Some comments on the stability theory of mhd shock waves

The stability (evolutionarity) problem for a kind of MHD shock waves is discussed in this paper. That is to solve the interaction problem of MHD shock waves with (2-dimensional) oblique incident disturbances. In other words, the result of gasdynamic shocks is generalized to the case of MHD shocks. The previous conclusion of stability theory of MHD shock waves obtained from the solution of interaction problem of MHD shock wave with (one-dimensional) normal shock wave is that only fast and slow shocks are stable, and intermediate shocks are unstable. However, the results of this paper show that when the small disturbances are the Alfven waves a new stability condition which is related to the parameters in front of and behind the shock wave is derived. When the disturbances are entropy wave and fast and slow magneto acoustic waves the stability condition is related to the frequency of small disturbances. As the limiting ease, i. e. when a normal incident (reflection, refraction) is consid...更多ered, the fast and slow shocks are unstable. The results also show that the conclusion drawn by Kontorovich is invalid for the stability theory of shock waves.

On the non-linear stability theory of spinning solid bodies with liquid-filled cavities and its application to the columbus problem

In this paper, we first present a system of differential-integral equations for the largedisturbance to the general case that any arbitrarily shaped solid body with a cavity contain-ing viscous liquid rotates uniformly around the principal axis of inertia, and then develop aweakly non-linear stability theory by the Lyapunov direct approach. Applying this theoryto the Columbus problem, we have proved the consistency between the theory and Kelvin'sexperiments.

Modal stability theory

This article contains a review of modal stability theory. It covers local stability analysis of parallel flows including temporal stability, spatial stability, phase velocity, group velocity, spatio-temporal stability, the linearized Navier-Stokes equations, the Orr-Sommerfeld equation, the Rayleigh equation, the Briggs-Bers criterion, Poiseuille flow, free shear flows, and secondary modal instability. It also covers the parabolized stability equation (PSE), temporal and spatial biglobal theory, 2D eigenvalue problems, 3D eigenvalue problems, spectral collocation methods, and other numerical solution methods. Computer codes are provided for tutorials described in the article. These tutorials cover the main topics of the article and can be adapted to form the basis of research codes. Copyright © 2014 by ASME.

Hydrodynamic stability theory of double ablation front structures in inertial confinement fusion.

The aim of inertial confinement fusion is the production of energy by the fusion of thermonuclear fuel (deuterium-tritium) enclosed in a spherical target due to its implosion. In the direct-drive approach, the energy needed to spark fusion reactions is delivered by the irradiation of laser beams that leads to the ablation of the outer shell of the target (the so-called ablator). As a reaction to this ablation process, the target is accelerated inwards, and, provided that this implosion is sufficiently strong a symmetric, the requirements of temperature and pressure in the center of the target are achieved leading to the ignition of the target (fusion). One of the obstacles capable to prevent appropriate target implosions takes place in the ablation region where any perturbation can grow even causing the ablator shell break, due to the ablative Rayleigh-Taylor instability. The ablative Rayleigh-Taylor instability has been extensively studied throughout the last 40 years in the case where the density/temperature profiles in the ablation region present a single front (the ablation front). Single ablation fronts appear when the ablator material has a low atomic number (deuterium/tritium ice, plastic). In this case, the main mechanism of energy transport from the laser energy absorption region (low density plasma) to the ablation region is the electron thermal conduction. However, recently, the use of materials with a moderate atomic number (silica, doped plastic) as ablators, with the aim of reducing the target pre-heating caused by suprathermal electrons generated by the laser-plasma interaction, has demonstrated an ablation region composed of two ablation fronts. This fact appears due to increasing importance of radiative effects in the energy transport. The linear theory describing the Rayleigh-Taylor instability for single ablation fronts cannot be applied for the stability analysis of double ablation front structures. Therefore, the aim of this thesis is to develop, for the first time, a linear stability theory for this type of hydrodynamic structures.

Relevance of local parallel theory to the linear stability of laminar separation bubbles

Laminar separation bubbles are thought to be highly non-parallel, and hence global stability studies start from this premise. However, experimentalists have always realized that the flow is more parallel than is commonly believed, for pressure-gradient-induced bubbles, and this is why linear parallel stability theory has been successful in describing their early stages of transition. The present experimental/numerical study re-examines this important issue and finds that the base flow in such a separation bubble becomes nearly parallel due to a strong-interaction process between the separated boundary layer and the outer potential flow. The so-called dead-air region or the region of constant pressure is a simple consequence of this strong interaction. We use triple-deck theory to qualitatively explain these features. Next, the implications of global analysis for the linear stability of separation bubbles are considered. In particular we show that in the initial portion of the bubble, where the flow is nearly parallel, local stability analysis is sufficient to capture the essential physics. It appears that the real utility of the global analysis is perhaps in the rear portion of the bubble, where the flow is highly non-parallel, and where the secondary/nonlinear instability stages are likely to dominate the dynamics.

Lyapunov theory for zeno stability

Zeno behavior is a dynamic phenomenon unique to hybrid systems in which an infinite number of discrete transitions occurs in a finite amount of time. This behavior commonly arises in mechanical systems undergoing impacts and optimal control problems, but its characterization for general hybrid systems is not completely understood. The goal of this paper is to develop a stability theory for Zeno hybrid systems that parallels classical Lyapunov theory; that is, we present Lyapunov-like sufficient conditions for Zeno behavior obtained by mapping solutions of complex hybrid systems to solutions of simpler Zeno hybrid systems defined on the first quadrant of the plane. These conditions are applied to Lagrangian hybrid systems, which model mechanical systems undergoing impacts, yielding simple sufficient conditions for Zeno behavior. Finally, the results are applied to robotic bipedal walking. © 2012 IEEE.

Effects of wall temperature on boundary layer stability over a blunt cone at Mach 7.99

Effects of wall temperature on stabilities of hypersonic boundary layer over a 7-degree half-cone-angle blunt cone are studied by using both direct numerical simulation (DNS) and linear stability theory (LST) analysis. Four isothermal wall cases with Tw/T0= 0.5, 0.7, 0.8 and 0.9, as well as an adiabatic wall case are considered. Results of both DNS and LST indicate that wall temperature has significant effects on the growth of disturbance waves. Cooling the surface accelerates unstable Mack II mode waves and decelerates the first mode (Tollmien–Schlichting mode) waves. LST results show that growth rate of the most unstable Mack II mode waves for the cases of cold wall Tw/T0=0.5 and 0.7 are about 45% and 25% larger than that for the adiabatic wall, respectively. Numerical results show that surface cooling modifies the profiles of rdut/dyn and temperature in the boundary layers, and thus changes the stability haracteristic of the boundary layers, and then effects on the growth of unstable waves. The results of DNS indicate that the disturbances with the frequency range from about 119.4 to 179.1 kHz, including the most unstable Mack modes, produce strong mode competition in the downstream region from about 11 to 100 nose radii. And adiabatic wall enhances the amplitudes of disturbance according to the results of DNS, although the LST indicates that the growth rate of the disturbance of cold wall is larger. That because the growth of the disturbance does not only depend on the development of the second unstable mode.

Stability of Hypervelocity Boundary Layers

The early stage of laminar-turbulent transition in a hypervelocity boundary layer is studied using a combination of modal linear stability analysis, transient growth analysis, and direct numerical simulation. Modal stability analysis is used to clarify the behavior of first and second mode instabilities on flat plates and sharp cones for a wide range of high enthalpy flow conditions relevant to experiments in impulse facilities. Vibrational nonequilibrium is included in this analysis, its influence on the stability properties is investigated, and simple models for predicting when it is important are described.

Transient growth analysis is used to determine the optimal initial conditions that lead to the largest possible energy amplification within the flow. Such analysis is performed for both spatially and temporally evolving disturbances. The analysis again targets flows that have large stagnation enthalpy, such as those found in shock tunnels, expansion tubes, and atmospheric flight at high Mach numbers, and clarifies the effects of Mach number and wall temperature on the amplification achieved. Direct comparisons between modal and non-modal growth are made to determine the relative importance of these mechanisms under different flow regimes.

Conventional stability analysis employs the assumption that disturbances evolve with either a fixed frequency (spatial analysis) or a fixed wavenumber (temporal analysis). Direct numerical simulations are employed to relax these assumptions and investigate the downstream propagation of wave packets that are localized in space and time, and hence contain a distribution of frequencies and wavenumbers. Such wave packets are commonly observed in experiments and hence their amplification is highly relevant to boundary layer transition prediction. It is demonstrated that such localized wave packets experience much less growth than is predicted by spatial stability analysis, and therefore it is essential that the bandwidth of localized noise sources that excite the instability be taken into account in making transition estimates. A simple model based on linear stability theory is also developed which yields comparable results with an enormous reduction in computational expense. This enables the amplification of finite-width wave packets to be taken into account in transition prediction.

Studies of stability of fluid flowa using variational methods

The study of stability problems is relevant to the study of structure of a physical system. It 1S particularly important when it is not possible to probe into its interior and obtain information on its structure by a direct method. The thesis states about stability theory that has become of dominant importance in the study of dynamical systems. and has many applications in basic fields like meteorology, oceanography, astrophysics and geophysics- to mention few of them. The definition of stability was found useful 1n many situations, but inadequate in many others so that a host of other important concepts have been introduced in past many years which are more or less related to the first definition and to the common sense meaning of stability. In recent years the theoretical developments in the studies of instabilities and turbulence have been as profound as the developments in experimental methods. The study here Points to a new direction for stability studies based on Lagrangian formulation instead of the Hamiltonian formulation used by other authors.

Stability for impulsive control systems

In this paper we extend the notion of the control Lyapounov pair of functions and derive a stability theory for impulsive control systems. The control system is a measure driven differential inclusion that is partly absolutely continuous and partly singular. Some examples illustrating the features of Lyapounov stability are provided.

Direct and adjoint global stability analysis of turbulent transonic flows over a NACA0012 profile

In this work, various turbulent solutions of the two-dimensional (2D) and three-dimensional compressible Reynolds averaged Navier?Stokes equations are analyzed using global stability theory. This analysis is motivated by the onset of flow unsteadiness (Hopf bifurcation) for transonic buffet conditions where moderately high Reynolds numbers and compressible effects must be considered. The buffet phenomenon involves a complex interaction between the separated flow and a shock wave. The efficient numerical methodology presented in this paper predicts the critical parameters, namely, the angle of attack and Mach and Reynolds numbers beyond which the onset of flow unsteadiness appears. The geometry, a NACA0012 profile, and flow parameters selected reproduce situations of practical interest for aeronautical applications. The numerical computation is performed in three steps. First, a steady baseflow solution is obtained; second, the Jacobian matrix for the RANS equations based on a finite volume discretization is computed; and finally, the generalized eigenvalue problem is derived when the baseflow is linearly perturbed. The methodology is validated predicting the 2D Hopf bifurcation for a circular cylinder under laminar flow condition. This benchmark shows good agreement with the previous published computations and experimental data. In the transonic buffet case, the baseflow is computed using the Spalart?Allmaras turbulence model and represents a mean flow where the high frequency content and length scales of the order of the shear-layer thickness have been averaged. The lower frequency content is assumed to be decoupled from the high frequencies, thus allowing a stability analysis to be performed on the low frequency range. In addition, results of the corresponding adjoint problem and the sensitivity map are provided for the first time for the buffet problem. Finally, an extruded three-dimensional geometry of the NACA0012 airfoil, where all velocity components are considered, was also analyzed as a Triglobal stability case, and the outcoming results were compared to the previous 2D limited model, confirming that the buffet onset is well detected.

Hegemony Without Stability: The fiscal and political vulnerabilities of monetary union. ACES Cases No. 2011.2

The crisis has forced the Euro area to establish an emergency fund that supports member states experiencing a sovereign debt crisis. The difficulties of coming up with such a fund for Greece and other Euro area members stands in marked contrast to the balance of payments support that non-Euro members like Hungary received, swiftly and quietly. In order to solve this puzzle, we first establish the difference between EU interventions and IMF programs and, second, trace the evolution of crisis management with France and Germany in the lead. The lens of hegemonic stability theory suggests that the Franco-German leadership is too weak to provide stability and the extensive use of conditionality is one symptom of this weakness. Providing incentives for cooperation "after hegemony" (Keohane) is the unresolved issues troubling the monetary union. Its dominant powers must acknowledge that markets perceive monetary union to be already politically more integrated than its lack of fiscal integration suggests.

Output feedback control of discrete-time systems in networked environments

This correspondence paper addresses the problem of output feedback stabilization of control systems in networked environments with quality-of-service (QoS) constraints. The problem is investigated in discrete-time state space using Lyapunov’s stability theory and the linear inequality matrix technique. A new discrete-time modeling approach is developed to describe a networked control system (NCS) with parameter uncertainties and nonideal network QoS. It integrates a network-induced delay, packet dropout, and other network behaviors into a unified framework. With this modeling, an improved stability condition, which is dependent on the lower and upper bounds of the equivalent network-induced delay, is established for the NCS with norm-bounded parameter uncertainties. It is further extended for the output feedback stabilization of the NCS with nonideal QoS. Numerical examples are given to demonstrate the main results of the theoretical development.

Finitary Abstract Elementary Classes

The research in model theory has extended from the study of elementary classes to non-elementary classes, i.e. to classes which are not completely axiomatizable in elementary logic. The main theme has been the attempt to generalize tools from elementary stability theory to cover more applications arising in other branches of mathematics. In this doctoral thesis we introduce finitary abstract elementary classes, a non-elementary framework of model theory. These classes are a special case of abstract elementary classes (AEC), introduced by Saharon Shelah in the 1980's. We have collected a set of properties for classes of structures, which enable us to develop a 'geometric' approach to stability theory, including an independence calculus, in a very general framework. The thesis studies AEC's with amalgamation, joint embedding, arbitrarily large models, countable Löwenheim-Skolem number and finite character. The novel idea is the property of finite character, which enables the use of a notion of a weak type instead of the usual Galois type. Notions of simplicity, superstability, Lascar strong type, primary model and U-rank are inroduced for finitary classes. A categoricity transfer result is proved for simple, tame finitary classes: categoricity in any uncountable cardinal transfers upwards and to all cardinals above the Hanf number. Unlike the previous categoricity transfer results of equal generality the theorem does not assume the categoricity cardinal being a successor. The thesis consists of three independent papers. All three papers are joint work with Tapani Hyttinen.

An optimal dynamic inversion-based neuro-adaptive approach for treatment of chronic myelogenous leukemia

Combining the advanced techniques of optimal dynamic inversion and model-following neuro-adaptive control design, an innovative technique is presented to design an automatic drug administration strategy for effective treatment of chronic myelogenous leukemia (CML). A recently developed nonlinear mathematical model for cell dynamics is used to design the controller (medication dosage). First, a nominal controller is designed based on the principle of optimal dynamic inversion. This controller can treat the nominal model patients (patients who can be described by the mathematical model used here with the nominal parameter values) effectively. However, since the system parameters for a realistic model patient can be different from that of the nominal model patients, simulation studies for such patients indicate that the nominal controller is either inefficient or, worse, ineffective; i.e. the trajectory of the number of cancer cells either shows non-satisfactory transient behavior or it grows in an unstable manner. Hence, to make the drug dosage history more realistic and patient-specific, a model-following neuro-adaptive controller is augmented to the nominal controller. In this adaptive approach, a neural network trained online facilitates a new adaptive controller. The training process of the neural network is based on Lyapunov stability theory, which guarantees both stability of the cancer cell dynamics as well as boundedness of the network weights. From simulation studies, this adaptive control design approach is found to be very effective to treat the CML disease for realistic patients. Sufficient generality is retained in the mathematical developments so that the technique can be applied to other similar nonlinear control design problems as well.