Group coordination and cooperative control of steered particles in the plane

The paper overviews recent and ongoing efforts by the authors to develop a design methodology to stabilize isolated relative equilibria in a kinematic model of identical particles moving in the plane at unit speed. Isolated relative equilibria correspond to either parallel motion of all particles with fixed relative spacing or to circular motion of all particles about the same center with fixed relative headings. © Springer-Verlag Berlin Heidelberg 2006.

Oscillator models and collective motion: Splay state stabilization of self-propelled particles

This paper presents a Lyapunov design for the stabilization of collective motion in a planar kinematic model of N particles moving at constant speed. We derive a control law that achieves asymptotic stability of the splay state formation, characterized by uniform rotation of N evenly spaced particles on a circle. In designing the control law, the particle headings are treated as a system of coupled phase oscillators. The coupling function which exponentially stabilizes the splay state of particle phases is combined with a decentralized beacon control law that stabilizes circular motion of the particles. © 2005 IEEE.

Low-order modelling of the response of ducted flames in annular geometries

The adoption of lean premixed prevaporised combustion systems can reduce NOx emissions from gas turbines, but unfortunately also increases their susceptibility to thermoacoustic instabilities. Initially, acoustic waves can produce heat release fluctuations by a variety of mechanisms, often by perturbing the equivalence ratio. If correctly phased, heat release fluctuations can subsequently generate more acoustic waves, which at high amplitude can result in significant structural damage to the combustor. The prediction of this phenomenon is of great industrial interest. In previous work, we have coupled a physics based, kinematic model of the flame with a network model to provide the planar acoustic response necessary to close the feedback loop and predict the onset and amplitude of thermoacoustic instabilities in a lab-scale, axisymmetric single burner combustor. The advantage of a time domain approach is that the modal interaction, the influence of harmonics, and flame saturation can be investigated. This paper extends this approach to more realistic, annular geometries, where both planar and circumferential modes must be considered. In lean premixed prevaporised combustors, fluctuations in equivalence ratio have been shown to be a dominant cause of unsteady combustion. These can occur, for example, due to velocity perturbations in the premix ducts, which can lead to equivalence ratio fluctuations at the fuel injectors, which are subsequently convected downstream to the flame surfaces. Here, they can perturb the heat release by locally altering the flame speed, enthalpy of combustion, and, indirectly, the flame surface area. In many gas turbine designs, particularly aeroengines, the geometries are composed of a ring of premix ducts linking a plenum and an annular combustor. The most unstable modes are often circumferential modes. The network model is used to characterise the flow response of the geometry to heat fluctuations at an appropriate location, such as the fuel injectors. The heat release at each flame holder is determined in the time domain using the kinematic flame model derived, as a function of the flow perturbations in the premix duct. This approach is demonstrated for an annular ring of burners on a in a simple geometry. The approach is then extended to an industrial type gas turbine combustor, and used to predict the limit cycle amplitudes. Copyright © 2012 by ASME.

Nonlinear phenomena in thermoacoustic systems with premixed flames

Nonlinear analysis of thermoacoustic instability is essential for prediction of frequencies, amplitudes and stability of limit cycles. Limit cycles in thermoacoustic systems are reached when the energy input from driving processes and energy losses from damping processes balance each other over a cycle of the oscillation. In this paper an integral relation for the rate of change of energy of a thermoacoustic system is derived. This relation is analogous to the well-known Rayleigh criterion in thermoacoustics, but can be used to calculate the amplitudes of limit cycles, as well as their stability. The relation is applied to a thermoacoustic system of a ducted slot-stabilized 2-D premixed flame. The flame is modelled using a nonlinear kinematic model based on the G-equation, while the acoustics of planar waves in the tube are governed by linearised momentum and energy equations. Using open-loop forced simulations, the flame describing function (FDF) is calculated. The gain and phase information from the FDF is used with the integral relation to construct a cyclic integral rate of change of energy (CIRCE) diagram that indicates the amplitude and stability of limit cycles. This diagram is also used to identify the types of bifurcation the system exhibits and to find the minimum amplitude of excitation needed to reach a stable limit cycle from another linearly stable state, for single- mode thermoacoustic systems. Furthermore, this diagram shows precisely how the choice of velocity model and the amplitudedependence of the gain and the phase of the FDF influence the nonlinear dynamics of the system. Time domain simulations of the coupled thermoacoustic system are performed with a Galerkin discretization for acoustic pressure and velocity. Limit cycle calculations using a single mode, as well as twenty modes, are compared against predictions from the CIRCE diagram. For the single mode system, the time domain calculations agree well with the frequency domain predictions. The heat release rate is highly nonlinear but, because there is only a single acoustic mode, this does not affect the limit cycle amplitude. For the twenty-mode system, however, the higher harmonics of the heat release rate and acoustic velocity interact resulting in a larger limit cycle amplitude. Multimode simulations show that in some situations the contribution from higher harmonics to the nonlinear dynamics can be significant and must be considered for an accurate and comprehensive analysis of thermoacoustic systems. Copyright © 2012 by ASME.

On kinematic relaxation and deposition of water droplets in the last stages of low pressure steam turbines

In the first part of the paper steady two-phase flow predictions have been performed for the last stage of a model steam turbine to examine the influence of drag between condensed fog droplets and the continuous vapour phase. In general, droplets due to homogeneous condensation are small and thus kinematic relaxation provides only a minor contribution to the wetness losses. Different droplet size distributions have been investigated to estimate at which size inter-phase friction becomes more important. The second part of the paper deals with the deposition of fog droplets on stator blades. Results from several references are repeated to introduce the two main deposition mechanisms which are inertia and turbulent diffusion. Extensive postprocessing routines have been programmed to calculate droplet deposition due to these effects for a last stage stator blade in three-dimensions. In principle the method to determine droplet deposition by turbulent diffusion equates to that of Yau and Young [1] and the advantages and disadvantages of this relatively simple method are discussed. The investigation includes the influence of different droplet sizes on droplet deposition rates and shows that for small fog droplets turbulent diffusion is the main deposition mechanism. If the droplets size is increased inertial effects become more and more important and for droplets around 1 μm inertial deposition dominates. Assuming realistic droplet sizes the overall deposition equates to about 1% to 3% of the incoming wetness for the investigated guide vane at normal operating conditions. Copyright © 2013 by Solar Turbines Incorporated.