Periodic axisymmetric vortex breakdown in a cylinder with a rotating end wall

When the fluid inside a completely filled cylinder is set in motion by the rotation of the bottom end wall, steady and unsteady axisymmetric vortex breakdown is possible. The onset of unsteadiness is via a Hopf bifurcation.

Figure 1 is a perspective view of the flow inside the cylinder where marker particles have been released from an elliptic ring concentric with the axis of symmetry near the top end wall. This periodic flow corresponds to a Reynolds number Re=2765 and cylinder aspect ratio H/R=2.5. Neighboring particles have been grouped to define a sheet of marker fluid and the local transparency of the sheet has been made proportional to its local stretching. The resultant dye sheet takes on an asymmetric shape, even though the flow is axisymmetric, due to the unsteadiness and the asymmetric release of marker particles.When the release is symmetric, as in Fig. 2, the dye sheet is also symmetric. These two figures are snapshots of the dye sheet after three periods of the oscillation (a period is approximately 36.3 rotations of the end wall). Figure 3 is a cross section of the dye sheet in Fig. 2 after 26 periods of the oscillation. Here only the marker particles are shown. They are colored according to their time of release, the oldest being blue, through green and yellow, and the most recently released being red. Comparison with Escudier's experiment shows very close agreement.

The particle equations of motion correspond to a Hamiltonian dynamical system and an appropriate.

The objective of this appendix is to provide an overview of the vehicles which perform rendezvous and capture in the ISS scenario. Of interest are the features of these vehicles which are most important for the implementation of the rendezvous trajectories and for the mating process. These are:

• masses and inertias of the vehicles;

• actuation means and force/torque capabilities;

• location of thrust engines;

• features of the vehicle geometry related to rendezvous and capture issues, such as size and shape of main body and appendages;

• type and location of mating devices;

• type of rendezvous sensors;

It is not the intention to give here a detailed and exhaustive description of all these vehicles, which may anyway undergo changes in the course of their development.

General information on the various vehicles can be obtained from the NASA, NASDA and ESA web-sites. Detailed information on design and history of the Russian vehicles can be obtained from NASAs ‘Mir hardware heritage’ (Portree 1995). Information on all aspects of space stations can be found in Messerschmid & Bertrand (1999). Some information contained in this appendix has been extracted from technical reports and specifications of the International Space Station Programme for the Station and its visiting vehicles (NASA 1999), and some has been obtained by verbal communication from specialists involved in the development of the various spacecraft.

The objective of this chapter is to provide a basic understanding of the dynamic and kinematic processes which are taking place during docking or berthing of two vehicles, and to give an overview of the design principles used for docking and berthing mechanisms. Design driving requirements for these mechanisms are briefly discussed, and an overview of existing mechanism developments is given. The dynamic processes of contact and capture at docking are discussed using a simple model of an equivalent mass, which represents the masses of both spacecraft plus a central attenuation system. Basic functional concepts of the design elements used for shock attenuation, capture, structural connection and sealing are discussed at the end of the chapter.

Basic concepts of docking and berthing

The main tasks and issues arising during docking and berthing have already been addressed in section 2.5. Definitions of the terms ‘docking’ and ‘berthing’ have been given in chapter 1. For completeness of this chapter, these key definitions shall be recalled here.

• As a general term for the process of achieving contact, capture and connection, the term mating is used. This includes the two cases ‘docking’ and ‘berthing’.

• The term docking is used for the case where the GNC system of the chaser controls the required vehicle state parameters necessary to ensure that its capture interfaces enter into those of the target vehicle, and where the capture location is also the location for structural connection.

• […]

The objective of this chapter is to explain the requirements for trajectory safety, to discuss the causes for trajectory deviations due to the orbital environment and to imperfections and errors of the onboard system, and to investigate the possibilities of employing protection against trajectory deviations. The discussions concerning trajectory deviations and trajectory safety concentrate on the rendezvous phases, since the mission phases of launch and phasing are generally controlled by operators or computer functions on ground. In the rendezvous phases the two spacecraft are relatively close together, their orbital planes are well aligned and the trajectory of the chaser, by definition, leads toward the target, so that any deviation from the planned trajectory can potentially lead to a collision, directly or after one or more orbital revolutions.

Trajectory safety – trajectory deviations

Rendezvous and docking is in fact a ‘planned collision’ of two spacecraft, which is controlled by considering the geometric location of the contact points on the two vehicles and the linear velocities and angular rates at contact. To achieve the contact conditions within the allowed margins, the trajectories have to be maintained within close tolerances prior to contact. Any deviation from such tolerances may lead either to a loss of the rendezvous and mating opportunity or even to the danger of collision of the two spacecraft at unsuitable points and dynamic conditions, with the risk of serious damage. For this reason, rendezvous operations, and all functions and systems involved in them, have to comply with failure tolerance and safety requirements.

In this chapter the basic equations for the calculation of orbits and trajectories are given, and the properties of the most important types of trajectories used in rendezvous missions are discussed. In sections 3.1 and 3.2 the reference frames are defined and the laws of motion in elliptic and circular orbits in the ‘orbital plane’ coordinate frame are addressed. Equations of motion, expressed in this frame, are conveniently used during launch and phasing operations. In sections 3.3 and 3.4, the trajectories between chaser and target vehicle which are used in the far and close range rendezvous approaches are discussed. They are treated as relative trajectories in the ‘local orbital frame’ of the target. Only the ideal undisturbed trajectories are looked at in this chapter, and the necessary velocity changes, or continuous forces to be applied and the resulting position changes, are derived for ideal cases. The major sources of trajectory disturbances are addressed in chapter 4.

Reference frames

The purpose of this section is to define the coordinate frames used in this book for the description of the orbital motion, for absolute and relative trajectory and attitude motions and for the relations of these motions to geometric features on the spacecraft. Each frame Fi is defined by its origin Oi and a set of three orthogonal vectors a1, a2, a3.

The material presented in this book provides a general overview of the major issues related to the development of automatic rendezvous and docking systems, without restricting the discussion to any particular project. It is intended to explain the general principles, and examples of actual developments are included only to demonstrate these general principles. Because of the large number of aspects to be discussed, the depth of discussion of each single issue will necessarily be limited and cannot go further than an introduction.

The information presented is based on the experience of the author, gained during his work with the European Space Agency (ESA), where, between 1981 and 1998, he was responsible for the development of rendezvous and docking technology. ESA has conducted a comprehensive development programme, within which it has awarded to European industry a large number of study and development activities to prepare the rendezvous and docking techniques and technology, first for the Hermes–Columbus Free-Flyer scenario, which was abandoned in 1992, and thereafter for the ATV–ISS scenario. The Automated Transfer Vehicle (ATV) is one of Europe's contributions to the International Space Station (ISS) Programme. In this context, the two largest technology development activities, among many others, were:

• the Rendezvous and Docking Pre-Development Programme for Hermes–Columbus (1989–1993),

• the ATV Rendezvous Pre-Development (1994–1998).

The design and development of the automatic rendezvous control system of the ATV, for which these two activities formed the basis, are driven to a large extent by the interfaces and requirements given by the ISS.

The purpose of this chapter is to give the reader a short overview of the different phases of a rendezvous approach and to describe the major issues of these phases. It is hoped that it will be easier, after familiarisation with the basic concept of a rendezvous mission, for the reader to put the information given in the subsequent chapters into their proper context. For this reason, some of the information provided in more detail in the later chapters had to be duplicated in condensed form here.

A rendezvous mission can be divided, as indicated in figure 2.1, into a number of major phases: launch, phasing, far range rendezvous, close range rendezvous and mating. During these phases, the kinematic and dynamic conditions that will eventually allow the connection of the chaser to the target spacecraft are successively established. In the following sections of this chapter an overview of the objectives, the end conditions to be achieved and the trajectory implementation possibilities of each of those phases will be given. This includes a rough order of magnitude of the major performance values which the guidance, navigation and control system of the chaser will have to achieve. For completeness, a short section on departure has been added, which addresses the issues and constraints of separation from and moving out of the vicinity of the target station. The mission phases between mating and departure and after departure are not addressed as they are both, in objective and concept, fully independent of the rendezvous mission.

The major natural and technical features and constraints which (along with trajectory safety) are the driving forces behind the design of the approach strategy will be discussed in this chapter. The consequences on trajectory elements and approach strategy for the various natural and technical issues will be indicated. Trajectory safety remains the over-riding requirement; this always has to be kept in mind when discussing all other potential design drivers. Three examples of approach strategies with different constraints are discussed at the end of the chapter, for which, within the context of a complete approach scenario, a detailed explanation of the rationale behind the choice of trajectory elements of the different rendezvous phases is provided.

Overview of constraints on the approach strategy

The most important disturbance which has to be taken into account in the launch strategy is the drift of nodes due to the J2-effect, described in section 4.2.2. Because of the difference in orbital altitude, this drift will be different for chaser and target over the duration of the approach. The difference will therefore have to be compensated for by corrective measures during launch and phasing. The phasing strategy is mainly driven by the difference in position between the target station and the chaser vehicle after launch and by the required arrival time at the target.

The subject of this chapter is the discussion of the measurement principles of sensors for relative navigation, required in the far and close range rendezvous phases to measure the relative state between the chaser and target vehicles. In the rendezvous phases proper (see figure 2.1), the accuracy of absolute navigation will no longer be sufficient. With one exception, sensor principles for absolute navigation will not be discussed here, since the measurement principles for absolute attitude and absolute position for spacecraft applications can be considered well-known. Measurement and control of absolute attitude is a feature of practically every spacecraft. Onboard measurement of absolute position is required, e.g., in Earth observation missions, where receivers for satellite navigation and for ground-based radio-positioning systems, e.g. DORIS (Carrou 1995), are accommodated on the spacecraft. In most other missions, absolute orbit and position determination is usually done by observations from ground, since, in the majority of cases, mission requirements do not justify the accommodation of an absolute position sensor aboard the spacecraft.

The above-mentioned exception, to be described in this chapter, comprises the basic functional principles of absolute position measurement by satellite navigation. At the time of writing, GPS and GLONASS are the satellite navigation services used, and, for the purpose of rendezvous navigation, the navigation results of, e.g., GPS receivers w.r.t. an Earth-fixed coordinate frame, are termed absolute GPS.

This chapter addresses the tasks and responsibilities of all parties outside the automatic onboard system involved in the control of a rendezvous mission. It looks at the hierarchy of authority, the support functions required and the constraints imposed by the communication links. Tasks and design principles of support tools for human operators are indicated.

As discussed already in chapter 6, in an Earth orbit there is no need to conduct the rendezvous and docking process fully autonomously. The interaction by external operators is, on the contrary, always desirable, when this will reduce the complexity of the system and increase safety and success probability. On the other hand, because of the limitations of the communication links, the complete control of the rendezvous and docking process cannot be performed entirely from ground. For this reason, the onboard control system of unmanned spacecraft must be able to perform automatically in the vicinity of the target vehicle the control tasks discussed in chapter 6:

• the control of the spacecraft state (attitude angles, position, velocities and angular rates);

• the sequencing of manoeuvres and modes at the right time and points of a trajectory;

• the detection of, and recovery from, anomalies and failures;

• in the case of docking, sequencing and control of mating operations.

A number of high level control tasks can be performed better by remote human operators, who can contribute the human capabilities of recognition and assessment of unpredicted situations, together with the much larger resources for information gathering and data processing than are available to the onboard system.