ILASS Americas 20th Annual Conference on Liquid Atomization and Spray Systems, Chicago, IL, May 2007

On Simulating Primary Atomization Using the Refined Level Set Grid Method

M. Herrmann∗

Center for Turbulence ResearchStanford University, Stanford, CA 94040

AbstractTo simulate primary atomization, one has to track the position of the phase interface accurately, handlelarge numbers of topology changes and drops, treat the singular force of surface tension in an accurate andstable manner, and ensure grid-independent numerical results. To address all of these challenges we presenta balanced force Refined Level Set Grid (RLSG) method for collocated, unstructured finite volume flowsolver grids that can be coupled to a Lagrangian spray model. Special emphasis is placed on the accuratetreatment of surface tension forces, since during the atomization of liquid jets by coaxial fast-moving gasstreams, the details of the formation of small-scale drops from aerodynamically stretched out ligaments aregoverned by capillary forces [1]. Several different generic verification examples are presented, discussing theaccuracy, volume preservation, and grid-convergence properties of the balanced force RLSG method.

∗Corresponding Author

IntroductionThe simulation of the primary atomization pro-

cess of liquid jets and sheets is a challenging numer-ical problem. One not only has to track the positionof the liquid/gas interface and handle a large num-ber of topology changes, but one also has to accountfor the fact that on scales typically associated withatomizers, the phase interface is a discontinuity andthe surface tension force represents a singular force.Treating the surface tension force numerically in astable and accurate manner is of crucial importance,since, for example, during the atomization of liquidjets by coaxial fast-moving gas streams, the detailsof the formation of small-scale drops from aerody-namically stretched out ligaments are governed bycapillary forces [1].

From a numerical point of view, surface tensionposes a unique challenge since it is a singular force,active only at the location of the phase interfacewhere material properties, like density and viscos-ity, change discontinuously. One of the prerequisitesfor correctly treating surface tension forces is there-fore the ability to locate the position of the phaseinterface accurately. To this end, several phase in-terface tracking schemes exist for fixed grid flowsolvers, among them the marker method [2], theVolume-of-Fluid (VoF) method [3], and the level setmethod [4]. Here we will use a variant of the levelset method, termed Refined Level Set Grid (RLSG)method [5, 6]. It handles all topology changes auto-matically, allows for easy grid convergence studies ofthe phase interface representation, and ensures goodfluid volume conservation properties. Furthermore,the RLSG approach can provide a straightforwardinterface to couple the tracked representation of thephase interface during primary atomization to a La-grangian point particle description during secondaryatomization. Such a coupling is a prerequisite tosimulate the atomization process as a whole and tohandle the vast number of generated drops in an ef-ficient manner.

Different strategies exist to discretize the sur-face tension force once the location of the phase in-terface is known. The most commonly used methodis due to Brackbill et al. [7] called Continuum Sur-face Force (CSF). Here, the ideally singular surfacetension force is spread into a narrow band surround-ing the phase interface by the use of regularizeddelta functions. These can take the form of a dis-crete derivative of a Heaviside scalar, i.e., the volumefraction, in VoF methods, or spread out delta func-tions, like the popular cosine approximation due toPeskin [8] in level set methods. Especially in levelset methods, the use of spread out delta functions

can be problematic, since convergence under gridrefinement is only guaranteed for certain, not com-monly employed delta function approximations [9].An alternative to the CSF method is the GhostFluid Method (GFM) [10] that aims to apply thejump conditions and surface tension force as singularsource terms within the context of finite differenceschemes.

Both the CSF and the GFM method, however,are prone to generating unphysical flows, so-calledspurious currents, near the location of the phase in-terface when surface tension forces are present. Inthe canonical test cases of an equilibrium columnand an equilibrium sphere, these velocity errors cangrow unbounded very fast, unless they are artificiallydamped by introducing viscosity. The amplitude ofthe spurious currents when damped by viscosity isof the order of u ∼ 0.01σ/µ for VoF and level setmethods and u ∼ 10−5σ/µ for marker methods [11],where σ is the surface tension coefficient and µ is theviscosity. Thus, numerical simulations are limited bya critical Laplace number, La = σρR/µ2, where ρis the density and R is a characteristic phase inter-face radius of curvature, since for large La, i.e., largeσ, spurious currents start to dominate the physicalflow [11].

The reason for the occurance of spurious cur-rents is twofold. The major reason is a discreteimbalance between the surface tension force andthe associated pressure jump across the phase in-terface [12]. The second source of error is dueto errors associated with evaluating phase interfacecurvature. To address the former source of error,Young et al. [13] proposed a modification to theprocedure of Kim and Choi [14] to regain discreteconsistency. However, they were using the CSFmethod with smeared out delta functions in a levelset context and, hence, the exact discrete balancewas not achieved. Francois et al. [12] proposed a so-called “balanced force algorithm” for VoF schemeson structured Cartesian meshes that discretely bal-ances the surface tension force and the associatedpressure jump across the interface. In that paper,the discrete evaluation of the delta function as thederivative of the volume fraction scalar naturally re-sults in the discrete balance when following a similarapproach to the one proposed in Young et al. [13].The approach by Francois et al. [12] eliminates spu-rious currents up to machine precision zero, if theinterface curvature is prescribed exactly.

Different strategies exist to increase the accu-racy of curvature evaluation. For VoF methods, theheight-function approach [15] allows second-orderconverging curvature calculation. However, the re-

quired stencil sizes are large and thus problematicfor interfaces close to each other. For level set meth-ods, curvature at the node location can be calcu-lated with high-order accuracy, however, the phaseinterface curvature is approximated to first order atmost, due to the fact that nodal location and phaseinterface position typically do not coincide.

In this paper, we will extend the balanced forcealgorithm of Francois et al. [12] and Young et al. [13]to unstructured flow solver grids using the RLSGlevel set method to track the phase interface. Toachieve second-order converging curvature evalua-tion, an interface projected curvature evaluationmethod is proposed. The performance of the result-ing balanced force RLSG method is demonstratedanalyzing equilibrium columns and spheres on struc-tured and unstructured flow solver grids. Then, cou-pling of the RLSG method to a Lagrangian pointparticle representation of small scale drops is brieflydiscussed. Finally, to demonstrate the capability ofthe new method in complex flows and to analyze gridconvergence behavior, a Rayleigh-Taylor instabilityis presented.

Governing equationsThe equations governing the motion of an un-

steady, incompressible, immiscible, two-fluid systemare the Navier-Stokes equations,

∂u

∂t+ u · ∇u = −1

ρ∇p+

1ρ∇ ·(µ(∇u +∇Tu

))+g +

1ρT σ , (1)

where u is the velocity, ρ the density, p the pres-sure, µ the dynamic viscosity, g the gravitationalacceleration, and T σ the surface tension force whichis non-zero only at the location of the phase interfacexf . Furthermore, the continuity equation results ina divergence-free constraint on the velocity field,

∇ · u = 0 . (2)

The phase interface location xf between the twofluids is described by a level set scalar G, with

G(xf , t) = 0 (3)

at the interface, G(x, t) > 0 in fluid 1, and G(x, t) <0 in fluid 2. Differentiating Eq. (3) with respect totime yields the level set equation,

∂G

∂t+ u · ∇G = 0 . (4)

Assuming ρ and µ are constant within each fluid,density and viscosity at any point x can be calcu-

lated from

ρ(x) = H(G)ρ1 + (1−H(G))ρ2 (5)µ(x) = H(G)µ1 + (1−H(G))µ2 , (6)

where indices 1 and 2 denote values in fluid 1, re-spectively 2, and H is the Heaviside function. FromEq. (3) it follows that

δ(x− xf ) = δ(G)|∇G| (7)

with δ the Dirac delta function. Furthermore, the in-terface normal vector n and the interface curvatureκ can be expressed in terms of the level set scalar as

n =∇G|∇G|

, κ = ∇ · n . (8)

Using Eqs. (7) and (8), the surface tension force T σ

can thus be expressed as

T σ(x) = σκδ(x− xf )n = σκδ(G)|∇G|n= σκδ(G)∇G , (9)

with σ the surface tension coefficient.

Numerical methodIn this section, we first briefly summarize the

RLSG method used to solve the level set equa-tion and discuss how the RLSG level set solution iscoupled to structured and unstructured flow solvergrids. Next, the level set-based balanced force algo-rithm for unstructured flow solver grids is presentedand the performance of the resulting method is il-lustrated using the canonical test cases of equilib-rium columns and spheres prescribing curvature ex-actly. Then, the method to calculate second-orderconverging interfacial curvatures is outlined. Follow-ing, results are presented for curvature evaluation ofcolumns and spheres on structured and unstructuredflow solver grids. Finally the coupling procedureof the RLSG tracked phase interface to Lagrangianpoint particle methods is outlined.

Refined Level Set Grid methodIn the RLSG method, all level set-related equa-

tions are evaluated on a separate, equidistant Carte-sian grid using a dual-narrow band methodology forefficiency. This so-called G-grid is overlaid onto theflow solver grid, which can be either structured orunstructured. Details on the method when used inconjunction with a structured, equidistant Cartesianflow solver grid and its performance in generic ad-vection test cases can be found in [6, 16]. To a cer-tain extent the RLSG method is similar to the re-cently proposed Narrow-Band Locally Refined Level

Set (NBLR-LS) approach [17]. However, the latteris limited to Cartesian grids, whereas the former candeal with arbitrary unstructured flow solver meshes.In the following, we will discuss only recent modi-fications to the RLSG method not contained in theaforementioned publications.

Re-initializationFor reasons of numerical accuracy, one would

like to maintain G away from the interface G = 0as smooth a field as possible. Sussman et al. [4]proposed defining the level set scalar away fromthe interface to be a signed distance function, i.e.,|∇G| = 1. Since solution of Eq. (4) will not maintainthis property, a re-initialization procedure has to beapplied to force G 6= 0 back to |∇G| = 1. Severaldifferent strategies exist to achieve this, the one usedin this work is based on the PDE re-initialization bySussman et al. [4] using the modified signed func-tion due to Peng et al. [18]. Unfortunately, it iswell known that repeated application of the PDE-based re-initialization will inadvertently move theG = 0 isosurface and hence will not conserve fluidvolume and thus fluid mass. It is therefore desir-able to limit the application of the re-initializationprocedure to situations where the divergence fromG being a signed distance function would adverselyimpact numerical accuracy by using an appropriatetrigger criterion.

Here we will use a slight modification to the cri-terion proposed by Gomez et al. [17]. The PDE-based re-initialization procedure is applied only if

max(|∇G|) > αmax or min(|∇G|) < αmin , (10)

evaluated inside the transport band T -band (see [6,16] for definition of the individual bands). Also,Eq. (10) is used as a convergence criterion for thepseudo-time iteration of the re-initialization, whilestill limiting the maximum number of iteration stepsto nmax = CdN /hG, where dN is the width of the re-initialization band (N -band), C is the CFL-number,and hG is the grid cell size of the G-grid. In the re-sults presented in this paper we use αmax = 2 andαmin = 10−4, resulting in typically 1-2 pseudo-timeiteration steps until convergence is reached, shouldre-initialization be triggered.

Coupling to flow solverIn the Navier-Stokes equation, the position of

the phase interface influences two different terms.The first term is due to Eqs. (5) and (6), since H(G)is a function of the position of the phase interface.For finite volume formulations, the volume fraction

Vcv,iG!iG

ViG

!iG

Vcv

!cv

G-grid volume integral flow solver

Figure 1. Volume integration for unstructured flowsolver grid cells.

ψcv of control volume cv is defined as

ψcv = 1/Vcv∫Vcv

H(G)dV , (11)

with Vcv the volume of the control volume cv. Inthe RLSG method, the above integral is evaluatedon the G-grid as

1/Vcv∫Vcv

H(G)dV =

∑iGVcv,iGψiG∑iGVcv,iG

, (12)

where Vcv,iG is the joined intersection volume of theG-grid cell iG and the flow solver control volume cv(see Fig. 1), and the G-grid volume fraction ψiG iscalculated using an analytical formula [19],

ψiG = f(GiG ,∇GiG) . (13)

The joined intersection volumes Vcv,iG are calcu-lated using CHIMPS [20], employing a Sutherland-Hodgman clipping procedure [21] to calculate the in-tersection volume between a Cartesian grid cell andconvex tetra-, penta-, and hexahedra.

The second term that is a function of the in-terface position is the surface tension force term,Eq. (9). This term could be calculated first on theG-grid, using a smeared out version of the delta func-tion δε, and then volume averaged to the flow solvergrid,

T σcv= 1/Vcv

∑iG

Vcv,iGT σiG

=

∑iGVcv,iGσκiGδε(GiG)(∇G)iG∑

iGVcv,iG

.(14)

However, as will be seen later, this formulation isinconsistent with the balanced force algorithm. In-stead, only the interface curvature is transferredfrom the G-grid to the flow solver grid,

κcv =

∑iGVcv,iGδiGκiG∑iGVcv,iGδiG

, (15)

where δiG = 0 if ψiG = 0 or ψiG = 1, and δiG = 1otherwise. The use of δiG ensures that κ is treatedas a surface quantity and not a volume quantity.

In order to couple the level set equation, Eq. (4),to the Navier-Stokes equation, uiG has to be calcu-lated from ucv. Again the CHIMPS infrastructureis used and either tri-linear or C1, isotropic tri-cubicinterpolation [22] is employed. It should be pointedout that strictly speaking neither one of these veloc-ity interpolations can maintain a smooth curvaturefield under G-grid refinement. To achieve this prop-erty, kint defined as

kint = ∇ · (∇(uint · n)) , (16)

where uint is the interpolated velocity onto the G-grid and n is the interface normal vector, would haveto be continuous when switching between neighbor-ing interpolation cells. Clearly, for tri-linear inter-polation, this is not the case and even the isotropictri-cubic interpolation of Lekien et al. [22] does notguarantee this property, since neither ∂,xx nor ∂,yynor ∂,zz are kept continuous between neighboringinterpolation cells. Constructing an interpolationscheme that fulfills the above condition will be partof future work.

To achieve second-order accuracy in time, thelevel set equation is solved staggered in time withrespect to the Navier-Stokes equations.

Balanced force algorithm

The solution method of the Navier-Stokes equa-tions is based on the fractional-step method for col-located variables on unstructured grids described inMahesh et al. [23]. In the following, only the part ofthe algorithm that ensures discrete balance betweensurface tension forces and pressure gradient forces isoutlined. It is based on the balanced force methodfor Volume of Fluid methods on collocated Cartesiangrids [12].

For simplicity, we will omit the viscous term inthe following discussion. The term is fully imple-mented and solved for in flux form, with the viscos-ity at the cell face calculated by the harmonic meanof the centroid viscosities of the two control volumescv and nbr sharing the face,

µf =2µcvµnbrµcv + µnbr

. (17)

The cell centroid values are calculated by

µcv = ψcvµ1 + (1− ψcv)µ2 . (18)

The algorithm then reads

Vcvu∗i,cv − uni,cv

∆t+

∑f

un+1/2f

un+1/2i,cv + u

n+1/2i,nbr

2Af

= Vcvg + VcvFn+1/2i,cv (19)

un+1i,cv − u∗i,cv

∆t= − 1

ρn+1/2cv

∂pn+1/2

∂xi, (20)

where Af is the face area, uf the face normal veloc-ity, Fi,cv the density weighted surface tension forcedefined below, and superscripts denote time levels.

To define the force Fn+1/2i,cv at the control volume

centroid, we first need to define the surface tensionforce at the cell face,

Tn+1/2σf

= σκn+1/2f (∇ψ)n+1/2

f . (21)

Here, the face curvature is calculated from the cen-troid curvature, Eq. (15),

κn+1/2f =

αn+1/2cv κ

n+1/2cv + α

n+1/2nbr κ

n+1/2nbr

αn+1/2cv + α

n+1/2nbr

(22)

with

αn+1/2cv =

{1 : 0 < ψ

n+1/2cv < 1

0 : otherwise(23)

and

(∇ψ)n+1/2f = (ψn+1/2

nbr − ψn+1/2cv )/|scv,nbr| . (24)

Here, scv,nbr is the vector connecting the cv and nbrcontrol volume centroids. Then, Fn+1/2 at the facebecomes

Fn+1/2f = Tn+1/2

σf/ρn+1/2f , (25)

with ρn+1/2f = (ρn+1/2

cv + ρn+1/2nbr )/2 and the centroid

densities calculated from

ρn+1/2cv = ψn+1/2

cv ρ1 + (1− ψn+1/2cv )ρ2 . (26)

Finally, Fn+1/2f defined at the cell face needs to be

transferred to the control volume centroid. It is cru-cial that for this, one uses exactly the same operationthat is used for transferring (∂p/∂n)f to (∂p/∂xi)cvin the pressure corrector step. Here we use a face-area weighted least-squares method [23] by minimiz-ing

εcv =∑f

(Fn+1/2i,cv ni,f − Fn+1/2

f

)2

Af . (27)

After solving Eq. (19) to obtain u∗i,cv, the cellface normal velocities u∗f are calculated,

u∗f =12(u∗i,cv + u∗i,nbr

)ni,f −

12

∆t(Fn+1/2i,cv + F

n+1/2i,nbr

)ni,f +

∆tFn+1/2f . (28)

This is essentially a modification of the procedureby Kim and Choi [14], first proposed by Young etal. [13]. To correct the face intermediate face ve-locities u∗f to be divergence free, we then solve thefollowing variable coefficient Poisson system,

∑f

1

ρn+1/2f

∂pn+1/2

∂nAf =

1∆t

∑f

u∗fAf , (29)

and then apply the correction

un+1f = u∗f −∆tPf , (30)

with

Pf =1

ρn+1/2f

(∇pn+1/2)f

=1

ρn+1/2f

pn+1/2nbr − pn+1/2

cv

|scv,nbr|. (31)

Next, the centroid-based density weighted pres-sure gradient Pcv is calculated from the face-baseddensity weighted gradient Pf using the same face-area weighted least-squares method employed in cal-culating Ff (see Eq. 27),

εcv =∑f

(Pi,cvni,f − Pf )2Af . (32)

Finally, the control volume centroid velocity iscorrected (cf. Eq. 20),

un+1i,cv = u∗i,cv −∆tPi,cv , (33)

concluding the flow solver time step.

Exact curvature equilibrium inviscid column andsphere

To illustrate the performance of the balancedforce algorithm, we analyze the canonical test casesof the equilibrium inviscid column and sphere. Inthis case, the surface tension forces should exactlybalance the pressure jump across the phase interface,resulting in the column and sphere remaining per-fectly at rest. We employ the test case parameterssuggested by Williams et al. [24] and used by Fran-cois et al. [12]: a column (or sphere), of radius R = 2is placed at the center of an 8x8(x8) domain. Thesurface tension coefficient σ is set to 73, resultingin a theoretical pressure jump across the interfaceof ∆pex = 36.5 for the column and ∆pex = 73 forthe sphere. The density inside the column/sphereis set to ρ1 = 1 and the density outside the col-umn/sphere ρ2 is varied. Equidistant Cartesian andunstructured prism and tetrahedral flow solver grids

ρ1/ρ2 L∞(u) Ekin E(∆pmax) E(∆ppart)1 6.76e-20 6.86e-40 4.14e-16 3.41e-15

103 3.59e-17 6.28e-35 9.90e-14 9.21e-13105 9.93e-19 2.26e-40 3.16e-16 1.58e-151010 9.77e-19 1.74e-40 7.30e-16 1.19e-15

ρ1/ρ2 L∞(u) Ekin E(∆pmax) E(∆ppart)1 1.82e-16 1.06e-32 1.82e-16 4.58e-13

103 2.55e-16 7.84e-36 2.55e-16 1.97e-15105 5.43e-16 8.43e-38 5.43e-16 2.29e-151010 3.93e-18 3.39e-39 3.93e-18 7.40e-15

Table 1. Errors in velocity and pressure after singletime step for varying density ratio in the inviscid,equilibrium column test case using exact curvatureand h = hG = 0.2; Cartesian flow solver grid (top),prism flow solver grid (bottom).

are tested. The flow solver grid is characterized bythe characteristic grid size h, whereas the equidis-tant Cartesian G-grid size is denoted by hG.

The error in pressure is measured in two differ-ent ways [12],

E(∆pmax) = max(pcv)−min(pcv)−∆pex (34)

E(∆ppart) = pcv|r≤R/2 − pcv|r≥3R/2

−∆pex , (35)

where the bar indicates an arithmetic average overall control volumes fulfilling the given condition.

Table 1 summarizes the errors in velocity, pres-sure, and kinetic energy Ekin for the column aftera single time step of size ∆t = 10−6 for varyingdensity ratios if the exact curvature of the columnis used for κiG in Eq. (15). As can be seen, bothon the Cartesian and the unstructured prism flowsolver grid, errors are machine precision zero, evenfor extremely large density ratios.

Table 2 summarizes the same quantities in thesphere test case. Again, both on the Cartesian andthe tetrahedral flow solver grid, machine precisionzero errors are achieved for varying density ratios, ifthe exact curvature is employed.

Thus, provided that the exact curvature isknown, the balanced force algorithm results in ma-chine zero spurious currents, even in the inviscidcase. However, the exact curvature is rarely known,instead it has to be evaluated and is prone to errors.These curvature errors are then the sole source oferror for spurious currents.

Curvature evaluationAs noted in the previous section, only curvature

errors result in spurious currents when employingthe balanced force algorithm. Hence the task of min-

ρ1/ρ2 L∞(u) Ekin E(∆pmax) E(∆ppart)1 5.49e-17 6.95e-34 6.57e-14 1.19e-12

103 1.15e-17 1.40e-36 1.15e-14 9.41e-14105 1.21e-16 3.30e-39 5.30e-15 1.38e-141010 5.47e-16 8.24e-36 3.54e-13 2.74e-13

ρ1/ρ2 L∞(u) Ekin E(∆pmax) E(∆ppart)1 3.31e-16 8.20e-33 1.31e-11 1.25e-14

103 1.07e-15 5.96e-35 4.81e-14 1.50e-15105 2.58e-15 2.76e-36 1.42e-14 1.02e-151010 9.95e-11 2.73e-35 2.40e-15 5.60e-16

Table 2. Errors in velocity and pressure after singletime step for varying density ratio in the inviscid,equilibrium sphere test case using exact curvatureand h = hG = 0.2; Cartesian flow solver grid (top),tetrahedral flow solver grid (bottom).

R

R1

G=0

G=R1-RhG

Figure 2. Inherent phase interface curvature errorwhen evaluating curvature at nodes. Curvature isdetermined to be κ = 1/R1 = 1/(R+O(hG)) insteadof κ = 1/R.

imizing spurious currents is equivalent to increasingthe accuracy of curvature evaluation. In standardlevel set methods [25], curvature is evaluated at G-node locations by discretizing Eq. (8),

κ =G2,xx(G2

,y +G2,z) +G,yy(G2

,x +G2,z) +G,zz(G2

,xG2,y)

(G2,x +G2

,y +G2,z)3/2

−2G,xyG,xG,y +G,xzG,xG,z +G,yzG,yG,z

(G2,x +G2

,y +G2,z)3/2

, (36)

typically using a 27-point stencil. It is important topoint out that this approach approximates the cur-vature of the G-isosurface that passes through thenodal point itself. It is therefore, at best, a first-order approximation to the curvature of the phaseinterface, which can be a distance hG away fromnodes directly adjacent to the interface (see Fig. 2).Figures 3 and 4 demonstrate this first-order conver-gence rate under G-grid refinement for both the col-umn and the sphere test case using either Cartesianflow solver grids (column and sphere), unstructured

10-6

10-5

10-4

10-3

10-2

10-1

0.01 0.1hG0.01 0.1hG

10-6

10-5

10-4

10-3

10-2

10-1

0.01 0.1hG0.01 0.1hG

Figure 3. Initial column curvature errors underG-grid refinement; flow solver grid h = 0.2; Carte-sian flow solver grid (left), prism flow solver grid(right); nodal curvature (circles), direct front cur-vature (squares), Chopp front curvature (triangles),first- and second-order convergence (dashed lines).

prism grids (column), or tetrahedral grids (sphere)with h = 0.2.

Since the root cause of the first-order conver-gence rate is the fact that curvature is not calcu-lated at the interface itself, different approaches canbe taken to overcome this problem. Introducing apolynomial representation of the interface in termsof interface-based coordinates is a viable approach intwo dimensions, but becomes cumbersome in threedimensions. Here, we will follow an alternative ap-proach using the fact that a quantity defined only onthe interface itself, like curvature, can be distributedto the whole computational domain in a meaningfulway by solving

∇κ · ∇G = 0 . (37)

This effectively sets κ constant in the front normaldirection. Note that due to Eq. (15), Eq. (37) needsto be solved only for G-nodes adjacent to the inter-face. The problem is therefore similar to determin-ing the initially accepted values in the Fast MarchingMethod [26]. For this purpose, Chopp [27] devel-oped a Newton’s method that determines the near-est point on the interface (called “base-point” in the

10-6

10-5

10-4

10-3

10-2

10-1

0.01 0.1hG0.01 0.1hG

10-6

10-5

10-4

10-3

10-2

10-1

0.01 0.1hG0.01 0.1hG

Figure 4. Initial sphere curvature errors under G-grid refinement; flow solver grid h = 0.2; Cartesianflow solver grid (left), tetrahedral flow solver grid(right); nodal curvature (circles), direct front cur-vature (squares), Chopp front curvature (triangles),first- and second-order convergence (dashed lines).

following) for a given node in two dimensions. Themethod relies on approximating the level set scalarwithin each computational cell close to the interfaceby a bi-cubic spline. For this purpose, G need notbe a distance function. We have extended Chopp’smethod to three dimensions using C1, isotropic tri-cubic interpolations [22]. We typically find the base-point within 2–4 Newton iterations. However, thealgorithm can find base-points that lay outside ofthe considered G-grid cell for which the tri-cubic in-terpolation is valid. In this case, the base-point isrejected, unless none of the alternative eight G-cellsthe node belongs to yields a valid base-point. Oncethe base-point coordinates have been determined,the base-point’s curvature is calculated by tri-linearinterpolation using the surrounding nodal curvaturevalues. Using Eq. (37), the nodal curvature is thenset equal to its base-point’s curvature.

The resulting curvature errors under G-grid re-finement using Chopp’s method are shown in Figs.3 and 4. They show second-order convergence, andeven on coarse grids, Chopp’s method yields morethan one order of magnitude better curvature esti-mates than the nodal-based evaluation. The draw-

10-10

10-8

10-6

10-4

10-2

0.01 0.1hG

10-10

10-8

10-6

10-4

10-20.01 0.1hG

column column

0.01 0.1hG0.01 0.1hG

sphere sphere

Figure 5. Equilibrium inviscid column and spherevelocity (circle) and pressure (triangle) errors after 1time step under G-grid refinement; flow solver gridh = 0.2, density ratio ρ1/ρ2 = 103; from top leftto bottom right: column Cartesian flow solver grid,column prism flow solver grid, sphere Cartesian flowsolver grid, and sphere tetrahedral flow solver grid;dashed lines mark second-order convergence.

back of Chopp’s method is that for complex interfacegeometries in three dimensions, the Newton algo-rithm does not always converge. The method thuslacks the stability required for complex interface ge-ometries typically found in liquid/gas flows. Thus,the following method is proposed as an alternative.

Assuming that G is smooth in the vicinity of thephase interface, the base-point xB for a given nodexG close to the interface can be explicitly calculatedfrom

xB = xG − dn = xG −G

|∇G|∇G|∇G|

, (38)

where all gradients are calculated using central dif-ferences. This approach is termed direct front cur-vature in the following. It gives good base-point es-timates only for nodes close to the interface. How-ever, due to the way Eq. (15) is evaluated, κ needsto be calculated only on nodes close to the interface,making the direct front curvature method viable. Asbefore, once base-points have been determined, theircurvature is again calculated using tri-linear interpo-

lation from the surrounding nodal curvature values.Then, according to Eq. (37), the curvature values ofnodes are set to their respective base-points’ curva-ture values. Figures 3 and 4 also include the cur-vature errors calculated by the direct method. Ascan be seen, they are virtually indistinguishable fromthe values obtained using Chopp’s method yieldingsecond-order convergence.

Comparing the obtained curvature errors tothose calculated by Francois et al. [12], both Chopp’sand the direct method give curvature errors an ap-proximate factor of 5 lower than the 7x3 stencilheight function method employed in that paper.While a height function approach could be employedin the RLSG method as well, since volume fractionsψ are readily available, the effective G-stencil neededwould be 9x5x5 (cf. Eq. 11), as compared to 4x4x4in the direct front curvature method. Smaller sten-cil sizes are especially important for complex inter-face geometries, since both the height function andall level set curvature methods are based on the as-sumption that all ψ and G values in the stencil relateto one continuous interface segment only. Auxiliary,non-contiguous, interface segments inside the sten-cil can introduce significant errors. Hence, smallerstencil sizes are preferred to limit these errors.

In the following, we will employ the direct frontcurvature method to calculate nodal curvature val-ues on the G-grid. Figure 5 shows the errors in ve-locity and pressure after a single time step of size∆t = 10−6, using ρ1 = 1 and ρ2 = 10−3 and refiningthe resolution hG of the G-grid. As expected, dueto the balanced force algorithm, errors in curvatureevaluation result in errors in velocity and pressure,showing the same second-order convergence behav-ior (cf. Figs. 3 and 4).

Coupling to Lagrangian spray modelAtomization typically produces a vast number

of both large and small scale drops. Resolving thegeometry by tracking the phase interface associatedwith each of the resulting drops quickly becomes pro-hibitively expensive, such that a different numeri-cal description has to be employed. An alternativeapproach is to introduce simplifying assumptionsconcerning drop size and shape and treat all dropssmaller than a cut-off length scale in a point parti-cle, Lagrangian frame. One of the typical prerequi-site of such standard spray models is that the dropsize be smaller that the flow solver grid size. Sincethe RLSG approach can resolve and track sub-flowsolver sized liquid structures, all transferred dropstaken from the level set tracked representation andinserted into the Lagrangian spray model represen-

10-10

10-8

10-6

10-4

0 0.1 0.2 0.3 0.4 0.5t 0 0.1 0.2 0.3 0.4 0.5t

10-10

10-8

10-6

10-4

0 0.1 0.2 0.3 0.4 0.5t 0 0.1 0.2 0.3 0.4 0.5t

Figure 6. Temporal evolution of kinetic energy forequilibrium inviscid column (top) and sphere (bot-tom) for 500 time steps under G-grid refinement:hG = 0.4 (circle), hG = 0.2 (triangle), hG = 0.1(box); flow solver grid h = 0.4, density ratio ρ1/ρ2 =103; Cartesian flow solver grids (top left and bottomleft), prism flow solver grid (top right), and tetrahe-dral flow solver grid (bottom right).

tation can fulfill this prerequisite. The details of therequired broken-off drop identification and removalalgorithm are presented in [16]. Drop transfer is ini-tiated if a separated liquid structure has a liquidvolume

VD ≤ Vcv , (39)

and its shape is nearly spherical,

rmax ≤ α(

34πVD

)1/3

, (40)

with typically α = 2 and rmax the maximum dis-tance of the liquid structure’s surface to it’s centerof mass. Further details concerning the applicationof the Lagrangian spray model coupling procedurecan be found in [28].

ResultsLong-time evolution of the equilibrium inviscid col-umn and sphere

As shown in the previous section, errors in cur-vature evaluation result in spurious currents that aresmall, but non-zero. Thus, the long-time behavior

column column

10-3

10-2

10-1

100

0.1 1hG

10-3

10-2

10-1

1000.1 1hG

sphere sphere

0.1 1hG0.1 1hG

Figure 7. Equilibrium inviscid column and spheremaximum velocity error during 500 time steps underG-grid refinement; flow solver grid h = 0.4, densityratio ρ1/ρ2 = 103; from left to right: column Carte-sian flow solver grid, column prism flow solver grid,sphere Cartesian flow solver grid, and sphere tetra-hedral flow solver grid; dashed lines mark second-order convergence.

of the equilibrium inviscid column and sphere is ofinterest, since errors might accumulate and result inlarge erroneous velocities.

Figure 6 shows the temporal evolution of the ki-netic energy in the computational domain for boththe inviscid column and sphere on Cartesian and un-structured flow solver grids. The flow solver charac-teristic grid size is h = 0.4 in all simulations, ρ1 = 1,ρ2 = 10−3, σ = 73, and the fixed time step sizeis ∆t = 10−3. As observed by Francois et al. [12],the column seems to enter an oscillatory mode thatappears quite stable on a Cartesian flow solver grid(top left), but shows a slight growth on the prismflow solver grid (top right). The inviscid sphere re-sults, on the other hand, do not exhibit such a clearperiodic behavior. In the Cartesian flow solver gridcase, different periods seem to be superposed, andthe unstructured tetrahedral grid shows an increasein kinetic energy without reaching a periodic state.This is due to the fact that the unstructured gridlacks the symmetry of the Cartesian grids of theflow solver and the G-grid. This symmetry seems to

0 1-2

-1

0

1

2t = 0.5 t = 0.6 t = 0.7 t = 0.8 t = 0.9

Figure 8. Rayleigh-Taylor instability interfaceshapes for reference solution, h = hG = 1/512 .

initiate a periodic oscillation instead of a constantgrowth in spurious currents and is thus beneficial inthis particular test case, but not indicative of themethod’s performance in a more general setting. Inthe general case, one can expect a growth of thekinetic energy along the lines of the unstructuredgrid results, necessitating at later times the use ofviscous dissipation to control the spurious currents.However, the balanced force method exhibits verylow levels of spurious currents and indeed, this levelcan be made even smaller if the G-grid is refinedto increase the accuracy of the interface curvatureevaluation. Figure 7 shows the convergence ratesfor the maximum velocity error under G-grid refine-ment, using a flow solver resolution of h = 0.4. Closeto second-order convergence can be observed both onstructured and unstructured flow solver grids.

Rayleigh-Taylor instabilityTo demonstrate the performance of the pro-

posed method, the complex flow of a Rayleigh-Taylor instability is computed. This is a commontest problem analyzed by a variety of different meth-ods [17, 29, 30, 31]. A heavy fluid, ρ1 = 1.225, µ1 =0.00313, is placed above a light fluid, ρ2 = 0.1694,µ2 = 0.00313, inside a domain of size 1x4. The inter-face between the two fluids is placed in the middle ofthe domain and is perturbed by a cosine wave of am-plitude 0.05. The gravity constant is set to g = 9.81.We set the time step size constant to ∆t = 2.5 ·10−4

and simulate up to t = 0.9. Figure 8 shows theinterface shape at different instances in time for aCartesian flow solver grid of h = 1/512 and a G-grid of hG = 1/512. As will be demonstrated below,

-1.0

-0.5

0

0.5-1.5

-1.0

-0.5

1

hG=1/64

t=0.6 t=0.7

t=0.8 t=0.9

1/128

1/128

1/128

1/128

1/256

1/256

1/512

1/512

1/256

1/256

1/512

1/512hG=1/64

hG=1/64

hG=1/64

Figure 9. Rayleigh-Taylor instability interfaceshapes under G-grid refinement hG = 1/64, 1/128,1/256, and 1/512 (left to right in each group) at t =0.6, 0.7, 0.8, and 0.9 (top left to bottom right) andflow solver grid h = 1/64. Thin line denotes refer-ence solution.

this grid resolution ensures grid converged resultsand thus will be used as a reference solution in thefollowing.

Figures 9–11 show the interface shape for dif-ferent flow solver and G-grid resolutions. Using thecoarsest flow solver grid of h = 1/64 presented inFig. 9, one can already notice deviations from thereference solution at early times. While the stemand bubble shape is well captured, the fine scale ge-ometry of the side arms is not well maintained. Upto t = 0.8, there appears almost no difference be-tween the results using hG ≥ 1/128. This indicatesthat the deviations from the reference solution aredue to errors in the flow representation and not dueto errors in the interface tracking scheme. However,at t = 0.9, the very fine connecting bridge at theside arms can only be maintained by hG = 1/512.Note that except for the difference in the details ofthe connecting bridge, the larger scale geometric fea-tures are consistent between different G-grid resolu-tion with hG ≥ 1/128.

At a flow solver grid of h = 1/128 presented inFig. 10, virtually no difference can be discerned be-tween the reference solution and G-grid resolutionsof hG ≥ 1/128 up to t = 0.8. At t = 0.9, how-ever, the thin connecting bridge is only supportedby hG = 1/512. Comparing the results at t = 0.9of h = 1/128 to those of h = 1/64 (cf. Fig. 9), thefiner grid flow solver results capture the shape of theinterface significantly better. This indicates that theflowfield is well resolved by the h = 1/128 grid.

This observation is further substantiated by re-fining the flow solver grid further to h = 1/256, pre-sented in Fig. 11. Virtually no difference to the

-1.0

-0.5

0

0.5-1.5

-1.0

-0.5

1

hG=1/128

t=0.6 t=0.7

t=0.8 t=0.9

hG=1/128

hG=1/128

hG=1/128

1/256 1/512 1/256 1/512

1/256 1/512 1/256 1/512

Figure 10. Rayleigh-Taylor instability interfaceshapes under G-grid refinement hG = 1/128, 1/256,and 1/512 (left to right in each group) at t = 0.6,0.7, 0.8, and 0.9 (top left to bottom right) and flowsolver grid h = 1/128. Thin line denotes referencesolution.

-1.0

-0.5

0

hG=1/256

t=0.6

1/512

t=0.7

hG=1/256 1/512

0.5-1.5

1

0.5 1

t=0.8

hG=1/256 1/512

t=0.9

hG=1/256 1/512

Figure 11. Rayleigh-Taylor instability interfaceshapes under G-grid refinement hG = 1/256 and1/512 (left to right in each group) at t = 0.6, 0.7, 0.8,and 0.9 (left to right) and flow solver grid h = 1/256.Thin line denotes reference solution.

reference solution can be discerned, with the excep-tion of hG = 1/256 at t = 0.9, where again thecomplete fine connecting bridge is not supported bythat G-grid resolution. Nonetheless, those parts ofthe bridge that can be maintained by the grid are inexcellent agreement with the reference solution.

To ascertain the volume/mass conservationproperties of the method, Fig. 12 depicts the nor-malized volume error, defined as

EV (t) =|∑cv ψcv(t)Vcv −

∑cv ψcv(t = 0)Vcv|∑

cv ψcv(t = 0)Vcv.

(41)Except for hG = 1/512, all solutions show an in-crease in error at late times. This is due to thedisappearance of the thin connecting bridge. Also,

10-7

10-6

10-5

10-4

10-3

10-2

0 0.3 0.6 0.9t

10-7

10-6

10-5

10-4

10-3

10-2

0 0.3 0.6 0.9t

h = 1/64 h = 1/128

h = 1/256 h = 1/512

0 0.3 0.6 0.9t 0 0.3 0.6 0.9t

Figure 12. Rayleigh-Taylor instability normalizedvolume error, h = 1/64, h = 1/128,h = 1/256,and h = 1/512 (from top left to bottom right) forhG = 1/64 (open circle), hG = 1/128 (filled trian-gle), hG = 1/256 (open box), and hG = 1/512 (filledcircle).

at constant flow solver grid size h and G-grid re-finement, the volume errors converge to a non-zerovalue. This indicates that this remaining volume er-ror is not due to errors in level set transport but ismost likely due to the fact that uiG used to solve thelevel set transport equation is not divergence free.Note that the G-grid converged volume error errordecreases when increasing the flow solver grid res-olution. Thus, the remaining volume error appearsto be due to the following two reasons. First, uiGis based on interpolation of the flow solver cell cen-ter velocities ucv which, in a collocated scheme, arenot guaranteed to be divergence free. Second, theinterpolation scheme itself can add additional diver-gence to the interpolated velocities uiG . While theformer inconsistency is hard to address in a collo-cated unstructured flow solver, except by an addi-tional projection-correction of the cell center veloc-ities used for interpolation, the latter inconsistencycan be addressed by improved, divergence preservinginterpolation schemes. Nonetheless, the observedvolume errors on fine G-grids are very small and wellwithin acceptable limits.

SummaryA balanced force RLSG method has been pre-

sented for structured and unstructured flow solvergrids. The method ensures machine precision zerospurious currents for arbitrary density ratios if thecurvature can be evaluated exactly. Spurious currentmagnitude is directly related to errors in the evalu-ation of the interface curvature. To minimize spuri-ous currents in actual applications, a robust second-order converging curvature evaluation scheme hasbeen presented that significantly reduces spuriouscurrents compared to the traditional first-order con-verging curvature evaluation schemes. The perfor-mance and good volume conservation properties ofthe RLSG method have been demonstrated using theRayleigh-Taylor instability. The RLSG method suc-cessfully addresses the numerical challenges presentin primary atomization simulations. It treats thephase interface as a discontinuity in the finite volumesense, handles the singular surface tension force in anaccurate and stable manner, automatically enablestopology changes, and presents an efficient methodto study grid dependency. Coupled to a Lagrangianspray model describing the secondary atomizationprocess, the RLSG method can be employed totrack the phase interface during primary atomiza-tion, thereby simulating the atomization process asa whole [28].

AcknowledgmentsThe author would like to thank Olivier Des-

jardins, Dokyun Kim, Guillaume Blanquart, andFrank Ham for many fruitful discussions. The workwas supported by the Department of Energy’s ASCprogram.

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