Numerical simulation of three-dimensional bubble oscillations · Numerical simulation of...

Numerical simulation of three-dimensionalbubble oscillations

C. PozrikidisDepartment of Mechanical and Aerospace EngineeringUniversity of California, San DiegoLa Jolla, CaliforniaUSA

Abstract

The radial, axisymmetric, and three-dimensional shape oscillations of compressiblebubbles with negligible density, suspended in an effectively inviscid ambient liq-uid, are reviewed and discussed with emphasis on the mathematical modeling andnumerical simulation of the finite-amplitude motion. A generalized representationof the flow in the ambient liquid is presented, wherein the disturbance potential dueto the bubble oscillations is described in terms of (a) an interfacial distribution ofpotential dipoles amounting to a double-layer harmonic potential, and (b) a pointsource situated inside the bubble accounting for changes in the bubble volume.In the numerical implementation, the interface is discretized into a surface gridof quadratic elements whose nodes are convected either with the liquid velocityor with the normal component of the liquid velocity. The strength density of thedouble-layer potential is interpolated isoparametrically based on the interfacial grid,and calculated by solving an integral equation of the second kind using an iterativemethod. Once the solution is available, the tangential and normal components of theinterfacial velocity are evaluated, respectively, in terms of tangential derivatives ofthe harmonic or vector potential. The latter is evaluated as a single-layer potentialin terms of the strength of the interfacial vortex sheet underlying the double-layerpotential. Numerical instabilities are filtered out by smoothing the interfacial dis-tribution of the potential by Fourier–Legendre spectrum truncation. The method isapplied to simulate the simultaneous volume and shape oscillations of stationaryand moving bubbles. The results confirm the occurrence of resonance and illustratesalient features of the three-dimensional nonlinear motion.

www.witpress.com, ISSN 1755-8336 (on-line) WIT Transactions on State of the Art in Science and Engineering, Vol 6, © 2005 WIT Press

doi:10.2495/1-85312-785-X/02

44 Instability of Flows

1 Introduction

Gas bubbles are elementary units of evaporating liquids and dispersed gases, servingas vehicles for material transfer and species transport in a broad range of applica-tions. In practice, bubbles arise spontaneously from the capillary instability andbreakup of larger volumes of vapor or gas emerging from nozzles, or else are gen-erated by homogeneous and heterogeneous chemical reactions at nucleation sites.Specific examples where bubble dynamics plays an important role in an overallprocess can be found in the fields of fuel combustion, ocean and coastal hydro-dynamics, industrial aeration, foam mechanics, fluidized-bed reaction, boiling andheat exchange.

Research in the broad field of bubble dynamics seeks to describe how bub-bles behave under a given set of conditions, and how the bubble motion affectsthe hydrodynamics, acoustics, rheology, and dynamics of the microstructure offoamy and bubbly liquids. The expansion, shrinkage, oscillations, and interactionof gas and vapor bubbles have been studied extensively in the context of two-phaseflow, micromechanics of bubbly liquids, bubble cavitation and collapse near rigidand compliant boundaries, and noise production due to volume oscillations of anengulfed or entrained volume of gas.

Various aspects of the general problem have been reviewed periodically in thepast 30 years by Harper [18], Plesset and Prosperetti [39], Blake and Gibson[9], Feng and Leal [15], Ohsaka and Trinh [37], and Magnaudet and Eames [26]among other authors. The diversity of the general problem is due, to a large extent,to the coupling of three elementary types of motion: volume oscillations, shapeoscillations, flow-induced deformation, and deformation due to bubble-boundaryand inter-particle interactions. When resonance occurs between volume and shapemodes under specific conditions, a bubble acts as a significant source of acous-tic noise.

In spite of the voluminous literature on the general subject of bubble dynam-ics, several important aspects remain unresolved or poorly understood. By way ofillustrating the state of the art, laying out the physical foundation, and identifyingimportant unsolved problems, in Sections 2–6 we review the background theory ofvolume oscillations, shape oscillations, resonance between the volume and shapemodes, oscillations of moving and rising bubbles, and the self-propulsion of vibrat-ing bubbles.

Numerical simulation has provided us with a fruitful venue for studying thedynamics of the nonlinear motion. Boundary-integral methods for simulating bub-ble motion and oscillations are reviewed in Section 7. Recently, a generalizedvortex/boundary-element method was developed for simulating the three-dimensional oscillations of inviscid drops and compressible bubbles in potentialflow [42, 44, 45]. The foundation and implementation of the method are discussedin Sections 8 and 9. Numerical simulations of the oscillations of stationary andrising bubbles are presented in Sections 10 and 11, and a further discussion ofselected physical, theoretical, and computational aspects is given in the conclu-ding Section 12.


Instability of Flows 45

2 Volume oscillations of a spherical bubble

Consider a gas or vapor bubble whose size is so small that the effects of gravity arenegligible across the bubble diameter, and the bubble surface remains spherical at alltimes. The equilibrium bubble pressure, ps

B , and far-field pressure, ps∞, established

in the absence of fluid motion, are related by

psB = ps

∞ +2 γas, (1)

where a is the bubble radius, γ is the surface tension, and the subscript or subscripts denotes the steady state. If either the bubble pressure, or the ambient pressure,or both are changed, the bubble will execute radial volume oscillations around thesteady state. The velocity field of the time-dependent motion in the exterior ofthe bubble may be represented in terms of a point source with a varying strengthsituated at the bubble center.

Analysis shows that, if viscous effects in the ambient liquid are negligible, theevolution of the the bubble radius is governed by Rayleigh’s equation

ad2a

dt2+

32

(dadt

)2

+2γρa

− pB − p∞ρ

= 0, (2)

(e.g., [39, 40]). The bubble pressure evolves according to the polytropic equationof state

pB − pV

pB(t = 0) − pV=(vB(t = 0)

vB

)λ

, (3)

where vB is the bubble volume, and pV is the partial vapor pressure of the ambientliquid inside the bubble, assumed to remain constant during the motion. The expo-nent λ is determined by the prevailing thermodynamic conditions; under adiabaticconditions, λ is equal to the ratio of the specific heats under constant pressure orvolume; for an ideal gas, λ = 1.4. In the case of a pure gas bubble, correspondingto pV = 0, eqn (2) takes the specific form

ad2a

dt2+

32

(dadt

)2

+2γρ

(1a

− α

as

)− α ps

∞ − p∞ρ

= 0, (4)

where α ≡ (as/a)3λ.

To study the bubble dynamics subject to a forced perturbation in the pressure atinfinity, we may set

p∞ = ps∞ [1 + εp cos(Ωt)], (5)

where εp is a dimensionless amplitude, and Ω is the angular frequency of the forcingfunction.

Unforced perturbations occur when the far-field pressure is kept at the steady-state level, p∞ = ps

∞. Classical analysis shows that small-amplitude oscillations



about the steady state due to an impulsive disturbance occur at the natural angularfrequency ω0, given by

ω20 =

3λρa2

s

ps∞ + (3λ− 1)

2γρa3

s

= [3λ (δ + 2) − 2]γ

ρa3s

, (6)

where

δ ≡ asps∞γ

(7)

is the reduced far-field pressure at steady state (e.g., [39]). For a bubble of radiusas = 1 mm, surface tension γ = 72 dyne/cm corresponding to the air-water interfaceat 25C, and far-field pressure of ps

∞ = 1 atm = 1.0133 × 106 dyne/cm2, we findδ = 1408. At this large value of δ, the effect of surface tension is negligible,and the term immediately after the first equal sign in the middle of expression (6)dominates.

More comprehensive evolution equations for the bubble radius that account forthe effects of viscous dissipation and compressibility, as well as generalized equa-tions of state for compressible bubbles that account thermal effects are available(e.g., [17]). For example, a generalization of eqn (2) is provided by the Kellerequation

(1 − 1

c

dadt

)a

d2a

dt2+

32

(1 − 1

3cdadt

) (dadt

)2

+1ρ

(1 +

1c

dadt

+a

c

ddt

) (2γa

+4µa

dadt

− pB + p∞

)= 0, (8)

where c is the speed of sound in the liquid, and µ is the viscosity of the liquid [21].Setting c = ∞ and µ = 0, we recover Rayleigh’s equation, eqn (2).

A generalization of the polytropic equation of state (3) is

pB − pV

pB(t = 0) − pV=(vB(t = 0) − vBs

/8.863

vB − vBs/8.863

)λ

, (9)

where the coefficient 8.86 corresponds to the hard-core van der Waals radius.

3 Shape oscillations of a spherical bubble

A bubble may undergo shape oscillations, sometimes called “asymmetric oscil-lations”, that are complementary to the radial volume oscillations discussed inSection 2. The shape of a nearly spherical bubble subject to a normal-mode pertur-bation is described by

r = a[1 + εmj T

|m|j (cos θ) cos(mϕ) sin(ωjt)

], (10)

where a is the mean bubble radius, (r, θ, ϕ) are spherical polar coordinates withorigin at the center of the undeformed bubble, j = 2, 3, . . ., is the azimuthal wave



number, m = 0,±1,±2, . . . ,±j is the meridional wave number, and εmj is adimensionless coefficient whose magnitude is small compared to unity. The depen-dence of the angular frequency, ωj , on the meridional wave number, m, has beendeliberately suppressed in hindsight.

The modified Legendre functions of degree j and orderm, denoted by T |m|j , are

given by

T|m|j (cos θ) =

12

[2j + 1π

(j − |m|)!(j + |m|)!

]1/2

P|m|j (cos θ), (11)

where P |m|j are the associated Legendre functions, and P 0

j are the Legendre poly-nomials. The computation of the modified and associated Legendre functions isdiscussed in Appendix A.

To first order in the oscillation amplitude εmj , changes in the bubble volumeinduced by the shape oscillations do not arise, and the bubble radius a may beassumed to be constant. Linear analysis for irrotational flow in the absence ofsignificant viscous effects shows that the square of the angular frequency ωj isgiven by

ω2j = (j2 − 1)(j + 2)

γ

ρ a3 , (12)

independent of the meridional wave number, m ([24], p. 475). The distribution ofthe harmonic potential over the bubble surface corresponding to the shape describedby eqn (10) is

φ = −εmjωja

2

j + 1T

|m|j (cos θ) cos(mϕ) cos(ωjt)

= −εmj[γa

ρ

(j − 1)(j + 2)j + 1

]1/2

T|m|j (cos θ) cos(mϕ) cos(ωjt). (13)

It is instructive to compare the exact relation (12) with the predictions of asimplified model based on two-dimensional standing capillary waves. The squareof the frequency of these waves is given by

Ω2(k) =γ k3

ρ, (14)

where k = 2π/L is the wave number, andL is the spatial period (e.g., [40], p. 464).Setting L = 2πa/j, whereupon k = j/a, we obtain the estimate

ω2j γ

ρ a3 j3, (15)

which is a fair approximation of the right-hand side of eqn (12). For j = 2, theprediction (15) is lower by 33% from the exact value.



Viscous effects cause the amplitude of the oscillations to decrease in time. Athigh Reynolds numbers, we obtain the exponential decay

εmjεmj (t = 0)

= exp(−βjt), (16)

where the damping coefficient βj is given

βj = (j + 2)(2j + 1)µ

ρa2 , (17)

and µ is the liquid viscosity ([24], p. 475).

4 Resonance of volume and shape nodes

Combining eqns (6) and (12), we find that the ratio of the angular frequencies ofthe volume and shape oscillations is given by

χ ≡ ω0

ωj=[

3λ (δ + 2) − 2(j2 − 1)(j + 2)

]1/2

, (18)

for j = 2, 3, . . . . Solving for the reduced far-field steady-state pressure δ, we find

δ =13λ

[ χ2(j2 − 1)(j + 2) + 2 ] − 2. (19)

Resonance between the jth shape mode and the radial mode occurs when eqn (19)is satisfied with χ being either an integer or a rational number belonging to theinfinite set: 1/3, 1/2, 2/3, . . ..

First subharmonic resonance occurs when χ = 2. For δ = 1265 and 1663, cor-responding to an air bubble of radius 1 mm suspended in water under atmosphericpressure, and λ = 1.25, resonance occurs at the high-order shape modes j = 10and 11. Conversely, resonance of the j = 2 mode occurs when δ = 11.333, corre-sponding to a tiny air bubble of radius 8 µm. Ohsaka and Trinh [37] observed moderesonance on a bubble suspended by ultrasonic radiation in the j = 4, 5, and 6axisymmetric modes, m = 0. In these experiments, forced radial oscillations wereinduced by acoustic forcing at selected frequencies.

From the viewpoint of stability, resonance occurs when volume oscillationsbecome unstable to shape modes excited when the amplitude of the interfacialmotion exceeds a critical threshold [15]. In particular, volume oscillations of suf-ficiently high amplitude cause the excitation of surface modes whose natural fre-quency is closest to half the frequency of the volume oscillations. To see this more



clearly, we consider the linear evolution equation for the amplitude of the shapemodes at high Reynolds numbers,

d2εmjdt2

+Adεmjdt

−B εmj = 0, (20)

where

A = 3a

a+ 2(j + 2)(2j + 1)

µ

ρa2 , (21)

B = (j − 1)(a

a− (j + 1)(j + 2)

γ

ρa3 − 2 (j + 2)µa

ρa3

), (22)

and a dot denotes a derivative with respect to time (e.g., [16, 17, 38]).Assuming now that the bubble radius varies sinusoidally in time with angular

frequency Ω, we set

a = a [1 + ε0 cos(Ωt)], (23)

where a is the mean bubble radius. Linearizing eqn (22) with respect to ε0, wederive the more compact form

d2Emj

dt2+ 2βj

dEmj

dt+[ω2

j + ε0 Ω2θj cos(Ωt)]Em

j = 0, (24)

where βj is the damping coefficient given in eqn (17), and we have defined

Emj ≡ a2/3 εmj , θj ≡ j +

12

+ 3(ωj

Ω

)2(25)

[16, 17]. In the absence of viscous dissipation, βj = 0, eqn (24) reduces toMathieu’s equation, and this clearly suggests the possibility of parametric exci-tation of the surface modes.

McDougald and Leal [27, 28] performed numerical simulations of axisymmetricbubble oscillations about the spherical or prolate equilibrium shape – the latter dueto an imposed pressure field – and found that the small-deformation theory fur-nishes reasonable estimates for the onset of shape oscillations at first subharmonicresonance, χ = 2. More recently, the present author [44] performed numerical sim-ulations of three-dimensional shape oscillations and demonstrated the coupling ofradial and three-dimensional shape modes, as will be discussed in Section 10.

5 Oscillations of rising and convected bubbles

When a gas or vapor bubble rises due to gravity or otherwise translates relative tothe surrounding liquid with velocityU at high Reynolds numbers,Re ≡ ρUa/µ, the



free surface deforms in response to the dynamic pressure field due to the relativemotion. In the definition of the Reynolds number, ρ is the liquid density, µ isthe liquid viscosity, and a the equivalent bubble radius defined by the equationvB = 4πa3/3, where vB is the bubble volume. The deformed bubble shape isdetermined by the Weber number, We ≡ ρ a U2/γ, expressing the magnitude ofthe dynamic pressure relative to capillary pressure due to the surface tension, γ.

If We is sufficiently small and gravitational effects are negligible across thebubble diameter, the bubble obtains a nearly spheroidal shape of revolution withrespect to the direction of motion. As the Weber number is raised, the bubbletransforms into a spherical cap and possibly develops a skirted shape (e.g., [4, 33,50]). Such shapes have been computed by several authors using asymptotic andnumerical methods based on boundary-integral formulations [5, 12, 18, 32, 36].Because the velocity of a rising bubble is a function of the a priori unknownbubble shape that is determined by the Weber number, it must be either assumed orcomputed as part of the solution.

Laboratory and theoretical studies have shown that when Re and We fall ina certain regime in parametric space, a path instability occurs in which a nearlyspherical or ellipsoidal bubble – but not a spherical-cap bubble – strays away fromthe rectilinear path and exhibits a zig-zag or spiraling helical motion (e.g., [11, 26]).Moreover, depending on its volume, a bubble may initially rise in the zig-zagpath and then exhibit a helical path, but not vice-versa. Recent experiments havesuggested that spherical bubbles whose volume falls in a certain range rising inclean water exhibit a zig-zag path, whereas ellipsoidal bubbles exhibit a spiralpath [51].

Although a physical reason for the path instability has not been identified withabsolute certainty, there is ample evidence to suggest that shedding of rotationalfluid from an attached or detached wake developing behind the bubble is the mostprobable cause, as first proposed by Saffman [46]. Support for this explanation isprovided by the observation that the purity of the liquid determining the mobil-ity of the interface plays an important role in the critical Reynolds number forunstable motion. More recent flow visualization studies by Lunde and Perkins [25]have indicated that the helical path is associated with the instability of a steadywake, whereas the zig-zag path is associated with the ejection of hairpin vortices.

In an effort to probe the physical origin of the path instability, Hartunian and Sears[20] performed an experimental investigation accompanied by a simplified stabilityanalysis of the rectilinear motion of a spherical bubble. More recently, Meiron [30]and Feng [13] studied the stability of non-spherical bubbles and demonstrated that,in contrast to the conclusions of the earlier work, a path instability associated withshape oscillations does not arise. Moreover, their analysis revealed that the degen-eracy of the frequency of shape oscillations with respect to the azimuthal modes isremoved when the unperturbed bubble shape is non-spherical. The deviation fromsphericity was found to reduce the frequency of the oscillation.A heuristic argumentfor this behavior can be made by referring to eqn (14), and observing that deviationfrom sphericity increases the circumferential arc length for a given bubble volume,and thereby reduces the wave number k.



Lunde and Perkins [25] observed shape oscillations of rising bubbles in theaxisymmetric 2-0 mode wherein the bubble shape alternates between an oblate anda prolate spheroidal shape, as well as in the three-dimensional 2-2 mode whereinthe bubble shape alternates between two triaxial ellipsoidal shapes. The authorsattributed the oscillations to mode excitation due to the shedding of vorticity behindthe bubble. The frequency of the oscillations was found to be in good agreementwith theoretical predictions for stationary oscillating bubbles. Thus, vortex sheddingappears to be significant only insofar as to initiate normal modes that subsequentlyevolve as though the ambient liquid were truly inviscid.

Numerical simulations of finite-amplitude axisymmetric and three-dimensionalbubble oscillations coupling volume and surface modes were conducted recentlyby the present author [45], as will be discussed in Section 11. A theoretical orcomputational analysis of the stability and nonlinear dynamics of the wake formingbehind a rising bubble is not available.

6 Self-propulsion of vibrating bubbles

When a small bubble is irradiated with sound, it is observed to exhibit an erratic,Brownian-like motion. Benjamin and Strasberg [7, 48] proposed that the physicalreason is the parametric excitation of shape modes. The ability of a deforming bodyto propel itself by means of surface deformation in potential flow was subsequentlyconfirmed by Saffman [47], and further analyzed by Benjamin and Ellis [6] usingthe Hamiltonian formulation for irrotational flow in a perfect liquid.

The analysis of Benjamin and Ellis [6] culminated in an equation of motionfor the bubble volume centroid, xc, involving the Kelvin impulse, I, and dipolecoefficient, d,

d(vB xc)dt

= −I − 4π d, (26)

where

xc ≡ 1vB

∫∫∫Bubble

x dV, I ≡ −∫∫

Bubble

φ n dS, (27)

and φ is the harmonic potential of the irrotational flow in the bubble exterior. Notethat the instantaneous structure of the flow must be available before the bubblemotion can be computed by integrating in time eqn (26).

Benjamin and Ellis [6] derived an evolution equation for the dipole coeffi-cient induced by the deformation of a nearly spherical bubble, and demonstratedthat self-propulsion is accountable wholly to interactions of the adjacent j andj + 1 surface modes discussed in Section 3. These conclusions were based onthe assumption that changes in the amplitude of the bubble radius are of secondorder in the amplitude of the shape modes, which is justified only for a narrowrange of prospective applications. In a related effort, Miloh and Galper [34] ana-lyzed the self-propulsion of a smooth deformable body in an inviscid liquid subjectto arbitrarily prescribed temporal shape variations.



Mei and Zhou [29] used the method of multiple time scales to study the reso-nance of forced volume oscillations and the two adjacent surface modes. Numericalsolutions of the differential equations governing the amplitude of the three modesrevealed that chaotic oscillations and erratic drift may occur, in agreement withthe predictions of Benjamin and Ellis [6]. Feng and Leal [14] remarked that theanalysis of Mei and Zhou [29] overlooks the coupling between bubble translationand deformation, in that the surface nodes are assumed to evolve as though the bub-ble were stationary, and presented a comprehensive analysis incorporating viscousdamping in high-Reynolds-number flow. Still, their analysis is restricted to circum-stances where translational motion arises only as the result of coupling betweentwo adjacent shape modes, both of which are excited by the volume oscillations.

7 Boundary-integral methods

The vast majority of numerical studies on bubble deformation and oscillations havebeen conducted using the boundary-integral method for axisymmetric potentialflow. Examples include the simulations of the interaction between two coaxialcavitation bubbles near a rigid boundary by Blake et al. [10], and the more recentsimulations of the oscillation of solitary bubbles coupling radial and axisymmetricsurface modes by McDougald and Leal [27, 28].

The numerical method implemented in these studies relies on the boundary-integral representation for the harmonic potential in the ambient liquid originatingfrom Green’s third identity. In this approach, the interfacial distribution of thenormal derivative of the potential is found by solving a Fredholm integral equationof the first kind using boundary-element methods, and the interfacial distribution ofthe potential is updated using Bernoulli’s equation for the exterior irrotational flow.The bubble pressure evolves according to an equation of state that is appropriatefor the bubble constitution and reflects the assumed thermodynamic conditions.The onset of numerical instabilities, invariably occurring in problems of interfacialpotential flow, can be prevented by smoothing the interfacial potential, the bubbleshape, or both, using either local point averaging or global Fourier filtering.

Numerical methods for simulating three-dimensional flow have been imple-mented by several authors, as reviewed by Blake et al. [8], Wang [49], andHarris [19]. As in the case of axisymmetric flow, Green’s third identity is typicallyinvoked to derive an integral equation of the first kind for the normal componentof the interfacial liquid velocity. However, because of the coarser structure of thethree-dimensional grid and inaccuracies in evaluating the single- and double-layerharmonic potential over the three-dimensional interfacial elements, the numericalerror is significantly more pronounced than that for axisymmetric flow, even inthe absence of surface tension. Moreover, because an efficient and general-purposemethod for smoothing interfacial irregularities is not available, simulations are car-ried out only for a limited length of time that depends on the particular problemunder consideration.

As an example, Wang [49] studied the non-axisymmetric collapse of a gas bubblenear a planar surface that is inclined at a certain angle with respect to the direction of



gravity, neglecting the effects of surface tension. In the boundary-element imple-mentation, the interface was discretized into flat triangular elements, all surfacevariables were approximated with linear functions over each element, and the lin-ear system descending from the integral equation of the first kind for the interfacialdistribution of the normal component of the velocity was solved by Gauss–Seideliteration.

Harris [19] emphasized that, as the level of discretization is refined, the inte-gral equation of the first kind originating from Green’s third identity becomesill-conditioned. To circumvent this difficulty, an alternative integral formulationresulting in a hyper-singular integral equation was developed, and numerical simu-lations of the growth and collapse of a gas bubble near a flat surface in the absenceof surface tension were carried out using the Galerkin boundary-element method(e.g., [43]).

8 Double-layer representation and generalized vortex methods

The search for an accurate and efficient numerical method that is capable of simu-lating three-dimensional interfacial motions for an extended period of time, whilealso accounting for the effect of surface tension, has led to an ingenious dual rep-resentation conceived by Baker and coworkers [1–3].

In this approach, the harmonic potential in the ambient liquid is representedby an interfacial distribution of potential dipoles complemented by a point sourcesituated inside the bubble; the latter is included to account for possible changes in thebubble volume. From a mathematical standpoint, the presence of the point sourcecompletes the deficient range of the double-layer potential for exterior flow ([31],p. 172). The representation is finalized in a way that preserves linearity, by settingthe strength of the point source proportional to the surface integral of the density ofthe double-layer potential. Once the strength density of the double-layer potentialis available, the tangential and normal components of the interfacial velocity can beevaluated, respectively, in terms of tangential derivatives of the harmonic and vectorpotential. A key idea is that the latter can be expressed as a single-layer potentialin terms of the strength of the interfacial vortex sheet underlying the double-layerpotential.

One important advantage of the generalized vortex method, as compared to meth-ods based on Green’s third identity, is that the strength density of the double-layerpotential is found by solving an integral equation of the second kind, which canbe done accurately and economically using iterative methods based on successivesubstitutions. In fact, comparisons show that, when the domain of solution is nearlyspherical, results obtained by the double-layer representation are more accurate byone order of magnitude than those obtained by the boundary-integral method basedon Green’s third identity. However, for large deformations from the spherical shapeand for coarse discretizations, the boundary-integral method appears to be morereliable.



8.1 Mathematical formulation

We consider potential flow past or due to the expansion, shrinkage, or oscillationsof a gas or vapor bubble occupied by a fluid with negligible density and viscosity,suspended in an effectively inviscid and incompressible ambient fluid. The irro-tational velocity field outside the bubble, u, can be expressed as the gradient ofthe single-valued potential φ, u = ∇φ. The continuity equation requires that thepotential satisfy Laplace’s equation,

∇2φ = 0. (28)

In the first step of the mathematical formulation, the harmonic potential φ isexpressed in terms of: (a) a specified incident potential of interest,φ∞, (b) a double-layer potential defined over the bubble surface, D, and (c) a point source situatedinside the bubble at the point xs,

φ(x0) = φ∞(x0) +∫∫

D

q(x) n(x) · ∇G(x,x0) dS(x)

− s(q)G(x0,xs), (29)

where q is the a priori unknown density of the double-layer potential, n is the unitvector normal to the bubble surface pointing toward the exterior, s(q) is the strengthof the point source, and G(x,x0) is the Green or Neumann function of Laplace’sequation in three dimensions (e.g., [40], p. 516). If the bubble is suspended in avirtually infinite fluid, G(x,x0) is the free-space Green function, whereas if thebubble is suspended in a semi-infinite fluid bounded by an impenetrable plane wall,G(x,x0) is the Neumann Green’s function whose gradient vanishes over the wall.

To remove the ambiguity in the definition of the strength of the point sourcewhile preserving linearity, we set

s(q) ≡ −β(t)( 4πSD

)1/2∫∫

D

q dS, (30)

where SD is the surface area of the bubble surface, and β is an arbitrary dimen-sionless coefficient.

Taking now the limit of eqn (29) as the point x0 approaches the bubble surfacefrom the exterior, and expressing the limit of the double-layer potential in terms ofits principal value, denoted by PV , we derive an integral equation of the secondkind for q,

φ(x0) = φ∞(x0) +12q(x0) +

∫∫ PV

D

q(x) n(x) · ∇G(x,x0) dS(x)

− s(q)G(x0,xs). (31)

The surface potential on the left-hand side is considered to be a known. Rearranging,



we derive the standard form

q(x0) = 2

[−∫∫ PV

D

q(x) n(x) · ∇G(x,x0) dS(x)

+ s(q)G(x0,xs) + φ(x0) − φ∞(x0)

]. (32)

The homogeneous integral equation corresponding to eqn (32) is

ψ(x0) = −2∫∫ PV

D

ψ(x) n(x) · ∇G(x,x0) dS(x) + 2 s(ψ)G(x0,xs), (33)

where ψ(x) is an eigenfunction. When the last term on the right-hand side of eqn(33) is absent, the simplified homogeneous equation has a constant eigensolution,and the corresponding inhomogeneous equation, eqn (32), has an infinite numberof solutions that differ by an arbitrary constant. Since including the point-sourceterm disqualifies the constant eigensolution, the presence of the point source in thecompound integral representation to account for bubble expansion or shrinkage isimperative, in agreement with physical intuition.

The spectrum of eigenvalues of the compact integral operator expressed by theright-hand side of eqn (33) is not known for arbitrary bubble shapes. Insights can begained by considering a spherical bubble of radius a, placing the point source at thebubble center, and assuming that the eigenfunction ψ a the constant value, c. Usingthe definition of the point-source strength, s, shown in eqn (30), we find s(c) =−βc4πa. Substituting this expression along with G(x0,xs) = 1/(4πa) in (33),we find c = (1− 2β)c, which shows that a nontrivial constant eigenfunction existsonly when β = 0. A necessary but not sufficient condition for the integral operatoron the right-hand side of eqn (33) to be a contraction mapping is |1 − 2β| < 1 or−1 < 1 − 2β < 1, which mandates that 0 < β < 1.

Alternatively, the velocity field associated with the double-layer potential can beexpressed as the curl of a vector potential, A, and the velocity can correspondinglybe expressed in the form

u(x0) = ∇0φ∞(x0) + ∇0 × A(x0) − s(q) ∇0G(x0,xs), (34)

where the gradient ∇0 involves derivatives with respect to the position of the eval-uation point x0. The vector potential is given by the integral representation

A(x0) =∫∫

D

ζ(x)G(x,x0) dS(x), (35)

where ζ is the strength of the vortex sheet corresponding to the double-layer poten-tial, given by

ζ = n × ∇q, (36)

(e.g., [40], p. 503). The right-hand side of eqn (36) involves only tangential deriva-tives of the density distribution q.



8.2 Algorithm and implementation

The evolution of the bubble interface is simulated according to the following steps:

Step 1: Specify the initial shape of the interface, interfacial distribution of theharmonic potential φ, bubble pressure, and pressure at infinity.

Step 2: Solve the integral equation, eqn (32), for the double-layer density, q.

Step 3: Evaluate the harmonic potential φ over the interface using eqns (29)and (30).

Step 4: Compute the tangential component of the velocity over the interface, givenby

n × (u × n) = P · u, (37)

in terms of surface derivatives of φ, where P ≡ I − nn is the tangentialprojection operator, and I is the unit matrix.

Step 5: Compute the strength of the vortex sheet ζ from eqn (36).

Step 6: Compute the interfacial distribution of the vector potential A fromeqn (35).

Step 7: Compute the normal component of the interfacial velocity, given byn (n · u), in terms of the normal component of the incident flow, tangentialderivatives of the vector potential A, and the normal component of the pointsource corresponding to the second term on the right-hand side of eqn (29).

Step 8: Advance the position of marker points distributed over the interface eitherwith the fluid velocity or with the normal component of the fluid velocity. Inthe first case, the marker points are Lagrangian point particles of the exteriorliquid.

Step 9: Update the harmonic potential at the interfacial marker points using theunsteady Bernoulli equation for irrotational flow, taking into considerationthat the pressure of the liquid at the bubble surface is equal to the bubblepressure pB reduced by the capillary pressure 2 γ κm, where γ is the surfacetension and κm is the interface mean curvature; for a spherical bubble ofradius a, κm = 1/a (e.g., [40], p. 489). The precise form of Bernoulli’sequation will be discussed later in this section.

Step 10: Update the bubble pressure according to a polytropic equation of state.

Step 11: Update the pressure at infinity according to a specified protocol.



The rate of change of the potential following Lagrangian point particles movingwith the liquid velocity involved in Step 9 is given by

DφDt

=12

|u|2 +2γρκm − pB − c

ρ+ g · x, (38)

where D/Dt is the material derivative, ρ is the liquid density, c is Bernoulli’sconstant, and g is the acceleration of gravity. On the other hand, the rate of changeof the potential following marker points moving with the normal component of thefluid velocity, is given by

dφdt

= |u · n|2 − 12

|u|2 +2γρκm − pB − c

ρ+ g · x. (39)

When gravitational effects are insignificant across the bubble diameter, the lastterms of the right-hand sides of eqns (38) and (39) are absent. If, in addition, theflow tends to become quiescent at infinity, Bernoulli’s constant c is equal to theliquid pressure far from the bubble, c = p∞. On the other hand, if the flow far fromthe bubble executes steady streaming motion with uniform velocity U, Bernoulli’sconstant is given by

c = p∞ +12ρ |U|2. (40)

In the numerical implementation [44, 45], the bubble surface is discretized intoan unstructured grid of six-node curved triangular elements, and all geometricalvariables and surface functions are approximated with quadratic functions over eachtriangle with respect to local triangle coordinates (e.g., [41]). The initial interfacialgrid is generated by successively subdividing each face of a regular octahedron intofour descendant elements, and then projecting the nodes onto the sphere. Significantimprovements in accuracy can be achieved by averaging the normal vector and meancurvature at each node over the elements sharing the node, and then describing thedistribution of the normal vector and mean curvature over each element by quadraticinterpolation.

The three scalar components of the tangential gradient of the potential φ can becomputed by solving a system of three linear equations for the components of thesurface gradient ∇sφ = P · ∇φ at each local node of a triangle,

∂φ

∂ξ=∂x∂ξ

· ∇sφ,∂φ

∂η=∂x∂η

· ∇sφ, n · ∇sφ = 0, (41)

where (ξ, η) are local curvilinear triangle coordinates. To improve the accuracy,the components of the surface gradient are averaged at each node over all triangleshosting each node. A similar method is used for the computation of the normalcomponent of the curl of the vector potential, A, which is defined in terms oftangential derivatives.

An important feature of the numerical method is the numerical solution of theintegral equation, eqn (32), by the method of successive substitutions. To illustrate



the effect of the dimensionless parameter β and location of the point source, weconsider a spherical interface of radiusa centered at the origin, and solve the integralequation for a null distribution of the surface potential φ, and a uniform initial guessfor the double-layer density q. Numerical experimentation confirms the theoreticalpredictions discussed in the paragraph following eqn (33). Each time an iterationis carried out, the error is multiplied by the factor 1 − 2β, provided that allowanceis made for the spatial discretization error [44]. Moreover, the rate of convergenceis fastest when the point source is placed at the center of the spherical bubble.

8.3 Accuracy

A main task of the generalized vortex method is the computation of the normalderivative of the harmonic potential over the bubble surface in terms of tangentialderivatives of the vector potential. Previous authors have computed the normalderivative directly by solving an integral equation of the first kind originating fromGreen’s third identity.

To compare the performance of the two methods, the author implemented aboundary-element collocation method for solving the integral equation of the firstkind [44]. In the numerical procedure, all geometrical variables and surface func-tions are approximated with quadratic functions with respect to the triangle coordi-nates over the boundary elements, and the integral equation is applied at the surfacenodes to produce a system of linear equations for the nodal values of the normalderivative. For simplicity, the linear system is finally solved by the method of Gausselimination (e.g., [41]).

As a benchmark, we consider Laplace’s equation for a harmonic function, φ, inthe exterior of a prolate spheroid with major and minor axes as and bs, orientedalong the x axis. In the problem statement, we specify that φ has the constant valueφs over the surface of the spheroid and decays to zero at infinity. Expressing theLaplacian operator in prolate spheroidal coordinates, and applying the method ofseparation of separation of variables (e.g., [24], p. 140), we find that the surfacedistribution of the outward normal derivative is given by

n · ∇φ = −φs

asF (X), (42)

for −as ≤ x ≤ as, where X = x/as,

F (X) =2 ζs

log ζs−1ζs+1

1√(ζ2

s − 1)(ζ2s −X2)

, (43)

and ζs = as/√a2

s − b2s. As bs tends to as, ζs tends to infinity, the function F (X)tends to unity, and eqn (42) yields the expected result for the sphere, n · ∇φ =−φs/as.

Numerical testing showed that, for bs/as = 0.90, the RMS error of the func-tion F (X) evaluated at the nodes and computed by the double-layer formulationwith the bubble surface discretized into 32, 128, and 512 elements defined by 66,



Figure 1: Nodal values of the functionF (X) defined in eqn (43) for a spheroid withaspect ratio bs/as = 2/3, and surface discretization into 512 boundaryelements. The circles correspond to the indirect method, and the squarescorrespond to the direct method based on Green’s third identity.

258, and 1026 nodes is, respectively, 3.3 × 10−3, 3.9 × 10−4, and 3.9 × 10−5.When the solution is computed using the direct formulation based on Green’s thirdidentity, the corresponding error is 1.1 × 10−2, 1.4 × 10−3, 1.7 × 10−4. Thus,results obtained by the completed double-layer representation are more accurateby one order of magnitude. The CPU time required for computing the solution withthe direct formulation is, respectively, 2 s, 13 s, and 8 min on a 600 MHz INTEL

processor running LINUX. The CPU time required by the completed double-layerrepresentation depends on the accuracy of the initial guess. In a dynamical simula-tion, the converged solution at the previous time step is used as the initial guess atthe next step, and only a few iterations are necessary, at a cost that is less than onetenth of that incurred by the direct method.

As the spheroid aspect ratio is increased, the accuracy of the direct method basedon Green’s third identity becomes comparable to, and then superior to that, ofthe indirect method. For example, when bs/as = 2/3, the RMS error for sur-face discretization into 32, 128, and 512 elements is, respectively, 1.2 × 10−2,1.5 × 10−3, 1.7 × 10−4 for the indirect method, and 1.2 × 10−2, 1.6 × 10−3,2.0 × 10−4 for the direct method. Figure 1 shows numerical results for the func-tion F (X) obtained with the finest discretization into 512 elements. The circlescorrespond to the indirect method, the squares correspond to the direct method,and the solid line represents the exact solution. In this case, the main advantage ofthe indirect method is the lower demand in CPU time. For bs/as = 0.5 and 512boundary elements, the RMS numerical error is 3.8×10−4 for the indirect methodand 2.8 × 10−4 for the direct method. The deterioration in the performance of the



indirect method is due to inaccuracies in the numerical computation of the tangen-tial derivatives of the vector potential required for the computation of the normalderivative of the harmonic potential.

9 Surface smoothing

Numerical simulations based on the generalized vortex method survive only for alimited amount of time that depends on the type of motion, level of discretization,and size of the time step. Grid distortion due to Lagrangian drifting of the sur-face nodes combined with the magnifying action of the Rayleigh-Taylor instabilityduring certain stages of the motion unavoidably lead to failure. To carry out simula-tions for an indefinite period of time, smoothing of the surface potential is required.Blake et al. [8] review methods for smoothing a function defined over a three-dimensional surface by function interpolation and approximation from scattereddata. A different method of smoothing over a closed surface based on spectrumtruncation was recently developed and implemented by the present author [42, 45].

To implement the method, we introduce a set of orthonormal, complex surfaceharmonic functions over the surface of the unit sphere,

Φmj (θ, ϕ) ≡ T

|m|j (cos θ) exp(−imϕ), (44)

where θ is the azimuthal angle,ϕ is the meridional angle, i is the imaginary unit, andT

|m|j are the modified Legendre functions of degree j and orderm given in eqn (11).

The modified Legendre functions are designed such that the functions Φmj (θ, ϕ)

comprise an orthonormal set with unit weighting function over the surface of theunit sphere,

∫∫Unit Sphere

Φ|m1|j1

Φ|m2|∗j2

dS = δj1j2 δm1m2 , (45)

where an asterisk denotes the complex conjugate, and δkl is Kronecker’s delta.Any nonsingular complex function f(θ, ϕ) defined over the surface of the unit

sphere may be expressed as an infinite Fourier–Legendre series in terms of theorthonormal basis functions defined in eqn (44),

f(θ, ϕ) =∞∑

j=0

j∑m=−j

cjmΦmj (θ, ϕ), (46)

where cjm = 12 (ajm + i bjm), are complex coefficients, and ajm, bjm are the

corresponding real coefficients. Using the orthogonality condition (45), we findthat the complex coefficients are given by

cjm =∫∫

Unit Sphere

f(θ, ϕ)(Φ|m2|j2

)∗ dS. (47)

If the function f(θ, ϕ) is real, then cj,−m = c∗jm and accordingly, aj,−m = ajm



and bj,−m = −bjm. Expansion (46) may then be recast into the form

f(θ, ϕ) =12a00 +

∞∑j=1

[12aj0 T

0j (cos θ)

+j∑

m=1

Tmj (cos θ) (ajm cosmϕ+ bjm sinmϕ)

], (48)

where the real Fourier–Legendre coefficients are given by

ajm =∫∫

Unit Sphere

f(θ, ϕ) Tmj (cos θ) cosmϕ dS,

bjm =∫∫

Unit Sphere

f(θ, ϕ) Tmj (cos θ) sinmϕ dS. (49)

Smoothing by spectrum truncation is implemented by truncating the infinite upperlimit of summation on the right-hand side of eqn (48) with respect to j to a certainfinite value N , leaving (N + 1)2 real coefficients on the right-hand side.

In the numerical method, the integrals on the right-hand side of eqns (49) are com-puted by discretizing the surface of the unit sphere into a grid of curved triangles, asdiscussed in Section 8, and then integrating numerically over the individual curvedtriangles using the 12-point Gaussian quadrature for the triangle (e.g., [41]). Inpractice, smoothing is implemented by multiplying the vector containing the nodalvalues of the surface function by a fixed smoothing matrix that is prepared before-hand by the method of impulses. In this method, the value of a surface function isset to unity at one cardinal node and to zero at all other nodes, the spectrum of thenumerically generated surface field is computed and truncated, as described previ-ously in this section, and the resulting vector of node values is placed at the columnof the smoothing matrix corresponding to the cardinal node. The process is thenrepeated sequentially for all nodes. Numerical testing has shown that, when the unitsphere is discretized into 512 curved triangles defined by 1026 nodes, the low-endfamily of the Fourier–Legendre coefficients are recovered up to j = 0, . . . , 8 witha less than 0.5% error.

To smooth a function over an arbitrary simply connected closed surface, weproject it onto the unit sphere by mapping the individual nodes, carry out spectrumexpansion and series truncation on the unit sphere, and then transfer the smoothednodal values back onto the actual surface.

10 Oscillations of stationary bubbles

In this section, we discuss the combined volume and shape oscillations of a sta-tionary bubble, as reviewed in Sections 2–4. The discussion is based on numericalsimulations conducted by the generalized vortex method discussed in Sections 8and 9 [45].



Figure 2: Oscillations in the axisymmetric, j = 2, m = 0 mode for ε02 = 0.15 andεp = 0.0. (a) Oscillation of the x axis of the bubble for different protocolsof smoothing, and (b) corresponding oscillations of the bubble volume.

Figure 2(a) illustrates the oscillations of the x axis of the bubble, denoted by b,reduced by the initial equilibrium bubble radius a0, for the axisymmetric, j = 2,m = 0 mode. The exponent λ introduced in eqn (3) is set to the value of 1.25. Atthe origin of computational time, the bubble surface is spherical, and the surfacepotential is perturbed from the null distribution to the normal mode distributionexpressed by eqn (13) with ε02 = 0.15. The pressure at infinity is fixed at thesteady-state value P s

∞, that is, εp in eqn (5) is set to zero. The reduced equilibriumpressure at infinity is δ ≡ asp

s∞/γ = 10, which is near the critical value for perfect

resonance, 11.333. The dimensionless time is defined as t′ ≡ tγ1/2/(ρ1/2a3/20 ).

In this simulation, the interface was discretized into 512 curved triangles de-fined by 1026 nodes, and time integration was carried out by the second-orderRunge–Kutta method with a constant time step ∆t′ = 0.05. The marker pointsdefining the boundary elements are convected with the fluid velocity on the side



of the liquid. According to linear theory, the ratio b/a0 oscillates around the meanvalue of unity, with dimensionless period T ′ = 1.814 and reduced amplitude0.0946, as indicated by the horizontal and vertical lines in Fig. 2(a).

The circles in Fig. 2(a) show results obtained without smoothing, illustrating thebreakdown of the simulation at an early stage of the motion due to the onset of anumerical instability. The solid, dotted, dashed, and long dashed lines show resultsof simulations where the surface potential is smoothed after each time step, withthe Fourier–Legendre spectrum truncated at the upper limit jmax = 2, 4, 8, and 16,as discussed in Section 9. The dot-dashed line shows the results of a simulationwith jmax = 8 and the surface potential smoothed after every 10 steps. The sim-ulation with jmax = 8 and smoothing done after each step is stable and faithfullyreproduces the nature of the nonlinear motion. Figure 2(b) presents correspondingresults for the bubble volume reduced by the initial equilibrium volume. Accordingto linear theory for radial oscillations, the reduced period of volume oscillations isT ′ = 0.958, as indicated by the vertical lines in Fig. 2(b).

The solid lines in Fig. 3 extend the dashed lines of Fig. 2 to longer times, spann-ing nearly five periods of the shape oscillation and ten periods of the volume oscil-lation. This simulation requires nearly two days of CPU time on a 1.7 GHz Intelprocessor running Linux. The results show that, over the time period of the simu-lation, the amplitude of the x axis of the bubble remains constant in time, whereasthe amplitude of the volume oscillation increases and then saturates. The dashedlines in Fig. 3 correspond to an initial condition where both the shape and the volumemode are perturbed with equal amplitudes, ε02 = 0.10 and εp = 0.10. In this case,the amplitude of the shape oscillations grows in time, whereas the amplitude ofthe volume oscillations slowly decays.

Figure 4 presents results of a simulation for the non-axisymmetric, j = 2,m = 1 mode, for ε12 = 0.25, εp = 0.10, λ = 1.25, and reduced pressure atinfinity δ = 10. The solid, dotted, dashed, and dot-dashed lines in Fig. 4(a) illus-trate, respectively, the evolution of the Fourier–Legendre coefficients a00, a20, a21,and a22 corresponding to the distance of a point on the bubble surface from thebubble centroid, normalized with respect to the initial bubble radius a0. Thesecoefficients were computed using eqn (49), where θ and ϕ are the initial merid-ional and azimuthal angles of the nodes, while the integration is performed overthe instantaneous bubble surface.

The results show that subharmonic resonance causes the periodic transfer ofenergy between the volume mode represented by a00 and the dominant surfacemode corresponding to a21. Figure 4(b) shows interfacial contours in the xy planeat reduced times t′ = 16.8 and 18.0, and Fig. 4(c) shows a three-dimensionalperspective of the bubble surface at time t′ = 18.0, illustrating the structure ofthe interfacial grid after several cycles of oscillation.

Figure 5 shows corresponding results for the non-axisymmetric, j = 2, m = 2mode, for ε22 = 0.25, εp = 0.10, λ = 1.25, and reduced equilibrium pressure atinfinity, δ = 10. In this mode, the bubble surface alternates between two triaxialellipsoids whose major axes lie in the equatorial xy or xz plane. The behaviorof the Fourier–Legendre coefficients displayed in Fig. 5(a) is nearly identical to



Figure 3: Oscillations in the axisymmetric, j = 2, m = 0 mode, for (a) ε02 = 0.15and εp = 0.0 (solid lines), and (b) ε02 = 0.10 and εp = 0.10 (dashed line).

that depicted in Fig. 4(a) for the j = 2, m = 1 mode. Thus, the degeneracy of themotion in the linear regime of small-amplitude motion persists at finite amplitudes.Figure 5(b) shows a three-dimensional illustration of the interfacial shape at timet′ = 5.985.

11 Oscillations of moving bubbles

Consider a bubble that rises in a quiescent ambient liquid or else translates relativeto a stationary or moving ambient liquid at high Reynolds numbers, and assumethat the bubble surface is and remains spherical at all times due to the action ofsurface tension. In a frame of reference moving with the bubble, the liquid far fromthe bubble executes uniform streaming motion with velocity U.

Elementary analysis shows that, if viscous effects are negligible and the flow canbe assumed to be irrotational, the velocity field around the bubble is given by the



Figure 4: Oscillations in the non-axisymmetric, j = 2,m = 1 mode, for ε12 = 0.25and εp = 0.10. (a) Evolution of the Fourier–Legendre coefficients; (b)traces of the interface in the xy plane at reduced times t′ = 16.8 and18.0; (c) a three-dimensional perspective of the bubble surface at timet′ = 18.0.

gradient of the harmonic potential

φ(x) = U · x(1 +

12a3

r3

), (50)

where a is the bubble radius, x = x − xc is the distance of the field point x from thebubble center xc, and r = |xc| (e.g., [40]). If gravitational effects are insignificant,the pressure far from the bubble is given by

p∞ = c− 12ρ |U|2, (51)

where c is Bernoulli’s constant, as shown in eqn (40).However, because the pressure varies over the exterior surface of the bubble

according to Bernouli’s equation, the interface cannot remain spherical at all



Figure 5: Oscillations in the non-axisymmetric, j = 2,m = 2 mode, for ε22 = 0.25and εp = 0.10; (a) evolution of the Fourier–Legendre coefficients, and (b)three-dimensional illustration of the bubble surface at time t′ = 5.985.

times, and will deform by an amount that is determined by the Weber numberWe ≡ ρa|U|2/γ, where γ is the surface tension. Deformation may cause the bub-ble to translate as a whole along the x axis with a velocity that depends on theWeber number. Changes in the bubble volume and pressure are determined by themagnitude of the dimensionless Bernoulli constant ac/γ.

To be more specific, let us assume that the bubble translates along the x axis, andrefer to a frame of reference where the spherical bubble is stationary, whereuponthe incident flow is given by U = −Uxex. If the initial bubble shape is sphericaland the initial distribution of the potential is given by eqn (50), the bubble willdeform to an oblate spheroidal shape while translating and exhibiting axisymmet-ric shape oscillations in the j = 2, m = 0 mode. In real life, the shape oscillationswill eventually be dampened due to viscous dissipation, and the bubble will obtaina steady oblate spheroidal shape. Moore [35] showed that, when the deviation fromthe spherical shape is small, to first order in We, the minor and major axes of the



bubble are given by ax = a (1 − ε) and aσ = a (1 + 12ε), where

ε =316

We. (52)

The corresponding bubble axes ratio is

χ ≡ aσ

ax 1 +

932

We+O(We2). (53)

Subsequently, Moore [36] derived the improved nonlinear inverse relation

We 2χ3 + χ− 2

χ4/3 (χ2 − 1)3[ χ2 arcsecχ−

√χ2 − 1 ]2, (54)

which has been shown to provide accurate predictions for Weber numbers on theorder of unity [32].

In the numerical simulations, we consider a bubble with a spherical initial shapeof radius a, and specify the initial distribution of the surface potential,

φ(x) = U · x(

1 +12a3

r3

)

− εmj

[γa

ρ

(j − 1)(j + 2)j + 1

]1/2

T|m|j (cos θ) cos(mϕ), (55)

where εmj is the dimensionless amplitude and T|m|j are the modified Legendre

functions. In the simulations discussed in this section conducted by the presentauthor using the generalized vortex method [45], the bubble surface was dis-cretized into 512 elements defined by 1026 nodes, the polytropic exponent λ wasset to 1.25, and time integration was performed by the second-order Runge–Kuttamethod with a constant reduced time step ∆t′ = 0.02, where the dimensionlesstime, indicated by a prime, is defined as t′ = t (γ/ρa3)1/2. Unless specified other-wise, smoothing is implemented by truncating the infinite upper limit of summationon the right-hand side of eqn (48) with respect to j to the value of 8.

Figure 6 illustrates the oscillations of a bubble in the axisymmetric j = 2,m = 0mode, forWe= 0.25 and ε02 = 0.25. The initially upward curves in Fig. 6(a) representthe bubble axis along the x axis of symmetry, defined as half the maximum bubblesize in this direction, reduced by the initial equivalent bubble radius. The initiallydownward curves represent the bubble axis normal to the x axis, reduced by theinitial equivalent bubble radius. In the simulations represented by the solid anddotted lines, the marker points move with the velocity normal to the interface; inthe case of the solid line, smoothing is not enabled, whereas in the case of the solidline, smoothing is applied after each time step. In the simulations represented by thedashed line and circular symbols, the marker points move with the liquid velocity;in the case of the circular symbols, smoothing is not enabled, and in the case of thedashed line, smoothing is applied after each time step. While the agreement between



Figure 6: Axisymmetric oscillations of a bubble in the j = 2, m = 0 mode, forWe = 0.25 and ε02 = 0.25, computed with different options. (a) Evolutionof the reduced x axis, and (b) deformed grid developing when the nodesmove with the fluid velocity.

all simulations is excellent during the initial stage of the motion, only the simulationrepresented by the solid line is able to survive for an extended period of time. Inparticular, because the interfacial marker points are convected toward the rear of thebubble under the influence of the streaming flow, severe grid distortion occurs whenthe nodes are identified with material point particles moving with the fluid velocity,and the computation breaks down at an early stage of the simulation. Figure 6(b)depicts a deformed grid at the time when the simulation fails, corresponding to thedashed line in Fig. 6(a).

Moore’s [35] small-deformation theory predicts that, in the absence of oscil-lations, a bubble moving at We = 0.25 obtains a spheroidal shape, presentlyreferred to as a “Moore spheroid”, with minimum axis ax = 0.9531 a, maximumaxis aσ = 1.0234 a, and axes ratio aσ/ax = 1.0234, where a is the equivalentbubble radius. Consider a Moore spheroid, and assume that the distribution of the



Figure 7: Axisymmetric oscillations of a bubble in the axisymmetric j = 2, m =0 mode. The heavy solid lines correspond to the Moore spheroid forWe = 0.25. (a) Bubble axis in the x direction, (b) bubble axis in thetransverse direction, and (c) bubble volume reduced by the initial value.

potential over the interface is described by eqn (55) with ε02 = 0. The heavy solidlines in Fig. 7(a, b) show the evolution of the longitudinal and transverse bubbleaxes, ax and aσ , reduced by the initial bubble equivalent radius a0. The heavy solidline in Fig. 7(c) shows the evolution of the bubble volume. In this simulation, themaker points move normal to the interface to ensure the regularity of the grid foran extended period of time. The numerical results reveal that the bubble exhibitssmall-amplitude axisymmetric periodic oscillations about the Moore spheroid.

To demonstrate the effect of the Weber number on the finite-amplitude motion,we consider the evolution of an initially spherical bubble, and set the initial distri-bution of the potential over the bubble surface as shown in eqn (55) with ε02 = 0.15.The solid and dashed lines in Fig. 7 show results of simulations, respectively, forWe = 0 and 0.25. In fact, the dashed line in Fig. 7(a) extends the solid line dis-played earlier in Fig. 6(a) to longer times.



Consider first the oscillations of a stationary bubble immersed in a quiescentfluid, corresponding toWe = 0. According to linear theory, the x axis of the bubblereduced by the mean bubble radius oscillates around the mean value of unity withdimensionless period T ′ = 1.814 and reduced amplitude 0.0946, as shown by thehorizontal and vertical lines in Fig. 7(a), while the transverse axis oscillates with a180 phase and half the reduced amplitude 0.0473, as shown by the horizontal andvertical lines in Fig. 7(a). The bubble volume exhibits simultaneous oscillationswith dimensionless period T ′ = 0.322, as shown by the vertical lines in Fig. 7(b).These predictions are borne out by the numerical results shown with the solid lines inFig. 7. Note that, even though the volume mode is not excited in the initial condition,volume oscillations do arise because of nonlinear interactions. Comparing the solidwith the solid lines in Fig. 7(a, b), we see that raising the Weber number increasesthe period, and thus reduces the frequency of the shape oscillations, in agreementwith theoretical predictions on the small-amplitude motion [13, 30].

Figure 8(a) shows the evolution of the x position of the bubble centroid forWe = 0.25, corresponding to the dashed lines in Fig. 7. The results show that abubble that would be stationary if it were spherical, moves opposite to the direc-tion of the oncoming flow. Stated differently, a rising bubble will slow down asa result of the deformation. The net motion effectively alters the Weber num-ber of the flow by an amount that must be found as part of the solution. Fig-ure 8(b) displays the bubble interface as described by the boundary-element gridat time t′ = 3.95. The incident streaming flow is directed from left to right towardthe negative direction of the x axis.

Similar results are obtained for the three-dimensional, j = 2, m = 2 mode,where the bubble shape alternates between two triaxial ellipsoids whose majoraxes lie in the equatorial xy or xz plane. Figure 9(a–c) shows the evolution ofthe bubble axes, defined as half the maximum bubble size in the correspondingdirection reduced by the initial bubble radius, for ε22 = 0.20, and We = 0 (solidlines) or 0.25 (dashed lines); in the second case, the bubble translates along the zaxis. The numerical results are consistent with the predictions of the linear theoryfor small-amplitude shape and volume oscillations represented by the vertical andhorizontal lines in Fig. 9(b, c). According to the small-amplitude linear theory, thex axis of the bubble described in Fig. 9(a) remains unchanged. The simulation ofthe j = 2, m = 2 mode oscillation reveals that, as in the case of the axisymmetricj = 2, m = 0 mode, raising the Weber number increases the period, and thusreduces the frequency of the shape oscillations. In the case of a translating bubble,oscillations occur about a mean spheroidal shape with two major axes perpendicularto the direction of the motion.

Figure 10(a) displays the oscillations in the bubble volume excited by nonlinearinteractions, and Fig. 10(b) shows the evolution of the z position of the bubblecentroid for We = 0.25, corresponding to the dashed lines in Fig. 9. The resultsshow that a rising bubble slows down as a result of the deformation. Finally,Fig. 10(c) displays the bubble surface, as described by the boundary-element grid attime t′ = 3.95. The incident streaming flow is directed from top to bottom towardthe negative direction of the z axis.



Figure 8: Axisymmetric oscillations of a translating bubble in the j = 2, m = 0mode, for We = 0.25, corresponding to the simulation represented bythe dashed lines in Fig. 7. (a) Reduced x position of the bubble centroid,x′ = x/a, and (b) instantaneous bubble surface at time t′ = 3.95; theincident streaming flow is directed from right to left toward the negativedirection of the x axis.

12 Discussion

The completed double-layer representation combined with the generalized vortexmethod provides us with an efficient algorithm for simulating three-dimensionalpotential flow due to the oscillations of, or past compressible bubbles. Numer-ical instabilities are a serious impediment to the longevity of the simulationsover an extended period of time, but smoothing by spectrum truncation removes



Figure 9: Oscillations of the bubble axes in the j = 2, m = 2 mode, for We = 0(solid lines) or 0.25 (dashed lines).

the irregularities while retaining the fundamental features of the nonlinearmotion. The superiority of the generalized vortex method, as compared to directmethods based on Green’s third identity, lies in its ability to yield the instantaneousflow by solving an integral equation of the second kind, which can be done accu-rately and economically by the method of successive substitutions. An importantaspect of the boundary-element implementation is the option for the boundary-element nodes to be convected normal to the interface with the liquid velocity andtangentially with an arbitrary component, thereby ensuring the regularity of theinterfacial grid for an extended period of time. Though regridding is an alterna-tive, (e.g., [23]), numerical error due to interpolation in some parametric spaceor directly in three dimensions in the absence of smoothing or regularization is aserious impediment.

In the computations discussed in this chapter, the boundary-element method hasbeen applied with success to simulate bubble oscillations in infinite flow. Althoughnot tested, the method is likely to perform equally well in the case of flow thatis bounded by an impenetrable boundary, such as flow in a semi-infinite domain



Figure 10: Oscillations of a translating bubble in the j = 2, m = 2 mode, corre-sponding to the simulation represented by the dashed lines in Fig. 9. (a)Oscillations in the bubble volume, (b) reduced z position of the bub-ble centroid, z′ = z/a, and (c) instantaneous bubble surface at timet′ = 3.98. The incident streaming flow is directed from top to bottomtowards the negative direction of the z axis.

bounded by a plane wall. The only necessary modification is the substitution of thefree-space Green’s function with a Neumann function appropriate for the geometryof the flow under consideration. A library of Green and Neumann functions isavailable as a companion of text of boundary-element methods [43].

The completed double-layer representation relies on a point source situated insidethe bubble to account for changes in the bubble volume. In the case of volumeoscillations of small or moderate amplitude, the point source may safely be placedat the bubble centroid. However, in the case of a collapsing bubble, the sponta-neous development of a liquid jet transforms the bubble into a toroidal ring, andthis may cause the interface to cross the surface centroid at some point duringthe evolution. When this occurs, a different strategy of completion is requiredinvolving, for example, point-source rings or point sources situated at strategicpositions inside the bubble.



Appendix A: Computation of the associated and modifiedLegendre functions

The associated Legendre functions, Pmj , where j and m are positive integers, may

be computed using the recursion relations

Pmj (x) =

2j − 1j −m

x Pmj−1(x) − j +m− 1

j −mPm

j−2(x),

Pmj (x) =

1√1 − x2

[(j +m− 1)(j +m)

2j + 1Pm−1

j−1 (x) (56)

− (j −m+ 1)(j −m+ 2)2j + 1

Pm−1j+1 (x)

],

(e.g., [22], p. 870), which provide us with a basis for the algorithm:

P 00 (x) = 1,

P 01 (x) = x,

DO j = 2, . . .

P 0j (x) =

2j − 1j

x P 0j−1(x) − j − 1

jP 0

j−2(x)

END DO

DO m = 1, . . .

P 0j (x) =

2j − 1j

x P 0j−1(x) − j − 1

jP 0

j−2(x)

END DO

DO m = 1, . . .

Pmm−1(x) = 0

Pmm (x) =

2√1 − x2

[m(2m− 1)

2m+ 1Pm−1

m−1 (x) − 12m+ 1

Pm−1m+1 (x)

]

DO j = m+ 1, . . .

Pmj (x) =

2j − 1j −m

x Pmj−1(x) − j +m− 1

j −mPm

j−2(x)

END DO

END DO (57)



The formula inside the first DO loop is the recursive expression of the Legendrepolynomials. The first few Legendre functions Pm

j (cos θ) are given by

P 00 = 1 −− −−P 0

1 = cos θ P 11 = sin θ −−

P 02 = 1

2 (3 cos2 θ − 1) P 12 = 3

2 sin 2θ P 22 = 3 sin2 θ

= 32 (1 − cos 2θ)

P 03 = 1

2 cos θ (5 cos2 θ − 3) P 13 = 3

8 (sin θ + 5 sin 3θ) P 23 = 15

4 ×(cos θ − cos 3θ)

P 33 = 15

4 (3 sin θ − sin 3θ)

(58)

The modified Legendre functions are defined in terms of the associated Legendrefunctions as shown in eqn (11). Using this definition, we derive the counterparts ofthe recursive relations (56),

Tmj (x) = A1

2j − 1j −m

x Tmj−1(x) −A2

j +m− 1j −m

Tmj−2(x),

Tmj (x) =

1√1 − x2

[B1

(j +m− 1)(j +m)2j + 1

Tm−1j−1 (x) (59)

−B2(j −m+ 1)(j −m+ 2)

2j + 1Tm−1

j+1 (x)],

where

A1 =(

2j+12j−1

j−mj+m

)1/2,

A2 =(

2j+12j−3

j−mj+m

j−m−1j+m−1

)1/2,

B1 =(

2j+1(2j−1)(j+m)(j+m−1)

)1/2,

B2 =(

2j+1(2j+3)(j−m+1)(j−m+2)

)1/2.

(60)

The algorithm for computing the modified Legendre functions is a slight modifica-tion of algorithm (57).

References

[1] Baker, G.R., Generalized vortex method for free-surface flow problems,Wavesin Fluid Interfaces, MRC, University of Wisconsin, 1983.

[2] Baker, G.R., Meiron, D.I. & Orszag, S.A., Generalized vortex methods forfree-surface flow problems. Journal of Fluid Mechanics, 123, 477–501, 1982.

[3] Baker, G.R., Meiron, D.I. & Orszag, S.A., Boundary integral methods foraxisymmetric and three-dimensional Rayleigh–Taylor instability problems.Physica, 12D, 19–31, 1984.



[4] Batchelor, G.K., The stability of a large gas bubble rising through liquid.Journal of Fluid Mechanics, 184, 399–422, 1987.

[5] Benjamin, T.B., Hamiltonian theory for motion of bubbles in an infinite liquid.Journal of Fluid Mechanics, 2, 118–128, 1987.

[6] Benjamin, T.B. & Ellis, A., Self-propulsion of asymmetrically vibrating bub-bles. Journal of Fluid Mechanics, 212, 65–80, 1990.

[7] Benjamin, T.B. & Strasberg, M., Excitation of oscillations in the shape ofpulsating gas bubbles: theoretical work (abstract.). Journal of the AcousticalSociety of America, 30, 697, 1958.

[8] Blake, J.R., Boulton-Stone, J.M. & Tong, R.P., Boundary integral methodsfor rising, bursting and collapsing bubbles, ed. H. Power, BE Applicationsin Fluid Mechanics, Southampton, Computational Mechanics Publications,1995.

[9] Blake, J.R. & Gibson, D.C., Cavitation bubble near boundaries. AnnualReview of Fluid Mechanics, 19, 99–123, 1987.

[10] Blake, J.R., Robinson, P.B., Shima, A. & Tomita,Y., Interaction of two cavita-tion bubbles with a rigid boundary. Journal of Fluid Mechanics, 255, 707–721,1993.

[11] Duineveld, P.C., Rise velocity and shape of bubbles in pure water. Journal ofFluid Mechanics, 292, 325–332, 1995.

[12] El Sawi, M., Distorted gas bubbles at large Reynolds number. Journal of FluidMechanics, 62, 163–183, 1974.

[13] Feng, J.Q., The oscillations of a bubble moving in an inviscid fluid. SIAMJournal on Applied Mathematics, 52, 1–14, 1992.

[14] Feng, Z.C. & Leal, L.G., Translational instability of a bubble undergoingshape oscillations. Physics of Fluids, 7, 1325–1336, 1995.

[15] Feng, Z.C. & Leal, L.G., Nonlinear bubble dynamics. Annual Review of FluidMechanics, 29, 201–243, 1997.

[16] Hsieh, D.-Y. & Plesset, M.S., Rectified diffusion of mass into gas bubbles.Journal of the Acoustical Society of America, 33, 206–215, 1961.

[17] Hao, Y. & Prosperetti, A., The effect of viscosity on the spherical stability ofoscillating gas bubbles. Physics of Fluids, 11, 1309–1317, 1999.

[18] Harper, J.F., The motion of bubbles and drops through liquids. Advances inApplied Mechanics, 12, 59–129, 1972.

[19] Harris, P.J., An investigation into the use of the boundary integral method tomodel the motion of a single gas or vapour bubble in a liquid. EngineeringAnalysis with Boundary Elements, 28, 325–332, 2004.

[20] Hartunian, R.A. & Sears, W.R., On the instability of small gas bubbles movinguniformly in various liquids. Journal of Fluid Mechanics, 3, 27–47, 1957.

[21] Keller, J.B. & Miksis, M., Bubble oscillations of large amplitude. Journal ofthe Acoustical Society of America, 68, 628–633, 1980.

[22] Korn, G.A. & Korn, T.M., Mathematical Handbook for Scientists and Engi-neers, McGraw Hill: New York, 1961.



[23] Kwak, S. & Pozrikidis, C., Adaptive triangulation of evolving, closed or opensurfaces by the advancing-front method. Journal of Computational Physics,145, 61–88, 1998.

[24] Lamb, H., Hydrodynamics, Cambridge University Press: Cambridge, 1932.[25] Lunde, K. & Perkins, R., Shape oscillations of rising bubbles. Applied Scien-

tific Research, 58, 387–408, 1998.[26] Magnaudet, J. & Eames, I., The motion of high-Reynolds number bubbles

in inhomogeneous flows. Annual Review of Fluid Mechanics, 32, 659–708,2000.

[27] McDougald, N.K. & Leal, L.G., Numerical study of the oscillations of a non-spherical bubble in an inviscid, incompressible liquid. Part I: free oscillationsfrom non-equilibrium initial conditions. International Journal of MultiphaseFlow, 25, 887–919, 1999.

[28] McDougald, N.K. & Leal, L.G., Numerical study of the oscillations of a non-spherical bubble in an inviscid, incompressible liquid. Part II: the response toan impulsive decrease in pressure. International Journal of Multiphase Flow,25, 921–941, 1999.

[29] Mei C.C. & Zhou, X., Parametric resonance of a spherical bubble. Journal ofFluid Mechanics, 229, 29–50, 1991.

[30] Meiron, D.I., On the stability of gas bubbles rising in an inviscid fluid. Journalof Fluid Mechanics, 198, 101–114, 1989.

[31] Mikhlin, S.G., Integral Equations and their Applications to Certain Problemsin Mechanics, Mathematical Physics, and Technology, Pergamon Press: NewYork, 1957.

[32] Miksis, M.J., Vanden-Broeck, J.M. & Keller, J.B., Axisymmetric bubble ordrop in a uniform flow. Journal of Fluid Mechanics, 108, 89–100, 1981.

[33] Miksis, M.J., Vanden-Broeck, J.M. & Keller, J.B., Rising bubbles. Journal ofFluid Mechanics, 123, 31–41, 1982.

[34] Miloh, T. & Galper, A., Self-propulsion of general deformable shapes in aperfect fluid. Proceedings of the Royal Society of London Series A, 442, 273–299, 1993.

[35] Moore, D.W., The rise of a gas bubble in a viscous liquid. Journal of FluidMechanics, 6, 113–130, 1959.

[36] Moore, D.W., The velocity of rise of distorted gas bubbles in a liquid of smallviscosity. Journal of Fluid Mechanics, 23, 749–766, 1965.

[37] Ohsaka, K. & Trinh, E.H., Resonant coupling of oscillating gas or vaporbubbles in water: An experimental study. Physics of Fluids, 12, 281–288,2000.

[38] Plesset, M.S. & Mitchell, T.P., On the stability of the spherical shape of a vaporcavity in a liquid. Quarterly of Applied Mathematics, 8, 419–430, 1956.

[39] Plesset, M.S. & Prosperetti, A., Bubble dynamics and cavitation. AnnualReview of Fluid Mechanics, 9, 145–185, 1977.

[40] Pozrikidis, C., Introduction to Theoretical and Computational Fluid Dynam-ics, Oxford University Press: New York, 1997.



[41] Pozrikidis, C., Numerical Computation in Science and Engineering, OxfordUniversity Press: New York, 1998.

[42] Pozrikidis, C., Three-dimensional oscillations of inviscid drops induced bysurface tension. Computers and Fluids 30, 417–444, 2001.

[43] Pozrikidis, C., A Practical Guide to Boundary-Element Methods with theSoftware Library BEMLIB, Chapman & Hall/CRC: Boca Raton, 2002.

[44] Pozrikidis, C., Numerical simulation of three-dimensional bubble oscillationsby a generalized vortex method. Theoretical and Computational Fluid Dynam-ics, 16, 133–150, 2002.

[45] Pozrikidis, C., Three-dimensional oscillations of rising bubbles. EngineeringAnalysis with Boundary Elements, 28, 315–323, 2004.

[46] Saffman, P.G., On the rise of small air bubbles in water. Journal of FluidMechanics 1, 249–275, 1956.

[47] Saffman, P.G., The self-propulsion of a deformable body in a perfect fluid.Journal of Fluid Mechanics 28, 385–389, 1967.

[48] Strasberg, M. & Benjamin, T.B., Excitation of oscillations in the shape ofpulsating gas bubbles: experimental work (abstract.). Journal of the AcousticalSociety of America, 30, 697, 1958.

[49] Wang, Q.X., The evolution of a gas bubble near an inclined wall. Theoreticaland Computational Fluid Dynamics, 12, 19–51, 1998.

[50] Wegener, P.P. & Parlange, J.Y., Spherical-cap bubbles. Annual Review of FluidMechanics, 4, 79–100, 1973.

[51] Wu, M. & Gharib, M., Experimental studies on the shape and path instabilityof small air bubbles rising in clean water. Physics of Fluids, 7, L49–L52, 2002.


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