dissipative members. There is another natural supposition that there are some details to be taken into account when carrying out limiting process from the real fluid to the ideal one. Therefore, the issue can't be closed. A more detailed study of this problem has not only academic interest but also applications. The possibility of the stationary flow configuration dependence on the initial conditions deserves further consideration. In this case, some new questions arise: what these conditions must be, and what the dynamics of attaining the stationary
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conditions is. Other alternatives involve flow instability and lack of stationary conditions. They are also of practical interest. This raises the problem of classical jet hydrodynamics, i.e. the problem of jets collision without the complete stagnation point [36], which involves inner constant pressure zone formation (stagnation region). This problem does not have a unique solution, with its free parameter being the pressure within the zone, or (which is its equivalent) dimensions of the zone. As for the stationary flow the parameters on the infinity do not depend on this zone existence or its dimensions, of primary interest is the study of the flow from the stagnation region, with nonstationary parameters of the incoming flow. Smoothing effect of this region on the development of various disturbances should be expected, particularly, smoothing of Relay–Taylor instability for thin liners. Another aspect of studying jet flows with stagnation regions is the study of the flows with supersonic velocities of the incoming flow. This is closely connected with the problem of jet-formation criterion. Main results on the jetformation criteria at this flow velocities are as follows [24]: at the collision angle less than critical, jetless configuration is realized; at greater angles the jet is formed, which spills in radial direction. The characteristic property of the flow, in case of jet formation, is existence of the detached shock wave. In case of the flow with stagnation region, the flow turns smoothly, without jump. A problem of the shock wave front and its existence in this configuration should be mentioned. In any case, tensions, which cause jet destruction in radial direction are likely to be smaller in stagnation region flows. The study of non-stationary supersonic flows in this configuration is of interest due to the fact that the jetformation criteria offered before [24] have been obtained for the case of stationary flow. One more base model problem is that of the collision of fluid jets, consisting of two or more layers. It was considered in works [33] and [34]. It was noted that in case of the equality of both layers Bernoulli constants, the solutions practically coincide with the classical one. Depending on the layers thickness ratio the following stationary configurations of flows are realized: with two-layers slug and a uniform jet; with uniform slug and two-layer jet; with uniform jet and slug, consisting of different materials. One might expect that, if compressibility (i.e. the difference in the sound speed of layers materials) is taken into account, the problem solutions will be considerably different from that for the case of the solid jet with the same value of Bernoulli constant. This problem solution is not evident, especially in case of supersonic flow velocity, at least in one layer.
flow with uniform jet and slug, consisting of different materials, is impossible, as in this case, there would be pressure break at the stagnation point. On physics grounds, there are no stationary conditions in case of boundary line being close to manifold line of flow. Stationary flow conditions are possible only if there is a stagnation region, whose pressure value is (in this case) determined unambiguously. The alternative offered here is formation of non-stationary periodic flow regime. Of special interest is the study of the flow at supersonic velocity of at least one of the layers. Thus, we've got used to the cumulative charges in which the liner is established with the complete opening angle smaller than 1801 (usually 30–1601), and it is compressed due to the effect of the explosive detonation, forming thin cumulative jet and massive slug containing a great part of the liner mass. Basic processes of cumulative jet formations are described in Lavrentiev–Birkhoff theory as a model of plane stationary collision of the incompressible liquid jets at the angle smaller than 1801. It is well known and confirmed by the centenary practice of cumulative charges designing. In cumulation conditions, without complete stagnation point, involving formation of the inner zone of constant pressure, which is to be studied, a new fundamental quality appears: the jet velocity, mass and pulse increase manifold (Figs. 2–5). It has been by far considered impossible [38,39]. 5. Numerical experiments Numerical experiments on formation of shaped charges were performed based on numerical schemes and technology developed in [35–41]. Along with classical conical lainer additional massive metallic body was used to make the shape as shown in Fig. 2. The angle of a cumulative layer is 501, material – aluminum, thickness 0.5 mm, the shaping cylindrical iron body has a diameter 8 mm and thickness 1 mm, external iron cover is 2 mm thick. High explosive is hexogen 1.66 g/ mm2. Diameter of the charge 40 mm. Three microseconds after detonation initiation detonation wave compresses the layer and it begins sliding along the shaping body. The angle of material converging to the center of symmetry approaches 1801. Radial contracting
4. Technology for hyper-cumulative jet formations For practical purposes, definition of the problem with different values of Bernoulli constant (in each layer at different infinity velocities [34]) seems to be more topical. This problem is much more complex even in case of the ideal incompressible liquid. It is evident that the stationary
Fig. 2. Initial configuration: 1 – detonator, 2 – detonation products, 3 – high explosive charge, 4 – additional shaping body, 5 – cumulative layer, 6 – external cover.
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Fig. 3. Deformation of materials 3 μs after initiating explosion: pressure maps and velocity profiles (Vz ) and (Vr) along the crossing coordinate lines.
Fig. 4. Deformation of materials 4.6 μs after initiating explosion: pressure maps and velocity and pressure profiles (Vz) and (p) along the crossing coordinate lines.
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Fig. 6. X-ray patterns of cumulative aluminum jets from identical charges (conical liner, 441) [41]. Lower figure: classical scheme, upper figure: prototype of hypercumulative flow. In upper X-ray pattern the jet mass is considerably enlarged.
By the time 4.6 μs collapse of aluminum cover layer took place (Fig. 4). Cumulative jet was formed characterized by tip velocity 14.65 km/s. The shaping body continues its motion and compresses the slug moving in the axial direction at 1.15 km/s. Velocity of shaping body axial motion is 2.98 km/s, the aluminum layer is contracting with velocity 2.83 km/s. Maximal pressure is 57.8 GPa. Fig. 5 illustrates the development of the process by 7.6 μs time after detonation initiation. By this time maximal velocity of a jet, which contributes to its elongation, slightly decreases (13.5 km/s). The velocity gradient continues jet stretching. Cut of the slow part of a jet should be performed based on the needed properties: velocity, mass and shape. Maximal velocity of the slug is 4 lm/s, while its minimal velocity is 400 m/s. By this time slug perforated the additional shaping body and practically destroyed it. 6. Experimental investigations Experimental validation of the developed theoretical results was performed based on techniques described in [41]. Fig. 6 illustrates X-ray patterns of cumulative aluminum jets from identical charges (conical liner, 441). Lower figure corresponds to classical scheme of cumulative jet formation, upper figure represents the prototype of hypercumulative flow. In upper X-ray pattern the jet mass is considerably enlarged. Comparison of results shows, that basic mechanism of cumulative jet formations were described correctly by our model. 7. Discussions
Fig. 5. Preliminary modeling of cumulative jet under hypercumulation conditions without complete stagnation point involving formation of the inner zone of constant pressure. Liner material: aluminum. Velocity distribution along the axis is shown: maximal velocity of a jet is 13.5 km/s, minimal velocity of slug is 400 m/s. Slug mass/jet mass ratio is far below unity.
velocity is 1434 km/s, horizontal velocity is 3259 km/s due to the piston effect of expanding reaction products pushing the shaping cylinder.(Fig. 3)
Thus, according to the hydrodynamic theory of cumulative jet formations in cumulative charges of Lavrentiev–Birkhoff, the jet velocity rises with the decrease of the liner opening angle, and simultaneous increase of the slug mass. In classical cumulative charges, high mass and velocity values of the jet in “low” liners are unattainable. Maximal jet velocity for the given material is limited by gas-dynamic limit due to the jet material destruction by “inner explosion” resulting from the liner material collision at the charge symmetry axis.
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Under the new hypercumulative conditions of cumulative jet formation without complete stagnation point involving formation of the inner zone of constant pressure (dead region), the jet mass is always greater than that of the slug. As a result of gas-dynamic properties redistribution in the flow turn area, the conditions for cumulative jet destruction due to the “inner explosion” are eliminated, and the jet velocity may considerably exceed gas-dynamic limit at the jet material density of the order of that of the liner material, that previously was considered impossible. Smoothing effect of this zone on the development of different types disturbances are likely to be expected, particularly, smoothing of Rayleigh–Taylor unsteadiness for thin liner. Therefore, whereas the effect of cumulative charge parameters and loading conditions of the liner (in classical Lavrentiev–Birkhoff scheme) on cumulative jets characteristics is quite predictable, the effect of sizes, parameters and geometric position of the flow off-axis stagnation region on instability development and jet-formation parameters is of considerably non-linear and non-monotonic character. According to the preliminary research [40,41], such region existence brings about both non-stability of the initially steady flows and stability of the non-stable ones, as well as the jet pulse and velocity growth or decrease.
8. Conclusions The fundamental new feature of hypercumulative jet formation was studied. It was proved both theoretically and experimentally that a hypercumulative jet has a higher thickness and mass. The basic conditions for the hypercumulative jet formation are the following: the liner material flow turn should proceed until material collapse on the symmetry axis, with the collapse angle of cumulative liner material being more than 1801. That was unattainable in the known and used cumulative flow schemes. This makes it possible controlling jet parameters (increase of mass, radius, and pulse as compared with the schemes known in world practice) in extremely wide ranges as well as opening new opportunities for technologies and techniques. The proposed method opens wide possibilities for laboratory testing of shielding materials under real conditions of collisions with space debris fragments. References [1] V.A. Chobotov (Ed.), Orbital Mechanics, 2nd ed. AIAA Education Series, Washington, D.C., 1996. [2] N.N. Smirnov (Ed.), Space Debris Hazard Evaluation and Mitigation, Taylor and Francis Publication, London, 2002. (208 pp.). [3] N.N. Smirnov, Evolution of orbital debris in space, Adv. Mech. 1 (2) (2002) 37–104. [4] N.N., Smirnov, A.B., Kiselev, A.I., Nazarenko, Mathematical modeling of space debris evolution in the near Earth orbits, Vestnik Moskovskogo Universiteta Ser. 1 Matematika Mekhanika, Moscow University Mechanics Bulletin, 4, Allerton Press, 2002, pp. 33–41. [5] N.N. Smirnov, A.I. Nazarenko, A.B. Kiselev, Modelling of the Space Debris Evolution Based on Continua Mechanics, European Space Agency, 2001, 391–396 (Special Publication) ESA SP). [6] N.N. Smirnov, V.F. Nikitin, A.B. Kiselev, Peculiarities of Space Debris Production in Different Types of Orbital Breakups, European Space Agency, 1997, 465–470 (Special Publication) ESA SP).
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Hypervelocity fragment formation technology for ground-based laboratory tests V.F. Minin, I.V. Minin n, O.V. Minin Siberian State Geodesic Academy, Novosibirsk, Russia
a r t i c l e i n f o
abstract
Article history: Received 11 July 2014 Received in revised form 19 July 2014 Accepted 21 July 2014 Available online 30 July 2014
Cumulative jet formation was regarded aimed at the formation of hypervelocity fragments up to 14 km/s for the investigation of space debris effects on shielding screens. The basic physical problems of jet formation process are analyzed. It is shown that in process of realization of hyper-cumulation conditions for jet formation without complete stagnation point involving formation of the inner zone of constant pressure (dead zone), the flow mass is always greater than slug mass. That opens wide the possibilities for increasing fragment mass in laboratory tests. & 2014 IAA. Published by Elsevier Ltd. All rights reserved.
Keywords: Hypervelocity impact Explosion Shaped charge Cumulative jet Hyper-cumulative jet
1. Introduction Space debris environment is a serious hazard to space missions [1–3]. Thus effective protection measures should be undertaken to increase survivability of space structures. That turns to be of a tremendous importance for developing the systems containing constellations of low Earth orbiting satellites as a space segment [4,5]. Developing the concepts of such systems it is necessary to take into consideration the space debris environment the systems will operate in. The speed of the strike of an object placed on LEO and element of space debris depending on the collision angle might be as high as 15 km/s [4,5]. The problem of space debris evolution is governed by processes of debris reproduction as a result of different break-ups and high-speed collisions on LEO [6–8]. Secondly, the problem of spacecrafts shielding from small particles could be solved protecting them by fluid-filled containments, which consume the impact energy and redistribute momentum [7]. n
Corresponding author. Tel.: þ7 983 3032239; fax: þ7 383 3433937. E-mail address: [email protected] (I.V. Minin).
http://dx.doi.org/10.1016/j.actaastro.2014.07.027 0094-5765/& 2014 IAA. Published by Elsevier Ltd. All rights reserved.
Evaluation of the results of hypervelocity impacts of space debris fragments on space vehicles of different structures were performed in many papers [8–10]. Those research are of great importance for developing strategies for shielding space structures, or recommending avoidance maneuvers. However, all those complex models of hypervelocity impacts need verification in experiments. Experimental data for hypervelocity impacts for velocities higher than 10 km/s and fragment size larger than 10 mm is hardly available in laboratories. The available experimental accelerators (light gas guns, explosive shaped charges, rail guns, electrostatic guns, etc.) could accelerate up to 14 km/s only very small fragments (less than 1 mm). The present paper investigates basic physical principles of cumulative jet formations and suggests the technology making it possible essentially increasing the mass of accelerated fragment by making use of a principle of hyper-cumulation. 2. Physics of cumulative jet formations Cumulative jets may be obtained in process of inertial collapse toward the symmetry axis of the conical metal
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liner. Conical liner collapse may result either from the explosion product effect of the explosive charge, with the explosive adjacent to the liner from outside, or from ablative pressure due to the high-power pulse of laser emitter affecting the liner outer surface. The first experimental theoretical research of cumulation effects [11–19] showed that the model problem whose solution is to contribute to understanding of the cumulative charge principles, is a problem of incompressible liquid jets collision. On the basis of this model the first methods were developed, which allowed for making sufficiently correct calculation of cumulative charges characteristics. Basic results of non-stationary theory of jet-formation are achieved by applying the law of conservation of mass, pulse and energy to some liner elements. Each element is considered independently, in the corresponding coordinate system. In [14], the experimental check of the calculation on such non-stationary cumulation theory was conducted. A good agreement between the theory and the experiment was stated. There are several hydrodynamic effects and properties of the materials, which may cause departure from the simple pattern of quasi-stationary jet formation: the effect of gradients caused by the cone convergence, time changes of the incoming flow velocity, phase velocity and pressure at stagnation points, shearing tension effect on the throwing process, and the effect of strength and viscosity on cumulative jet-formations. Strength and viscosity cause hysteresis of the process of compression-unloading in the zone of the flow turn. Therefore jet velocities experimentally fixed in works [16,20] are lower than theoretical ones, with this difference growing with reduction of the throwing angle. The model, taking into account viscosity [20] effect, predicts analogous dependencies, that is the main argument in favor of the determinative contribution of viscosity to the observed departure. Strength forces also prevent conical surface from deformation along its axis. In [21] some results of comparison of calculation and experimental data on the cylinder liner expansion in process of explosive charge detonation are presented. Analogous deformation model was used in calculation. Viscosity dissipation turned out to be able to efficiently impede the liner at its initial velocity of 110 m/s, max, with final deformation being no more than 15–20%. High velocities caused liner deformation, with characteristic deformations in jet-forming process being 7–10 times as large. Experiments [20] state metal viscosity reduction by a factor of 104 with deformation rate growing up to the values characteristic for shock wave stressing (108 s 1). It is noted that viscosity effect on conical liner collapse is by a factor of 102 or 103 less and can be neglected everywhere except the area adjoining the cone base, where the explosive layer is thinner. The formulae of the cumulation theory (for example, [13]) do not have any limits for maximum velocity achievable for cumulative jets. It rises limitlessly with approaching collapse angle of conical liner, but in the corresponding experiments cumulative jet was not revealed. Moreover, it was found out that the jet is not formed at the angles of collapse less than some critical one, whose values depend on the materials of cumulative liner. Relying on the result of works [16,22] it was established that this limitation is a matter of principle, being related to the
compressibility of the conical liner material (the problem of the compressed liquid plane jets collision was considered). It is well known in gas dynamics that for the flow with the velocity of the free line of flow there is a maximal rotation angle at which the attached shock wave may exist. It is also true for the substance, which is not an ideal gas. For axisymmetric flows, it is much more difficult to theoretically estimate compressibility effect. It was mentioned in [23] that jetless flow is impossible in case of cone collapse. But as it was noted in [24], by more detailed analysis it was found out that this conclusion refers to the conical flow, that is to the flow whose properties are constant along the rays originating from the cone vertex. There is no reason to think the flows are conical in case of cumulative charge liner collapse. In works [24,25] mathematical modeling techniques were used to reveal jet formation criteria. The results [25] for supersonic conditions agree with papers [16,22]. Numerical investigations of axisymmetric flows in inert and chemically reacting mixtures showed, that cumulative jet formations due to geometrical factors plays an important role in detonation onset [26–29]. More detailed research, using mathematical modeling made it possible for the authors [24] to formulate the following specified criteria of jet formation:
Jet formation is to take place at the flow velocities (in
the reference system related to the critical point) which are less than the sound velocity; jet formation takes place also at supersonic velocities of the flow, if the angle of collision is large enough for the incoming flux jump not to contact with the collision point; and the jets, forming at supersonic velocities of collision, atomize in radial direction; at subsonic velocity jets remain continuous.
Qualitative results in revealing jet-formation criteria are the same for both plane and axisymmetric cases. However, it is important that in axial symmetry for jetless configuration, there appeared a jet in the calculation, whose mass was max. 2% greater than that of the analogous jet in case of incompressible liquid. The authors of the work of reference made an assumption, that when revealing jet-formation criteria, such jet formation may be neglected. The authors have not come to the certain conclusion concerning the appearance of jet in axisymmetric case. It may be explained either by the numerical effect or physical reality reflection. The effect of jet appearance in axisymmetric case for “jetless” configuration was explained in works [35,36]. Using strict mathematical methods the authors generalized Walsh criterion for axisymmetric case and formulated criteria of jet-formation in this geometry. It was noted by the authors [24] that numerical experiments did not model the desired conditions for the flow in all cases. The offered criteria of jet formation relate to the steady flow. In case of the plane flow, the steadiness was achieved. But in configuration used in calculations for axisymmetric case stationary conditions are unachievable, as the mass of the inward flow is continually growing. The
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authors [24] consider, that for the exact assessment of the criteria, in case of axial symmetry, some additional calculations with specified initial statement are required. It was considered in [35,36]. Preliminary numerical research on the jet-formation beginning process conducted by the authors showed [37] that in the turn zone the flow character is considerably nonstationary, wave and quasiperiodic. Such behavior is characteristic for the viscous medium. At the initial time, when the gradients are large, the substance heating is maximal and there is an increase of the characteristic disturbance propagation velocities. At the stationary stage, the heating of the substance due to viscosity is lower, the sound speed is also lower, and, consequently, the oscillation period is to be somewhat longer. In other words, this feature of the flow near the certain point is caused by dissipative processes, with oscillation amplitude decreasing with the increase of viscosity. The period slightly grows in spite of the fact that with growing viscosity the substance heating also grows, resulting in the sound speed increase. After the shock wave departure from the symmetry axis, pulsations (whose period is equal to that of the pressure oscillations at the critical point (Fig. 1)) start propagating both in the direction of the slug and in that of the cumulative jet.
3. Peculiarities of converging flows In preliminary numerical research of the flow it was detected that after the shock wave reflection from the symmetry axis two additional pressure maxima, diverging
Fig. 1. Successive steps of cumulative jet formation. Iso-levels of pressure are shown. The jet pressure pulsations are clearly seen.
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across the plane, are formed. At some time the distance between the two maxima reaches its highest point, and then the maxima begin converging and join at the time corresponding to the first peak of the pressure pulsation. After that, the pressure pulsations begin propagating up the flow. Their oscillation period is the same as that at the critical point. This pattern of the flow contradicts the generally accepted Lavrentiev theory, according to which maximal pressure is to be observed at the flow turning point on the symmetry axis. Pressure variations in the region of the flow turn results in the velocity value fluctuation along the jet, as well as fluctuation of its surface. Thus, we can come to the conclusion, that the models using stationary flow cannot be applied for calculation. Since the time of classical works of Lavrentyev and Taylor the problem on symmetric collision of jets, or problems of conical liner collapsing have been taken as a model problem whose solution allowed for understanding basic laws of cumulative charges. With some modifications due to the necessity of taking into account the rate of liner elements collapse along the generatrix, this model is used as a basis for most of the engineering approaches for calculating cumulative charges characteristics. It is logical to assume that some other classical schemes of jets collision studied in ideal fluid dynamics may serve as a proper basis for describing the corresponding physical process, i.e. the opposite direction of research is offered to be chosen: from mathematical abstraction to the real phenomenon. An interesting problem of the classical ideal fluid dynamics is that of a glancing collision of different thickness jets [30]. The point is, the conservation laws may work at different angles of outgoing jets, i.e. there is no unique solution. There are two view points concerning this problem: (1) there exists a unique solution of this problem, and different variants of correlation, closing the system of conservation laws are offered; (2) there is no unique solution, and this non-uniqueness testifies either to the dependence of the stationary flow on the solutions on the initial conditions, or to the solutions instability. Experimental research of water jets collisions at the velocities of several m/s [31,32] confirms existence of the unique solution. It is revealed in the papers [31,32] that the stationary flow configurations are non-symmetric (relative to the velocity sign change), whereas equations of motion do have this symmetry. It may be concluded that in these experiments, the stationary flow configuration is considerably affected by dissipative processes in the flow turn zone, i.e. to describe these experiments some other equations are required, those taking into account dissipative members. There is another natural supposition that there are some details to be taken into account when carrying out limiting process from the real fluid to the ideal one. Therefore, the issue can't be closed. A more detailed study of this problem has not only academic interest but also applications. The possibility of the stationary flow configuration dependence on the initial conditions deserves further consideration. In this case, some new questions arise: what these conditions must be, and what the dynamics of attaining the stationary
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conditions is. Other alternatives involve flow instability and lack of stationary conditions. They are also of practical interest. This raises the problem of classical jet hydrodynamics, i.e. the problem of jets collision without the complete stagnation point [36], which involves inner constant pressure zone formation (stagnation region). This problem does not have a unique solution, with its free parameter being the pressure within the zone, or (which is its equivalent) dimensions of the zone. As for the stationary flow the parameters on the infinity do not depend on this zone existence or its dimensions, of primary interest is the study of the flow from the stagnation region, with nonstationary parameters of the incoming flow. Smoothing effect of this region on the development of various disturbances should be expected, particularly, smoothing of Relay–Taylor instability for thin liners. Another aspect of studying jet flows with stagnation regions is the study of the flows with supersonic velocities of the incoming flow. This is closely connected with the problem of jet-formation criterion. Main results on the jetformation criteria at this flow velocities are as follows [24]: at the collision angle less than critical, jetless configuration is realized; at greater angles the jet is formed, which spills in radial direction. The characteristic property of the flow, in case of jet formation, is existence of the detached shock wave. In case of the flow with stagnation region, the flow turns smoothly, without jump. A problem of the shock wave front and its existence in this configuration should be mentioned. In any case, tensions, which cause jet destruction in radial direction are likely to be smaller in stagnation region flows. The study of non-stationary supersonic flows in this configuration is of interest due to the fact that the jetformation criteria offered before [24] have been obtained for the case of stationary flow. One more base model problem is that of the collision of fluid jets, consisting of two or more layers. It was considered in works [33] and [34]. It was noted that in case of the equality of both layers Bernoulli constants, the solutions practically coincide with the classical one. Depending on the layers thickness ratio the following stationary configurations of flows are realized: with two-layers slug and a uniform jet; with uniform slug and two-layer jet; with uniform jet and slug, consisting of different materials. One might expect that, if compressibility (i.e. the difference in the sound speed of layers materials) is taken into account, the problem solutions will be considerably different from that for the case of the solid jet with the same value of Bernoulli constant. This problem solution is not evident, especially in case of supersonic flow velocity, at least in one layer.
flow with uniform jet and slug, consisting of different materials, is impossible, as in this case, there would be pressure break at the stagnation point. On physics grounds, there are no stationary conditions in case of boundary line being close to manifold line of flow. Stationary flow conditions are possible only if there is a stagnation region, whose pressure value is (in this case) determined unambiguously. The alternative offered here is formation of non-stationary periodic flow regime. Of special interest is the study of the flow at supersonic velocity of at least one of the layers. Thus, we've got used to the cumulative charges in which the liner is established with the complete opening angle smaller than 1801 (usually 30–1601), and it is compressed due to the effect of the explosive detonation, forming thin cumulative jet and massive slug containing a great part of the liner mass. Basic processes of cumulative jet formations are described in Lavrentiev–Birkhoff theory as a model of plane stationary collision of the incompressible liquid jets at the angle smaller than 1801. It is well known and confirmed by the centenary practice of cumulative charges designing. In cumulation conditions, without complete stagnation point, involving formation of the inner zone of constant pressure, which is to be studied, a new fundamental quality appears: the jet velocity, mass and pulse increase manifold (Figs. 2–5). It has been by far considered impossible [38,39]. 5. Numerical experiments Numerical experiments on formation of shaped charges were performed based on numerical schemes and technology developed in [35–41]. Along with classical conical lainer additional massive metallic body was used to make the shape as shown in Fig. 2. The angle of a cumulative layer is 501, material – aluminum, thickness 0.5 mm, the shaping cylindrical iron body has a diameter 8 mm and thickness 1 mm, external iron cover is 2 mm thick. High explosive is hexogen 1.66 g/ mm2. Diameter of the charge 40 mm. Three microseconds after detonation initiation detonation wave compresses the layer and it begins sliding along the shaping body. The angle of material converging to the center of symmetry approaches 1801. Radial contracting
4. Technology for hyper-cumulative jet formations For practical purposes, definition of the problem with different values of Bernoulli constant (in each layer at different infinity velocities [34]) seems to be more topical. This problem is much more complex even in case of the ideal incompressible liquid. It is evident that the stationary
Fig. 2. Initial configuration: 1 – detonator, 2 – detonation products, 3 – high explosive charge, 4 – additional shaping body, 5 – cumulative layer, 6 – external cover.
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Fig. 3. Deformation of materials 3 μs after initiating explosion: pressure maps and velocity profiles (Vz ) and (Vr) along the crossing coordinate lines.
Fig. 4. Deformation of materials 4.6 μs after initiating explosion: pressure maps and velocity and pressure profiles (Vz) and (p) along the crossing coordinate lines.
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Fig. 6. X-ray patterns of cumulative aluminum jets from identical charges (conical liner, 441) [41]. Lower figure: classical scheme, upper figure: prototype of hypercumulative flow. In upper X-ray pattern the jet mass is considerably enlarged.
By the time 4.6 μs collapse of aluminum cover layer took place (Fig. 4). Cumulative jet was formed characterized by tip velocity 14.65 km/s. The shaping body continues its motion and compresses the slug moving in the axial direction at 1.15 km/s. Velocity of shaping body axial motion is 2.98 km/s, the aluminum layer is contracting with velocity 2.83 km/s. Maximal pressure is 57.8 GPa. Fig. 5 illustrates the development of the process by 7.6 μs time after detonation initiation. By this time maximal velocity of a jet, which contributes to its elongation, slightly decreases (13.5 km/s). The velocity gradient continues jet stretching. Cut of the slow part of a jet should be performed based on the needed properties: velocity, mass and shape. Maximal velocity of the slug is 4 lm/s, while its minimal velocity is 400 m/s. By this time slug perforated the additional shaping body and practically destroyed it. 6. Experimental investigations Experimental validation of the developed theoretical results was performed based on techniques described in [41]. Fig. 6 illustrates X-ray patterns of cumulative aluminum jets from identical charges (conical liner, 441). Lower figure corresponds to classical scheme of cumulative jet formation, upper figure represents the prototype of hypercumulative flow. In upper X-ray pattern the jet mass is considerably enlarged. Comparison of results shows, that basic mechanism of cumulative jet formations were described correctly by our model. 7. Discussions
Fig. 5. Preliminary modeling of cumulative jet under hypercumulation conditions without complete stagnation point involving formation of the inner zone of constant pressure. Liner material: aluminum. Velocity distribution along the axis is shown: maximal velocity of a jet is 13.5 km/s, minimal velocity of slug is 400 m/s. Slug mass/jet mass ratio is far below unity.
velocity is 1434 km/s, horizontal velocity is 3259 km/s due to the piston effect of expanding reaction products pushing the shaping cylinder.(Fig. 3)
Thus, according to the hydrodynamic theory of cumulative jet formations in cumulative charges of Lavrentiev–Birkhoff, the jet velocity rises with the decrease of the liner opening angle, and simultaneous increase of the slug mass. In classical cumulative charges, high mass and velocity values of the jet in “low” liners are unattainable. Maximal jet velocity for the given material is limited by gas-dynamic limit due to the jet material destruction by “inner explosion” resulting from the liner material collision at the charge symmetry axis.
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Under the new hypercumulative conditions of cumulative jet formation without complete stagnation point involving formation of the inner zone of constant pressure (dead region), the jet mass is always greater than that of the slug. As a result of gas-dynamic properties redistribution in the flow turn area, the conditions for cumulative jet destruction due to the “inner explosion” are eliminated, and the jet velocity may considerably exceed gas-dynamic limit at the jet material density of the order of that of the liner material, that previously was considered impossible. Smoothing effect of this zone on the development of different types disturbances are likely to be expected, particularly, smoothing of Rayleigh–Taylor unsteadiness for thin liner. Therefore, whereas the effect of cumulative charge parameters and loading conditions of the liner (in classical Lavrentiev–Birkhoff scheme) on cumulative jets characteristics is quite predictable, the effect of sizes, parameters and geometric position of the flow off-axis stagnation region on instability development and jet-formation parameters is of considerably non-linear and non-monotonic character. According to the preliminary research [40,41], such region existence brings about both non-stability of the initially steady flows and stability of the non-stable ones, as well as the jet pulse and velocity growth or decrease.
8. Conclusions The fundamental new feature of hypercumulative jet formation was studied. It was proved both theoretically and experimentally that a hypercumulative jet has a higher thickness and mass. The basic conditions for the hypercumulative jet formation are the following: the liner material flow turn should proceed until material collapse on the symmetry axis, with the collapse angle of cumulative liner material being more than 1801. That was unattainable in the known and used cumulative flow schemes. This makes it possible controlling jet parameters (increase of mass, radius, and pulse as compared with the schemes known in world practice) in extremely wide ranges as well as opening new opportunities for technologies and techniques. The proposed method opens wide possibilities for laboratory testing of shielding materials under real conditions of collisions with space debris fragments. References [1] V.A. Chobotov (Ed.), Orbital Mechanics, 2nd ed. AIAA Education Series, Washington, D.C., 1996. [2] N.N. Smirnov (Ed.), Space Debris Hazard Evaluation and Mitigation, Taylor and Francis Publication, London, 2002. (208 pp.). [3] N.N. Smirnov, Evolution of orbital debris in space, Adv. Mech. 1 (2) (2002) 37–104. [4] N.N., Smirnov, A.B., Kiselev, A.I., Nazarenko, Mathematical modeling of space debris evolution in the near Earth orbits, Vestnik Moskovskogo Universiteta Ser. 1 Matematika Mekhanika, Moscow University Mechanics Bulletin, 4, Allerton Press, 2002, pp. 33–41. [5] N.N. Smirnov, A.I. Nazarenko, A.B. Kiselev, Modelling of the Space Debris Evolution Based on Continua Mechanics, European Space Agency, 2001, 391–396 (Special Publication) ESA SP). [6] N.N. Smirnov, V.F. Nikitin, A.B. Kiselev, Peculiarities of Space Debris Production in Different Types of Orbital Breakups, European Space Agency, 1997, 465–470 (Special Publication) ESA SP).
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