H. Bruce Stewart

Two-phase flow: Models and methods

Abstract A variety of two-phase flow models can be derived following a few basic principles, which are here illustrated which no more generality than is essential. Among the models derived is one already widely used in applications, even though it is ill-posed in the sense of Hadamard. Final assessment of such models remains a distant goal, but will clearly involve numerical solutions; several methods in current use are discussed with a guide to selecting the one appropriate to a particular problem.

NONLINEAR RESONANCE IN BASIN PORTRAITS OF TWO COUPLED SWINGS UNDER PERIODIC FORCING

A model of a simple electric power supply network involving two generators connected by a transmission network to a bus is studied by numerical simulation. In this model, the bus is supposed to maintain a voltage of fixed amplitude, but with a small periodic fluctuation in the phase angle. In such a case, traditional analysis using direct methods is not applicable. The frequency of the periodic fluctuation is varied over a range of values near a nonlinear resonance of the two-generator network. When the bus fluctuation frequency is away from resonance, the system has several attractors; one is a small-amplitude periodic oscillation corresponding to synchronized, quasi-normal operation (slightly swinging), while others are large amplitude periodic oscillations which, if realized, would correspond to one or both generators operating in a desynchronized steady state. When the bus fluctuation frequency approaches resonance, a new periodic attractor with large amplitude oscillations appears. Although it does correspond to a synchronized steady state, this attractor has a disastrously large amplitude of oscillation, and represents an unacceptable condition for the network. Basin portraits show that this resonant attractor erodes large, complicated regions of the basin of the safe operating condition. Under conditions of small periodic fluctuation in bus voltage, this basin erosion would not be detected by traditional analysis using direct methods. Further understanding of such complicated basin structures will be essential to correctly predict the stability of electric power supply systems.

Bifurcations in a system described by a nonlinear differential equation with delay

Computer simulations of a nonlinear differential equation with time delay have been carried out to determine the possible steady states over a wide range of parameter values. A variety of nonlinear phenomena, including chaotic attractors and multiple coexisting attractors, are observed. Precision of the solutions is verified by means of evaluating the computational error at each time step. A number of bifurcations are observed, and the involvement of unstable periodic orbits is confirmed. The phase space of the system is infinite dimensional, but nonetheless all the bifurcation phenomena observed, including the blue sky disappearance (boundary crisis) of a chaotic attractor, show geometric structures which are consistent with familiar low-dimensional center-manifold descriptions.

Fractional step methods for thermohydraulic calculation

Abstract A method of fractional steps is used to extend a semi-implicit finite-difference technique for multidimensional two-phase flow calculations. This extension permits time step size to be chosen independent of convection velocities in one space direction, and still resolves multidimensional coupled sonic and phase exchange effects implicitly by forming a simple Poisson problem for pressure. Because pure time-splitting by physical phenomena was found unsuited to thermal hydraulics problems, a stabilizing corrections method was chosen. An application in nuclear reactor safety analysis is demonstrated.

Chaos, dynamical structure, and climate variability

Deterministic chaos in dynamical systems offers a new paradigm for understanding irregular fluctuations. Techniques for identifying deterministic chaos from observed data, without recourse to mathematical models, are being developed. Powerful methods exist for reconstructing multidimensional phase space from an observed time series of a single scalar variable; these methods are invaluable when only a single scalar record of the dynamics is available. However in some applications multiple concurrent time series may be available for consideration as phase space coordinates. Here we propose some basic analytical tools for such multichannel time series data, and illustrate them by applications to a simple synthetic model of chaos, to a low‐order model of atmospheric circulation, and to two high‐resolution paleoclimate proxy data series.

BASIN CONFIGURATION OF A SIX-DIMENSIONAL MODEL OF AN ELECTRIC POWER SYSTEM

As part of an ongoing project on the stability of massively complex electrical power systems, we discuss the global geometric structure of contacts among the basins of attraction of a six-dimensional dynamical system. This system represents a simple model of an electrical power system involving three machines and an infinite bus. Apart from the possible occurrence of attractors representing pathological states, the contacts between the basins have a practical importance, from the point of view of the operation of a real electrical power system. With the aid of a global map of basins, one could hope to design an intervention strategy to boot the power system back into its normal state. Our method involves taking two-dimensional sections of the six-dimensional state space, and then determining the basins directly by numerical simulation from a dense grid of initial conditions. The relations among all the basins are given for a specific numerical example, that is, choosing particular values for the parameters in our model.

Application of fixed point theory to chaotic attractors of forced oscillators

A review of the structure of chaotic attractors of periodically forced second-order nonlinear oscillators suggests that the theory of fixed points of transformations gives information about the fundamental topological structure of attractors. First a simple extension of the Levinson index formula is proved. Then numerical evidence is used to formulate plausible conjectures about absorbing regions containing chaotic attractors in forced oscillators. Applying the Levinson formula suggests a fundamental relation between the number of fixed points or periodic points in a section of the chaotic attractor on the one hand, and a topological invariant of an absorbing region on the other hand.

On the mathematics of multifield flow

Abstract Equations of motion for multifield flow are defined; their solutions depend continuously on the initial data in the class of functions with finite resolution.