ILASS Americas 17th Annual Conference on Liquid Atomization and Spray Systems, Arlington, VA, May 2004

A Level Set/Vortex Sheet Method for Modeling Phase Interface DynamicsDuring Primary Breakup

M. Herrmann∗

Center for Turbulence ResearchStanford University, Stanford, CA 94305

AbstractThe atomization of turbulent liquid jets and sheets can usually be divided into two subsequent processes: theprimary breakup, where the jet or sheet initially breaks up into both large and small scale structures, followedby the secondary breakup, where these structures continue to break up into ever smaller drops, forming aspray. Considering numerical grid resolutions typical for Large Eddy Simulations (LES), the large scale liquidstructures during primary breakup are well resolved, the liquid volume fraction can be of order one, and thephase interface can have any arbitrary complex shape. To correctly account for these characteristics, thephase interface has to be explicitly tracked on the numerically resolved scale and all phase interface dynamicson the subgrid scales have to be modeled. To derive such a Large Surface Structure (LSS) model, a levelset/vortex sheet method is proposed. The level set/vortex sheet method tracks the phase interface by a levelset scalar, so that topological changes of the interface, like breaking and merging, are handled automatically.Assuming inviscid fluids, the dynamics of the phase interface can be described by a vortex sheet locatedat the phase interface. The evolution equation for the vortex sheet strength then contains explicit localsource terms for the physical processes at the phase interface, namely stretching terms, a surface tensionterm, and terms accounting for the differences in fluid density. The proposed level set/vortex sheet methodthus provides a framework for the derivation of the primary breakup LSS subgrid models. Some preliminaryresults, namely the Kelvin-Helmholtz instability in the linear regime, the oscillations of liquid columns andspheres, and the three-dimensional breakup of a liquid/gas surface and a liquid sheet are presented in thispaper.

IntroductionAtomization processes play an important role

in a wide variety of technical applications and nat-ural phenomena, ranging from inkjet printers, gasturbines, direct injection IC-engines, and cryogenicrocket engines to ocean wave breaking and hy-drothermal features. The atomization process of liq-uid jets and sheets is usually divided into two consec-utive steps: the primary and the secondary breakup.During primary breakup, the liquid jet or sheet ex-hibits large scale coherent structures that interactwith the gas phase and break up into both large andsmall scale drops. During secondary breakup, thesedrops break up into ever smaller drops that finallymay evaporate.

Usually, the atomization process occurs in a tur-bulent environment, involving a wide range of timeand length scales. Given today’s computational re-sources, the direct numerical simulation (DNS) ofthe turbulent breakup process as a whole, resolvingall physical processes, is impossible, except for somevery simple configurations. Instead, models describ-

∗Corresponding Author

/∆x > 1`

/∆x < 1`/∆x << 1`

secondary breakup

primary breakup

Θl = O(1)

Θl < 1Θl << 1

Figure 1. Breakup of a liquid jet.

ing the physics of the atomization process have to beemployed. Various models have already been devel-oped for the secondary breakup process. There, itcan be assumed that the characteristic length scale `of the drops is much smaller than the available gridresolution ∆x and that the liquid volume fraction ineach grid cell Θl is small, see Fig. ??. Furthermore,assuming simple geometrical shapes of the individ-

ual drops, like spheres or ellipsoids, the interactionbetween these drops and the surrounding fluid canbe taken into account. Statistical models describingthe secondary breakup process in turbulent environ-ments can thus be derived [?, ?, ?, ?, ?].

However, the above assumptions do not holdtrue for the primary breakup process. Here, theturbulent liquid fluid interacts with the surround-ing turbulent gas-phase on scales larger than ∆x,resulting in highly complex interface dynamics andindividual grid cells that can be fully immersed inthe liquid phase, compare Fig. ??. An explicittreatment of the phase interface and its dynamicsis therefore required. To this end, we propose tofollow in essence a Large Eddy Simulation (LES)type approach: all interface dynamics and physicalprocesses occurring on scales larger than the avail-able grid resolution ∆x shall be fully resolved and alldynamics and processes occurring on subgrid scalesshall be modeled. The resulting approach is calledLarge Surface Structure (LSS) model.

In order to develop such a LSS model for the tur-bulent primary breakup process, one potential ap-proach is to start off from a fully resolved descriptionof the interface dynamics using the Navier-Stokesequations and include an additional source termin the momentum equation due to surface tensionforces [?]. In order to track the location, motion, andtopology of the phase interface, the Navier-Stokesequations are then coupled to one of various possi-ble tracking methods, for example marker particles[?, ?, ?], the Volume-of-Fluid method [?, ?, ?], or thelevel set method [?, ?, ?]. Then, introducing ensem-ble averaging or spatial filtering results in unclosedterms that require modeling [?, ?]. Unfortunately,the derivation of such closure models is not straight-forward and, hence, has not been achieved yet. Thisis in part due to the fact that, with the exception ofthe surface tension term, all other physical processesoccurring at the phase interface itself, like for exam-ple stretching, are not described by explicit sourceterms. Instead, they are hidden within the interde-pendence between the Navier-Stokes equations andthe respective interface tracking equation. Thus, aformulation containing the source terms explicitlycould greatly facilitate any attempt to derive theappropriate closure models.

To this end, a novel three-dimensional Eulerianlevel set/vortex sheet method is proposed. Its ad-vantage is the fact that it contains explicit sourceterms for each individual physical process that oc-curs at the phase interface. It thus constitutes apromising framework for the derivation of the LSSsubgrid closure models.

This paper is divided into four parts. First,the underlying governing equations of the levelset/vortex sheet method for three-dimensional two-phase interface dynamics are presented. Second,the numerical methods employed to solve the levelset/vortex sheet method are discussed. Then, simu-lation results for the growth of the Kelvin-Helmholtzinstability in the linear regime and the oscillationof liquid columns and spheres are compared totheory in order to validate the proposed method.Then, preliminary three-dimensional results, i.e. thebreakup of a randomly perturbed liquid surface andsheet are presented. Finally, conclusions are drawnand an outlook is given.

The Level Set/Vortex Sheet methodThe objective of the level set/vortex sheet

method is to describe the dynamics of the phase in-terface Γ between two inviscid, incompressible fluids1 and 2, as shown in Fig. ??. In this case, the veloc-ity ui on either side i of the interface Γ is determinedby the incompressible Euler equations, given here indimensionless form,

∇ · ui = 0 , (1)∂ui

∂t+ (ui · ∇)ui = − 1

ρi∇p , (2)

subjected to the boundary conditions at the inter-face Γ,

[(u1 − u2) · n]∣∣∣Γ

= 0 (3)

[n× (u2 − u1)]∣∣∣Γ

= η (4)

[p2 − p1]∣∣∣Γ

=1

Weκ (5)

and at infinity,

limy→±∞

ui = ±u∞ . (6)

Here, n is the interface normal vector, η is the vortexsheet strength, and κ is the local curvature of Γ. TheWeber number is defined as

We = ρrefu2ref/σLref , (7)

where σ is the surface tension coefficient and ρref ,uref , and Lref are the reference density, velocity, andlength, respectively. An interface subjected to theabove boundary conditions is called a vortex sheet[?].

The partial differential equation describing theevolution of the vortex sheet strength η can be de-rived by combining the Euler equations, Eqs. (??)

fluid 1

G= G0

G< G0

x

y

z

G > G0

nt1

t2

¡

fluid 2

Figure 2. Phase interface definition.

and (??), with the boundary conditions at the inter-face, Eqs. (??)-(??), resulting in [?]

∂η

∂t+ u · ∇η = −n× [(η × n) · ∇u]

+n [(∇u · n) · η]

+2(A + 1)

We(n×∇κ)

+2An× a . (8)

Here, A = (ρ1−ρ2)/(ρ1 +ρ2) is the Atwood numberand a is the average acceleration of fluid 1 and fluid2 at the interface. The major advantage of Eq. (??),as compared to a formulation based on the Eulerequations, is the fact that Eq. (??) contains explicitlocal individual source terms on the right-hand sidedescribing the physical processes at the interface.These are, from left to right, two stretching termsS, a surface tension term T σ, and a density differ-ence term.

In addition to the evolution of the local vortexsheet strength, Eq. (??), the location and motion ofthe phase interface itself has to be known. To thisend, vortex sheets are typically solved by a bound-ary integral method within a Lagrangian frameworkwhere the phase interface is tracked by marker par-ticles [?, ?, ?, ?, ?]. Marker particles allow for highlyaccurate tracking of the phase interface motion in aDNS. However, the introduction of ensemble aver-aging and spatial filtering of the interface topologyis not straightforward and hence a strategy for thederivation of appropriate LSS subgrid closure mod-els is not directly apparent.

Level sets, on the other hand, have been suc-cessfully applied to the derivation of closure modelsin the field of premixed turbulent combustion [?, ?].Thus, instead of using marker particles to describethe location and motion of the phase interface, here,the interface is represented by an iso-surface of thelevel set scalar field G(x, t), as shown in Fig. ??.

Setting

G(x, t)|Γ = G0 = const , (9)

G(x, t) > G0 in fluid 1, and G(x, t) < G0 in fluid 2,an evolution equation for the scalar G can be derivedby simply differentiating Eq. (??) with respect totime,

∂G

∂t+ u · ∇G = 0 . (10)

This equation is called the level set equation [?]. Us-ing the level set scalar, geometrical properties of theinterface, like its normal vector and curvature, canbe easily expressed as

n =∇G

|∇G|, κ = ∇ · n . (11)

Strictly speaking, Eqs. (??) and (??) are validonly at the location of the interface itself. However,to facilitate the numerical solution of both equa-tions in the whole computational domain, η is re-distributed by setting it constant in the interfacenormal direction,

∇η · ∇G = 0 , (12)

and G is reinitialized by defining it to be a distancefunction away from the interface,

|∇G|∣∣∣G 6=G0

= 1 . (13)

Equations (??) and (??) are coupled by the self-induced velocity u of the vortex sheet. To calculateu, the vector potential ψ is introduced,

∆ψ = ω . (14)

Here, the vorticity vector ω is calculated following avortex-in-cell type approach [?, ?]

ω(x) =∫

V

η(x′)δ(x−x′)δ (G(x′)−G0) |∇G(x′)|dx′ ,

(15)where δ is the delta-function. Then, u can be calcu-lated from

u(x) =∫

V

δ(x− x′) (∇×ψ) dx′ . (16)

In summary, Eqs. (??) and (??) together withEqs. (??) - (??) describe the three-dimensional two-phase interface dynamics and constitute the levelset/vortex sheet method.

Numerical MethodsThe system of equations describing the phase in-

terface dynamics is solved on an equidistant, Carte-sian grid. However, instead of solving the equationsthroughout the whole computational domain, a tubeapproach is employed to significantly speed up thecomputations. Following arguments by Peng et al.[?], five distinct tubes around the G = G0 level set,called I-, I2- B-, T -, and N -tube are introduced.The condition for a grid node (i, j) to belong to oneor more specific tubes is given by

(i, j) ∈

I if |Gi,j | ≤ αI∆xI2 if |Gi,j | ≤ 2αI∆xB if |Gi,j | ≤ αB∆xT if |Gi,j | ≤ αT ∆xN if (i0, j0) ∈ T with

(17)

i0 ∈ [i− 3, i + 3] , j0 ∈ [j − 3, j + 3] ,

with ∆x being the grid size and typically αI = 4,αB = αI + 3, and αT = αB + 3. The use of therespective tubes will be discussed in the followingsections.

Convective TermsThe level set equation, Eq. (??), is a Hamilton-

Jacobi equation. In this work, a third order WENOscheme for Hamilton-Jacobi equations [?] is used. ARoe flux with local Lax-Friedrichs entropy correc-tion (Roe-LLF) [?, ?] is employed to solve both thelevel set equation, Eq. (??), and the convective termof the η-equation, Eq. (??). Integration in time isperformed by a third order TVD Runge-Kutta timediscretization [?].

Solution of the convective terms is restricted tothe T -tube, where u in Eqs. (??) and (??) is re-placed by [?]

ucut = c(G)u , (18)

with the cut-off function

c(G) =

1 : αG≤αB

(αG−αT )2(2αG+αT−3αB)(αT−αB)3 : αB<αG≤αT

0 : αG>αT

(19)and αG = |G|/∆x. This ensures that no artificialoscillations are introduced at the T -tube boundaries.

Source TermsBoth the stretching terms S as well as the sur-

face tension term T σ in the η-equation, Eq. (??),are integrated in time within the convective TVDRunge-Kutta scheme. Since the source terms havephysical meaning only at the interface itself, theirevaluation is limited to the I-tube. For numerical

reasons, these values are then redistributed to theN -tube by solving

∇S · ∇G = 0 , ∇T σ · ∇G = 0 . (20)

Reinitialization

The reinitialization procedure employed here tosolve Eq. (??) has originally been proposed by Suss-man et al. [?], where the Hamilton-Jacobi type equa-tion

∂H(x, t∗)∂t∗

+

S (H(x, t∗)) (|∇H(x, t∗)| − 1) = 0 (21)H(x, t∗ = 0) = G(x, t)−G0 (22)

is solved for all (i, j) ∈ N until

||H(x, t∗)−H(x, t∗−∆t∗)||∞ < ε1 , (i, j) ∈ B (23)

with typically ε1 = 10−3∆x. In Eq. (??), S(H) isan approximation to the sign function. The properchoice of this approximation is crucial to minimizeundesired movement of the H(x, t∗) = 0 interface[?] while solving Eq. (??). Here, we will use theapproximated sign function [?],

S(H) =H√

H2 + |∇H|2(∆x)2(24)

with a second order central difference approximationfor ∇H. Equation (??) is solved by the third orderWENO scheme but employing a Godunov insteadof a Roe-LLF flux function. Again, the solution intime is advanced by a third order TVD Runge-Kuttascheme.

Reinitialization is limited to the N -tube whichis three cells larger in each direction than the T -tube. In [?], an extension of the T -tube by onlyone cell in each direction was proposed. However, itwas found that this still might introduce fluctuationsin the solution. Hence an extension by three cells isemployed here. The convergence criterium, Eq. (??),is evaluated within the B-tube only.

Redistribution

A Fast Marching method is employed to solvethe redistribution equations (??) and (??). Here, weuse the standard second order method as describedin detail in [?, ?]. Redistribution is performed in theN -tube only.

Employing a Hamilton-Jacobi equation basedredistribution as proposed in [?] instead of the FastMarching method proved to introduce too muchundesired tangential transport in the redistributedscalars, η, S, and T σ. Hence, this method has notbeen used here.

0

0.5

1

1.5

2

2.5

3

3.5

10 100 1000 ooWe

w

Figure 3. Growth rates w of the Kelvin-Helmholtzinstability in the linear regime, level set/vortex sheetmethod (•) and linear theory (line).

Velocity Calculation MethodsIn order to solve the Poisson equations for the

vector potential, Eq. (??), the vorticity at each gridnode in the computational domain has to be calcu-lated by a numerical version of Eq. (??). Approx-imating the delta function by a smoothed version,the vorticity, theoretically located solely on the in-terface, is in effect spread out onto the neighboringgrid nodes, thereby prescribing a constant, non-zerolocal shear layer thickness. Hence, this approachis similar to the vortex-in-cell method that spreadsthe vorticity of Lagrangian vortex particles to theirsurrounding grid nodes [?, ?]. Quite a number ofspreading functions have been proposed to this end[?]. Here, we will use

δ(x− x(s)) = δε(x− x(s))δε(y − y(s))δε(z − z(s)) (25)

δ(G(x′)−G0) = δε(G(x′)−G0) (26)

with the numerical delta function δε as proposed byPeskin [?]

δε(x) =

12ε

[1 + cos

(πx

ε

)]: |x| ≤ ε

0 : |x| > ε(27)

and the spreading parameter ε set to ε = αI∆x.The integration in Eq. (??) is performed by first

evaluating

Ω(x′, t) = η(x′, t)δ (G(x′)−G0) |∇G(x′)| (28)

0

1

2

3

4

5

1 2 3 4 5 6

T

n

We = 100

We = 10

0

1

2

3

4

5

1 2 3 4 5 6

T

n

We = 100

We = 10

Figure 4. Oscillation period T of liquid columns(left) and spheres (right) as a function of modenumber n for varying Weber numbers We, levelset/vortex sheet method (•) and linear theory (line).

for all cells within the I-tube and then integrating

ω(x, t) =∫

V ′δ(x− x′)Ω(x′, t)dx′ . (29)

for all cells within the I2-tube by a simple midpointrule, with V ′ being the area, where |x − x′| ≤ ε,|y − y′| ≤ ε, and |z − z′| ≤ ε.

The Poisson equations for the vector potential,Eq. (??), are solved by the package FISHPACK [?]throughout the whole computational domain em-ploying appropriate boundary conditions.

Finally, the calculation of the vortex sheet in-duced velocity u from the vector potential, Eq. (??),is again a two-step process. First, an initial velocityU is calculated at each grid node by second ordercentral differences of ψ. Second, an interpolationstep using the same numerical integration methodand δ-function has to be employed,

u(x, t) =∫

V ′δ(x− x′)U(x′, t)dx′ , (30)

evaluated within the T -tube. Note that strictlyspeaking, this interpolation step is not fully consis-tent with the spreading step, Eqs. (??) and (??),because a matching δ(G) term in Eq. (??) cannotbe defined in Eq. (??).

ResultsIn order to both validate the proposed level

set/vortex sheet method and to demonstrate its abil-ity to perform DNS of the primary breakup process,the results of three different test cases are presented.First, the level set/vortex sheet method is used tocalculate the growth rates of the two-dimensionalKelvin-Helmholtz instability in the linear regime.Then, the oscillation periods of liquid columns and

We = ooWe = 100We = 30We = 13

s/L s/L s/L s/L

-3.10-3

0.100

3.10-3

0 0.2 0.4 0.6 0.8 1

-1.10-3

0.100

1.10-3

0 0.2 0.4 0.6 0.8 1

-3.10-4

0.100

3.10-4

0 0.2 0.4 0.6 0.8 1

-7.10-4

0.100

7.10-4

0 0.2 0.4 0.6 0.8 1

Figure 5. Stretching term S (solid line) and tension term Tσ (dashed line) at t = 0.5, We = 13, 30, 100,and ∞ (from left to right).

spheres are calculated and compared to theoreti-cal results. Finally, the temporal evolution of arandomly perturbed three-dimensional surface andsheet are presented.

Kelvin-Helmholtz Instability in the LinearRegime

The objective of this test problem is to validatethe level set/vortex sheet method by comparing thecalculated growth rates of the Kelvin-Helmholtz in-stability in the linear regime to those obtained bylinear theory.

The initial conditions for the level set scalar Gin two dimensions are given by

G(x, t = 0) =

y −A0 sin(

2π

B

[x−A0 sin

2π

Bx

]), (31)

with A0 = 1 · 10−5 and B = 1. The initial vortexsheet strength distribution is calculated from

η(x, t = 0) =w(We)

w(We = ∞)(ηVS(x, t = 0)− η∗) + η∗, (32)

where w(We) is the growth rate predicted by lineartheory [?], η∗ = −1, and ηVS is given by

ηVS(x, t = 0) =η∗√

1 + 4πA0B cos 2π

B x + 2[2πA0

B cos 2πB x

]2 . (33)

Periodic boundary conditions are employed at theleft and right domain boundary. The velocity at theupper and lower boundary is set to

u(x, y = ±0.5) = (±0.5, 0)T , (34)

respectively. The simulations are performed in a B×B box on an equidistant 256 × 256 Cartesian grid.The spreading parameter is set to ε = 4/256.

Figure ?? compares the calculated growth ratesw,

w =1t1

∫ t1

0

w(t)dt , t1 = 0.5 . (35)

Although the level set/vortex sheet methodmarginally over-predicts the linear growth rate,agreement between simulation results and linear the-ory is very good. The reason for the slight over-prediction is not directly apparent, but it is mostlikely due to the lack of a consistent level set basedinterpolation step to calculate the vortex sheet in-duced velocity u, Eq. (??).

Figure ?? shows the distribution of the stretch-ing term S and the tension term Tσ along the nor-malized arc length s/L at t = 0.5. For We = 13,both terms almost balance, resulting in a smallgrowth rate w. For increasing Weber numbers, thestretching term S starts to dominate the surfacetension term T , resulting in larger growth rates w.For We = ∞, secondary instabilities of higher wavenumbers emerge in the distribution of the stretchingterm. This is consistent with linear theory, since theKelvin-Helmholtz instability is an ill-posed problemin the absence of surface tension forces. The forma-tion of the Moore singularity [?] in the limit of We →∞ is avoided here, because the level set/vortex sheetmethod introduces two types of desingularizationinto the governing equations. For one, an under-lying numerical grid is employed, thereby limitingthe numerical evaluation of any gradients. Further-more and more importantly, the spreading param-eter ε introduces a constant, non-zero shear layerthickness, thereby mimicking the desingularizationeffects of viscosity [?]. Still, in the limit of We →∞,disturbances of higher wave number caused by nu-merical errors have significantly higher growth ratesthan the initial disturbance, thereby becoming dom-inating for later times.

Oscillating Columns and SpheresTo further validate the proposed level set/vortex

sheet method, the calculated oscillation periods T ofliquid columns and spheres of mean radius R = 0.25,center xc = (0.5, 0.5, 0.5), amplitude A0 = 0.05R,and Atwood number A = 0 are compared to theo-retical results [?]. The initial vortex sheet strength

Figure 6. Distribution of the surface tension term T σ in the x-direction (top), y-direction (left), andz-direction (right) for the mode n = 5 oscillating sphere at t = 0 and We = 10.

in both cases is set to

η(x, t = 0) = 0 . (36)

All calculations are performed in a unit sized boxresolved by an equidistant Cartesian grid of 128×128and 128× 128× 128 nodes, respectively.

Figure ?? depicts on the left the oscillation pe-riod T of liquid columns for varying mode numbers nand two different Weber numbers. The correspond-ing results for the oscillating spheres are shown onthe right. As can be clearly seen, agreement betweensimulation and theory is very good.

Figure ?? shows the distribution of the surfacetension term T σ of the η-equation (??) in the x-, y-,and z-direction,

T σ =2(A + 1)

We(n×∇κ) , (37)

for the oscillating sphere with n = 5 and We = 10at t = 0. As the shape of the sphere indicates, T σ

in the x-direction is a factor of roughly four higherthan T σ in the other two directions, leading to thepredominant oscillation in the y-z-plane.

Liquid Surface and Sheet BreakupTo demonstrate the capability of the proposed

level set/vortex sheet method to simulate the pri-mary breakup process, the temporal evolution ofboth a randomly perturbed liquid surface and sheetare simulated. In the case of the liquid surface, theon average flat interface located at z = 0 is per-turbed in the z-direction by a Fourier series of 64sinusoidal waves in both the x- and y-direction withrandom amplitude 0 < A0 < 0.01 and random phaseshift. In the case of the liquid sheet, the two on av-erage flat interfaces are located at z = −C/2 andz = +C/2 and are again perturbed by two Fourier

series of 64 sinusoidal waves. The thickness of theliquid sheet is set to C = 0.1.

The initial vortex sheet strength for the liquidsurface is set to

η(x, t = 0) = (−1, 0, 0)T (38)

and to

η(x, t = 0) =

(−1, 0, 0)T : z > 0( 1, 0, 0)T : z ≤ 0 ,

(39)

in the liquid sheet case. Both, the surface andthe sheet simulation were performed in a x- and y-direction periodic box of size [0, 1] × [0, 1] × [−1, 1]resolved by a Cartesian grid of 64×64×128 equidis-tant nodes. In both simulations, the Atwood num-ber is A = 0. The Weber number in the surfacesimulation is We = 500 and the Weber number inthe sheet simulation based on the sheet thickness isWeC = 100.

As depicted in Fig. ??, the liquid surfaceshows an initial growth of two-dimensional Kelvin-Helmholtz instabilities (t = 1). These continue togrow (t = 3) and form three-dimensional structures(t = 5) resulting in elongated fingers (t = 6.5) thatfinally initiate breakup (t = 8.0).

The liquid sheet, depicted in Fig. ??, againexhibits the initial formation of two-dimensionalKelvin-Helmholtz instabilities (t = 1) that continueto grow (t = 3) until the liquid film gets too thin andruptures (t = 5). Individual fingers are formed thatextend mostly in the transverse direction (t = 8) andcontinue to break up into individual drops of varyingsizes (t = 12).

Conclusions and OutlookA Eulerian level set/vortex sheet method has

been presented that allows for the three-dimensional

Figure 7. Temporal evolution of the three-dimensional liquid surface breakup, A = 0, We = 500.

Figure 8. Temporal evolution of the three-dimensional liquid sheet breakup, A = 0, WeC = 100.

calculation of the phase interface dynamics betweentwo inviscid and incompressible fluids. Results ob-tained with the proposed method for the growth ofthe Kelvin-Helmholtz instability in the linear regimeand for the oscillation periods of liquid columns andspheres show very good agreement with theoreticalpredictions. Furthermore, the applicability of themethod to the primary breakup process has beendemonstrated by simulations of the breakup of botha liquid surface and a liquid sheet.

The proposed level set/vortex sheet method hasthe advantage that it allows for the detailed studyof each individual physical process occurring at thephase interface, because they appear as explicit localsource terms in the governing equations. Couplingof the level set/vortex sheet method to an outsideturbulent flow field will allow for DNS of the pri-mary breakup process to help identify characteris-tic regimes of turbulent primary breakup and theirdominant physical processes. The level set/vortexsheet method thus provides a promising frameworkfor the derivation of LSS subgrid models.

AcknowledgmentsThe support of the German Research Founda-

tion (DFG) is gratefully acknowledged.

Nomenclature

A Atwood numberA0 initial amplitudea average accelerationB size of the computational domainC liquid sheet thicknessG level set scalar` integral length scalen mode numbern normal vectorp pressureS stretching termst timeT σ surface tension termu velocity∆x grid sizew growth rateWe Weber numberδ delta functionε spreading parameterη vortex sheet strengthκ curvatureψ vector potentialρ densityσ surface tension coefficientΘl cell liquid volume fractionω vorticity vector

Subscriptsref reference quantities

SuperscriptsT transpose

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