ILASS Americas 27th Annual Conference on Liquid Atomization and Spray Systems, Raleigh, NC, May 2015

Direct numerical simulation of surface tension effects on interface dynamics inturbulence

J.O. McCaslin∗ and O. DesjardinsSibley School of Mechanical and Aerospace Engineering

Cornell UniversityIthaca, NY 14853 USA

AbstractThe interaction between turbulence and surface tension is studied through direct numerical simulation ofa canonical multiphase flow. An initially flat interface is inserted into a triply periodic box of decayinghomogeneous isotropic turbulence, simulated for a variety of turbulent Reynolds and Weber numbers onmesh sizes of 5123 and 10243. Unity density and viscosity ratios are used in order to isolate the interactionbetween fluid inertia and the surface tension force. Interface height correlations and liquid volume fractionvariance spectra are used to study the spatial scales of corrugations on the interface. A case with zero surfacetension is first considered, yielding a passive interface that moves materially with the fluid. The powerspectral density of the liquid volume fraction variance follows a κ−1 scaling, where κ is the wavenumber,which is consistent with dimensionality arguments. In the presence of surface tension, this corrugationspectrum follows a κ−1 scaling for large scales, but then deviates at a length scale wich corresponds to thecritical radius. A spectral analysis of liquid volume fraction variance transfer is conducted, shedding lighton the role played by surface tension in this process. Results will be used to deduce important ramificationsfor sub-grid scale models in large-eddy simulations of liquid-gas flows.

∗Corresponding Author: jom48@cornell.edu

Introduction

Interactions between turbulence and immiscibleinterfaces are ubiquitous in both natural environ-ments and engineering applications, such as the dy-namics of the upper ocean and primary liquid atom-ization in combustion devices. Liquid fueled com-bustion is accomplished through a variety of ways,such as pressure injection in Diesel engines or coaxialair-blast injectors in aircraft engines. A detailed de-scription of primary atomization has remained elu-sive, due in part to insufficient understanding of howtopologically complex interfaces modulate their sur-rounding flow fields. Primary air-blast atomizationhas been understood in recent years as a mechanis-tic progression of linear instabilities that destabilizethe liquid core [1, 2, 3]. However, while useful incertain canonical flows, in particular in mostly par-allel and laminar flows, this sort of approach is lessuseful for describing atomizing liquids in complexgeometries or in the presence of turbulence. Theassumptions of linear stability analysis break downin presence of a significant non-zero normal velocityat the interface. Full characterization and subse-quent modeling of liquid-gas turbulent flows in suchcomplex scenarios requires a more general theory ofturbulent atomization. Though numerical studies inthis direction are limited, some progress has beenmade. Turbulence-interface interactions have beenstudied in two-dimensional simulations via an in-terfacial particle-level set method [4], highlightingthe interplay between surface tension and baroclin-ity near the interface, either of which can enhanceturbulent kinetic energy (TKE) dissipation in thisregion. Trontin et al. [5] isolate the interaction be-tween fluid inertia and surface tension by simulatinga slab of fluid in a box of three-dimensional decayinghomogeneous isotropic turbulence (HIT) with unitydensity and viscosity ratios, examining anisotropiceffects of surface tension on surrounding turbulence.A general theory of turbulent atomization must en-compass these types of effects, as well as other phys-ical processes by which a phase interface modulatesa surrounding turbulent flow field.

The goal of the present study is to furtherthe development of turbulent atomization theoryby studying numerical simulations of canonicalliquid-gas turbulent configurations, isolating differ-ent physical mechanisms present in the atomizationprocess. As a starting point, we isolate the inter-play between a turbulent fluid and surface tensionby inserting an interface into a box of homogeneousisotropic turbulence. Short-time deformation of aninitially flat interface in a field of decaying homo-geneous isotropic turbulence (HIT) is studied in ef-

fort to develop a statistical description of how sur-face tension impacts the resulting corrugations thatform on the interface. Results show that interfa-cial corrugations follow classical turbulent scalingsup to a surface tension-defined length scale, afterwhich interface statistics are controlled by surfacetension. The surface tension length scale extractedfrom the simulations agrees very well with the criti-cal radius [6, 7].

Direct numerical simulations

The equation for the velocity fluctuation u′ sim-ulated herein is written as

∂u′

∂t+ u′ · ∇u′ =− ∇p

ρ+ ν∇2u′

+Au′ +σ

ργ δ(x− xΓ)n ,

(1)

where p is the pressure, ρ is the density, ν is thekinematic viscosity, γ is the curvature of the inter-face, δ is the Dirac delta function, xΓ is the pointon the interface Γ closest to x, n is the unit vec-tor normal to Γ, and the forcing term Au′ acts as asource term of turbulent kinetic energy. The inclu-sion of the last term in Eq. (1) accounts for the singu-lar surface tension force at the interface. The forc-ing coefficient A = 1/3τ was introduced by Lund-gren [8] and dictates the large-eddy turnover timeof the flow field, τ . Rosales and Meneveau [9] haveshown that this linear forcing methodology yieldsturbulence statistics that are nearly indistinguish-able from band-limited forcing in spectral space.

Generation of homogeneous isotropic turbulence

To generate a field of statistically stationary ho-mogeneous isotropic turbulence (HIT), Eq. (1) isfirst evolved with σ = 0 within the NGA code [10]in a triply periodic box of size (2π)3. The re-sulting Taylor-microscale Reynolds number Reλ =urmsλg/ν is based on the root-mean-square velocityurms and the Taylor microscale λg =

√10(η2L)1/3,

where η is the Kolmogorov length scale and L is thecharacteristic length scale of the large eddies. Thislength scale L = k3/2/ε is defined in terms of theTKE k, written as k = 3u2

rms/2, and ε, which is thedissipation rate of TKE. All values of Reλ consid-ered in the present study exhibit a classical −5/3inertial subrange.

Turbulent interfacial flow

Once the HIT becomes statistically stationary,the forcing coefficient A in Eq. (1) is set to zero andthe HIT decays with an interface distribution thatis initially flat. Because ρ and ν are uniform every-where, the only discontinuity is in the pressure field,

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and we write the jump in pressure [p]Γ as [p]Γ = σγ.The ghost fluid method [11] is used to handle thepressure jump across the interface, and details re-garding the implementation of Eq. (1) (for the moregeneral case of variable density) are provided by Des-jardins et al. [10].

The recent volume-of-fluid (VOF) method ofOwkes and Desjardins [12] is coupled to NGA [10]in order to capture the interface. The semi-Lagrangian, flux-based VOF scheme provides dis-crete consistency between mass and momentumtransport through geometric calculation of flux vol-umes. The resultant scheme provides second or-der accuracy in space and time, exact conservationof mass and momentum everywhere in the domain,conservation of kinetic energy away from the inter-face, and is combined with a second order curvaturecalculation.

Uniform density and viscosity ratios utilizedherein isolate the interplay between the inertial con-tent of the turbulent fluid and the surface tension ofthe interface. We have run with different values ofσ, leading to a range of turbulent Weber numbers,defined as Weλ = ρu2

rmsλg/σ. Table 1 summarizesthe simulations.

Table 1. Parameters that characterize theHIT/interface simulations.

Case Mesh Reλ Weλ0 10243 313 ∞1 10243 313 8.472 10243 313 1.363 10243 313 0.224 5123 194 8.475 5123 194 1.366 5123 194 0.22

The impact of surface tension on the interface

The four cases shown in Fig. 1 are Case 0 –3. Case 0 corresponds to Weλ = ∞ (σ = 0), forwhich the interface moves materially with the fluid.It is observed from Fig. 1 that the large-scale varia-tions of the interface are similar for each case, whilesmall scale interfacial corrugations become increas-ingly suppressed as Weλ decreases. The same obser-vation can be made for cases with Reλ = 194.

The spectral footprint of surface tension

The interfacial feature suppression by surfacetension that is evident in Fig. 1 is quantified by defin-ing α(x) as the volume fraction of fluid 1, i.e., thefluid beneath the initially flat interface. The vol-ume fraction fluctuation α′ is computed as α′(x) =

α(x) − 〈α(x)〉x,z, where 〈∗〉x,z denotes an averagewith respect to coordinates x and z. Denoting y asthe direction normal to the initially flat interface,we compute the auto-covariance of α′(x, y = 0, z)(i.e., at the location of the initially flat interface)as Rα(rx, rz) = 〈α′(x, 0, z)α′(x + rx, 0, z + rz)〉x,z.Taking the Fourier transform of Rα gives the liquidvolume fraction variance spectrum Rα, written as

Rα(κx, κz) =

1

2π

∫ ∫Rα(rx, rz)e

−i(κxrx+κzrz) drxdrz ,(2)

where κx and κz are the wavenumbers in x and z,respectively. Figure 2(a) shows plots of the com-

pensated spectrum κRα(κ) for cases 0 – 3 plottedas a function of κη, where κ = (κ2

x + κ2y)1/2 and η

is the Kolmogorov scale of the initial turbulent flowfield. Note that the result looks the same for cases4 – 6 (at a lower Reλ), so only the high Reλ resultsare shown for clarity. For case 0 in which the in-terface moves materially with the fluid (no surface

tension), κRα is quite flat for roughly two decadesbefore transport due to low mesh resolution affectsthe spectrum. This is due to the fact that even inthe absence of physical surface tension, our VOFtransport scheme at low mesh resolution introducesa numerical surface tension. This flat region meansthat Rα ∼ κ−1, which is consistent with the findingsof Lesieur and Rogallo [13], who reported a range ofκ−1 scaling for the variance of passive scalars in tur-bulence, followed by κ−5/3 scaling. The extent towhich the passive temperature field of Lesieur andRogallo [13] experiences a κ−1 depends on the rel-ative decay of temperature variance versus kineticenergy. In the case of α with σ = 0, however, thereis no analog to diffusion, and Rα ∼ κ−1 for all re-solved scales.

The spectra for cases 1 and 2 also show a κ−1

region, but the presence of surface tension is evidentthrough the distinct change in scaling once a limit-ing wavenumber κσ is reached. Once κ ≈ κσ, Rα as-sumes a κ−2 scaling, as evidenced by the clear κ−1 ofthe compensated spectra for κ > κσ. In case 3, sur-face tension is so large that surface corrugations aresuppressed on even the largest scales, and Rα ∼ κ−2

for all scales. Note that we also observe the sametwo-slope type of behavior in wave spectra (definedsimilarly to the wave spectra of Phillips [14]) andsurface density variance spectra.

We observe that the critical wavenumber κσ ispredicted very well by κσ = 2π/lσ, where

lσ ∼(σ3

ρ3ε2

)1/5

(3)

3

(a) Case 0 (b) Case 1

(c) Case 2 (d) Case 3

Figure 1. Topology of the interface after half a large-eddy turnover time. Color indicates velocity magnitudeof the surrounding turbulent flow, ranging from 0 (red) to maximum value (white).

is the critical length scale below which interfacialcorrugations are suppressed by surface tension [6, 7].The critical length scale lσ is extracted for all sim-ulations in Table 1 and compared to the predictedvalue from Eq. (3), and good agreement is observedin Fig. 2(b).

Interface curvature statistics

Surface tension suppresses interface corruga-tions and therefore acts to prevent regions of highcurvature. The probability density function (PDF)of maximum curvature γmax is shown in Fig. 3(a) for

cases 0 – 6, defined as γmax = max(|γ1|, |γ2|), whereγ1 and γ2 are the two principal curvatures. PDFswere also computed for the curvedness, defined as√

(γ21 + γ2

2)/2 [15], but the difference was negligible.The peak for each PDF in Fig. 3(a) shifts to the

left as Weλ decreases, due to the prevention of highcurvatures by surface tension. The curvature dis-tribution decreases beyond the peak value rapidlyfor the lowest Weber number, which is in qualita-tive agreement with the relatively flat interface inFig. 1(d). Curvatures approaching 1/∆x (where ∆xis the mesh spacing) are orders of magnitude more

4

10−2

10−1

100

10−2

10−1

100

101

102

κR

α(κ

)

κη

(a) κRα for the Reλ = 313 cases. Case 0 (thick solid); case1 (dashed); case 2 (dash-dotted); case 3 (dotted). Thin hor-izontal lines highlight regions ∼ κ0; the thin diagonal lineshows κ−1 for comparison.

0 0.5 1 1.50

0.5

1

1.5

2

2.5

3

3.5

lσ

lDNS

σ

(b) Predicted lσ compared to measured lDNSσ : case 1 (circle);

case 2 (diamond); case 3 (square); case 4 (triangle); case 5(pentagram); case 6 (hexagram); best fit (line).

Figure 2. Liquid volume fraction variance spectrafor Cases 0 – 3 and comparison between predictedand measured lσ.

probable for case 0 than for all cases with surfacetension, as no physical mechanism prevents interfa-cial features on the scale of the mesh size to develop.The PDFs look similar for each pair of cases with thesame Weλ, indicating that the principal curvaturedistribution is not a strong function of Reλ.

Interestingly, the PDFs for all cases show rea-sonable collapse on one another when normalizingγmax by the critical length scale, as seen in the PDFsof lσγmax in Fig. 3(b). Cases 3 and 6 with the lowestWeλ peak near lσγmax ≈ 3, while cases 1, 2, 4, and5 peak near lσγmax ≈ 2. The collapse of the peakvalues of γmax near lσ supports the idea that turbu-

10−2

100

102

10−8

10−6

10−4

10−2

100

PDF

γmax

(a) Unnormalized curvature γmax

10−2

100

102

10−8

10−6

10−4

10−2

100

PDF

lσγmax

(b) Normalized curvature lσγmax

Figure 3. PDFs of maximum principal curvature:case 0 (solid); case 1 (thick dashed); case 2 (thickdash-dotted); case 3 (thick dotted); case 4 (thindashed); case 5 (thin dash-dotted); case 6 (thin dot-ted).

lence corrugates the interface by wrinkling and fold-ing it to smaller and smaller scales, creating largerand larger curvatures, until the critical length scaleis reached, and surface tension acts against furtherwrinkling and folding.

Liquid volume fraction variance transfer

We study the effect that surface tension has onthe transfer of liquid volume fraction variance bydecomposing the α field as α = α + α′, where αis the result of filtering α on a scale ∆ and α′ isthe sub-filter part of α. This allows us to study thefiltered quantity part as a function of the filter size∆. The filtered quantity α is obtained by taking the

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convolution of α with the kernel g:

α(x) =1

V

∫Vα(x′) g (|x− x′|) dx′ . (4)

We write the α transport equation as

∂α

∂t+∇ · (uα) = 0 , (5)

where u·∇α = ∇·(uα) has been used due to ∇·u =0 (i.e., solenoidal velocity field). Applying the filterin Eq. (4) to this equation and different forms of itleads to

∂α2

∂t+∇ ·

(uα2

)= −∇ · (2αRα)− Tα (6)

∂α′2

∂t+∇ ·

(uα′2

)= ∇ · (2αRα −Rα2) + Tα (7)

Rα = uα− uα

Rα2 = uα2 − uα2

Tα = −2Rα · ∇α ,

where Eq. (6) governs the resolved variance α2 and

Eq. (7) governs the sub-filter variance α′2 = α2−α2.Inspecting Eqs. (6) and (7), we see that first

term on the RHS of each equation is a conserva-tive diffusion term. The last term Tα, which ap-pears with opposite sign in each equation, is non-conservative and represents the transfer rate of αvariance from the resolved to sub-filter scales. Fig-ure 4(a) shows the globally averaged 〈Tα〉 for eachReλ = 313 case. For case 0 with σ = 0, 〈Tα〉 isconstant and goes to zero for ∆ ≈ η, indicating thatthe transfer decay is controlled by lack of mesh res-olution at the Kolmogorov scale. For cases 1 – 3,however, it is clear that η does not control the onsetof 〈Tα〉 decay. Figure 4(b) shows that 〈Tα〉 beginsto decay near a value of ∆/lσ ≈ 1, indicating thattransfer of α variance is suppressed near the crit-ical length scale. Note that this trend cannot beobserved for case 3, because lσ ∼ L and there isno well-defined large length scale that is sufficientlylarger than lσ.

Backscatter of α variance by surface tension

The role played by surface tension in the α vari-ance and kinetic energy transfer process can be fur-ther understood by examining the transfer termsthat arise from a model velocity field induced purelyby surface tension. Noting that the instantaneousacceleration of fluid due to surface tension can bewritten as aσ = σγ∇α, we can write such a modelvelocity as

umod =

∫ t0+∆t

t0

aσ dt′ ≈ σγ∇α∆t , (8)

0 50 100 150 2000

0.05

0.1

0.15

0.2

〈Tα〉

∆/η

(a) Filter size ∆ normalized by η

0 1 2 3 4 5 60

0.05

0.1

0.15

0.2

〈Tα〉

∆/lσ

(b) Filter size ∆ normalized by lσ

Figure 4. 〈Tα〉 for the Reλ = 313 cases. Case 0(solid line); case 1 (dashed line); case 2 (dash-dottedline); case 3 (dotted line).

where umod(t0) = 0 and ∆t is small such that thechange in γ and ∇α over ∆t is negligible. Themean α variance transfer term 〈Tmod

α 〉 is computedfrom umod for cases with Reλ = 313 and shown inFig. 5 (normalized by ∆t, which is taken to be thesimulation time step). Negative values indicate abackscatter of α variance, i.e., a transfer from sub-filter to resolved scales. Plotting the transfer as afunction of ∆/lσ shows that the backscatter is max-imum near the critical length scale, as shown in fig-ure 6(a). The α variance term has units of (time)−1,and a collapse of the data is observed when 〈Tmod

α 〉is made dimensionless by the surface tension timescale τσ =

√l3σ/σ, as shown in figure 6(b).

Conclusion and future direction

We have constructed a canonical multiphase tur-bulence test case that allows us to isolate the in-

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0 50 100 150 200−12

−10

−8

−6

−4

−2

0⟨

Tmod

α

⟩

∆/η

Figure 5. α variance transfer term computed fromumod for Reλ = 313 cases: case 0 (solid line); case1 (dashed line); case 2 (dash-dotted line); case 3(dotted line).

terplay between surface tension and fluid inertia.This approach allows us to statistically examine theeffects of surface tension on the corrugations thatform on the interface. Through spectra of liquidvolume fraction fluctuations and integrated interfaceheights, it is observed that interface corrugations aregreatly suppressed on length scales smaller than thecritical radius [6, 7].

Probability density functions of principal curva-ture are shown over a parameter space of Reynoldsnumber and Weber number. All curves exhibit apeak value at the critical radius, demonstrating therobustness of this length scale as a delineation be-tween interface corrugations generated by turbu-lence and those suppressed by surface tension. Therole of surface tension in the cascade of liquid vol-ume fraction variance is explored through a filteringapproach, highlighting the importance of the criticallength scale as a source of backscatter. Work cur-rently in development includes conducting a similarfiltering approach for the fluid momentum, revealingan analogous source of backscatter for kinetic energyfrom sub-filter to resolved scales. Future work willinclude a priori testing of sub-grid scale models forthe advancement of enhanced large-eddy simulationsof liquid-gas turbulent flows.

References

[1] L Raynal, E Villermaux, JC Lasheras, andEJ Hopfinger. 11th Symposium on TurbulentShear Flows, 3:27.1–27.5, 1997.

[2] P Marmottant and Emmanuel Villermaux.Journal of Fluid Mechanics, 498:73–111, 2004.

0 1 2 3 4 5 6−12

−10

−8

−6

−4

−2

0

⟨

Tmod

α

⟩

∆/lσ

(a) Filter size ∆ normalized by lσ

0 1 2 3 4 5 6−2

−1.5

−1

−0.5

0

⟨

Tmod

α

⟩

τσ

∆/lσ

(b) Collapse of curves when normalized by time scale τσ =√l3σ/σ

Figure 6. α variance transfer term computed fromumod for Reλ = 313 cases: case 1 (dashed line); case2 (dash-dotted line); case 3 (dotted line).

[3] Fares Ben Rayana, Alain Cartellier, and EmilHopfinger. Proceedings of the international con-ference on Liquid Atomization and Spray Sys-tems (ICLASS), pp. 1–8, Kyoto, Japan, 2006.

[4] Zhaorui Li and Farhad a. Jaberi. Physics ofFluids, 21(9):095102, 2009.

[5] P. Trontin, S. Vincent, J.L. Estivalezes, andJ.P. Caltagirone. International Journal ofMultiphase Flow, 36(11-12):891–907, November2010.

[6] AN Kolmogorov. Doklady Akademii NaukSSSR, 66:825–828, 1949.

[7] J Hinze. AIChE Journal, 1(3):289–295, 1955.

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[8] T. S. Lundgren. Center for Turbulence ResearchAnnual Research Briefs, pp. 461–473, 2003.

[9] Carlos Rosales and Charles Meneveau. Physicsof Fluids, 17(9):095106, 2005.

[10] Olivier Desjardins, Guillaume Blanquart, Guil-laume Balarac, and Heinz Pitsch. Journalof Computational Physics, 227(15):7125–7159,July 2008.

[11] Ronald P. Fedkiw, Tariq D. Aslam, Barry Mer-riman, and Stanley Osher. Journal of Compu-tational Physics, 152(2):457–492, July 1999.

[12] Mark Owkes and Olivier Desjardins. Journal ofComputational Physics, 270:587–612, 2014.

[13] Marcel Lesieur and Robert Rogallo. Physics ofFluids A: Fluid Dynamics, 1(4):718, 1989.

[14] O. M. Phillips. The dynamics of the upperocean. Cambridge University Press, 1977.

[15] Jan J Koenderink and Andrea J van Doorn. Im-age and Vision Computing, 10(8):557–564, Oc-tober 1992.

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