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Simulation and analysis of a 23-mm HEI projectile hydrodynamic ram experiment
International Journal of Impact Engineering 22 (1999) 981}997
Simulation and analysis of a 23-mm HEI projectile hydrodynamic ram experiment Charles E. Anderson, Jr.*, T.R. Sharron, James D. Walker, Christopher J. Freitas Mechanical and Materials Engineering Division, Southwest Research Institute, 6220 Culebra Road, P.O. Drawer 28510, San Antonio, TX 78228-0510, USA Received 3 September 1998; received in revised form 3 August 1999
Abstract An Eulerian wavecode was used to simulate the impact, penetration, and detonation of a 23-mm high-explosive projectile into a water-"lled tank. The pressure}time response is compared with results from an experiment conducted by Lundstrom and Andersen (Symp. on Shock and Wave Propagation, Fluid}Structure Interaction, and Structural Responses, 1989). Computed peak pressures and impulses compare well with the experimental values. 1999 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction Hydrodynamic ram refers to the process whereby momentum and kinetic energy from a projectile are transferred through a #uid-"lled container to the walls of that container. Additionally, some projectiles contain high explosives and the detonation of the explosive substantially increases the total energy. The liquid e$ciently couples the kinetic and explosive energy of the projectile to the structure and greatly increases the magnitude of the damage sustained. Although considerable experimental work has been conducted to investigate and quantify damage from hydrodynamic ram, little numerical simulation work has been performed to examine the entire problem because of issues related to numerical accuracy, and the very disparate length and time scales of the problem. Projectile penetration through a fuel tank shell, travel through the #uid, detonation of the high explosive, propagation of the shock wave and loading of the structure, and subsequent structural response is a di$cult problem to simulate numerically because of disparate length and time scales.
* Corresponding author. Tel.: #1-210-522-2313; fax: #1-210-522-5122. E-mail address: [email protected] (C.E. Anderson, Jr.) 0734-743X/99/$ - see front matter 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 0 4 6 - 9
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A typical high-explosive incendiary (HEI) projectile is 23}30 mm in diameter, whereas the fuel tank itself is one to two orders of magnitude larger. The projectile penetration velocity is on the order of 0.7 mm/ls; the projectile travels for approximately 200 ls prior to detonation of the high explosive. The shock resulting from detonation of the explosive travels at a velocity of 1.5}2.0 mm/ls (the detonation wave travels 7}8 mm/ls), with a pulse width on the order of 100}300 ls. For purposes of discussion, the initial shock interacts with the wall several hundred microseconds after detonation, but the resultant structural response is at least several milliseconds in duration. To achieve a numerically accurate solution then requires "ne spatial and temporal resolution to resolve all pertinent scales, and thus, the calculation of the entire problem is computationally intensive. The type of computer programs that can be used to simulate high explosive detonation, material compressibility, propagation of shocks, and nonlinear material response are typically referred to as hydrocodes. The word `hydrocodea is a contraction of `hydrodynamic computer codea, and it is a misnomer since hydrocodes rigorously calculate elastic}plastic material response including strain hardening, rate e!ects, thermal softening, and to various degrees of "delity, failure. Hydrocodes provide solutions to the three conservation equations, coupled with material response descriptions (equation of state and constitutive models) in either Lagrangian (material) or Eulerian (spatial) coordinates [1]. An explicit time integration scheme is used to advance the solution in time, with the time step interval subject to a modi"ed Courant stability criterion [1]. Numerical simulations have hardly been used to simulate hydrodynamic ram. One of the earliest examples is by Kimsey, reported in Zukas et al. [2]. Simpli"ed analytical/numerical models have been developed but these have had limited predictive capability due to shortcomings in the modeling approximations used [3]. Although there may have been studies in the intervening years, little-to-no documentation exists. There are several reasons for this observation. Kimsey used the Lagrangian "nite element code EPIC [4] for his study, and grid distortion e!ectively terminated the problem. Eulerian codes, on the other hand, are not subject to the problem of grid distortion, but prior to the 1980s, Eulerian codes were only "rst-order accurate in the advection and interface reconstruction algorithms, and were therefore very di!usive. The large disparity in length and time scales precluded the very "ne gridding required for "rst-order accurate codes to replicate hydrodynamic ram experimental results; it was not until the late 1980s that second-order advection was added to a new generation of Eulerian programs [5}7]. The objective of this study was to quantify how well state-of-the-art numerical simulations could reproduce the response of a #uid-"lled structure to hydrodynamic ram. Freitas et al. [3,8], conducted a literature survey to compile experimental data that could be useful in numerical validation. One problem from this literature survey was selected for in-depth numerical studies, the impact and subsequent detonation of a 23-mm HEI projectile into a water-"lled tank instrumented with pressure gages.
2. The experiment Lundstrom and Andersen [9] report data from experiments where HEI rounds were shot into a rectangular tank "lled with water. Both 23-mm and 30-mm HEI rounds were used in the
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Fig. 1. Fluid pressure histories from Lundstrom}Andersen experiment [9].
experimental series. The focus here is the results of the 23-mm experiment. The 23-mm round impacted the tank at a speed of 701 m/s. The round had a time delay between fuse activation at impact and detonation of the explosive. The time delay is captured in the pressure}time histories as the time di!erence between a small pressure rise due to projectile impact and the large pressure spike that results from detonation of the high explosive (see Fig. 1). The tank was constructed of reinforced 0.635-cm thick steel side walls, 91.4 cm in width. The tank was open on the top, and the water depth was 94.0 cm. The entry panel and exit panels were 0.3175-cm thick steel, and the HEI round impacted normal to the panel surface. Fluid pressure data were reported; the tank was instrumented with four Kistler 603H piezoelectric accelerationcompensated pressure gages. The gages were mounted at the end of pipes that were suspended from a shock-isolated beam. All gages were located in the vertical plane of the projectile path, and faced the entrance wall of the tank. Fig. 2 provides a schematic of gage location relative to the shotline. Gage locations, with the x-coordinate measured from the front of the tank and the y-coordinate
The experiments were incorrectly identi"ed in Ref. [9]; the data from Shot 4 are for the 23-mm experiment.
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Fig. 2. Schematic of pressure gage location. Solid circles denote pressure gages. The arrow indicates the projectile shotline.
measured from the centerline, were: Gage A (25.40, 27.90); Gage B (40.64, 17.78); Gage C (57.15, 7.62); and Gage D (71.12, !2.54). All measurements are in centimeters. The square symbol on the shotline denotes the approximate location * as estimated from the numerical simulations * of the center of the explosive-"lled cavity when detonation occurred. The pressure}time histories of the four gages are shown in Fig. 1.
3. The computational model 3.1. Description of the 23-mm HEI projectile In devising the numerical model for the 23-mm HEI projectile, we are concerned with the "delity of the model with respect to parameters important to penetration and high explosive energy release. Other details, for the purposes of this study, are not considered as important, e.g., the actual workings of the fuzing mechanism. Thus, the 23-mm HEI projectile is idealized for the speci"c numerical simulations for which it is to be used. Some information about the 23-mm projectile is provided in Ref. [10]. The projectile of interest is the ZU-23 HE/I/T, which is denoted here as the 23-mm HEI for simplicity. Jane's contains little information concerning details of the interior of the projectile. For these details, information is taken from Zabel and Riegel [11]; the drawing of the 23-mm HEI is shown in Fig. 3. Jane's reports the total mass of the projectile to be 185 g. Czarnecki measured 180.6 g for the MG-25 projectile [12]. The walls around the explosive are 3-mm thick in the model; in the actual 23-mm round, the wall thickness varies between approximately 2.75 and 4 mm. The details of the wall design, e.g., the rotating band, the bourrelet, crimping grooves, etc., are not warranted in the model since the focus is on the response of the fuel tank. So long as the overall characteristics of the projectile are modeled su$ciently accurately, the speci"c details should not matter. Dimensions and materials for the model are compared to the actual round in Table 1.
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Fig. 3. Drawing of Russian 23-mm projectile (MG 25) [11].
Table 1 Physical characteristics of the 23-mm HEI projectile
Length Diameter Wall thickness Total mass Explosive Explosive mass Casing material
Actual round
Model
100 mm 23 mm (rotating band) 2.75}4.00 mm 180}185 g RDX 13.28 g 1018 steel
100 mm 23 mm 3.00 mm 182.5 g C-4 (91% RDX) 14.4 g 1006 steel
The HE mass has been measured and found to be 13.28 g of RDX (MG-25) [12]. The JWL equation of state [13] was used to model explosive energy release and subsequent expansion of the detonation products. JWL constants exist for C-4, which is 91% RDX [14]. The explosive mass was estimated, using model dimensions, to be 13.2 g. The mass was then increased by approximately 9% to 14.4 g to account for the di!erences in energy content between 100% RDX and C-4. Johnson}Cook constitutive constants [15] exist for 1006 steel, which is in the same family as the 1018 steel. The initial #ow stress for the 1006 steel was estimated to be 350 MPa. As already discussed, the details of the fuze assembly were not modeled. Instead, the fuze assembly was designated as aluminum, but a very soft aluminum to re#ect that it has very little rigidity. The tracer cavity, while most probably containing something like phosphorous, was modeled as an inert plastic.
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3.2. The computational model The nonlinear, large deformation Eulerian wavecode CTH [5] was used to simulate the experiment. The 2-D cylindrically symmetric option of CTH was used to model the problem. CTH uses a van Leer algorithm for second-order accurate advection [16] that has been generalized to account for a non-uniform and "nite grid, and multiple materials. CTH also has an advanced material interface algorithm for the treatment of mixed cells [5]. The equation of state for water was examined by Kerley [17] and modi"ed for this study. The thermodynamic response of water was modeled with the Mie}GruK neisen equation of state (EOS). The Mie}GruK neisen EOS relates the pressure P, density o, and internal energy E by P(o, E)"P (o)#!(o)o[E!E (o)], (1) 0 0 where P (o) and E (o) are the pressure and internal energy on some reference curve, and the 0 0 GruK neisen parameter ! is assumed to depend only on the density o, and is de"ned as
1 *P !" o *E
(2)
! o !" , o
(3)
. M The dependence of the GruK neisen parameter on density is given by
where the constant ! is the GruK neisen parameter at the initial density o . The dependence of the internal energy on temperature is de"ned by an analogous expression to Eq. (1): E(o, ¹)"E (o)#C [¹!¹ (o)], (4) 0 4 0 where ¹ (o) is the temperature along the reference curve, and the speci"c heat C is assumed to be 0 4 a constant. The reference curve is taken as the Hugoniot. The relationship between the shock velocity (u ) and particle velocity (u ) is determined experimentally, and is assumed to "t a quadratic relationship u "c #s u #s u, (5) where c is the adiabatic sound speed and s and s are additional material constants. Using Eq. (5), the Hugoniot reference states can be written as: P "o c gF(g), & E "c gF(g), & F(g)"4+(1!s g)#((1!s g)!4c s g,\, E c dF(g) dg, ¹ (g)"¹ eC E# eC E e\C Eg & 2C dg 4 g"1!o /o. M
(6a) (6b) (6c) (6d) (6e)
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The expression for ¹ (g) cannot be integrated in closed form, but a polynomial expansion in terms & of g permits the integral to be evaluated; the expansion was carried out to a "fth-order polynomial in g. The six EOS constants are o , c , s and s , C , and C . For water, these were determined to be 4 o "1.0 g/cm; c "1.48 mm/ls; s "1.984; s "!0.0966 ls/mm; C "0.48; C "4.28;10 4 erg/g/eV. The Hugoniot "t was taken from Frank and Gathers [18]. Parameters C and C were 4 selected to give the best possible agreement with thermal expansion and heat capacity data [19]. Square zoning was used for all computations. In the "rst simulation, 4 zones were used to resolve the projectile radially (across the projectile radius); the zone dimension was 0.2875 cm on a side. A total of 184,869 zones and 56 Mbytes of memory were required for the simulation. In the second simulation, the grid resolution was doubled. The radius of the projectile was resolved with 8 zones; the zone dimension was 0.1438 cm on a side. Now a total of 726,660 zones and 121 Mbytes of memory were required to resolve the problem domain. These two sets of simulations will be referred to as the 4-zone simulation and the 8-zone simulation, respectively.
4. Numerical simulation of the experiment The 23-mm HEI projectile has a time-delay fuze. The projectile position necessary to give the correct arrival times of the detonation shock at the four gage locations was estimated from a preliminary simulation run. The location of the projectile (center of the HE) at the time of detonation is denoted by the square symbol in Fig. 2. The detonation delay time was then determined from the time it takes for the projectile to penetrate and transit to the correct detonation location. It is estimated that the explosive detonated approximately 200 ls after impact. The calculation proceeded from impact (t"0) for 200 ls. At 200 ls, the explosive was initiated and the result of energy release and shock propagation calculated. Pressures were monitored in the simulations at the locations of the experimental gages. Computer-generated material plots are shown in Fig. 4. The disparate length scales for the problem are evident in Fig. 4(a). A close-up view of the projectile at time t"0 ls is shown in Fig. 4(b). Fig. 4(c) shows the projectile and penetration cavity in the water just prior to explosive detonation; the `ripplinga of the steel case is an artifact of numerical inaccuracies associated with the treatment of mixed cells combined with the interface reconstruction algorithm. Fig. 4(d) shows the response approximately 400 ls after detonation (600 ls after impact). The explosive detonation products, the `detonation cavitya, and outward bulging of the entrance plate can be observed in Fig. 4(d). The pressure}time results at the four gage locations (see Fig. 2) are shown in Fig. 5. The peak pressures are slightly higher in the 8-zone simulation than for the 4-zone simulation. The largest disparity in the two calculations is the peak pressure at gage A, which di!ers by approximately 17%; the other calculated pressures di!er by less than 10%. The cause of the time delay between arrival of the explosive shocks is discussed below. The predicted peak pressures are approximately the same as the experimentally measured peak pressures for gage locations A and C, but the experimental pressures are larger than the numerical ones for gage locations B and D. The pressures and impulses (the areas under pressure-time curves) will be discussed later in the paper.
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Fig. 4. Material plots of simulation results.
The detonation times, penetration velocities, and projectile positions for the two simulations are compared in Table 2. The detonation time is the delay between impact of the projectile and explosive initiation. The projectile velocity and position are given in columns 3 and 4, respectively, at the time of explosive initiation. (The position of the nose at time t"0 ls is 0.0 cm, i.e., the nose is in contact with the tank wall.) The distances between the center of the explosive and the four gage locations are given in the last four columns of Table 2.
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Fig. 5. Comparison of pressure pro"les from numerical simulations.
The projectile does not decelerate as rapidly for the "ner-zoned problem. The velocity}time histories of the leading edge of the explosive cavity for the two zonal resolutions are plotted in Fig. 6. The deceleration of the projectile is approximately the same for the "rst 50 ls, but between 60 and 70 ls, the 4-zone simulation undergoes a very rapid deceleration. In contrast, the deceleration for the 8-zone simulation is much more constant. In the actual experiment, the projectile penetrates through the tank wall and into the #uid as a rigid body. In the 4-zone simulation, the cylindrical body `deformsa, and thus there is a larger decelerating force on the projectile. This is a direct consequence of averaging of material properties within mixed cells (computational cells that contain more than one material). This e!ect is a well-known problem in the application of Eulerian formulations to rigid-body penetration, e.g., see Ref. [20]. Increasing the zonal resolution decreases the percentage of mixed cells, and thus decreases their in#uence on the numerical solution. Thus, there is little mushrooming of the projectile nose, and less retarding force on the projectile, in the "ner-zoned simulation, resulting in a more realistic projectile velocity history. Since the projectile in the 8-zone simulation does not decelerate as rapidly as the 4-zone simulation per unit time, the projectile travels a greater distance per unit time. The detonation location (center of the explosive) was determined to be approximately 5.4 cm into the tank based on
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Table 2 Computational results: projectile Simulation
4-zone 8-zone
Det.time (ls)
200 175
Proj.vel. (m/s)
490 640
HE pos. (cm)
5.36 5.36
Distance to gages (cm) A
B
C
D
34.4 34.4
39.5 39.5
52.3 52.3
65.8 65.8
Center of HE at detonation. At detonation, with the distance determined from the center of the HE to the gage.
Fig. 6. Velocity versus time as a function of grid resolution.
a comparison of the 4-zone results to those of the experiment. The time of arrivals for the impact shock and the detonation wave at the four pressure gages for the 4-zone simulation are within 6% of the experimentally measured values (see Table 3). Therefore, for the projectile in the 8-zone simulation to be at the correct location for detonation, the denotation time must be 175 ls after impact. The peak pressures and the time of arrivals (TOA) are summarized in Table 3. Three TOAs are given: TOA-1 is the time of arrival of the impact shock, TOA-2 is the time of arrival of the detonation wave, and TOA-3 is the time of the peak pressure. The pressures listed in Table 3 are the peak pressures. The times of arrival for the impact shocks are within approximately 10 ls for the two simulations, as seen in Fig. 5. However, the times of arrival of the detonation waves are o!set from each other. The center of the explosive cavity for the 8-zone simulation reaches the prescribed 5.4-cm detonation location 25 ls sooner than it does in the 4-zone simulation because of the di!erences in `penetrationa velocities. The o!set between the arrival of the detonation waves shown in Fig. 5 is this 25 ls.
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Table 3 Computational and experimental results: peak pressures and times of arrival Gage position Simulation
4-zone 8-zone Experiment
A
B
TOA-1 (ls)
Press (MPa)
TOA-2 (ls)
TOA-3 (ls)
TOA-1 (ls)
Press (MPa)
TOA-2 (ls)
TOA-3 (ls)
240 248 255
33.9 40.9 41.4
385 363 348
398 371 355
282 292 300
35.7 36.1 51.7
411 389 385
423 400 390
Gage Position Simulation
4-zone 8-zone Experiment
C
D
TOA-1 (ls)
Press (MPa)
TOA-2 (ls)
TOA-3 (ls)
TOA-1 (ls)
Press (MPa)
TOA-2 (ls)
TOA-3 (ls)
370 383 390
23.7 24.7 28.3
506 569 465
506 488 472
457 470 487
17.0 18.9 32.4
574 559 560
596 575 565
TOA-1: Time of arrival of impact shock; TOA-2: Time of arrival of detonation wave; TOA-3: Time of peak pressure.
In general, the arrival times and the times of the peak pressures from the 8-zone simulation are in slightly better agreement with the experimental results than those for the 4-zone simulation. The rise time of the detonation wave shock is much steeper for the 8-zone simulation, a consequence of the "ner zoning. The fact that the arrival times of the impact shock and detonation wave agree quite well with those of the experiment implies that the detonation times and detonation location (i.e., the position of the projectile at detonation) are approximately correct. The 8-zone simulation results are compared to the experimental data in Fig. 7 . The numerical pressure traces are very similar to the experimental pressures. Several points are worth noting. The pressure pro"les have some `structurea, as compared to a simple exponential-like decay obtained from a point charge in an in"nite domain. This structure is the result of multidimensional e!ects, and is present in the simulations and experimental traces, particularly for gage locations A and B. There is less structure in the experimental traces at gage locations C and D; similarly for the numerical results. The pressure `spikesa at late times are the result of re#ections from the container walls. The re#ected waves in the simulations arrive at times di!erent than those in the experiment because the square tank (experiment) was idealized as a cylinder for the simulations. Additionally, there is a free surface in the experiment and none in the simulations, but again, this is a late time e!ect. For the main pressure pulse, the simulated pressure} time pro"les provide very good qualitative and quantitative representations of the experimental data.
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Fig. 7. Comparison of 8-zone simulation pressure}time results with experimental data.
4.1. Peak pressure versus scaled distance Lundstrom and Andersen conducted only one 23-mm test, so there is no estimate of experimental reproducibility, including test-to-test variation. However, they did conduct two experiments using a 30-mm HEI projectile. A scaling relationship is required to compare the data from the 23-mm projectile and the 30-mm projectiles. Self-similar blast (shock) waves are produced at identically scaled distances when two explosive charges of similar geometry di!er only by the explosive charge weight [21]. The scaled distance Z is given by r , Z" =
(7)
where r is the distance and = is the explosive mass. Z has dimensional units of cm/g .
Technically, = should be the total energy of the explosive, but for a given type of chemical explosive, the energy is proportional to the total mass.
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Fig. 8. Peak pressure versus scaled distance from detonation.
A large number of 30-mm test data exist for peak pressure versus distance [8]. A power law of the form P"aZ@ was used to "t the scaled experimental data [8]. The coe$cients a and b were determined by a least-squares regression "t: P"856.4Z\ MPa.
(8)
The scaled experimental data, the results of the numerical simulations, and the regression "t are plotted in Fig. 8. The 30-mm data are shown as open symbols, and the 23-mm data are displayed as solid symbols. The 30-mm experimental data provide an indication of the test-to-test variability. The peak pressures as a function of scaled distance for the 4- and 8-zone simulations are plotted as the solid circles and diamonds, respectively, in Fig. 8. (The abscissa values for the 4-zone and 8-zone simulations have been separated slightly so that the `dataa points are visible in the graph; the pressures have not been altered.) The dashed lines in Fig. 8 represent $5 MPa about the power law curve "t. For the most part, the simulation results lay within this `scatter banda. A simulation of a static arena test was conducted for a pre-test prediction [22]. Free-"eld pressures were calculated at 15.24 and 30.48 cm; these correspond to scaled distances of 6.44 and 12.88 cm/g , respectively. These two points are plotted in Fig. 8 as open squares (the simulation was conducted for 8 zones across the radius of the projectile). These two points are calculated for a static situation * the projectile is not moving * but they agree quite well with the other scaled data. As already mentioned, the peak pressures for the 8-zone simulation agree very well with the 23-mm projectile experimental data at gage locations A and C. The measured pressures at gage
There is some ambiguity of whether to use 13.2 g of RDX explosive or 14.4 g of C-4 explosive. However, since the explosive weight is taken to the one-third power, this uncertainty amounts to less than 3%.
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locations B and D for the 23-mm projectile are larger than those from the calculations, and larger than what would be expected from the regression "t. The measured results for the two 30-mm experiments at gage locations B and D also lay above the regression curve. The peak pressure at gage D is larger than at gage C for test 3, similar to the 23-mm test. Thus, there appears to be some geometric and/or impedance e!ect in the experiment * for these two gages * that is not being replicated in the idealized cylindrical representation of the rectangular tank. 4.2. Scaled impulse versus scaled distance The impulse * the area under the pressure}time curve * is of considerable interest since it is the impulse that provides the loading function for structural response. Analogous to the pressure, there is a similarity relationship for the impulse for similar explosive charges [21]. The scaled impulse is given by
R P(q)dq I " . (9) IM " = = The impulses for the pressure peaks were calculated for the experimental data of Ref. [9]. The impulses were calculated only for the primary pressure pro"le. The scaled impulses versus scaled distances are plotted in Fig. 9. The scaled impulses for the two nominally identical 30-mm experiments are essentially equivalent at each of the gage locations. The results for the 23-mm experiment are shown as the solid squares. Scaled impulses were also calculated for the 4- and 8-zone simulations. A distinction between the two simulations is that the pressure goes through a rarefaction phase for the 8-zone case (see Fig. 5). For the 4-zone simulation, the pressure at a gage location does not quite return to zero, a consequence of zonal resolution of the pressure "eld. Regardless, the results of the numerical simulations are nominally the same.
Fig. 9. Scaled impulse versus scaled distance.
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It is interesting to observe that the impulse at gage location B is higher than the impulse at gage A for all experiments and the simulations. However, the simulations do not capture the magnitude of the elevated value observed in the experiment. Except for gage location B, the numerical simulations are in good to excellent agreement with the experimental data. The larger impulses for the experiment at gages A and B are due, in part, to a larger amplitude spike near the end of the main pressure pulse. This spike is captured in the simulations, but it is not as large as in the experiments. The solid line in Fig. 9 is an empirically based curve from Cole [23]: 4470 IM " Z
r Z" , =
(10)
where the 4470 and 0.89 are constants for TNT. I has units of kg/(m s g ). TNT has less speci"c energy than RDX or C-4, thus, the empirical curve lays below the experimental data and the simulation results. Although it is possible in principle to adjust Eq. (10) for C-4 explosive using a TNT equivalency, such a procedure is often inaccurate. The point here is that the data and simulation results are consistent with this empirical "t to TNT data, i.e., the curve lays below results for a more energetic explosive. A conclusion can be reached concerning zonal resolution for the HEI projectile simulation. The "ne-zone simulation * 8 zones across the projectile radius * provided approximately the same answers as the more coarsely zoned problem (4 zones across the projectile radius). The biggest di!erence between the two simulations is penetration of the projectile through the #uid: penetration was retarded due to projectile deformation in the 4-zone simulation and not in the 8-zone simulation. This unrealistic deformation, as already mentioned, is a direct consequence of a numerical di!usion e!ect due to mixed cells in a poorly resolved Eulerian simulation. However, after detonation, the peak pressures at the four gage locations were approximately equal, with the largest discrepancy being at the closest gage. Presumably, the "ner zoned simulation more accurately resolved the shock front and thus the peak pressure. The 4- and 8-zone speci"c impulses were in agreement to within 6% for the "rst three gages. The larger relative error between the two simulations for gage D is probably due to the lower resolved pressure of the initial impact shock in the 4-zone simulation (see Fig. 4). Therefore, the 4-zone simulation * for a reasonable degree of accuracy * was numerically resolved for structural response calculations. We note, in making these comparisons, that square zoning was used throughout the computational grid. Square zoning minimizes numerical truncation error, and the good results obtained for the 4-zone calculation almost certainly are due in part to using square zoning.
5. Summary and conclusions The Eulerian wavecode CTH was used to investigate the impact, penetration, and detonation of a 23-mm high explosive projectile into a water-"lled tank, simulating an experiment by Lundstrom and Andersen [9]. In the experiment, pressures were monitored at four gage locations. A numerical model for the 23-mm Russian HEI projectile was developed. The total mass of the projectile model is in good agreement with the mass reported in Jane's [10], and the mass obtained from an
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independent measurement [12]. The explosive mass, based on the dimensions of the cavity provided by Zabel and Riegel [11], is estimated to be 13.2 g, also in excellent agreement with Ref. [12]. The explosive mass was increased to 14.4 g of C-4 explosive to provide the equivalent explosive energy of 13.2 g of RDX. Two simulations were conducted, with the second simulation having twice the zonal resolution of the "rst. At two gage locations, the overall pressure}time pro"les and predicted peak pressures from the simulations are in very good agreement with experiment. For whatever reason, the peak numerical pressures at the other two gage locations are lower than those observed in the experiment. The results for the "ner zoned simulation, particularly the peak pressure of gage A, are in slightly better agreement with the experimental data. However, given that the "ner zoned simulation took eight times longer to run (&40 days as compared to &5 days), the slight di!erences in the answers do not warrant the large di!erence in run times. The scaled impulses versus scaled distance for the simulations agree well with those from the experiment at two gage locations, and are consistent with an empirically based formula for TNT. At the other two gage locations, the calculated impulses are approximately 20% less that that obtained from the experiment. The impulse provides the loading function for subsequent structural response. Thus, it has been demonstrated that this "rst-principles modeling approach can be used to estimate fairly accurate the loading function for structural response calculations. We distinguish the results reported herein from work by other researchers where the emphasis is on the prediction of structural response and subsequent damage, e.g., Refs. [24,25]. In Refs. [24,25], the explosive charge was static (stationary), and in Ref. [25] the authors state that the explosive mass had to be increased to 53 g to obtain #uid pressures in agreement with test results. The current simulation results * without recourse to arti"cially increasing the explosive mass * are entirely consistent with the 23-mm experimental data and appropriately scaled 30-mm HEI data from the literature. Additional work, in which numerical pressure}time pro"les are compared to those from another set of experiments, further demonstrate that the wave pro"les are accurately reproduced [26]. Thus, it would appear that "rst-principles numerical simulations can predict the pressure and impulse loadings in a hydrodynamic ram event.
Acknowledgements This work was conducted under contract F33615-95-C-3408 with Air Force Research Laboratory (AFRL), Wright-Patterson AFB, OH. The authors wish to thank Dr. Greg Czarnecki of AFRL for his support and encouragement during this study. The authors would like to acknowledge and express their thanks for the technical support provided by D. L. Goodlin of SwRI during this study.
The problems were run on an Hewlett-Packard 735 workstation.
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References [1] Anderson Jr. CE. An overview of the theory of hydrocodes. Int J Impact Engng 1987;5:33}59. [2] Zukas JA, Nicholas T, Swift HF, Greszczuk LB, Curran, DR. Impact dynamics. New York, NY: Wiley, 1982. [3] Freitas CJ, Anderson CE, Jr, Walker JD, Little"eld DL, Hydrodynamic ram: a benchmark suite, Structures under extreme loading conditions*1996, PVP}vol. 325. 1996. New York, NY: ASME, 1996. p. 63}74. [4] Johnson GR, Colby DD, Vavrick DJ. Three-dimensional computer code for dynamic response of solids to intense impulsive loads. Int J Num Meth Engng 1979;14:1865}71. [5] McGlaun JM, Thompson SL, Elrick MG. CTH: a three-dimensional shock wave physics code. Int J Impact Engng 1990;10:351}60. [6] Cagliostro DJ, Mandell DA, Schwalbe LA, Adams TF, Chapyak EJ. MESA 3-D calculations of armor penetration by projectiles with combined obliquity and yaw. Int. J. Impact Engng. 1990;10:81}92. [7] Trucano TG, McGlaun JM. Hypervelocity impact calculations using CTH: case studies. Int J Impact Engng 1990;10:601}13. [8] Freitas CJ, Anderson CE, Jr., Goodlin DL. Hydrodynamic ram: a review of experimental data for use in validation of numerical simulations, SwRI Report 06-7365/001, Southwest Research Institute, San Antonio, TX 78228, prepared for Wright Laboratory, Wright-Patterson AFB, OH, April, 1997. [9] Lundstrom EA, Andersen T. Hydraulic ram model for high explosive ammunition. Symposium on Shock and Wave Propagation, Fluid}Structure Interaction, and Structural Responses, ASME Pressure Vessels and Piping Conference, Honolulu, HI, 1989. [10] Hogg IV. Jane's directory of military small arms ammunition. New York, NY; Jane's Publishing Inc. 1985. [11] Zabel PH, Riegel JP, III. Characterization of close-in blast parameters for 23-mm high explosive incendiary-tracer projectile. SwRI Report 02-6291, performed for Naval Weapons Center, China Lake, CA, September 1981. [12] Czarnecki G. Air Force Research Laboratory, Wright-Patterson AFB, OH, private communication, 1999. [13] Lee EL, Hornig HC, Kury JW. Adiabatic expansion of high explosive detonation products. UCRL-50422, Lawrence Radiation Laboratory, University of California, Livermore, CA, 1968. [14] Dobratz BM. LLNL explosives handbook: properties of chemical explosives and explosive simulants. UCRL52997, Lawrence Livermore National Laboratory, Livermore, CA, 1981. [15] Johnson GR, Cook WH. A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures. Proceedings 7th international symposium on ballistics, The Hague, The Netherlands, 1983: 541-7. [16] van Leer B. Towards the ultimate conservative di!erence scheme IV. A new approach to numerical convection. J Comp Phys 1977;23:276}99. [17] Kerley GJ. On calculation of underwater explosions: a sensitivity study. KPS96-9, Kerley Publishing Services, prepared for Southwest Research Institute, San Antonio, TX, September 1996. [18] Frank AM, Gathers GR. Shock pressure determination in detonator wires. In: Schmidt SC, Johnson JN, Davison LW, editors. Shock compression of condensed matter*1989. Amsterdam: North-Holland, 1990. p. 759}62. [19] Kerley GI. Annotated bibliography, containing more than 120 references on the properties of water and ice. [20] Walker JD, Anderson Jr. CE. Multi-material velocities for mixed cells. In: Schmidt SC, Shaner JW, Samara GA, Ross M, editors. High pressure science and technology*1993, vol. 2. Woodbury, NY: AIP Press, 1994. p. 1773}7. [21] Baker WE, Westine PS, Dodge FT. Similarity methods in engineering dynamics: theory and practice of scale modeling. Rochelle Park, NJ: Hayden Book Company, 1973. [22] Freitas CJ, Anderson CE, Jr, Sharron TR. Pretest predictions of 23-mm HEI Arena Test, summary report. SwRI Letter Report, prepared for the Air Force Research Laboratory, Wright-Patterson AFB, OH, March, 1998. [23] Cole RH. Underwater explosions. Dover, New York, NY; 1965. [24] Schmidt RM, Voss ME, Kari SE, Shrader JE. Explosive loading of lightweight aircraft structures. Structures under extreme loading conditions*1995, PVP-vol. 299. New York: ASME, 1995. p. 107}16. [25] Lewin L, Kari S. Simulation of aircraft wing response to high-energy explosive projectile. ITEA J 1997;18(3):22}8. [26] Freitas CJ, Anderson CE, Jr, Walker JD. A numerical study of a hydrodynamic ram benchmark experiment. In: Structures under extreme loading conditions * 1998, PVP-vol. 361. New York, NY: ASME, July 1998. pp. 139}56.
Simulation and analysis of a 23-mm HEI projectile hydrodynamic ram experiment Charles E. Anderson, Jr.*, T.R. Sharron, James D. Walker, Christopher J. Freitas Mechanical and Materials Engineering Division, Southwest Research Institute, 6220 Culebra Road, P.O. Drawer 28510, San Antonio, TX 78228-0510, USA Received 3 September 1998; received in revised form 3 August 1999
Abstract An Eulerian wavecode was used to simulate the impact, penetration, and detonation of a 23-mm high-explosive projectile into a water-"lled tank. The pressure}time response is compared with results from an experiment conducted by Lundstrom and Andersen (Symp. on Shock and Wave Propagation, Fluid}Structure Interaction, and Structural Responses, 1989). Computed peak pressures and impulses compare well with the experimental values. 1999 Published by Elsevier Science Ltd. All rights reserved.
1. Introduction Hydrodynamic ram refers to the process whereby momentum and kinetic energy from a projectile are transferred through a #uid-"lled container to the walls of that container. Additionally, some projectiles contain high explosives and the detonation of the explosive substantially increases the total energy. The liquid e$ciently couples the kinetic and explosive energy of the projectile to the structure and greatly increases the magnitude of the damage sustained. Although considerable experimental work has been conducted to investigate and quantify damage from hydrodynamic ram, little numerical simulation work has been performed to examine the entire problem because of issues related to numerical accuracy, and the very disparate length and time scales of the problem. Projectile penetration through a fuel tank shell, travel through the #uid, detonation of the high explosive, propagation of the shock wave and loading of the structure, and subsequent structural response is a di$cult problem to simulate numerically because of disparate length and time scales.
* Corresponding author. Tel.: #1-210-522-2313; fax: #1-210-522-5122. E-mail address: [email protected] (C.E. Anderson, Jr.) 0734-743X/99/$ - see front matter 1999 Published by Elsevier Science Ltd. All rights reserved. PII: S 0 7 3 4 - 7 4 3 X ( 9 9 ) 0 0 0 4 6 - 9
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A typical high-explosive incendiary (HEI) projectile is 23}30 mm in diameter, whereas the fuel tank itself is one to two orders of magnitude larger. The projectile penetration velocity is on the order of 0.7 mm/ls; the projectile travels for approximately 200 ls prior to detonation of the high explosive. The shock resulting from detonation of the explosive travels at a velocity of 1.5}2.0 mm/ls (the detonation wave travels 7}8 mm/ls), with a pulse width on the order of 100}300 ls. For purposes of discussion, the initial shock interacts with the wall several hundred microseconds after detonation, but the resultant structural response is at least several milliseconds in duration. To achieve a numerically accurate solution then requires "ne spatial and temporal resolution to resolve all pertinent scales, and thus, the calculation of the entire problem is computationally intensive. The type of computer programs that can be used to simulate high explosive detonation, material compressibility, propagation of shocks, and nonlinear material response are typically referred to as hydrocodes. The word `hydrocodea is a contraction of `hydrodynamic computer codea, and it is a misnomer since hydrocodes rigorously calculate elastic}plastic material response including strain hardening, rate e!ects, thermal softening, and to various degrees of "delity, failure. Hydrocodes provide solutions to the three conservation equations, coupled with material response descriptions (equation of state and constitutive models) in either Lagrangian (material) or Eulerian (spatial) coordinates [1]. An explicit time integration scheme is used to advance the solution in time, with the time step interval subject to a modi"ed Courant stability criterion [1]. Numerical simulations have hardly been used to simulate hydrodynamic ram. One of the earliest examples is by Kimsey, reported in Zukas et al. [2]. Simpli"ed analytical/numerical models have been developed but these have had limited predictive capability due to shortcomings in the modeling approximations used [3]. Although there may have been studies in the intervening years, little-to-no documentation exists. There are several reasons for this observation. Kimsey used the Lagrangian "nite element code EPIC [4] for his study, and grid distortion e!ectively terminated the problem. Eulerian codes, on the other hand, are not subject to the problem of grid distortion, but prior to the 1980s, Eulerian codes were only "rst-order accurate in the advection and interface reconstruction algorithms, and were therefore very di!usive. The large disparity in length and time scales precluded the very "ne gridding required for "rst-order accurate codes to replicate hydrodynamic ram experimental results; it was not until the late 1980s that second-order advection was added to a new generation of Eulerian programs [5}7]. The objective of this study was to quantify how well state-of-the-art numerical simulations could reproduce the response of a #uid-"lled structure to hydrodynamic ram. Freitas et al. [3,8], conducted a literature survey to compile experimental data that could be useful in numerical validation. One problem from this literature survey was selected for in-depth numerical studies, the impact and subsequent detonation of a 23-mm HEI projectile into a water-"lled tank instrumented with pressure gages.
2. The experiment Lundstrom and Andersen [9] report data from experiments where HEI rounds were shot into a rectangular tank "lled with water. Both 23-mm and 30-mm HEI rounds were used in the
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Fig. 1. Fluid pressure histories from Lundstrom}Andersen experiment [9].
experimental series. The focus here is the results of the 23-mm experiment. The 23-mm round impacted the tank at a speed of 701 m/s. The round had a time delay between fuse activation at impact and detonation of the explosive. The time delay is captured in the pressure}time histories as the time di!erence between a small pressure rise due to projectile impact and the large pressure spike that results from detonation of the high explosive (see Fig. 1). The tank was constructed of reinforced 0.635-cm thick steel side walls, 91.4 cm in width. The tank was open on the top, and the water depth was 94.0 cm. The entry panel and exit panels were 0.3175-cm thick steel, and the HEI round impacted normal to the panel surface. Fluid pressure data were reported; the tank was instrumented with four Kistler 603H piezoelectric accelerationcompensated pressure gages. The gages were mounted at the end of pipes that were suspended from a shock-isolated beam. All gages were located in the vertical plane of the projectile path, and faced the entrance wall of the tank. Fig. 2 provides a schematic of gage location relative to the shotline. Gage locations, with the x-coordinate measured from the front of the tank and the y-coordinate
The experiments were incorrectly identi"ed in Ref. [9]; the data from Shot 4 are for the 23-mm experiment.
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Fig. 2. Schematic of pressure gage location. Solid circles denote pressure gages. The arrow indicates the projectile shotline.
measured from the centerline, were: Gage A (25.40, 27.90); Gage B (40.64, 17.78); Gage C (57.15, 7.62); and Gage D (71.12, !2.54). All measurements are in centimeters. The square symbol on the shotline denotes the approximate location * as estimated from the numerical simulations * of the center of the explosive-"lled cavity when detonation occurred. The pressure}time histories of the four gages are shown in Fig. 1.
3. The computational model 3.1. Description of the 23-mm HEI projectile In devising the numerical model for the 23-mm HEI projectile, we are concerned with the "delity of the model with respect to parameters important to penetration and high explosive energy release. Other details, for the purposes of this study, are not considered as important, e.g., the actual workings of the fuzing mechanism. Thus, the 23-mm HEI projectile is idealized for the speci"c numerical simulations for which it is to be used. Some information about the 23-mm projectile is provided in Ref. [10]. The projectile of interest is the ZU-23 HE/I/T, which is denoted here as the 23-mm HEI for simplicity. Jane's contains little information concerning details of the interior of the projectile. For these details, information is taken from Zabel and Riegel [11]; the drawing of the 23-mm HEI is shown in Fig. 3. Jane's reports the total mass of the projectile to be 185 g. Czarnecki measured 180.6 g for the MG-25 projectile [12]. The walls around the explosive are 3-mm thick in the model; in the actual 23-mm round, the wall thickness varies between approximately 2.75 and 4 mm. The details of the wall design, e.g., the rotating band, the bourrelet, crimping grooves, etc., are not warranted in the model since the focus is on the response of the fuel tank. So long as the overall characteristics of the projectile are modeled su$ciently accurately, the speci"c details should not matter. Dimensions and materials for the model are compared to the actual round in Table 1.
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Fig. 3. Drawing of Russian 23-mm projectile (MG 25) [11].
Table 1 Physical characteristics of the 23-mm HEI projectile
Length Diameter Wall thickness Total mass Explosive Explosive mass Casing material
Actual round
Model
100 mm 23 mm (rotating band) 2.75}4.00 mm 180}185 g RDX 13.28 g 1018 steel
100 mm 23 mm 3.00 mm 182.5 g C-4 (91% RDX) 14.4 g 1006 steel
The HE mass has been measured and found to be 13.28 g of RDX (MG-25) [12]. The JWL equation of state [13] was used to model explosive energy release and subsequent expansion of the detonation products. JWL constants exist for C-4, which is 91% RDX [14]. The explosive mass was estimated, using model dimensions, to be 13.2 g. The mass was then increased by approximately 9% to 14.4 g to account for the di!erences in energy content between 100% RDX and C-4. Johnson}Cook constitutive constants [15] exist for 1006 steel, which is in the same family as the 1018 steel. The initial #ow stress for the 1006 steel was estimated to be 350 MPa. As already discussed, the details of the fuze assembly were not modeled. Instead, the fuze assembly was designated as aluminum, but a very soft aluminum to re#ect that it has very little rigidity. The tracer cavity, while most probably containing something like phosphorous, was modeled as an inert plastic.
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3.2. The computational model The nonlinear, large deformation Eulerian wavecode CTH [5] was used to simulate the experiment. The 2-D cylindrically symmetric option of CTH was used to model the problem. CTH uses a van Leer algorithm for second-order accurate advection [16] that has been generalized to account for a non-uniform and "nite grid, and multiple materials. CTH also has an advanced material interface algorithm for the treatment of mixed cells [5]. The equation of state for water was examined by Kerley [17] and modi"ed for this study. The thermodynamic response of water was modeled with the Mie}GruK neisen equation of state (EOS). The Mie}GruK neisen EOS relates the pressure P, density o, and internal energy E by P(o, E)"P (o)#!(o)o[E!E (o)], (1) 0 0 where P (o) and E (o) are the pressure and internal energy on some reference curve, and the 0 0 GruK neisen parameter ! is assumed to depend only on the density o, and is de"ned as
1 *P !" o *E
(2)
! o !" , o
(3)
. M The dependence of the GruK neisen parameter on density is given by
where the constant ! is the GruK neisen parameter at the initial density o . The dependence of the internal energy on temperature is de"ned by an analogous expression to Eq. (1): E(o, ¹)"E (o)#C [¹!¹ (o)], (4) 0 4 0 where ¹ (o) is the temperature along the reference curve, and the speci"c heat C is assumed to be 0 4 a constant. The reference curve is taken as the Hugoniot. The relationship between the shock velocity (u ) and particle velocity (u ) is determined experimentally, and is assumed to "t a quadratic relationship u "c #s u #s u, (5) where c is the adiabatic sound speed and s and s are additional material constants. Using Eq. (5), the Hugoniot reference states can be written as: P "o c gF(g), & E "c gF(g), & F(g)"4+(1!s g)#((1!s g)!4c s g,\, E c dF(g) dg, ¹ (g)"¹ eC E# eC E e\C Eg & 2C dg 4 g"1!o /o. M
(6a) (6b) (6c) (6d) (6e)
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The expression for ¹ (g) cannot be integrated in closed form, but a polynomial expansion in terms & of g permits the integral to be evaluated; the expansion was carried out to a "fth-order polynomial in g. The six EOS constants are o , c , s and s , C , and C . For water, these were determined to be 4 o "1.0 g/cm; c "1.48 mm/ls; s "1.984; s "!0.0966 ls/mm; C "0.48; C "4.28;10 4 erg/g/eV. The Hugoniot "t was taken from Frank and Gathers [18]. Parameters C and C were 4 selected to give the best possible agreement with thermal expansion and heat capacity data [19]. Square zoning was used for all computations. In the "rst simulation, 4 zones were used to resolve the projectile radially (across the projectile radius); the zone dimension was 0.2875 cm on a side. A total of 184,869 zones and 56 Mbytes of memory were required for the simulation. In the second simulation, the grid resolution was doubled. The radius of the projectile was resolved with 8 zones; the zone dimension was 0.1438 cm on a side. Now a total of 726,660 zones and 121 Mbytes of memory were required to resolve the problem domain. These two sets of simulations will be referred to as the 4-zone simulation and the 8-zone simulation, respectively.
4. Numerical simulation of the experiment The 23-mm HEI projectile has a time-delay fuze. The projectile position necessary to give the correct arrival times of the detonation shock at the four gage locations was estimated from a preliminary simulation run. The location of the projectile (center of the HE) at the time of detonation is denoted by the square symbol in Fig. 2. The detonation delay time was then determined from the time it takes for the projectile to penetrate and transit to the correct detonation location. It is estimated that the explosive detonated approximately 200 ls after impact. The calculation proceeded from impact (t"0) for 200 ls. At 200 ls, the explosive was initiated and the result of energy release and shock propagation calculated. Pressures were monitored in the simulations at the locations of the experimental gages. Computer-generated material plots are shown in Fig. 4. The disparate length scales for the problem are evident in Fig. 4(a). A close-up view of the projectile at time t"0 ls is shown in Fig. 4(b). Fig. 4(c) shows the projectile and penetration cavity in the water just prior to explosive detonation; the `ripplinga of the steel case is an artifact of numerical inaccuracies associated with the treatment of mixed cells combined with the interface reconstruction algorithm. Fig. 4(d) shows the response approximately 400 ls after detonation (600 ls after impact). The explosive detonation products, the `detonation cavitya, and outward bulging of the entrance plate can be observed in Fig. 4(d). The pressure}time results at the four gage locations (see Fig. 2) are shown in Fig. 5. The peak pressures are slightly higher in the 8-zone simulation than for the 4-zone simulation. The largest disparity in the two calculations is the peak pressure at gage A, which di!ers by approximately 17%; the other calculated pressures di!er by less than 10%. The cause of the time delay between arrival of the explosive shocks is discussed below. The predicted peak pressures are approximately the same as the experimentally measured peak pressures for gage locations A and C, but the experimental pressures are larger than the numerical ones for gage locations B and D. The pressures and impulses (the areas under pressure-time curves) will be discussed later in the paper.
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Fig. 4. Material plots of simulation results.
The detonation times, penetration velocities, and projectile positions for the two simulations are compared in Table 2. The detonation time is the delay between impact of the projectile and explosive initiation. The projectile velocity and position are given in columns 3 and 4, respectively, at the time of explosive initiation. (The position of the nose at time t"0 ls is 0.0 cm, i.e., the nose is in contact with the tank wall.) The distances between the center of the explosive and the four gage locations are given in the last four columns of Table 2.
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989
Fig. 5. Comparison of pressure pro"les from numerical simulations.
The projectile does not decelerate as rapidly for the "ner-zoned problem. The velocity}time histories of the leading edge of the explosive cavity for the two zonal resolutions are plotted in Fig. 6. The deceleration of the projectile is approximately the same for the "rst 50 ls, but between 60 and 70 ls, the 4-zone simulation undergoes a very rapid deceleration. In contrast, the deceleration for the 8-zone simulation is much more constant. In the actual experiment, the projectile penetrates through the tank wall and into the #uid as a rigid body. In the 4-zone simulation, the cylindrical body `deformsa, and thus there is a larger decelerating force on the projectile. This is a direct consequence of averaging of material properties within mixed cells (computational cells that contain more than one material). This e!ect is a well-known problem in the application of Eulerian formulations to rigid-body penetration, e.g., see Ref. [20]. Increasing the zonal resolution decreases the percentage of mixed cells, and thus decreases their in#uence on the numerical solution. Thus, there is little mushrooming of the projectile nose, and less retarding force on the projectile, in the "ner-zoned simulation, resulting in a more realistic projectile velocity history. Since the projectile in the 8-zone simulation does not decelerate as rapidly as the 4-zone simulation per unit time, the projectile travels a greater distance per unit time. The detonation location (center of the explosive) was determined to be approximately 5.4 cm into the tank based on
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Table 2 Computational results: projectile Simulation
4-zone 8-zone
Det.time (ls)
200 175
Proj.vel. (m/s)
490 640
HE pos. (cm)
5.36 5.36
Distance to gages (cm) A
B
C
D
34.4 34.4
39.5 39.5
52.3 52.3
65.8 65.8
Center of HE at detonation. At detonation, with the distance determined from the center of the HE to the gage.
Fig. 6. Velocity versus time as a function of grid resolution.
a comparison of the 4-zone results to those of the experiment. The time of arrivals for the impact shock and the detonation wave at the four pressure gages for the 4-zone simulation are within 6% of the experimentally measured values (see Table 3). Therefore, for the projectile in the 8-zone simulation to be at the correct location for detonation, the denotation time must be 175 ls after impact. The peak pressures and the time of arrivals (TOA) are summarized in Table 3. Three TOAs are given: TOA-1 is the time of arrival of the impact shock, TOA-2 is the time of arrival of the detonation wave, and TOA-3 is the time of the peak pressure. The pressures listed in Table 3 are the peak pressures. The times of arrival for the impact shocks are within approximately 10 ls for the two simulations, as seen in Fig. 5. However, the times of arrival of the detonation waves are o!set from each other. The center of the explosive cavity for the 8-zone simulation reaches the prescribed 5.4-cm detonation location 25 ls sooner than it does in the 4-zone simulation because of the di!erences in `penetrationa velocities. The o!set between the arrival of the detonation waves shown in Fig. 5 is this 25 ls.
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Table 3 Computational and experimental results: peak pressures and times of arrival Gage position Simulation
4-zone 8-zone Experiment
A
B
TOA-1 (ls)
Press (MPa)
TOA-2 (ls)
TOA-3 (ls)
TOA-1 (ls)
Press (MPa)
TOA-2 (ls)
TOA-3 (ls)
240 248 255
33.9 40.9 41.4
385 363 348
398 371 355
282 292 300
35.7 36.1 51.7
411 389 385
423 400 390
Gage Position Simulation
4-zone 8-zone Experiment
C
D
TOA-1 (ls)
Press (MPa)
TOA-2 (ls)
TOA-3 (ls)
TOA-1 (ls)
Press (MPa)
TOA-2 (ls)
TOA-3 (ls)
370 383 390
23.7 24.7 28.3
506 569 465
506 488 472
457 470 487
17.0 18.9 32.4
574 559 560
596 575 565
TOA-1: Time of arrival of impact shock; TOA-2: Time of arrival of detonation wave; TOA-3: Time of peak pressure.
In general, the arrival times and the times of the peak pressures from the 8-zone simulation are in slightly better agreement with the experimental results than those for the 4-zone simulation. The rise time of the detonation wave shock is much steeper for the 8-zone simulation, a consequence of the "ner zoning. The fact that the arrival times of the impact shock and detonation wave agree quite well with those of the experiment implies that the detonation times and detonation location (i.e., the position of the projectile at detonation) are approximately correct. The 8-zone simulation results are compared to the experimental data in Fig. 7 . The numerical pressure traces are very similar to the experimental pressures. Several points are worth noting. The pressure pro"les have some `structurea, as compared to a simple exponential-like decay obtained from a point charge in an in"nite domain. This structure is the result of multidimensional e!ects, and is present in the simulations and experimental traces, particularly for gage locations A and B. There is less structure in the experimental traces at gage locations C and D; similarly for the numerical results. The pressure `spikesa at late times are the result of re#ections from the container walls. The re#ected waves in the simulations arrive at times di!erent than those in the experiment because the square tank (experiment) was idealized as a cylinder for the simulations. Additionally, there is a free surface in the experiment and none in the simulations, but again, this is a late time e!ect. For the main pressure pulse, the simulated pressure} time pro"les provide very good qualitative and quantitative representations of the experimental data.
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Fig. 7. Comparison of 8-zone simulation pressure}time results with experimental data.
4.1. Peak pressure versus scaled distance Lundstrom and Andersen conducted only one 23-mm test, so there is no estimate of experimental reproducibility, including test-to-test variation. However, they did conduct two experiments using a 30-mm HEI projectile. A scaling relationship is required to compare the data from the 23-mm projectile and the 30-mm projectiles. Self-similar blast (shock) waves are produced at identically scaled distances when two explosive charges of similar geometry di!er only by the explosive charge weight [21]. The scaled distance Z is given by r , Z" =
(7)
where r is the distance and = is the explosive mass. Z has dimensional units of cm/g .
Technically, = should be the total energy of the explosive, but for a given type of chemical explosive, the energy is proportional to the total mass.
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993
Fig. 8. Peak pressure versus scaled distance from detonation.
A large number of 30-mm test data exist for peak pressure versus distance [8]. A power law of the form P"aZ@ was used to "t the scaled experimental data [8]. The coe$cients a and b were determined by a least-squares regression "t: P"856.4Z\ MPa.
(8)
The scaled experimental data, the results of the numerical simulations, and the regression "t are plotted in Fig. 8. The 30-mm data are shown as open symbols, and the 23-mm data are displayed as solid symbols. The 30-mm experimental data provide an indication of the test-to-test variability. The peak pressures as a function of scaled distance for the 4- and 8-zone simulations are plotted as the solid circles and diamonds, respectively, in Fig. 8. (The abscissa values for the 4-zone and 8-zone simulations have been separated slightly so that the `dataa points are visible in the graph; the pressures have not been altered.) The dashed lines in Fig. 8 represent $5 MPa about the power law curve "t. For the most part, the simulation results lay within this `scatter banda. A simulation of a static arena test was conducted for a pre-test prediction [22]. Free-"eld pressures were calculated at 15.24 and 30.48 cm; these correspond to scaled distances of 6.44 and 12.88 cm/g , respectively. These two points are plotted in Fig. 8 as open squares (the simulation was conducted for 8 zones across the radius of the projectile). These two points are calculated for a static situation * the projectile is not moving * but they agree quite well with the other scaled data. As already mentioned, the peak pressures for the 8-zone simulation agree very well with the 23-mm projectile experimental data at gage locations A and C. The measured pressures at gage
There is some ambiguity of whether to use 13.2 g of RDX explosive or 14.4 g of C-4 explosive. However, since the explosive weight is taken to the one-third power, this uncertainty amounts to less than 3%.
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locations B and D for the 23-mm projectile are larger than those from the calculations, and larger than what would be expected from the regression "t. The measured results for the two 30-mm experiments at gage locations B and D also lay above the regression curve. The peak pressure at gage D is larger than at gage C for test 3, similar to the 23-mm test. Thus, there appears to be some geometric and/or impedance e!ect in the experiment * for these two gages * that is not being replicated in the idealized cylindrical representation of the rectangular tank. 4.2. Scaled impulse versus scaled distance The impulse * the area under the pressure}time curve * is of considerable interest since it is the impulse that provides the loading function for structural response. Analogous to the pressure, there is a similarity relationship for the impulse for similar explosive charges [21]. The scaled impulse is given by
R P(q)dq I " . (9) IM " = = The impulses for the pressure peaks were calculated for the experimental data of Ref. [9]. The impulses were calculated only for the primary pressure pro"le. The scaled impulses versus scaled distances are plotted in Fig. 9. The scaled impulses for the two nominally identical 30-mm experiments are essentially equivalent at each of the gage locations. The results for the 23-mm experiment are shown as the solid squares. Scaled impulses were also calculated for the 4- and 8-zone simulations. A distinction between the two simulations is that the pressure goes through a rarefaction phase for the 8-zone case (see Fig. 5). For the 4-zone simulation, the pressure at a gage location does not quite return to zero, a consequence of zonal resolution of the pressure "eld. Regardless, the results of the numerical simulations are nominally the same.
Fig. 9. Scaled impulse versus scaled distance.
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It is interesting to observe that the impulse at gage location B is higher than the impulse at gage A for all experiments and the simulations. However, the simulations do not capture the magnitude of the elevated value observed in the experiment. Except for gage location B, the numerical simulations are in good to excellent agreement with the experimental data. The larger impulses for the experiment at gages A and B are due, in part, to a larger amplitude spike near the end of the main pressure pulse. This spike is captured in the simulations, but it is not as large as in the experiments. The solid line in Fig. 9 is an empirically based curve from Cole [23]: 4470 IM " Z
r Z" , =
(10)
where the 4470 and 0.89 are constants for TNT. I has units of kg/(m s g ). TNT has less speci"c energy than RDX or C-4, thus, the empirical curve lays below the experimental data and the simulation results. Although it is possible in principle to adjust Eq. (10) for C-4 explosive using a TNT equivalency, such a procedure is often inaccurate. The point here is that the data and simulation results are consistent with this empirical "t to TNT data, i.e., the curve lays below results for a more energetic explosive. A conclusion can be reached concerning zonal resolution for the HEI projectile simulation. The "ne-zone simulation * 8 zones across the projectile radius * provided approximately the same answers as the more coarsely zoned problem (4 zones across the projectile radius). The biggest di!erence between the two simulations is penetration of the projectile through the #uid: penetration was retarded due to projectile deformation in the 4-zone simulation and not in the 8-zone simulation. This unrealistic deformation, as already mentioned, is a direct consequence of a numerical di!usion e!ect due to mixed cells in a poorly resolved Eulerian simulation. However, after detonation, the peak pressures at the four gage locations were approximately equal, with the largest discrepancy being at the closest gage. Presumably, the "ner zoned simulation more accurately resolved the shock front and thus the peak pressure. The 4- and 8-zone speci"c impulses were in agreement to within 6% for the "rst three gages. The larger relative error between the two simulations for gage D is probably due to the lower resolved pressure of the initial impact shock in the 4-zone simulation (see Fig. 4). Therefore, the 4-zone simulation * for a reasonable degree of accuracy * was numerically resolved for structural response calculations. We note, in making these comparisons, that square zoning was used throughout the computational grid. Square zoning minimizes numerical truncation error, and the good results obtained for the 4-zone calculation almost certainly are due in part to using square zoning.
5. Summary and conclusions The Eulerian wavecode CTH was used to investigate the impact, penetration, and detonation of a 23-mm high explosive projectile into a water-"lled tank, simulating an experiment by Lundstrom and Andersen [9]. In the experiment, pressures were monitored at four gage locations. A numerical model for the 23-mm Russian HEI projectile was developed. The total mass of the projectile model is in good agreement with the mass reported in Jane's [10], and the mass obtained from an
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independent measurement [12]. The explosive mass, based on the dimensions of the cavity provided by Zabel and Riegel [11], is estimated to be 13.2 g, also in excellent agreement with Ref. [12]. The explosive mass was increased to 14.4 g of C-4 explosive to provide the equivalent explosive energy of 13.2 g of RDX. Two simulations were conducted, with the second simulation having twice the zonal resolution of the "rst. At two gage locations, the overall pressure}time pro"les and predicted peak pressures from the simulations are in very good agreement with experiment. For whatever reason, the peak numerical pressures at the other two gage locations are lower than those observed in the experiment. The results for the "ner zoned simulation, particularly the peak pressure of gage A, are in slightly better agreement with the experimental data. However, given that the "ner zoned simulation took eight times longer to run (&40 days as compared to &5 days), the slight di!erences in the answers do not warrant the large di!erence in run times. The scaled impulses versus scaled distance for the simulations agree well with those from the experiment at two gage locations, and are consistent with an empirically based formula for TNT. At the other two gage locations, the calculated impulses are approximately 20% less that that obtained from the experiment. The impulse provides the loading function for subsequent structural response. Thus, it has been demonstrated that this "rst-principles modeling approach can be used to estimate fairly accurate the loading function for structural response calculations. We distinguish the results reported herein from work by other researchers where the emphasis is on the prediction of structural response and subsequent damage, e.g., Refs. [24,25]. In Refs. [24,25], the explosive charge was static (stationary), and in Ref. [25] the authors state that the explosive mass had to be increased to 53 g to obtain #uid pressures in agreement with test results. The current simulation results * without recourse to arti"cially increasing the explosive mass * are entirely consistent with the 23-mm experimental data and appropriately scaled 30-mm HEI data from the literature. Additional work, in which numerical pressure}time pro"les are compared to those from another set of experiments, further demonstrate that the wave pro"les are accurately reproduced [26]. Thus, it would appear that "rst-principles numerical simulations can predict the pressure and impulse loadings in a hydrodynamic ram event.
Acknowledgements This work was conducted under contract F33615-95-C-3408 with Air Force Research Laboratory (AFRL), Wright-Patterson AFB, OH. The authors wish to thank Dr. Greg Czarnecki of AFRL for his support and encouragement during this study. The authors would like to acknowledge and express their thanks for the technical support provided by D. L. Goodlin of SwRI during this study.
The problems were run on an Hewlett-Packard 735 workstation.
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