Numerical simulation of water jet load induced by spherical bubble collapse under underwater explosion

In this study, a numerical method combining the HLLC solver and FV-WENO scheme is proposed to accurately simulate the behavior of multiphase compressible fluids and investigate the formation process of water jets. The HLLC solver approximates the conservation equations of the fluid by using numerical fluxes from the approximate Riemann problem, while the FV-WENO scheme improves the accuracy of the numerical solution by employing the weighted essentially non-oscillatory (WENO) reconstruction method. The combination of these methods allows for a more accurate simulation of the behavior of multiphase compressible fluids, leading to an accurate depiction of the formation process of water jets. By simulating the interaction between underwater shock waves from explosions and single or multiple spherical bubbles, we study the generation, development, and collapse of water jets during the bubble collapse process. By revealing the characteristics of the load during the formation of water jets and the interaction mechanisms between bubbles, our study provides support for further research on water jet phenomena. The formation process of water jets is of great significance for understanding the propagation of underwater shock waves and structural damage. Through this study, we delve into the mechanism of water jet formation, providing a new perspective on the propagation of underwater shock waves and structural damage. Our research results demonstrate that the application of the HLLC solver and FVWENO scheme in water jet simulation effectively improves the accuracy and precision of the numerical solution.


Introduction
The phenomenon of bubble formation is widely observed in various fields such as the motion of underwater propellers, bubble curtains, shockwave lithotripsy, and wastewater treatment.The study of bubble dynamics has always been a subject of considerable interest.Since Rayleigh [1] initiated the study of spherical bubbles, a large number of scholars have conducted extensive research on bubble theory, experiments, and numerical simulations.One of the significant characteristics of underwater bubbles under transient loads such as shock waves is the formation of water jets during the collapse process.These high-speed water jets can cause substantial damage to underwater structures.The investigation of the collapse phenomenon of underwater bubbles directly through experimental or theoretical methods poses significant challenges.However, numerical simulation methods can approximate the process of bubble collapse to a certain extent, thereby garnering widespread attention.The numerical simulation of bubble dynamics provides an effective approach for understanding the complex processes involved in bubble formation and collapse.This method allows for the approximation of the various stages of bubble dynamics, including the initial formation, growth, and eventual collapse of the bubble.By using numerical simulations, researchers can analyze the effects of different parameters on bubble dynamics, such as pressure, temperature, and fluid properties.The application of numerical simulation methods in the study of bubble dynamics has led to significant advancements in various fields.For instance, in the field of underwater propeller motion, understanding bubble dynamics can help improve the efficiency and performance of propellers.Similarly, in the field of wastewater treatment, knowledge of bubble dynamics can assist in the development of more effective treatment processes.Despite the challenges associated with the study of bubble dynamics, the continued interest and research in this field highlight its importance and potential for further advancements.As numerical simulation methods continue to evolve and improve, they will undoubtedly play an increasingly important role in the study of bubble dynamics.
Currently, the most mature method for the simulation of bubble motion both domestically and internationally is the boundary element method, as demonstrated by the works of Blake [2,3] , Best [4,5] , Wang [6] , and Zhang [7] .The primary advantage of the boundary element method is its lower computational load, and it provides high accuracy in simulating the micro-amplitude and slow motion of bubbles.However, when transient effects such as shock waves and vortices appear in the flow field, the potential flow theory assumption model in the boundary element method fails.
In order to accurately capture the complex motion of the bubble interface during underwater motion and the propagation of shock waves in the flow field, it is necessary to establish a compressible fluid model and be able to analyze and deal with the gas-liquid two-phase flow that occurs in the flow field.Therefore, for the simulation of the bubble collapse phenomenon underwater with strong shock waves such as shock waves, it is essential to establish a multiphase compressible fluid model for analysis.In the literature [8,9] , a two-phase compressible fluid model based on the γ interface capture has been established.This model is discretely solved by using the fifth-order WENO reconstruction and the HLLC approximate Riemann solver to solve the control equations.This establishes a complete solution method.This method has proven to be effective in capturing the complex dynamics of bubble motion and the propagation of shock waves in underwater environments.Moreover, it provides a more comprehensive and accurate approach to modeling and simulating multiphase compressible fluid dynamics, particularly in scenarios involving strong shock waves Therefore, this study utilized the aforementioned method to conduct a numerical simulation of the water jet phenomenon induced by the collapse of a spherical bubble under the action of an underwater explosion shock wave.An initial introduction to the numerical method was given, followed by the utilization of this method to simulate the shock wave stage load in the near field of an underwater explosion and compare it with experimental results.On this basis, the processes of a single bubble and double bubbles being impacted by the underwater explosion shock wave were simulated.The generation, development, and collapse of the water jet caused by the bubble collapse were introduced, and the load curve of the water jet generated during the bubble water jet collapse process was obtained.The results of this study can provide support for in-depth research on water jets.To elaborate, understanding the behavior of water jets induced by bubble collapse under an underwater explosion shock wave is crucial in various fields。

Numerical model
For the motion of multiphase fluid, assuming each phase is inviscid and compressible, neglecting surface tension and phase transitions between different phases, the governing equations of the fluid can be unifiedly represented as follows [10] : Where ρ represents the density of the fluid medium, p represents the pressure in the flow field, u, v represent the components of fluid velocity in x and y directions.E is the total energy per unit volume, ( ) The stiffened gas equation of state has been widely used to simulate multiphase flow problems involving shock wave propagation.The interface position of the multiphase flow is captured by different stiffened parameters within the fluid cells.The stiffened gas equation of state is defined as follows [11] : ( ) γ π = are two phase stiffness parameters.For ideal gases, =1.4,=0 γ π ; for water media, =5.5, =4.92 8 e Pa γ π . Since there is convective motion at the interface of multiphase flow, the fluid stiffness parameters a and b must satisfy the following convective equation [9][10] 0 The equation ( 1) is combined with equations (3) (4) to form the governing equations for compressible multiphase flow.Equation ( 1) is a conservation equation that ensures the conservation of important physical quantities such as density, momentum, and energy in the flow field.Equation (4), on the other hand, is a non-conservation equation primarily used to determine the position of the interface between the two phases.

Discrete method
Consider a two-dimensional system of conservation equations.

( ) ( )
The main steps of the 5th-order WENO reconstruction and HLLC approximate Riemann solver are as follows: ① Compute the integral averages , i j q of the two-dimensional grid.
② Perform a 5th-order WENO reconstruction along the y-direction to obtain high-order approximations of the integral averages at the points located above and below , +1/ 2 i j q ± .This reconstruction is based on two templates, one biased to the left and the other biased to the right.
③ Similarly, perform a 5th-order WENO reconstruction along the x-direction to obtain high-order approximations of the integral averages +1/ 2, i j q ± at the points located to the left and right.This reconstruction is also based on two templates, one biased to the left and the other biased to the right.④ Use the reconstructed values obtained in the previous steps as input for the HLLC approximate Riemann solver to calculate the numerical flux.

Preliminary verification of shock wave load of underwater explosion
For the simulation of underwater explosions, the detonation process of the explosive is ignored, and the initial explosive charge is directly approximated as an instantaneous expansion of highly compressed, homogeneous mass gas [14] .This simplifies the simulation of the early shock wave stage of underwater explosions into a typical gas-liquid two-phase fluid discontinuity problem for solution.
The following simulation depicts the shock wave load at a distance of 0.6m from the explosive charge when 1g of TNT is detonated underwater at a depth of 1m.The initial radius of the explosive charge is 5.27mm.The axisymmetric model used is shown in Fig. 1, where the left side represents a solid wall condition and the other boundaries are transparent boundaries.The x-axis represents the axial direction, and the r-axis represents the radial direction.
The initial parameters of flow field are: To compare with experimental results, underwater explosion tests were conducted using detonators with 1g of TNT charge.A steel plate was placed 0.6m below the explosive charge, and the pressure signals in the flow field were recorded using PVDF pressure sensors.Fig. 2 shows a comparison of the pressure-time curves at the center point of the wall.From Figure 2, it can be observed that the numerical simulation results are close to the experimental measurements in terms of peak pressure, duration, and other parameters.The slight differences that exist are mainly due to the ideal gas assumption for the explosive charge and the neglect of the detonation process, which introduce certain errors in the propagation of pressure in the flow field.However, under near-field explosion conditions, these approximations can be made.Similar methods were employed for the analysis of the underwater shock wave and the interaction process with a spherical bubble caused by the explosive charge.

Simulation of water jet induced by single bubble under explosive shock wave loading
The schematic diagram of the interaction between an explosive shock wave and a single spherical bubble is shown in Figure 3.The initial radius of the explosive bubble is r0 = 0.0527 m, corresponding to a 1 kg TNT charge detonated underwater at a depth of 5m.The spherical bubble is located at a horizontal distance of L = 0.The evolution process of the interaction between the shock wave and the spherical bubble within the initial 2.59 ms is shown in Figure 4.The spherical shock wave generated by the underwater explosive encounters the spherical bubble during propagation.Due to the inconsistency between the density distribution gradient near the bubble interface and the direction of pressure propagation, according to the vorticity theorem, counterclockwise vorticity will be generated on the upper half of the bubble on the left side, while clockwise vorticity will be generated on the lower half of the bubble on the left side, causing the spherical bubble to start inwardly collapsing.At 0.12 ms, the explosive shock wave is approaching the bubble, and at 0.34ms, the explosive shock wave has passed through half of the bubble volume.During this process, the shock wave begins to compress the bubble, causing the bubble to gradually collapse and intensify the degree of collapse.At 1.13 ms, the shock wave has completely passed through the bubble and continues to propagate to the right.At this time, the volume on the left side of the bubble has almost disappeared, and the bubble collapses violently towards the right side.At this moment, it can be observed from Figure 4 that the flow field on the left half of the bubble is still in a high-pressure region, which will continue to compress the bubble until a water jet moving to the right is formed.At 1.57 ms, the water jet formed on the left side of the bubble has completely penetrated the right bubble wall, causing the complete collapse of the bubble and the generation of vortices, and a local high pressure is formed near the impact point of the water jet and the bubble.At 1.78ms, due to the continued movement of the vortices, the bubble reaches its minimum volume, and the previously generated local pressure zone from the water jet begins to radiate to the surroundings.From 1.85ms to 2.59ms, with the continuous movement of the vortices, the bubble starts to undergo further fragmentation, fusion, and expansion.During the expansion process, it can be observed that high-pressure regions in the flow field begin to appear in the right fluid area of the bubble.From the figure, it can be seen that the spherical bubble undergoes the typical process of bubble collapse, collapse, and expansion.Combined with Figure 4, it can be observed that the minimum volume of the bubble does not occur at the moment when the bubble starts to collapse at t =1.57ms, but approximately appears at the moment between 1.75ms and 1.8ms.This is due to the occurrence of vortices and the inertial effects of the fluid.
Figure 5(b) shows the pressure-time curve at the selected pressure measurement point at the moment when the bubble starts to collapse in Figure 4.The analysis is conducted under different conditions with and without the presence of the bubble.From Figure 5(b), it can be seen that when there is no bubble on the right side of the flow field, the pressure at the measurement point reaches a peak value of 58MPa at t = 0.4 ms, which is the pressure when the shock wave propagates freely in water to the measurement point.When a spherical bubble with a radius of 0.2 m exists, due to the obstruction of the shock wave propagation and the appearance of water jets, the peak pressure value of 130 MPa is reached at t = 1.7 ms, which is about 124% higher than the peak pressure of 58 MPa without the bubble, and the duration of the pressure also increases significantly.If there is a structure at the measurement point, and both the peak pressure and impulse on the structure increase significantly compared to the case without the bubble, the destructive effect on the structure will be more pronounced.

Simulation of water jet induced by double bubbles under explosive shock wave loading
The schematic diagram of the interaction between explosion shock wave and double bubbles is shown in fig.6.The initial parameters of TNT, spherical bubble and water are the same as above, in which the geometric shape and state parameters of two bubbles (r1 = r2 = 0.2m) are exactly the same, and the spacing is set as 0 1 /r =2.5 D 。 Figure 6.Schematic of double bubble model Figure 7 shows the evolution of the interaction between explosion shock wave and double bubbles.As can be seen from figure 7, the shock wave generated by the underwater explosion first acts on the first bubble to cause it to collapse (0.34~1.78 ms), and the high-pressure zone produced by the induced water jet continues to compress the second bubble (2.08~2.78ms), resulting in the collapse of the second bubble around 2.78 ms.
Figure 8 shows the equivalent radius of two bubbles and the pressure time history curve at the impact point of the water jet, respectively.It can be seen from figure 8 (a) that the collapse time and the minimum equivalent radius of the first bubble are basically the same as those in figure 5 (a), indicating that the existence of the second bubble has little effect on the deformation process of the first bubble.The second bubble collapses at about 2.8ms time, and the minimum volume is larger than the minimum volume of the first bubble.
It can be seen from Fig. 8 (b) that the first bubble reaches the peak pressure 105MPa at the time of 1.65 ms, which is about 19% lower than the 130 MPa peak pressure in figure 5 (b), indicating that the presence of the second bubble weakens the intensity of the water jet formed by the collapse of the first bubble.The second bubble reaches the peak pressure 92 MPa at 2.9 ms, which is about 12% lower than the peak pressure 105 MPa produced by the first bubble water jet.Although the peak pressure produced by the two water jets has not reached the yield limit of the material, the loading time is basically maintained above 0.5ms, and the damage to the underwater structure caused by the impulse cannot be ignored.indicating that the existence of the second bubble has little effect on the deformation process of the first bubble.The second bubble collapses at about 2.8 ms time, and the minimum volume is larger than the minimum volume of the first bubble.It can be seen from fig. 8 (b) that the first bubble reaches the peak pressure 105 MPa at the time of 1.65 ms, which is about 19% lower than the 130 MPa peak pressure in figure 5 (b), indicating that the presence of the second bubble weakens the intensity of the water jet formed by the collapse of the first bubble.The second bubble reaches the peak pressure 92 MPa at 2.9 ms, which is about 12% lower than the peak pressure 105 MPa produced by the first bubble water jet.Although the peak pressure produced by the two water jets has not reached the yield limit of the material, the loading time is basically maintained above 0.5 ms, and the damage to the underwater structure caused by the impulse cannot be ignored.

Conclusion
In this paper, the multiphase compressible fluid model is used to simulate the water jet phenomenon caused by explosive shock wave and double bubble collapse.the model involves the coupling between explosive gas, water and bubbles and other multiphase materials.it has a high degree of nonlinearity and complexity.On the premise of preliminary verification of the load calculation accuracy of underwater explosion shock wave, by calculating the interaction process of explosion shock wave with single and double bubbles, the following main research results are obtained.The main results are as follows: (1) The whole process of spherical bubble collapse under the action of underwater explosion shock wave is accurately captured, and the distribution region of water jet load and typical pressure time history curve are obtained.
(2) Due to the existence of the second bubble, the peak load pressure of the first bubble water jet decreases by about 19%; Under the condition 0 1 /r =2.5 D , the peak pressure of the second water jet is about 12% lower than that of the first water jet.
(3) Although the peak pressure of the two water jet loads has not reached the yield stress of the material, the characteristic time of the water jet load is long, and the damage effect on the underwater structure cannot be ignored.
the specific internal energy of the fluid per unit mass.The parameter n is related to the model being used: when n = 1, it represents a two-dimensional planar model; when n = 2, it represents a two-dimensional axisymmetric model, where the velocity v represents the radial velocity component.When n = 1 and n = 0, it degenerates into a one-dimensional planar model; when n = 2 and n = 0, it degenerates into a one-dimensional spherically symmetric model.

Figure 1 .
Figure 1.Schematic of numerical model for underwater explosion near the wall.

Figure 2 .
Figure 2. Comparison of the pressure curve between simulation and experiment.
6 m and has a radius of r1 = 0.2 m.Other initial parameters include: [Please provide the additional initial parameters.

Figure 3 .
Figure 3. Schematic diagram of signal bubble model.

Figure 4 .
Figure 4. Interface location(up) and pressure contours(down) for signal bubble collapses

Figure 5 (
Figure5(a) shows the time history curve of the equivalent radius of the bubble during the process of the explosive shock wave.From the figure, it can be seen that the spherical bubble undergoes the typical process of bubble collapse, collapse, and expansion.Combined with Figure4, it can be observed that the minimum volume of the bubble does not occur at the moment when the bubble starts to collapse at t =1.57ms, but approximately appears at the moment between 1.75ms and 1.8ms.This is due to the occurrence of vortices and the inertial effects of the fluid.Figure5(b) shows the pressure-time curve at the selected pressure measurement point at the moment when the bubble starts to collapse in Figure4.The analysis is conducted under different conditions with and without the presence of the bubble.From Figure5(b), it can be seen that when there is no bubble on the right side of the flow field, the pressure at the measurement point reaches a peak value of 58MPa at t = 0.4 ms, which is the pressure when the shock wave propagates freely in water to the measurement point.When a spherical bubble with a radius of 0.2 m exists, due to the obstruction of the shock wave propagation and the appearance of water jets, the peak pressure value of 130 MPa is reached at t = 1.7 ms, which is about 124% higher than the peak pressure of 58 MPa without the bubble, and the duration of the pressure also increases significantly.If there is a structure at the measurement point, and both the peak pressure and impulse on the structure increase significantly compared to the case without the bubble, the destructive effect on the structure will be more pronounced.

Figure 5 .
Figure 5.Time history of bubble average radius and pressure on the point impacted by water-jet

Figure 7 .
Figure 7. Interface location and pressure contours for double bubble collapsesFigure8shows the equivalent radius of two bubbles and the pressure time history curve at the impact point of the water jet, respectively.It can be seen from figure8(a) that the collapse time and the minimum equivalent radius of the first bubble are basically the same as those in figure5(a),

Figure 8 .
Figure 8.Time history of bubble average radius(left) and pressure(right) on the point impacted by water-jet