Numerical study on the cavity dynamics for vertical water entries of twin spheres

Xu Wng , Xujin Lyu ,*, Ruisheng Sun , Dongdong Tng

a School of Energy and Power Engineering, Nanjing University of Science and Technology, Nanjing, 210094, China

b Shanghai Institute of Spacecraft Equipment, Shanghai, 200240, China

Keywords:Twin water entries Side-by-side Cavity Numerical simulation

ABSTRACT In this study, a three dimensional (3D) numerical model of six-degrees-of-freedom (6DOF) is applied to simulate the water entries of twin spheres side-by-side at different lateral distances and time intervals.The turbulence structure is described using the shear-stress transport k-ω (SST k-ω) model, and the volume of fluid (VOF) method is used to track the complex air-liquid interface.The motion of spheres during water entry is simulated using an independent overset grid.The numerical model is verified by comparing the cavity evolution results from simulations and experiments.Numerical results reveal that the time interval between the twin water entries evidently affects cavity expansion and contraction behaviors in the radial direction.However, this influence is significantly weakened by increasing the lateral distance between the two spheres.In synchronous water entries, pressure is reduced on the midline of two cavities during surface closure, which is directly related to the cavity volume.The evolution of vortexes inside the two cavities is analyzed using a velocity vector field,which is affected by the lateral distance and time interval of water entries.

1.Introduction

Water entry is a transient process characterized by instability and nonlinearity.A complex multiphase flow occurs when an object passes through a free surface and enters water,which typically involves cavity formation, cavity elongation, and cavity collapse.Over the past century, the water entry problem has been studied extensively for its wide range of engineering applications such as naval weapons [1], ship industry [2], aerospace [3], and ballistic characteristics underwater[4-8].During the water entry,it is likely that the violent collision between the object and the water surface will affect the trajectory of the object and even damage its structural components.Thus,the interaction between the object and the air-water multiphase flow during water entry is one of the research hotspots.

Worthington and Cole [9] presented an in-depth analysis and discussion of the water entry problem,established an experimental study system,and reported the phenomena of water splashing and surface closure.Based on their pioneering work,experiments were performed to explore how cavities, jets, and splashes are formed and evolved[10-13].With the development of high-speed camera technology, different experimental methods have been applied to investigate the mechanism of fluid-object interaction.Objects of various shapes were used in water entry experiments, such as spheres, cylinders, wedges, droplets, and even bullets [14-19].With advanced computational technology,flow characteristics and complex turbulence behaviors that are difficult to observe through experiments can now be captured using Computational Fluid Dynamics (CFD) code.This numerical simulation tool integrates the turbulence models and the Reynolds Average Navier-Stokes equations to simulate the details of flow structures during water entry,and the overset grid technology was commonly adopted to simulate objects motion [20,21].

Previous research on water entry has largely focused on single objects.Only a few studies addressed the problem of multi-object water entry.During the process of multi-object water entry, violent interactions occur and are reflected in the evolution of cavity coupling and fluid dynamics interferences [22,23].Using highspeed photography, Yun and Lyu [24-27] evaluated the water entry of two tandem spheres.They found that the dynamic and cavitation behaviors of the second sphere was pronouncedly affected by the bubble wake of the first cavity.A non-dimensional parameter named "matryoshka number" was also proposed to describe cavity regimes and drag reduction.It was found that the impact force acting on the second sphere depends on the first cavity and the relative timing of the impact of the second sphere[28].

Different from the disturbance of the water entry of tandem objects, that of side-by-side objects is primarily caused by the adjacent object, where one object influences the two-phase flow pattern and movement of the other one.During the past few years,increasing attention has been paid to this field.Wang and Wang[29] numerically investigated the effect of various hydrodynamic parameters on the twin cylinders entering water side-by-side,and used the finite volume method with Roe's approximate Riemann solver to analyze the additional force.The potential theory was introduced to build a 3D calculation model for the cavities [30].Cylinders, wedges and spheres were used to perform the twin water entries side-by-side, and the cavity evolution, dynamic characteristics and interaction between the objects were studied under different conditions [31-34].The formation and development of twin cavities were also explored by the experimental method.However,the mechanisms of cavity dynamics in the twin water entries remain unclear,given the limits of the view field and measurement methods.To reveal the characteristics of the multiphase flow field,our study investigates the vertical water entries of twin spheres side-by-side using numerical method.A 3D simulation model is established based on the six-degree-of-freedom(6DOF) model, shear-stress transport (SST k-ω) model, and overset grid technology.At a velocity of 14.8 m/s,the combined effects of lateral distance and time interval between the two spheres on hydrodynamics are discussed.

The reminder of this paper is organized as follows: Section 2 introduces the numerical method; Section 3 discusses the effects of lateral distance and time interval between the two spheres on cavity dynamics during water entry; Section 4 summarizes the study.

2.Numerical configuration

This study utilizes ANSYS Fluent,a commercial CFD software,for simulations.It uses an explicit volume of fluid (VOF) model to describe the gas-liquid interface.Detailed CFD configurations are introduced below, including the governing equations, VOF model,turbulence model and numerical setup.

2.1.Numerical method

The water entry process involves a complex two-phase flow accompanied by motions at the solid-liquid-gas interface.The continuity equation and momentum conservation equation of the multiphase flow are

where t is time, xiis the Cartesian coordinate component, uiis the velocity component in Cartesian coordinate(i,j=1,2,3),ρmis the gas-liquid mixture density, μmis the dynamic viscosity, and μt=ρmCμk2/ε is the turbulence viscosity coefficient.

The SST k-ω turbulence model is selected for the water-entry simulation, which can effectively predict wall-bounded flow and free shear flow and is highly nonlinear,suitable for solving complex water-entry processes.The expression is

2.2.Meshing and calculation conditions

A 3D computational domain for vertical twin water entries is constructed, as shown in Fig.1.The cylindrical domain has both height and diameter of 30D, where the height of air and water is 10D and 20D, respectively.When the sphere contacts the free surface,the time point and vertical location are t=0 and y=0.For consistency with the experiment, the diameter and density of the spheres are D = 14.7 mm and ρs= 7778.6 kg/m3, and the initial velocity U0=14.8 m/s.

Fig.1.Illustration of the computational domain for twin water entries.The top boundary is defined as pressure inlet and the surrounding and bottom boundaries as the wall surfaces.

In the simulations,an overset grid technology is used to capture the 6DOF motions of twin spheres.The whole fluid field is divided into several sub-computational domains.The computation among these sub domains is independent of each other, which are overlapping, covering and nesting among grids.The grids of the background and overset regions are generated by Fluent Meshing and ICEM CFD, respectively, as shown in Fig.2(a).To improve the grid quality and capture the evolution details of the multiphase flow, grid refinement is conducted around the air-water free surface and along the trajectories of spheres.To ensure the accuracy of numerical simulation,the height of the first grid near the spheres is set to 0.015 mm,with y+<5 for the wall,as shown in Fig.2(b).In this study, the effects of lateral distance between the two spheres (a)and time interval (Δ t) between the two water entries are considered.As shown in the sketch, the time interval is controlled by adjusting the initial height of the left sphere (Δ y = U0×Δ t), and the spheres move downward at the same velocity as the initial one.

The time-dependent governing equations for vertical water entries of twin spheres are discretized using the finite volume method.The VOF method is applied to capture the motion of the air-water surface.The PRESTO format is adopted to discretize the spatial pressure field, and the Coupled scheme used to solve the pressure-velocity coupling algorithms.The second-order upwind scheme is adopted for the convection terms of separate flows.The time step is set at 1×10-5s.During each time step,the iteration is conducted for 20 times to ensure accuracy and convergence.

2.3.Verification and validation

For a typical case (a = 1.5D and Δ t = 0), a grid-independence study is carried out for three different grid levels, namely 3.27 million (Coarse), 5.86 million (Medium), and 7.75 million (Fine)grids.From Coarse to Medium,the cavity diameter(DC)at t=2 ms changes significantly.However, from Medium to Fine, the numerical results only vary slightly with an increase in grid number, as shown in Fig.3(a).Grid independence results for twin water entries at different time intervals and lateral distances further support the choice of Medium as the grid level.A similar study on the time step independence is conducted(Fig.3(b))and a time step of 1×10-5s is chosen for subsequent simulations.

To validate the numerical method applied in this study, this section compares the numerical results with those from the experiments by Wang and Lyu[34].Similar to the definition by Wang and Lyu, the sphere entering water first is designated as Sphere I and the second one as Sphere II.The cavities caused by the two spheres are defined as Cavity I and Cavity II respectively.The time point when Sphere I reaches the quiescent water surface is t1=0 and that for Sphere II is t2=0.U10and U20represent the respective initial velocities of Sphere I and Sphere II upon water entry.It should be noted that the time point at which the spheres reach the quiescent water surface in synchronous twin water entries is defined as t = 0.

Fig.2.Grids of the computational domain: (a) Details of near-wall grids; (b) Near-wall grids treated by y+<5 for accurate simulation.

Fig.3.(a)Grid independence and(b)time step independence studies for three different grid levels and time steps.The grid level of Medium and time step of 1×10-5 s is chosen for capturing the cavity shape.

Fig.4.Comparison of experimental and simulated cavity evolution results: (a) The synchronous water entries(Δ t=0 ms);(b)Successive water entries are at a=2.5D(Δ t = 3 ms).

For the lateral distance of a=2.5D,the cavity evolution for twin water entries both synchronous and successive are demonstrated in Fig.4,with numerical simulation and experimental results being compared.Due to the limitations in the accuracy of the setup and initial disturbance during the launch of the experiment, there is a small time interval between the two spheres upon reaching the free surface,resulting in a slight difference in the diameter and length of the two cavities.The numerical results for both synchronous and successive water entries show good agreements with those from experiments.Due to the complex flow structure of consecutive water entries, the simulated and experimental results of cavity diameter differ slightly.Nevertheless, the simulation is able to capture the characteristics of cavity evolution rather accurately.As shown in Fig.5, the cavity length varies during both synchronous and consecutive water entries, with good agreements between simulations and experiments.Therefore, it is believed that the numerical method is accurate and suitable for the investigation of twin water entries in this study.

3.Results and discussion

3.1.Synchronous water entries

To understand the synchronous water entries of twin spheres,the cavity evolution and the multiphase flow field of the synchronous water-entry scenario are depicted.The influence of lateral distance between the two spheres,i.e.,a=1.5D,2.5D,and 3.5D,on cavity evolution, pressure distribution and velocity field are also discussed in detail to reveal the flow characteristics.

3.1.1.Cavity evolution

Fig.6 illustrates the cavity evolution in synchronous water-entry scenario with different lateral distances.The two spheres impact water and step into the open-cavity stage simultaneously.The inner sides of the splash crowns induced by the water entries are pushed together to the middle, resulting in the two cavities expanding axially and radially.The radial expansion,however,is limited by the smaller lateral distance between the two cavities, leading to two asymmetric cavities with straight inner profiles, as shown in Figs.6(d1)-(e1).From the time point t = 6-8 ms, the cavity diameter and length increase as spheres move downward, and splash turns inward from outward during contraction.The splash along the midline moves upward in the meantime.The closure of the splash crown is mainly driven by two factors, namely the surface tension of the cavity, and the pressure difference between the inside and outside of the cavity [35].

Fig.5.Comparison of cavity length results of(a)synchronous water entries and(b)successive water entries from experiments and simulations at a=2.5D.LC2 denotes the length of Cavity II.

Fig.6.Cavity evolution during synchronous water entries of twin spheres at different lateral distances.Formation, development, contraction, and closure of the cavities are compared in three cases: (a) a = 1.5D; (b) a = 2.5D; (c) a = 3.5D.

The splashes above the two cavities tend to accumulate along the midline, forming an enhanced jetting, as shown in Figs.6(c1)-(e1).The splashes move upward,causing a slight delay of surface closure.The adjacent limitation between the cavities is weakened as the lateral distance is increased to 2.5D.The splash dome moves toward each cavity axis, and two upward jets are induced upon surface closure.As the lateral distance increases to 3.5D, the influence of adjacent limitations on cavity formation,elongation and contraction drastically decreases.Despite the cavities are still slightly asymmetric,their evolution is highly similar to that of a single water entry.

To quantify the effect of lateral distance on cavity shape during synchronous water entries, time histories of cavity length LCand diameter DCare presented.The effect is rarely observed for cavity length,yet it is evident for cavity diameter,as shown in Fig.7(b).As the lateral distance increases, the cavity has a more obvious radial expansion as adjacent limitation is weakened.When the lateral distance reaches a = 3.5D, the time history of cavity diameter is close to that for a single water entry, indicating that this is almost the critical point in determining whether the cavity evolution in synchronous water-entry is disrupted.The difference between diameters of 3.5D and 2.5D tends to decrease at t=10 ms,indicating that the occurrence time of contraction is earlier in the case of large lateral distance than in the narrow one.Typical cavity diameters at t = 5 ms and 10 ms for three depths of H = D, 2D and 4D with different lateral distances are illustrated in Fig.8.In the initial stage of water entry, higher velocity of the sphere leads to the larger cavities and the stronger interaction in shallow water.However,as the spheres travel downwards, its velocity gradually decreases while the water pressure increases, which limits the cavity expansion in deep water.As the lateral distance increases from 1.5D to 3.5D,the cavity diameter in shallow water(H= D) differs more dramatically, while it changes little at deeper depths.Thus, the limitation of lateral distance on radial expansion is more obvious in the cavity tail.

3.1.2.Flow field characteristics

Fig.7.Time histories of (a) cavity length and (b) diameter at different lateral distances.The depth is set at H = D.

Fig.8.Variation in cavity diameter at H = D, 2D and 4D for three lateral distances at (a) t = 5 ms and (b) t = 10 ms.

The pressure field distribution at different lateral distances is displayed in Fig.9, which stays symmetric throughout the entire process of synchronous water-entry.At t=4 ms,both cavities are in the opening stage, and high-pressure regions are formed at the heads of the spheres after impacting the water.At a = 1.5D, the pressure fields produced by the two spheres are found to interfere and overlap,creating a large area of high pressure.The overset area gradually decreases as the lateral distance increases due to the weakening of the interaction.As the velocity declines during water entry, the high-pressure area decreases, as shown in Figs.9(a2)-(c2).In addition, two detection lines are placed at the center of the two cavities and at the top of the sphere,respectively,to investigate the pressure distribution of the synchronous waterentry case, which will be discussed in the following.

Clearly,splashes above the water surface are thrown over to the domes, and thus both cavities are closer to the surface seal.An evident low-pressure region appears around the cavities at t=8 ms in all three cases, which tends to expand as the lateral distance increases.The low-pressure region is caused by airflow into the cavity behind the sphere, similar to that reported by Gilbarg and Anderson [11].To further describe the expansion of the lowpressure region, the cross-sections of cavities with pressure distribution and velocity vector at a depth of H = 4D are shown in Figs.9(a3)-(c3).In terms of the velocity vector,the fluid around the cavities is always expelled, decreasing the pressure of the surrounding area.Low pressure appears both inside and around the cavities at t=6 ms,with the low-pressure area gradually increasing with the cavity expansion.

The pressure on the detection line near the head of the sphere is shown in Fig.10.It is clear that the high pressure created by the violent collision between the sphere and the water sharply decreases along the y-axis in the negative direction.Due to water resistance, the spheres’ velocity gradually decreases, and the peak pressure acting on the spheres tends to be reduced.However, the vertical distribution of the pressure seems insensitive to the lateral distance between spheres.For different distances, the pressure at the head of the spheres is almost identical when the spheres arrive at different depths.

Fig.9.Pressure fields of synchronous water entries at different lateral distances at t = 4 ms and 8 ms: (a1) a = 1.5D, t = 4 ms; (b1) a = 2.5D, t = 4 ms; (c1) a = 3.5D,t=4 ms;(a2)a=1.5D,t=8 ms;(b2)a=2.5D,t=8 ms;(c2)a=3.5D,t=8 ms;(a3)a = 3.5D, t = 6 ms; (b3) a = 3.5D, t = 7 ms; (c3) a = 3.5D, t = 8 ms.

Fig.11 shows the pressure on the midline of two cavities at t=2 ms, 4 ms, 6 ms and 8 ms upon impacting the water.In the case of a=1.5D,the pressure where the cavities exist is lower than that in other cases and less than the atmospheric pressure(101.325 kPa).It is also observed that the pressure values are closer in the early stage of water entry when a = 2.5D and 3.5D.In addition, the pressure increases significantly when the midline crosses the high-pressure region,resulting in turning points on the pressure curves.It is clear that the smaller the lateral distance, the stronger the pressure on the midline influenced by the high-pressure region.For the lateral distance a = 1.5D, the pressure peak on the midline is obviously higher than the other two cases, leading to crossing points among the three curves.As the surface seal happens from 6 to 8 ms,there is a significant pressure drop on the midline,which is also affected by the lateral distance.Fig.11(d)illustrates a maximum reduction of about 5 kPa on the midline in the cases of 2.5D and 3.5D.Although the distance between the midline and the cavity is larger in the case of a = 3.5D than in a = 1.5D, the pressure of the three cases is similar.The reason is that the pressure drop is driven by the cavity volume during the surface seal.The greater the lateral distance,the weaker the adjacent limitation between the two cavities.As a result, the cavity volume is larger at a lateral distance of a = 3.5D when surface seal occurs, resulting in a lower pressure inside the cavities, the same as that reported by Abelson [36].Then, the midline pressure decreases further due to the diffusion of the lowpressure region, resulting in a lower pressure than the one with a lateral distance of a = 1.5D.

Fig.10.Pressure distribution at the head of spheres at different lateral distances.As the spheres move downwards, the peak pressure gradually decreases.

The velocity field for synchronous water entries at a = 2.5D is shown in Fig.12.It is found that kinetic energy of the spheres is gradually transferred to the surrounding water as the spheres descend.Two parts of water are expelled by the two spheres,resulting in opposite flows in the middle of the two spheres, as shown in Fig.12(a).Then, part of the water moves upward and a higher splash is formed in the middle of the two cavities.For synchronous water entries, the distribution of the velocity filed inside the cavity is symmetric around the midline yet asymmetric around the cavity axis.During cavity formation and elongation,some external air flows into the two cavities, forming a highvelocity region around the cavity opening, as shown in Fig.12(a)and (b).The unstable splashes in the middle of the two cavities form vortexes, which are basically symmetrically distributed.

As the cavities begin to contract at t = 8 ms,the splash crowns created by the twin water entries shrink toward the midline, narrowing the cavity's open area.Meanwhile, airflow is continuously entrained into cavities and accelerated around the contracting cavity opening.A large high-velocity region is then formed in the tail of the cavities.The flow pattern inside the cavity is significantly affected during surface closure, as illustrated in Fig.12(c).The asymmetric evolution of the splashes induces a backward-facing step inside the cavities, forming two large-scale vortexes in the backflow region, as illustrated in Fig.13.The cavities are closed at t = 10 ms and the high-velocity region is significantly reduced.Then,the vortexes in the two cavities gradually dissipate as energy is consumed.

3.2.Effects of time interval and lateral distance on successive water entries

Fig.11.Pressure distribution in the midline of the two cavities at (a) t = 2 ms, (b) t = 4 ms, (c) t = 6 ms and (d) t = 8 ms.Near the head of the sphere, the pressure increases.

Fig.12.Velocity field evolution during the cavity neck-off process at a=2.5D.The high-velocity region and vortexes are marked with black squares:(a)t=4 ms;(b)t=6 ms;(c)t = 8 ms; (d) t = 10 ms.

Fig.13.Vortexes in cavities.During contraction, air flow near the cavity opening is accelerated, forming a larger vortex in the recirculation region.

The interactions between two cavities of successive water entries at a lateral distance of a=1.5D,2.5D and 3.5D are discussed.In addition, the structural characteristics of pressure and velocity based on the evolution of the flow field are compared and analyzed.

3.2.1.Cavity evolution

With the effect of the time interval,the trailing sphere is met by a more complex flow field generated by cavity evolution after the primary water entry, such as cavity formation, elongation,contraction and closure.The morphology of the primary cavity will directly influence the evolution of the cavity as the trailing sphere is immersed in water.Time intervals of Δ t=1.5 ms,3 ms and 4.5 ms are applied to illustrate the evolution of cavities during successive water entries at a typical lateral distance,a=1.5D.The influence of time interval is reflected from different stages, including cavity opening, deformation, and closure.In the case of Δ t = 1.5 ms, the water entry of the trailing sphere squeezes Cavity I, as shown in Figs.14(a1) and (b1).As discussed in the previous section, Cavity I will cause a low-pressure region in the flow field.During cavity elongation,the pressure differential induces a significant expansion of Cavity II.In Figs.14(c1)-(e1), the cavity wall tends to expand toward the low-pressure region, leading to a large Cavity II.Meanwhile,Cavity I becomes slender as it is continuously squeezed by the expanding Cavity II, as demonstrated in Fig.14(f1).

Fig.14.Cavity evolution of twin water entries with different time intervals at a = 1.5D in the xoy plane.The first sphere to reach the free surface is Sphere I and the other one is Sphere II: (a) Δ t = 1.5 ms; (b) Δ t = 3 ms; (c) Δ t = 4.5 ms.

Fig.15.Time histories of cavity diameter at the depth of H=D for Δ t=1.5 ms,3 ms,4.5 ms at a = 1.5D.

As the time interval increases to 4.5 ms,Cavity II expands rapidly during the open-cavity stage below the water surface.In Fig.14(e3),the inner wall of Cavity II continues to expand until it ruptures the wall of Cavity I at t2=6 ms.Afterwards,the rupture increases under the compression of Cavity II and eventually leads to a pinch-off,as shown in Fig.14(f3).During the rupture,the cavity length increases gradually while the diameter of Cavity II hardly changes.Fig.15 shows the time histories of the diameter of Cavity II at the depth of H=D for three time intervals,i.e.,Δ t=1.5 ms,3 ms and 4.5 ms.It is evident that the diameter at Δ t = 4.5 ms is significantly larger than those of the other two cases,getting the highest growth rate.Due to the formation and expansion of the rupture,this growth rate decreases sharply after t = 6 ms.Clearly, the cavity diameter at Δ t=1.5 ms decreases after t=7 ms.One possible reason is that the energy accumulated by the expansion of Cavity II causes a redound between the two cavities and then squeezes Cavity II.

The cavity expansion and rupture are presented from a side view (rotated about 50°around the y-axis) and a top view (in the xoz plane) for the case of Δ t = 4.5 ms, as shown in Fig.16.At t2=4 ms,a bulge to the right side appears on Cavity II,which can be observed in the top view.Later, Cavity II exhibits a crack at t2= 5 ms.To clearly present this phenomenon, the outlines of the two cavities are marked in yellow solid line and red dash-line, as shown in Fig.16(b).There is a thin layer of water sandwiched between the walls of the two cavities at t2=4 ms,indicating that the cavity walls have not collided.After that, the bulge continues to expand to the right, pushing the water layer aside, colliding with the inner wall of Cavity I and then hitting the wall on the other side.Finally,the crack is opened to form a rupture at t2=6 ms.As a result of the unstable cavity, the rupture grows rapidly, leading to the pinch-off of Cavity I.To clearly demonstrate the process of rupture evolution, the three-dimension velocity distribution of the cavity wall is detailed in Fig.19.During the cavity expansion, the wall of Cavity II has a higher velocity under the influence of the lowpressure region, and a crack appears on the bulge of Cavity II.The expansion velocity of Cavity II decreases significantly upon collision of the cavities.Under the effect of the surface tension, the cavity wall surrounded by the rupture appears to curl and expands in around at a high velocity, making the rupture area gradually increase.

Fig.16.Expansion of Cavity II from (a) side view and (b) top view at a = 1.5D.The growth of the rupture is marked with orange circles.

To uncover the interesting and significant interaction for the case of Δ t=4.5 ms,the cavity evolution with two different lateral distances of a=2.5D and 3.5D are studied.In the case of a=2.5D,only small deformations are observed at the water surface during the initial stage of water entry,as shown in Fig.17(c1).At t2=5 ms,significant expansion and depression occur for the two cavities,making them highly asymmetrical around the cavity axis.At t2=7 ms,although the deformation is similar to that in the case of a=1.5D,the walls of the cavities fail to collide,indicating that the water pressure in the middle of two cavities limits the expansion of Cavity II significantly.Moreover, this limitation is enhanced when the lateral distance increases to 3.5D.The evolution of Cavity II is hardly influenced by the primary cavity and remains symmetric,as illustrated in Fig.17(b2).

Fig.18 illustrates the diameter of Cavity II at the depth of H=D below the water surface for three different lateral distances.Evidently,spheres with smaller lateral distances(such as a=1.5D)are more sensitive to the effects of water-entry time intervals.When two spheres successively enter the water at Δ t=4.5 ms,the diameter of Cavity II increases by 74.5% at t = 4 ms and 114.2% at t = 8 ms, compared to synchronous water entries.However, a noticeable reduction in diameter growth is observed at the lateral distance of 2.5D, indicating that the effect of the time interval on the diameter of Cavity II diminishes rapidly.As the lateral distance increases to 3.5D, there is only an increase of 4.3% and 10.2% in diameter for Δ t = 0 ms and Δ t = 4.5 ms respectively.

Fig.17.Comparison of cavities with lateral distances at (a) a = 2.5D and a = (b) 3.5D.Cavity evolution for the case of Δ t = 4.5 ms is depicted in detail.

Fig.18.Effects of time interval and lateral distance between the two spheres on the diameter of Cavity II at (a) t2 = 4 ms and (b) t2 = 8 ms.

3.2.2.Flow field characteristics

During consecutive water entries, the asymmetric characteristics of the trailing sphere during cavity evolution are strongly influenced by Cavity I.The presence of Cavity I induces a different flow field structure on both sides of Cavity II.Fig.19 shows the pressure distribution for the consecutive water-entry case with a lateral distance of a = 1.5D.As the time interval increases, the overlap area of the high-pressure region decreases compared to that in the synchronous water-entry case.Cavity I makes the pressure field to be distributed asymmetrically at the head of the trailing sphere.No apparent changes are observed in the pressure distribution inside Cavity I as the time interval increases.The pressure on the midline when the trailing sphere reaches the water surface is shown in Fig.19(d)for three time intervals of Δt=1.5 ms,3 ms and 4.5 ms.The overlap and interference between the two highpressure regions result in a higher pressure on the midline at the time interval of 1.5 ms.As the time interval increases to 3 ms, the pressure on the midline is mainly affected by the low-pressure region caused by the primary cavity, and it is close to atmosphere pressure(101.325 kPa).Obviously,the low pressure on the midline is related to the length of Cavity I.When the time interval increases to 4.5 ms, the pressure approaches atmosphere pressure over a longer distance.

Fig.19.Pressure distribution at a = 1.5D with different time intervals: (a) Δ t = 1.5 ms, a = 1.5D; (b) Δ t = 3 ms, a = 1.5D; (c) Δ t = 4.5 ms, a = 1.5D.

Fig.20.Pressure distribution at Δ t = 4.5 ms with different lateral distances: (a) a = 1.5D, Δ t = 4.5 ms; (b) a = 2.5D, Δ t = 4.5 ms; (c) a = 3.5D, Δ t = 4.5 ms.

Fig.21.Structural characteristics of the velocity field for consecutive water entries with a lateral distance of a=1.5D and time interval of Δ t=4.5 ms:(a)t2=3 ms;(b)t2=5 ms;(c) t2 = 7 ms; (d) t2 = 9 ms.

Fig.22.Velocity scalar and vector distribution at a = 2.5D and Δ t = 4.5 ms: (a) t2 = 2 ms; (b) t2 = 3 ms; (c) t2 = 5 ms; (d) t2 = 7 ms; (e) t2 = 9 ms.

Fig.20 demonstrates the pressure field for three different lateral distances with the same time interval of 4.5 ms.Due to the weakened adjacent limitation, the high-pressure region gradually loses its asymmetry as the lateral distance increases.The pressure distribution along the midline is shown in Fig.20(d).At the lateral distance of 1.5D, the pressure on the midline is close to the atmospheric pressure under the influence of Cavity I.As the distance increases to 2.5D and 3.5D,higher pressure appears on the midline due to the weak interaction between the twin water entries.

In Fig.21, a lateral distance of a = 1.5D and a time interval of 4.5 ms are adopted to study the velocity field characteristics of consecutive water entries.A high-velocity region is formed around the cavity opening are due to the air entrainment, similar to the case in synchronous water entries.The splash caused by the collision of cavity is pushed to the center of the opening cavities and leading disturbance to the airflow directly.Due to the contraction of the cavity opening,the airflow enters Cavity II at a higher velocity,resulting in a large high-velocity region.A distinct airflow deflection is found inside the cavity due to the radial expansion of Cavity II.It is clear that a large-scale vortex is formed inside Cavity II,whose location differs from that in a synchronous water-entry.As the surface closure occurs, the air entrance channel is closed and the airflow speed inside Cavity II drops significantly.

Fig.22 depicts the velocity field at a = 2.5D to highlight the influence of lateral distance on the flow field of consecutive water entries.Prior to the rapid expansion of Cavity II, there are two vortexes inside Cavity I below the water surface, and the velocity field is almost symmetrically distributed.As Cavity II begins to expand rapidly, only one vortex is left due to the compression of Cavity I.Different from the case with a lateral distance of a=1.5D,the velocity field distribution in Cavity II is almost symmetric,and two vortexes are observed inside the cavity.As Cavity II gradually closes, the two vortexes gradually dissipate and the high-velocity region inside the cavity decreases.

4.Conclusions

An investigation of the side-by-side vertical water entries of twin spheres is presented in this paper using a CFD method, in which the effects of lateral distance and time interval on the waterentry behaviors are analyzed in detail.Main findings of this work are summarized as follows:

(1) As for synchronous water entries of twin spheres, the evolution of cavities is influenced by each other,forming a pair of cavities that are symmetrical about the midline.The adjacent limitation is weakened by increasing the lateral distance between the two spheres.The pressure fields produced by the two spheres interfere and overlap each other.Besides,the pressure in the center of the two cavities is related to the cavity volume and influenced by the lateral distance.As the distance increases to 3.5D, a maximum pressure drop of 5 kPa is found on the midline upon the cavity surface closure.

(2) For the case of a = 1.5D and Δ t = 4.5 ms, Cavity II clearly expands under the influence of the low-pressure region,and finally results in a rupture on Cavity I.Once the rupture forms, the cavity wall around the rupture will immediately curl and expand in around at a high velocity under the effect of surface tension.As the lateral distance between the two spheres increases, the effect of time interval on cavity expansion gradually diminishes, and Cavity II remains symmetrical, similar to that in single water entry.

(3) In the case of synchronous water entries, the distribution of the velocity field inside the cavity is rather symmetric around the midline yet asymmetric around the cavity axis.Due to the contraction of the splashes, backward flow is induced inside the cavities, resulting in two large vortexes.As for consecutive water entries with a time interval of Δ t = 4.5 ms, the airflow inside Cavity II is significantly deflected by cavity expansion and forms a large vortex.Under the influence of Cavity II's expansion,one of the vortexes in Cavity I disappears as the lateral distance increases.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

The present work is financially supported by China Academy of Launch Vehicle Technology(Grant No.CALT-2022-03);Science and Technology on Underwater Information and Control Laboratory(Grant No.2021-JCJQ-LB-030-05).