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Received：2016 - 10 - 31Supported by：National Natural Science Foundation of China (51479041, 51279038); Young Scientist Fund of the National Natural

Science Foundation of China (11402143); Training Program of Young Teachers in Colleges and Universities ofShanghai City (A1-2035-14-0010-18); Open Fund of the Key Laboratory of Sustainable Exploitation of OceanicFisheries Resources (A1-0203-16-2007-5)

Authors: YE Xi, male, born in 1987, Ph.D., engineer. Research interest: vibration noise and shock of ships. E-mail: yexi0527@163.comCHU Wenhua（Corresponding author）, female, born in 1986, Ph.D., lecturer. Research interest: fluid dynamics. E-mail:whchu@shou.edu.cnCHEN Lin, male, born in 1987, master, engineer. Research interest: vibration noise and shock of ships. E-mail:1021763445@qq.comZHANG Aman, male, born in 1981, Ph.D., professor. Research interest: explosion and shock resistance of ships. E-mail:zhaman@163.com

CHINESE JOURNAL OF SHIP RESEARCH，VOL.12，NO.5，OCT 2017DOI：10.3969/j.issn.1673-3185.2017.05.011Translated from：YE X，CHU W H，CHEN L，et al. Interaction between bubble near free surface and shock wave［J］. Chinese

Journal of Ship Research，2017，12（5）：90-96.

http：// english.ship-research.com

0 Introduction

In the field of ship and ocean engineering, the in⁃teraction among shock wave, bubble, and free sur⁃face often appears, e.g., the interaction between theshock wave due to underwater explosion or structurereflection and the bubble produced by explosion,and the interaction between the shock wave and thecavitation generated near the free surface in thecourse of ship navigation (there is a large amount ofcavitation in the ship wake). Such interaction will

change the propagation characteristics of shock waveand affect the motion characteristics of bubble, lead⁃ing to the change in the load characteristics of shipstructure. Therefore, the analysis of the interactionamong shock wave, free surface, and bubble is ofgreat significance for deep study of the damage mech⁃anism of ship structure in underwater explosion con⁃dition.

The discontinuity of flow field caused by the medi⁃um difference between the two sides of the bubble in⁃terface in the water will bring some difficulties to the

Interaction between bubble near free surface andshock wave

YE Xi1, CHU Wenhua2,3,4, CHEN Lin1, ZHANG Aman5

1 Marine Design and Research Institute of China, Shanghai 200011, China2 College of Marine Sciences, Shanghai Ocean University, Shanghai 201306, China

3 National Engineering Research Center for Oceanic Fisheries, Shanghai Ocean University, Shanghai 201306, China4 Key Laboratory of Sustainable Exploitation of Oceanic Fisheries Resources, Ministry of Education, Shanghai

Ocean University, Shanghai 201306, China5 School of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Abstract: [Objectives] This paper presents research into the interaction among a free surface, shock wave and bubble.[Methods] Based on the Discontinuous Galerkin method and combined with the Level Set method and Real Ghost Flu⁃id method, the characteristics of shock wave propagation and bubble motion are analyzed, and the generation and evolu⁃tion of waves in fluid fields are described in detail. [Results] The results show the presence of complex waves in the flu⁃id field after the interaction, including multiple rarefaction waves and shock waves. The collapse speed of the bubblesslows down with the existence of the free surface, while the incident shock wave accelerates the collapse speed of thebubble and increases the upwarp motion of the free surface. [Conclusions] The conclusions drawn from this paper canbe used as reference points for further research into the damage mechanisms of ship structures subjected to underwaterexplosions.Key words: bubble; free surface; shock wave; Discontinuous Galerkin methodCLC number: U661.144

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numerical simulation. Johnsen et al. [1-2] adopted thevolume fraction method and finite volume method toreconstruct the Weighted Essentially Non-Oscillato⁃ry (WENO) format, and calculated the bubble evolu⁃tion in water under the action of strong shock waveand weak shock wave, taking into account the casethat there was a wall in the downstream of the bub⁃ble. Wang and Guan [3] adopted the Level Set meth⁃od, the Ghost fluid method and the finite volumemethod to solve the Euler equation set, and analyzedthe interaction between the shock wave and bubblein the water. Besides, the free Lagrange method [4],the boundary element method [5], the DiscontinuousGalerkin method [6-10], etc. are also applicable to thesolutions of this physical phenomenon. However,there has been no study on the interaction amongfree surface, bubble, and shock wave at present.

In this paper, based on the Discontinuous Galer⁃kin method and combined with the Level Set meth⁃od [11], the Real Ghost Fluid method [12], and the Adap⁃tive Mesh Refinement (AMR) technique [13], the inter⁃actions among underwater bubble, shock wave, andfree surface are studied, the effects of free surface onthe bubble motion and flow field characteristics areanalyzed, and the generation and evolution of newwave system re described in detail.1 Basic theory and method

As shown in Fig. 1, the calculation model adoptsthe zor axisymmetric coordinate system. In the initialstate, the bubble is relatively stationary with the freesurface. The radius of the bubble is R, and the initialdistance between the bubble center and the free sur⁃face is d f . The wavefront of the shock wave with theintensity of M is parallel to the free surface, and hasa distance of ds from the bubble center. It shocksthe bubble from bottom up. In this paper, all parame⁃ters are treated as dimensionless. The reference pa⁃rameters are the initial radius of the bubble, the stan⁃dard atmospheric pressure, and the air density. Sincethe characteristics of shock wave in the flow field aremainly studied in this paper, we assume that the flowfield is viscous and initially irrotational, and de⁃scribe the flow field with the axisymmetric Eulerequation as follows:

¶E¶t

+ ¶F¶z

+ ¶G¶r

= S （1）where

E = [ ]ρ ρu ρv ΣT

F = [ ]ρu ρu2 + p ρuv u( )Σ + pT

G = [ ]ρv ρuv ρv2 + p v( )Σ + pT

S = - 1z [ ]ρu ρu2 ρuv u( )Σ + p

T（2）

In Eqs. (1) and (2), t is the time; ρ is the density ofthe flow field; p is the pressure of the flow field ; uand v are the velocities along the z-and r-axes;Σ = ρein + 0.5ρ( )u2 + v2 is the total energy (the sumof the internal energy and kinetic energy), where ein

is the internal energy per unit mass.

The ideal gas state equation[14] is adopted to de⁃scribe the interior of bubble as follows:

p = ( )γg - 1 ρein （3）where γg = 1.4 is the specific heat ratio of gas.

The Tait state equation [6] is adopted to describethe external flow field of the bubble as follows:

p = ρein( )N - 1 - NPw （4）where N and Pw are medium constants, N =7.15, and Pw = 3.31 × 108 Pa.

We adopt the Discontinuous Galerkin method tosolve the Euler equation. The semi-discrete schemeis

åk = 0

pm

ΩΦ lΨk dΩdEk

dt=

Ωé

ëêê

ù

ûúúF

æ

èçç

ö

ø÷÷å

k = 0

pm

EkΨk

¶Φ l

¶z+ G

æ

èçç

ö

ø÷÷å

k = 0

pm

EkΨk

¶Φ l

¶rdΩ -

¶Ωé

ëêê

ù

ûúúF

æ

èçç

ö

ø÷÷å

k = 0

pm

EkΨk nez + G

æ

èçç

ö

ø÷÷å

k = 0

pm

EkΨk ner Φ ldA （5）

where pm is the maximum order of polynomials; Ωis the element area; Φ l (l = 01pm ) and Ψk

(k = 01pm ) are trial functions; Ek is the coef⁃ficient of Ψk in the kth-order polynomial; ne

z and ner

are the normal vectors of the element boundary; A isthe element boundary.

We select the Legendre orthogonal polynomial asthe trial function. Then, the variable E in the flowfield is

E ( )zr t = åk = 0

pm

Ek ( )t Ψk ( )zr （6）

Fig.1 Numerical model

Free surface

Bubble

WavefrontShock intensity M

ds

df

r R

zo

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Eq. (5) adopts the third-order Runge-Kutta ex⁃plicit format for the time stepping. TheHarten-Lax-van Leer-contact (HLLC) format [15] isadopted to calculate the numerical fluxes of F andG . The slope limiter [16] is used to restrain thenon-physical oscillation at discontinuity due to thehigh precision discrete format. To improve the calcu⁃lation accuracy and efficiency, we adopt the AMRtechnology to build the multilayer grid (see Fig. 2),and locally densify the position where the physicalparameters such as the pressure and density in theflow field are discontinuous.

2 Validation by numerical method

Fig. 3 shows the result comparison for numericalcalculation about the shock wave – bubble interac⁃tion in water. The medium interface has large densi⁃ty and pressure differences between the two sides ofthe interface, a large intensity of shock wave, and ahigh requirement for calculation stability and preci⁃sion of the discontinuous capture algorithm. The cal⁃culation domain is 24 mm × 24 mm (dimensionalphysical parameters are still used here so as to becompared with the results in Reference[17-19]). Thebottom grid number is 150 × 150, and we adoptthree-layer AMR grids. The cylindrical bubble radi⁃us is 3 mm, the bubble center is located at (12 mm,12 mm), and the Mach number of shock wave is1.72. The density of the bubble is 1 kg/m3, the pres⁃sure is 1 bar, and the gas constant is γg = 1.4; thedensity of the water outside the bubble is 1 000 kg/m3,the pressure is 1 bar, the medium constant is N =4.4, and Pw = 6 × 108 Pa. The distance between theinitial position of the shock wave and the bubble cen⁃ter is 54 mm. The left boundary of the calculation do⁃main is the inlet boundary, and the right, upper, andlower boundaries are transmitting boundaries. In this

paper, the calculation results will be compared withthose in Reference [17-19]. Though both the bottomgrid number and the AMR optimization layer num⁃ber are smaller than those of Reference [17], the in⁃teraction between the bubble and shock wave in wa⁃ter can still be well simulated.

3 Numerical calculation and anal-ysis

For the sake of simplification, we suppose that SWiis the incident wave; SWb is the shock wave due tothe underwater explosion; SWir is the reflected waveof SWi on the free surface; SWir* is the reflected wave

Fig.2 Diagram for AMR multiple levels grid

3

2

1

0

-1

-2

-3

z

-3 -2 -1 0 1 2 3r

（a）Distribution of pressure and Mach number(calculated results of Ref. [17])

（b）Distribution of pressure (calculated results of present paper)Fig.3 Results comparison for numerical calculation about

shock-bubble interaction in water

3.5 μs3.1 μs 3.7 μs

3.8 μs

1.8 μs1.1 μs 2.1 μs

3.4 μs3.1 μs 3.66 μs

3.85 μs

1.6 μs1.0 μs 2.2 μsMach

Pressure

TransmittedshockIncidentshock

Reflectedrarefactionwave

Water jet

Irregularshockrefraction

Blast wave

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of SWi on the upstream surface of the bubble; SWirr isthe reflected wave of SWi on the downstream surfaceof the bubble; SWj is the shock wave generated whena toroidal bubble is generated; SWjr is the reflectedwave of SWj on the free surface; and SWt is the trans⁃mitted shock wave inside the bubble.

First, we consider the interaction between theshock wave near the free surface and the bubble withbalanced initial internal and external pressures. Theinitial distance between the bubble center and thefree surface is d f = 1.5, and the distance betweenthe wavefront of the shock wave and the bubble cen⁃ter is ds = 2. The calculation domain is 3 × 6, andthree-layer AMR grids are adopted. The number ofthe bottom layer grids is 120 × 240. The lowerboundary is the inlet boundary condition, the upperboundary is the outlet boundary condition, and theleft and right boundaries are the wall boundary con⁃ditions. The physical parameters in each medium aredenoted by [ ρ , p , u , v , γ ]. Then, the internal pa⁃rameters of the bubble are [1, 1, 0, 0, 1.4], and theparameters of the water domain in the external flow

field are [1 000, 1, 0, 0, 7.15]. The parameters of theair domain are the same as those of the internal pa⁃rameters of the bubble. The intensity of the shockwave is M = 1.1. The interaction between the shockwave and the bubble near the free surface is shownin Fig. 4. Since the cavitation model has not beenused in the flow field, negative pressure exists.

From Figs. 4(a) and 4(b), we can see that, SWi canbe divided into three parts after it propagates to thebubble surface: one part is reflected to form rarefac⁃tion wave; another part transmits to the interior of thebubble and makes the interior gas of the bubblemove at a high speed; and the remaining part bypass⁃es the bubble and continues to propagate.

From Figs. 4(c) and 4(d), we can see that, whenSWi reaches the free surface, on one hand, it will beincident into air from water. Since the characteristicimpedance of water is higher than that of air, SWi isreflected back as the form of rarefaction wave, andforms SWir; on the other hand, the reflected SWir willact on the surface of the bubble, and will be also inci⁃dent into air from water, and thus, the rarefaction

（f）t=1.128

（a）t=0.193 （b）t=0.509 （c）t=0.677

（d）t=0.864 （e）t=1.042Fig.4 Shock-bubble interaction near free surface for M=1.1（In figures：left are pressure distribution，

and right are Mach number field distribution）

0 300 600 900 1 200 1 500Pressure

Mach0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.453

2

1

0

-1

-2

z

-3 -2 -1 0 1 2 3r

0 300 600 900 1 200 1 500Pressure

Mach0 0.2 0.4 0.6 0.8 13

2

1

0

-1

-2

z

-3 -2 -1 0 1 2 3r

-400 -20 360 740 1 120 1 500Pressure

Mach0 0.2 0.4 0.6 0.8 1 1.2 1.43

2

1

0

-1

-2

z

-3 -2 -1 0 1 2 3r

-700 -60 580 1 220 1 860 2 500Pressure

Mach0 0.4 0.8 1.2 1.6 23

2

1

0

-1

-2

z

-3 -2 -1 0 1 2 3r

-700 3 440 7 580 11 720 15 860 20 000Pressure

Mach0 0.5 1 1.5 2 2.53

2

1

0

-1

-2

z

-3 -2 -1 0 1 2 3r

-1 500 600 2 700 4 800 6 900 9 000Pressure

Mach0 0.3 0.6 0.9 1.2 1.5 1.83

2

1

0

-1

-2

z

-3 -2 -1 0 1 2 3r

SWSWjj

SWSWjrjr

SWSWjj&&SWSWbb

SWSWjjSWSWirrirr

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wave SWir will be reflected back in the form of shockwave and forms SWirr. Similarly, between the bubbleand free surface, shock wave and rarefaction wavewill continue to propagate back and forth. At this mo⁃ment, upwarp will appear on the free surfaces of thetwo sides under the effect of the shock wave, whilethe upstream surface of the bubble will be indentedto form a jet near which there is a local high pressurearea.

From Figs. 4(e) and 4(f), we can see that, at thismoment, the jet formed by the bubble induced byshock wave has broken through the downstream sur⁃face to form a toroidal bubble, and a new shock waveSWj forms and propagates around. Since the intensityof SWj is far higher than that of SWi, the negativepressure caused by the rarefaction wave SWjr, whichis formed by the reflection of SWj on the free surface,is far higher than that of SWir. At this moment, up⁃warp will appear on the center of free surface underthe action of SWj. Meanwhile, there are some localhigh pressure regions in the upstream of the toroidalbubble, which is the result of the intersection be⁃tween the local high pressure generated upstreamwhen the bubble is forming a jet and the SWj propa⁃gating upstream. Moreover, when SWi and SWj act onthe free surface, the shock wave with low intensitytransmits to the gas phase, which will cause the gason the other side of the surface to move.

Fig. 5 shows the pressure distribution at r = 0 anddifferent moments. The time points of intersection be⁃tween the one-way arrow and each curve in Fig. 5(a)from left to right are t = 0.402, 0.702, 0.772, 0.816,0.852, 0.916, 1.030, 1.145, respectively; those inFig. 5(b) are t1 = 0.702, t2 = 0.744, t3 = 0.775, t4 =0.809, t5 = 0.863, t6 = 0.954, t7 = 0.982, t8 = 1.016,respectively. When SWi reaches the free surface, itwill be reflected back as the form of rarefaction waveSWir, and forms local negative pressure. From Fig. 5(b),we can see that, when the rarefaction wave SWir actson the downstream surface of the bubble, it will bereflected back as the shock wave SWirr and changethe flow field to positive pressure state. Meanwhile,SWirr will reach the free surface again, be reflectedback as rarefaction wave on the downstream surfaceof the bubble, and be reflected as the form of shockwave finally. At this moment, the shock wave SWt in⁃side the bubble is close to the downstream surface ofthe bubble. In Fig. 5(b), the curve at t8 is just the lo⁃cal high pressure generated near the shock positionafter the bubble jet impacts the downstream surface,where the shock wave SWj forms. When SWj acts on

the free surface, a corresponding rarefaction waveSWjr will also form, and local negative pressure alsoforms. Since the intensity of SWj is far higher thanthat of SWi, the negative pressure induced by SWjr isalso far higher than that of SWir, as shown in Fig. 5(a)(t = 1.145).

Fig. 6 shows the dynamic characteristics of bubblewith and without free surface. When the rarefactionwave reflected by the free surface acts on the surfaceof the bubble, the downstream interface of the bub⁃ble will move downstream, which results in the in⁃crease in the bubble volume at the corresponding mo⁃ment and the longer time to form the minimum vol⁃ume. Since the action time of the reflected wave isshort, its effect is also small. Since the jet is formedby the upstream surface of the bubble back to thefree surface, the effect of the reflected wave on thefree surface on the jet is small. Therefore, whetherthere is free surface or not, the velocity of the jet ver⁃tex is basically the same.

The multipoint explosions underwater are com⁃mon. While a bubble is being generated, shock wavewill propagate around. The shock wave will couplewith other shock waves propagating in other direc⁃tions, and then affect the bubble motion and flowfield characteristics. The interaction between the

（a）Overall distribution diagram

（b）Partial enlarged detailFig.5 Pressure distribution at r=0

10 000

8 000

6 000

4 000

2 000

0

Pressu

re

-3 -2 -1 0 1 2 3z

After shock generated by jetimpact on free surface

Pressu

re

0.8 0.9 1 1.1 1.2 1.3 1.4 1.5z

1 000800600400200

0-200-400-600-800

t8 t1 t5

t7

t2

t4t3

t6

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shock wave (M = 1.1) and the bubble with initial in⁃ternal high pressure near the free surface is shown inFig. 7. The calculation domain is 8 × 12. Four-layerAMR grids are adopted, in which the number of bot⁃tom layer grids is 120 × 180. The medium parameterof the bubble interior is [1 630, 1 500, 0, 0, 1.25],and the other parameters are the same as the previ⁃ous content.

From Fig. 7(a), we can see that, the shock waveSWb released by the bubble and the incident waveSWi are superimposed at the intersection to form thehigh pressure zone. After SWi contacts with the up⁃stream surface of the bubble, it is reflected andforms the rarefaction wave SWir*, which is superim⁃posed with the previous high pressure zone to form ahigh pressure band zone and continues to propagatearound. In the region between the bubble and thefree surface, SWb is reflected as the rarefaction waveat the free surface, and then is reflected as the shockwave at the downstream surface of the bubble. Thisprocess is repeated, so that interval shock wave re⁃gions and rarefaction wave regions are formed. At theinitial time, in addition to releasing the shock waveSWb into the surrounding area, the bubble also formscohesive rarefaction wave inside of itself, so that a

high pressure zone forms in the bubble center final⁃ly , as shown in see Fig. 7(b).

Fig. 8 shows the shape change of the bubble underdifferent initial conditions (the initial pressure insidethe bubble is 1 500) and the free surface with time.Compared with the condition without incident shockwave, under the action of SWi, the bubble volume atthe same moment decreases. While the bubble is ex⁃panding, it drifts towards the free surface, and obvi⁃ous upwarp motion happens on both sides of the freesurface. When the intensity of the incident shockwave SWi is large, the upwarp motion of the free sur⁃face is more obvious, and a sharp peak is rapidlyformed on the downstream surface of the bubble andmerges with the free surface.4 Conclusions

In this paper, the Discontinuous Galerkin methodis adopted, combined with the Level Set method andthe Real Ghost Fluid method, to solve the interactionbetween the bubble near free surface and the shockwave. The motion characteristics of the bubble, thegeneration and propagation processes of variousshock waves and rarefaction waves, and the physical

（a）Bubble volume

0 0.2 0.4 0.6 0.8 1 1.2t

4

3

2

1

0

Volum

e

Without free surfaceWith free surface

（b）Jet speedFig.6 Dynamics of bubble with and without free surface

0 0.2 0.4 0.6 0.8 1 1.2t

6543210

Veloci

ty

With free surfaceWithout free surface

（a）t=0.34

（b）t=0.68

0 1 000 2 000 2 200

4

2

0

-2

-4

z

0 0.1 0.2 0.3 0.4 0.5

PressureMach

-4 -2 0 2 4r

SWSWirir**Superposition zoneof shock waves

-300 700 1 700 1 800

4

2

0

-2

-4

z

0 0.2 0.4 0.6 0.8 1

PressureMach

-4 -2 0 2 4r

Fig.7 Shock-bubble interaction with initial high innerpressure near free surface for M=1.1（In figures：leftare pressure distribution，and right are Mach numberfield distribution）

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parameters such the pressure and velocity in the flowfield are analyzed in detail. The main conclusionsare as follows:

1) During the interaction process among the bub⁃ble, shock wave, and free surface, there are multiplecomplex rarefaction wave system and shock wave sys⁃tem.

2) When the internal and external pressures of thebubble are initially balanced, the bubble collapsesinto a toroidal bubble under the action of the inci⁃dent shock wave. The intensity of the shock wavegenerated in this process is far higher than that ofthe incident shock wave. Under the action of thisshock wave, an upward bulge will be formed on thecenter of the free surface. When there is free surface,the motion of the bubble is similar to that in freefield. However, under the action of the rarefactionwave reflected by the free surface, the collapse speeddecreases.

3) When there is a certain initial pressure insidethe bubble, the shock wave released when the bub⁃

ble begins to move couples with the incident shockwave to form more complex flow field distribution.Under the action of the incident shock wave, the bub⁃ble does not form a toroidal bubble. Compared withthe condition without incident wave, the bubbledrifts towards the free surface, its volume decreases,and the upwarp motion of the free surface increases.References［1］ JOHNSEN E，COLONIUS T. Shock-induced collapse

of a gas bubble in shockwave lithotripsy［J］. The Jour⁃nal of the Acoustical Society of America，2008，124（4）：

2011-2020.［2］ JOHNSEN E. Numerical simulation of non-spherical

bubble collapse：with applications to shockwave litho⁃tripsy［D］. California：California Institute of Technolo⁃gy，2007.

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（a）M=1.1

（b）M=0

Fig.8 Shape of bubble and free surfacewith different initial conditions

（c）M=1.3

4

2

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-4 -2 0 2 4r

t = 2.38

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-2

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-4 -2 0 2 4r

t = 2.39

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-2

z

-4 -2 0 2 4r

t = 1.26

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近自由液面气泡与冲击波的相互作用

叶曦 1，初文华 2，3，4，陈林 1，张阿漫 5

1 中国船舶及海洋工程设计研究院，上海 2000112 上海海洋大学 海洋科学学院，上海 201306

3 上海海洋大学 国家远洋渔业工程技术研究中心，上海 2013064 上海海洋大学 大洋渔业资源可持续开发教育部重点实验室，上海 201306

5 哈尔滨工程大学 船舶工程学院，黑龙江 哈尔滨 150001

摘 要：［目的目的］为了研究自由液面、气泡与冲击波三者之间的相互作用，［方法方法］基于间断迦辽金法，结合

Level Set与 Real Ghost Fluid 方法，分析复杂流场中冲击波传播特性及气泡运动特性，描述流场内各种波系的生

成与发展过程。［结果结果］结果表明：在相互作用过程中，流场生成的复杂波系中包含多个稀疏波和冲击波。自由

液面减缓了气泡的溃灭速度，而入射冲击波则加快了气泡的溃灭速度，并使自由液面的上拱运动增大。［结论结论］

所得结果可为水下爆炸对舰船结构的毁伤机理提供参考。

关键词：气泡；自由液面；冲击波；间断迦辽金法

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