Aerospace Science and Technology, 1997, no 3, 191-202

Abstract

R&urn6

Numerical Simulation of the Influence of Vibrations on a Free Surface Flow Under Microgravity

P. Ballereau cl), C. Gelas c2), M. Salatin (I), J. J. Sibilla (I)

(‘1 Institut a&otechnique de Saint-Cyr, 15, rue Marat, 78210 Saint-Cyr-lkole, France. (‘1 Centre National d ‘ktudes Spatiales, 18, avenue kdouard-Belin, 31055 Toulouse, France.

Manuscript received July 20, 1995; accepted March 22, 1996.

Ballereau P., Gelas C., Salaiin M., Sibilla .I. J., Aerospace Science and Technology, 1997, no 3, 191-202.

The goal of the present study is to have a better understanding of the behaviour of propellant in the “surface tension” tanks of satellites. In this paper, we present “ARGO”, a finite element code which models a free surface flow subject to small vibrations under microgravity.

This software is based on the time-dependent linear three-dimensional theory of inviscid fluids. The comparison with analytical results for simple cases helps to understand the behaviour of the capillary waves, whose velocity and wavelength are linked to the depth of liquid. Moreover, this software includes a modal analysis part which contributes to forecasting the fluid instabilities in presence of perturbations due either to pressure variations at the tank exit, or to structural movements governed by a Mathieu’s equation.

During development of this software, the numerical results have been compared with experimental results performed on the ground, principally photographs. The limits of the software and future developments are presented.

Keywords: Microgravity - Flow - Free surface - Vibrations - Simulation.

Simulation numhrique de l’influence des vibrations sur un Ccoulement ?I surface libre en microgravitk Dans le cadre des efforts mis en ceuvre en vue d’une meilleure comprehension du comportement des egols dans les reservoirs << B tension superficielle >> alimentant les sous-systemes propulsifs des satellites, cet article presente un logiciel denomme << ARGO >> qui modclise par une methode aux elements finis la reponse d’un Ccoulement presentant une surface libre lorsque cet Ccoulement est soumis B des petites excitations vibratoires.

Ce logiciel est Btabli a partir d’une modelisation mathematique en theorie linearisee d’un Ccoulement instationnaire tridimensionnel non visqueux, valid6 dans des cas simples de solutions analytiques connues ; il aide a comprendre l’tvolution des rides dont la cClCritC et la longueur d’onde sont likes a la profondeur du liquide. D’autre part, ce logiciel comporte un calcul modal qui contribue a prevoir les instabilites du fluide soumis a des perturbations; celles-ci sont dues, soit aux fluctuations de pression en sortie du reservoir, soit au mouvement d’ensemble de la structure du reservoir regi par une equation de Mathieu.

Tout au tours du developpement de ce logiciel, les rtsultats numeriques sont confront& a des resultats experimentaux obtenus en gravite terrestre, principalement sous forme de photographies. Les limites du logiciel et les perspectives d’evolution sont ainsi identifiees.

Mots-cl& : Microgravite - Ecoulement - Surface libre - Vibrations - Simulation.

Aerospace Science and Technology, 0034.1223, 97/031$ 7.00/O Gauthier-Villars

192 P. Ballereau et al.

I - INTRODUCTION

In liquid propellant systems, the propellant is stored in tanks from which it is ejected to the engines. The so-called “surface tension” tank [ 11 schematically illustrated in Figure 1 has a cylindrical/spherical shape and uses the capillarity force, predominant in orbit. The aim of our work was to gain a better understanding of how the propellant behaves in this type of tank, especially along the vanes, generally close to the walls, which guide the liquid to a reserve containing a metallic sponge.

After analysis of the problem, a mathematical model and a physical model were constructed and a software code called ARGO was developed [2, 3, 41.

Taward the exterior of the tank

Fig. 1 - Surface tension tank.

II - MATHEMATICAL MODEL

The flow is schematically illustrated in Figure 2. The propellant wettens the vane and wall of the

tank. The vibrations affecting the fluid are due to

Entrance

Exit

Fig. 2 - Modelling of a free surface flow.

fluid pressure fluctuations in the exit of the vane/wall system where the fluid is flowing and/or to structural vibrations of the vane/wall system due to the satellite.

11.1 - Simplifying Assumptions

The main simplifying assumptions used in the mathematical model were as follows:

- The flow is ii-rotational and the fluid is inviscid (i.e. there is no boundary layer) and incompressible; the flow can therefore be considered as a potential flow.

- The perturbations are limited to “small vibrations” in the neighbourhood of equilibrium of the free surface (waves with an infinitesimal amplitude)

- The vane and wall are completely rigid - The fluid/vane and fluid/wall contact angles

remain constant throughout the fluid movement. And the fluid can slip along the vane and wall.

Remark on the contact angle: An analysis of the literature shows that the dynamic behaviour of the contact angle is a complex problem for a real fluid. Yves Ousset suggested a way of investigating the dynamic behaviour of the contact angle and the influence of the speed of movement of the fluid/gas/solid contact line [5]. Other authors describe work conducted at molecular level and continuum level, In particular, Pierre-Gilles de Gennes [6] made a synopsis of these problems in a paper where he indicated in particular that macroscopic slip originates in several different phenomena occurring on smaller scales: surface roughness, fluid viscosity, chemical contamination, inhomogeneities in the solid surface and matter dissolved in the fluid (surface-active agents, polymers, etc.)

A few remarks about the scope of validity of this assumption will be given during the discussion on experimental validation of the numerical model.

II.2 - Local Equations

The fluid movement is governed by local equations. These equations, resulting from a synopsis of the work conducted by the Russian school (Moiseyev) on the linear theory of small oscillations of a fluid mass subjected to the action of surface tensions, were presented by Zeytounian [7] in the 1960s and more recently by other authors, especially Morand and Ohayon [S]. Two problems are discussed below: the unsteady problem and the spectral problem.

11.2,1 - Unsteady Problem Below, we consider a velocity potential cp

fluctuating around the mean flow potential (p. The fluid is considered potential. Considering the mass conservation principle, the

analysis of the disturbed flow requires solving a boundary problem for Laplace’s equation in fluid volume Ro presented in Figure 3 in the case of a rectilinear vane.

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Fig. 3 - Fluid volume Ro (rectilinear vane).

Equation in @id volume !&I

Ap = 0 in s2a, (1)

where fla is the domain occupied by the fluid at equilibrium.

We define surface dRa, boundary of domain Ra, by:

where C,: fluid entrance

C,: fluid exit c,: fluid contact area with the solid structures

(wall and vane) Fe: free surface y. (X’o = yo) denotes the contact line of the three

media: gas/fluid/solid. 6 denotes the unit normal pointing outward from

surface X20. Unit vector $ gives the direction of the acceleration

incurred by the vane/wall system in motion. This acceleration may be due to a combination of gravity and structural movements.

According to the equivalence principal, it can be considered that the acceleration applied to the vane/wall system in motion in a reference frame assumed inertial can be represented by a time- dependent gravity field applied to the fluid with the vane/wall system motionless.

In the mathematical analysis given below, the fluid motion is determined assuming the vane/wall system to be in an inertial reference frame and subjected to a mean gravity which can be zero or the Earth’s gravity or any other gravity (in all cases, the position of equilibrium of the free surface must be determined before calculation). 1997, no 3

Two cases can occur:

- There may be no fluctuations around the mean gravity. This is the case in our application for surface movements due to operation of the sponge; or

- There are fluctuations around the mean value. In our application, this corresponds to surface movements due to satellite vibrations.

Boundary Conditions

A. - On the walls c,:

Since the motionless impermeable walls impose v’.n’= 0

dp 0 z=. (2)

This is a homogeneous Neumann type boundary condition.

B. - In the entrance C,:

where V, (t) are velocity fluctuations in the entrance with respect to time t. This is an inhomogeneous Neumann type boundary condition.

C. - In the exit C,:

cp = cpw (4)

This condition actually corresponds to imposing a pressure p(t) in the exit, since p is related to cp by the linearised Euler equation expressing the momentum conservation principle for a potential flow.

It is a an inhomogeneous Dirichlet type boundary condition.

D. - On free surface l?a:

The scalar field of normal velocities n defined on equilibrium surface l?a gives us a kinetic representation at all times of the deformed area denoted I’. This description is obtained by solving the equation on surface r0 associated with its boundary conditions.

194 P. Ballereau et aZ.

Equation on free surface ITO

Integrating the linearised Euler equation yields the unsteady Bernoulli equation.

where: p: fluid density 6P: capillary pressure jump on either side of the

free surface g: acceleration in a gravity field 6~: fluid height depending on the direction of the

gravity vector. Differentiating with respect to time yields:

d2P %P GF+ at

-+pgg=o. (7)

Since:

and considering unit vector L showing the direction of the gravity field, then:

At time t, Laplace’s law applies for the instanta- neous curve radii R& and Rb:

(P+sP)-P,,,=cr (&+&) onr. (13)

This yields an expression for the differential pressure SP:

6P=u($+$/J -k}. (14)

We then need to calculate the term:

This term is expressed as a function of the normal displacement field [ of surface Ia, by the following linear approximation [X] :

($+&) -j$= -j&-&t- (16)

equation (8) becomes:

We can therefore write:

a2 P dSP ~ __- p at2 + at

pg (F CpX) = 0 on Ia. (10)

This last equation can be expressed in (cp, v) by introducing the pressure condition on the free surface.

The pressure jump on either side of the fluid surface is expressed in terms of the main curve radii R, and R, of the surface separating the two fluids, in static equilibrium, by the Young-Laplace law [9], in any point of the deformed free surface I, i.e.:

where: P ext: gas pressure above the liquid C: surface tension. The pressure jump at equilibrium can be expressed

by the following equation:

-is - Pext = CJ & on r0,

where 2- denotes the mean curvature of the surface .Ro at eqmhbrium.

(15)

where a, is the Laplacian in a three-axis reference system related to free surface Ia.

If we consider the normal velocity q = 2 = z + a.+ [ and neglect the convective term by applying the small perturbation hypothesis, this yields:

g = -n(.&v+Asi). (18)

Substituting equation (18) in equation (10) then gives us the equation in (cp, q) that we were looking for:

a2 P P--O ( &+b > - pg%f.k= 0. (1% 0

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Boundary conditions imposed on the contact angle 11.2.2 - Spectral Problem

Equation (19) must be supplemented by boundary conditions on boundary X0, i.e. on contact line 70.

The contact angle condition is expressed:

The spectral problem is determined from the unsteady problem by setting the harmonic equation:

a2YJ -IZ at2 --ziT cp,

v9

where up, is the geodesic normal to y. when differentiating with respect to time (it is recalled that the contact angle is assumed constant):

(21)

This is therefore a homogeneous Neumann type boundary condition.

The boundary problem is then described by the system of two coupled equations in (cp, v):

IN FLUID VOLUME 0”

lb=01 (1)

with the following boundary conditions:

dP 3n=O on C P

h

F-L= -v, (t) on 1,

cp= v(t) on C,

% dn = rl (t) on r0

ON FREE SURFACE r.

(2)

(3)

(4)

(5)

pg$o(++4sn) -P!l(t)ib.G=o (19)

with the following boundary condition

87 -=O on70 874,

(contact angle condition) (21)

We must set the initial conditions for these two equations and their boundary conditions. 1997, no 3

then by cancelling the right-hand side of the equations.

III - NUMERICAL MODEL

It was decided to search for an approximated solving method after formulating the equation variationally, in order to allow development of a discretisation model for these equations using Gale&in’s finite element method [lo].

The elements selected are linear prismatic elements. This means they are isoparametric elements with six nodes. There is only one degree of freedom per node in the discretised domain R. (which is the velocity potential, a scalar variable). Surface I’0 is covered by a set of quadrangles called finite boundary elements.

The solving method selected for the unsteady model is a direct method. As for the eigenvalues, they are calculated by Rayleigh’s quotient method with translations used after reduction to a higher Hessenberg form. The associated eigenvectors are calculated by the inverse iterated power method.

IV - VALIDATION AND USE OF THE ARGO PROGRAMME

Before using the ARGO programme operationally, we confronted it with analytic solutions and experimental data.

IV.1 - Validation of the ARGO Programme

These experiments were conducted on a physical model subjected to the Earth’s gravity, illustrated in Figure 4.

Three fluids were tested: water, special petrol (“essence G”) distilled between 45°C and 60°C and ethanol.

The fluids were placed in a prismatic trough (1) with a rectangular or triangular section. The waves were created by a wave generator (2). A clock (8) triggered a camera (9) so as to visualise the shape of the free surface at different known times after starting the wave generator. The experiment was conducted for various wave generator frequencies.

Figure 5 shows an example of the wave motion.

196 P, Ballereau et al.

Celerity x Numerical values . Water A Essence G

1) Trongh 2) Wavcgenerator

3) Tension generator 4) Current-amplifier

5) Loudspeaker G).Accelerometer

7) Spectrum analyser 8) Clock

9) Camera

0 1 2 3 4 5 Wave-Yength ?J&

Fig. 6. - Comparison of experimental and numerical results to analytical results.

Fig. 4. - Experimental set up.

Fig. 5. - Wave motion (ethanol).

For several frequencies, the wavelengths measured during these experiments and those obtained by numerical simulation were compared with the values obtained by a 2D analytic model [9]. The analytic model shows that these surface wave are dispersive. The velocity c and the wavelength X are related by the following equation:

c2= (g+F)tanhy.

In the case of a liquid height h greater than the wavelength of the waves, the equation simplifies to:

2= (C+T).

It can be represented by the curve given in Figure 6, showing the numerical and experimental values.

Remarks

- The experiments conducted with special petrol and ethanol give a clearer image of the wave generated than those conducted with water.

The clearer image may be due to the fact that the contact angle remains constant during movement in the case of the special petrol and ethanol. Since the viscosity of water is between that of the other two fluids, the phenomenon can be attributed to chemical contamination by the fats which may be dissolved in the special petrol and ethanol but not in the water.

- Above 15 to 20 Hz, the experiments are difficult to conduct. But the waves are not very troublesome in a fuel tank, since they are rapidly damped over short distances.

It can be seen that the experimental values and the numerically calculated values agree well with the analytic model.

In the comparison between the numerical model and analytic model, the validation was also made after separating the capillarity and gravity effects.

IV.2 - Operational Use of the ARGO Code

We chose a number of results already obtained during industrial use of the ARGO code with or without gravity for inclusion in this paper.

IV.2.1 - Use in Unsteady State

The first example selected is that of wave motion under microgravity, shown in Figure 7 at time t = 0.2 s for an excitation frequency of 20 Hz. The photograph shows that the wave propagation velocity

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Fig. 7. - Evolution of the wave under microgravity.

and wavelength are different along the walls than in the middle.

Remark: The wave amplitudes are highly exagger- ated as they are in all the illustrations below.

The second example with a dihedral under gravity corresponds to a vane and fluid geometry similar to that of the initial vane/wall system. This geometry is illustrated in Figure 8.

The fluid is “arranged” in the right dihedral formed by the vane and wall. Its surface of equilibrium is flat.

Figure 9 shows propagation of a wave for two excitation frequencies, 10 Hz and 40 Hz.

It can be seen that propagation is rectilinear for the excitation frequency of 10 Hz, whereas at 40 Hz,

a) f = 10 Hz (L = 200 mm)

Fig. 8. - Geometry used

the wave propagates more slowly (with a smaller wavelength) along the sides than in the middle.

As in the case of microgravity, this phenomenon may be due to the fact that the ratio of the liquid depth to the wavelength is smaller near the edges, causing a modification of the propagation speed and wavelength. These “edge waves” have been described by Lamb [ 1 I]. This phenomenon becomes negligible at a distance such that its projection on the inclination exceeds the wavelength. Analytic solutions have been demonstrated for certain special cases, such as that of the right dihedron [ 11, 31.

IV.2.2 - Use for Modal Analysis

Another example given in Figure 10 shows the eigenmodes obtained for a 300 mm diameter tank and a vane length of 100 mm.

An example is also given for the right dihedral under the Earth’s gravity (Figure 8 for L = 50 mm).

b) .f = 40 Hz (L = 60 mm)

Fig. 9. - Motion of a wave subjected to the Earth’s gravity.

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198 P. Ballereau et al.

dn Frequency (Hz) ,-

35

30

25

20 -u-- m= 15 --U- m=l

+ m=2 10 4 m=3

5

fc 1 2 3 4 5 6 7 8 9 10 Longitudinal axis(n)

Fig. 10. - Eigenmodes of the curved vane (4th eigenmode f = 4.64 Hz, C. L. Dirichlet-Neumann). Fig. 11. - Spectrum under terrestrial conditions

Calculation of the first 40 eigenmodes then their identification from an (m, n) model of nodal lines allows characterisation of the frequency spectra. These modes are shown in the graph of Figure 11.

The modes whose eigenfrequencies are located on abscissa n = 0 (denoted fc) define exclusively transverse standing waves. These are the cut- off frequencies. For m = 0 and m = 1, the associated spectra are asymptotic when the frequency is increased. The lower part of the (m = 1) spectrum defines the modes with a central modal line. The asymptotic part of these two spectra corresponds to edge wave type modal forms described analytically

by Lamb, which actually correspond to local modes [l, 31.

Various eigenforms are given in Figure 12 for the case of a right dihedral subjected to the Earth’s gravity.

Another example is given in Figure 13. It shows a “local mode” under microgravity.

To conclude, an understanding of this modal scheme can help predict the fluid behaviour in the low frequency dynamic environment due to the sponge.

Fig. 12. - Eigenmodes under the Earth’s gravity.

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Fig. 13. - Local mode under microgravity.

IV.2.3 - Prediction of the Parametric Instabilities of the Fluid

Before going into the general problem of fluid vibration under microgravity, it is important to examine the motion of a small perturbation.

The small perturbation selected corresponds to an initial velocity of 1 mm/s normal to the surface with zero acceleration. It is located on a node in the middle of the mesh.

Figure 14 shows the surface deformation at time t = 10 ms.

In the case of structural vibration of the satellite, which actually corresponds to a variation of g over time as was explained above, a mathematical analysis shows that the free surface movement due to such vibrations can be stable or unstable [ 12, 13, 14, 151.

In effect, the fluid movement is governed by a Hill- Mathieu type parametric equation. The movement is physically triggered by a spurious perturbation which is damped or amplified.

In order to determine the frequencies corresponding to these instabilities, the fluid is agitated in a specific direction by wobbulation, i.e. by a sinusoidal signal with a constant amplitude but whose frequency varies over time. This signal deforms the surface. To analyse the behaviour of the system, the signal representing the variation over time of the normal displacement in different points of the free surface is collected.

The software allows any point to be chosen. The point selected is shown in Figure 15. The two vibration directions selected, longitudinal and transverse, are also indicated.

A frequency representation of the output signal can be obtained by a Fourier transform. The resulting amplitude spectrum is given in Figure 16.

Fig. 1.5 ~ Direction of the vibrations.

Amplitude(m) c Hanning window

xlOY

14 1,4

12 13 078 08 034

02 0

0 5 10 15 20 25 30 35 40 45 50

Frequency (Hz)

Fig. 14. - Movement of a small perturbation. Fig. 16. - Spectrum analysis (longitudinal vibrations).

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200 P. Ballereau et al.

In this case, the vibration occurs in the longitudinal direction of the vane, the initial velocity in the middle of the surface is 1 mm/s normal to the surface, and the input signal corresponds to an acceleration with an amplitude of 1 m/s2 and a frequency varying linearly between 2.1 and 72.1 Hz. It can be seen that the spectrum exhibits a large number of peaks.

The fluid motion can be visualised for excitation frequencies that are the same as the peak frequencies. An example showing the variation over time of the shape of the free surface for a sinusoidal excitation frequency of 10 Hz corresponding to a high peak is given in Figure 17.

It can be seen that the wave amplitude increases as the system accumulates energy.

t = 0.25 s

t = 0,60 s t = 0,65 s

Fig. 17. - Instabilities in the movement of the surface (longitudinal vibrations).

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Another example is chosen for a movement in the traverse direction of the vane. The spectrum is given in Figure 18.

Amplitude (m) Rectangular window

x106

136

1,4

12

1.0

078

026

034

02

0

r

!!iJ Ll

0 5 10 15 20 25 30 35 40 45 50

Frequency (Hz)

Fig. 18. - Spectrum analysis (transverse vibrations)

Two cases are considered. Excitation is by sinusoidal accelerations at a frequency of 22 Hz where there are no peaks and 34 Hz where there is a high peak. The results obtained are shown in Figure 19.

It can be seen that no instabilities occur for 22 Hz, contrary to 34 Hz.

For each of the cases examined where instability exists, the frequency resulting from the signal

f=22Hz f=34Hz

(t = 1.82 s, i.e. 40 wave generator periods). (t = 0.294 s, i.e. 10 wave generator periods).

corresponding to the normal displacement in a point of the surface is the same as the excitation frequency. However, it would be interesting to continue investigating these phenomena related to the Hill-Mathieu equations, which can give rise to other domains corresponding to subharmonic instabilities.

In the real case where the tank vibrations are arbitrary but known, it would be possible to introduce this signal and observe its variation over time to determine whether the system is stable or unstable.

V - CONCLUSION

The ARGO code is designed to numerically model free surface flows under microgravity subjected to vibrations.

The numerical model was confronted with a 2D analytic model and experimental data on a physical model “under the Earth’s gravity”.

Validation was carried out for the surface tension term which is predominant under microgravity and for the gravity term which represents the structural vibrations of the satellite since it varies over time.

The software has already been used experimentally: - To facilitate interpretation of surface wave

behaviour which is relatively complicated even when the wave amplitude is infinitesimal, since the waves are dispersive. In addition, the ratio of the wavelength to the fluid depth is involved in the dispersion equation

- In the case of the vanes installed in the tank, this allowed us to understand that the waves could propagate either rectilinearly or more slowly along the edges then in the centre

Fig. 19. - Instabilities in the movement of the surface (transversal vibrations).

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202 P. Ballereau et al.

- It can help predict certain risks which can be caused by operation of the sponge by calculating the eigenmodes

- Finally, frequencies corresponding to Hill-Mathieu type instabilities which can occur in case of structural vibration of the satellite can be determined.

REFERENCES

[l] Ducret E. - L’etude des reservoirs a tension superficielle de la flottaison neutre au vol spatial. Proceeding of the$rst European Symposium “Fluids in Space”, Ajaccio, France, 18-22 novembre 1991, ESA SP-353 (June 1992).

[2] Ballereau P. - Impact des vibrations sur un Ccoulement capillaire. Rapport &al de la com- mande 840/2/91/CNES/OO87. Note no 225/93, Institut Aerotechnique de Saint-Cyr, 1993.

[3] Sibilla J. J. - Reponse d’un Ccoulement capillaire presentant une surface libre soumis a de petites perturbations : Amelioration, Validation et Experimentation du code << ARGO >>, Memoire d’ingenieur CNAM, 1994.

[4] Ballereau P. - Amelioration de la simulation numerique de l’impact des vibrations sur un Ccoulement capillaire possedant une surface libre, Rapport final de la

commande no 84O/CNES/93/0868/00. Note no 257/94, Institut Aerotechnique de Saint-Cyr, 1994.

[5] Ousset Y. - Etat actuel de la recherche sur la dynamique des fluides en rotation, 1’” partie et 2” partie, ESA Journal, 1977, 1.

[6] de Gennes P. G. - Wetting: Statics and Dynamics, Review of Modern Physics, July 1985, 57, no 3, Part 1.

[7] Zeytounian. - Mecanique des fluides fondamentale, Springer-Verlag, 199 1.

[8] Morand H. J. P., Ohayon R. - Interaction fluide- structure, Col. Recherches en Mathematiques Appli- quees, RMA 23, Masson, Paris, 1992.

[9] Brun. - MCcanique des fluides, tomes 1 et 3, Dunod, 1970.

[lo] Dhatt G., Touzot G. - Une presentation de la methode des elements finis. Col. Universite de Compiegne, Maloine Cd., 1984.

[ll] Lamb H. - Hydrodynamics, 5th ed., Cambridge University Press, 1924.

[12] Benjamin, Ursell - The Stability of the Plane free Surface of a Liquid in Vertical Periodic Motion, Proc. Roy. Sot., 1954.

[13] Drazin, Reid. - Hydrodynamic Stability, Cambridge University Press, 1989.

[14] Soutif M. - Vibrations, propagation, diffusion, Col. Dunod Universite, Dunod, Paris, 1970.

[15] Berge, Pomeau, Vidal. - L’ordre dans le chaos, Hermann, 1992.

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