Feedback control of combustion oscillations in combustion chambers

Model-based algorithms are generally employed in active control of combustion oscillations. Since practical combustion processes consist of complex thermal and acoustic couplings, their accurate models and parameters may not be obtained in advance economically, a model free controller is necessary for the control of thermoacoustic instabilities. Active compensation based control algorithm is applied in the suppression of combustion instabilities. Tuning the controller parameters on line, the amplitudes of the acoustic waves can be modulated to desired values. Simulations performed on a control oriented, typical longitudinal oscillations combustor model illustrate the controllers’ capability to attenuate combustion oscillations.     

Stabilization by Slow Diffusion in a Real Ginzburg-Landau System

The Ginzburg-Landau equation is essential for understanding the dynamics of patterns in a wide variety of physical contexts. It governs the evolution of small amplitude instabilities near criticality. It is well known that the (cubic) Ginzburg-Landau equation has various unstable solitary pulse solutions. However, such localized patterns have been observed in systems in which there are two competing instability mechanisms. In such systems, the evolution of instabilities is described by a Ginzburg-Landau equation coupled to a diffusion equation. In this article we study the influence of this additional diffusion equation on the pulse solutions of the Ginzburg-Landau equation in light of recently developed insights into the effects of slow diffusion on the stability of pulses. Therefore, we consider the limit case of slow diffusion, i.e., the situation in which the additional diffusion equation acts on a long spatial scale. We show that the solitary pulse solution of the Ginzburg-Landau equation persists under this coupling. We use the Evans function method to analyze the effect of the slow diffusion and to show that it acts as a control mechanism that influences the (in)stability of the pulse. We establish that this control mechanism can indeed stabilize a pulse when higher order nonlinearities are taken into account.     

Zigzag and Breakup Instabilities of Stripes and Rings in the Two-Dimensional Gray–Scott Model

Two different types of instabilities of equilibrium stripe and ring solutions are studied for the singularly perturbed two‐component Gray–Scott (GS) model in a two‐dimensional domain. The analysis is performed in the semi‐strong interaction limit where the ratio O(?^?2) of the two diffusion coefficients is asymptotically large. For ?→ 0 , an equilibrium stripe solution is constructed where the singularly perturbed component concentrates along the mid‐line of a rectangular domain. An equilibrium ring solution occurs when this component concentrates on some circle that lies concentrically within a circular cylindrical domain. For both the stripe and the ring, the spectrum of the linearized problem is studied with respect to transverse (zigzag) and varicose (breakup) instabilities. Zigzag instabilities are associated with eigenvalues that are asymptotically small as ?→ 0 . Breakup instabilities, associated with eigenvalues that are O(1) as ?→ 0 , are shown to lead to the disintegration of a stripe or a ring into spots. For both the stripe and the ring, a combination of asymptotic and numerical methods are used to determine precise instability bands of wavenumbers for both types of instabilities. The instability bands depend on the relative magnitude, with respect to ?, of a nondimensional feed‐rate parameter A of the GS model. Both the high feed‐rate regime A=O(1) , where self‐replication phenomena occurs, and the intermediate regime O(?^1/2) ?A?O(1) are studied. In both regimes, it is shown that the instability bands for zigzag and breakup instabilities overlap, but that a zigzag instability is always accompanied by a breakup instability. The stability results are confirmed by full numerical simulations. Finally, in the weak interaction regime, where both components of the GS model are singularly perturbed, it is shown from a numerical computation of an eigenvalue problem that there is a parameter set where a zigzag instability can occur with no breakup instability. From full‐scale numerical computations of the GS, it is shown that this instability leads to a large‐scale labyrinthine pattern.     

Stability of Front Solutions of the Bidomain Equation

The bidomain model is the standard model describing electrical activity of the heart. Here we study the stability of planar front solutions of the bidomain equation with a bistable nonlinearity (the bidomain Allen‐Cahn equation) in two spatial dimensions. In the bidomain Allen‐Cahn equation a Fourier multiplier operator whose symbol is a positive homogeneous rational function of degree two (the bidomain operator) takes the place of the Laplacian in the classical Allen‐Cahn equation. Stability of the planar front may depend on the direction of propagation given the anisotropic nature of the bidomain operator. We establish various criteria for stability and instability of the planar front in each direction of propagation. Our analysis reveals that planar fronts can be unstable in the bidomain Allen‐Cahn equation in striking contrast to the classical or anisotropic Allen‐Cahn equations. We identify two types of instabilities, one with respect to long‐wavelength perturbations, the other with respect to medium‐wavelength perturbations. Interestingly, whether the front is stable or unstable under long‐wavelength perturbations does not depend on the bistable nonlinearity and is fully determined by the convexity properties of a suitably defined Frank diagram. On the other hand, stability under intermediate‐wavelength perturbations does depend on the choice of bistable nonlinearity. Intermediate‐wavelength instabilities can occur even when the Frank diagram is convex, so long as the bidomain operator does not reduce to the Laplacian. We shall also give a remarkable example in which the planar front is unstable in all directions.© 2016 Wiley Periodicals, Inc.     

Exact boundary controllability of vibrating non-classical Euler–Bernoulli micro-scale beams

This study investigates the exact controllability problem for a vibrating non-classical Euler–Bernoulli micro-beam whose governing partial differential equation (PDE) of motion is derived based on the non-classical continuum mechanics. In this paper, it is proved that via boundary controls, it is possible to obtain exact controllability which consists of driving the vibrating system to rest in finite time. This control objective is achieved based on the PDE model of the system which causes that spillover instabilities do not occur.     

Chaos control for numerical instability of first order reliability method

The HL-RF algorithm of the first order reliability method (FORM) is a kind of popular iterative algorithm for solving the reliability index in structural reliability analysis and reliability-based design optimization. However, there are the phenomena of convergence failure such as periodic oscillation, bifurcation and chaos in the FORM for some nonlinear problems. This paper suggests a novel method to overcome the numerical instabilities of HL-RF algorithm of FORM based on the principle of chaos control. The essential causes of chaotic dynamics for numerical instabilities including periodic oscillation and chaos of iterative solutions of FORM are revealed. Moreover, the geometrical properties of periodic oscillation of the iterative formulas derived from the FORM and performance measure approach are analyzed and compared. Finally, the stability transformation method (STM) of chaos feedback control is proposed to implement the convergence control of FORM. Several numerical examples with explicit or implicit HL-RF iterative formulas illustrate that the STM is effective, simple and versatile, and can control the periodic oscillation, bifurcation and chaos of the FORM iterative algorithm.     

Primary and secondary instabilities in soft bilayered systems

Wrinkling phenomena emerging from mechanical instabilities in inhomogeneously compressed soft bilayered systems can evoke a wide variety of surface morphologies. Applications range from undesired instabilities in engineering structures such as sandwich panels, via fabricating surfaces with controlled buckling patterns of unique properties, to wrinkling phenomena in living matter such as lungs, mucosas, and brain convolutions. While moderate compression evokes periodic sinusoidal wrinkles, higher compression induces secondary instabilities - the surface bifurcates into increasingly complex morphologies. Periodic wrinkling has already been extensively studied, but the rich pattern formation in the highly nonlinear post-buckling regime remains poorly understood. Here, we establish a computational model of differential growth to explore the evolving buckling pattern of a growing layer bonded to a non-growing substrate. Our model provides a mechanistic understanding of growth-induced primary and secondary instabilities. We show that amongst all possible secondary bifurcations, the mode of period-doubling is energetically favorable. We experimentally validate our numerical results by examining buckling of a compressed polymer film on a soft foundation. Our computational studies have broad applications in the microfabrication of distinct surface patterns as well as in the morphogenesis of living systems, where growth is progressive and the formation of structural instabilities is critical to biological function. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On Nonlinear Rayleigh-Taylor Instabilities

Some of the mathematical properties of the interface between two incompressible inviscid and immiscible fluids with different densities under the influence of a constant gravity field 9 are investigated. The purpose of this paper is to prove that linearly unstable modes for Rayleigh-Taylor instabilities give birth to nonlinear instabilities for the full nonlinear system. The main ingredient is a general instability theorem in an analytic framework which enables us to go from linear to nonlinear instabilities.     

Linear and nonlinear instabilities of a granular bed: Determination of the scales of ripples and dunes in rivers

Granular media are frequently found in nature and in industry and their transport by a fluid flow is of great importance to human activities. One case of particular interest is the transport of sand in open-channel and river flows. In many instances, the shear stresses exerted by the fluid flow are bounded to certain limits and some grains are entrained as bed-load: a mobile layer which stays in contact with the fixed part of the granular bed. Under these conditions, an initially flat granular bed may be unstable, generating ripples and dunes such as those observed on the bed of rivers. In free-surface water flows, dunes are bedforms that scale with the flow depth, while ripples do not scale with it. This article presents a model for the formation of ripples and dunes based on the proposition that ripples are primary linear instabilities and that dunes are secondary instabilities formed from the competition between the coalescence of ripples and free surface effects. Although simple, the model is able to explain the growth of ripples, their saturation (not explained in previous models) and the evolution from ripples to dunes, presenting a complete picture for the formation of dunes.     

Analytical studies on the instabilities of heterogeneous intelligent traffic flow

It has been widely reported in literature that a small perturbation in traffic flow such as a sudden deceleration of a vehicle could lead to the formation of traffic jams without a clear bottleneck. These traffic jams are usually related to instabilities in traffic flow. The applications of intelligent traffic systems are a potential solution to reduce the amplitude or to eliminate the formation of such traffic instabilities. A lot of research has been conducted to theoretically study the effect of intelligent vehicles, for example adaptive cruise control vehicles, using either computer simulation or analytical method. However, most current analytical research has only applied to single class traffic flow. To this end, the main topic of this paper is to perform a linear stability analysis to find the stability threshold of heterogeneous traffic flow using microscopic models, particularly the effect of intelligent vehicles on heterogeneous (or multi-class) traffic flow instabilities. The analytical results will show how intelligent vehicle percentages affect the stability of multi-class traffic flow.     

Prediction of the bed-load transport by gas–liquid stratified flows in horizontal ducts

Solid particles can be transported as a mobile granular bed, known as bed-load, by pressure-driven flows. A common case in industry is the presence of bed-load in stratified gas–liquid flows in horizontal ducts. In this case, an initially flat granular bed may be unstable, generating ripples and dunes. This three-phase flow, although complex, can be modeled under some simplifying assumptions. This paper presents a model for the estimation of some bed-load characteristics. Based on parameters easily measurable in industry, the model can predict the local bed-load flow rates and the celerity and the wavelength of instabilities appearing on the granular bed.     

Stability analysis of an interface between immiscible liquids in Hele-Shaw flow in the presence of a magnetic field

Hydrodynamic instabilities may occur when a viscous fluid is driven by a less viscous one through a porous medium. These penetrations are common in enhanced oil recovery, dendrite formation and aquifer flow. Recent studies have shown that the use of magnetic suspensions allow the external control of the instability. The problem is nonlinear and some further improvements of both theory and experimental observations are still needed and continue being a current source of investigation. In this paper we present a generalized Darcy law formulation in order to examine the growth of finger instabilities as a magnetic field is applied to the interface between the fluids in a Hele-Shaw cell. A new linear stability analysis is performed in the presence of magnetic effects and provides a stability criterion in terms of the non-dimensional physical parameters of the examined flow and the wavenumber of the finger disturbances. The interfacial tension inhibits small wavelength instabilities. The magnetic field contributes to the interface stability for moderate wavelength as it is applied parallel to the liquid-interface. In particular, we find an explicit expression, as a function of the susceptibility, for a critical angle between the interface and the magnetic field direction, in which its effect on the interface is neutral. We have developed a new asymptotic solution for the flow problem in a weak nonlinear regime. The first correction captures the second order nonlinear effects of the magnetic field, which tends to align the fingers with the field orientation and have a destabilizing effect. The analysis predicts that the non-linear effects at second order can counterbalance the first order stabilizing effect of a parallel magnetic field which results in a loss of effectiveness for controlling the investigated finger instabilities. The relevant physical parameters for controlling these finger instabilities are clearly identified by our non-dimensional analysis.     

Stability of anisotropic electroactive polymers with application to layered media

The stability of anisotropic electroactive polymers is investigated. A general criterion for the onset of instabilities under plane-strain conditions is introduced in terms of a sextic polynomial whose coefficients depend on the instantaneous electroelastic moduli. In a way of an example, the stable domains of layered neo-Hookean dielectrics are determined. It is found that depending on the direction of the electrostatic excitation field relative to the lamination direction, the critical stretch ratios at which instabilities may occur can be either larger or smaller than the ones for the purely mechanical case.     

Resonance-sensitivity in dynamic Hopf bifurcations under fluid loading

Under steady fluid loading, elastic structures are liable to exhibit dynamic bifurcations to limit cycles: such unimodal instabilities are referred to as galloping while such multimodal instabilities are referred to as flutter. The trace of limit cycles energing from the critical equilibrium state can be either super-critical and stable, in analogy with a stable symmetric static bifurcation, or sub-critical and unstable, in analogy with an unstable symmetric static bifurcation. Galloping of a bluff body in a steady flow can be of the unstable type, and we might expect some form of imperfection sensitivity, although in contrast to static bifurcations, a Hopf bifurcation is actually topologically stable under the operation of a single control parameter: the form of the Hopf bifurcation cannot be rounded off or destroyed by imperfections as in the static case. However, since the dynamic instabilities are associated with a well defined and non-zero circular frequency we might expect the failure ‘load’ to be sensitive to resonant periodic forcing, and this is here shown to be the case, with a two-thirds power law sensitivity analogous to the static cusp.The conclusion is drawn that the concept of structural stability, vital as it is to good mathematical modelling, must be examined with care, particular attention being given to any restrictions on the class of allowable perturbations.     

Blending Modified Gaussian Closure and Non-Gaussian Reduced Subspace Methods for Turbulent Dynamical Systems

Turbulent dynamical systems are characterized by persistent instabilities which are balanced by nonlinear dynamics that continuously transfer energy to the stable modes. To model this complex statistical equilibrium in the context of uncertainty quantification all dynamical components (unstable modes, nonlinear energy transfers, and stable modes) are equally crucial. Thus, order-reduction methods present important limitations. On the other hand uncertainty quantification methods based on the tuning of the non-linear energy fluxes using steady-state information (such as the modified quasilinear Gaussian (MQG) closure) may present discrepancies in extreme excitation scenarios. In this paper we derive a blended framework that links inexpensive second-order uncertainty quantification schemes that model the full space (such as MQG) with high order statistical models in specific reduced-order subspaces. The coupling occurs in the energy transfer level by (i) correcting the nonlinear energy fluxes in the full space using reduced subspace statistics, and (ii) by modifying the reduced-order equations in the subspace using information from the full space model. The results are illustrated in two strongly unstable systems under extreme excitations. The blended method allows for the correct prediction of the second-order statistics in the full space and also the correct modeling of the higher-order statistics in reduced-order subspaces.     

Stability of anisotropic electroactive polymers with application to layered media

The stability of anisotropic electroactive polymers is investigated. A general criterion for the onset of instabilities under plane-strain conditions is introduced in terms of a sextic polynomial whose coefficients depend on the instantaneous electroelastic moduli. In a way of an example, the stable domains of layered neo-Hookean dielectrics are determined. It is found that depending on the direction of the electrostatic excitation field relative to the lamination direction, the critical stretch ratios at which instabilities may occur can be either larger or smaller than the ones for the purely mechanical case.     

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

In this paper we introduce a conceptual model for vegetation patterns that generalizes the Klausmeier model for semi-arid ecosystems on a sloped terrain (Klausmeier in Science 284:1826–1828, 1999). Our model not only incorporates downhill flow, but also linear or nonlinear diffusion for the water component. To relate the model to observations and simulations in ecology, we first consider the onset of pattern formation through a Turing or a Turing–Hopf bifurcation. We perform a Ginzburg–Landau analysis to study the weakly nonlinear evolution of small amplitude patterns and we show that the Turing/Turing–Hopf bifurcation is supercritical under realistic circumstances. In the second part we numerically construct Busse balloons to further follow the family of stable spatially periodic (vegetation) patterns. We find that destabilization (and thus desertification) can be caused by three different mechanisms: fold, Hopf and sideband instability, and show that the Hopf instability can no longer occur when the gradient of the domain is above a certain threshold. We encounter a number of intriguing phenomena, such as a ‘Hopf dance’ and a fine structure of sideband instabilities. Finally, we conclude that there exists no decisive qualitative difference between the Busse balloons for the model with standard diffusion and the Busse balloons for the model with nonlinear diffusion.     

Stability analysis of model problems for elastodynamic boundary element discretizations

In the literature there is growing evidence of instabilities in standard time-stepping schemes to solve boundary integral elastodynamic models [1]–[3]. In this article we use three distinct model problems to investigate the stability properties of various discretizations that are commonly used to solve elastodynamic boundary integral equations. Using the model problems, the stability properties of a large variety of discretization schemes are assessed. The features of the discretization procedures that are likely to cause instabilities can be established by means of the analysis. This new insight makes it possible to design new time-stepping schemes that are shown to be more stable. © 1996 John Wiley & Sons, Inc.     

Electro-thermo-capillary-convection in a square layer of dielectric liquid subjected to a strong unipolar injection

In this paper, the effect electric field on the flow induced by the combined buoyancy and thermocapillary forces is carried out. Calculations are performed for a strong unipolar injection (C = 10) and different values of Marangoni number (−10000 ≤ Ma ≤ 10000), thermal Rayleigh number (5000 ≤ Ra ≤ 50,000) and electric Rayleigh number (0 ≤ T ≤ 800). The Prandtl number (Pr) and the mobility parameter (M) are fixed at 116.6 and 49, respectively. These values correspond to the Silicone oil used as working liquid several practical applications. The full set of coupled equations: Navier–Stokes, Electro-hydrodynamic (EHD) and heat transfer equations are directly solved using stream function-vorticity formalism. Obtained results show that the electric forces can control the thermocapillary instabilities. According to the intensity and the direction of the applied electric forces, it is demonstrated that these instabilities can be accentuated, attenuated, or even completely eliminated.     

Reactive Infiltration Instabilities

When a fluid flow is imposed on a porous medium, the infiltrationflow may interact with the reaction-induced porosity variationswithin the medium and may lead to fingering instabilities. Anonlinear model of such interaction is developed and morphologicalinstability of a planar dissolution front is demonstrated usinga linear stability analysis of a moving-free-boundary problem.The fully nonlinear model is also examined numerically usingfinite-difference methods. The numerical simulations confirmthe predictions of linear stability theory and, more importantly,reveal the growth of dissolution fingers that emerge as a resultof these instabilities