Numerical Parametric Study of Hydrodynamic Ram

A numerical parametric study was conducted to better understand the Hydrodynamic Ram (HRAM) event. The model considered a projectile penetrating a box which contains water either partially or fully. A standard model was developed and validated against the available experimental data. Then, each parameter was varied individually to determine its effect during the HRAM event. The parameters considered were the water filling level in the box, its wall thickness, projectile impact velocity, projectile mass, impact angle, and projectile shape. The effect of each individual parameter was studied, and the effects of different parameters were compared. Then, an attempt was made to predict the combined effects of multi-variables. Even though the results and discussion are for the specific geometric and material data used in this study, the present findings are expected to provide valuable insights to the qualitative characteristics of the HRAM event.


B. OBJECTIVE
The objective of this research is to understand the FSI effect on a structure filled with a fluid (water) and subjected to a high velocity impact. Understanding the dynamic response of the fluid filled structure can provide important information for future HRAM designs. It is important to understand the conditions that cause the dynamic behavior, plastic strain, and deformation in the structure as well shock wave propagation in the fluid. Ultimately, the goal of the research is to provide insight into trends that can improve the technology related to HRAM in defense..

A. DESCRIPTION OF NUMERICAL MODELING
The numerical models were developed using the commercial finite-element code LS-DYNA v.971 [16]. In order to study the HRAM phenomenon, the ALE method was adopted to model the fluid inside a structure. This section describes the computer models in detail. The models represent the nominal or standard case based on which a parametric study was conducted. When a parametric study is presented, the selected variable is discussed regarding how it is changed from the nominal value.

B. STRUCTURE FINITE ELEMENT MODEL (BOX AND PROJECTILE)
A box structure was used for the present study. The symmetry of the structure under consideration allows modeling only a quarter of the whole box as shown in Fig. 1.
Since the nature of this simulation demands a very high mesh density, such a reduction in the model size is very desirable. The box is divided into three parts, the walls impacted by the projectile (entry and exit walls), the lateral wall and the PMMA(Polymethyl Methacrylate) window as studied in Ref. [12]. The geometric dimension of the box and its wall thickness are provided in Fig. 2. The impacted walls and the PMMA window were discretized by means of eightnode solid elements with reduced integration. A refined mesh was used around the impact zone, and a progressively coarser mesh was used as the distance to the striking point became larger. The impacted walls had five elements through the thickness, and the elements in the impact zone were 1 mm in the other two directions. Based on the mesh sensitivity study, the mesh size was considered appropriate to reproduce the behavior of the solids in the impacted zone. Four-noded shell elements were chosen to discretize the other wall in order to reduce the number of elements. Finally, the mesh of the box consisted of 25,300 elements as shown in Table 1. The material properties and parameters used are provided in Table 2. The Johnson-Cook strain hardening constitutive equation [17] was selected to model the aluminum of the box as the projectile impacts and penetrates the aluminum surface. The Johnson-Cook plasticity model is expressed as where y σ , p ε , and ε , and * The projectile is a solid sphere with its diameter 12.5 mm and mass 8g. It is made of steel as shown in Table 2, and the initial impact velocity is 900 m/s. It is divided into 1000 eight node solid elements. The projectile strikes the center of the entry wall at the normal direction.

C. FLUID FINITE ELEMENT MODEL
It is expected that the fluid inside the box undergoes large motions such that the Eulerian description is selected for fluid. As a result, a multi-material ALE formulation was chosen for the treatment of the fluid. Multi-material means that each element of the mesh has the ability to contain two or more materials, in this case water and air.
The fluid inside the box is discretized by means of eight-node solid elements.
Strictly, the fluid is discretized by means of an Eulerian mesh as shown in Fig 2. Modeling the air region is essential to allow the water to flow inside the box. This is only possible if the water and air meshes share the same nodes at their interface. The fluid inside the box and the surrounding air region has 96,000 elements, as listed in Table. 3. Table 4.
Mesh of the fluids in the ALE approach.  Where, S 1 , S 2 , S 3 is the slope of the u s − u p curve , D is the intercept of the u s − u p curve, γ 0 is the Gruneisen coefficient and a is the first volume correction to γ 0 .
Water was modeled using the Gruneisen equation of state as given below: where p , and E are the pressure and internal energy per initial volume; and Here V is the relative volume. All other material properties and coefficients are provided in Table 4. The air was modeled using the linear polynomial EOS(equation of state) as below: ( ) in which the coefficients are also given in Table 4.
where, C 4 = C 5 = γ − 1 and γ is The ratio of specific heats.

D. MODEL VALIDATION
In order to validate the computer model before the parametric study, the numerical results were compared to the experimental data [12]. The numerical model is the same as the nominal model described above. The projectile displacement was compared between the two results as shown in Fig.3. They agree very well. More refined meshes would improve the numerical results. However, based on the balance between the accuracy and computational time, the present mesh was decided to be used without any further refinement. Applying Newton's 2 nd law to a projectile with drag force yields Where, v is the projectile velocity, p m is the projectile mass, D C is the drag coefficient, p A is the cross-sectional area of the projectile, and f ρ is the fluid density. Assuming the drag coefficient is constant, Eq. (4) gives the following expression for a spherical projectile velocity in which o v is the initial velocity, t is time, p ρ is the projectile mass density, and p d is the projectile diameter. One more integration results in the projectile travelling distance as below where x is the travelling distance of the projectile. The results predicted using Eq. (6) is also plotted in Fig. 3, and the prediction agrees well with the experimental data. This suggests that a representative drag coefficient may be assumed constant throughout the projectile motion with reasonable accuracy.

III. NUMERICAL RESULTS AND DISCUSSION
Several different parameters were considered in the study. One parameter was varied at a time based on the nominal model as described in the last section. The parameters were the water filling level, wall thickness of structure, projectile impact velocity, projectile mass, impact angle, and projectile shape. In the following subsections, each parameter was discussed individually in the order as stated above.

A. WATER FILLING LEVEL
The water filling level was changed in the box. They were 0%, 25%, 40%, 50%, 60%, 75%, and 100%. Figure 4 shows the projectile motion along the z-axis inside the box with different water levels. The z-axis is along the height of the box. The lower part of the z-axis has water and the upper part has air for a partially filled box. When the water level is 50%, the air-water interface inside the box is at the level at which the projectile stoke the entry face. As a result, when the water level is less than 50%, the projectile does not get wet during its travel inside the box. When the water level is 50%, the projectile is wet partially in the beginning. Then, the pressure differential between the wet and dry surfaces of the projectile quickly pushed the projectile toward the air side (i.e., the positive z-direction in  projectile to travel through the 60% or 75% full box than the 100% full box in order to arrive at the exit wall. For the 50% full box, partial wetting in the beginning applied the asymmetric drag and pressure to the projectile so that the projectile was pushed out of water resulting in a constant velocity without drag at later times as shown in Fig. 5 which compares the velocities for different water filling levels.
Examining the projectile velocity at the exit wall, there is a steep drop in the velocity for the 50% or less full box as shown in Fig. 6. However, there is no sizable drop in the projectile velocity for the exit wall with the 60% or more water full box. That is because the high water pressure already results in damage in the exit wall before the projectile reaches it. Therefore, the projectile can easily pass through the exit wall without much loss in its speed.  between the last data points and the right-side vertical line in Fig. 7 indicates the hole radius induced by the projectile. The hole size was almost identical for all cases. However, the higher water level resulted in a larger gradient of deformation of the hole.
This suggests that the shock pressure in water is greater with a higher water level.
where h is the water level in fraction so that 1 h = for the completely full box. The equation is valid 0.6 h ≥ for the data fitting, and it shows that the increment in the peak pressure max p ∆ becomes smaller as the water level h increases. Figure 9(b) shows the location of pressure measurement. Strains at the exit wall were compared for two different water filling levels, 75% and 100%. Figure 11 shows that the higher water filling level significantly increases the strain at the exit wall resulting from the higher water pressure. The strain curves showed that the high strain level is achieved before the arrival time of the projectile at the exit wall. The high pressure shock wave resulted in high strains before the projectile. The strain was computed at the location which is 50 mm away from the center of the exit wall along the horizontal axis. Figure 11. Comparison of strains near the center of the exit wall for different water filling levels

B. WALL THICKNESS
The projectile nominal velocity is 900 m/s, and the nominal wall thickness of the box is 2.5 mm. It was changed to 1.5 mm and 3.5 mm, respectively. The box is full of water. As expected, the thicker wall results in a greater loss in the projectile velocity as it penetrates through the entry wall. However, once the projectile gets into the box, its velocity varies almost the same rate regardless of the wall thickness. Figure 12 shows the  which is shown in Fig. 13. The initial steep reduction in the velocity occurred during the penetration process, and the speed reduction is greater for the a faster speed than for the slower speed. However, the duration of the steep reduction in velocity is shorter for a  Table 5 compares the velocity loss during penetration into either entry or exit wall. The absolute magnitude in the velocity loss is greater for the higher impact speed, but the percentage reduction is smaller for the higher speed. For the exit wall penetration, the fluid pressure with high impact velocity (600 m/s or higher in Table   5) yields severe damage to the exit wall before the projectile reaches the exit wall. As a result, the velocity loss in the projectile is so mall for the exit wall. However, the projectile with the initial impact velocity 300 m/s shows a much greater loss in its speed because the fluid pressure loading is not large enough to yield any significant damage to the exit wall prior to the projectile impact.    (9) in which the kinetic energy is measured as N-m.
The deformed shapes of the entry and exit walls are plotted in Fig. 14. The entry wall has a smooth deformed shape and it is in the opposite direction to the projectile movement. The magnitude of the deformation is greater for the faster impact speed.
However, there is an ultimate deformed shape of the entry wall. As the impact speed increases, the incensement in the deformed shape becomes lesser and lesser approaching the maximum shape. On the other hand, the deformed shape at the exit face showed that the opening had steeper deformation with a higher impact velocity as shown in Fig.   14 The resultant residual plastic strains are compared in Fig. 15 for different initial impact velocities. The plots indicate that as the velocity becomes 900 m/s, the exit hole has tearing along the diagonal directions. When the velocity is 1200 m/s, the diagonal tearing is very significant and the plastic strain contours are changed to a square shape from a circular shape. The fluid shock pressures are plotted in Fig. 16 for different initial impact velocities. The plot for 900 m/s is in Fig. 10(c) so that it is omitted in Fig. 16 where p and v are the peak pressure and the impact velocity, respectively. Comparing Eqs. (7) and (10) suggests that the peak pressure can be represented approximately as a quadratic function of either the projectile impact velocity or the water level in the box.

D. PROJECTILE MASS
The projectile mass was varied. The original mass was 8g, and it was changed to 4g and 16g, respectively. When the mass became heavier, it took much shorter time to penetrate the impact wall. As a result, the velocity just after the entry wall penetration is higher for the greater mass so that it takes less time for the heavier projectile to pass through the box as seen in Fig. 17. The total loss in the linear momentum as well as kinetic energy was examined for the different projectile mass. Then, the effect of the mass change was included in Eqs. (8) and (9) where m is the projectile mass, and o m is the reference mass which is 8g for the above equations. Furthermore, if the wall thickness effect is included, the total loss in the linear momentum and kinetic energy may be expressed as where h is the wall thickness in terms of mm, and o h is the reference thickness of 2.5 mm. Tables 6 and 7 compare the numerical data to the predicted results using Eqs. (13) and (14). They agree well.    The strains at the entry wall were compared for different masses. The strain was calculated at the location which is 50 mm away from the center of the entry wall along the horizontal axis. The strain increases as the mass increased, as expected. The increment in the kind-of plateau strains for the change of the projectile mass from 4g to 8g is about the same as that for the change of the projectile mass from 8g to 16g. The pressure in water is not much different for different masses even though there is a measurable difference in their velocities inside water.

E. IMPACT ANGLE
The impact angle was varied to the water tank as sketched in Fig. 18. Figure 18.

Sketch showing impact angle
The original model had the impact angle θ = 0 ο , and it was changed to 30 ο and 45 ο , respectively. The projectile velocity is plotted in Fig. 19 for different impact angles while the water tank was filled partially or fully. The impact location was still at the center of the impact face. When the water level was 40%, the projectile just after penetration did not contact with water because the water level was low. As a result, the projectile velocity did not decrease for a short while until it plunged into the water inside the box. When the impact location was dry (i.e., water level of 40%), the reduction in the projectile velocity during the penetration process was smaller as compared to other cases with greater water fills. The kinetic energy consumption during penetration was also greater for the higher impact angle.
Furthermore, the travel distance through water is longer for the higher angle impact so that the projectile could not penetrate the back face as shown in Figure 19(b). The projectile velocity became zero after stopped by the exit face. However, for the 45 ο impact, the projectile inside the low fill tank shows initial deviation from the impact line followed by a later return toward the impact line as seen in Fig.   20(b).

F. PROJECTILE SHAPE
The projectile shape is sketched in Fig 21. In this parametric study, all dimensions were fixed as shown in the figure except for the length L which was varied such that the length to the diameter ratio / L D becomes 3, 4.5, and 6, respectively. Two sub-cases were considered. The first sub-case is all three projectiles have the same mass. In other words, the density of each projectile is different such that the mass becomes 8g as before even though the shapes are not spherical. The second sub-case has the constant density. In this sub-case, the projectiles of / L D = 3, 4.5, and 6 have the masses of 8g, 11g, and 16g, respectively. As the projectiles had the same mass, the bullet shape projectiles had a smaller coefficient of drag than the spherical projectile because the diameter of the bullet shape is smaller than the spherical shape. As a result, the reduction in speed was much less for the bullet shape projectile. However, the coefficient of drag was the same for different ratios of / L D. When the density was constant, the longer projectile with the heavier mass has a smaller loss in speed. Hence, this sub-case is similar to the parametric study with the mass change as shown in Fig. 17. Comparing the plastic strain at the entry and exit walls, the longer projectile produced larger plastic strain than the shorter projectile as shown in Figure 23 when both projectiles had the same mass. When the projectile was longer and heavier, it certainly produced much greater plastic strains in both the entry and exit walls. On the other hand, the spherical shape projectile resulted in much larger plastic deformation than the bullet shape projectile even though they had the same mass. Therefore, the spherical shape projectile lost more speed during the penetration process. The effect of the projectile length on the trajectory deviation can be explained by examining the fluid pressure around the projectiles. Figure 25 shows the fluid pressure around the three bullet shape projectiles. The surface area in contact with the high fluid pressure is approximately the same for all projectiles. As a result, the shorter projectile is subjected to high pressure to most or all its surface area while the longer projectile has its partial surface area subjected to high pressure. Therefore, the shorter projectile is more prone to deviation from its intended trajectory.

IV. CONCLUSIONS
A series of parametric study was conducted for a HRAM event using a model which had an aluminum box filled with water and subjected to projectile penetration.
The study was undertaken using a numerical technique which coupled the Lagrangian and Eulerian formulations. A base model was analyzed and validated against the experimental data. Then, each parameter was varied one by one individually. Those parameters were the water filling level, wall thickness of structure, projectile impact velocity, projectile mass, impact angle, and projectile shape. The results and discussion are for the specific geometric and material data used in this paper. However, it is expected that they provide some valuable insight to the qualitative characteristics of the HRAM event. Some of the major findings from this specific parametric study are summarized below.
Both the water level inside the box and the projectile velocity influenced the pressure loading in the water, which also affected the penetration and deformation in both the entry and exit walls. The change in either the water level or the projectile velocity resulted in a comparable change in the exit wall deformation. The peak pressure can be represented approximately as a quadratic function of either the projectile impact velocity or the water level in the box.
While the 50% water level yielded larger plastic strain at the bottom side than the top side of the entry wall, all other water levels yielded almost the same plastic deformation at the top and bottom sides. The exit wall had higher plastic strains than the entry wall consistently regardless of water level, impact velocity, projectile mass and shape. This suggests that even though the impact velocity is much greater on the entry wall than the exit wall, the fluid shock pressure produces greater plastic deformation in the exit wall. If either the impact speed of the spherical projectile is 900 m/s with the water filling level 60% or the impactor velocity is greater than 500 m/s with full water, fluid shock pressure results in severe plastic deformation to the exit wall before the projectile reaches the wall. Therefore, there is a minimal loss in the projectile velocity as the projectile penetrates the exit wall.
Plastic deformation in both entry and exit walls was more significant for the spherical shape projectile than the bullet shape projectile. Therefore, the latter had a very minor loss in its speed during the penetration process of the entry wall. When the impact speed was 1200 m/s, there were cracks and plastic deformation along the diagonal directions. Therefore, the contours of the plastic strains at the exit wall became a squarelike shape for 1200 m/s while those were circular shapes for the lower speeds.
The total loss in the linear momentum of the projectile during the HRAM process (i.e. from initial impact to final exit out of the box) can be expressed using the following equation.
where all the variables were defined in the last section. Those equations stated above were obtained for the geometric and material data used in this study.
The coefficient of drag force could be assumed constant during the HRAM event.
For the spherical projectile, the coefficient was 0.6. The spherical projectile trajectory is deviated from the impact line if either the water level is not full or the initial impact is not normal to the entry wall. For the bullet shape projectiles, the shorter and lighter bullet was more prone to trajectory deviation, as expected. This could be explained by examining the fluid pressure profiles around the projectiles.
In general, the findings stated above were obtained for the models studied here.
However, they can provide insight to qualitative characteristics of the HRAM event to enhance further understanding of the process. Surely, more extensive parametric studies are needed for further investigation.