The Dynamic Behaviour of Fluids in Microgravity

W. Teichert

Product Centre Engineering, IKOSS GmbH, Stuttgart, Germany

M. Klein

Structures and Mechanisms Division, ESTEC, Noordwijk, The Netherlands

The increasingly large volumes of fluid that have to be carried aboard satellites designed for long operating lifetimes, but which often must also deliver high-accuracy pointing, mean that special attention has to be paid to the behaviour of fluids in low gravity. Several novel methods are being used at ESTEC to model the influence of the fluids on the satellite's dynamic behaviour in orbit.

Low gravity and fluid dynamics

There are several methods for analysing the dynamic behaviour of fluids under normal gravity conditions on Earth. A common assumption in such methods is that the free surface of the fluid is flat and at right angles to the direction in which gravity is acting, i.e. normal to the gravity vector. In this case, surface tension does not contribute to the fluid's dynamic behaviour. However, this assumption is no longer valid if gravity is significantly reduced, as is the case for a satellite in orbit.

An indicator of the importance of the surface tension is the so-called 'Bond number', which is a measure of the relative magnitudes of the gravitational and capillary forces. It is proportional to the gravity level, the fluid density and the square of the characteristic length of the fluid's free surface, and is inversely proportional to the surface tension of the fluid. If the Bond number is much greater than 1, the surface- tension effects can be neglected; if it is much less than 1, the gravity forces can be neglected.

To take an example, the Bond number for water in a 2.7 mm- diameter tube subject to Earth's gravity is 1, and the fluid surface in such a tube is curved. If the tube diameter is much smaller, gravity forces have no significant influence on the fluid, and the water would not flow out when the tube is turned upside down.

The gravity level on an orbiting spacecraft is very low. On a spacecraft like ESA's Eureca retrievable platform, for example, drag results in a residual acceleration of just 5x10 (exp -6) m/s 2 . Under such conditions, a fluid in a typical 1 m-diameter tank will have a Bond number of less than 1, which means that the free surface will be curved and surface tension has to be considered when modelling the fluid. In some cases, it might be possible to neglect gravity effects in the analysis, depending on the tank's diameter and the type of fluid that it holds. This would be true, for example, for a 0.6 m tank carrying nitrous oxide, for which the Bond number would be 0.1 under a Eureca-type residual acceleration.

Free-surface shapes in low gravity

Considering only surface-tension effects, the free-surface shape can be determined analytically for simple tank geometries. If the tank wall is a shell of revolution, the free-surface shape is spherical. The tank geometry, fluid volume and contact angle between tank wall and free surface are required to determine the shape of the fluid. The contact angle is mainly a property of the fluid and distinguishes wetting fluids such as water, and non-wetting fluids such as Mercury. The value of this angle is also influenced by the surface material of the tank, and it is usually difficult therefore to ascertain the contact angle exactly.

Figure 1. Typical contact angles for wetting (e.g. water) and non-wetting (e.g. mercury) fluids

So far we have been considering only tanks with an axi- symmetric inner surface. Tanks for microgravity applications often contain baffles and fluid-management devices and their effects need to be considered as a next step.

Fluid sloshing frequencies

The fluid sloshing modes of tanks subject to Earth's gravity exhibit frequencies that are usually far below the structural resonance frequencies. The first sloshing mode in a typical spacecraft tank has a frequency around 1 Hz. The frequency is approximately proportional to the square root of the gravity, and therefore decreases if the gravity level is reduced. If the Bond number becomes small, the frequency's dependence on the surface-tension increases. Analytical solutions are possible for simple problems of limited engineering value, but numerical methods are needed to model complex engineering problems.

Fluid sloshing mode frequencies in microgravity are generally very low, typically in the range 0.1 to 0.01 Hz. These low- frequency modes could interfere with the structural bending modes of large solar arrays, or with the satellite's Attitude and Orbit Control System (AOCS). As a result, such fluid effects in low gravity need to be determined as one of the more critical inputs when analysing the dynamic control of satellites.

Analysis methods

Appropriate analysis capabilities have been established at ESTEC to support evaluation of the performances of satellites with stringent pointing requirements. Facilities are available for the generation of a simplified dynamic model (few degrees of freedom) of the propellant fluid in low gravity for coupling with satellite mass and stiffness matrices as input to the satellite AOCS model. The analysis facility is based on the boundary-element program RAYON, and employs standard graphical software for model visualisation (PATRAN) and a general-purpose finite-element program (ASKA) for establishing the satellite's mass and stiffness matrices.

The analysis is conducted in three steps. As a first step, the shape of the fluid's free surface is determined. Subsequently, the mass and stiffness matrices are evaluated. Finally, the fluid modes and frequencies are computed and simplified models are derived, assuming a rigid tank. At present, the analysis of the tank geometry is limited to cylindrical tanks with elliptical end caps, a spherical tank being a special case of such a geometry.

Free-surface shape determination
As mentioned earlier, the two shape extremes are the flat free surface obtained when the surface tension is negligible, and the spherical surface in zero gravity. Numerical analysis is necessary when gravity and surface tension are taken into account simultaneously, and in most cases a non-linear iteration process is required to derive the free-surface shape. The latter is discretized using finite elements, and this idealization is employed to describe the boundary of the fluid.

Figure 2 shows the different shapes of the free surface at different Bond numbers for fluid contained in a cylindrical tank. It is possible in the analysis to define an offset angle between the tank's axis and gravity vector. The surface shapes displayed in the right-hand figure are for an offset angle of 5 degrees.

Figure 2. Meridian of the free-surface for different Bond numbers, in a cylindrical tank

Once the free surface and its contact line with the tank wall have been established, it is easy to determine the fluid boundary necessary to establish the fluid dynamics.

Generation of fluid mass and stiffness matrices
To account for the effect of the fluid on the satellite, the fluid mass and stiffness matrices are generated and implemented in the satellite mathematical model. In general, the latter is built up using the finite-element method (FEM). Similar finite-element representation of the fluid would involve a significant mesh- generation effort. Boundary-element techniques, however, facilitate the generation of the fluid matrices. A number of difficulties related to the usage of the boundary-element method (BEM) for this type of fluid processing have been solved. The combination of FEM and BEM techniques arrived at has been employed successfully to represent satellite propellant tanks, which have then been coupled into the satellite mathematical model for pointing-performance evaluations.

As an example of the analysis possibilities, Figure 3 shows the first mode of vibration of the fluid in a satellite tank.

Figure 3. The first fluid sloshing mode at 0.02 Hz

Derivation of dynamically equivalent simple models
The importance of each mode of vibration depends on the load generated at the tank interface with the satellite. The interface force depends on the effective modal mass matrix of the vibrational modes. It is possible to derive from the latter single-degree-of-free-dom models which have equivalent dynamic behaviours in terms of tank interface loads.

Mass-spring and pendulum models are two types of models which can be inserted directly into the satellite AOCS model (Fig. 4).

Figure 4. The mass-spring and pendulum models

The simplified mass-spring models are determined by the vibrating mass and its location, the spring stiffness, and the residual mass and its location. The vibration characteristics of the pendulum model are determined by the pendulum length rather than the spring stiffness, and can be used for non-zero-gravity conditions.

Conclusion

The analysis of fluid effects in satellite propellant tanks that has been presented very briefly here can be employed to support stability and pointing-performance evaluations for spacecraft susceptible to fluid-driven perturbations in the microgravity environment.

The latter include missions such as Artemis, with its large low-frequency solar arrays, microgravity and observation-type platforms like Eureca and the Polar Platform, respectively, and technology-demonstration satellites such as Sloshsat (an ESA technology demonstration project), the results from which will be used to verify the analytical predictions against in-flight measurements.

Further work is planned to extend the evaluation capabilities that have been summarised above to complex tank geometries and to cover the effects of in-tank baffles and fluid-management devices.