Stationary Disturbances Research Articles

Persistence of the laminar regime in a flat plate boundary layer at very high Reynolds number

Starting from the Navier-Stokes and the continuity equations of a viscous incompressible fluid, a statistical theory is developed for the prediction of transition and breakdown to turbulence in a laminar boundary layer exposed to small, statistically stationary axisymmetric disturbances. The transport equations for the statistical properties of the disturbances are closed using the two-point correlation technique and invariant theory. By considering the local equilibrium to exist between production and viscous dissipation, which forces the energy of the disturbances in the boundary layer to be lower than that of the free stream, the transition criterion is formulated in terms of the anisotropy of the disturbances and a Reynolds number based on the intensity and the length scale of the disturbances. The transition criterion determines conditions that guarantee maintenance of the laminar flow regime in a flat plate boundary layer. It is shown that predictions of the transition onset deduced from the transition criterion yield the critical Reynolds number, which is in good agreement with the experimental data obtained under well-controlled laboratory conditions reported in the literature. For the preferred mode of the axisymmetric disturbances, for which the intensity of the disturbances in the stream wise direction is larger than in the other two directions, the analysis shows that the anisotropy increases the critical Reynolds number. Theoretical considerations yield the quantitative estimate for the minimum level of the anisotropy of the free stream required to prevent transition and breakdown to turbulence. The numerical databases for fully developed turbulent wall-bounded flows at low and moderate Reynolds numbers were utilized to demonstrate the stabilizing and destabilizing role of the anisotropy in the disturbances on the development of the transition process in wall-bounded flows. The stabilizing role of increased anisotropy in a free stream on the boundary layer development was successfully tested experimentally in a large wind tunnel by maintaining the stable laminar regime in a flat plate boundary layer up to (Rex)T

4·106.

Experiments on secondary instability of streamwise vortices in a swept-wing boundary layer

A detailed experimental study on the formation of crossflow vortex mode packets and their high-frequency secondary instability in a swept-wing boundary layer was carried out. Stationary vortex packets are most likely to be generated under natural flight conditions and transition to turbulence is quickest within these disturbances. In the present experiments, different methods of controlled excitation are used so that the crossflow vortex packets are generated by surface-roughness elements and by localized continuous suction. It is found that as the stationary disturbance reaches a significant amplitude, of about 10% of the free-stream velocity, while being below the saturation level, high-frequency secondary instabilities start to grow. Influence of the crossflow vortex packet magnitude on the development of the secondary instability is investigated in detail and below its threshold the crossflow vortex packet was found to be nearly neutrally stable. By studying the unstable packets, the frequency of natural secondary perturbations was identified and the travelling disturbances were forced in a controlled manner by periodic blowing–suction applied locally under the stationary vortex. Two modes of secondary instability were found to develop and the preferred mode was dependent on the properties of the primary stationary disturbance. Additionally, the underlying physics of the process of nonlinear formation and development of the vortices in the boundary layer is clarified. It was observed that the large-amplitude co-rotating vortices may interact, thus reducing their amplitude. Also a large-scale excitation by an isolated roughness element produced two individual stationary crossflow vortex packets at its tips, each with different preferred secondary instability modes.

Receptivity of stationary transient disturbances to surface roughness

Recent experiments on transient disturbance growth in boundary layers indicate that disturbances generated by surface roughness undergo suboptimal growth. The implication is that the receptivity mechanism governing the distribution of disturbance energy among the continuous spectrum of damped Orr–Sommerfeld/Squire modes produces energy distributions that are significantly different from the theoretical optimum. Experiments presented here are intended to investigate how the amplitude and diameter of cylindrical roughness elements arranged in a spanwise array affect various features of transient growth. The objective is to infer how or to what extent the roughness features affect the continuous spectrum and to use this information as a foundation for future receptivity models. The results show that the energy of stationary disturbances varies as Rek2 and that the streamwise distance over which the disturbances grow increases slightly with increasing Rek. As the roughness diameter is varied, dramatic changes in the qualitative nature of the resulting transient growth occur. Both the variation of the growth length with Rek and the behavioral changes with roughness diameter indicate that the energy distribution among the continuous modes is a strong function of roughness features and that an accurate and sophisticated receptivity model will be necessary to accurately predict transient growth.

Responses of discretized systems under narrow band nonstationary random excitations. Part 1: linear problems

The extended stochastic central difference (ESCD) method is proposed as a viable alternative for computing linear responses of discretized multi-degrees-of-freedom (mdof) systems under narrow band stationary and nonstationary random disturbances. The method provides a means of controlling the center frequencies and bandwidths of narrow band stationary and nonstationary random excitation processes. It is suitable for larger-scale random response analysis of complicated structures idealized by the finite element method. Its additional important feature is that application of normal mode or complex normal mode analysis or direct numerical integration algorithms such as the fourth-order Runge–Kutta scheme is not required. Examples, including one of flow-induced vibration of a pipe containing a moving fluid are included to demonstrate: (1) the capability of the proposed method and difference between responses of discretized systems under narrow band and wide band random excitations, and (2) its accuracy and efficiency by way of comparison to the Monte Carlo simulation data. Generalization of the ESCD method for computation of responses of nonlinear mdof systems is presented in a companion paper.

A joint monitoring scheme for automatically controlled processes

It has been proposed to integrate statistical process control techniques with Automatic Process Control (APC) or engineering process control schemes to ensure continuous quality improvement. The data streams collected from an APC-controlled process have been shown to be critically dependent on the process settings. This paper proposes a univariate approach which combines the information provided by both the process output and input control actions based on a uniformly most powerful test. The proposed control chart can be customized to be sensitive to large or small shifts in a linearly-controlled process with an optimal efficiency in either transient or steady-state conditions. The presented numerical study is restricted to consider only APC-controlled stationary disturbances.

Optimal Disturbances in Swirling Flows

The mechanism of optimal disturbances, which are composed of both growing and decaying modes, in swirling flows (with an axial velocity component) is explored. Two characterizing cases are considered: stationary disturbances in rotating pipe flow and temporal disturbances in a trailing vortex

Frequency Domain Gaussian Estimation of Temporally Aggregated Cointegrated Systems

This paper discusses the joint estimation of the long run equilibrium coefficients and the parameters governing the short run dynamics of a fully parametric cointegrated system formulated in continuous time. The model allows the stationary disturbances to be generated by a stochastic differential equation system and for the variables to be a mixture of stocks and flows. We derive a precise form for the exact discrete analogue of the continuous time model in triangular error correction form, which acts as the basis for frequency domain Gaussian estimation of the unknown parameters using discrete time data. We formally establish the order of consistency and the asymptotic sampling properties of such an estimator. The function of the data that estimates the cointegrating parameters is shown to converge at the rate of the sample size to a mixed normal distribution, while that estimating the short run parameters converges at the rate of the square root of the sample size to a limiting normal distribution.

Smart panel with multiple decentralized units for the control of sound transmission. Part I: theoretical predictions

This is the first of three companion papers that summarize the theoretical and experimental work carried out to develop a prototype smart panel with 16 decentralized vibration control units for the reduction of sound radiation/transmission. The smart panel is made of a thin aluminium plate with 16 closely spaced accelerometer sensor and piezoceramic actuator transducer pairs connected by single-channel velocity feedback controllers (i.e., active damping units). In this paper a preliminary theoretical study is carried out to assess the behaviour of the smart panel when it is mounted on the top of a rectangular cavity with rigid walls. The smart panel is excited either by the acoustic field produced in the cavity by a monopole source or by a transverse point force. The simulations carried out have shown that for both the acoustic and the force sources, good reductions of the averaged kinetic energy or total sound power radiation can be achieved within a band 0–2kHz. The theoretical study is preceeded by a general review of the development of smart panels for the control of sound radiation/transmission. In particular, the various approaches developed for the design of sensors and actuators are analyzed with reference to the control of tonal disturbances using feed-forward controllers and the control of stationary random disturbances using feedback controllers.

Smart panel with multiple decentralized units for the control of sound transmission. Part I: theoretical predictions

This is the first of three companion papers that summarize the theoretical and experimental work carried out to develop a prototype smart panel with 16 decentralized vibration control units for the reduction of sound radiation/transmission. The smart panel is made of a thin aluminium plate with 16 closely spaced accelerometer sensor and piezoceramic actuator transducer pairs connected by single-channel velocity feedback controllers (i.e., active damping units). In this paper a preliminary theoretical study is carried out to assess the behaviour of the smart panel when it is mounted on the top of a rectangular cavity with rigid walls. The smart panel is excited either by the acoustic field produced in the cavity by a monopole source or by a transverse point force. The simulations carried out have shown that for both the acoustic and the force sources, good reductions of the averaged kinetic energy or total sound power radiation can be achieved within a band 0– 2 kHz . The theoretical study is preceeded by a general review of the development of smart panels for the control of sound radiation/transmission. In particular, the various approaches developed for the design of sensors and actuators are analyzed with reference to the control of tonal disturbances using feed-forward controllers and the control of stationary random disturbances using feedback controllers.

The flow instability over the infinite rotating disk

The hydrodynamic instability of the three-dimensional boundary layer on a rotating disk introduces a periodic modulation of the mean flow in the form of stationary cross flow vortices. The instability labeled Type II by Faller occurs first at lower Reynolds number than that of well known Type I instability. Detailed numerical values of the amplification rates, neutral curves and other characteristics of the two instabilities have been calculated over a wide range of parameters. Presented are the neutral stability results concerning the two instability modes by solving the appropriate linear stability equations reformulated not only by considering whole convective terms but also by correcting some errors in the previous stability equations. The present stability results agree with the previously known ones within reasonable limit. Consequently, the flow is found to be always stable for a disturbance whose dimensionless wave number is greater than 0.75. Some spatial amplification contours have been computed for the stationary disturbance wave, whose azimuth angle e=11.29° to 15° and for the moving disturbance wave, whose azimuth angle e=12.5° to 15°. Also, some temporal amplification contours have been computed for the stationary disturbance wave, whose azimuth angle e=11.29° to 15° and for the moving disturbance wave, whose azimuth angle e=12° to 15°. The flow instability was observed by using a white titanium tetrachloride gas over rotating disk system. When the numerical results are compared to the present experimental data, the numerical results agree quantitatively, indicating the existence of the selective frequency mechanism.

Inertia–Gravity Wave and Neutral Eady Wave Trains Forced by Directionally Sheared Flow over Isolated Hills

Abstract Analytical solutions are obtained to the linearized equations describing a particular class of directionally sheared flow over isolated hills or ridges. The flows are characterized by constant buoyancy frequency and vertical wind shear, though the wind direction changes with height due to the presence of a constant, horizontal wind component normal to the wind shear vector. The inclusion of a constant Coriolis parameter permits the formation of inertia wave trains and neutral baroclinic wave trains. A particular focus of this study is the nature of the selective critical level absorption process that results from varying wind direction with height in the presence of an azimuthal spectrum of forced wave modes. It will be shown that the stationary disturbance generated by circular and elliptical hills of mesoscale dimensions comprises an inertia wave train that trails downwind at all heights in the form of a horizontal wind perturbation, and a baroclinic wave train that extends in the direction of ...

Transient growth of stationary disturbances in a flat plate boundary layer

Theoretical and direct numerical simulation models of transient algebraic growth in boundary layers have advanced significantly without an adequate, parallel experimental effort. Experiments that feature disturbances excited by high levels of freestream turbulence or distributed surface roughness show behavior consistent with optimal-disturbance theories but cannot address the theories’ key predictions concerning the growth and decay of disturbances at specific spanwise wavenumbers. The present experiment seeks to provide such data for a flat plate boundary layer using a spanwise roughness array to excite controlled stationary disturbances. The results show that although general trends and qualitative behaviors are correctly captured by optimal-disturbance theories, significant quantitative differences exist between the theories’ predictions and the current experimental measurements. Discrepancies include the location of the wall-normal disturbance profiles’ maxima and the streamwise location of the maximum energy growth. While these discrepancies do not argue against the validity of transient-growth theory in general, they do indicate that correct modeling of receptivity to realistic disturbances is critical and that realistic stationary disturbances can exhibit strongly nonoptimal behavior.

Validation of the Natural-Laminar-Flow Design for the National Experimental Supersonic Transport.

The natural-laminar-flow (NLF) design for the wing of the national experimental supersonic transport (NEXST) aims to suppress crossflow (CF) instability near the leading edge. We computationally investigate the growth of various stationary CF disturbances on the outer wing of the NEXST-1 model using the prediction system of boundary-layer transition. According to the N factors obtained, the growth of those disturbances is completely suppressed around the designed angle of attack, which shows that the NLF design for the NEXST-1 wing is valid.

FEEDFORWARD AND FEEDBACK CONTROL OF SOUND AND VIBRATION—A WIENER FILTER APPROACH

This paper considers the active control of stationary random disturbances in single-input–single-output vibrating and acoustical systems. Both feedforward and feedback control are studied using a deterministic approach, where white noise is replaced by an impulse so that the active control of a stationary random disturbance is transformed to the active control of an impulse response. A theoretical framework using the Wiener filter to study both systems is described where internal model control is used to convert a feedback system into feedforward architecture. The effects of the constraint of causality on the performances of feedforward and feedback control of random disturbances are discussed using two simple examples; a minimum phase single-degree-of-freedom vibrating system and a non-minimum phase acoustical system.

Nonlinear evolution of a layered stratified shear flow

Abstract We investigate numerically and theoretically the nonlinear evolution of a parallel shear flow at moderate Reynolds number which has embedded within it a mixed layer of intermediate fluid. The two relatively thin strongly stratified density interfaces are centered on the edges of the shear layer. We are particularly interested in the development of primary and secondary instabilities. We present the results of a stability analysis which predicts that such flows may be unstable to stationary vortical disturbances which are a generalization of an inviscid instability first considered by G.I. Taylor. We investigate the behavior of these “Taylor billows” at finite amplitude through two-dimensional numerical simulations. We observe that the braid regions connecting adjacent primary Taylor billows are susceptible to secondary, inherently two-dimensional instabilities. We verify that these secondary instabilities, which take the form of small elliptical vortices, arise due to a local intensification of the spanwise vorticity in the braid region.

Polynomial H2-Optimization of Sampled-Data Ship Control Systems in Continuous Time

The problem of optimal digital ship course stabilization under stationary stochastic disturbances is considered. The chosen cost function consists of a weighted sum of average (with respect to continuous time) variances of the controlled variable (yaw angle) and control signal (rudder angle). A polynomial design method is developed on the basis of the parametric transfer functions method for direct design of optimal digital controller.

Spatial theory of optimal disturbances in a circular pipe flow

A spatial theory of linear transient growth for disturbances in a circular pipe is presented. Following from the consideration of a signaling problem, the spatial development of disturbances downstream of a source may be presented as a sum of decaying eigenmodes. Therefore, the problem of optimal disturbances in the pipe flow may be considered as an initial value problem on the subset of the downstream decaying eigenmodes, and a standard optimization procedure may be applied for evaluation of the optimal transient growth. Examples are presented for spatial transient growth of axisymmetric and nonaxisymmetric disturbances. It is shown that stationary disturbances may achieve more significant transient growth than nonstationary ones. The maximum of the transient growth exists at azimuthal index m=1 for stationary perturbations, whereas nonstationary perturbations may achieve their maxima at higher azimuthal indices.

Planar velocity measurements in compressible mixing layers

In the present study, we use particle image velocimetry (PIV) to obtain planar, two-component velocity fields in two-dimensional, turbulent mixing layers at convective Mach numbers Mc of 0.25, 0.63, and 0.76. The experiments are performed in a large-scale blowdown wind tunnel, with high-speed free-stream Mach numbers up to 2.25 and shear-layer Reynolds numbers up to 106. Specific issues relating to the application of PIV to supersonic flows are addressed. The instantaneous data are analysed to produce maps of derived quantities such as vorticity, and ensemble averaged to provide turbulence statistics. The results show that compressibility introduces marked changes in the disposition of instantaneous velocity gradients within the layer, and hence in the vorticity field. In particular, peak transverse vorticity values are seen to be confined to thin streamwise sheets under compressible conditions, with little transverse communication. The location of these sheets near the lab-frame sonic velocity suggests a sensitivity of the compressible layer to stationary disturbances. Turbulence statistics derived from the planar velocity measurements confirm previous observations of strong suppression of transverse velocity fluctuations and primary Reynolds stress as Mc increases between 0.25 and 0.76.

A Study of Ship's Guidance Control by Time-Varying LQ Controller

In this paper, a ship's tracking controller using an optimal tracking algorithm by the Bryson-Ho algorithm is proposed. In this system, the optimal controller by using the algorithm of time-varying linear-quadratic controller (TVLQC) steers a rudder so that the ship navigates on the desired track. When stationary disturbances like wind or current occur, a virtual track route in which the true desired track is appropriately shifted, is introduced. In the latter part of this paper, as a tracking by TVLQC in low speed a tracking using bow thruster is discussed.The possibilities of the proposed algorithm are tested in simulations and actual ship's tracking experiments using an actual ship.

Anisotropy-based performance analysis of linear discrete time invariant control systems

The anisotropic norm of a linear discrete-time-invariant system measures system output sensitivity to stationary Gaussian input disturbances of bounded mean anisotropy. Mean anisotropy characterizes the degree of predictability (or colouredness) and spatial non-roundness of the noise. The anisotropic norm falls between the H