Amplification Of Disturbances Research Articles

Numerical study of laminar-turbulent transition in particle-laden channel flow

We present direct numerical simulations of subcritical transition to turbulence in a particle-laden channel flow, with particles assumed rigid, spherical, and heavier than the fluid. The equations describing the fluid flow are solved with an Eulerian mesh, whereas those describing the particle dynamics are solved by Lagrangian tracking. Two-way coupling between fluid and particles is modeled with Stokes drag. The numerical code is first validated against previous results from linear stability: the nonmodal growth of streamwise vortices resulting in streamwise streaks is still the most efficient mechanism for linear disturbance amplification at subcritical conditions as for the case of a single phase fluid. To analyze the full nonlinear transition, we examine two scenarios well studied in the literature: (1) transition initiated by streamwise independent counter-rotating streamwise vortices and one three-dimensional mode and (2) oblique transition, initiated by the nonlinear interaction of two symmetric oblique waves. The threshold energy for transition is computed, and it is demonstrated that for both scenarios the transition may be facilitated by the presence of particles at low number density. This is due to the fact that particles may introduce in the system detrimental disturbances of length scales not initially present. At higher concentrations, conversely, we note an increase of the disturbance energy needed for transition. The threshold energy for the oblique scenario shows a more significant increase in the presence of particles, by a factor about four. Interestingly, for the streamwise-vortex scenario the time at which transition occurs increases with the particle volume fraction when considering disturbances of equal initial energy. These results are explained by considering the reduced amplification of oblique modes in the two-phase flow. The results from these two classical scenarios indicate that, although linear stability analysis shows hardly any effect on optimal growth, particles do influence secondary instabilities and streak breakdown. These effects can be responsible of the reduced drag observed in turbulent channel flow laden with heavy particles.

Worst-case amplification of disturbances in inertialess Couette flow of viscoelastic fluids

AbstractAmplification of deterministic disturbances in inertialess shear-driven channel flows of viscoelastic fluids is examined by analysing the frequency responses from spatio-temporal body forces to the velocity and polymer stress fluctuations. In strongly elastic flows, we show that disturbances with large streamwise length scales may be significantly amplified even in the absence of inertia. For fluctuations without streamwise variations, we derive explicit analytical expressions for the dependence of the worst-case amplification (from different forcing to different velocity and polymer stress components) on the Weissenberg number ($\mathit{We}$), the maximum extensibility of the polymer chains ($L$), the viscosity ratio and the spanwise wavenumber. For the Oldroyd-B model, the amplification of the most energetic components of velocity and polymer stress fields scales as${\mathit{We}}^{2} $and${\mathit{We}}^{4} $. On the other hand, the finite extensibility of polymer molecules limits the largest achievable amplification even in flows with infinitely large Weissenberg numbers: in the presence of wall-normal and spanwise forces, the amplification of the streamwise velocity and polymer stress fluctuations is bounded by quadratic and quartic functions of$L$. This high amplification signals low robustness to modelling imperfections of inertialess channel flows of viscoelastic fluids. The underlying physical mechanism involves interactions of polymer stress fluctuations with a base shear, and it represents a close analogue of the lift-up mechanism that initiates a bypass transition in inertial flows of Newtonian fluids.

Shear flow energy redistribution stipulated by the internal-gravity wavy structures in the dissipative ionosphere

The linear mechanism of generation, intensification and further nonlinear dynamics of internal gravity waves (IGW) in stably stratified dissipative ionosphere with non-uniform zonal wind (shear flow) is studied. In case of the shear flows the operators of linear problem are non-selfadjoint, and the corresponding Eigen functions – nonorthogonal. Thus, canonical – modal approach is of less use studying such motions. Non-modal mathematical analysis becomes more adequate for such problems. On the basis of non-modal approach, the equations of dynamics and the energy transfer of IGW disturbances in the ionosphere with a shear flow is obtained. Exact analytical solutions of the linear as well as the nonlinear dynamic equations of the problem are built. The increment of shear instability of IGW is defined. It is revealed that the transient amplification of IGW disturbances due time does not flow exponentially, but in algebraic – power law manner. The effectiveness of the linear amplification mechanism of IGW at interaction with non-uniform zonal wind is analyzed. It is shown that at initial linear stage of evolution IGW effectively temporarily draws energy from the shear flow significantly increasing (by an order of magnitude) own amplitude and energy. With amplitude growth the nonlinear mechanism turns on and the process ends with self-organization of nonlinear solitary, strongly localized IGW vortex structures (the monopole vortex, the transverse vortex chain or the longitudinal vortex street). Accumulation of these vortices in the ionospheric medium can create the strongly turbulent state.

Characteristics of nonlinear evolution of wavepackets in boundary layers

The nonlinear evolution of a finite-amplitude disturbance in a 3-D supersonic boundary layer over a cone was investigated recently by Liu et al. using direct numerical simulation (DNS). It was found that certain small-scale 3-D disturbances amplified rapidly. These disturbances exhibit the characteristics of second modes, and the most amplified components have a well-defined spanwise wavelength, indicating a clear selectivity of the amplification. In the case of a cone, the three-dimensionality of the base flow and the disturbances themselves may be responsible for the rapid amplification. In order to ascertain which of these two effects are essential, in this study we carried out DNS of the nonlinear evolution of a spanwise localized disturbance (wavepacket) in a flat-plate boundary layer. A similar amplification of small-scale disturbances was observed, suggesting that the direct reason for the rapid amplification is the three-dimensionality of the disturbances rather than the three-dimensional nature of the base flow, even though the latter does alter the spanwise distribution of the disturbance. The rapid growth of 3-D waves may be attributed to the secondary instability mechanism. Further simulations were performed for a wavepacket of first modes in a supersonic boundary layer and of Tollmien-Schlichting (T-S) waves in an incompressible boundary layer. The results show that the amplifying components are in the band centered at zero spanwise wavenumber rather than at a finite spanwise wavenumber. It is therefore concluded that the rapid growth of 3-D disturbances in a band centered at a preferred large spanwise wavenumber is the main characteristic of nonlinear evolution of second mode disturbances in supersonic boundary layers.

Networks of Historical Contingency in Earth Surface Systems

Earth surface systems (ESS) are characterized by various degrees of historical contingency, which complicates efforts to relate observed features and phenomena to environmental controls. This article provides a conceptual framework for understanding and assessing historical contingency in ESS that is based on algebraic graph theory. ESS are conceptualized as consisting of components (e.g., climate, topography, and lithology) observed or inferred at time periods. Each component at each time period represents a node of a network or graph, and interactions among components constitute the links or edges. Four indexes are applied: the S-metric, which indicates the extent to which observations of part of the network (e.g., topographic changes between two time periods) are likely to represent the dynamics of the network as a whole; spectral radius, which measures coherence and potential amplification of changes or disturbances; Laplacian spectral radius, an index of the relationship between network stability and time steps and an indication of path dependence; and algebraic connectivity, which measures the inferential synchronizability. For each of these, an index on a 0–1 scale is developed, which represents high and minimum levels of historical contingency for a given n, q. These are applied to several archetypal graph structures that represent various forms of historical contingency in the geosciences and to two specific case studies involving Quaternary evolution of fluvial systems in Texas and Kentucky.

Sensitivity of linear stability analysis of measured separated shear layers

Linear stability analysis is commonly performed on measured mean velocity profiles in experimental studies on transitional separation bubbles; however, the results of such an analysis are sensitive to both the analysis approach and to experimental data scatter. In this investigation, the sensitivity of linear stability analysis of measured separated shear layer profiles is examined in order to identify reliable analysis approaches and to quantify the uncertainty in the results due to experimental data scatter and the method of analysis. Analysis approaches are compared based on their predictions for a simulated velocity profile without data scatter, profiles with simulated data scatter, and several published experimental separated shear layer profiles. Both viscous and inviscid analyses are considered. The predicted maximum disturbance amplification rates between the viscous and inviscid solutions are found to agree to within 15% for several measured separated shear layer profiles. It is shown that, for the level of data scatter typical of hot-wire measurements over flat plates and airfoils, the measured mean velocity profile should not be used directly in the stability equation, and therefore curve fitting the base velocity profile is necessary. Three curve fits are identified that provide stability predictions with relatively low sensitivity to mean profile data scatter. Stability predictions obtained using these fits are found to be more sensitive to the type of curve fit used in the analysis than to experimental data scatter, with the predicted maximum disturbance growth rate and corresponding frequency varying by approximately 35% between fits. The results of this investigation provide a means of estimating the uncertainty in linear stability predictions for measured separated shear layer profiles.

An intuitive approach for feedback active noise controller design

A simple and effective approach for designing feedback controllers is proposed based on analyzing the waterbed effect of the feedback active noise control system. By flattening the noise amplification due to the waterbed effect in the whole disturbance amplification frequency band intuitively, the designed feedback controller can suppress the disturbance amplification to almost the minimum value for the given disturbance attenuation and ensure good stability of the system. Being compared with the existing approaches, the main advantage of the proposed approach is its simplicity for implementation, which avoids complicated weight function selections and the numerical optimizations of the existing approaches. Detailed design procedures and implementation of the digital controller are illustrated, simulation and experiment results are provided to show the effectiveness of the proposed approach.

Effects of End Plates and Blockage on Low-Reynolds-Number Flows Over Airfoils

Measurements on airfoils at low Reynolds numbers can have a strong dependence on the experimental setup as a result of the sensitivity of the transitioning separated shear layer that develops over the model. In this investigation, the effects of two aspects of the experimental setup, namely end plates and test-section blockage, on lowReynolds-numberairfoilexperiments areexploredthroughmeasurementsonaNACA0018airfoilmodelatachord Reynolds number of 100,000. The improvement in mean spanwise uniformity with end plates installed is quantified and demonstrates the importance of using end plates in two-dimensional airfoil experiments at low Reynolds numbers. Consistent with previous studies on the use of end plates on circular cylinder models, it is found that mean quantities measured on the center-span plane are least sensitive to end-plate spacing for spacings greater than roughly 7 times the projected model thickness. It is shown that the end-plate configuration affects vortex-shedding characteristics and disturbance amplification in the separated shear layer. Blockage effects are investigated by comparing measurements before andafter adaptive-wall test-section streamlining forsolid-blockage ratios between 4 and 8%. These blockage ratios are shown to cause errors in lift as high as 9% of the maximum lift and 3.5% in the wake vortex-shedding frequency. The results of this investigation can be used to estimate the effects of blockage on flow development in low-Reynolds-number airfoil experiments. It is demonstrated that a common blockagecorrectionmethodcanaccuratelycorrectliftmeasurementsformoderateblockagesinlow-Reynolds-numberairfoil experiments for conditions under which a separation bubble forms over the model.

Spatial linear disturbances in a plane wall jet

A two-dimensional direct numerical simulation study of the linear instability in a laminar plane wall jet is presented. The evolution of the wall jet disturbances is in reasonable agreement with predictions by spatial linear stability theory only with regard to the wavelength and the amplitude shape of the disturbance, whereas significant differences in the linear growth rate are noticed. As a consequence, the “stable island” on the instability map based on linear stability theory turns out to be connected with the outer stable region in the simulations, thus taking the form of a “stable peninsula”. The failure of the theory is attributed to the rapid streamwise spread and decay of the wall jet, which is incompatible with the assumption of parallel flow. We also assess the maximum possible transient linear amplification of two-dimensional disturbances in the plane wall jet, using the concept of optimal initial disturbances. The transient energy growth relies on the Orr mechanism, and the upper bound of the disturbance energy increases linearly in time for the present flow configuration. The optimal disturbances exhibit local maxima near the edge of the jet and close to the wall, where sites of effective receptivity are hence expected. We find that the outer region of the plane wall jet is more receptive to time periodic forcing than the inner region.

Parametric study of separation and transition characteristics over an airfoil at low Reynolds numbers

Time-resolved surface pressure measurements are used to experimentally investigate characteristics of separation and transition over a NACA 0018 airfoil for the relatively wide range of chord Reynolds numbers from 50,000 to 250,000 and angles of attack from 0° to 21°. The results provide a comprehensive data set of characteristic parameters for separated shear layer development and reveal important dependencies of these quantities on flow conditions. Mean surface pressure measurements are used to explore the variation in separation bubble position, edge velocity in the separated shear layer, and lift coefficients with angle of attack and Reynolds number. Consistent with previous studies, the separation bubble is found to move upstream and decrease in length as the Reynolds number and angle of attack increase. Above a certain angle of attack, the proximity of the separation bubble to the location of the suction peak results in a reduced lift slope compared to that observed at lower angles. Simultaneous measurements of the time-varying component of surface pressure at various spatial locations on the model are used to estimate the frequency of shear layer instability, maximum root-mean-square (RMS) surface pressure, spatial amplification rates of RMS surface pressure, and convection speeds of the pressure fluctuations in the separation bubble. A power-law correlation between the shear layer instability frequency and Reynolds number is shown to provide an order of magnitude estimate of the central frequency of disturbance amplification for various airfoil geometries at low Reynolds numbers. Maximum RMS surface pressures are found to agree with values measured in separation bubbles over geometries other than airfoils, when normalized by the dynamic pressure based on edge velocity. Spatial amplification rates in the separation bubble increase with both Reynolds number and angle of attack, causing the accompanying decrease in separation bubble length. Values of the convection speed of pressure fluctuations in the separated shear layer are measured to be between 35 and 50% of the edge velocity, consistent with predictions of linear stability theory for separated shear layers.

Connect & Drive: design and evaluation of cooperative adaptive cruise control for congestion reduction

Road throughput can be increased by driving at small inter-vehicle time gaps. The amplification of velocity disturbances in upstream direction, however, poses limitations to the minimum feasible time gap. This effect is covered by the notion of string stability. String-stable behavior is thus considered an essential requirement for the design of automatic distance control systems, which are needed to allow for safe driving at time gaps well below 1 s. Using wireless inter-vehicle communications to provide real-time information of the preceding vehicle, in addition to the information obtained by common Adaptive Cruise Control (ACC) sensors, appears to significantly decrease the feasible time gap, which is shown by practical experiments with a test fleet consisting of six passenger vehicles. The large-scale deployment of this system, known as Cooperative ACC (CACC), however, poses challenges with respect to the reliability of the wireless communication system. A solution for this scalability problem can be found in decreasing the transmission power and/or beaconing rate, or adapting the communications protocol. Although the main CACC objective is to increase road throughput, the first commercial application of CACC is foreseen to be in truck platooning, since short distance following is expected to yield significant fuel savings in this case.

Low Frequency Vibration Detection and Compensation in Hard Disk Drive

The paper proposes a control scheme for detecting and compensating the low-frequency vibrations in hard disk drive systems. A vibration detector is proposed, utilizing the down-sampled position error signal (PES). Due to the performance limitation in loop shaping, attenuating low-frequency vibration is usually penalized by high frequency disturbance amplification. This paper proposes an adaptive scheme to intelligently adjust a vibration reduction factor, such that when vibration level is low, the servo loop maintains its normal operation mode, and little or no compensation is carried out. Experiments on hard disk drives on a vibration shaker validated the effectiveness of the proposed compensation scheme.

Nonmodal amplification of stochastic disturbances in strongly elastic channel flows

Nonmodal amplification of stochastic disturbances in elasticity-dominated channel flows of Oldroyd-B fluids is analyzed in this work. For streamwise-constant flows with high elasticity numbers μ and finite Weissenberg numbers We, we show that the linearized dynamics can be decomposed into slow and fast subsystems, and establish analytically that the steady-state variances of velocity and polymer stress fluctuations scale as O ( We 2 ) and O ( We 4 ) , respectively. This demonstrates that large velocity variance can be sustained even in weakly inertial stochastically driven channel flows of viscoelastic fluids. We further show that the wall-normal and spanwise forces have the strongest impact on the flow fluctuations, and that the influence of these forces is largest on fluctuations in the streamwise velocity and the streamwise component of the polymer stress tensor. The underlying physical mechanism involves polymer stretching that introduces a lift-up of flow fluctuations similar to vortex tilting in inertia-dominated flows. The validity of our analytical results is confirmed in stochastic simulations. The phenomenon examined here provides a possible route for the early stages of a bypass transition to elastic turbulence and might be exploited to enhance mixing in microfluidic devices.

Worst-case amplification of disturbances in inertialess flows of viscoelastic fluids

AbstractAmplification of deterministic disturbances in inertialess channel flows of viscoelastic fluids is studied by analyzing the frequency responses from spatio-temporal body forces to the velocity fluctuations. In strongly elastic flows, we show that velocity fluctuations can exhibit significant amplification even in the absence of inertia. Our analysis demonstrates that streamwise-constant disturbances are the most amplified. Explicit expressions are established for the worst-case amplification of velocity fluctuations arising from different components of the body forces. These show that amplification from the wall-normal and spanwise forces to the streamwise velocity component scales quadratically with the Weissenberg number. The underlying physical mechanism involves stretching of polymer stress fluctuations by a background shear, which is a close analog of the vortex titling mechanism that is responsible for amplification in inertial flows of Newtonian fluids.

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Disturbance evolution in a Mach 4.8 boundary layer with two-dimensional roughness-induced separation and shock

A numerical investigation of the disturbance amplification in a Mach 4.8 flat-plate boundary layer with a localized two-dimensional roughness element is presented. The height of the roughness is varied and reaches up to approximately 70% of the boundary-layer thickness. Simulations are based on a time-accurate integration of the compressible Navier–Stokes equations, with a small disturbance of fixed frequency being triggered via blowing and suction upstream of the roughness element. The roughness element considerably alters the instability of the boundary layer, leading to increased amplification or damping of a modal wave depending on the frequency range. The roughness is also the source of an additional perturbation. Even though this additional mode is stable, the interaction with the unstable mode in the form of constructive and destructive interference behind the roughness element leads to a beating and therefore transiently increased disturbance amplitude. Far downstream of the roughness, the amplification rate of a flat-plate boundary layer is recovered. Overall, the two-dimensional roughness element behaves as disturbance amplifier with a limited bandwidth capable of filtering a range of frequencies and strongly amplifying only a selected range.

Fast Large-Eddy Simulation of Low Reynolds Number Flows over a NACA0025

Fast Large-Eddy Simulation of Low Reynolds Number Flows over a NACA0025

Model Reduction of the Nonlinear Complex Ginzburg–Landau Equation

Reduced-order models of the nonlinear complex Ginzburg-Landau (CGL) equation are computed using a nonlinear generalization of balanced truncation. The method involves Galerkin projection of the nonlinear dynamics onto modes determined by balanced truncation of a linearized system and is compared to a standard method using projection onto proper orthogonal decomposition (POD) modes computed from snapshots of nonlinear simulations. It is found that the nonlinear reduced- order models obtained using modes from linear balanced truncation capture very well the transient dynamics of the CGL equation and outperform POD models; i.e., a higher number of POD modes than linear balancing modes is typically necessary in order to capture the dynamics of the original system correctly. In addition, we find that the performance of POD models compares well to that of balanced truncation models when the degree of nonnormality in the system, in this case determined by the streamwise extent of a disturbance amplification region, is lower. Our findings therefore indicate that the superior performance of balanced truncation compared to POD/Galerkin models in capturing the input/output dynamics of linear systems extends to the case of a nonlinear system, both for the case of significant transient growth, which represents a basic model of boundary layer instabilities, and for a limit cycle case that represents a basic model of vortex shedding past a cylinder.

Steady and unsteady Görtler boundary-layer instability on concave wall

The subject of the present combined experimental and theoretical investigation is the steady and unsteady linear Görtler instability. The majority of previous experiments were devoted to the steady Görtler vortices, despite the unsteady ones are also observed in real transitional flows. Moreover, even for the steady Görtler vortices no quantitative agreement between the experimental and theoretical linear-stability characteristics was obtained, especially for disturbance amplification rates. The experimental difficulties were connected, in particular, with a rather poor accuracy of measurements at zero disturbance frequency, a possible influence of nonlinearity, and an admixture of non-modal (transient) growth mechanism. All these difficulties have been overcome in the experimental part of the present study by means of: ( i) tuning-out of the exact zero frequency of Görtler vortices and working, instead, with quasi-steady perturbations of very low frequencies, ( ii) performing measurements at low disturbance amplitudes, and ( iii) minimization and careful estimation of the disturbance-source near-field by means of utilizing a special controlled disturbance source and performing special numerical computations for exact experimental conditions. A detailed study of all linear-stability characteristics for essentially unsteady Görtler vortices was performed in this paper as well. The results are obtained in a range of Görtler numbers 13 ≲ Gö ≲ 17.3, frequency parameters F = 0.56–22.70, and spanwise wavelength parameters Λ = 149–775 (close to the most amplified Görtler modes). Appropriate calculations based on locally-parallel and non-local non-parallel linear-stability theories were performed and compared quantitatively with experimentally obtained linear-stability characteristics. For the first time all stability characteristics measured for steady Görtler vortices (in quasi-steady regimes) are found to agree very well with those calculated for the most amplified first discrete-spectrum mode of the linear Görtler-instability problem. Similar good agreement is obtained for essentially unsteady Görtler vortices. The roles of effects of the base-flow non-parallelism and the disturbance-source near-field are examined.

On vortex shedding from an airfoil in low-Reynolds-number flows

Development of coherent structures in the separated shear layer and wake of an airfoil in low-Reynolds-number flows was studied experimentally for a range of airfoil chord Reynolds numbers, 55 × 103 ≤ Rec ≤ 210 × 103, and three angles of attack, α = 0°, 5° and 10°. To illustrate the effect of separated shear layer development on the characteristics of coherent structures, experiments were conducted for two flow regimes common to airfoil operation at low Reynolds numbers: (i) boundary layer separation without reattachment and (ii) separation bubble formation. The results demonstrate that roll-up vortices form in the separated shear layer due to the amplification of natural disturbances, and these structures play a key role in flow transition to turbulence. The final stage of transition in the separated shear layer, associated with the growth of a sub-harmonic component of fundamental disturbances, is linked to the merging of the roll-up vortices. Turbulent wake vortex shedding is shown to occur for both flow regimes investigated. Each of the two flow regimes produces distinctly different characteristics of the roll-up and wake vortices. The study focuses on frequency scaling of the investigated coherent structures and the effect of flow regime on the frequency scaling. Analysis of the results and available data from previous experiments shows that the fundamental frequency of the shear layer vortices exhibits a power law dependency on the Reynolds number for both flow regimes. In contrast, the wake vortex shedding frequency is shown to vary linearly with the Reynolds number. An alternative frequency scaling is proposed, which results in a good collapse of experimental data across the investigated range of Reynolds numbers.