COUPLED EULERIAN-LAGRANGIAN SIMULATIONS OF AN UNDER-WATER EXPLOSION NEAR A FLOATING TARGET

M.J. Smith1, J.J. Lee2

1DRDC - Atlantic Research Centre, P.O. Box 1012, Dartmouth, Nova Scotia, B2Y 3Z7, Canada; 2DRDC - Suffield Research Centre, P.O. Box 4000 Stn Main, Medicine Hat, Alberta, T1A 8K6,

Canada

ABSTRACT An experiment involving a 1.1 gram explosive charge detonated in water below a floating steel box is simulated with the coupled Chinook/LS-Dyna code using a two-way coupling scheme in which fluid pressures and structural veloci-ties are exchanged at the fluid-structure interface. The predicted expansion and collapse of the underwater explosion gas bubble is compared with high-speed photography of the experiment, and the predicted response of the box is compared to measurements obtained with digital image correlation (DIC). Simulations are performed for three dif-ferent target configurations: a rigid-fixed target with all degrees of freedom fixed; a rigid-free target, in which the rigid structure can undergo vertical rigid-body motion; and a responding target capable of rigid-body motion and deformation. The predicted gas bubble collapse and jetting behavior agrees satisfactorily with experimental observa-tions. Introducing target motion and deformation in the simulations shortens the bubble period and lessens the water jet impact loading on the target. The amplitude of the predicted response of the target plate during the shock and bubble response phases agrees well with DIC measurements of the target plate displacement, although there is some discrepancy in the timing of the peaks of the response.

INTRODUCTION The objective of the present work is to validate a coupled Eulerian-Lagrangian solution method for simulation of underwater explosions in close proximity to floating targets. In this study, “close proximity” refers to explosions that are close enough to a target that the hydrodynamic processes of the explosion are significantly affected by the target and its response behavior. For underwater explosions, this typically occurs for standoff distances less than the fully expanded diameter of the explosion gas bubble. The accuracy of the solution method in two areas is of par-ticular interest where comparisons with experimental observations can be made: (1) the time-history response at the center of the target plate; and (2) the gas bubble behavior, especially the bubble collapse and water jetting.

Compressible hydrodynamic calculations have been used to model underwater explosions since Mader first used a particle-in-cell method to simulate bubble collapse in 2D [1]. Since then, hy-drocodes for underwater explosion simulations are most commonly based on Coupled Eulerian-Lagrangian (CEL) or Arbitrary Lagrangian Eulerian (ALE) formulations [2]. However, validation against experimental measurement is still lacking in the open literature, especially with regard to gas bubble collapse behavior and water jetting. Incompressible hydrodynamic methods, original applied by Blake et al [3] to transient cavities and by Best [4] to explosion bubbles, have had considerable success in predicting gas bubble dynamics in close proximity to rigid and deforming targets. However, the underlying incompressible equations are inherently unable to predict the shock wave propagation and interaction with a target, making these methods less useful for mili-tary applications.

DRDC-RDDC-2016-N043

© Her Majesty the Queen in Right of Canada, as represented by the Minister of National Defence, 2016

Originally published: Malcolm J. Smith and Julian J. Lee, Coupled Eulerian-Lagrangian simulations of an underwater explosion near a floating target, Proceedings of the 24th International Symposium of Military Aspects of Blast and Shock, Halifax, Nova Scotia, 18–23 September 2016, Paper No: P37.

SOLUTION METHOD The CEL solution method used in this study couples Chinook, an Eulerian blast code developed by Lloyd’s Applied Technology Group [5], with LS-Dyna, an explicit finite element analysis code developed by Livermore Software Technology Corp. [6]. The coupling takes place at the fluid-structure interface (FSI), defined as the intersection of one or more LS-Dyna parts (groups of shell elements) with the fluid grid. The matrix of possible coupling options is given in Table 1. Two-way coupling is necessary in UNDEX simulations since the velocity response of the target strongly influences the pressure field in the water adjacent to the FSI to the extent of producing regions of cavitation. In small deformation coupling, Chinook calculations are based on the orig-inal position of the FSI regardless of target deformations or rigid-body motions. In large-deformation coupling method the position of the FSI is updated at set intervals throughout the simulation according to deformations and rigid-body motions. Further details on the coupling methods can be found in Gregson [7] and Alexander [8].

Table 1: Chinook/LS-Dyna coupling options

One-way coupling Two-way coupling Small Deformation pressure pressure, velocity Large Deformation pressure, displacement pressure, displacement, velocity

A two-stage solution methodology is used for reasons of computational efficiency. The first stage is a short-duration simulation of the shock wave propagation and initial gas bubble expansion from a time immediately after detonation until just before the shock wave reaches the target plate. This stage does not require the presence of the target and can therefore be performed with a high-ly refined grid in 1D or 2D, which limits the dispersion of the shock wave. In the simulations pre-sented here, the initial stage is a 2D axisymmetric simulation initiated with a small volume of high-pressure explosion products of identical volume and mass as the undetonated explosive. The second stage is a 3D coupled analysis initiated by mapping of the field variables from the first stage on to a 3D grid. However, in cases where the target is axisymmetric, a 2D coupled analysis may be used throughout the entire simulation without the necessity of mapping. For the present study, the two stage approach is used in which 3D coupled simulations are performed using one-quarter models of the target and the surrounding fluid volume.

Coupled simulations are performed by simultaneously executing a modified version of LS-Dyna version 971 and Chinook version 211. Coupling is invoked on the LS-Dyna side through the *USER_LOADING keyword; and on the Chinook side using the *FSI keyword. Data is ex-changed at a user-defined coupling interval which can be varied throughout the duration of the simulation. For the simulations performed for this study, a constant coupling interval of 2 μs is used for all simulations.

EXPERIMENT A series of underwater explosion experiments were conducted in the 50-m UNDEX pond at DRDC – Suffield Research Centre (SRC) in 2014 involving a floating steel box target made from 4.76 mm thick A569 steel with a length, breadth and depth of 1 m, 0.5 m, and 0.3 m, respectively [9]. The box floated in freshwater with a draught of approximately 0.1 m. The experiment mod-elled in the presented study was a Reynolds RP-83 detonator containing 1.1 g of RDX submerged

directly below the centre of the bottom plate at a distance of 0.179 m, approximately equivalent to the radius of the explosion gas bubble in the free field. The response of the bottom plate was measured from the surface using a digital image correlation system (Fig. 1a). This consisted of: a randomized speckle pattern applied to upper surface of the bottom plate (Fig. 1b); two Phantom 7.3 high-speed video cameras; spotlights mounted on a rigid bar and suspended from supports on a floating platform; and the ARAMIS data processing system developed by GOM. Another Phan-tom 7.3 high-speed camera was submerged to photograph the underwater explosion at a distance. Two PCB Piezotronics 138A01 pressure gauges were suspended from the corners of one of the shorter edges of the box to a depth of 1 m and 1.5 m, respectively. A Neptune TR-2-5000 surface gauge was positioned on the underside of the bottom plate of the box, 250 mm from the centre along the long axis of the plate. The entire experiment was conducted at the UNDEX pond at SRC, which has a diameter of 50 m and a depth of approximately 8 m.

(a) (b)

Figure 1 a) Digital image correlation arrangement using high-speed video cameras; b) Floating steel box deployed in the water.

MODELS Fluid models

The first stage (free-field propagation) of the simulation was performed with a 2D axisymmetric model in which the 1.1 g detonator is represented as a cylinder of detonated explosion products with a radius of 3.375 mm and length of 18.6 mm. The cylinder is oriented with its axis of revo-lution on the y-axis of symmetry of the model, as shown in Fig. 2. The explosion products are modelled using the Jones-Wilkins-Lee (JWL) equation of state [10-12], in which pressure P is related to the specific volume v by the following relation:

𝑃𝑃 = 𝐴𝐴 �1 − 𝜔𝜔𝑅𝑅1𝑣𝑣

� 𝑒𝑒−𝑅𝑅1𝑣𝑣 + 𝐵𝐵 �1 − 𝜔𝜔𝑅𝑅2𝑣𝑣

� 𝑒𝑒−𝑅𝑅2𝑣𝑣 + 𝜔𝜔𝜌𝜌0𝐸𝐸𝑣𝑣

(1)

with the parameters given in Table 2. The surrounding water is modelled with the Tait equation of state [13], in which P is related to density 𝜌𝜌 by

𝑃𝑃 = 𝑃𝑃0 + 𝐵𝐵((𝜌𝜌/𝜌𝜌0)𝑛𝑛 − 1) (2)

with the parameters defined in Table 2. The field pressures are initialized to hydrostatic pressure, with atmospheric pressure of 101.3 kPa at 0.3 m above the centre of the charge, which corre-

sponds to the approximate charge depth of the experiment. The grid cells in the 2D range from 0.5 mm within the footprint of the charge, increasing to 1 mm at a distance of 0.25 m from the detonation point. The 2D simulation has a total duration of 0.095 ms, at which time the shock front is slightly less than 0.179 m from the point of detonation. The field variables are saved to restart files at set intervals during the simulation.

Table 2: Equation of state and material parameters

Tait equation (water) JWL (RDX explosive) Target material (A569 steel) ρ0 1000.0 kg/m3 ρ 1659 kg/m3 E 207 GPa P0 101,325 Pa A 495.1 GPa 𝜌𝜌𝑠𝑠 7850 kg/m3 B 3000 atm B 7.21 GPa 𝜎𝜎y0 425 MPa n 7.14 R1 4.387 C 40.0 m 5.00 R2 0.9954 p 5.0 αc 0.05 ω 0.3469 E0 354 kJ/kg E 5877.9 kJ/kg

Figure 2: Axisymmetric 2D fluid grid showing refined region near the origin (left), and grid detail near the charge (right).

The second stage (fluid-structure interaction) of the simulation was performed with a 3D quarter model of the fluid domain, initialized with the field variables of the 2D simulation at 0.095 ms after detonation. The 3D grid shown in Fig. 3 is conical with a radius of 40 m and a maximum water depth 7.3 m. The grid cell sizes range from 2.2 mm, near the point of detonation to 2 m near the outer boundaries of the model. Within the volume of the expanded gas bubble, the cell size varies from 2.2 mm up to 6 mm. Next to the face of the bottom plate of the target, the cell size ranges from 6 mm to 12 mm. The water depth at the centre is 7.3 m. The upper layer of the grid (between z = 0.3 m and z = 2.5) is filled with air, as is the inside of the target box structure, described below. Because of the mixed-cell formulation used in Chinook, the boundaries between different materials are generally not cleanly defined. Instead, transition regions containing cells with more than one material (e.g., air and water, or water and explosion products) cause the

boundaries between materials to be fuzzy. Although the transition regions can be avoided through the use of a material-interface tracker at the free surface and the bubble water boundary, this comes with a prohibitive computational penalty and is therefore not employed in the present study.

Figure 3: 3D fluid quarter-model of test pond (top), with detail view of mesh near target struc-ture (lower left), and gas bubble at time of mapping (0.095 ms). Water is coloured red; air or gas is blue.

For all of the grids developed, perfectly reflecting boundary conditions are assumed for the flat and sloping bottom of the pond, the outermost radial boundary, and the upper surface of the air layer. There is little possibility of shock wave reflections from the pond bottom influencing the simulations because of the pond geometry and dimensions, and the dispersion of the shock through progressively larger grid cells near the boundary. The xz and yz coordinate planes in the

3D grids are modelled as reflecting symmetry planes. All fluid grids were developed using the Pointwise software, version 17.3 R2C4 [14].

Structural models

Structural models used in the 3D simulations are quarter-models of the box structure, shown in Fig. 4, built from regular meshes of LS-Dyna’s Hughes-Liu 4-node shell elements with an ele-ment edge length of 7 mm and a thickness of 4.76 mm. Three different versions of the box model were developed, all of which are identical except for the following varying levels of rigidity and restraint: a rigid-fixed model in which all nodes of the box are fixed; a rigid-free model in which the box is unrestrained but in which the elements are rigid (LS-Dyna material model *MAT_RIGID); and a responding model, in which the box is completely unrestrained. For the latter, the *MAT_PLASTIC_KINEMATIC material model is used in which

𝜎𝜎y = 𝜎𝜎y0 �1 + ��̇�𝜀𝐶𝐶�1/𝑝𝑝

� (3)

where 𝜎𝜎y0 is the static yield stress of the material, and C and p are the Cowper-Symonds coeffi-cients, with the material properties listed in Table 2.

Figure 4: LS-Dyna mesh of one-quarter of steel box target structure.

RESULTS The simulations listed in Table 3 were performed using the two stage simulation methodology described above. All simulations use the identical 2D and 3D fluid grids for the first and second stage simulation, and differ only in the restraint and rigidity of the target structure. Simulations were performed with small deformation two-way coupling only, as large deformation coupling simulations terminated after only about 3 ms. The small deformation coupling simulation with the responding target terminated prematurely at 40 ms, whereas the other two simulations ran successfully to the termination time of 65 ms.

Table 3: Summary of simulations

Structural model

Coupling method Duration

Rigid-fixed Two way, small deformation 65 ms Rigid-free Two way, small deformation 65 ms Responding Two way, small deformation 40 ms

Rigid-fixed model

This is a two-way coupled simulation in which no target motion occurs. The sequence of images in Fig. 5 shows the evolution of the gas bubble from the time of maximum expansion at approxi-mately 16.095 ms to just after the water jet impact at 34.595 ms. Clearly visible is the impinge-ment of the collapsing bubble from below by the water jet, and the change in the bubble topology from spheroidal to toroidal. Fig. 6 shows the time-histories of the pressure and impulse predicted at the centre of the plate showing the first bubble collapse at 33.15 ms and the second bubble col-lapse at 60.26 ms. The first and second bubble collapses are associated with changes in impulse of about 10 kPa-s and 3 kPa-s, respectively.

Figure 5: Density contour plots showing bubble collapse and jetting, rigid-fixed case.

Figure 6: Time histories of pressure and impulse at the centre of the target plate, rigid-fixed case.

Rigid-free model

This is a two-way coupled simulation in which vertical rigid body motion of the target can occur but no elastic or plastic deformation occurs. The sequence of images in Fig. 7 shows the evolu-tion of the gas bubble from the time of maximum expansion at approximately 14.595 ms to just after the bubble collapse at 30.095 ms. The water jet formation is now less visible, and the bub-ble-collapse period has shortened. The latter is also visible in the pressure time history shown in Fig. 8, which shows the first and second pulses occurring at 29.38 and 53.09 ms, respectively. It also shows that the pressure and change in impulse associated with the second pulse is now larger than that of the first pulse, while the overall change in impulse is only about half that of the rigid-fixed case. The rigid body displacement of the model is shown in Fig. 9, showing clearly defined phases of displacement due to the shock (first peak), the first bubble pulse (second peak), and the second bubble pulse (third peak).

Responding model

This simulation represents the experimentally tested case. Fig. 10 shows a side-by-side compari-son of the gas bubble in the experiment and the simulation. Considering first the sequence of photographs from the experiment, the bubble has a slightly irregular shape at 1.00 ms as it begins to form, and some bulk cavitation from shock rarefaction is visible as a milky region at the sur-face. At approximately 17.00 ms, the bubble has reaches a maximum size, and the surface insta-bilities have died down giving it a smooth surface. At 26.25 ms the bubble is in the middle of its contraction phase as it moves towards the bottom of the box. At 30.00 ms a strong bubble jet has formed and impinges on the box. At 32.50 ms, the bubble has fully collapsed against the box surface. Pressure measurements from free-field gauges indicate that the peak of the acoustic pulse generated by the bubble collapse occurs at 30.5 ms.

Figure 9: Vertical displacement of the target structure, rigid-free case.

The contour plots of the simulation show that at 1.1 ms, the bubble is very smooth and spherical in shape. At 17.1 ms, the bubble is already past its point of maximum expansion, and is begin-ning to contract, with considerable mixing on the upper surface of the bubble. At 26.1 ms, the bubble has contracted considerably and lost much of its spherical shape. At 30.1 ms, the bubble is past the point of maximum contract, and the water jet has formed but not quite reached the target. At 32.6 ms, the remnants of the bubble are re-expanding with an elongated shape. As in the rigid-free case, the bubble jetting is not as apparent as in the rigid-fixed simulation. But the simulation contours generally agree well with the experiment photographs. The pressure and impulse time histories at the centre of the target plate (Fig. 11) confirm the first bubble collapse occurring at 29.13 ms, about 1.4 ms before the measured peak of the acoustic pulse.

Between the shock pulse and the bubble collapse, the pressure is predominantly at the cutoff pressure indicating cavitation induced by the coupling of the velocity of the bottom plate to the adjacent fluid velocity. This can also be seen immediately below the target plate in the contour plots. The large spike at 27.12 ms is due to closure of this cavitation likely induced by the rising pressure of the collapsing bubble. The change in impulse due to the first bubble collapse is about one-third of what is predicted in the rigid-free case and less than one-quarter of what is predicted in the rigid-fixed case. This is a clear indication of the relative weakness of the bubble pulse when the target is allowed to move and deform.

The predicted response at the centre of the target plate is compared to the response measured by DIC in Fig. 12. The agreement is quite close during the initial shock portion of the response, and the simulation accurately predicts the maximum displacement of about 17 mm. The amplitude of the response to the bubble collapse is also accurately predicted, as is the amplitude of the reversal immediately after bubble collapse. However, the timing of the peak is off by 3-4 ms, a signifi-cantly longer discrepancy than the 1.4 ms difference between predicted and measured bubble col-lapse times. This appears to be at least partly due to the slower rise time in the measured response of the plate. The reasons for this difference need to be investigated further.

Table 4: Characteristics of the first bubble collapse with different target plates

Target model Peak Pressure (MPa)

Δ Impulse (kPa-s)

Bubble period (ms)

Rigid-fixed 38.45 9.90 33.15 Rigid-free 30.39 3.17 29.38 Responding 9.42 2.16 29.13

Similarly, Table 4 shows that the bubble period is also highly sensitive to target motion and de-formation. The rigid-body may be generating a more expansive flow field that affects the pres-sure distribution around the bubble during the collapse phase, which can be an important factor in the bubble jet formation. Because the velocity coupling at the FSI lowers the pressure immediate-ly below the target and generates regions of cavitation observed in both simulations and experi-ments, it is conjectured that the shortening of the bubble period is similar to that seen for bubbles near free surfaces. Following Snay [15], the period of the first bubble pulse near a free surface is given by,

𝑇𝑇1 = 𝐾𝐾5𝑊𝑊1/3

𝑍𝑍15/6 �1 − α𝐴𝐴1𝑑𝑑1� (4)

where 𝑊𝑊 is the charge mass, 𝑍𝑍1 is the hydrostatic head, 𝑑𝑑1 is the average depth, 𝐴𝐴1 is the bubble radius at maximum expansion, 𝐾𝐾5 is an empirical constant, and α is the surface correction param-eter. Using α = 0.2, as recommended in [15], 𝑑𝑑1= 0.3 m, and 𝐾𝐾5 = 2.232 for the RDX charge (SI units), the bubble period is 28.96 ms with the free surface correction, and 32.84 ms without the free surface correction (i.e., the normal free-field bubble collapse period). These values agree well with the bubble periods predicted for the responding and rigid-fixed target models, respec-tively, suggesting that the cavitation next to a target surface produces something similar to a free surface effect. However, the bubble collapse for a responding target in this case is upwards, to-ward the target; whereas studies of explosion bubbles near a free surface show that jetting is downward [16,17]. The measured bubble period of 30.5 ms is also significantly less than the the-oretical free-field value, indicating that the bubble period shortening is supported by the experi-mental results but is not as pronounced as in the Chinook predictions.

CONCLUSION The simulations performed with coupled Chinook/LS-Dyna capture the main features of the bub-ble collapse and jetting observed in the floating box experiment. The predicted bubble periods are found to be quite sensitive to the target motion and deformation, and the pressure and impulse loads on a responding target are found to be less than one-quarter of that seen on an identical rig-id-fixed target. The predicted displacement response of the responding target plate agrees well with measurements with digital image correlation. These predictions used the small-displacement coupling method, in which the original target position is used to determine the loading on the tar-get. The good agreement suggests that the large-displacement coupling method is not needed for standoff distances of one bubble radius or greater.

The premature failures of some of the simulations attempted for this study, including the re-sponding model simulation which failed near 40 ms, were caused by numerical instability in the iterative algorithm used to compute the material density fractions in the mixed material cells.

Plans have been made to replace this algorithm with one better suited to underwater explosion problems.

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