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Liapunov stability theory applied to class of partial differential equations, and generation of contraction groups related to equivalent norms
Liapunov stability theory generalized and applied to class of partial differential equations
Linear stability theory results are presented for flight and low-disturbance wind tunnels, at low and high speeds, and for T-S, Gortler, and cross-flow modes. Results show that, in the absence of Morkovin 'bypasses', transition at low background disturbance levels corresponds to N-factors of the order of 9 to 11, provided that phenomena such as curvature effects are included. The results point to a wider range of applicability for the e exp N method than previously conjectured, and suggest that the e exp N method can be used to parametrize transition predictions as a function of parameters which affect the mean flow profiles for such purposes as LFC and minimum drag design.
Stability theory applied to static and dynamic stability problems of electric power systems
Stability theory application to control, circuit theory and aerospace systems
The fundamentals of stability theory, its chief results, and the physical mechanisms at work are presented. The stability theory of the laminar boundary determines whether a small disturbance introduced into the boundary layer will amplify or damp. If the disturbance damps, the boundary layer remains laminar. If the disturbance amplifies, and by a sufficient amount, then transition to turbulence eventually takes place. The stability theory establishes those states of the boundary layer which are most likely to lead to transition, identifys those frequencies which are the most dangerous, and indicates how the external parameters can best be changed to avoid transition.
Aspects of the linear compressible stability theory are considered, with special attention given to the inviscid theory and the additional solutions that arise when there is a region of supersonic flow relative to the phase velocity. For the case of highly cooled flat-plate boundary layers at Mach number 5.8, the unstable region is found to include supersonic outgoing waves. A previously unknown neutral incoming wave has also been identified. An example of viscous multiple solutions is provided, and calculations of higher viscous discrete modes and the compressible counterpart of the Squire mode are presented. It is noted that compressible stability theory is missing a firm connection with boundary-layer transition.
Liapunov stability theory applied to control system, noting stability of function and stochastic differential equations
Colloid stability theory has been applied to protein crystallization and predicts a narrow range of conditions under which crystals can be grown without the agglomeration of protein molecules (colloids) in the bulk solution. It also predicts a critical electrolyte concentration above which agglomeration will always occur. Using this theory, the rapid protein agglomeration occurring during Schlieren experiments as well as a terminal crystal size effect in a fixed container were explained. Following this concept, the supposed 'terminal' crystal size has been at least doubled.
Paper briefly summarizes the current status of linear stability theory as applied to laminar flow control for aerodynamics. Results indicate that the conventional 'N factor' method of correlating stability theory and transition has a broad application range, including low- and high-speeds, two- and three-dimensional mean flow and TS, Gortler and crossflow disturbance modes. Linear theory is particularly applicable to the laminar flow control problem as, for system efficiency, control must be exercised and disturbances maintained in the linear regime. Current areas of concern for LFC, which require further stability theory research, include TS-crossflow interaction, combined disturbance fields (roughness, waviness, noise) and suction-induced disturbances. Some results on wave-interactions are presented.
Linear stability theory for chemically reacting (equilibrium and nonequilibrium) flows is applied to study the stability of hypersonic boundary layers under the equilibrium assumption. The results for Mach 15 flow indicate that the second mode instability shifts to lower frequencies, as compared to the perfect gas results. The peak second mode growth rate is also increased for the equilibrium gas model. However, the results for Goertler instability show very little real gas effect. The present theory has been applied to a sphere-cone flight transition experiment performed at freestream Mach of about 20.
The purpose of this thesis is to investigate an extension of mu theory for robust control design by considering systems with linear and nonlinear real parameter uncertainties. In the process, explicit connections are made between mixed mu and absolute stability theory. In particular, it is shown that the upper bounds for mixed mu are a generalization of results from absolute stability theory. Both state space and frequency domain criteria are developed for several nonlinearities and stability multipliers using the wealth of literature on absolute stability theory and the concepts of supply rates and storage functions. The state space conditions are expressed in terms of Riccati equations and parameter-dependent Lyapunov functions. For controller synthesis, these stability conditions are used to form an overbound of the H2 performance objective. A geometric interpretation of the equivalent frequency domain criteria in terms of off-axis circles clarifies the important role of the multiplier and shows that both the magnitude and phase of the uncertainty are considered. A numerical algorithm is developed to design robust controllers that minimize the bound on an H2 cost functional and satisfy an analysis test based on the Popov stability multiplier. The controller and multiplier coefficients are optimized simultaneously, which avoids the iteration and curve-fitting procedures required by the D-K procedure of mu synthesis. Several benchmark problems and experiments on the Middeck Active Control Experiment at M.I.T. demonstrate that these controllers achieve good robust performance and guaranteed stability bounds.
The consistency of second-order closure models with results from hydrodynamic stability theory is analyzed for the simplified case of homogeneous turbulence. In a recent study, Speziale, Gatski, and MacGiolla Mhuiris showed that second-order closures are capable of yielding results that are consistent with hydrodynamic stability theory for the case of homogeneous shear flow in a rotating frame. It is demonstrated in this paper that this success is due to the fact that the stability boundaries for rotating homogeneous shear flow are not dependent on the details of the spatial structure of the disturbances. For those instances where they are -- such as in the case of elliptical flows where the instability mechanism is more subtle -- the results are not so favorable. The origins and extent of this modeling problem are examined in detail along with a possible resolution based on rapid distortion theory (RDT) and its implications for turbulence modeling.
The linear stability of fully three-dimensional supersonic boundary layers formed over swept-wing configurations is investigated using a modified version of the linear stability code COSAL. Configurations studied include a highly swept leading-edge model to be utilized for transition studies in the LARC Low-disturbance Mach 3.5 Pilot Tunnel. The model is a representation of the leading edge of a laminar flow control wing for the F-16XL aircraft. In addition, the region over a laminar flow control glove fitted on the midportion of an F-16XL wing was studied. For each configuration, estimates of the location of the onset of transition were computed using linear stability theory and the e exp N method. The effectiveness of suction in stabilizing the boundary layer over the F-16XL wing glove was also investigated.
Applications of Liapunov stability theory to control systems
Increased understanding in recent years of boundary layer transition has been made possible by the development of strongly nonlinear stability theories. After some twenty or so years when nonlinear stability theory was restricted to the application of the Stuart-Watson method (or less formal amplitude expansion procedures), there now exist strongly nonlinear theories which can describe processes which have an 0(1) effect on the basic state. These strongly nonlinear theories and their possible role in pushing theoretical understanding of transition ever further into the nonlinear regime are discussed.
The paper summarizes the state-of-the-art for application of stability theory to laminar flow control using suction, wall temperature and/or favorable pressure gradient ('natural laminar flow'). Discussions include current LFC problem areas requiring stability analyses, methods of relating stability theory to transition with results from data and theory comparisons available thus far, and a summary of low disturbance data available for theory calibration on swept wings. Critical issues highlighted are problems peculiar to suction LFC on high performance transonic wings and application of the e-to-the-n-power method to both low and high speed flight data.
It is found that under mild assumptions, feedback system stability can be concluded if one can 'topologically separate' the infinite-dimensional function space containing the system's dynamical input-output relations into two regions, one region containing the dynamical input-output relation of the 'feedforward' element of the system and the other region containing the dynamical output-input relation of the 'feedback' element. Nonlinear system stability criteria of both the input-output type and the state-space (Liapunov) type are interpreted in this context. The abstract generality and conceptual simplicity afforded by the topological separation perspective clarifies some of the basic issues underlying stability theory and serves to suggest improvements in existing stability criteria. A generalization of Zames' (1966) conic-relation stability criterion is proved, laying the foundation for improved multivariable generalizations of the frequency-domain circle stability criterion for nonlinear systems.