High-amplitude Disturbances Research Articles

ЗАДАЧА О НЕЛИНЕЙНОМ ФИЛЬТРАЦИОННОМ ПОЛЕ ДАВЛЕНИЯ В СРЕДЕ СО СЛАБОСЖИМАЕМЫМ СКЕЛЕТОМ

The solution of the problem of the pressure field during filtration of a compressible fluid in a porous medium with an incompressible skeleton in the presence of high-amplitude disturbances is presented. The equation describing pressure changes during field development takes into account the compressibility of the fluid and is presented in a nonlinear form. The known dependences of the density of the filtered medium on the pressure are approximated by a linear function. The movement is assumed to be one-dimensional and horizontal. The porosity, density and permeability of the porous medium skeleton, as well as the viscosity of the filtered medium are considered constant. The solution of the problem is found using an asymptotic expansion by a formal parameter added in the problem as a factor to the compressibility of the fluid. An approximate analytical solution of the nonlinear problem of the filtration pressure field in the zero and first approximations is found. The zero and first coefficients are represented by solutions of quasi-stationary equations, in which time is included as a parameter through the dimensions of the perturbation zone determined by the law of conservation of mass. It is established that taking into account the nonlinearity leads to a decrease in the size of the zone of disturbances of the pressure field. An approach to determining the upper boundary of the perturbation zone in nonlinear problems of this kind, which is based on the use of conservation laws, is proposed. It is shown that the special case of the zero approximation coincides with the solution of the linear problem obtained by the method of changing successive stationary states. The expressions found expand the possibilities of studying high-amplitude filtration processes, and the proposed approach removes the limitations of classical approaches associated with neglecting the dependence of fluid density on pressure in the divergent term of the continuity equation. The method used makes it possible to construct analytical expressions for decomposition coefficients orders of magnitude higher than the first, in addition, it creates the possibility of studying the contribution of nonlinearity caused by the dependence of permeability and viscosity on pressure.

Effect of unsteady supersonic flow on detonation under different hot jet initiation conditions

To understand the actual operation of detonation engines, it is of great significance to simulate the detonation characteristics after being affected by unsteady supersonic flow. Adaptive mesh refinement method and single-step chemical reaction model were used for simulation calculations. The Navier–Stokes equations and hybrid sixth-order weighted essentially non-oscillatory centered difference scheme was adopted in this study. Three different amplitudes of disturbances (0.1, 0.2, and 0.3) and two hot jet injection times were analyzed. The results showed that a moderate disturbance can improve the stability of detonation wave under the ideal hot jet injection time. It allows detonation wave to stay in the combustion chamber for a longer time, which is beneficial for the release of thrust performance. Although the stability of detonation wave is significantly different between the low and high-amplitude disturbances, the residence time of detonation wave is close to the steady flow case under ideal initiation conditions. When the ignition time is insufficient, successful detonation can occur with low and medium-amplitude disturbances. However, the ignition delay was significantly prolonged with medium-amplitude disturbance. There was also greater instability, which is detrimental to the thrust release. When the inflow has a high-amplitude disturbance, the detonation wave fails to ignite due to insufficient energy.

Multilevel Feature Extraction Method for Adaptive Fault Diagnosis of Railway Axle Box Bearing

Railway axle box bearing fault signal contains high Q-factor resonance and low Q-factor transient shock components with periodic transient shock features that can characterize bearing faults. However, extracting fault features is usually difficult due to noise, transmission paths, and high-amplitude accidental shocks. Therefore, to address the abovementioned problems, the multilevel feature extraction method for adaptive fault diagnosis of railway axle box bearing was proposed. This paper used the maximum second-order cyclostationary blind convolution (CYCBD) to weaken the influence of disturbances, enhancing weak transient shock components. Resonant sparse decomposition(RSSD) is used for the selection of quality factor Q due to its excellent fault feature separation property. Considering the strict fault frequency component periodicity in the envelope spectrum, the envelope spectrum multipoint kurtosis (ESMK) is proposed as a metric. The Q factor is optimized using the gray wolf optimization algorithm (GWO) to obtain an adaptive sparse decomposition method (GWO-RSSD) for extracting bearing transient fault shocks to eliminate the signal effects of high-amplitude disturbance shocks and background noise. The simulation results and measured signal analysis of railway axle box bearing show that ESMK can effectively measure periodic transient shocks under strong shock interference compared with the MED-RSSD method. Thus, GWO-RSSD can adaptively separate the optimal low-Q resonance components, verifying the methods’ effectiveness and superiority.

Numerical study of detonation propagation under the action of supersonic pulsating flow in expanding combustor

In this study, the effect of a velocity-pulsating stoichiometric hydrogen-oxygen mixture entering an expanding combustion chamber on the detonation wave is investigated. The hybrid sixth-order weighted essentially non-oscillatory centered difference scheme was used to solve the Navier–Stokes equations with a one-step two species chemistry model. The results verify that the induction zone is extended under the influence of pulsating flow. The discontinuous structure is formed in the flow field. A lot of unburned jets between the wavefront and the reaction zone were generated, which resulted in insufficient energy release. Coupled with the weakening of the wave strength by the Prandtl–Meyer (P-M) expansion fan, the detonation wave show different degrees of attenuation. The decay of the detonation wave shows different patterns at different amplitudes and periods. When the velocity amplitude is 0.3VCJ, the decay rate of detonation wave increases with the increase of the disturbance period. This attenuation law is different from the amplitude of 0.1VCJ and 0.2VCJ. This is because the discontinuous structure area formed by high-amplitude disturbance is large. Therefore, the longer its existence time is, the more serious the detonation wave attenuation is.

Film thickness, interfacial waviness and heat transfer during downflow condensation of R134a

• Condensation tests are performed with R134a in a 3.38 mm tube. • Simultaneous film thickness and heat transfer measurements are carried out. • The experimental results of R134a are compared to those of R245fa. • The interfacial waves inception is delayed for R134a due to the high vapor density. • Models for condensation heat transfer, film thickness and void fraction are assessed. Coupled measurements of liquid film thickness and heat transfer during condensation inside minichannels are rare in the literature due to the technical difficulties in performing accurate and non-intrusive measures. The availability of such experimental data is fundamental for better understanding the occurring physical mechanisms and for the proper design of heat exchangers. To cover this gap, in the present work liquid film thickness and heat transfer coefficients have been simultaneously measured during vertical downflow condensation inside a 3.38 mm inner diameter channel. Condensation tests have been run with refrigerant R134a at mass flux between 30 and 150 kg m −2 s −1 and 30 °C saturation temperature. The instantaneous liquid film thickness and interfacial waves are detected by combining two complementary optical methods: shadowgraph technique (applied to a sequence of flow pattern images recorded by a high speed camera) and chromatic confocal imaging. In the data analysis, the experimental results of R134a have been discussed in comparison to R245fa condensation data taken during a previous experimental campaign at much lower pressure. On average, R134a displays higher liquid film thickness than R245fa and the appearance of high-amplitude disturbance waves, which promote the thinning of the liquid film and enhance the heat transfer, is detected at higher mass flux for R134a as compared to R245fa. The liquid film thickness is found to decrease with the increasing mass flux for both fluids, except from 30 to 50 kg m −2 s −1 due to the transition between shear stress-driven and gravity-driven downflow condensation. The accuracy of several well-known models for the prediction of heat transfer coefficient, film thickness and void fraction has been assessed with comparison to the experimental data of both R134a and R245fa.

Robustness of Vibrational Control in the Presence of Additive Disturbances

Vibrational control seeks to stabilize an unstable system by judiciously injecting a state-free high-frequency dither. This article presents some robustness properties of vibrational control with respect to additive disturbances in a nonlinear system. We assume that, without disturbances, the appropriately averaged system is regionally asymptotically stable. Using perturbation techniques, our first result shows that the stabilization realized through vibrational control is robust with respect to additive disturbances of sufficiently small amplitude. Indeed, the perturbed system of vibrational control has a robustness feature similar to input-to-state stability in a local region. In the case of periodic disturbances, our second result indicates that vibrational control naturally dampens disturbances of sufficiently high frequency, which allows for relatively high amplitude disturbances. A tight bound for the effect of such periodic disturbances on the ultimate deviation of states from the desired equilibrium is presented. Simulation results from a planar manipulator support the theoretic analysis.

Contribution to the Optimization of the Transient Stability of an Electric Power Transmission Network Using a Universal Power Flow Compensator Controlled by a Three-Stage Inverter

The use of an electrical network as close as possible to its limits can lead to its instability in the event of a high amplitude disturbance. The damping of system oscillations can be achieved by conventional means of voltage and speed regulation but also by FACTS (Flexible AC Transmission Systems) devices, which are increasingly used in power networks. In this work, optimal control coordination between a hybrid power flow controller and a three-level inverter was used to improve the transient stability of a transmission line. The UPFC is a combination of a serial compensator (SSSC) and a parallel compensator (STATCOM) both connected to a DC-LINK DC bus. The SSSC acts as a voltage source for the network and injects a voltage that can be adjusted in phase and amplitude in addition to the network voltage; the STATCOM acts as a current source. The approach used is tested in the Matlab Simulink environment on a single machine network. Optimal controller tuning gives a better transient stability improvement by reducing the transport angle oscillations from 248.17% to 9.85%.

Weakly-nonlinear interactions of broadband Tollmien–Schlichting waves in a non-self-similar boundary layer on an airfoil

Abstract The present investigation is devoted to initial nonlinear stages of laminar–turbulent transition initiated by broadband disturbances in an essentially non-self-similar boundary layer, developing on a two-dimensional airfoil. The measurements are performed at fully controlled disturbance conditions with respect to the distribution of the initial spectra of excited 3D (in general) Tollmien–Schlichting (TS) waves. The main attention is devoted to the investigation of resonant (and non-resonant) interactions and amplification of broadband low-amplitude 3D instability waves in the presence (and in absence) of one or two primary harmonic 2D TS-waves. The broadband, “noise-like” TS-waves are composed of controlled perturbations, pseudo-random in time and space, with a continuous frequency–wavenumber spectrum. These perturbations simulate typical background disturbances usually observed at so-called “natural” disturbance conditions. The harmonic primary TS-wave simulates a primary instability mode, which has been formed by the linear instability mechanism. In fact, this paper represents the third step in a set of detailed investigations (Wurz et al., 2012 [1] ) and Wurz et al., 2012 [2] ) of weakly-nonlinear stages of transition performed in the same base flow (on an airfoil) but at increased complexity of initial disturbance conditions. In contrast to the studies of periodic resonant regimes described in refs. Wurz et al. (2012) [1] and Wurz et al. (2012) [2] , no specific (preinstalled) disturbance frequencies or/and spanwise wavenumbers were excited in the boundary layer (except for the primary wave). So, the flow was free to amplify almost any mode of the frequency–wavenumber spectrum by means of any available mechanisms of TS-wave interaction. Such disturbance conditions are the closest to the so-called “natural” ones, i.e. to those observed typically for uncontrolled external perturbations. Four main groups of regimes of disturbance excitation were studied: (i) two harmonic fundamental waves and a pair of frequency detuned subharmonics, (ii) the low-amplitude broadband disturbances only and (iii) the low-amplitude broadband disturbances together with harmonic 2D primary TS-waves, and (iv) the high-amplitude broadband disturbances only. It is found that in case (i) the superposition of the resonances takes place in a nonlinear way and joint efforts of the two fundamental waves are similar to a fundamental-mode amplitude enhancement, which always leads to the increase of the subharmonic amplification rates. In case (ii) the perturbations develop downstream in agreement with the linear stability mechanism. In case (iii) strong resonant interactions between the broadband perturbations and the harmonic primary TS-wave is observed in a wide frequency range. In case (iv) very similar strong resonant interactions are observed between 2D and 3D spectral modes starting already at significantly lower amplitudes of the 2D modes in the frequency range of the primary fundamental modes. In cases (i), (iii) and (iv) the amplification is double-exponential (exponent in exponent), similar to that found in isolated tuned and detuned resonances of deterministic modes in papers Wurz et al. (2012) [1] and Wurz et al. (2012) [2] . For all resonantly interacting disturbances the phase synchronism conditions are found to be satisfied. Based on the cumulative results obtained in the framework of this intensive investigation, the most important observation is the predominant role of resonant TS-wave interactions in the non-self-similar boundary layer developing on a natural laminar flow airfoil section.

Receptivity to Freestream Acoustic Noise in Hypersonic Flow over a Generic Forebody

Direct numerical simulations of the Navier–Stokes equations have been performed to investigate the receptivity and breakdown mechanisms in a Mach 6 flow over a generic forebody geometry with freestream acoustic disturbances. The simulations are based on transition experiments carried out in April 2015 in the Boeing/U.S. Air Force Office of Scientific Research Mach 6 facility at Purdue University. A three-dimensional model for both fast and slow freestream acoustic waves with multiple frequencies and spanwise wave numbers has been adopted in the numerical simulations, for which high-amplitude disturbances have been considered in order to simulate noisy wind-tunnel conditions. The numerical results revealed similarities in comparison to the experimental observations, especially when slow acoustic waves were considered as freestream disturbances. In particular, slow acoustic waves have been found to induce the breakdown process via crossflow instabilities located in the off-centerline region, with formation of streamwise streaks. Fast acoustic waves, in contrast, appeared more efficient in inducing earlier nonlinear growth through destabilization of the boundary layer along the symmetry plane of the body.

Enhancement of disturbance wave amplification due to the intrinsic three-dimensionalisation of laminar separation bubbles

ABSTRACTPrevious studies demonstrated that laminar separation bubbles (LSBs) in the absence of external disturbances or forcing are intrinsically unstable with respect to a three-dimensional instability of centrifugal nature. This instability produces topological modifications of the recirculation region with the introduction of streamwise vorticity in an otherwise purely two-dimensional time-averaged flows. Concurrently, the existence of spanwise inhomogeneities in LSBs have been reported in experiments in which the amplification of convective instability waves dominates the physics. The co-existence of the two instability mechanisms is investigated herein by means of three-dimensional parabolised stability equations. The spanwise waviness of the LSB on account of the primary instability is found to modify the amplification of incoming disturbance waves in the linear regime, resulting in a remarkable enhancement of the amplitude growth and a three-dimensional arrangement of the disturbance waves in the aft portion of the bubble. Present findings suggest that the oblique transition scenario should be expected in LSBs dominated by the convective instability, unless high-amplitude disturbances are imposed.

Nonlinear Interactions Between the Fundamental and Higher Harmonics of Self-Excited Combustion Instabilities

ABSTRACTMeasurements of self-excited combustion instability were performed in a lean-premixed, swirl-stabilized, gas turbine combustor in order to characterize the interaction between the fundamental and higher harmonics of dynamic pressure, heat release rate, and acoustic velocity perturbations. The formation of higher harmonics, their interactions with the fundamental, and the influence of these interactions on nonlinear combustion processes are highlighted in this article. A detailed analysis of the fundamental and higher harmonics of the pressure signal reveals that for a small amplitude disturbance the second harmonic of the pressure signal is governed by a quadratic function. For high amplitude disturbances, the measured data deviate from the power law dependence; this is a manifestation of nonlinear energy transfer toward higher harmonic components. The disturbance amplitude at which the inception of the non-quadratic dependence occurs coincides with the transition from linear to nonlinear heat release response. In particular, the higher harmonic characteristics are significantly altered by varying swirler location, which is known as a primary source of vortical disturbance in swirl-stabilized combustion systems. As a result, for certain inlet conditions, the amplitude of the higher harmonic of acoustic velocity is even higher than the amplitude of the fundamental, and this leads to multiple-frequency excitation. The system then evolves toward a periodic state that is completely different from that of a system with single frequency excitation.

Experimental investigation of a low-amplitude transition in pipe flow

The vast majority of experiments on transition in pipe flow have been concerned with high-amplitude disturbance (of of the centerline velocity) at Reynolds numbers around 2000 and above. In this experimental study, we concentrate on the transition process in water pipe flow where the level of the disturbance is at least an order of magnitude smaller. We follow the sequence of transitional events while varying gradually the disturbance level from a laminar state to an almost fully turbulent one, using flow visualization, pressure-drop and hot-wire measurements. This is accomplished by injection of a very small diameter jet perpendicular to the main stream at a controllable flow rate, in a downstream distance where the flow is approximately fully developed. The injection flow rate normalized by the main stream rate is of . With increasing injection flow rates the friction coefficient (λ) increases along with changes in the nature of the flow structures. The transition begins with the generation of a streamwise counter-rotating vortex pair, followed by the formation of a packet of hairpins and their breakdown. As the injection level is increased, the separation distance between two consecutive bursts (wavelength) decreases and the breakdown to turbulence begins further upstream and consequently the value of the friction coefficient λ increases. Above and below a certain threshold of the disturbance level, turbulence is either triggered or the flow is relaminarized, respectively. Finally, it is established that the evolution of the packet of hairpins is a key element during this transition scenario, and is well explained by the three-element model recently proposed by Cohen et al ().

Enhanced instability of supersonic boundary layer using passive acoustic feedback

Tripping of boundary-layers from laminar to turbulent flow, which may be needed in specific practical applications, requires introducing high amplitude disturbances into the boundary layers without large drag penalties. As a possible improvement on fixed trip devices, a technique based on feedback instability is demonstrated in the present contribution for a Mach 1.5 boundary layer. The compressible Navier-Stokes equations are solved directly using a high-order (fifth-order in space and third-order in time) finite difference method. The geometries tested include wall-mounted rectangular blocks and off-surface cylinders. For a particular combination of an upstream cylinder and downstream block, a feedback loop is observed to provide more robust self-amplification of disturbances than a cylinder alone, whereas blocks by themselves only enhance a convective instability. A Fourier analysis and an impulse response simulation are used to confirm the presence of an acoustic feedback loop for the cylinder-block configuration. A parametric study of the size of the cylinder, the height, and the location of the rectangular block is also provided. It is observed that the feedback strength is not monotonically related to the height and location of the rectangular block.

A Nonsingular Terminal Sliding Mode Approach Using Adaptive Disturbance Observer for Finite-Time Trajectory Tracking of MEMS Triaxial Vibratory Gyroscope

This paper develops a nonsingular terminal sliding mode controller (NTSMC) with adaptive disturbance observer (ADOB) for finite-time trajectory tracking of a MEMS triaxial vibratory gyroscope, which has parameter variations and external high-amplitude disturbance. A novel sliding mode controller with adaptive disturbance observer is designed to reconfigure the parameter variations and external high-amplitude disturbance and reduce the chattering phenomenon on the sliding surface greatly through setting the switching gain in control signal as a smaller value. The disturbance adaptive law is derived to guarantee the stability of the closed-loop adaptive control system via the Lyapunov criterion. The simulation results are performed to verify the effectiveness of the presented schemes.

Nonlinear vibration suppression of flexible structures using nonlinear modified positive position feedback approach

This paper introduces Nonlinear Modified Positive Position Feedback (NMPPF) control approach for nonlinear vibration suppression at primary resonance. Nonlinearity in the system is due to large deformations caused by high-amplitude disturbances, while this control approach is applicable to all types of nonlinearities in resonant structures. NMPPF controller consists of a resonant second-order nonlinear compensator, which is enhanced by a lossy integrating compensator. The two compensators create a combination of exponential and periodic control inputs, which needs innovative time scaling for using the Method of Multiple Scales to obtain the analytical solution of the closed-loop system. The results of the analytical solution for the closed-loop NMPPF controller are presented and compared with the result of the conventional PPF controller. Effects of the control parameters on the system response are comprehensively studied by parameter variations. The approximate solution is then verified using numerical simulations. According to the results, the NMPPF controller provides a higher level of suppression in the overall frequency domain, as the peak amplitude at the neighborhood frequencies of the primary mode is reduced by 44 %, compared to the PPF method. The tunable control parameters also give more flexibility to create the expected type of system response.

Numerical simulation of vortex-induced vibration of two circular cylinders of different diameters at low Reynolds number

Two-degree-of-freedom Vortex-Induced Vibration (VIV) of two rigidly coupled circular cylinders of different diameters at a low Reynolds number of 250 is investigated numerically. While the diameter ratio and the mass ratio are kept constant, the study is focused on the effect of the position angle of the small cylinder on the lock-in regime of the VIV. Simulations are carried out for position angles α of the small cylinder ranging from 0° to 180° with an interval of 22.5° and the reduced velocities ranging from 1 to 15 with an increment of 1. In order to find the effect of the gap between the two cylinders on the vibration, two gap-to-diameter ratios (0 and 0.2) are considered. It is found that compared with a single cylinder case, the lock-in regime of the reduced velocity is widened significantly when the position angle of the small cylinder is α = 0°, 22.5°, 90°, or 112.5°. Pulsed beating phenomenon characterized by regular vibration with occasional high-amplitude disturbances at regular or irregular intervals is observed at G = 0 and α = 90°. At α = 135°, more than one lock-in regimes are observed in the computed range of reduced velocity for both gaps (G = 0 and 0.2). Setting a small gap (gap-to-diameter ratio of 0.2) between the two cylinders mitigates the vibration by narrowing the lock-in regime and reducing the vibration amplitude.

Acoustic characteristics of Swedish dorsal fricatives

The identity of [“hooktop heng”], the “voiceless dorso-palatal/velar fricative” or “simultaneous” [“esh”] and [x], is controversial (IPA 2009: 140, 203). The present study attempts to acoustically differentiate [“esh”], [x], and (non-labialized) [“hooktop heng”]. A female speaker of Stockholm Swedish produced VCV nonsense words with balanced V and alternating C. Windowed spectra were obtained and averaged at various locations in the fricatives. [“esh”] can be reliably differentiated from [x] and [“hooktop heng”] based on center of gravity measures in the middle of the fricative. However, center of gravity differences between [x] and [“hooktop heng”] are not revealing, confirming Lindblad's x-ray tracings (1980: 89). The most robust acoustic difference between the two dorsal fricatives is the presence of erratic, high-amplitude disturbances during [x]. These may be attributed to sudden explosions of saliva and ephemeral contact with the uvula, more likely for slightly-retracted [x] than for [“hooktop heng”]. It seems unlikely that [“hooktop heng”] is produced with a second simultaneous constriction that is of acoustic relevance. International Phonetic Association. 1999. Handbook of the International Phonetic Association. Cambridge: Cambridge University Press. Lindblad, P. 1980. Svenskans sje- och tje-ljud i ett Allmänfonetisk perspektiv. Travaux de l'Institut de linguistique de Lund 16. Lund: LiberLäromedel/Gleerup.

Signal Feature Extraction From Microbarograph Observations Using the Hilbert–Huang Transform

The Hilbert-Huang transform (HHT) is a relatively new time-frequency analysis tool. We present a new signal feature extraction technique based on HHT. This technique is used to extract diurnal and semidiurnal tides from atmospheric pressure data obtained from a microbarograph stationed at the Arecibo Observatory. Observation of seasonal variations of semidiurnal tides is possible due to the high precision offered by this technique. Furthermore, we apply the signal extraction procedure to isolate and remove high-amplitude disturbance signals from the time-series signal. This is demonstrated by extracting a hurricane event from the pressure data. The superior capabilities of the HHT-based technique to analyze time-varying signals compared to traditional linear techniques such as the wavelet transform and the fast Fourier transform are demonstrated.

Experimental investigations of rotating flow instabilities in the last stage of a low-pressure model steam turbine during windage

Experimental investigations on two different three-stage low-pressure (LP) model turbines operated on steam during windage are presented. At very low flows, strain gauges identified that the two turbines exhibit different dynamic stress characteristics of the last-stage rotor blades. The first turbine experienced the highest vibrations from aerodynamic excitations at the resonant frequency of the second mode in a narrow operating range around 13 per cent of the design flow, whereas the second turbine experienced no resonance up to the fourth mode and has the highest dynamic loading at zero flow. Specially developed high-response total pressure probes have been used to determine the cause of the aerodynamic excitation from traverses of the unsteady pressure field at three planes in the last stage. The experimental data for both turbines show that unsteady pressure disturbances steadily grow when the flow is reduced below 25 per cent of design flow and that the excitations are strongest in the axial gap between the guide vane and the rotor of the last stage near the outer casing. Detailed analysis shows that high-amplitude disturbances occur at distinct frequencies and that these rotate in the circumferential direction at a fraction of the rotor speed. Comparison of the pressure signals measured at two circumferential locations on the casing of the second turbine confirmed the characteristic frequency pattern to be a so called ‘rotating instability’. This unsteady phenomenon arising from the tip leakage flow has previously been observed in axial flow fans and compressors and is demonstrated for the first time here in a turbine operating at very low flow rates.

“Anomalous” nonlinear wave phenomena in a supersonic boundary layer

Nonlinear evolution of high-amplitude periodic disturbances in a boundary layer on a flat plate for Mach numberM=2 is studied. An anomalous downstream evolution of the disturbances is found, quasi-two-dimensional disturbances being most unstable. The obtained phase velocities of the waves are 30–40% greater than the phase velocities of the Tollmien-Schlichting waves. The nonlinear evolution of vortex waves is accompanied by an increase in steady disturbances from the source of controlled vibrations. High-frequency disturbances decay, and a periodic wave train degenerates downstream into a quasiharmonic wave train.