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Results of numerical modelling and experimental activities in preparation of the maxus-5 experiment
Available online at www.sciencedirect.com
PERGAMON
Acta Astronautica 53 (2003) 299-308 www.elsevier.com/locate/actaasrro
RESULTS OF NUMERICAL MODELLING AND EXPERIMENTAL PREPARATION OF THE MAXUSEXPERIMENT
ACTIVITIES
IN
Edmondo Bassano, Dario Castagnolo, Raimondo Fortezza MARS Center s.c.a r.1. Via E. Gianturco 31, I-80146 Napoli, Italy. [email protected] Abstract The present paper reports on the research activities in b&h experimental and numerical field carried out at MARS center for the realization of an experiment on the thermo-solutal-capillary migration of a dissolving drop, composed by a liquid binary mixture having a miscibility gap. The experiment will be carried out onboard of the sounding rocket MAXUS 5, by using the INEXMAM facility. The main goals of the analysis are diagnostic equipment ensuring that the performanres cope with the stringent resolution requirements dictated by the physics and assessing of the experiment time profile. 8 2003 International Astronautical Federation. Published by Elsevier Science Ltd. All rights reserved. INTRODUCI7ON Two liquid binary mixtures constituted by the same components exhibit a miscibility gap when they coexist, in a given range of temperatures, in two different phases having different composition. A miscibility gap in the liquid phase is found in several different physical systems: metal alloys, binary mixtures of organic liquids, sulphides and silicates systems, glasses and liquid crystals, and many industrial applications and processes are based on miscibility gap and related phenomena. Microgravity experiments on miscibility gap were conducted for more than two decades and flew on almost every kind of microgravity platform’. Research activities in both experimental and numerical field are carried out at MARS center for the realization of an experiment on the thermosplutal-capillary migration of a dissolving drop, composed by a liquid binary mixture having a miscibility gap. The experiment will be hosted onboard the MAXUS 5 sounding rocket, by using the INEX-MAM facility’.2*3” Since capillary migration of drops is strongly affected, on Fzuth, by buoyancy due to both different densities of the mixture components and Boussinesq effects, the dissolution due to a miscibility gap and its influence on migration can be profitably studied in microgravity.
Theoretical models for calculating the equilibrium curve are available and are particularly helpful for ideal solutions, but differences in atomic radius and bonding5 are responsible of possibly large discrepancies between real and ideal behavior, thus the cyclohexane-methanol equilibrium phase diagram, shown in fig. I, is given by points. It is supposed that equilibrium always occurs at the binodal line. THE MAXUS 5 EXPERIMENT The idea is to inject a drop of a pure component in a matrix of the other component, that tills a closed squared based cavity and has a linearly stratified temperature. The upper and lower walls of the cavity are held at constant different temperatures. Since the system is in microgravity, only interfacial tension gradient effects occur. The drop should migrate towards the hot upper wall because of thermal Marangoni effect and dissolve because the system is in a region of the phase diagram where ir shall form a single phase. The liquids selected for the sounding rocket experiment and used in numerical simulations are cyclohexane (CsH1& - methanol (CH30H). This couple of liquids has been selected because: i) the components have nearly the same density; ii) are transparent, allowing either the direct visualization of the drop migration and deformation either the use of optical diagnostics devices; iii) their mixture has a critical temperature of 45.7 “C, slightly larger than the ambient one and compatible with the INJZXMAM facility thermal requirements4. The facility diagnostic technique is described in Bassanoet al.‘. NUMERICAL
MODELING
Some of the restrictive hypotheses usually adopted in the classical theoretical and numerical literature on thermo-capillary migration of bubbles and drops, removed in the present approach, were discussed in Bassano et a1.le3together with modem CFD techniques for multiphase flows modelling. With respect to the previous 2D-planar works, the present code upgrade is represented by the axis-
(@94-5765/03/$- seefront matter Q 2003 International Astronautical Federation.Published by Elscvier ScienceLtd. All rights reserved. doi: 10.1016/S0094-5765(03)00145-O
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symmetric formulation and integration of the relative motion equations. This aJloWS to correctly describe the physics of the migration process preserving the real topology, geometry and dynamics of the thermo-fluid dynamic field occuring during experiment on ground and in space. In the present paper the drop interface evolution is modeled by the level set technique, in which the interface is represented by the embedding of the zero level set of a scalar function defined within the whole computational domain. The level-set equation is written in this upgraded model in .a different form suitable for interface speed extension. An extension algorithm was implemented to better simulate the interface dynamics and localized to reduce run-times. The weight of the Dirac delta function appearing in the momentum production is extended as discussed in*. A major &dification made in the present analysis, greatly enhancing the solution accuracy, is a different formulation of the energy equation. In past works the internal energy was used as dependent energy variable, having a discontinuity at the interface due to the specific heat coefftcient jump. This choice was responsible for a poor accuracy of the calculation of the energy equation, which in turns gave temperature profiles with kinks not located on interface but slightly outside. Also a too small energy flux entering within the drop, a crowding of isotherms around it, and ultimately a tendency of the drop surface to become isotherm during migration was observed. These problems, which are much more relevant in the axissymmetric case, have been solved by using as independent variable the temperature, which is continuous at interface and has a discontinuity only in the normal derivative. THE LEVEL SET METHOD The level set method for advancing fronts was proposed by Osher and Sethiat?. The front, in our case the interface y(t), is represented at each instant of time t by the zero level set of a smooth scalar function $. Let x(f) be a representation of y(t) and Vi (t ) = -i(t). The evolution equation for $t is described by the scalar equation of HamiltonJacoby type t$, +vi .Vtp=@,
+v,$q=o
(1)
53 (2003) 299-308
for this equation states that, if at the initial time @ is constant along the curve Y moving in normal direction with speed Vii, = n .V;. , it will be constant there also at later times. In writing the second equality above, we have exploited the fact that the unit vector n normal to Y. being @ constant along it, can be written as n = V@//pd Note that the dynamics of ‘y is fully described by
V, only. We assume that t$ is positive outside the drop and negative inside so that n points outside. MODEL ASSUMF’TJONS The cavity containing the drop and the external liquid matrix is cylindrical. In the following V indicates the nabla operator, p the average normal stress, p the dynamic viscosity, V the kinematic viscosity, c, the specific heat coefficient, CX the thermal diffusivity, h the thermal conductivity,
‘Z?FT+
(T),c (T)).
At the critical point (dc/dT = Oin fig. I), the different phases merge in a single phase and Q,, = 0. Since the equilibrium concentrations depend on the temperaturc one can assume’ d = 6,, (T - T,,). According to experimental data 8,s < 0. Since cyclohexane and methanol have nearly the same densities the difference p, - p2 is neglected. Mass conservation implies the continuity of the mass flow rate ti through the interface ril=ti*
={n*p(v-v;)}
(2)
where { f}= f ’ - f - is the jump operator and the superscript “+” indicates the domain towards which n points. b
E. Bassano et al. /Acta Astronautica
53 (2003) 299-308
Table 1 Properties of cyclohclane and methanol
Fig 3 Temperalum isolioa at start-up. t = 1.47 I for R=O.ZScm. t=2.94 s for RE0.5.0.75.
Table 2 External sod internal Reynolds sod Marangoni numbers bascdoothe~oftable 1 forthreetadii.
60 E
40
1 30 I- 20
0
0.2
01
ssamld
0.6
0.6
20
30
40 so 40 10 so 90 tba WI Fig. 4. Drop vertical velocity vc-rsos time. Triangle R=O.25 cm. circle R&5 cm, square R&75 cm
10 0.0
to
19
-colK-
Fig. I Cyclobcxaowncthaool
phase diagram.
6
5
4
4
3
3
2
2
1
1
0
0
0.6
t
0
.-
0
a 0.6
I
Fig. 2 Concenttation isolines at start-up, t = 14.7 s for R=O.25cm. 1=10.3 s for R=O.S.0.75
-I 1s
o,=
05
1
116
Fig. 5 Wall temperature. gradient perturbations for UK three radix of table 2.
E. ~a~sano et al. /Acta Astronautica 53 (2003) 29%308
302
hypothesis P, = P2 the nod wmpomnt of the velocity is continuous, but differs from the interface normal speed appearing in the level set equation (1). that is given by DUG
t0
the
of the Heavyside step function h(e). In the present case the non-dimensional motion equations can be written a.8 v*v=o
n.V, =V,= n.V-F ( I
(3)
(4)
It is useful to write down explicitly the local mass fraction flu* through the interface
f * = {F(n . pDVc)}/(c}
+(VVr)lRe-VV]+
V, +Vp = Vb(VV
(7)
This expression holds also in the case of a discontinuous density. In the present problem the mass flow rate through the interface is given by rh = {n * pDVc}/{c}
(6)
(5)
where an overbar indicates quantities evaluated at the opposite side of the interface, and comparing it with the mass flux (4). The above formulae show that terms occurring in rir appear also in f * but are differently weighted. Thus the two fluxes may or may not have the same sign, depending on the actual values of the concentration at the opposite side of the interface. On the cylindrical cavity walls the velocity vanishes. The top and bottom walls are held at constant different temperatures, T, and Tb respectively, with T, > Tb, the lateral wall is adiabatic, there is no diffusion of species through the cavity walls, i.e. C, = 0. At the initial time the velocity is zero everywhere, the temperature is a linear function of the axial coordinate, the drop is spherical with radius R and its centre is located on the symmetry axis at an ordinate ye. The concentrations along the drop inner and outer faces are fixed according to the equilibrium curve. The initial discontinuity between the inner [outer] drop surface and the interior [exterior] is stnnotbed in few cells.
[(T -T, - We,/We)kn - V,Tln
.Vh)/We,
T, +v.(Tv)-V*(hVT)/c,Ma
= (8)
- T{c, )iz(n sVh)/c, c, +V@-DVc/ReSc]=O
(9)
$, + v$7tf( = 0
(10)
where the superscript “T” indicates transposition, V, = V - n iI/& is the surface nabla operator, k the interface curvature and the adimensional density is omitted bellg uniformly equal to one. The non-dimensional characteristic Schmidt SC, Reynolds Re, Marangoni Ma, Weber We, tltertno-solutal Weber We, numbers ate given by
Sc=v,/D,
Re=RV,/v,
We = VFprR/6,
Ma=RV,/a, (11) We, = Vfp,RA$,,(T, (12)
The reference quantities are Y = lb,~~,/~,
4 = R
f, = R/V,
P, = PXZ
T, = (q -TV )R/ H
(13)
and c,,,D,,h,,p,,p,
(14) are those of the external
fluid. One has We, = Re. Terms weighted by Mach and Ecltert numbers are neglected. ‘Ihe sign, Heavyside and Dirac functions are mollified as
TfIE MOTION EOUATIONS The motion equations are formulated in a standard imbedded boundary approach by defining the dependent variables in the whole domain in terms
0-C-E
sgn,(@)=-1
I$>E
sgn,(@)=l
E. Bassano et al. /Acta Astronautica 53 (2003) 299-308
(16) (17)
where E = mh . Tbe drop volume and centre of mass ordinate are given by the integrals ( m = l/2 )
vol = 273(1-
00
h,)xdxdy
(18)
It is worth noting that for pure Marangoni convection,(i.e. in absence of dissolution), the r.h.s. of equation (8) vanishes. THE NUMERICAL SOLUTION METHOD The system of motion equations in conservation form was solved by a finite,volume approach on a MAC (Markers And Cells) staggered grid. The fluxes were evahmted by EN0 (Essentially NonOscillatory) upwind biased three points formulae, diffusive fluxes with second order centered formulae. The ordinary projection method was used to determine the velocity field. The pressure Poisson equation was solved with a SOR (Successive Over Relaxations) algorithm. The space semi-discretised equations were advanced in time with a first order Euler explicit time stepping. The trace k of the curvature tensor is calculated with second order centered formulae in terms of the level-set function derivatives as
Details on the calculation of the concentration field can be found in Bassano et al’.‘,‘. NUMERICAL RESULTS The MEX-MAM cell is 3 cm wide, 3 cm deep and 6 cm long in the temperature gradient direction. The 2D axis-symmetric simulation domain, corresponds to the right half of an axial planar section of a cylindrical cavity having a radius of 3 cm and a height of 6 cm.
303
At the initial time, t = 0 s. the centre of drop is located on the symmetry axis. We considered three different drop radii: R = 0.25,O.S and 0.75 cm. The drop centre distance from the lower wall is I cm. The cold temperature is set to 15 “C, the hot one to 55 “C. Accordingly the imposed temperature gradient is 6.66 “C/cm. Initially the temperature profile is linearly stratified, tire drop is of pure methanol, the surrounding matrix of pure cyclohexane, both phases are quiescent. The properties of the fluids used for simulations are reported in table 1. Non-dimensional characteristic numbers depending only on fluid properties (Prandtl and Schmidt) are also reported in the same table. Internal and external Reynolds and Marangoni numbers, depending on R. are listed in table 2. The computationat grid in the axial semi-plane is composed by 60 x 240 square cells, 80 x 320 square cells are used for the smaller drop. Preliminary tests showed that used grids are sufficiently refined for large and medium drops, while grid refinement is probably required for the small drop, but has not been pursued here due to the large computational times required. During the initial start-up phase of the drop motion, which is similar for all cases, both internal and external liquids are dragged along the drop surface by the interface tension gradient. The internal fluid dragged along the interface is warmer than the drop bulk, which is wrapped and tends to stay cold due to the large internal Peclet number. When the inner surface driven flow reaches the rear pole, it rises along the symmetry axis, reaches the front region and then flows back, creating a ring vortex inside the drop as evidenced by the concentration isolines of fig. 2a,b,c (the radius increases from left to right). At the same time the external fluid dragged along the external side of the drop creates a warm wake behind the drop, entraining the cold bottom region of the cavity fig. 3a,b,c (the radius increases from left to right). After a sudden acceleration, due to the rise of surface tension forces, the drop velocity decreases, since the isotherms wrap around the drop, whose surface tends to become isotherm. The drop vertical velocity versus time is plotted in fig. 4. After the initial deceleration occurring whatever is the radius during the first 10 seconds, the medium and large drops start newly to accelerate and their speed increases until the drop stops at the hot wall of the cavity. Particularly, the medium drop reaches a larger velocity having a longer path to migrate before touching the hot wall. On the contrary, foi the small drop, the initial deceleration creates a
304
E. Bassano et al. /Acta Astronautica
53 (2003) 299-308
#~o.,~~... 0.6 ...-.. .....i... i..i____. ...__. .____ ii.._... .ii..{ ...{...j...$...i......: ..; __.___ j__..__ i______ i_____ IJO
t
: : 3 . . . . . i . . . . . . j,. ___... i .._ ! :
0.2
J_._. j.._.. j_____ -:____. (..._ .jb; _: 1 0.2
I
I 21 0.2l oz 0.23
0.24
f
81 I s I, 0.15 066 0.27 023 0 a 0.3 m&n fig. 9 mdiaI cottc.ontratioa distribution for R=O.zS cm along three diffcnnt radial di4dons at t = 1.47s.
02
021
024
fig. 6. Drop shapes during ttw ascent. Initial time h1.47 s. AI = 4.41 s for R=O.25 cm. At = 1.47 s for R=O.5.0.75.
0.22
OP
0.25 Rdin
026
, 0.n
I 0.a
I 0.26
03
Fig. IO radial concentrstion distribution for R=OZ cm along three different radial din*ims at I = 23.5 s.
W’ 1 ( ; I”I 0.8 . ..I .. ....(...... $ .. .. . .f.- . . ..!......!...... + . ... . j .. . .... .. ...: : : ..f . . .. . .. . ... ..i . .._ : : j : : . j j .+ .. .. . .. . .. . ... .. .../ :i :j. : Fig. 7 concentration isolincsdwing arcent. t = 58.8 s for R&25 em, t = 20.6 s for R = 0.5.0.75 cm.
: : ! 0 . . . . . . ~......!.......: . a2
on
0.22
i ; i : : : . . . . . . >. . . . . . . . . . . : : : : I I , 1 oz3 014 025 026
: 4 0.27
: I 0.a
: * 0.26
0.3
Fig. I I radial concmhadon distribution for R=OZi cm along thnxdiffemumdialdinxthnsatt=Ms.
*.
E. Bassuno et al. /Acta Astronautica
wave in the front of the drop that destroys the stratitied linear profile and determines a drop slowing, as shown in fig. 5a, which illustrates the temperature distribution. This behavior is due to the fact that the small drop cools more quickly than larger drops and its surface becomes nearly isotherm. At later times the temperature ceases to decrease monotonically along the surface of the small drop and the interface tension maximum ceases to be located on the symmetry axis. This leads to the -inversion of the diion Of the interface tension near the interface fore pole. The inverted flow region growf and gives rise to a second. counter-rotating, and vortex within drop. The outer fluid in the f&t of the drop is pushed up towards the hot wall and the inner fluid is pushed back down towards the drop centre, as indicated by the isotherms of fig 5a. The occurrence of the reverse fld~ in front of the smaller drop has been observed both on differently refined grids and with different stencils for the convective terms so it does ‘not seem to be a numerical artifact. As shown by fig 5a.b.c. (the radius increases from left to right), the stratified temperature profile near the lateral wall is left almost unperturbed by the small drop, is slightly perturbed by the medium drop, is highly distorted by capillary convection around the largest drop. ‘Iberefore the largest drop migrates subjected to a non-uniform temperature gradient. During its ascent every drop warms up, as shown in fig. 5a.b.c. The smaller is the radius the warmer is the drop and the faster is reduction of the thermal wake behind it. No noticeable drop shrinking can be observed during the drops ascent as shown in fig. 6, where the drop shapes are reported at constant intervals of time from bottom to top of each picture. Besides the thermal Marangoni convection, the concentration field is also affected by dissolution. According to the initial conditions and binodal curve, the concentration d ecreases going from the inner phase to the inner s&face of the drop and still decreasesgoing from the outer surface towards the external phase. The internal vertical flow that establishes afIer the start-up, is characterized by concentration values lower than those of the ring vortex core both near the drop surface and along the axis. Obviously this is due to the fact that the low concentration fluid had been convected along the interface, where the concentration was fixed by the binodal curve, as shown by fig Za.b,c. The outer flow drags a more concentrated fluid in the wake. In the first part of the drop rise, the concentration field follows the qualitative evolution of the
53 (2003) 299-308
305
thermal field. as shown in figs 2,3, even if the diffusion of matter is sensibly lower than that of energy, Schmidt numbers being huger than Prandtl numbers. It must be noted that the less concentrated fluid rising along the drop axis is convected downward by the vortex ring before reaching the fore pole and the interface After a complete turn of the fluid along the drop surface and axis, the inner low (outer high) concenaation regions become larger due to the depletion (enrichment) of the fluid along the interface, as one can see e.g. in fig. 7. As shown in the same figun, the mass fraction of the external phase changes slightly and in the wake only, due to the very low diffusion coefficient and to the low values of the concentration along the left branch of the binodal curve of fig. 1. The mass fraction variations are instead appreciable within the drop. Here the concentration changes from the initial unit value to values lower than 0.7 at the drop inner surface. The unit concentration persists in a region near the ring vortex core. Differently from the case of the mass flow through the interface discussed below, no conclusions on the sign of the concentration flux, given by eq. (5), can be drawn a priori, since the two terms in the r.h.s. are about the same order of magnitude at the initial time. The positiveness of the concentration flux through the interface during migration is revealed by the already noted increase of the concentration outside the drop and in the wake. However, it is important to underline that, according to eq.s (4, 5). dissolution can also occur while the volume increases, i.e. partial mass flux can be positive even if the mass flux is negative. In the present problem the volume flow through the drop interface is proportional to the mass flow, the phases being isodense and the flow incompressible. The local mass flow through the interface ti, according to eq. (4), is proportional to the difference of the concentration normal derivatives and inversely proportional to the concentration jump, which depends on the local temperature and goes to zero at the critical point. Thus, according to fig. 1, the denominator of (4) is always negative, the concentration of methanol being larger in the inner phase. An increase of temperature can cause an increase of the intensity of ti but not a sign change. At initial time the concentration normal derivatives are both negative. The initial conditions and the binodal curve make the inner normal derivative, in modulus. fairly larger than the external one, of about an order of magnitnde. This leads to the conclusion that at least for short times the drop volume shall iucrease. in absence 01
306
E. Bassano et al. /Acta Astronautica
convection, volume variation should occur in times of the order of the diffusive characteristic time R’/D and then the volume should increase for quite long times. variation volume relative The drop vol /voZ(O)- 1 is plotted versus time in fig. 8. where the full symbols refer to dissolution and the empty symbols refer to absence of didution. The volume variation in the second case (only Marangoni migration) should be xero rather than 4%. This indicates that a grid refinement is still neceauuy for small drops. Medium and large drops grow slightly and slowly while smaller drops grow more and faster. Nevertheless one can observe that the dissolution is tesqonsible of .a~ increase of ,diqrensions. At startup the droR cold core is wrapped by the warmer fluid draga by interface tension along its surface. Since the energy exchange by heat between the rising drop and the surrounding phase is very small, the drop tends to stay cold. Accordingly, the external isotherms encountered by the drop during the ascent sligl$y penetrate ,wi@n the drop and fold over it. The drop pushes the isotherms up and constrains them in a narrow thermal boundary layer, surrounding the drop and having a steep normal temperature gradient, as shown in fig.s 2,4. Therefore the temperature on the drop surface does not change appreciably during the rise and remains lower than the one corresponding to the unperturbed stratified profile at the actual height. Correspondingly the concentration jump across the interface (c+ -c- ), appearing in the denominator of eq. (4). neither goes to zero nor decreases appreciably and the mass flow rate ti remains bounded also in the upper part of the cavity, where the unpetturbed temperature is higher than the critical one. In fig 9-12 are plotted the concentration profiles in planes passing through the cell centre nearest to the drop centre of mass and inclined at 90”. 0”. -90” w.r.t. the horizontal for the. small drop at different increasing times, The plots show a neighborhood of the interface. The line connecting the inner and outer concentration is added by the visualization software but it must not be considered as a part of the profile~since in our calculation the concentration jump is sharp. The normal derivatives (the drop is nearly spherical) are given by the profile slopes at the sides of the interface rcprescnted by the almost vertical segment. In the present cast eq. (4) and (5) can be written, respectively, as
53 (2003) 299-308
iI =pD(-Ic;I+Ic;I)/Ic--c+I
(21)
f’
(22)
= pD(-
c+lc;l + c-Ic;I)/lc-
- c+l
where the normal coordinate n increases like the radial one. By comparing the inner and outer slopes at the interface, one deduces that the mass flow through the interface is large and negative for the horizontal profile. Since the same happens also for a large part of the radial profiles (not displayed in the figures), the global exiting mass flux (the integral of tit along the interface) is negative and the drop increases in vohtme, at large times the increase rate diiishes as shown in figs 7 and 12. Since c- >> C+ and the inner normal derivative is smali iu both poles f9r the small drop, accorping to eq. (22). the methanol fhrx is positive there and a methanol pluthe develops at both poles as shown in fig. 13. while it is present only at the rear Role for medium and large drops. In addition in the case of small drops, even if energy diffusion is quite limited, it is nevertheless sufficient to reduce drop temperature, due to the reduced drop dimensions. The migration of the small drop is also slower and the drop has a larger time to adapt itself to the external conditions. The increase of the drop surface temperature, through the concentration jump at the interface, also contributes to a volume variation faster than that of bigger drops. Finally it must bc considered that a given volume variation, when normalized w.r.t the initial volume, increases with decreasing the initial dimensions. ERIMENTAL
ACMVEES
Experiments have been carried out in preparation of the MAXUS-S mission. The optical diagnostic system has been described in the previous paper?.A Wollaston shearing interferometer will be used for the determination of the refraction index gradient field caused by concentration gradients. The choice of this intereferometric set-up is m&ivatcd by the need to measure h&h local concentration gradients, and by the fact that this technique is very &bie and does not suffer some g-jitters’ expected on MAXUSJ. During ground-based aotiv$ies, drops (3-S@ of methanoi injected in the cyclohexane matrix and kept anchored to a capillary showed occurrence of ’These j-gitters are caused by the presence of other experiments using centrifuges.
E. Bassano et al. /Acta Astronautica 53 (2003) 299-308
02
021
022
023
024
025 nayJ*
0.x
0.27
0s
0.29
2
0.3 Ran
Fig. 13 radial concmtration distriLwtion for R&25 cm along thm dierent radial directions at t = 16.2s.
Fig. 12 radial concentration distribution for R425 cm along rhnc different radial directions at t = 72.1 s.
0
10
307
20
30
40
50
60
70
80
90
time [sl Fig. 8. Normalized drop volume variation defined as (voUvol(o)-I) , vol(0) is rhe initial (1=0) drop volume. Triangle R&25 cm. circle R=O.5 cm. square R=O.75 cm. Open symbols cornspond IO presence of both Mamngoni migration and dissolution. Filled symbols to migration without dissolution.
308
E. Bassano et al. / Acta Astronautica
solutal buoyant instabilities. The frequeocy and the critical radius for the onset of these instabilities are strongly depodent 00 the temperature of the system, and on the initial volume. Results will be presentedin a forthcoming paper. Both the numerical simulation and the ground based experimental act&es allowed us to define the flight experimental sequence. Once the facility is in microgravity (100 seconds after MAXUS-S launch), the curtain system allows one to have a stable and constant temperature gradient, and there~orc to inject drop under controlled tempemtum conditions. Du&g the fast nm a methanol drop of diameter 0 = 5 mm is fad in the cyclohexane matrix. In order to study the occmmm~ of plume observed on ground and eventuaJly the onset of its instability, the drop is kft anchord to the injection needle for two minuk The drop is then released by activating a solenoid. Tim migration and the further dissolution are studied until the drop reaches the hot wall. The two successive runs foresee tire injection of two larger drops (0 = 10.15 mm). The idea is to study again if any plume develops, the migration for larger radius, and. evemually. the interaction between dissolving drops. The interferometric data will be recorded on board and then available atIer the mission for correlation with the numerical results. Since these results predict the formation of a wake along the migration path, the shearing direction has been set from the left to the right flange. CONCLUSIONS In the present paper we report on the ongoing
53 (2003) 299-308
research activities for the preparation of an experiment on Marangoni migration of a dissolving drop to be carried out onboard the MAXUSsounding rocket. The already developed 2D planar version of the numerical code has been upgraded to treat axis-symmetric geometries. Many improvements of the code have also been implemented to increase accuracy and to reduce computational load. REFERENCES 1. Bassano E., Castagnolo D.: “Marangom migration of a methanol drop in cyclohexane matrix in a closed cavity”. Accepted for publication on Microgravity Sci. and Tech. 2. Bassano E.: “Numerical simulation of thermosohnal-capillary migration of a dissolving drop in a cavity”. Submitted to Int. J. of Nwn Mel forfluids. 3. Bassano E., Castagnolo D., Albano F.: Thefmocapillaty two-phase flow of a binary mixture drop exhibiting a miscibility gap. IAFOl-5.4%). 52nd IA% Congress, Tolouse, France, 1-J Oct. 2001. 4. MARS Center, “Bread-Board Activities Test Report for Drop Dissolution and Migration Experiment”, MARS Internal Report, 2001. 5. Predel B., Ratke L.. Fredrikmon H., “Fluid science and material science in space”, Walter H.U. Ed., Springer, Berlin 1987. 6. Qsher S, Sethian J. Fronts propagating with - curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. J. Camp. Phys. 1988; 79; 12-49.
PERGAMON
Acta Astronautica 53 (2003) 299-308 www.elsevier.com/locate/actaasrro
RESULTS OF NUMERICAL MODELLING AND EXPERIMENTAL PREPARATION OF THE MAXUSEXPERIMENT
ACTIVITIES
IN
Edmondo Bassano, Dario Castagnolo, Raimondo Fortezza MARS Center s.c.a r.1. Via E. Gianturco 31, I-80146 Napoli, Italy. [email protected] Abstract The present paper reports on the research activities in b&h experimental and numerical field carried out at MARS center for the realization of an experiment on the thermo-solutal-capillary migration of a dissolving drop, composed by a liquid binary mixture having a miscibility gap. The experiment will be carried out onboard of the sounding rocket MAXUS 5, by using the INEXMAM facility. The main goals of the analysis are diagnostic equipment ensuring that the performanres cope with the stringent resolution requirements dictated by the physics and assessing of the experiment time profile. 8 2003 International Astronautical Federation. Published by Elsevier Science Ltd. All rights reserved. INTRODUCI7ON Two liquid binary mixtures constituted by the same components exhibit a miscibility gap when they coexist, in a given range of temperatures, in two different phases having different composition. A miscibility gap in the liquid phase is found in several different physical systems: metal alloys, binary mixtures of organic liquids, sulphides and silicates systems, glasses and liquid crystals, and many industrial applications and processes are based on miscibility gap and related phenomena. Microgravity experiments on miscibility gap were conducted for more than two decades and flew on almost every kind of microgravity platform’. Research activities in both experimental and numerical field are carried out at MARS center for the realization of an experiment on the thermosplutal-capillary migration of a dissolving drop, composed by a liquid binary mixture having a miscibility gap. The experiment will be hosted onboard the MAXUS 5 sounding rocket, by using the INEX-MAM facility’.2*3” Since capillary migration of drops is strongly affected, on Fzuth, by buoyancy due to both different densities of the mixture components and Boussinesq effects, the dissolution due to a miscibility gap and its influence on migration can be profitably studied in microgravity.
Theoretical models for calculating the equilibrium curve are available and are particularly helpful for ideal solutions, but differences in atomic radius and bonding5 are responsible of possibly large discrepancies between real and ideal behavior, thus the cyclohexane-methanol equilibrium phase diagram, shown in fig. I, is given by points. It is supposed that equilibrium always occurs at the binodal line. THE MAXUS 5 EXPERIMENT The idea is to inject a drop of a pure component in a matrix of the other component, that tills a closed squared based cavity and has a linearly stratified temperature. The upper and lower walls of the cavity are held at constant different temperatures. Since the system is in microgravity, only interfacial tension gradient effects occur. The drop should migrate towards the hot upper wall because of thermal Marangoni effect and dissolve because the system is in a region of the phase diagram where ir shall form a single phase. The liquids selected for the sounding rocket experiment and used in numerical simulations are cyclohexane (CsH1& - methanol (CH30H). This couple of liquids has been selected because: i) the components have nearly the same density; ii) are transparent, allowing either the direct visualization of the drop migration and deformation either the use of optical diagnostics devices; iii) their mixture has a critical temperature of 45.7 “C, slightly larger than the ambient one and compatible with the INJZXMAM facility thermal requirements4. The facility diagnostic technique is described in Bassanoet al.‘. NUMERICAL
MODELING
Some of the restrictive hypotheses usually adopted in the classical theoretical and numerical literature on thermo-capillary migration of bubbles and drops, removed in the present approach, were discussed in Bassano et a1.le3together with modem CFD techniques for multiphase flows modelling. With respect to the previous 2D-planar works, the present code upgrade is represented by the axis-
(@94-5765/03/$- seefront matter Q 2003 International Astronautical Federation.Published by Elscvier ScienceLtd. All rights reserved. doi: 10.1016/S0094-5765(03)00145-O
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symmetric formulation and integration of the relative motion equations. This aJloWS to correctly describe the physics of the migration process preserving the real topology, geometry and dynamics of the thermo-fluid dynamic field occuring during experiment on ground and in space. In the present paper the drop interface evolution is modeled by the level set technique, in which the interface is represented by the embedding of the zero level set of a scalar function defined within the whole computational domain. The level-set equation is written in this upgraded model in .a different form suitable for interface speed extension. An extension algorithm was implemented to better simulate the interface dynamics and localized to reduce run-times. The weight of the Dirac delta function appearing in the momentum production is extended as discussed in*. A major &dification made in the present analysis, greatly enhancing the solution accuracy, is a different formulation of the energy equation. In past works the internal energy was used as dependent energy variable, having a discontinuity at the interface due to the specific heat coefftcient jump. This choice was responsible for a poor accuracy of the calculation of the energy equation, which in turns gave temperature profiles with kinks not located on interface but slightly outside. Also a too small energy flux entering within the drop, a crowding of isotherms around it, and ultimately a tendency of the drop surface to become isotherm during migration was observed. These problems, which are much more relevant in the axissymmetric case, have been solved by using as independent variable the temperature, which is continuous at interface and has a discontinuity only in the normal derivative. THE LEVEL SET METHOD The level set method for advancing fronts was proposed by Osher and Sethiat?. The front, in our case the interface y(t), is represented at each instant of time t by the zero level set of a smooth scalar function $. Let x(f) be a representation of y(t) and Vi (t ) = -i(t). The evolution equation for $t is described by the scalar equation of HamiltonJacoby type t$, +vi .Vtp=@,
+v,$q=o
(1)
53 (2003) 299-308
for this equation states that, if at the initial time @ is constant along the curve Y moving in normal direction with speed Vii, = n .V;. , it will be constant there also at later times. In writing the second equality above, we have exploited the fact that the unit vector n normal to Y. being @ constant along it, can be written as n = V@//pd Note that the dynamics of ‘y is fully described by
V, only. We assume that t$ is positive outside the drop and negative inside so that n points outside. MODEL ASSUMF’TJONS The cavity containing the drop and the external liquid matrix is cylindrical. In the following V indicates the nabla operator, p the average normal stress, p the dynamic viscosity, V the kinematic viscosity, c, the specific heat coefficient, CX the thermal diffusivity, h the thermal conductivity,
‘Z?FT+
(T),c (T)).
At the critical point (dc/dT = Oin fig. I), the different phases merge in a single phase and Q,, = 0. Since the equilibrium concentrations depend on the temperaturc one can assume’ d = 6,, (T - T,,). According to experimental data 8,s < 0. Since cyclohexane and methanol have nearly the same densities the difference p, - p2 is neglected. Mass conservation implies the continuity of the mass flow rate ti through the interface ril=ti*
={n*p(v-v;)}
(2)
where { f}= f ’ - f - is the jump operator and the superscript “+” indicates the domain towards which n points. b
E. Bassano et al. /Acta Astronautica
53 (2003) 299-308
Table 1 Properties of cyclohclane and methanol
Fig 3 Temperalum isolioa at start-up. t = 1.47 I for R=O.ZScm. t=2.94 s for RE0.5.0.75.
Table 2 External sod internal Reynolds sod Marangoni numbers bascdoothe~oftable 1 forthreetadii.
60 E
40
1 30 I- 20
0
0.2
01
ssamld
0.6
0.6
20
30
40 so 40 10 so 90 tba WI Fig. 4. Drop vertical velocity vc-rsos time. Triangle R=O.25 cm. circle R&5 cm, square R&75 cm
10 0.0
to
19
-colK-
Fig. I Cyclobcxaowncthaool
phase diagram.
6
5
4
4
3
3
2
2
1
1
0
0
0.6
t
0
.-
0
a 0.6
I
Fig. 2 Concenttation isolines at start-up, t = 14.7 s for R=O.25cm. 1=10.3 s for R=O.S.0.75
-I 1s
o,=
05
1
116
Fig. 5 Wall temperature. gradient perturbations for UK three radix of table 2.
E. ~a~sano et al. /Acta Astronautica 53 (2003) 29%308
302
hypothesis P, = P2 the nod wmpomnt of the velocity is continuous, but differs from the interface normal speed appearing in the level set equation (1). that is given by DUG
t0
the
of the Heavyside step function h(e). In the present case the non-dimensional motion equations can be written a.8 v*v=o
n.V, =V,= n.V-F ( I
(3)
(4)
It is useful to write down explicitly the local mass fraction flu* through the interface
f * = {F(n . pDVc)}/(c}
+(VVr)lRe-VV]+
V, +Vp = Vb(VV
(7)
This expression holds also in the case of a discontinuous density. In the present problem the mass flow rate through the interface is given by rh = {n * pDVc}/{c}
(6)
(5)
where an overbar indicates quantities evaluated at the opposite side of the interface, and comparing it with the mass flux (4). The above formulae show that terms occurring in rir appear also in f * but are differently weighted. Thus the two fluxes may or may not have the same sign, depending on the actual values of the concentration at the opposite side of the interface. On the cylindrical cavity walls the velocity vanishes. The top and bottom walls are held at constant different temperatures, T, and Tb respectively, with T, > Tb, the lateral wall is adiabatic, there is no diffusion of species through the cavity walls, i.e. C, = 0. At the initial time the velocity is zero everywhere, the temperature is a linear function of the axial coordinate, the drop is spherical with radius R and its centre is located on the symmetry axis at an ordinate ye. The concentrations along the drop inner and outer faces are fixed according to the equilibrium curve. The initial discontinuity between the inner [outer] drop surface and the interior [exterior] is stnnotbed in few cells.
[(T -T, - We,/We)kn - V,Tln
.Vh)/We,
T, +v.(Tv)-V*(hVT)/c,Ma
= (8)
- T{c, )iz(n sVh)/c, c, +V@-DVc/ReSc]=O
(9)
$, + v$7tf( = 0
(10)
where the superscript “T” indicates transposition, V, = V - n iI/& is the surface nabla operator, k the interface curvature and the adimensional density is omitted bellg uniformly equal to one. The non-dimensional characteristic Schmidt SC, Reynolds Re, Marangoni Ma, Weber We, tltertno-solutal Weber We, numbers ate given by
Sc=v,/D,
Re=RV,/v,
We = VFprR/6,
Ma=RV,/a, (11) We, = Vfp,RA$,,(T, (12)
The reference quantities are Y = lb,~~,/~,
4 = R
f, = R/V,
P, = PXZ
T, = (q -TV )R/ H
(13)
and c,,,D,,h,,p,,p,
(14) are those of the external
fluid. One has We, = Re. Terms weighted by Mach and Ecltert numbers are neglected. ‘Ihe sign, Heavyside and Dirac functions are mollified as
TfIE MOTION EOUATIONS The motion equations are formulated in a standard imbedded boundary approach by defining the dependent variables in the whole domain in terms
0-C-E
sgn,(@)=-1
I$>E
sgn,(@)=l
E. Bassano et al. /Acta Astronautica 53 (2003) 299-308
(16) (17)
where E = mh . Tbe drop volume and centre of mass ordinate are given by the integrals ( m = l/2 )
vol = 273(1-
00
h,)xdxdy
(18)
It is worth noting that for pure Marangoni convection,(i.e. in absence of dissolution), the r.h.s. of equation (8) vanishes. THE NUMERICAL SOLUTION METHOD The system of motion equations in conservation form was solved by a finite,volume approach on a MAC (Markers And Cells) staggered grid. The fluxes were evahmted by EN0 (Essentially NonOscillatory) upwind biased three points formulae, diffusive fluxes with second order centered formulae. The ordinary projection method was used to determine the velocity field. The pressure Poisson equation was solved with a SOR (Successive Over Relaxations) algorithm. The space semi-discretised equations were advanced in time with a first order Euler explicit time stepping. The trace k of the curvature tensor is calculated with second order centered formulae in terms of the level-set function derivatives as
Details on the calculation of the concentration field can be found in Bassano et al’.‘,‘. NUMERICAL RESULTS The MEX-MAM cell is 3 cm wide, 3 cm deep and 6 cm long in the temperature gradient direction. The 2D axis-symmetric simulation domain, corresponds to the right half of an axial planar section of a cylindrical cavity having a radius of 3 cm and a height of 6 cm.
303
At the initial time, t = 0 s. the centre of drop is located on the symmetry axis. We considered three different drop radii: R = 0.25,O.S and 0.75 cm. The drop centre distance from the lower wall is I cm. The cold temperature is set to 15 “C, the hot one to 55 “C. Accordingly the imposed temperature gradient is 6.66 “C/cm. Initially the temperature profile is linearly stratified, tire drop is of pure methanol, the surrounding matrix of pure cyclohexane, both phases are quiescent. The properties of the fluids used for simulations are reported in table 1. Non-dimensional characteristic numbers depending only on fluid properties (Prandtl and Schmidt) are also reported in the same table. Internal and external Reynolds and Marangoni numbers, depending on R. are listed in table 2. The computationat grid in the axial semi-plane is composed by 60 x 240 square cells, 80 x 320 square cells are used for the smaller drop. Preliminary tests showed that used grids are sufficiently refined for large and medium drops, while grid refinement is probably required for the small drop, but has not been pursued here due to the large computational times required. During the initial start-up phase of the drop motion, which is similar for all cases, both internal and external liquids are dragged along the drop surface by the interface tension gradient. The internal fluid dragged along the interface is warmer than the drop bulk, which is wrapped and tends to stay cold due to the large internal Peclet number. When the inner surface driven flow reaches the rear pole, it rises along the symmetry axis, reaches the front region and then flows back, creating a ring vortex inside the drop as evidenced by the concentration isolines of fig. 2a,b,c (the radius increases from left to right). At the same time the external fluid dragged along the external side of the drop creates a warm wake behind the drop, entraining the cold bottom region of the cavity fig. 3a,b,c (the radius increases from left to right). After a sudden acceleration, due to the rise of surface tension forces, the drop velocity decreases, since the isotherms wrap around the drop, whose surface tends to become isotherm. The drop vertical velocity versus time is plotted in fig. 4. After the initial deceleration occurring whatever is the radius during the first 10 seconds, the medium and large drops start newly to accelerate and their speed increases until the drop stops at the hot wall of the cavity. Particularly, the medium drop reaches a larger velocity having a longer path to migrate before touching the hot wall. On the contrary, foi the small drop, the initial deceleration creates a
304
E. Bassano et al. /Acta Astronautica
53 (2003) 299-308
#~o.,~~... 0.6 ...-.. .....i... i..i____. ...__. .____ ii.._... .ii..{ ...{...j...$...i......: ..; __.___ j__..__ i______ i_____ IJO
t
: : 3 . . . . . i . . . . . . j,. ___... i .._ ! :
0.2
J_._. j.._.. j_____ -:____. (..._ .jb; _: 1 0.2
I
I 21 0.2l oz 0.23
0.24
f
81 I s I, 0.15 066 0.27 023 0 a 0.3 m&n fig. 9 mdiaI cottc.ontratioa distribution for R=O.zS cm along three diffcnnt radial di4dons at t = 1.47s.
02
021
024
fig. 6. Drop shapes during ttw ascent. Initial time h1.47 s. AI = 4.41 s for R=O.25 cm. At = 1.47 s for R=O.5.0.75.
0.22
OP
0.25 Rdin
026
, 0.n
I 0.a
I 0.26
03
Fig. IO radial concentrstion distribution for R=OZ cm along three different radial din*ims at I = 23.5 s.
W’ 1 ( ; I”I 0.8 . ..I .. ....(...... $ .. .. . .f.- . . ..!......!...... + . ... . j .. . .... .. ...: : : ..f . . .. . .. . ... ..i . .._ : : j : : . j j .+ .. .. . .. . .. . ... .. .../ :i :j. : Fig. 7 concentration isolincsdwing arcent. t = 58.8 s for R&25 em, t = 20.6 s for R = 0.5.0.75 cm.
: : ! 0 . . . . . . ~......!.......: . a2
on
0.22
i ; i : : : . . . . . . >. . . . . . . . . . . : : : : I I , 1 oz3 014 025 026
: 4 0.27
: I 0.a
: * 0.26
0.3
Fig. I I radial concmhadon distribution for R=OZi cm along thnxdiffemumdialdinxthnsatt=Ms.
*.
E. Bassuno et al. /Acta Astronautica
wave in the front of the drop that destroys the stratitied linear profile and determines a drop slowing, as shown in fig. 5a, which illustrates the temperature distribution. This behavior is due to the fact that the small drop cools more quickly than larger drops and its surface becomes nearly isotherm. At later times the temperature ceases to decrease monotonically along the surface of the small drop and the interface tension maximum ceases to be located on the symmetry axis. This leads to the -inversion of the diion Of the interface tension near the interface fore pole. The inverted flow region growf and gives rise to a second. counter-rotating, and vortex within drop. The outer fluid in the f&t of the drop is pushed up towards the hot wall and the inner fluid is pushed back down towards the drop centre, as indicated by the isotherms of fig 5a. The occurrence of the reverse fld~ in front of the smaller drop has been observed both on differently refined grids and with different stencils for the convective terms so it does ‘not seem to be a numerical artifact. As shown by fig 5a.b.c. (the radius increases from left to right), the stratified temperature profile near the lateral wall is left almost unperturbed by the small drop, is slightly perturbed by the medium drop, is highly distorted by capillary convection around the largest drop. ‘Iberefore the largest drop migrates subjected to a non-uniform temperature gradient. During its ascent every drop warms up, as shown in fig. 5a.b.c. The smaller is the radius the warmer is the drop and the faster is reduction of the thermal wake behind it. No noticeable drop shrinking can be observed during the drops ascent as shown in fig. 6, where the drop shapes are reported at constant intervals of time from bottom to top of each picture. Besides the thermal Marangoni convection, the concentration field is also affected by dissolution. According to the initial conditions and binodal curve, the concentration d ecreases going from the inner phase to the inner s&face of the drop and still decreasesgoing from the outer surface towards the external phase. The internal vertical flow that establishes afIer the start-up, is characterized by concentration values lower than those of the ring vortex core both near the drop surface and along the axis. Obviously this is due to the fact that the low concentration fluid had been convected along the interface, where the concentration was fixed by the binodal curve, as shown by fig Za.b,c. The outer flow drags a more concentrated fluid in the wake. In the first part of the drop rise, the concentration field follows the qualitative evolution of the
53 (2003) 299-308
305
thermal field. as shown in figs 2,3, even if the diffusion of matter is sensibly lower than that of energy, Schmidt numbers being huger than Prandtl numbers. It must be noted that the less concentrated fluid rising along the drop axis is convected downward by the vortex ring before reaching the fore pole and the interface After a complete turn of the fluid along the drop surface and axis, the inner low (outer high) concenaation regions become larger due to the depletion (enrichment) of the fluid along the interface, as one can see e.g. in fig. 7. As shown in the same figun, the mass fraction of the external phase changes slightly and in the wake only, due to the very low diffusion coefficient and to the low values of the concentration along the left branch of the binodal curve of fig. 1. The mass fraction variations are instead appreciable within the drop. Here the concentration changes from the initial unit value to values lower than 0.7 at the drop inner surface. The unit concentration persists in a region near the ring vortex core. Differently from the case of the mass flow through the interface discussed below, no conclusions on the sign of the concentration flux, given by eq. (5), can be drawn a priori, since the two terms in the r.h.s. are about the same order of magnitude at the initial time. The positiveness of the concentration flux through the interface during migration is revealed by the already noted increase of the concentration outside the drop and in the wake. However, it is important to underline that, according to eq.s (4, 5). dissolution can also occur while the volume increases, i.e. partial mass flux can be positive even if the mass flux is negative. In the present problem the volume flow through the drop interface is proportional to the mass flow, the phases being isodense and the flow incompressible. The local mass flow through the interface ti, according to eq. (4), is proportional to the difference of the concentration normal derivatives and inversely proportional to the concentration jump, which depends on the local temperature and goes to zero at the critical point. Thus, according to fig. 1, the denominator of (4) is always negative, the concentration of methanol being larger in the inner phase. An increase of temperature can cause an increase of the intensity of ti but not a sign change. At initial time the concentration normal derivatives are both negative. The initial conditions and the binodal curve make the inner normal derivative, in modulus. fairly larger than the external one, of about an order of magnitnde. This leads to the conclusion that at least for short times the drop volume shall iucrease. in absence 01
306
E. Bassano et al. /Acta Astronautica
convection, volume variation should occur in times of the order of the diffusive characteristic time R’/D and then the volume should increase for quite long times. variation volume relative The drop vol /voZ(O)- 1 is plotted versus time in fig. 8. where the full symbols refer to dissolution and the empty symbols refer to absence of didution. The volume variation in the second case (only Marangoni migration) should be xero rather than 4%. This indicates that a grid refinement is still neceauuy for small drops. Medium and large drops grow slightly and slowly while smaller drops grow more and faster. Nevertheless one can observe that the dissolution is tesqonsible of .a~ increase of ,diqrensions. At startup the droR cold core is wrapped by the warmer fluid draga by interface tension along its surface. Since the energy exchange by heat between the rising drop and the surrounding phase is very small, the drop tends to stay cold. Accordingly, the external isotherms encountered by the drop during the ascent sligl$y penetrate ,wi@n the drop and fold over it. The drop pushes the isotherms up and constrains them in a narrow thermal boundary layer, surrounding the drop and having a steep normal temperature gradient, as shown in fig.s 2,4. Therefore the temperature on the drop surface does not change appreciably during the rise and remains lower than the one corresponding to the unperturbed stratified profile at the actual height. Correspondingly the concentration jump across the interface (c+ -c- ), appearing in the denominator of eq. (4). neither goes to zero nor decreases appreciably and the mass flow rate ti remains bounded also in the upper part of the cavity, where the unpetturbed temperature is higher than the critical one. In fig 9-12 are plotted the concentration profiles in planes passing through the cell centre nearest to the drop centre of mass and inclined at 90”. 0”. -90” w.r.t. the horizontal for the. small drop at different increasing times, The plots show a neighborhood of the interface. The line connecting the inner and outer concentration is added by the visualization software but it must not be considered as a part of the profile~since in our calculation the concentration jump is sharp. The normal derivatives (the drop is nearly spherical) are given by the profile slopes at the sides of the interface rcprescnted by the almost vertical segment. In the present cast eq. (4) and (5) can be written, respectively, as
53 (2003) 299-308
iI =pD(-Ic;I+Ic;I)/Ic--c+I
(21)
f’
(22)
= pD(-
c+lc;l + c-Ic;I)/lc-
- c+l
where the normal coordinate n increases like the radial one. By comparing the inner and outer slopes at the interface, one deduces that the mass flow through the interface is large and negative for the horizontal profile. Since the same happens also for a large part of the radial profiles (not displayed in the figures), the global exiting mass flux (the integral of tit along the interface) is negative and the drop increases in vohtme, at large times the increase rate diiishes as shown in figs 7 and 12. Since c- >> C+ and the inner normal derivative is smali iu both poles f9r the small drop, accorping to eq. (22). the methanol fhrx is positive there and a methanol pluthe develops at both poles as shown in fig. 13. while it is present only at the rear Role for medium and large drops. In addition in the case of small drops, even if energy diffusion is quite limited, it is nevertheless sufficient to reduce drop temperature, due to the reduced drop dimensions. The migration of the small drop is also slower and the drop has a larger time to adapt itself to the external conditions. The increase of the drop surface temperature, through the concentration jump at the interface, also contributes to a volume variation faster than that of bigger drops. Finally it must bc considered that a given volume variation, when normalized w.r.t the initial volume, increases with decreasing the initial dimensions. ERIMENTAL
ACMVEES
Experiments have been carried out in preparation of the MAXUS-S mission. The optical diagnostic system has been described in the previous paper?.A Wollaston shearing interferometer will be used for the determination of the refraction index gradient field caused by concentration gradients. The choice of this intereferometric set-up is m&ivatcd by the need to measure h&h local concentration gradients, and by the fact that this technique is very &bie and does not suffer some g-jitters’ expected on MAXUSJ. During ground-based aotiv$ies, drops (3-S@ of methanoi injected in the cyclohexane matrix and kept anchored to a capillary showed occurrence of ’These j-gitters are caused by the presence of other experiments using centrifuges.
E. Bassano et al. /Acta Astronautica 53 (2003) 299-308
02
021
022
023
024
025 nayJ*
0.x
0.27
0s
0.29
2
0.3 Ran
Fig. 13 radial concmtration distriLwtion for R&25 cm along thm dierent radial directions at t = 16.2s.
Fig. 12 radial concentration distribution for R425 cm along rhnc different radial directions at t = 72.1 s.
0
10
307
20
30
40
50
60
70
80
90
time [sl Fig. 8. Normalized drop volume variation defined as (voUvol(o)-I) , vol(0) is rhe initial (1=0) drop volume. Triangle R&25 cm. circle R=O.5 cm. square R=O.75 cm. Open symbols cornspond IO presence of both Mamngoni migration and dissolution. Filled symbols to migration without dissolution.
308
E. Bassano et al. / Acta Astronautica
solutal buoyant instabilities. The frequeocy and the critical radius for the onset of these instabilities are strongly depodent 00 the temperature of the system, and on the initial volume. Results will be presentedin a forthcoming paper. Both the numerical simulation and the ground based experimental act&es allowed us to define the flight experimental sequence. Once the facility is in microgravity (100 seconds after MAXUS-S launch), the curtain system allows one to have a stable and constant temperature gradient, and there~orc to inject drop under controlled tempemtum conditions. Du&g the fast nm a methanol drop of diameter 0 = 5 mm is fad in the cyclohexane matrix. In order to study the occmmm~ of plume observed on ground and eventuaJly the onset of its instability, the drop is kft anchord to the injection needle for two minuk The drop is then released by activating a solenoid. Tim migration and the further dissolution are studied until the drop reaches the hot wall. The two successive runs foresee tire injection of two larger drops (0 = 10.15 mm). The idea is to study again if any plume develops, the migration for larger radius, and. evemually. the interaction between dissolving drops. The interferometric data will be recorded on board and then available atIer the mission for correlation with the numerical results. Since these results predict the formation of a wake along the migration path, the shearing direction has been set from the left to the right flange. CONCLUSIONS In the present paper we report on the ongoing
53 (2003) 299-308
research activities for the preparation of an experiment on Marangoni migration of a dissolving drop to be carried out onboard the MAXUSsounding rocket. The already developed 2D planar version of the numerical code has been upgraded to treat axis-symmetric geometries. Many improvements of the code have also been implemented to increase accuracy and to reduce computational load. REFERENCES 1. Bassano E., Castagnolo D.: “Marangom migration of a methanol drop in cyclohexane matrix in a closed cavity”. Accepted for publication on Microgravity Sci. and Tech. 2. Bassano E.: “Numerical simulation of thermosohnal-capillary migration of a dissolving drop in a cavity”. Submitted to Int. J. of Nwn Mel forfluids. 3. Bassano E., Castagnolo D., Albano F.: Thefmocapillaty two-phase flow of a binary mixture drop exhibiting a miscibility gap. IAFOl-5.4%). 52nd IA% Congress, Tolouse, France, 1-J Oct. 2001. 4. MARS Center, “Bread-Board Activities Test Report for Drop Dissolution and Migration Experiment”, MARS Internal Report, 2001. 5. Predel B., Ratke L.. Fredrikmon H., “Fluid science and material science in space”, Walter H.U. Ed., Springer, Berlin 1987. 6. Qsher S, Sethian J. Fronts propagating with - curvature-dependent speed: algorithms based on Hamilton-Jacobi formulation. J. Camp. Phys. 1988; 79; 12-49.