In this study, the Euler equations were solved using the Finite Volume Method (FVM) [31] . In this approach, the FVM solver solves flow field variables at each grid point for various time frames. The governing equations, such as the conservation of mass, momentum, and energy equations, were solved using the FVM approach. The equations are as follows:

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{v}{\int }\rho \text{d}V}+{\displaystyle \underset{A}{\int }\rho \left(u\cdot n\right)\text{d}A}=0$(1)

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{v}{\int }\rho u_i\text{d}V}+{\displaystyle \underset{A}{\int }\rho u_i\left(u\cdot n\right)\text{d}A}=-{\displaystyle \underset{A}{\int }p{n}_i\text{d}A}$(2)

$\frac{\text{d}}{\text{d}t}{\displaystyle \underset{v}{\int }\rho E\text{d}V}+{\displaystyle \underset{A}{\int }\rho E\left(u\cdot n\right)\text{d}A}=-{\displaystyle \underset{A}{\int }u\cdot pn\text{d}A}$(3)

where pressure, velocity, area, volume, density, and specific total energy are denoted as P, u, A, V, ρ, and E, respectively. The normal vector of each element face is represented by n. For an inviscid 2D compressible flow, we have

$\frac{\partial U}{\partial t}+\frac{\partial F\left(U\right)}{\partial x}+\frac{\partial G\left(U\right)}{\partial y}=0$(4)

For gas flow, the equation is given as follows,

$U=\left[\begin{array}{c}\rho \\ \rho u\\ \rho v\\ E\end{array}\right],F=\left[\begin{array}{c}\rho u\\ \rho u^2+p\\ \rho uv\\ \left(E+p\right)u\end{array}\right],G=\left[\begin{array}{c}\rho v\\ \rho uv\\ \rho v^2+p\\ \left(E+p\right)v\end{array}\right]$(5)

where u and v are flow velocities in the x and y directions, respectively.

Explosives were modeled using Jones, Wilkins, and Lee (JWL) equation of state. The equation is as follows:

$P=C_1\left(1-\frac{\omega }{R_1\upsilon }\right){\text{e}}^{-R_1\upsilon }+C_2\left(1-\frac{\omega }{R_2\upsilon }\right){\text{e}}^{-R_2\upsilon }+\frac{\omega E}{\upsilon }$(6)

where E and v are specific internal energy and specific volume, respectively. C₁, C₂, R₁, R₂, and ω are material constants. In this study, commercial hydrocode software is used, and values of all other constants are available in the hydrocode package [32] . Water and Air were modeled using Mie-Gruneisen (shock) and ideal gas equation of state and the equations are as follows:

$P=P_r\left({\epsilon }_v\right)+\frac{\Gamma \left(v\right)}{v}\left[e-e_r\right]$(7)

$P=\left(\gamma -1\right)\frac{\rho }{{\rho }_0}E$(8)

where Г(v) is Gruneisen gamma, which is the thermodynamic property, and γ is the specific heat ratio (γ = 1.4).

The computational domain is shown in Figure 1 . The domain is 12 m in length and 10 m in height. The domain is symmetric about the Y axes. The lower half is filled with water and the upper half is filled with air, which is separated by an interface. Three cases were analyzed in this study. Case 1: Underwater Explosion near the free surface (explosion in water); Case 2: Explosion above the free surface (explosion in the air); and Case 3: Explosion at air-water interface (explosion at the free surface). TNT is used as a cylindrical explosive with a charge weight of 10 kg. The Euler sub-grid is employed to study the flow properties. The grid is equally spaced, and the flow is allowed to pass through the material. The gauge points were fixed near the interface to monitor the pressure history. The numerical domain is modeled using a transmitting boundary condition. It reduces stress wave reflection from boundaries. The commercial software ANSYS workbench is used to model the simulations.

To validate the numerical model and perform a grid independence study, the computational domain of free field explosion is shown in Figure 2(a) . It is modeled with only water, and 10 kg of TNT is fixed at the center of the domain. The monitoring point was fixed at 1.5 and 2 m meters from the explosion. These domain dimensions and other boundary conditions remain the same as in Figure 1 . The grid independence study has been carried out for underwater explosions using five different grids of 0.3, 0.53, 0.768, 1.2, and 2.13 million cells, and each mesh corresponds to a physical length (Δx = Δy) of 20 mm, 15 mm, 12.5 mm, 10 mm, and 7.5 mm, respectively. These simulations were carried out on five different grids, as mentioned above. The flow field variables were monitored at the gauge point fixed at 1.5 m and 2 m above the explosion. The peak pressures measured at 1.2 and 2.13 million grid cells predict similar results, and the difference is negligible, as shown in Figure 3 . So, a further increase in mesh does not impact the results. Hence, considering the computational time, a physical length (Δx = Δy) of 10 mm and 1.2 million cells are used in this study.

Figure 5(a) shows the computational domain of Case 1: explosion in water. After the detonation of the charge, the spherical shock front radiates out from the center as shown in Figure 6(a) . The shock wave propagates and moves toward the free surface, as shown in Figure 6(b) . Once the initial shock front interacts with the water-air interface, the shock wave transmits into the air medium, and the compression shock wave reflects as a tensile reflected wave as shown in Figure 6(c) . At 1 atm and 15˚C, the acoustic impedance of air and water is 410 and 1.5 × 10⁶ kg∙m⁻²∙s⁻¹ respectively. These values show that the acoustic impedance of water is nearly 3500 times higher than that of air. Due to the huge difference in acoustic impedance between air and water, the transmitted shock is much weaker compared to the reflected rarefaction waves. Due to this reason, the transmitted shock is not visible in Figure 6(d) . The reflected shock wave has the same strength as the incident shock wave. Since water cannot withstand significant tension, the reflected rarefaction wave causes a sudden decrease in water pressure that results in cavitation, as shown in Figure 6(e) . As the propagation of the rarefaction wave continues, the pressures of the incident and reflected waves cancel each other, resulting in a very large cavitation zone beneath the free surface, as shown in Figure 6(e) . Further, the reflected waves interact with the gas bubble at the center of the explosion, which results in a secondary reflected shock wave as shown in Figure 6(f) .

The pressure time histories of monitoring points near the interface are depicted in Figure 7 . The monitoring points are fixed at 0.2 m, 0.4 m, and 0.6 m below the water-air interface, respectively. The results show that the incident shock front has a pressure ratio of nearly 3000 at monitoring point 1 and attenuates at points 2 and 3, respectively as shown in Figure 7 . At monitoring point 3, which is very close to the interface the incident shock strikes the interface and reflects as shown in Figure 7(c) . Because of the very large acoustic impedance difference between air and water, the pressure ratio of the reflected shock reduces abruptly to around −500. The other monitoring points also record similar variations in pressure. This large negative pressure results in a huge cavitation zone near the water-air interface for a few milliseconds. Further, the reflected wave interacts with the initial bubble, resulting in a secondary reflected shock wave, and cavitation due to these secondary reflections is not significant as shown in Figure 7(c) .

Figure 5(b) depicts the computational domain of Case 2: explosion in air. In this case, the explosion medium is air. Once the explosion ignites, the shock wave propagates in all directions as shown in Figure 8(a) . The density of air is 1.225 kg/m³, and the speed of sound propagation is approximately 340 m/s. So, the propagation of shock wave is slower in the air compared to that of water. Figure 8(b) depicts the shock wave approaching and interacting with the air-water interface. Due to the nature of water, the surface of the water almost acts like a solid wall. At time t = 0.6 ms, the shock wave impinges on the interface and attempts to transmit inside water and reflect in air, as shown in Figure 8(c) . Unlike an explosion in water, both the reflected and transmitted shock waves are compression waves in this case, so both can be seen in Figure 8(e) . The density of water is 998 kg/m³, and the speed of sound in water is nearly 1500 m/s. So once the shock interacts with the interface, it tries to reflect as a compression/shock wave. Also, it propagates inside the water as a compression/shock wave as shown in Figure 8(f) .

Figure 9 shows the pressure time histories of Case 2 Explosion in air. The monitoring points were fixed above and below the interface at a distance of 0.2 m, 0.4 m, and 0.6 m, respectively, as shown in Figure 5(b) . As the incident shock approaches the interface, the pressure ratio at monitoring point 1 is around 40 and attenuates further at the 2nd and 3rd monitoring points to 27.6 and 20.4, respectively as shown in Figure 9(b) and Figure 9(c) . Once the shock impinges on the air-water interface, the water almost acts like a solid body, so the reflected shock strength measured at monitoring point 3 is higher than that of the incident shock as shown in Figure 9(c) . The reflected shock attenuates gradually, and other monitoring points record lower values as shown in Figure 9 . In this case, the arrival of secondary shock wave reflection due to the presence of an air-water interface is shown in Figure 9(c) and it dissipates immediately due to its weak shock strength.

The monitoring points in Figure 10 recorded the pressure time histories of the transmitted shock wave in the water region. Once the incident shock interacts with the air-water interface, the monitoring points near the interface experience a considerable transmitted shock wave, as shown in Figure 10 . At monitoring point 1, the shock wave pressure ratio is around 153, and it attenuates gradually in monitoring points 2 and 3 as shown in Figure 10 . The shock wave propagates as a compression wave in the water medium, but once it crosses, it experiences a smaller negative pressure variation and the secondary shock waves as shown in Figure 10(c) . As per Classical Nucleation Theory (CNT), water has a strong, cohesive nature and can withstand a tensile stress or negative pressure of nearly 100 MPa [18] . as the negative pressure recorded at these monitoring points is very minimal and not able to withstand considerable time. Hence, the cavitation in these regions is neglected. In this case, both reflected shock and transmitted shock wave are significant. The magnitude of the transmitted shock wave is higher in water due to its acoustic impedance.

Figure 5(c) shows the computational domain of Case 3: Explosion at the interface. In this case, the explosive charge is fixed exactly at the interface. Once the explosion occurs, the shock wave approaches both air and water equally as shown in Figure 11(a) . As the explosion happens at the interface, it deforms the air-water interface and the acoustic impedance of water is very high; the shock wave that tries to enter water experiences a considerable reflection at the interface as shown in Figure 11(b) . At time t = 0.3 ms, the reflected shock from water joins the incident shock that is propagating in the air medium as shown in Figure 11(c) . Hence, initially, the incident shock wave propagation in both air and water is significant as shown in Figure 11(d) . Then the shock wave attenuates at a faster pace in air compared to water due to the nature of the respective medium as shown in Figure 11(e) . The shock front pressures for air and water are of different magnitudes due to their huge acoustic impedance differences. At time t = 1 ms, the magnitude of the shock front in the air side is weak compared to that of water as shown in Figure 11(f) .

Figure 12 shows the pressure time histories of monitoring points fixed near the interface. In this case, the monitoring point was fixed at 0.6 m, 0.8 m, and 1 m below and above the interface, respectively. Monitoring points were fixed a little far because the initial explosion has a very high magnitude compared to other gauge points. To avoid those initial effects, the monitoring points were fixed slightly away from the interface in this case. The shock wave propagates as a compression wave in the air medium, as shown in Figure 12(a) . It reaches a peak and drastically attenuates at all monitoring points, as shown in Figure 12(a) . In water, the shock wave also enters as a compression wave, as shown in Figure 12(b) . Once the shock crosses the monitoring point, it experiences secondary shock and it gradually attenuates. In this case, the monitoring points record small negative wiggles in Figure 12(b) but the strength of those negative pressure profiles is negligible.

Figure 13 shows the wave diagram of all three cases. In this analysis, the characteristics of shockwave propagation are discussed in detail. This graph shows that the region from 0 - 5 m in the x-axis is filled with water, 5 - 10 m is filled with air, and the water-air interface (free surface) was fixed at 5 m in all three cases. Whereas the explosion is fixed at 4 m (1 m below the interface) for Case 1, at 6 m (1 m above the interface) for Case 2, and at 5 m (at the air-water interface) for Case 3. Figure 13(a) depicts the wave diagram of an underwater explosion. In this case, TNT is fixed at 1 m below the interface in the water medium and explodes at time t = 0. The shock propagates in water and it approaches the water-air interface at time t = 0.5 ms. The shock wave reflects and transmits due to the effect of the interface. The difference in acoustic impedance between air and water is very high. Because of that the reflected expansion wave drops below the ambient pressure and results in cavitation and the transmitted shock is very minimal, as shown in Figure 13(a) . The strength of the reflected and incident shock wave is of equal magnitude; the reflected rarefaction wave propagates in water as does the incident shock wave, as shown in Figure 13(a) .

In Case 2, the explosion is fixed at 1 m above the interface in the air medium, as shown in Figure 13(b) . The remaining conditions are identical to those in Case 1. After the explosion, the shock wave moves toward the air-water interface (free surface). When the shock wave reaches the interface, part of it transmits into the water and the other part reflects in the air region, as shown in Figure 13(b) . Due to the cohesive nature of water, the reflected shock wave and transmitted shock wave are strong compression waves. The speed of sound in water is around 1500 m/s, so the transmitted shock wave propagates inside the water medium faster compared to that of the reflected wave in the air medium, as shown in Figure 13(b) . In Case 3, the explosion is fixed at the interface. After the explosion, the shock wave propagates in both regions as shown in Figure 13(c) . Initially, the shock wave propagated in the air at a faster pace. This could be caused by the reflected wave from the interface interacting with the incident wave in the air medium. But as time proceeds, the attenuation of shock wave in the air is higher than that of water. So, the shock gets weaker in magnitude and is not able to propagate a greater distance. Hence a significant bend in the curve was able to be noticed in the air region and it was compared against the imaginary line as shown in Figure 13(c) . The imaginary line is a shock wave propagation pattern without the effect of the interface. On the other hand, the shock wave in water experiences a significantly higher magnitude of shock wave due to its acoustic impedance. Furthermore, water has a much higher density and sound speed than air, allowing it to transmit shock wave at a much faster rate.

Figure 14 shows the comparison of shock wave attenuation in air, water, and interface explosions. The Euler number is used as the scaling parameter to compare the results of air and water. Density and speed of sound are the major parameters that differentiate air from water. To compare the results of air, water, and interface explosions, the Euler number is plotted against the distance to study the attenuation effect of the shock wave in different mediums of explosion. In all cases, once an explosion initiates from the center, the shock wave reaches its peak at each monitoring point and attenuates with respect to time. In this graph, the monitoring points were fixed at 1 m, 2 m, 3 m, and 4 m from the charge to measure the flow field variables, and the Euler number was calculated at each point. For water and air explosion, all four monitoring points are fixed at the water and air medium, respectively. In an interface explosion, four monitoring points each are fixed on both the air and water medium from the charge center.

In water explosion, the Euler number at the initial point is 154.49 followed by 83.4, 57.08, and 41.81 respectively. Meanwhile, in the case of air explosions, the Euler numbers were 10.59, 3.53, 1.56, and 1.05, respectively. The Euler number calculated at each location for water explosion (Case 1) is significantly higher than that of air explosion (Case 2). Also, the shock wave in water attenuates gradually, whereas, in the air, it reaches its minimum within a shorter time scale as shown in Figure 14 . This major difference is due to the cohesive nature of water and the acoustic impedance difference between the respective mediums. Interface explosions have both air and water sides, shock wave attenuation on the waterside (Case 3) is similar to water explosions (Case 1). The calculated Euler numbers are 89.69, 47.53, 32.99, and 25.11 respectively. The magnitude is considerably lower than in Case 1 due to the transmission of shock wave through the interface. However, the magnitude of the explosion on the air side of the interface (Case 3) is 20.07, 9.44, 5.14, and 3.00 respectively. It is slightly greater and gradually diminishes when compared to the air-side explosion (Case 2). This could be because the shock wave reflected from the water-air interface during an explosion joins the incident shock wave, resulting in a larger magnitude and slower attenuation than in an air explosion. Hence the graph shows the medium of the explosion near the interface has a significant impact on the shockwave propagation and attenuation characteristics.