Instability Of Wave Packets Research Articles

Perturbation-based active flow control in overexpanded to underexpanded supersonic rectangular twin jets

The effects of perturbation-based active flow control on supersonic rectangular twin jets (SRTJ) over a wide range of nozzle pressure ratios (NPR = 2.77 to 6.7, corresponding to fully expanded Mach numbers Mj = 1.3 to 1.9) were investigated. The aspect ratio and design Mach number for the bi-conic, converging-diverging nozzles were 2 and 1.5, respectively. The flow and acoustic fields of SRTJ are known to couple, often generating high near-field (NF) pressure fluctuations and elevated far-field (FF) noise levels. Large-scale structures (LSS), or equivalently instability waves or wave packets, are responsible for mixing noise, broadband shock-associated noise, screech and coupling. The primary objective of this research was to manipulate the development of LSS in this complex flow to better understand and mitigate their effects. The organization and passage frequency of the LSS were altered by excitation of instabilities over a wide range of frequencies and modes. Key findings include: (1) the screech mode of each jet was flapping along its minor axis; (2) the jets coupled, out-of-phase primarily in overexpanded cases and in-phase primarily in underexpanded cases, along the minor axis of the SRTJ; (3) coupling has significant effects on the NF pressure fluctuations, but only minor effect on the FF noise; (4) standing waves were observed only on the minor axis plane of the SRTJ; (5) altering or suppressing coupling can significantly reduce NF pressure fluctuations; (6) two high-frequency excitation methods proved effective in reducing the FF noise; and (7) nonlinear interactions between the screech tones and excitation input were observed in controlled cases in which screech was only partially suppressed.

Active control of jet-plate interaction noise for excited jets by plasma actuators

This paper examines the possibility of controlling jet-wing interaction noise by plasma actuators. The low-frequency part of the jet installation noise is considered as produced by the diffraction of instability wave packets developing in the jet shear layer. Thus, the main idea of noise control comes to the attempt to reduce the amplitude of these wave packets near the wing trailing edge. A simplified jet-plate configuration is studied experimentally for jet Mach numbers ranging from 0.4 to 0.6. To avoid additional complications related to the control of stochastic signals in the first step, the jet is excited by a loudspeaker at a frequency corresponding to Strouhal number 0.6. Acoustic forcing generates axisymmetric instability wave in the jet shear layer, which is the object of control. The control action is implemented by a high-frequency dielectric barrier discharge (HF DBD) plasma actuator with a ring-like electrode mounted inside the nozzle near the exit. It is demonstrated that installation noise can be significantly suppressed if the plasma actuator generates instability wave with the amplitude equal to that of the excited by the loudspeaker, but in antiphase to it, and vice versa, if these instability waves are in phase, installation noise increases by about 6 dB. The obtained results support the idea that low-frequency jet-wing installation noise can be controlled in a linear framework.

Unstable wave-packet perturbations in an advective Cahn-Hilliard process.

A study of the perturbation dynamics in a one-dimensional advective Cahn-Hilliard system, characterized by a nonvanishing driving force, is carried out to test the stability of a uniform basic solution. The linear stability of small-amplitude perturbations is analyzed both in the case of normal Fourier modes, with a given wave number, and in the case of wave packets localized in space. The dual nature of the instability, either of convective or absolute type, is studied, revealing that the driving force creates a gap between the parametric threshold to instability of normal modes and that to instability of wave packets. When the driving force is zero, also the gap between such thresholds disappears.

Experiments on localized secondary instability in bypass boundary layer transition

An experimental study on localized secondary instability of unsteady streamwise streaks in bypass boundary layer transition under an elevated level of free-stream turbulence has been carried out mainly using the particle image velocimetry (PIV) technique. Simultaneous orthogonal dual-plane PIV measurements were performed for a concurrent examination of the transitional flow features in both wall-normal and spanwise planes. These quantitative and simultaneous visualizations clearly show the wall-normal view of a low-speed streak undergoing sinuous/varicose motion in the spanwise plane. An oscillating shear layer in the wall-normal plane is found to be associated with the sinuous/varicose streak oscillation in the spanwise plane. Further, these measurements indicate that a localized secondary instability wavepacket can originate near the boundary layer edge. The time-resolved PIV measurements in the wall-normal plane clearly show how an instability develops on a lifted-up inclined shear layer and leads to flow breakdown. The estimated wavelength and convection velocity of such instabilities are found to compare well with those calculated from the one-dimensional linear stability analysis of the spatially averaged velocity profiles associated with the lifted-up shear layers. The time-resolved PIV measurements in the spanwise plane also facilitate quantitative visualizations of sinuous and varicose instabilities. These measurements experimentally confirm that a varicose instability at the juncture of an incoming high-speed streak and a downstream low-speed streak can eventually lead to the formation of lambda structures. The estimated convection velocity, wavelength and growth rate of these instabilities are found to be consistent with the numerical results reported in the literature. Moreover, the streak secondary instability is found to be apparent in the velocity contours, while the estimated streak amplitude is approximately 30 % of the free-stream velocity.

Modeling of coherent structures in a turbulent jet as global linear instability wavepackets: Theory and experiment

Perturbation wavepackets in a jet, induced by externally applied forcing, are computed from the linearized equations of motion. An experimentally measured mean flow serves as the base state for this linear analysis, and optimally amplified mono-frequency perturbations are identified, in the sense of maximal gain between the forcing energy input and the flow response energy. Sub-optimal orthogonal forcing/response structures are also discussed. Linear analysis results are then compared to measured perturbation wavepackets in the jet experiment. The study addresses the question to what extent the true dynamics of a turbulent jet can be represented by a model based on linear instability.

Modulation instability of pulsed radiation in an optical waveguide in the presence of the traveling refractive index wave

The conditions for the onset of modulation instability of wave packets in an optical waveguide in the presence of the traveling refractive index wave are investigated. An expression governing the growth rate of a small harmonic perturbation at the early development stage of the modulation instability is obtained. Based on numerical analysis, the behavior of the wave packet at the developed modulation instability stage is studied for different parameters of the waveguide and the refractive index wave.

On the accuracy of dynamic mode decomposition in estimating instability of wave packet

Lots of unstable flows in both nature and engineering pose multi-scale perturbations with infinitesimal initial amplitude, which compete and interact with each other during their unstable evolution. Dynamic mode decomposition (DMD) analysis can be used to extract these components’ temporal/spatial growth rate. Therefore, it is necessary to evaluate the accuracy performance and confidence limit of DMD algorithm in the circumstance of multi-scale instability wave packet. In the present study, we use a linear combination of a sinusoidal unstable wave and its high-order harmonics as a prototype, based on which an error analysis of DMD algorithm is taken. In first, different numerical algorithms of DMD analysis are compared in terms of both accuracy and efficiency. The accuracy evaluation of the classical DMD algorithm in a large parameter domain is followed. It is found that the superimposition of finer structures with less energy dominance might damage the estimation accuracy of the primary structures’ growth rate. Strong evidences suggest that even in a linear circumstance, resolving the dynamics of small-scale structures is comparably more difficult than that of the primary structures, i.e., DMD algorithm has a preference for structures with energetic dominance. Finally, the recommended thresholds for the sampling/discretizing parameters are summarized for practical usage.

Pressure fluctuations beneath instability wavepackets and turbulent spots in a hypersonic boundary layer

AbstractTo investigate the pressure-fluctuation field beneath turbulent spots in a hypersonic boundary layer, a study was conducted on the nozzle wall of the Boeing/AFOSR Mach-6 Quiet Tunnel. Controlled disturbances were created by pulsed-glow perturbations based on the electrical breakdown of air. Under quiet-flow conditions, the nozzle-wall boundary layer remains laminar and grows very thick over the long nozzle length. This allows the development of large disturbances that can be well-resolved with high-frequency pressure transducers. A disturbance first grows into a second-mode instability wavepacket that is concentrated near its own centreline. Weaker disturbances are seen spreading from the centre. The waves grow and become nonlinear before breaking down to turbulence. The breakdown begins in the core of the packets where the wave amplitudes are largest. Second-mode waves are still evident in front of and behind the breakdown point and can be seen propagating in the spanwise direction. The turbulent core grows downstream, resulting in a spot with a classical arrowhead shape. Behind the spot, a low-pressure calmed region develops. However, the spot is not merely a localized patch of turbulence; instability waves remain an integral part. Limited measurements of naturally occurring disturbances show many similar characteristics. From the controlled disturbance measurements, the convection velocity, spanwise spreading angle, and typical pressure-fluctuation field were obtained.

Instability of wave packets in nonlinear inhomogeneous waveguides

Conditions for modulation instability of a wave packet to occur in cubically nonlinear single- and double-mode waveguides with various distributions of dispersion parameters over the waveguide length are investigated. Analytical expressions are obtained for the integral gain increment and other characteristics governing the modulation instability dynamics. Features of the dynamics are revealed using numerical simulation.

Extreme waves in random crossing seas: Laboratory experiments and numerical simulations

[1] We present an experimental and numerical investigation on the statistical properties of the surface elevation in crossing sea conditions. Experiments are performed in a very large wave basin (70 m × 50 m × 3 m) and numerical results are obtained using a higher order method for solving the Euler equations. Both experimental and numerical results indicate that the number of extreme events depends on the angle between the two interacting systems. This outcome is supported by recent theoretical investigations which have highlighted that the instability of wave packets may be triggered by the nonlinear interactions between coexisting, non-collinear wave systems.

A Note on Noise Generation by Large Scale Turbulent Structures in Subsonic and Supersonic Jets

This paper describes a method to connect the measured spectral density in the acoustic far field with the wavenumber/frequency spectrum of the near field fluctuations that produce the noise. A relationship is first derived between the far field spectral density and the wavenumber/frequency spectrum of the pressure fluctuations on a cylindrical surface surrounding the jet in the near field. Measurements of the far field spectral density are then decomposed into contributions from the Large Scale Similarity (LSS) and Fine Scale Similarity (FSS) spectra. The near field wavenumber/frequency spectrum associated with the LSS spectral density alone is then determined. It is shown to have a very similar form for a range of jet operating conditions and Strouhal numbers. This form is consistent with an instability wave or wave packet model. Since both subsonic and supersonic jet exit conditions are examined, it is argued that the noise radiation in the peak radiation directions is controlled by the axial evolution of the turbulent large scale structures and is not associated with a combination of source convection and mean flow refraction effects.

Instability wave control in a subsonic round jet

It is generally accepted that one of the main sources of acoustic radiation from a turbulent jet is due to spatial instability wave packets propagating downstream within the jet. This approach has enabled to explain and predict the principal features of sound radiated by a supersonic jet. Therefore the problem of noise control for the jet could be considered as a problem of instability wave control. Instability wave developed in the tone‐excited jet issued from axisymmetric, semi‐infinite duct is considered. The flow field in the vicinity of the nozzle edge is investigated for the purpose of identifying instability wave parameters. Instability wave turned out to be damped completely on the condition that the amplitude and the phase of the external forcing are chosen in accordance with those of the instability wave. The robust strategy of instability wave experimental suppression based on the near field measurements is formulated. Similar problem in two‐dimensional case was recently analyzed in Kopiev & Fa...

Acoustic near field of a transonic instability wave packet

We consider the problem of acoustic radiation generated by a spatial instability wave on a weakly developing shear flow. Assuming a local WKBJ approximation for the instability wave near its maximum, we compute the acoustic pressure field by using a Fourier transform along the streamwise direction. When the instability wave is close to transonic near its maximum amplitude, approximations for this pressure field are obtained by a steepest descent method. A branch cut and several saddle points are shown possibly to contribute to the approximation. A detailed analysis of these contributions is provided. The modifications of the acoustic field when we pass from subsonic to supersonic are examined. In particular, the superdirective character of the acoustic field of subsonic instability waves and the directivity pattern of supersonic waves are shown to be both compatible with our mathematical description and associated with a single saddle-point contribution.The acoustic near field is also shown to possess a caustic around which a specific approximation is derived. In a large region of the physical space, the near field is composed of two saddle-point contributions. Close to the shear flow, one of these contributions degenerates into a branch-point contribution which always becomes dominant over the instability wave downstream of a location that is computed. An interesting phenomenon is observed in certain regions downstream of the maximum: the transverse behaviour of the instability wave has to be exponentially growing far from the shear layer to match the acoustic field. We demonstrate that this phenomenon neither requires a branch-point contribution nor a supersonic instability wave.

Modulation instability of wave packets in the presence of linear and nonlinear mode coupling

Conditions for modulation instability of a wave packet consisting of two codirected linearly coupled waves propagating in a light guide with Kerr-type nonlinearity are formulated. The development of the instability with regard to light guide parameters, linear and nonlinear mode coupling, and input power is analyzed for symmetric and antisymmetric light guide excitations. Unlike in single-mode light guides, here modulation instability may arise in the frequency ranges of normal material dispersion and at the zero perturbation frequency.

Aeroacoustics of Supersonic Jet Issued from Corrugated Nozzle: New Approach and Prospects

It is generally accepted that one of the main sources of acoustic radiation from a supersonic jet is due to spatial instability wave packets propagating downstream within the jet. This approach has enabled us to explain and predict the principal features of sound radiated by a supersonic jet with a circular nozzle. The aim of this present work is to generalize such an approach to jets with a circular nozzle deformed into a weakly corrugated (s-lobed) shape. This leads to the presence of two additional parameters, the lobe number (an integer) characterizing the corrugation wavelength and the corrugation amplitude. Their presence can be shown to lead to a resonant coupling between different instability waves which, in turn, can both intensify and suppress the aerodynamic noise. The assumption of a small corrugation amplitude allows us to consider the problem analytically using disturbance theory methods. The structure of the eigen-oscillations is determined at leading orders, where it is shown that differing azimuthal harmonics are coupled over a wide range of frequencies. The result of such couplings on the radiated sound is significant, even for a small corrugation amplitude. The above suggests that the effect of mode coupling, due to nozzle corrugation, could be used as a tool for noise control in supersonic jets.

Study of the Instability of Wave Packets in Fluidized-Bed Furnaces within the Framework of Ginzburg–Landau Equation

Study of the Instability of Wave Packets in Fluidized-Bed Furnaces within the Framework of Ginzburg–Landau Equation

Experimental study of acoustic radiation from a boundary layer transition region

Wall pressure fluctuations were measured on a rigid axisymmetric body in the CEPRA 19 low‐noise, anechoic wind tunnel, using flush‐mounted microphones placed from the laminar region to the fully turbulent boundary layer. Microphones placed in the laminar flow region are used to detect noise radiated from the transition region, which occurs naturally, without separation, under a slightly positive pressure gradient. Cross‐spectral analyses show upstream acoustic propagation in a very wide frequency band, 4–30 kHz, detected in the laminar region. A method of conditional analysis is then used to establish the sequence of events from the onset of near‐harmonic instability wave packets to the generation, about 10 ms later, of turbulent spots leading to the acoustic emission. This intermittent acoustic radiation is detected in the nearfield for wind velocities ranging from 20–70 ms. Farfield detection was not achieved probably because of instrument limitations and propagation effects. [Work supported by DRET, Direction des Recherches et Etudes Techniques.]

Sound Radiation by Instability Wavepackets in a Boundary Layer

Sound radiation by instability wavepackets evolving from general initial conditionsin a low‐Mach‐number, slightly unstable boundary layer is studied. The formulation can be viewed as an exact instance of Lighthill's [13] acoustic analogy. The original disturbance may be line‐ or point‐centered with an infinitesimal amplitude. The sound field is confined within an expanding sphere. The directivity of sound is beamed and points upstream. This “superdirectivity” is slowly enhanced in time whereas the sound level is slowly attenuated. Nonlinear processes, once they come into play, reverse the temporal behavior of the sound level and sound directivity. For the boundary‐layer profiles studied, nonlinearity causes a “burst,” in a finite time, of the sound field in the form of a dipole.

Local Instabilities of Quantum Trajectories in Non-Linear Dynamical Systems

Abstract Local instability of wave packets is related to an exponential increase of the second moments of the components ⊿Pi2 and ⊿Qi2 of the momentum and position operator as a function of the expectation values <Pi> and <Qi>• Equations of motion for ⊿Pi2 and ⊿Qi2 are derived. In the quasiclassical limit, the instability criterion approaches the Toda-Brumer criterion [1, 2] for classical trajectories.