Part 10

DELETERIOUS CONVECTIVE FLOW ARISING IN MICROGRAVITY EXPERIMENTS

Abstract. Residual gravity always exists at any space vehicle. The residual accelerations, which are taken into account in the calculations, correspond to the values of accelerations typical for an experiments on the satellite FOTON due to the atmospheric drag and the stabilizing rotation of the platform. The 3D calculations give that the maximum velocity of the flow, arising due to residual gravity is varying in the limits 0.2−4.05·10⁻³ mm/s. The presence of a single bubble in the experimental cell can cause strong flow due to the Marangoni force acting on the gas-liquid interface. 3D simulations have been done for a four different bubble models taking into account both effects: Marangoni and buoyancy. Analyzing the temperature distribution across the cell one can discover the nature of the deleterious flow.

Very often the efficiency of material processing and the quality of grown crystal strongly depends on the behavior of a liquid phase. It is thus of great importance to understand the fluid motions and related phenomena. But during the last decades more and more the space experiments are aimed to measure different phenomena, especially thermo-physics properties of fluids, (e.g. SCCO, DCCO, JET). In these experiments any convective flow in the experimental cell is deleterious for the desirable results. Surely, it is almost impossible to exclude totally gravity-induced convection, since there is always a residual gravity due to the atmospheric drag and the stabilizing rotation of the platform. But one can predict and calculate the possible influence of natural convection on the experimental data as the values of residual gravity are known. One of the first survey on this subject has been done by Ostrach [84] concerning space experiments on American vehicles. In 2D cylindrical case Dressler [28] has analytically obtained the formula for estimation transient convective velocities for low Rayleigh numbers. So, one of the aims of this study is to estimate more accurately the intensity of the flow induced by residual gravity. The effect of g-jitter due to vibration of payload itself is excluded.

Sometimes the experimental data show the presence of more intensive fluid motions, than

induced by pure buoyancy only. This can be due to Marangoni convection induced by the

unexpected presence of bubbles in the cell, and the results of current calculations show that it

may be so. The nature of the bubbles may be different. They may appear just after the injection

of liquid into the cell, or to be a result of chemical reactions in the system. The second goal

of this study is to estimate the magnitude of velocities, caused by the presence of bubbles in

order to predict their amplitudes and minimize their effects. It may be observed experimentally

that when values of the velocity in liquid are high enough, the influence of Coriolis force might

be not negligible. To clarify these points, the 3D code written for the cylindrical geometry was adapted for Cartesian co-ordinate system and validated.

The residual accelerations, which are taken into account in the calculations, correspond to the values of accelerations usually taking place on a platform with an altitude around 200-400 km, which is rotating around its main axis with a period of 600 s. These values are typical for an experiments on board the satellite FOTON. Although the residual acceleration provided by this vehicle is one of the best which is avaliable, the induced convection is not necessarily negligible for some class of investigations, e.g. when the characteristic time is large as in diffusion controlled phenomena. This is proved via simulating the TRAM experiment which was strongly influenced by the residual gravity.

MATHEMATICAL FORMULATION OF THE PROBLEM

Thermocapillary convection of an incompressible, Newtonian liquid with constant transport coefficients is considered. All the physical characteristics are taken constant, except the density which varies linearly with temperature in the buoyancy term, ρ = ρ

(1

(T

T

)) (Boussinesq approximation), where β is the thermal expansion coefficient, and surface tension σ(T ) at the gas-liquid interface, which taken as a linear function of the temperature σ = σ

σ

(T

T

).

The geometry of the cell is shown in Fig. 10.1. It reflects a real experimental situations, when observation is done through the window, which occupies only one part of the cell. Two vertical isothermal side-walls are kept at temperatures T

on the left and T

on the right, T

> T

. All other boundaries are assumed to be adiabatic.

The geometry of the cell is shown in Fig. 10.1. It reflects a real experimental situations, that observation is done through the window, which occupies only one part of the cell.

Figure 10.1: Geometry of the problem.

Here we take into account that the vector of the residual acceleration has components in any

spatial direction. The signs of g

are chosen such a way, that the direction of the component of

residual gravity is opposite to the direction of axis of the coordinates. In the Cartesian coordinate system the 3D non-dimensional Navier-Stokes, the energy, and the continuity equations are given by:

∂U

∂t + Γ

U ∂U

∂x + Γ

V ∂U

∂y + W ∂U

∂z =

Γ

∂P

∂x + ∆U + Gr

(Θ

x + 1) (10.1)

∂V

∂t + Γ

U ∂V

∂x + Γ

V ∂V

∂y + W ∂V

∂z =

Γ

∂P

∂y + ∆V + Gr

(Θ

x + 1) (10.2)

∂W

∂t + Γ

U ∂W

∂r + Γ

V ∂U

∂y + W ∂W

∂z =

∂P

∂z + ∆W + Gr

(Θ

x + 1) (10.3)

∂Θ

∂t + Γ

U ( ∂Θ

∂x

1) + Γ

V ∂Θ

∂y + W ∂Θ

∂z = 1

P r ∆Θ (10.4)

continuity equation Γ

∂U

∂x + Γ

∂V

∂y + ∂W

∂z = 0 (10.5)

where Laplacian is

∆ = Γ

∂

∂x

+ Γ

∂

∂y

+ ∂

∂z

Here the height of the cell L

is taken as a scale for the velocity, time and pressure U

= ν/L

, τ

= L

/ν, P

= ρ

U

. Let us note that L

- is the length of the cell in the direction of the temperature gradient, and L

- is the depth of the cell. Then introducing some characteristic values x

= L

, y

= L

, z

= L

two aspect ratios Γ

and Γ

appear in the system:

Γ

= L

L

, Γ

= L

L

The dimensionless temperature is introduces as Θ

= (T

T

)/∆T, here T

= T

, The linear temperature profile is subtracted from the total value Θ

to have zero boundary conditions for temperature in the direction of applied temperature gradient.

Θ = Θ

1 + x,

The equations (10.1)-(10.5) have to be solved together with the following boundary conditions:

no-slip condition on the rigid walls: V = 0,

the temperature is constant on hot and cold walls: Θ(x = 0) = Θ(x = 1) = 0, thermal adiabatic conditions are imposed on other walls: ∂Θ

∂ n = 0.

For the calculation of Marangoni convection from bubbles the stress balance on y = 0 will be used in corresponding paragraph.

The following dimensionless parameters arise in the equations:

P r = ν

k , Gr

= g

β∆T L

ν

, Re

= σ

∆T L

ρν

This mathematical approach for the choice of parameters does not reflect the real physics,

since the driving force for the flow is proportional to the ∆T /L

and not ∆T /L

. Therefore

Table 10.1: Physical properties of a mixture of Ethylene-glycol and water

T

ν β k σ

= dσ/dT ρ

C

m

/s 1/K m

/s N/mK kg/m

10 2.836

10

5.0

10

(1.1

1.3)

10

10

1058.4 20 2.34

10

5.03

10

(1.1

1.3)

10

10

1053.1

corrected Grashof and Reynolds numbers are introduced (without superscript):

Gr

= g

β∆T L

ν

L

L

₃

= Gr

/Γ

, (10.6)

Re = σ

∆T L

ρν

L

L

= Re

/Γ

. (10.7)

In the discussion of the experimental results the Marangoni number (M a = Re P r) and the Rayleigh number (Ra = Gr P r) are often used instead of Re and Gr. Therefore on the plots the values of Ma and Ra will be given.

The numerical results presented below correspond to physical values of a mixture of Ethylene- glycol and water. It is a typical mixture with viscosity around 2-3 cSt. The physical properties of this mixture are listed in Table 1.

The data, which are given in Table 10.1, are based on a PhD thesis [42]. As it follows from the table, viscosity of the liquid slightly depends upon the reference temperature. Taking into account uncertainty in the definition of thermal diffusion, it appears that a realistic value for Prandtl number is around P r = 20. This value will be used in all calculations.

The geometrical sizes of the considered cell are L

= 2

10

m, L

= 5

10

m, L

= 5

10

m. It gives the following aspect ratios: Γ

= L

/L

= 2.5, Γ

= L

/L

= 1.

NUMERICAL METHOD AND CODE VALIDATION

To solve the time-dependent Navier-Stokes, energy and continuity equations in Cartesian co-ordinate system (10.1 - 10.5), the same technique as for the case of cylindrical geometry is utilized (see Section3.4). The difference is that an explicit ADI method is applied for calculating the Poisson equation for pressure in the X,Y-directions. To solve Poisson equation in Z-direction, where all fields are uniform, the FFT (expansion only in cosine series) is used.

The calculation of the buoyancy convection is validated by quantitative comparison with the well-known De Valh Davis test [27]. To make a comparison a 2D stream function has been calculated from the 3D results in the middle of the cell in XZ-plane. One should expect a lower values of the maximal velocities in 3D case, as the influence of front and back side walls and possible motion in third direction are neglected in 2D case. Indeed, as it is written in Table 10.2, the maximal value of stream function, obtained from 3D results, is of about 4% less then the value for 2D results. We consider this favorable comparison as a proof of correctness of our code.

The Marangoni convection is validated by quantitative comparison of our 3D result with those of Saβ

[102]. The present calculation has been done on the uniform mesh 26

32

31 by described above method, and the calculation in [102] have been done on mesh 32

by multi grid method. The date from Table 1 in [102] have been recalculated in our dimensionless variables.

According the results in Table 10.3, the comparison shows good agreement. The difference

between the value for the velocity on the free surface is of 3%, whereas the differences between

Table 10.2: The values of

ψ

P r = 0.71, Γ

= 1, Γ

= 1 P r = 0.71, Γ

= 1, Γ

= 1

present work (3-D) 2-D ref.

2-D Vahl Davis

bench mark solution

Ra = 10

1.584 1.654 1.653

Table 10.3: Velocity and the temperature on the free surface at x = y = 0.5, z = 1. Results of 3D calculation: P r = 1, Γ

= 1, Γ

= 1

P r = 0.71, Γ

= 1, Γ

= 1

present work present work ref [102] ref [102]

U Θ

U Θ

Re = 10

49.392 0.696 51.0 0.655

the respective temperature fields is of 5.9%.

For higher values of Prandtl number P r = 20 the temperature near the cold corner at x = 1, z = 1 can not be completely resolved on the mesh 26

32

31 due to the spike of isotherms.

But the isotherms in the bulk are reliably calculated, which is proved by comparison with 2D solutions on fine grid.

PURE BUOYANT CONVECTION AS A RESULT OF SOME RESIDUAL ACCELERATIONS

Our attention is focused on the investigation of the convective flow, induced by residual gravity constant in time. The estimation of the convective deleterious effects is related to space experiments carried out on sounding rockets or satellites.

If the density gradient is parallel and opposite to the gravitational vector, the liquid remains in a state of unstable equilibrium until a critical density gradient is exceeded. If a density gradient (e.g. due to thermal effect) is normal to the one of the components of the gravity vector, flow results immediately. This motion appears in the form of cells or vortex rolls. As a rule, the residual gravity on space vehicle has components in all the three directions, therefore parasitic flow will always appear. To evaluate the magnitude of the velocity of this flow the present 3-D calculations are done for the amplitudes of acceleration existing on Foton-12 during the ESA experiments in FluidPac.

The present calculations are done for P r = 20, Γ

= 2.5, Γ

= 1 and the amplitude of acceleration is following [1, 120]:

g

= 3.8

10

g

, g

= 10

10

g

(10.8)

The simulation have been done for two different temperature gradients:

∆T = 10

C, and the basic parameters are taken from the Table 10.1 at the reference tem- perature T

= 10

C. It gives Ra

= 3.72, Ra

= 9.77.

∆T = 60

C, and the basic parameters are taken at the reference temperature T

= 20

C. It gives Ra

= 32.97, Ra

= 86.6.

(a) (b)

Figure 10.2: Velocity field (a) and temperature isolines (b) in horizontal XY-cross section for pure buoyant flow when Ra

= 32.97, Ra

= 86.6, Ra

= 0, (∆T /∆x = 3 K/mm).

The simulations show that the resulting flow appears in the form of symmetrical vortex rolls located in the center of the cell. The most intense motion is observed in the XY-plane in the middle of the deepness of the cell. Velocity field and temperature isotherms are shown in Fig. 10.2 in this cross section, at z = 0.5 for the ∆T = 60 K . To clarify the geometry, a small cell with the particular cross section is given on the right side of the plots. Along the hot wall the liquid is moving down, opposite to the direction of the gravity vector.

To have an idea about the magnitude of the velocity in space experiments, the values are given in m/s. The velocity is calculated as V = [V ]V

, where V

= 0.468

10

m/s. The maximum velocity V

= 4.05

10

m/s of the cellular flow in the XY-plane is observed near the rigid walls (x/L

0.25, x/L

0.75, z = 0.5).

In the case of pure buoyant flow the realistic maximal velocity in the cell and the velocity, which can be recorded through the experimental window on the top of the cell, are the same.

The small distortion of the temperature isolines is visible in the direction of applied temperature gradient: at the upper part the temperature isolines are inclined towards to the hot wall, see Fig. 10.2(b), in the bottom part to the opposite side.

The velocity field in an other cross section, shown in Fig. 10.3(a), consists of two vortexes,

center of which is shifted towards the cold wall. The maximum velocity in XZ-plane is two

orders of value smaller and achieves only 3.0

10

m/s. The distribution of temperature in this

cross section is practically linear. The isotherms in XZ-plane are almost parallel lines, therefore

to emphasize the change in temperature field the isolines of deviation from linear temperature

profile are shown in Fig. 10.3(b).

(a) (b)

Figure 10.3: Velocity field (a) and isolines of the deviation of the temperature from linear profile (b) in XZ-cross section for pure buoyant flow when Ra

= 32.97, Ra

= 86.6, Ra

= 0. The particular cross section is shown in the small cell.

Table 10.4: Numerical results for pure buoyancy induced flow

∆T = 10 K, ∆T = 60 K,

g

= 3.8

10

g, g

= 10

10

g, g

= 0.

Ra

3.72 32.97

Ra

9.77 86.6

Ra

0 0

V

m/s 0.56

10

4.05

10

The velocity flow in cross section perpendicular to the applied temperature gradient (YZ- plane), consists of four vortices, see Fig. 10.4(a), and the maximal velocity is almost one order of value less than in XY-plane, V

= 8.4

10

m/s. The most visible influence of the flow on the temperature field is observed in this plane, see Fig. 10.4(b). Due to the direction of gravity field, the liquid is hotter on the surface y = 0 than on the surface y = 1, although the thermally insulated boundary conditions are imposed on both walls.

The numerical results for the parametric study are summarized in Table 10.4. The magnitude of velocity, caused by static residual gravity, can achieve a few micron per second.

For set of parameters considered, the weak buoyancy induced flow can be caught by ther-

mocouples in the experimental cell. The positions of the ”numerical thermocouples” are given

by dotted lines in Fig. 10.2(b), they are placed in the middle of the height of the cell z = 0.5,

when y = 0.3 and y = 0.7 (see also Fig. 10.1). The temperature profiles T (x) along the ”ther-

mocouples” are given in Fig. 10.5 for buoyancy induced and mixed convection.

(a) (b)

Figure 10.4: Velocity field (a) and temperature isotherms (b) in YZ-cross section for pure buoyant flow when Ra

= 32.97, Ra

= 86.6, Ra

= 0. The particular cross section is shown in the small cell.

Knowledge of these dependencies is very important in experiments, as it allows us to have an idea about the nature of the flow in the cell. If the experimental flow is induced by buoyancy only, the thermocouples would record temperature distributions similar to those ones, shown in Fig. 10.5(a). The dotted line corresponds to the linear profile. Due to the symmetry of the flow, the profiles deviate in opposite directions from the linear one. The maximal deviation is observed in the central part of the cell, where velocities are higher. In case when ∆T = 10 K, the flow is so weak that isotherms demonstrate linear temperature distribution. With the increase of

∆T the symmetrical deviation of temperature profiles becomes more pronounced. The full lines in Fig. 10.5(a) correspond to the temperature profiles for ∆T = 60 K. It can happen that an unexpected gas bubble appears in the experimental cell. The temperature profiles for combined gravitational and Marangoni convection, shown in Fig. 10.5(b) are not symmetrical. Analyzing the shape of the profiles one can determine the side of the cell where this bubble is located. The detailed results about combined convection will be presented below.

It is expected that in future space experiments the orientation of the cell with respect to the axis of vehicle could be optimized. But due to the non-ideality of the system, it can happen, that the angle between the temperature gradient and the direction of the residual gravity is still varying between 1

3

.

The dependence of maximum velocity upon the value of angle is given in Table 10.5. With good accuracy it is a linear dependence. The maximum velocity is about 10

m/s.

COMBINED MARANGONI AND BUOYANT CONVECTION

Some space experiments registrate non-desirable flow much stronger, than those listed in Tables 10.4, 10.5. It can not be a result of buoyancy force. What are the other possible forces?

First of all, it can be Coriolis force, g-gitter and Marangoni force. Other possible sources of the

Figure 10.5: Temperature profiles along the numerical thermocouples, curve 1 corresponds to T(x, y=0.5, z=0.3) and curve 2 corresponds to T(x, y=0.5, z=0.7); (a) Pure buoyant flow. (b) Mixed buoyant and Marangoni convection.

Table 10.5: Numerical results for buoyancy induced velocities in future space experiments.

|g_∗|

=

,

=

g

= cos α

g

, g

= sin α

g

, g

= 0 α = 1

α = 2

α = 3

Ra

88.18 88.14 88.07

Ra

1.54 3.07 4.62

V

m/s 0.71

10

1.42

10

2.13

10

motion are not so obvious.

A few numerical runs have been made taking the Coriolis force into account. The analysis of the numerical results shows that the Coriolis force in combination with a buoyancy force does not change the structure of the flow, and the maximal values of velocities are varied for tens of percents. So, for the problem on hand, Coriolis force could be neglected as a counterpart of buoyancy flow.

A post-flight treatment of the results reveals the presence of the gas bubbles in the cell. The

Marangoni force, appearing in the bubble on the liquid-gas interface could be important in the

case of high temperature gradients. Therefore the values of the characteristic parameters are

calculated under assumption, that applied temperature difference is ∆T = 60 K. To model the

Marangoni flow it is assumed, that a set of bubbles is located on the rigid wall in the middle of

the XZ-plane. Bubbles can appear anywhere in the volume, but we consider the steady state,

when they achieved the rigid wall and formed one big bubble staying stuck to the wall (we

chose y = 0 location). Besides the Marangoni force acting on gas-liquid interface the same level

of residual gravity as in previous chapter is taken into account. The thickness of the bubble

is taken negligibly small, therefore the gas-liquid interface is located on the surface y = 0. At

any model described below the Marangoni force is acting only on the length of the bubble.

Correspondingly, the boundary conditions on the bubble are:

x

< x < x

, y = 0, z

< z < z

: ∂U

∂y = Re Γ

Γ

∂Θ

∂x

1

,

V = 0, (10.9)

∂W

∂y = Re Γ

∂Θ

∂z . Beyond this region either stress free conditions

0 < x < x

, x

< x < 1; y = 0; 0 < z < z

, z

< z < 1 :

∂U/∂y = 0, V = 0, ∂W/∂y = 0, (10.10) or no-slip conditions are imposed

0 < x < x

, x

< x < 1; y = 0; 0 < z < z

, z

< z < 1 :

U = V = W = 0. (10.11) A four different models are considered:

a) ”Large bubble”, when the size of the bubble is equal to the size of injection bubble;

b)”Small bubble”, when ”effective” size of the bubble is much smaller, but the same Marangoni number as in case (a)

c) Analyzing the temperature distribution in case (b) a new, more realistic Marangoni num- ber is chosen

d) The stress free conditions eq.(10.11) have been imposed beyond the bubble in all previous cases (a-c) on the surface y = 0. In this last case, we consider the wall around a bubble as a rigid one, eq.(10.12).

Case a.

Mathematically the shape of the bubble is described as a square with side length about δx = 12 mm, or in dimensionless values ∆x = x

x

= 0.6, ∆z = z

z

= 0.24. This size have been chosen after looking through some experimental records. At the moment of the filling of the cell by a liquid the injection bubble of this size have been observed. How to evaluate the Marangoni number on the bubble using the data of Table 10.1? The temperature drop per 1mm in the cell in x-direction is δT = ∆T /L

= 3 K/mm. For calculating the Marangoni number the linear temperature profile is assumed along the bubble, but the temperature drop is chosen smaller in magnitude δ

T = 1K/mm. Taking a characteristic size of bubble ∆x = 12mm, we calculate the temperature drop δx

δ

T on the length of the bubble and the Marangoni number is

M a = σ

∆T L

ρ

k ν

= σ

δx

δ

T

δx

ρ

k ν

= 74 000. (10.12)

The distribution of Marangoni force along the

is shown in Fig. 10.6(a). The same distribution is imposed in both directions x and z. It is a cosine dependence, and a maximal value is achieved only in one point. As it was mentioned above this Marangoni force acts only on the length equal to the size of the bubble. If we allow the maximal value of Marangoni force to act on the whole surface y = 0, non-locally, the flow will be turbulent. That is beyond experimental expectations.

Case b.

If the filling of the cell is properly done, the bubbles of smaller size can appear as

a result of some chemical reaction, e.g. corrosion. The second model is proposed, when the

(a) (b)

Figure 10.6: Distribution of the Marangoni number along the bubble. Large bubble case.

”effective” size of the new bubble is 5.5 mm, (∆x = 0.275, ∆z = 0.11.) The term ”effective” size is used due to the fact, that it can be a few tiny bubbles on the rigid wall. The temperature drop along the interface is δ

T = 2K/mm. Actually, it corresponds to the same value of Marangoni number M a = 74 000. But in this case of

the Marangoni distribution, shown in Fig. 10.6(b), is different. Unlike the previous cosine dependence, here the Marangoni force with a maximal value acts within some region in the center of XZ-plane.

The temperature fields in XY-plane for z = 0.5 and on the surface with gas-liquid interface y = 0 are shown in Fig. 10.7 for the ”large” bubble and, in Fig. 10.8 for the small bubble. The size of the bubble does not change the qualitative picture of temperature distribution in this plane. The temperature field in XY-plane, being uniform far away from the bubbles, becomes strongly distorted near the liquid-gas interface. The positions of the experimental thermocouples are shown by dotted line in Fig. 10.7, 10.8. The temperature profiles along the two different set of thermocouples are not the similar. They are shown for the large, Fig. 10.9(a), and small Fig. 10.9(b) bubbles, respectively. Unlike the case of pure buoyancy, both profiles are deviated from the linear one to the same side, below linear profile. The thermocouples, located far away from the bubbles, exhibit much less deviation from linear profile. The temperature behavior along the raw of thermocouples (curves 1)on the side of the gas-liquid interface, strongly depends upon the size of the bubble, although they correspond to the same Ma number. Definitely, analyzing the experimental records on thermocouples it is possible to determine on which side the bubbles are located. Comparing experimental and numerical results, it could be possible to evaluate the size of the bubbles.

Case c.

From the temperature distribution on the surface with gas-liquid interface y = 0,

(plane XZ in Fig. 10.7, 10.8), it follows that along the bubble the temperature distribution is

almost uniform and Marangoni number is really less than it was estimated from the first attempt.

Figure 10.7: Combined thermocapillary and buoyant convection. Large bubble case, 1.2mm, M a = 74 000. Temperature fields in mid-crossection Z = 25mm and on the liquid-gas interface where the bubble is situated.

In the both cases, despite of the big difference in size, the bubble is surrounded by the same isotherms: T /∆T = 0.72 and T /∆T = 0.8. Using this information, we have recalculate the value of Marangoni number. The numerically obtained temperature drop ∆T = 4.8K has been taken along the small bubble, which gives us a smaller Marangoni number M a = 32 300. A new set of 3D simulations has been performed for the small bubble with distribution of Marangoni force, equivalent to those, shown in Fig. 10.6(b).

The results of the 3D calculation have confirmed the correct choice of temperature drop.

The temperature fields for this case are shown in Fig. 10.10, and the temperature profiles along the thermocouples look almost the same as in the case of small bubble and M a = 74 000. The maximal velocity is reduced up to 0.756 mm/s and the maximal velocity visible in window is 0.03 mm/s.

Case d.

Within the considered range of parameters, the maximal velocity is much higher in combined

convection, than in the case of pure buoyant flow. The presence of Marangoni force completely

changes the structure of the flow field. The vortex flow in the plane ZY (at which the records are

done) is non-symmetrical, the center is strongly shifted towards the bubble. The maximum of

velocity is located near the surface of the bubble. Therefore the maximal velocity visible in the

experimental window is 20-30 times lower, than the absolute maximal value. But these values,

obtained in field of the view of the window, are coherent with some experimental records. The

numerical results for those 3 cases, described above, are summarized in Table 10.6.

Figure 10.8: Combined thermocapillary and buoyant convection. Small bubble case, 5.5mm, M a = 74 000. Temperature fields in mid-crossection Z = 25mm and on the liquid-gas interface where the bubble is situated.

(a) Large bubble case. (b) Small bubble case.

Figure 10.9: Combined Thermocapillary and Buoyant Convection. Small bubble case,

5.5mm. Temperature fields in mid-crossection Z = 25mm and on the liquid-gas interface

where the bubble is situated.

Table 10.6: Numerical results for combined Rayleigh and Marangoni convection.

=

=

=

=

[V]

Ra

= 32.97 Ra

= 86.6 Ra

= 0 Case a Case b Case c Case b/d Conditions

beyond the bubble stress free stress free stress free rigid wall Size of the bubble 12 mm 5.5mm 5.5 mm 5.5 mm

M a 74 000 74 000 32 300 74 000

V

mm/s in the bulk 1.32 1.847 0.756 0.994 V

mm/s in the view

through window 0.07 0.052 0.03 0.047

Figure 10.10: Combined thermocapillary and buoyant convection. Small bubble case,

5.5mm, M a = 32 300. Temperature fields in mid-crossection Z = 25mm and on the

liquid-gas interface where the bubble is situated.

Table 10.7: Comparison of the results for combined convection with different Rayleigh numbers.

Ma

=

=

big bubble, 12mm small bubble, 5.5mm Type of distribution of Ma

force along the bubble cosine step

Ra

67.66 0 67.66 0

Ra

67.66 88.19 67.66 88.19

V

in the cell 1.32 mm/s 1.34 mm/s 1.847 mm/s 1.93 mm/s V

in the field of view

through the window 0.07mm/s 0.05 mm/s 0.052 mm/s 0.05 mm/s

Comparing the order of velocities in Table 10.3 and Table 10.6, one can draw a conclusion that combined flow (Ma + Gr) is mainly Marangoni driven. Although, the set of calculations has been done when the residual gravity was directed only parallel to the temperature gradient.

In this case of pure buoyancy and such direction of residual gravity the rest state would be stable up to Ra

1700. The numerical data, summarized in Table 10.7, demonstrate a minor difference in the values of maximal velocity. The flow fields do not reveal any visible difference.

The analysis of the flow, visible in the experimental window, shows that some particles move according parabolic trajectories. Actually, such type of motion is observed on some experimental records.

CONCLUSIONS TO PART 10

The main goal of the microgravity experiment TRAMP is to prove the existence of the thermal pressure. Due to this phenomenon it is supposed to observe the motion of tiny particles in the direction of temperature gradient, e.g. from hot to cold wall. Actually, this kind of motion was not observed. Instead of this, the particles preferably move perpendicular to the temperature gradient according complex trajectories. To shed light on the possible forces, inducing such unexpected behavior, a 3-D numerical study has been performed.

Rayleigh convection.

Taking into account the level of the residual acceleration during the Foton-12 mission, the results of the 3-D simulations show that the velocity of the flow induced only by buoyancy is be very small. Depending upon the applied temperature difference ∆T = 10

60

C, the values of the maximal velocity are varying in the limits 0.6

4.05

10

mm/s.

In case of pure buoyant flow the simulations show that cellular flow must have a maximum of

velocity located about 4

5mm away from the hot and the cold walls. The center of the vortex

should be located in the center of the experimental window. Because of very small velocities and

symmetry of the flow, the distribution of the temperature along the two raws of thermocouples

should be almost linear. The experimentally recorded results exhibit much higher velocities

in comparison with a numerical values in the case of pure buoyancy. To estimate accurately

the experimental velocity is a difficult task, therefore a direct comparison of values of velocities

could has some uncertainty. But even the qualitative analysis of the pictures gives an idea

about the structure of the flow field. As the flow in TRAMP cell is not symmetrical and in the

field of view instead of a vortex flow, parabolic (or more complex) motions of the particles are

observed, it is a sign of the existence of Marangoni force on one of the walls of the experimental

cell. Experimental data from thermocouples confirm existence of another force of motion than buoyancy.

TRAMP-2.

It is supposed that in future space experiments (e.g. TRAMP-2 ) the only parasitic flow will be some buoyancy induced. The orientation of the cell could be optimized, but due to the non-ideality of the system, it can happen, that the angle between the temperature gradient and the direction of the residual gravity is still varying between 1

3

. 3D simulations show that it will result in convective flow with a maximal velocity 0.7−2.1·10

mm/s. Therefore it is worth to carry out space experiment under such conditions only in case, when the maximal velocity of the expected phenomenon is one order of value larger than the buoyancy induced flow e.g.

10

mm/s.

Combined convection.

After the processing of the experimental results a few gas bubbles were found in the system. 3D simulations have been done for a few cases taking into account both effects: Marangoni and buoyancy. There is some ambiguity in the modeling of Marangoni force from bubble. Calculations have been done for two different sizes of bubbles (free surfaces) in the middle of the XZ plane and 2 different Marangoni numbers. The first Marangoni number has been estimated as following: in the case of a linear temperature profile the temperature drop in z-direction is δ = 3 K/mm, choosing a characteristic size of bubble ∆x, we calculate the temperature drop ∆x

δ on the length of the bubble and as a result the Marangoni number is M a = 74 000. This Marangoni force is acting only on the length equal to the size of bubble.

The maximal velocity in this case is much higher, than in case of pure buoyant convection V

= 1.85 mm/s. But this maximum velocity is located near the surface of the bubble. The vortex flow in the plane ZY (at which the records are done) is non-symmetrical, the center is strongly shifted toward the bubble. Therefore the maximal velocity visible in the experimental window is much less, V

= 0.05

0.07 mm/s. This value is coherent with some experimental records. The simulations give, that the flow structure which can be visible in the experimental window, is the flow with a parabolic trajectories. Analyzing the temperature distribution as a post-fact we have found that the temperature distribution along the bubble is not linear, it is almost uniform. Taking this temperature drop on the size of bubble ∆x, we have recalculated the Marangoni number and a new set of calculations for M a = 32 300 were performed. The magnitude of the maximal velocities is almost twice less, V

= 0.756 mm/s in the cell and V

= 0.03 mm/s in the window. The flow structure remains the same.

Of course, the position and the size of the bubble are not known a priori. From the temper-

ature distributions along the thermocouples it is possible to say at which wall it is located and

approximately to estimate the size of the bubble. The maximal deviation of the temperature

from a linear profile in Z-direction is observed on the side, on which the bubble is located.