ILASS-Americas 30th Annual Conference on Liquid Atomization and Spray Systems, Tempe, AZ, May 2019

A Robust All-Mach Multiphase Flow Algorithm for High-Fidelity Simulationsof Compressible Atomization

M.B. Kuhn∗ and O. DesjardinsSibley School of Mechanical and Aerospace Engineering

Cornell UniversityIthaca, NY 14853 USA

AbstractHigh-fidelity simulations can accelerate understanding and illuminate important aspects of liquid atomiza-tion in highly compressible environments. For example, simulations can provide invaluable insights on thephysics of scramjet engine cold-start, thereby helping design successful injection strategies. For simulatingcompressible liquid-gas flows with topology changes, we have developed an all-Mach, compressible multiphaseflow solver that utilizes a low dissipation transport scheme to accurately represent turbulence in smooth,single-phase regions and applies a robust semi-Lagrangian scheme to handle discontinuous transport at in-terfaces and shocks. We employ a pressure projection scheme to avoid acoustic limitations on the time-stepsize. Within this framework, we focus on specific treatments of the volume fraction, pressure, and energyequations to improve stability.

∗Corresponding Author: mbk225@cornell.edu

Introduction

Simulating liquid atomization in compressibleenvironments provides unique challenges, as flowquantities can become highly discontinuous due toshocks and phase interfaces. However, this problemis of high relevance to the development of fuel deliv-ery systems for scramjet engines, for instance. Thefuel injection strategy heavily influences the atom-ization process, which contributes to the evaporationcharacteristics and spatial distribution of the result-ing spray, having consequences on the performanceof the engine [1]. Improving numerical capabilitiesto solve relevant flows, like a liquid jet in supersoniccrossflow, can add significant physical insight andintroduce design principles in a context where bothexperimental and computational studies have beenlimited.

Numerical techniques capable of simulatingcompressible atomization are sparse in the litera-ture, in comparison with incompressible atomiza-tion. Shukla et al. [2] combine a diffuse representa-tion of the interface with a density-sharpening tech-nique to significantly improve the results of shockinterface interaction problems, though the sharpen-ing technique is non-conservative. Garrick et al. [3]improve upon Shukla’s method, employing a mod-ification of the THINC (Tangent of Hyperbola forINterface Capturing) scheme from [4] and [5]. Thisstrategy is conservative, and they apply it to thesimulation of a liquid jet in supersonic crossflow.

Diffuse interface schemes are confined to spread-ing the interface over a handful of cells, which hasthe tendency to practically reduce resolution at theinterface. Jemison et al. [6] treats the phase in-terface sharply, representing it with a planar recon-struction within a single cell. The equations govern-ing density and energy are solved separately in eachphase, while there is a single momentum equationand a single pressure equation. Their scheme com-bines a directionally split semi-Lagrangian transportalgorithm with a pressure projection scheme adaptedfrom [7]. Finally, they show this method to be capa-ble of simulating a liquid jet in subsonic crossflow.

We improve upon the framework of [6], utiliz-ing an unsplit semi-Lagrangian transport scheme [8]along with a low-dissipation scheme adapted from[9] to better simulate turbulence. Moreover, we dis-cuss techniques for handling multiphase cells that wehave found to be critical for stability and accuracyin difficult problems like a liquid jet in supersoniccrossflow.

Numerical Framework

We solve the compressible Navier-Stokes equa-tions using a finite volume framework. Equations arewritten in each phase for α, ρα, ραE (i.e., volume,mass, and energy of each phase — the i subscript be-low represents each phase), and an additional equa-tion for mixture momentum ρu completes the sys-tem. It is fully given as

∂αi

∂t+∇ · (αiu) = αi∇ · u, (1)

∂ρiαi

∂t+∇ · (ρiαiu) = 0, (2)

∂ρiEiαi

∂t+∇ · (ρiEiαiu) +∇ · (pαiu)

= ∇ · (u · τ + k∇T ) + u · Fαi, (3)

and

∂ρu

∂t+∇ · (ρu⊗ u) +∇ · (pI) = ∇ · (τ ) + F , (4)

where

τ = µ

(∇u+∇uᵀ − 2

3∇ · uI

)(5)

is the viscous stress tensor, the mixture densityis ρ = α1ρ1 + α2ρ2, and the mixture energy isρE = α1ρ1E1 +α2ρ2E2. F contains all body forces,including gravity and surface tension. The mixtureviscosity µ is given by µ = µ1µ2/(µ1α2 + µ2α1).

We employ a fractional step approach to time in-tegration, dividing the process of time advancementinto three components. First, we calculate the con-vection of each variable, then we apply the viscousand forcing terms, and finally we solve for a pres-sure correction. Similar to [6], we use an additionalfixed-point iteration to reduce oscillatory behaviordue to the non-linearity of the momentum equation.

To solve for the convective terms, we use thevolumetric transport strategy discussed in [8]. Thisscheme is conservative and formally bounded. Toimprove the accuracy of the transport of continuousvariables, we use a linear reconstruction in each cell,limited by the minmod slope limiter for stability.

Viscous terms are discretized using second-orderfinite differences and calculated explicitly using sub-stepping for robustness. Surface tension is insertedas a pressure jump in the Helmholtz equation forpressure, as in [10]. Interface curvature is calculatedusing a quadratic least squares approach [11, 12].

We calculate the pressure by implicitly solving aHelmholtz equation, thereby eliminating the acous-tic CFL restriction. This is particularly advanta-geous for atomization problems, where the speed of

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sound in the liquid can be significantly higher thanthat of the gas and the associated CFL restrictionwould otherwise become a cost-intensive limit. Thepressure equation we solve is adapted from [7], andis written as

pn+1 − ρn+1 (c∗)2

∆t2 ∇ ·(∇pn+1

ρn+1

)= p∗ − ρn+1 (c∗)

2∆t∇ · u∗. (6)

This pressure is then used to correct the flow vari-ables. Only the pressure at n + 1 is employed inthe pressure correction. This is consistent with theequation from [7] and agrees with the common prac-tice of biasing the future pressure, like in [9, 13].

In applying this pressure projection approach toa multiphase problem, decisions have to be made re-garding how to calculate the advected pressure (p∗)and the bulk modulus (ρc2) in cells that containmore than one phase. The solution of the pressureequation applies to both phases, but the calcula-tion of the advected pressure and the bulk modulusmust come from some combination of both equationsof state, using information from both phases. Themethods used for multiphase cells can have signifi-cant consequences on accuracy and stability, so care-ful treatment is required. We discuss the handlingof these terms later.

Hybrid Framework

Although the unsplit sem-Lagrangian transportscheme is robust, it also introduces significant un-physical kinetic energy dissipation. In [14], we dis-cussed hybridization of our numerical solver by in-troducing centered schemes in regions that requireless robustness. Since then, we have improvedour approach by adapting concepts from [9]. Weuse their discretization of the momentum equationand the kinetic energy fluxes, integrated with semi-Lagrangian transport of the density and internal en-ergy. In situations of constant density and internalenergy, results are identical to those in [14], but thismodification adds robustness that was lacking whensolving flows with high density and internal energyvariations present.

Multiphase Treatment

Mixture Rules and Liquid Volume Fraction Calcula-tion

Although many aspects of our numerical ap-proach maintain a sharp interface between gas quan-tities and liquid quantities, the pressure projectionmethod requires there to be a single pressure field.Solving the Helmholtz equation requires a system-atic approach to designating a single advected pres-

sure and a single bulk modulus in cells that havetwo phases, where each phase has its own equationof state. We compare two approaches based on com-mon practices with a new approach that we havedesigned. All examples consider the stiffened gasequation of state, written as

p = (γ − 1)(ρE − 1

2ρu · u)− γp∞ (7)

Jemison et al. suggest using the equation of stateof the phase occupying the most volume in the cell,that is, the majority phase. However, this simpleapproach is dramatically inaccurate in the compress-ible atomization simulations we have performed.

A different straightforward approach to calculat-ing these quantities is to assume isobaric closure be-tween phases, and formulate an equation of state forthe mixture. This involves calculating mixture prop-erties based on the volume fraction and the proper-ties of each individual phase, shown below.

1

γ − 1=∑i

αi

γ − 1(8)

p∞ =γ − 1

γ

∑i

αiγip∞i

γi − 1(9)

These mixture properties can be used to calculatethe advected pressure and speed of sound in theHelmholtz equation. Although they do not use theserelations in the context of a pressure projectionscheme, these relations are consistent with [2] and[15]. This approach more closely aligns with a dif-fuse interface strategy.

A third approach is derived from the definitionof isentropic bulk modulus, and follows from thework of Alahyari Beig and Johnsen [15]. In the pres-ence of nonzero dilatation, they note that a sourceterm should be added to the volume fraction evolu-tion equation. In describing this approach, we referto the use of the volume fraction source term, whichthey derive, and corresponding mixture rules, whichwe put forth.

This source term takes into account the differ-ence in compressibility between phases when morethan one phase occupies a cell. In essence, dilatationshould apply more to the more compressible phasein a cell rather than apply to both phases equally.The volume fraction then becomes

∂αi

∂t+ u · ∇αi = Γii′∇ · u, (10)

where

Γii′ = αi

(K

Ki− 1

). (11)

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In this equation, K represents the isentropic bulkmodulus, so for an individual fluid, Ki = ρic

2i . The

isentropic bulk modulus of the mixture is defined as

1

K=∑i

αi

Ki. (12)

To calculate this term during the transport step, we

1. Use the equation of state to calculate the pres-sure in each phase

pni = EOSi(ρEi, ρi,u)

2. Primitively transport the pressure in eachphase, and the volume fraction

p∗i = pni −∆tu · ∇pi

α∗i = αni −∆tu · ∇αi

3. Calculate the bulk modulus of each phase usingthe transported pressure

Ki = (ρc2)∗i = EOSi(p∗i )

4. Calculate the bulk modulus of the mixture

K = ρc2 =

[∑i

α∗i(ρc2)∗i

]−1

5. Update the volume fraction

αn+1i = α∗i + Γii′∇ · u

Since this term modifies the volume fraction, it indi-rectly modifies all of the transported variables andcan significantly influence the behavior of quantitiesin interface cells, particularly when dilatation is sub-stantial. This formulation of the mixture bulk mod-ulus also points to using a different calculation ofmixture terms in the pressure equation. If we statethat the evolution equation for pressure of a phase iis

pn+1 = p∗i −∆t(ρc2)i∇ · un+1, (13)

and the bulk modulus of the mixture is defined asin Equation 12, then the advected pressure of eachphase must be combined in a consistent way to pro-duce the mixed equation

pn+1 = p∗ −∆t ρc2∇ · un+1. (14)

This relationship informs us that

p∗ = ρc2∑i

αi

(ρc2)ip∗i . (15)

Kinetic Energy Exchange

Although using this volume fraction source termand corresponding mixture rules can add robustness,there remain other issues to be addressed. Specialattention must be paid to the consequences of treat-ing each fluid separately but using a single velocityfor both fluids. In the process of convection, wetransport the momentum of each phase separately.After all variables have been transported, we use thefollowing relation to calculate the velocity in the cell,which is assumed to be the same in both phases. The∗ indicates the quantity calculated by the convectivestep.

u∗ =

∑αi(ρu)∗i∑αiρi

(16)

This effectively modifies the kinetic energy of eachphase, but the total energy variable has not beenmodified. Lei and Li [16] present a term that ap-proximates the kinetic energy exchange that takesplace between two phases when the velocity is “uni-formized”, as they put it.

Work Done by Pressure

The final area where problems can arise is incalculating the work done by pressure, which is theterm in the total energy equation denoted as ∇ ·(pu). To determine the proper treatment of thisterm, we look individually at the contributions fromthe internal energy equation and from the kineticenergy equation. For the sake of brevity, let us defineDfDt = ∂f

∂t +∇·(fu). The evolution of internal energyin each phase follows the equation

DρieiDt

= −pi∇ · ui. (17)

The mixture internal energy is defined as

ρe =∑i

αiρie, (18)

leading to the mixture evolution equation

Dρe

Dt= −

∑i

αipi∇ · ui. (19)

Since we assume that the pressure and velocity arecontinuous in both phases, this simplifies to

Dρe

Dt= −

∑i

αip∇ · u. (20)

The kinetic energy equation, however, is more com-plicated, since the velocity is shared between phases.

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The kinetic energy is defined as

ρk =∑i

1

2αiρi

[u2 + v2 + w2

]=∑i

αiρik. (21)

Rephrasing this into only conserved variables, thedefinition becomes

ρk =∑i

1

2

ρiρ2[(ρu)2 + (ρv)2 + (ρw)2

]. (22)

To derive the governing equation for kinetic energyin a phase, we apply the operator Df

Dt along with thechain rule.

D(ρk)iDt

=DρiDt

ρk +Dρ

Dt

(−ρiρ

2k

)+Dρu

Dt

ρiρu

+Dρv

Dt

ρiρv +

Dρw

Dt

ρiρw, (23)

The density transport terms are equal to zero, andthe momentum transport terms can be replaced bytheir equivalent pressure gradients.

D(ρk)iDt

= −∂p∂x

ρiρu− ∂p

∂y

ρiρv − ∂p

∂z

ρiρw (24)

D(ρk)iDt

= −ρiρu · ∇p (25)

This analysis indicates that the pressure termin the total energy equation should not simply bewritten

D(ρE)iDt

= −ρiρ∇ · (pu) (26)

or

D(ρE)iDt

= −∇ · (pu)

rather, it should instead be written

D(ρE)iDt

= −p∇ · u− ρiρu · ∇p (27)

In single-phase cells, this makes no difference.However, in multiphase cells, this alters the effectthat dilatation and pressure gradient have on thetotal energy of each phase. Discretized directly, thisformulation does not conserve the total energy of themixture, so we rephrase the expression as

D(ρE)iDt

= −p∇ · u− ρiρ

(∇ · (pu)− p∇ · u) , (28)

which conserves the total energy of the mixture.

Results and Discussion

In order to evaluate the accuracy and stabilityof these methods, we apply each method to threetest cases. Table 1 shows the five approaches thatwe evaluate. The first three schemes compare differ-ent mixture rules, the fourth introduces the kineticenergy exchange term from [16], and the fifth usesthe discretization of the work done by pressure thatwe have proposed in this work with (28). Note thatschemes 1-4 use the discretization of that term from[6], shown by (26).

Scheme Description1 Majority-phase equation of state2 Mixture rules from [2] and [15]3 Volume fraction source term with

associated mixture rules4 Scheme 3 with kinetic energy exchange5 Scheme 4 with proposed discretization

of the work done by pressure

Table 1. Numbering of schemes for reference

1D water-air shock-interface interaction

The first test case is from [2], where a water-airinterface is accelerated by a strong shock in water.This leads to a strong rarefaction wave reflected backinto the water, while a weak shock is transmittedthrough the air. We solve this problem with a gridof 400 points and a constant time step size of 3.8 µs.This test case does little to distinguish between thedifferent schemes, and it suggests that each schemeis reasonable.

Scheme 3, however, crashes on the fifth timestep and cannot complete the simulation. As theliquid momentum increases the velocity of the inter-face cell, the pressure in the gas dramatically drops,to the point that it becomes negative. The kineticenergy exchange term included in schemes 4 and 5is necessary for stability. Schemes 1 and 2, whichlack the kinetic energy exchange term, do not crash.We suspect that this is because they lack the sourceterm in the volume fraction equation. Without thisterm, the compressibility of each phase is enforcedless strictly. When the rarefaction wave begins inthe liquid, schemes 1 and 2 allow for more expansionof the liquid than the other schemes, which allowsfor changes in the neighboring gas to be less drastic,thus avoiding a crash. Despite this difference, thecurves are remarkably similar by the time shown inFigure 1, which highlights the subtlety involved withdeciding the optimal treatment of multiphase cells.

5

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

0.0 0.2 0.4 0.6 0.8 1.0

ρ(kg/m

3)

x (m)

0

20

40

60

80

100

120

140

160

180

0.0 0.2 0.4 0.6 0.8 1.0

u(m/s)

x (m)

0

5

10

15

20

25

0.0 0.2 0.4 0.6 0.8 1.0

p(kPa)

x (m)

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

α

x (m)

Figure 1. The density, velocity, pressure, and volume fraction for the one-dimensional water-air shock-interface interaction at a time of 1.5 ms. Schemes 1 and 2 ( and , respectively) are essentiallycoincident, and appear to be slightly smoother than schemes 4 and 5 ( and , respectively), butdifferences are slight.

1D rarefaction-droplet interaction

The next test case is the interaction between ararefaction wave, originating in the gas phase, witha one-dimensional “droplet”. During the simulation,we use the stiffened gas equation of state with theparameters listed in Table 2. The length of the do-main is 20 mm, and a droplet of width 0.3 mm isplaced at a position of 6.7 mm. The initial liquiddensity is 1000 kg/m3, and the initial gas density is1.2 kg/m3. The initial pressure is uniform at 101325Pa. The velocity is initialized with a discontinu-ity at 7.8 mm, where the velocity is 0 to the leftof the discontinuity and 224.786 m/s to the right ofthe discontinuity. This velocity discontinuity creates

a rarefaction wave that travels toward the droplet.This wave is partially transmitted to the droplet andinduces the droplet to move to the right as the gason the left side of the droplet expands. We use auniform grid of 401 points and a constant time stepof 0.18 µs.

γ p∞

Gas 1.4 0Liquid 4.4 6× 108 Pa

Table 2. Material properties used for each phase inthe second and third test cases. The expression forthe equation of state is shown by (9).

6

0

200

400

600

800

1000

6 8 10 12 14 16

998

999

1000

1001

1002

1003

6.7 6.8 6.9

ρ(kg/m

3)

x (mm)

123456789

101112

6 8 10 12 14 16

u(m/s)

x (mm)

20

30

40

50

60

70

80

90

100

110

6.0 8.0 10.0 12.0 14.0 16.0

−4000−2000

0200040006000

6.7 6.8 6.9

p(kPa)

x (mm)

0.0

0.2

0.4

0.6

0.8

1.0

6 8 10 12 14 16

α

x (mm)

Figure 2. The density, velocity, pressure, and volume fraction for the one-dimensional rarefaction-dropletinteraction at a time of 37.26 µs. The inset plots in the density and pressure are the liquid density andliquid pressure, respectively. The liquid pressure is calculated using the liquid equation of state in cells thatcontain liquid; it is not synonymous with the pressure calculated from the Helmholtz equation, shown in themain pressure plot. Schemes 1 and 2 ( and , respectively) are essentially coincident, and schemes3, 4, and 5 ( and , respectively), are essentially coincident as well.

This test case is of interest because it showcasesa wave originating in the gas phase impacting the liq-uid. Additionally, the response of the liquid phasecauses the droplet to move, and the combination be-tween the motion of the droplet and rarefaction waveelucidates differences between the schemes. As thedroplet moves slowly through the domain, contain-ing information from its response to the rarefactionwave, the differences between the schemes becomedistinguishable as artifacts in the pressure and ve-locity, shown in Figure 2. In addition to those quan-tities, significant differences arise in the liquid gasdensity and the liquid pressure, which are insets in

Figure 2.Using some assumptions, we can estimate the

expected variation in density and pressure in the liq-uid. Treating the liquid-gas interface as a stationarywall with an impedance mismatch, where the acous-tic impedance of each phase is equal to ρc, the initialpressure variation of -38.6 kPa caused by the rarefac-tion wave should almost double to -77.1 kPa in theliquid. Then, assuming the transport of the wave isisentropic, the expected density variation is -0.0003kg/m3. Though these numbers rely on approxima-tions and ignore the additional reflections that willoccur as time continues, they indicate the order of

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magnitude to expect from the simulation. At thetime step shown in Figure 2, schemes 1 and 2 led toliquid density variations of 2 kg/m3 and liquid pres-sure variations of 450 kPa, while schemes 3 through5 led to liquid density variations of -0.015 kg /m3

and liquid pressure variations of -50 kPa. In addi-tion to these errors in the liquid phase, the pressuresolution and velocity solution appear more oscilla-tory for schemes 1 and 2 than for the other schemes.

From this test case, it seems schemes 1 and2 allow the liquid density to deviate more than itshould, and they allow the pressure solution to de-viate strongly from the pressure calculated by theequation of state. This behavior contributes to inac-curate oscillations in the pressure and velocity, weak-ening the stability of the solver.

2D shock-water column interaction

Finally, we look at a test case closer to the in-tended application, which is atomization. This testcase is the breakup of a water column by a highspeed flow behind a shock wave of M=1.47, from[17]. However, we use a smaller water column withabout 6 cells/diameter, so that we can probe theaccuracy and stability of the method in handlingunder-resolved liquid structures which arise duringatomization processes. The square domain is 100mm long, and the shock is located at 48.4 mm. Thedroplet has a diameter of 1.6 mm and is located at50 mm. The uniform grid has 400 points in eachdirection, and we run the simulation at a convectiveCFL of 0.9. We incorporate additional physics, witha gas viscosity of 1.78×10−5 Pa s, a liquid viscosityof 1.137×10−3 Pa s, and a surface tension coefficientof 0.078 N/m.

As the shock impacts the droplet and atomizesit, we investigate the evolution of the liquid densityand liquid pressure. To get a sense of the variationof the liquid density, we plot its maximum and mini-mum obtained by each scheme over the course of thesimulation. Results from scheme 1 are shown in Fig-ure 3. This scheme yields grossly unrealistic results,with the maximum liquid density increasing seven-fold while the minimum liquid density plummets toalmost zero. This 2D flow reveals that scheme 1 failsto get the liquid to behave like a liquid, due to howthe liquid equation of state is only considered if theliquid takes up more than half of a cell.

In Figure 4, we compare the results from theother schemes. Schemes 3 through 5 outperformscheme 2 in maintaining a near-constant liquid den-sity as the simulation progresses. Although scheme2 does better than scheme 1, the density still variesrapidly and unrealistically.

0

1000

2000

3000

4000

5000

6000

7000

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

ρl(kg/m

3)

t (ms)

Figure 3. Liquid density bounds during course of2D shock-water column interaction, using scheme 1.The maximum liquid density at time t ( ) andthe minimum liquid density at time t ( ) evolveerratically, diverging significantly.

970

975

980

985

990

995

1000

1005

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

ρl(kg/m

3)

t (ms)

Figure 4. Liquid density bounds during course of2D shock-water column interaction, for schemes 2through 5 ( , , , and , re-spectively). Dashed lines indicate maxima, and solidlines indicate minima.

Next, we plot the evolution of the minimum liq-uid pressure, shown in Figure 5. We look at theminimum because when the pressure becomes toolow, the code can crash, so this quantity is rele-vant to stability. The variations in pressure fromscheme 2 correspond strongly with the variations indensity from Figure 4. Because of the “stiffness” in-troduced in the liquid equation of state, small vari-ations in liquid density can lead to huge variationsin liquid pressure. Schemes 3 through 5 steadily im-

8

prove on the curve from scheme 2, with each beingmore smooth and more horizontal than the previ-ous. Though this complicated test case is ill-suitedto determine concretely which treatment is the mostaccurate, it strongly suggests that scheme 5, our fulltreatment, is the most stable method for problemslike this one.

−100−90−80−70−60−50−40−30−20−10

010

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

minpl(M

Pa)

t (ms)

Figure 5. Minimum liquid pressure during courseof 2D shock-water column interaction, for schemes 2through 5 ( , , , and , respec-tively).

Summary and Conclusions

The treatment of the pressure, energy, and vol-ume fraction in multiphase cells can be highly in-fluential to the stability of a compressible, multi-phase flow solver. In a framework that considersa sharp interface between phases and a pressureprojection method, we have evaluated different ap-proaches to calculations in multiphase cells and haverecommended specific numerical improvements andinnovations. This new approach has enabled us tosimulate a liquid jet in supersonic crossflow, shownin Figure 6, with consistency and robustness that wecould not otherwise attain.

References

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[12] Emilie Marchandise, Philippe Geuzaine, Nico-las Chevaugeon, and Jean-Francois Remacle.Journal of Computational Physics, 225(1):949–974, 2007.

[13] Yucheng Hou. PhD thesis, University of Min-nesota, 2007.

[14] Michael B Kuhn and Olivier Desjardins. 14thTriennial International Conference on Liq-uid Atomization and Spray Systems, pp. 1–6,Chicago, IL, 2018.

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Figure 6. Simulation of a liquid jet in supersonic crossflow. Liquid interface is depicted using the 0.5isocontour of volume fraction. Center plane is colored by velocity magnitude, with 700 m/s as red and 0m/s as blue. Snapshot taken at 1.73 ms.

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