Stability analysis of linear continuous time- delay feedback systems
Linear continuous time delay feedback systems stability analysis, using root-locus and frequency response techniques
King-Smith, E. A.↗
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Results for “Linear stability analysis”
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A linear thermal stability analysis for the vertical structure of alpha model accretion disks
The linear equations that describe the thermal stability of the convective and radiative vertical structure of an alpha model accretion disk are derived. For a specific model chosen to be representative of a dwarf nova disk, it is found that the eigenfunction for the temperature shows large excursions in the zone of partial ionization, but that the eigenfunctions for the unstable fundamental mode perturbations of the vertical distance and flux are nearly constant over the vertical extent of the disk. The growth of any instability is thus nearly homologous and the stability criteria derived for vertically averaged structures represent an adequate approximation.
Saio, Hideyuki↗
Stability analysis of linear sampled-data time-delay feedback systems
Linear single loop sampled data time delay feedback systems stability analysis
King-Smith, E. A.↗
Linear Temporal Stability Analysis of a Low-Density Round Gas Jet Injected into a High-Density Gas
It has been observed in previous experimental studies that round helium jets injected into air display a repetitive structure for a long distance, somewhat similar to the buoyancy-induced flickering observed in diffusion flames. In order to investigate the influence of gravity on the near-injector development of the flow, a linear temporal stability analysis of a round helium jet injected into air was performed. The flow was assumed to be isothermal and locally parallel; viscous and diffusive effects were ignored. The variables were represented as the sum of the mean value and a normal-mode small disturbance. An ordinary differential equation governing the amplitude of the pressure disturbance was derived. The velocity and density profiles in the shear layer, and the Froude number (signifying the effects of gravity) were the three important parameters in this equation. Together with the boundary conditions, an eigenvalue problem was formulated. Assuming that the velocity and density profiles in the shear layer to be represented by hyperbolic tangent functions, the eigenvalue problem was solved for various values of Froude number. The temporal growth rates and the phase velocity of the disturbances were obtained. The temporal growth rates of the disturbances increased as the Froude number was reduced (i.e. gravitational effects increased), indicating the destabilizing role played by gravity.
Lawson, Anthony L.↗
Transition prediction using three dimensional stability analysis
Several methods of transition prediction by linear stability analysis are compared. The spectral stability analysis code SALLY is used to analyze flows over laminar flow control wings. It is shown that transition by the envelope method and a new modified wave packet method are comparable in reliability but that the envelope method is more efficient computationally.
Malik, M. R.↗
Stability Analysis of Bow Shocks
We present a linear stability analysis of bow shocks created by the interaction of a spherical wind moving with respect to its surrounding medium. The bounding shocks are assumed isothermal and with Mach number M = infinity. Following Soker (1990) we study the evolution of short wavelength perturbations. We find that the motion is unstable in this limit. Moreover, the ratio of the wind velocity v(sub w) to the star velocity v(sub *) characterizes the stability properties. Bow shocks with fast winds for which v(sub *)/v(sub w)<<1 are more stable than bow sho with slow winds i.e. v(sub *)/v(sub w)>>1.
Bow Shocks↗
Stability of elastic bending and torsion of uniform cantilever rotor blades in hover with variable structural coupling
The stability of elastic flap bending, lead-lag bending, and torsion of uniform, untwisted, cantilever rotor blades without chordwise offsets between the elastic, mass, tension, and areodynamic center axes is investigated for the hovering flight condition. The equations of motion are obtained by simplifying the general, nonlinear, partial differential equations of motion of an elastic rotating cantilever blade. The equations are adapted for a linearized stability analysis in the hovering flight condition by prescribing aerodynamic forces, applying Galerkin's method, and linearizing the resulting ordinary differential equations about the equilibrium operating condition. The aerodynamic forces are obtained from strip theory based on a quasi-steady approximation of two-dimensional unsteady airfoil theory. Six coupled mode shapes, calculated from free vibration about the equilibrium operating condition, are used in the linearized stability analysis. The study emphasizes the effects of two types of structural coupling that strongly influence the stability of hingeless rotor blades. The first structural coupling is the linear coupling between flap and lead-lag bending of the rotor blade. The second structural coupling is a nonlinear coupling between flap bending, lead-lag bending, and torsion deflections. Results are obtained for a wide variety of hingeless rotor configurations and operating conditions in order to provide a reasonably complete picture of hingeless rotor blade stability characteristics.
Hodges, D. H., Roberta.↗
Linear Stability Regime Analysis of the Compressible Reacting Mixing Layer
Previous investigations have shown that a compressible reacting mixing layer can develop two peaks in the mean density weighted vorticity profile. Linear stability analyses show that at these peaks two distinct 'outer' instability modes appear in addition to the more common central mode, which exists unaccompanied in incompressible nonreacting flows. The present study parametrically analyzes the effects of compressibility, heat release, stoichiometry, and density ratio on the amplification rate and obliquity of each stability mode. The mean profiles used in the spatial stability calculation are generated by self-similar solutions of the compressible boundary layer equations combined with the assumption of infinitely fast chemistry. It is shown that the influence of stoichiometry and density ratio on the peaks of the density weighted vorticity profile determines which modes will dominate. Of particular interest are the conditions where two modes are equally amplified, causing the mixing layer to develop into a 'colayer' structure.
Day, M. J.↗
The Harmonic Linearized Navier-Stokes Equations for Transition Prediction in Three-Dimensional Flows
The conventional method to predict the onset of laminar-turbulent transition in convectively unstable boundary-layer flows is based on the logarithmic amplification ratio, the so-called N-factor, of the linear instability waves. To calculate the N-factor, the flow variables are decomposed into a laminar basic state solution and the linear disturbances, which are assumed to be harmonic in time. The most commonly used linear stability analysis approaches include the locally parallel linear stability theory (LST) and the nonlocal, weakly nonparallel parabolized stability equations (PSE). However, these methods do not account for strong streamwise gradients that are encountered in several configurations of interest, such as those in the vicinity of roughness elements, steps, gaps, or corners. To compute the linear evolution of disturbances along such strongly nonparallel regions, the harmonic linearized Navier-Stokes equations (HLNSE) need to be solved. The discretization of the HLNSE for spanwise/azimuthally inhomogeneous laminar basic states yields a linear system of complex arithmetic with a leading dimension of the order of 10^(7) to 10^(8) even in relatively simple flows. A combined multithread and multiprocessor algorithm is implemented for the direct solution of such linear systems. Results for a supersonic boundary layer over a three-dimensional roughness patch show good agreement with experimental measurements when the evolution of the instability waves over the roughness patch is included via the HLNSE. Additionally, inflow-resolvent analysis based on the HLNSE for discrete-roughness-induced disturbances in the nose tip of a blunt cone at Mach 6 demonstrates the importance of including the disturbance amplification along the near vicinity of the roughness element and separation region.
Boundary Layer Stability↗
Numerical Simulations of Unsteady Low-Reynolds-Number Flows Over the APEX Airfoil
Laminar and transitional separation bubbles are an important feature of low-Reynolds-number flows over airfoils. The separation bubbles are unsteady and have a significant impact on the aerodynamic properties of the airfoils. In this paper unsteady low-Reynolds-number separated flows over the APEX airfoil are calculated using a Navier-Stokes solver. The numerical results show the presence of unsteady separation bubbles in the flowfield. An analysis of the numerical results shows that flowfield disturbances are amplified significantly in the separation bubble, leading to periodic vortex shedding. A linear stability analysis of the separated boundary layer is performed and the results show that the dominant wavenumber and frequency in the numerical simulations agree with the most unstable wavenumber and frequency from the linear stability analysis. The numerical results also show the growth and interaction of disturbance waves in the separation bubble. For transonic flows over the APEX airfoil, the calculations show that the presence of shocks causes significant changes in the separation location and consequently, the overall flowfield.
Tatineni, Mahidhar↗
The Harmonic Linearized Navier-Stokes Equations for Transition Prediction in Three-Dimensional Flows
The conventional method to predict the onset of laminar-turbulent transition in convectively unstable boundary-layer flows is based on the logarithmic amplification ratio, the so-called N-factor, of the linear instability waves. To calculate the N-factor, the flow variables are decomposed into a laminar basic state solution and the linear disturbances, which are assumed to be harmonic in time. The most commonly used linear stability analysis approaches include the locally parallel linear stability theory (LST) and the non-local, weakly nonparallel parabolized stability equations (PSE). However, these methods do not account for strong streamwise gradients that are encountered in several configurations of interest, as roughness elements, steps, gaps, or corners. To solve the linear evolution of disturbances along such strongly nonparallel regions, the harmonic linearized Navier-Stokes equations (HLNSE) need to be solved. The discretization of the HLNSE for spanwise/azimuthally inhomogeneous laminar basic states yields a linear system of complex arithmetic with a leading dimension of the order of 107 to 108. A combined multithread and multiprocessor algorithm is implemented for the direct solution of such linear system. Results for a supersonic boundary layer over a three-dimensional roughness patch show good agreement with experimental measurements when the evolution of the instability waves over the roughness patch is included via the HLNSE.
Boundary Layer Stability↗
Numerical Study of Buoyancy and Different Diffusion Effects on the Structure and Dynamics of Triple Flames
Numerical simulations of 2-D triple flames under gravity force have been implemented to identify the effects of gravity on triple flame structure and propagation properties and to understand the mechanisms of instabilities resulting from both heat release and buoyancy effects. A wide range of gravity conditions, heat release, and mixing widths for a scalar mixing layer are computed for downward-propagating (in the same direction with the gravity vector) and upward-propagating (in the opposite direction of the gravity vector) triple flames. Results of numerical simulations show that gravity strongly affects the triple flame speed through its contribution to the overall flow field. A simple analytical model for the triple flame speed, which accounts for both buoyancy and heat release, is developed. Comparisons of the proposed model with the numerical results for a wide range of gravity, heat release and mixing width conditions, yield very good agreement. The analysis shows that under neutral diffusion, downward propagation reduces the triple flame speed, while upward propagation enhances it. For the former condition, a critical Froude number may be evaluated, which corresponds to a vanishing triple flame speed. Downward-propagating triple flames at relatively strong gravity effects have exhibited instabilities. These instabilities are generated without any artificial forcing of the flow. Instead disturbances are initiated by minute round-off errors in the numerical simulations, and subsequently amplified by instabilities. A linear stability analysis on mean profiles of stable triple flame configurations have been performed to identify the most amplified frequency in spatially developed flows. The eigenfunction equations obtained from the linearized disturbance equations are solved using the shooting method. The linear stability analysis yields reasonably good agreements with the observed frequencies of the unstable triple flames. The frequencies and amplitudes of disturbances increase with the magnitude of the gravity vector. Moreover, disturbances appear to be most amplified just downstream of the premixed branches. The effects of mixing width and differential diffusion are investigated and their roles on the flame stability are studied.
Chen, Jyh-Yuan↗
Calculation of AGARD Wing 445.6 Flutter Using Navier-Stokes Aerodynamics
The flutter characteristics of the first AGARD standard aeroelastic configuration for dynamic response, Wing 445.6, are studied using an unsteady Navier-Stokes algorithm in order to investigate a previously noted discrepancy between Euler flutter characteristics and the experimental data. The algorithm, which is a three-dimensional, implicit, upwind Euler/Navier-Stokes code (CFL3D Version 2.1), was previously modified for the time-marching, aeroelastic analysis of wings using the unsteady Euler equations. These modifications include the incorporation of a deforming mesh algorithm and the addition of the structural equations of motion for their simultaneous time integration with the governing flow equations. In this paper, the aeroelastic method is extended and evaluated for applications that use the Navier- Stokes aerodynamics. The paper presents a brief description of the aeroelastic method and presents unsteady calculations which verify this method for Navier-Stokes calculations. A linear stability analysis and a time-marching aeroelastic analysis are used to determine the flutter characteristics of the isolated 45 deg. swept-back wing. Effects of fluid viscosity, structural damping, and number of modes in the structural model are investigated. For the linear stability analysis, the unsteady generalized aerodynamic forces of the wing are computed for a range of reduced frequencies using the pulse transfer-function approach. The flutter characteristics of the wing are determined using these unsteady generalized aerodynamic forces in a traditional V-g analysis. This stability analysis is used to determine the flutter characteristics of the wing at free-stream Mach numbers of 0.96 and 1.141 using the generalized aerodynamic forces generated by solving the Euler equations and the Navier-Stokes equations. Time-marching aeroelastic calculations are performed at a free-stream Mach number of 1.141 using the Euler and Navier-Stokes equations to compare with the linear V-g flutter analysis method. The V-g analysis, which is used in conjunction with the time-marching analysis, indicates that the fluid viscosity has a significant effect on the supersonic flutter boundary for this wing while the structural damping and number of modes in the structural model have a lesser effect.
Lee-Rausch, Elizabeth M.↗
Comparison of methods for prediction of transition by stability analysis
The paper compares several methods of transition prediction of linear stability analysis. The spectral stability analysis code SALLY is used to analyze flows over laminar flow control wings. It is shown that transition prediction by the envelope method and a new modified wave packet method are comparable in reliability, but that the envelope method is more efficient computationally. This is based on the results which show that the wave packet method provides N factors which are at best as consistent as those of the envelope method; since the wave packet method is at least 3 times as expensive as the envelope method, the latter is recommended for engineering design calculations.
Malik, M. R.↗
Perturbation Effects on a Supercritical C7H16/N2 Mixing Layer
A computational-simulation study has been presented of effects of perturbation wavelengths and initial Reynolds numbers on the transition to turbulence of a heptane/nitrogen mixing layer at supercritical pressure. The governing equations for the simulations were the same as those of related prior studies reported in NASA Tech Briefs. Two-dimensional (2D) simulations were performed with initially im posed span wise perturbations whereas three-dimensional (3D) simulations had both streamwise and spanwise initial perturbations. The 2D simulations were undertaken to ascertain whether perturbations having the shortest unstable wavelength obtained from a linear stability analysis for inviscid flow are unstable in viscous nonlinear flows. The goal of the 3D simulations was to ascertain whether perturbing the mixing layer at different wavelengths affects the transition to turbulence. It was found that transitions to turbulence can be obtained at different perturbation wavelengths, provided that they are longer than the shortest unstable wavelength as determined by 2D linear stability analysis for the inviscid case and that the initial Reynolds number is proportionally increased as the wavelength is decreased. The transitional states thus obtained display different dynamic and mixture characteristics, departing strongly from the behaviors of perfect gases and ideal mixtures.
Okongo'o, Nora↗
Stability analysis for linear systems with time delays
Time delays in the mathematical description of a physical system occur whenever the system is affected not only by conditions at the present time, but also by conditions which have occurred in the past. The tau-decomposition method, as refined by Lee and Hsu, is a method for studying the effects of time delay on the stability of retarded dynamical systems. In this paper, the method is extended to examine the stability and relative stability of: (1) retarded systems with many time delays; and (2) a class of neutral systems with one delay. Applications are included which illustrate the power of the technique.
Barker, L. K.↗
Magnetotail dynamics under isobaric constraints
Using linear theory and nonlinear MHD simulations, we investigate the resistive and ideal MHD stability of two-dimensional plasma configurations under the isobaric constraint dP/dt = 0, which in ideal MHD is equivalent to conserving the pressure function P = P(A), where A denotes the magnetic flux. This constraint is satisfied for incompressible modes, such as Alfven waves, and for systems undergoing energy losses. The linear stability analysis leads to a Schroedinger equation, which can be investigated by standard quantum mechanics procedures. We present an application to a typical stretched magnetotail configuration. For a one-dimensional sheet equilibrium characteristic properties of tearing instability are rediscovered. However, the maximum growth rate scales with the 1/7 power of the resistivity, which implies much faster growth than for the standard tearing mode (assuming that the resistivity is small). The same basic eigen-mode is found also for weakly two-dimensional equilibria, even in the ideal MHD limit. In this case the growth rate scales with the 1/4 power of the normal magnetic field. The results of the linear stability analysis are confirmed qualitatively by nonlinear dynamic MHD simulations. These results suggest the interesting possibility that substorm onset, or the thinning in the late growth phase, is caused by the release of a thermodynamic constraint without the (immediate) necessity of releasing the ideal MHD constraint. In the nonlinear regime the resistive and ideal developments differ in that the ideal mode does not lead to neutral line formation without the further release of the ideal MHD constraint; instead a thin current sheet forms. The isobaric constraint is critically discussed. Under perhaps more realistic adiabatic conditions the ideal mode appears to be stable but could be driven by external perturbations and thus generate the thin current sheet in the late growth phase, before a nonideal instability sets in.
Birn, Joachim↗
Laminar to Turbulence Transition in Boundary Layers due to Tripping Devices
Direct numerical simulations (DNS)and linear stability analysis were performed to identify the mechanisms by which two-and three-dimensional trips promote early transition in subsonic boundary layers over a flat plate. Investigations were conducted for two boundary layer transition trip configurations: one is a two-dimensional rectangular-shaped rod and the other is a three-dimensional zigzag tape. Reynolds numbers based on the trip heights are 683 and 300 for the two-and three-dimensional trips, respectively. Simulations with the two-dimensional trip showed that the flow separates at the top of the trip and forms a long recirculation zone downstream. However, the flow remains steady in the simulation without any external disturbances. Linear stability analysis showed that the inflectional profile in the recirculation zone is strongly unstable to frequencies in the range of 100-300 Hz. Simulations with the external disturbances forced at the frequency of 200 Hz showed that disturbances grow strongly inside the recirculation zone as observed in the Klebanoff1 experiment.
Transition↗