American Institute of Aeronautics and Astronautics1

Experimental Study of Subcritical to Supercritical JetMixing

S. A. Polikhov1 and C. Segal2

University of Florida, Gainesville, Florida, 32611, USA

A study of liquid injected into a gaseous surrounding under subcritical, transcritical andsupercritical conditions is presented. Significant laminarization of the jet under supercriticalconditions was observed. A linear stability analysis is performed to develop a distortionrelation for the viscous jet in the inviscid gaseous surrounding. The results indicate that thelinear stability analysis describes well the subcritical mixing but fails at trans andsupercritical conditions.

Nomenclature

σρ lll

l

ruWe

2

= Weber number

l

lJ

ru

ν=Re = jet Reynolds number

a = spatial disturbance magnitude [m]

ld = liquid jet diameter [m]

210 ,, III = modified Bessel function of first kind of zero, first and second order

10, KK = modified Bessel functions of second kind of zero and first order

ir jkkk += disturbance wave number [m-1]

lkdk 5.0= dimensionless disturbance wave number

ic k

Lπ= dimensionless (divided by jet diameter) characteristical length of disturbance growth rate

M = Mach numberp = pressure [atm]

cp = critical pressure [atm]

cr p

pp = reduced pressure

lr = liquid jet radius [m]

T = temperature [K]

cT = critical temperature [K]

bT = boiling temperature [K]

cr T

TT = reduced temperature

1 Graduate Research Assistant, MAE University of Florida, Gainesville FL 32611, Student Member AIAA.2 Associate Professor, MAE University of Florida, Gainesville FL 32611, Associate Fellow AIAA.

45th AIAA Aerospace Sciences Meeting and Exhibit8 - 11 January 2007, Reno, Nevada

AIAA 2007-569

Copyright © 2007 by C. Segal. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

American Institute of Aeronautics and Astronautics2

w = axial component of velocity [m/s]

lu = liquid injection velocity [m/s]

u = radial component of velocity [m/s]

δ = shear layer thickness [m]

lρ = liquid density [kg/m3]

gρ = gas density [kg/m3]

ν = cinematic viscosity [m2/s]σ = surface tension [N/m]ω = disturbance radial frequency [s-1]

ωωl

l

u

d5.0= dimensionless disturbance frequency

rc k

πλ = dimensionless (divided by jet diameter) disturbance wavelength

λ disturbance wavelength [m]

Subscriptsi = fluid index: 1=i - liquid, 2=i - gas

I. IntroductionThe problem of supercritical mixing attracts significant attention since numerous applications where supercritical

conditions are present abound. These include diesel engines, rocket engines main combustion chambers, etc. Theinverse problem of a supercritical jet injected in subcritical conditions also is present, for example, in a supersoniccombustion engine. In particular, the advancement of liquid propellant rocket technologies led to a significant se ofpressure increase in the combustion chamber and, in many applications, the pressure and temperature largely exceedthe critical values of one of the propellants.

In experiments on cryogenic oxygen jet mixing and combustion Mayer et al.1, 2 have shown that substantialdifferences between subcritical and supercritical mixing exist. At subcritical conditions typical jet atomization wasobserved. At these conditions, surface tension causes jet instability and promotes growth and development of

disturbances leading to formation of two-phase spray. At pressures exceeding 2Orp ≈ 1.2, a significant change of

mixing mechanisms was observed. The absence of surface tension caused the domination of the diffusion processesover the jet atomization. Later results using supercritical mixing visualization, as reported by several researchteams3,4,5,6,7 are similar in conclusions to those reported by Mayer et. al.1, 2

The processes taking place in the shear layer during supercritical mixing are not well understood. Yet, thequalitative picture of the thermodynamic transformations that an injected fluid undergoes during the process ofmixing can be illustrated with the simplified phase diagram shown in Error! Reference source not found.. Beforeentering the injecting nozzle, the fluid is at supercritical pressure but sub-critical temperature (point 1 in the phasediagram). The pressure drop across an injector nozzle is quite small therefore after injection the fluid remains atsupercritical pressure; the fluid, then, is warming up and evolves from its initial sub-critical thermodynamic state toa supercritical one (point 2 on the phase diagram). Most authors suggest that after transition to the supercritical state,the fluid further mixes with the surrounding gas; therefore its partial pressure drops below the critical value (point 3on the phase diagram). There is no clear opinion in the literature on the processes which take place during thesethermodynamic transformations although some points of overall agreement can be found. First of all, the enthalpy ofevaporation vanishes at these conditions; therefore the liquid-gas mixing process is likely to be controlled bydiffusion rather then evaporation8, 9. Additionally, as Woodward and Talley7 pointed out, during mixing between theliquid and the surrounding gas, the critical pressure can be several times higher than its value for a pure fluid. Thusfor binary systems, the critical pressure is not a fixed value but a dynamic parameter depending on local conditions.

Important arguments are presented in the literature regarding the shear layer development under supercriticalconditions and, as a result, on the jet break-up length. Chehroudi et al.10 suggest that due to the disappearance of thesurface tension and the evaporation enthalpy, mixing between injected fluid and surrounding gas exhibits gas-gasmixing-like behavior once the critical values are achieved. These authors support these conclusions with

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experimental results from Mayer et al.1,2 and their own experimental studies of liquid nitrogen injection into asupercritical nitrogen environment. Bellan et al.11,12,13,14 and Yang et al.15,16,17 presented computational analyseswhich suggested that as long as the initial gas/fluid density ratio is below 0.1, a supercritical jet exhibits significantdifferences from a submerged turbulent jet behavior. In particular, such a large density difference between the fluidand the surrounding gas leads to turbulence damping and inhibition of the mixing rate. Due to these factors, the jethas a much longer unmixed core length compared to the widely recognized and accepted turbulent gaseous jettheory summarized, for example, by Abramovich in his book on theory of the turbulent jets18. However,Abramovich also presented the theory of a heavy jet, injected into a lighter surrounding in the absence of surfacetension, e.g. a dust or droplet-laden jet injected into a gaseous environment which led to different mixing theory thanthe accepted gas-gas mixing. So far, this theory it has not been applied to the supercritical jet.

To bridge the difference between numerical simulations and qualitative assumptions that can be made from theavailable experimental results, Bellan et al.13 pointed out that almost all of available experimental information onsupercritical jet mixing was obtained using the shadowgraph technique; an experimental method that has severalinherent restricting features. First, it is integrative – the light has to pass through the entire jet, therefore the pictureis an average throughout the jet. Secondly, the shadowgraph is measuring not density but its gradient i.e. low densitybut highly turbulent regions can easily saturate the image. As a result, a relatively low-dense cloud of already mixedfluid which, indeed, exhibits gas-gas mixing features can hide the high density core in the center of the jet. Theseassumptions are confirmed by comparisons of X-ray and shadowgraph images of sub-critical, transcritical andsupercritical jet mixing conducted by Birk et al.19,20

The goal of the present work is to expand the existing database of reliable experimental measurements of densitydistribution during supercritical liquid/gas mixing as well as gain some insight into fundamental features ofsupercritical mixing process. Planar laser induced fluorescence (PLIF) was used to generate a section through themixing jet.

A description of the experimental setup - shown in Figure 2 - and the technique was given previously21; thereforeonly a brief description is included here. All experiments were done using a round liquid injector with diameter, dl =0.84 mm. The liquid was not preheated and the jet velocity was varied between 7 to 25 m/s. For these injectionvelocities, the Reynolds number varied from 11000 to 42000. Since the flow was laminar before entering theinjector, turbulence did not develop while the fluid was passing through the relatively short, 15.4 mm, injector tip.Nitrogen was used as the surrounding gas. FK-5-1-12 [CF3CF2C(O)CF(CF3)2] has been chosen as the injected fluid.The choice of this fluid was determined by safety reasons, good spectroscopic properties and has relatively lowcritical point - atmPcr 4.18= , KTcr 441= . The third harmonic, 355 nm, of Nd:Yag laser was used to excite the

fluorescence. Preliminary tests have shown that emission spectrum of FK-5-1-12 within 400 – 500 nm does notreveal significant dependence on pressure and temperature within a range of interest. Based on emission spectra anoptical filter with 420 nm centerline and 10nm FWHM width was incorporated into the CCD camera lens toeliminate any elastic scattering.

II. Experimental ResultsThe experimental test matrix is shown in Figure 1. Three different types of jet break-up were observed and are

marked in figure as the sub-critical, transitional and supercritical regimes.The sub-critical break-up regime was observed for relatively low temperatures and pressures. Surface tension

and inertia forces dominate under these conditions. The well documented first wind-induced, second wind-inducedand atomization break-up mechanisms are observed under these experimental conditions, depending on the injectionvelocity and surrounding gas density. Since the observation was conducted in the direct vicinity to the injector faceand the jet is initially laminar, droplet formation was rarely observed. However, the occasionally observed dropletshad an ellipsoid or round shape, i.e. the surface tension forces were relatively strong. The images of sub-critical jetbreak-up can be found in Figure 2 – 9. Both second wind-induced and atomization break-up mechanisms werecharacterized by well developed liquid surface. Pronounced ligament formation was observed under theseconditions. The density gradient tends to be the highest at these experimental conditions.

The images of transcritical jet break-up can be found in Figure 9 - 11. A significantly different jet break-up wasobserved in the trans-critical mixing region. The characteristic feature of this region is the apparent decreasedimportance of surface tension. This manifests through the smoothening of the liquid – gas interface. Ligamentsformation tends to decrease significantly. The ligaments shape is similar to descriptions available inliterature.1,2,3,4,11,13 Due to the decreased surface tension forces, ligaments have a “cluster” or “finger”-likeappearance. Once detached from the main jet body the liquid exhibits a dual behavior: in some experiments theformation of round liquid drops was observed (see Figure 0 for an example), while other experiments revealed a

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cluster-like droplet formation which has been reported in literature as a characteristic feature of supercriticalmixing1, 4, 6 (see, for an example Figure 7).

Finally with increased pressure and temperature, Figures 12 - 14, the jet behavior changes again. Occasionaldroplet formation was observed under these conditions only at relatively low temperatures. Density gradient valuesdecreased drastically and approached the values characteristic for a laminar jet at STP conditions. See for example acomparison of density gradient fields in Figure 2 and Figure 12. To authors best knowledge this type of a liquid/gasinterface behavior was not observed in experimental results currently available in the literature. It was howeverreported in numerical simulations conducted by Bellan et al.11, 12, 13, 14

Probably, this type of mixing could be explained through a similarity to a phenomenon well recognized inacoustics where a large discrepancy between two adjoined media’s acoustic impedance can decrease the energytransmitted by several orders of magnitude. Hence, in cases where effective transmission is desired, a third media,i.e. a matching layer is introduced. Arguably, in the case of sub-critical jet mixing, the situation can be explained asfollows: surface tension works as storage of potential energy, i.e. the surface tension works as a spring in the spring– mass system, or as a capacitor in the LC circuit. The presence of a characteristic disturbance wavelength is in agood agreement with this point of view. The characteristic disturbance waves correspond to the resonant frequenciesof the system which consists of liquid and gas inertia and surface tension forces. But once the surface tensionbecomes insufficient, some other means of energy transmission between liquid and gaseous phases have to manifestthemselves. The most obvious energy exchange mechanism is the Kelvin-Helmholtz instability. One might expectthat the jet mixing would eventually approach a liquid/liquid or a gas/gas like mixing at very high temperatures andpressures. But within the experimental conditions covered in this study the gas/liquid density ratio can be estimatedas 0.008 – 0.012 which means that momentum transfer across the liquid – gas border could be significantlyinhibited.

III. Analysis

To gain some insight into this jet behavior a linear stability analysis was performed. Since 1<<M the flowwas considered incompressible. Only axially symmetric disturbances are considered. The surrounding gas isassumed to be inviscid while the injected fluid has a viscosity. Jet velocity profile was assumed to be uniform which,as shown by Ibrahim et al.,22 leads to jet instability overprediction although still correct as an overall estimation.Linearized, equations of mass, r momentum and z momentum conservation look as follows:

( ) 01 =

∂∂+

∂∂

z

uru

rri

i (1)

−∇+

∂∂

−=∂∂+

∂∂

22

11

1r

uu

r

p

z

uu

t

u iii

iili

i νδρ

δ (2)

iii

lii w

z

p

z

wu

t

w 211

1∇+

∂∂

−=∂∂+

∂∂ νδ

ρδ (3)

The pressure and velocity disturbances are sought in the form:

( ) ( )( )tkzjrPp ii ω−= exp (4)

( ) ( )( )tkzjrUu ii ω−= exp (5)

( ) ( )( )tkzjrWw ii ω−= exp (6)

where k and ω represent the wave number and radial frequency respectively. Following an analysis similar toLin23 the dispersion equation can be formulated as:

( ) ( ) ( )( ) ( )

( ) ( )( ) ( ) ( )

( ) ( )( )

( )0

121

Re4

21)(

Re2)( 2

1

202

12

3

1

20

1

02

1

02

1

22

1

0 =−+

+−−−

++−+−We

kk

I

II

kI

Ik

kk

kI

kIkIk

kI

kIkj

kK

kKk

kI

kI

λλλλωω

ρρω (7)

where ( )Re22 kjk −−= ωλ . nn KI , represent here the modified Bessel functions of first and second kind

respectively. To simplify the algebraic manipulations wave number as well as radial frequency were transformed to

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nondimensional form ωωl

l

u

d5.0= and lkdk 5.0= . Only spatial disturbances were considered, since at large

Weber numbers spatial and temporal disturbances approaches lead the results with ( )2−WeO discrepancy22.

The problem was solved numerically. The root search is greatly simplified if one realizes that23

( )2−+−= WeOkr ω . The solution for jet injection at STP conditions shown in Figure 2 is plotted in Figure 13.

One can see that according to Figure 13 the fastest growing disturbances are expected to be

034.06.2 jk −= which corresponds to 26.1=cλ and 92=cL i.e. the most pronounced disturbance should

have the wavelength comparable to the jet diameter and it would to manifest itself within 90 jet diameters. It wasfound that the predicted instability parameters are in a good agreement with those observed experimentally. Thefollowing equation was used to estimate the surface tension:

972.0

10329.0

−=

cT

Tσ (8)

and, in cases where the temperature of the surrounding gas was higher when the boiling temperature of the jet atgiven surrounding gas pressure, the surface tension was estimated as

972.0

10329.0

−=

c

b

T

Tσ (9)

The comparison between the fastest growing disturbance estimations made via linear stability analysis and thatmeasured in the experiments on is given in Table 1 and Figure 16. Since the linear stability analysis failed to predictdisturbances for transitional and supercritical mixing those experiments are not included in the table. In particular,this analysis is considering the jet as a system which is withheld together via surface tension forces, therefore when

∞→We the system became unavoidably less stable.An estimation of the disturbance growth rate and wavelength under these circumstances can be evaluated via

considering the shear layer itself. According to the instability theories summarized by Chandrasekhar24, thedisturbance magnitude in this case can be expressed as:

( )

+≈ z

kaa 2

12

210 2exp

ρρρρ

π(10)

Here,λπ2=k . In the case of inviscid shear layer24 the shorter the disturbance wavelength the faster in grows.

Although in case of viscous shear layer Schlichting25 noted that the shortest possible disturbance wavelength is

defined by the thickness of the shear layer i.e. δλ ~c and shorter wavelength would be vanished due to viscosity

forces. Assuming m410−≈δ , 10≈cL which means that disturbances are growing much slower that predicted via

jet stability analysis.

IV. ConclusionA study of liquid jet injected into gaseous environment was undertaken under subcritical, transitional and

supercritical conditions. The results indicated the following:- First wind-induced, second wind-induced and atomization break-up mechanisms are observed under

subcritical conditions with variations depending on the injection velocity and surrounding gas density.- Pronounced ligament formation was observed under these conditions when the density gradient tends to be

the highest.- At transcritical conditions a decreased importance of surface tension is apparent which manifests through

the smoothening of the liquid – gas interface. Ligaments’ formation tends to decrease significantly. Theligaments shape is similar to descriptions available in literature described as “fingers” or “clusters”.

- Once detached from the main jet, packets of liquid exhibit a dual behavior: in some cases the formation ofround liquid drops was observed as in the subcritical case, while in other cases a cluster-like dropletformation was noticed, a feature characteristic to supercritical mixing.

- With increased pressure and temperature the jet behavior changed again with occasional droplet formationobserved under these conditions only at relatively low temperatures.

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- Density gradient values decreased drastically at supercritical conditions and approached the valuescharacteristic for a laminar jet at STP conditions; this type of a liquid/gas interface behavior was describedin previous computational results but not seen in previous experiments.

- A linear analysis of the disturbance wavelength corresponding to the resonant frequencies of the systemshowed good correlation with experimental results for subcritical mixing but not for the transcritical andsupercritical regimes.

AcknowledgmentsThis work is supported by NASA through CUIP grant. Mrs. Claudia Meyer, the Program Manager support is

greatly appreciated. We also would like to acknowledge the continuous support and advice given by Kevin Tucker,James Hulka and Douglas Talley.

References

1 Mayer W. O. H., Schik A. H. A., Vielle B., Chauveau C., Gokalp I., Talley D. G., and Woodward R. G. Atomization and break-up of cryogenic propellants under high pressure sub-critical and supercritical conditions. Journal of Propulsion and Power, Vol.14, No. 5, pp. 835 – 842, 19982 Mayer W., Tamura H., Propellant injection in a liquid oxygen/gaseous hydrogen rocket engine. Journal of Propulsion andPower, Vol. 12, No. 6, pp. 1137 – 1147, 19963 Chehroudi B., Talley D. and Coy E. Visual characteristics and initial growth rates of round cryogenic jets at subcritical andsupercritical pressures. Physics of fluids, Vol. 14, No. 2, pp. 850 – 861, 20024 Chehroudi B., Talley D. and Coy E. initial growth rate and visual characteristics of a round jet into a sub- to supercriticalenvironment of relevance to rocket, gas turbine and diesel engines. AIAA 1999 – 0206, 19995 Candel, S., Herding, G., Snyder, R., Scouflaire, P., Rolon, C., and Vingert, L., Experimental investigation of shear coaxialcryogenic jet flames. Journal of Propulsion and Power, vol. 14, No5, 826–834, 19986 Habiballah, M., Orain, M., Grisch, F., Vingert, L., and Gicquel, P. Experimental studies of high pressure cryogenic flames onthe mascotte facility. Combust. Sci. Tech., 177: 2139–2166, 20057 Oschwald M., Smith J. J., Braman R., Hussong J., Schik A., Chehroudi B., and Talley D., Injection of fluids into supercriticalenvironments. Combust. Sci. and Tech., No 178, pp. 49 – 100, 20068 Yang V., Hsiao G. C., Shuen J-S. and Hsieh K-Ch., Droplet behavior at supercritical conditions. Progress in astronautics andaeronautics, Vol. 166, pp. 413 – 437, 19969 Yang V., Lafon P., Hsiao G. C., Habiballah M. and Zhuang F-C., Liqud propellant droplet vaporization and combustion.Progress in astronautics and aeronautics, Vol. 200, pp. 295 – 321, 200410 Oschwald M., Smith J. J., Braman R., Hussong J., Schik A., Chehroudi B., and Talley D., Injection of fluids into supercriticalenvironments. Combust. Sci. and Tech., No 178, pp. 49 – 100, 200611 Bellan J. Supercritical (and sub-critical) fluid behavior and modeling: drops, streams, shear and mixing layers and sprays.Prog. Energy Combust. Sci. 26 pp. 329 – 366, 200012 Harstad K., Bellan J. An all-pressure fluid drop model applied to a binary mixture: heptane in nitrogen. Int. J. MultiphaseFlow, No 26, pp. 1675 – 1706, 200013 Bellan J. Theory, modeling and analysis of turbulent supercritical mixing. Combust. Sci. and Tech., No 178, pp. 253 – 281,200614 Okong’o N., Harstad K. and Bellan J. Direct numerical simulations of

22 / HO temporal mixing layers under supercritical

conditions. AIAA 2002 – 0779, 200215 Zong N., Yang V., Cryogenic fluids jets and mixing layers in trans-critical and supercritical environments. Combust. Sci. andTech., No 178, pp. 193 – 227, 200616 Yang V. Modelling of supercritical vaporization, mixing and combustion processes in liquid fueled propulsion systems.Proceedings of the Combustion Institute, Vol. 28, pp. 925 – 942, 200017 Zong N., Meng H., Hsieh S-Y. and Yang V., A numerical study of cryogenic fluid injection and mixing under supercriticalconditions. Physics of fluids, Vol. 16, No. 12, pp. 4248 – 4261, 200418 Abramovich G. N. The theory of turbulent jets. The M.I.T. press, 196319 Birk A. and McQuaid M. Deliberations of the dynamics and core structure of reacting sprays at elevated pressures. Progress inastronautics and aeronautics, Vol. 166, pp. 309 – 326, 199620 Birk A., McQuaid M. and Gross M. Liquid core structure of evaporating sprays at high pressures – flash X-ray studies.ICLASS – 94, 199421 Polikhov S. A. and Segal C., Two phase flow supercritical mixing. AIAA 2006 - 0756, 44th Aerospace Science Meeting. Reno,Nevada, 2006

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22 Ibrahim E. A., Marshall S. O., Instability of a liquid jet of parabolic velocity profile, Chemical Engineering Journal, 76, pp. 17-27, 200023 Lin S. P., Breakup of liquid sheets and jets., Cambridge University Press, 200324 Chandrasekhar S., Hydrodynamic and hydromagnetic stability, Dover Publications, New York, 198125 Schlichting H., Boundary-Layer Theory, Springer, Berlin, New York, 2000

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Table 1

Chamberpressure, atm.

( rP )

Chambertemperature, K

( rT )

Injectionvelocity, m/s cL (predicted)

Predicted

wavelength, cλMeasured

wavelength cλ

1.4 (0.054) 293 (0.66) 10.7 92 1.2 1.36.5 (0.35) 293 (0.66) 13.9 4.5 0.15 0.29.4 (0.51) 293 (0.66) 14.6 2.85 0.13 0.1

15.6 (0.84) 293 (0.66) 11.7 2.32 0.12 0.1521.6 (1.17) 293 (0.66) 15.7 1.44 0.095 0.1125.5 (1.38) 293 (0.66) 16.8 1.2 0.089 0.130.2 (1.64) 293 (0.66) 7.4 1.96 0.14 0.126.9 (0.375) 361 (0.81) 11.8 5.23 0.18 0.169.8 (0.53) 364 (0.82) 22.6 2.41 0.098 0.11

19.6 (1.06) 363 (0.82) 15.3 1.77 0.1 0.0927.1 (1.47) 365 (0.82) 11.4 1.63 0.11 0.1230.7 (1.66) 357 (0.80) 14.0 1.29 0.096 0.1134.3 (1.86) 360 (0.81) 13.3 1.23 0.096 0.097.3 (0.39) 407 (0.92) 9.4 5.71 0.20 0.23

13.4 (0.72) 404 (0.91) 14.1 2.59 0.11 0.0917.7 (0.96) 409 (0.92) 15.3 1.98 0.10 0.1223.5 (1.27) 416 (0.94) 15.3 1.61 0.096 0.086.4 (0.34) 451 (1.02) 14.2 4.83 0.17 0.15

11.3 (0.61) 452 (1.02) 13 3.34 0.13 0.1215.7 (0.85) 448 (1.01) 12.2 2.55 0.12 0.114.1 (0.22) 472 (1.07) 24.7 5.7 0.12 0.14

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Figure 0. Qualitative phase diagram to illustrate supercritical mixing.

Temperature

Pre

ssu

re

Vapor

Solid

Liquid

Supercritical fluid

Triple point

Criticalpoint

1 2

3

Chamber upthermocouple

Liquid injectionport

Gas injectionport

Gas temperaturethermocouple

Chamber pressuretransducer

Pressure relief valve

Figure 0. High pressure chamber overall view.

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Figure 2. Jet injection at STP conditions. Density field (left) and density gradient field (right).

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 0.5 1 1.5 2

Pr

Tr

''293 K'' ''363 K'' ''403 K'' ''443 K'' ''473 K'' ''523 K'' (Pcr, Tcr), (18.4 atm, 441 K)

sub-critical mixing region

transitionalmixing

supercritical mixing region

Figure 1. Test matrix.

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Figure 4. Density field (left) and density gradient field (right).

Figure 3. Density field (left) and density gradient field (right).

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Figure 6. Density field (left) and density gradient field (right).

Figure 5. Density field (left) and density gradient field (right).

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Figure 8. Density field (left) and density gradient field (right).

Figure 7. Density field (left) and density gradient field (right).

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Figure 10. Density field (left) and density gradient field (right).

Figure 9. Density field (left) and density gradient field (right).

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Figure 12. Density field (left) and density gradient field (right).

Figure 11. Density field (left) and density gradient field (right).

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0

0.05

0.1

0.15

0.2

0.25

0 0.05 0.1 0.15 0.2 0.25

Figure 14. Correlation between measured (vertical axis) and predicted (horizontal axis)disturbance wavelength.

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5

Figure 13. Spatial growth at STP conditions. smul /7.10= , 11500Re = , 7000=We , 4

1

2 1025.6 −⋅=ρρ .

k−

rk

.c measλ

c estλ