Amplification Of Disturbances Research Articles

Nonlinear Nonmodal Analysis of Hypersonic Flow over Blunt Cones

The linear amplification of modal disturbances that lead to boundary-layer transition in two-dimensional/axisymmetric hypersonic configurations is strongly reduced by the presence of a blunt nose tip, and the mechanisms underlying the low Mack’s second-mode N-factor values at the observed onset of transition over the cone frustum are currently unknown. Linear nonmodal analysis has shown that both planar and oblique traveling disturbances that peak within the entropy-layer experience appreciable energy amplification for moderate to large nose-tip bluntness. The present study extends the previous linear analysis by including the nonlinear effects. Specifically, the harmonic Navier–Stokes equations (HNSE) are solved with a fully implicit formulation and the Newton–Raphson method. The increased number of degrees of freedom for the nonlinear system presents difficulties for solution strategies based on direct solution of the linearized system. Such difficulties are overcome by using the generalized minimal residual method (GMRES) iterative method with a preconditioner corresponding to a simplified Jacobian without the cross-derivative terms. The HNSE solver is verified by comparing with nonlinear parabolized stability equation results for the nonlinear evolution of planar waves in an incompressible Blasius boundary layer and in a Mach 6 flow over a blunt cone. Finally, nonlinear nonmodal results are presented for planar traveling disturbances over a blunt cone configuration with reduced transition N factor as measured in wind-tunnel experiments. The nonmodal analysis demonstrates that entropy-layer disturbances generated close to the nose tip can seed the amplification of higher-frequency Mack’s second-mode instabilities farther downstream and hence can lead to a reduction in the transition N factor.

Enhancing vehicular platoon stability in the presence of communication Cyberattacks: A reliable longitudinal cooperative control strategy

This study proposes a reliable longitudinal distributed control strategy for connected and automated vehicles (CAVs) in the presence of communication cyberattacks. The proposed strategy is designed to utilize dependable measurements from onboard sensors to evaluate the reliability of vehicle-to-vehicle (V2V) information and enhance the platoon string stability in an integrated manner. Specifically, this paper proposes a real-time reliability evaluation mechanism for the ego vehicle based on sensing and communication information from multiple predecessors, employing an upper-tailed chi-squared test. The mechanism is then incorporated into a distributed multi-predecessor linear feedback controller to adaptively weigh the preceding vehicles’ information, ensuring robustness against cyberattacks and maintaining the string stability of the vehicular platoon. To better understand the whole framework, sufficient conditions of true string stability and pseudo string stability are theoretically derived, unveiling the disturbance amplification jointly impacted by measurement accuracy of onboard sensors, cyberattack severity in V2V communication, and control parameters. Numerical experiments are conducted to validate the controller across various types of communication cyberattacks. The results suggest that the proposed control strategy can significantly alleviate the impact of cyberattacks and simultaneously dampen traffic oscillations effectively.

Boundary-layer instability on a highly swept fin on a cone at Mach 6

The growth and characteristics of linear, oblique instabilities on a highly swept fin on a straight cone in Mach 6 flow are examined. Large streamwise pressure gradients cause doubly inflected cross-flow profiles and reversed flow near the wall, which necessitates using the harmonic linearized Navier–Stokes equations. The cross-flow instability is responsible for the most-amplified disturbances, however, not all disturbances show typical cross-flow characteristics. Distinct differences in perturbation structure are shown between small ( $\sim$ 3–5 mm) and large ( $\sim$ 10 mm) wavelength disturbances at the unit Reynolds number $Re' = 11 \times 10^6$ m $^{-1}$ . As a result, amplification measurements based solely on wall quantities bias a most-amplified disturbance assessment towards larger wavelengths and lower frequencies than would otherwise be determined by an off-wall total-energy approach. A spatial-amplification energy-budget analysis demonstrates (i) that wall-normal Reynolds-flux terms dictate the local growth rate, despite other terms having a locally larger magnitude and (ii) that the Reynolds-stress terms are responsible for large-wavelength disturbances propagating closer to the wall compared with small-wavelength disturbances. Additionally, the effect of free-stream unit Reynolds number and small yaw angles on the perturbation amplification and energy budget is considered. At a higher Reynolds number ( $Re' = 22 \times 10^6$ m $^{-1}$ ), the most-amplified wavelength shrinks. Perturbations do not behave self-similarly in the thinner boundary layer, and the shift in most-amplified wavelength is due to decreased dissipation relative to the lower-Reynolds-number case. Small yaw angles produce a streamwise shift in the boundary layer and disturbance amplification. The yaw results quantify a potential uncertainty source in experiments and flight.

Design of General Parametric Repetitive Control Using IIR Filter With Application to Piezo-Actuated Nanopositioning Stages

The achievable performance with repetitive control is limited due to its inherent sensitivity to the frequency shift away from the intended periodic frequencies and the undesired gain amplification of the aperiodic disturbances. To address these limitations, the paper proposed a general parametric repetitive control (GPRC) method based on the IIR filter with the features of low-order and excellent magnitude responses to result in better tracking performance in diverse applications. By analyzing its sensitivity function, it is found that the design of GPRC can be converted to the explicit parametric design of an IIR high pass filter. The controller design process and the stability analysis are presented in detail. To show the effectiveness of GPRC, comparative experiments are conducted via tracking sinusoids, triangular trajectories and other complex trajectories with multi-frequency components. The experimental results show that, in contrast with the conventional repetitive control (CRC) and a modified repetitive control (MRC), the GPRC exhibits excellent robustness against the frequency shift and advanced performance at the aperiodic frequencies. The tracking results of the sinusoids show that the maximum tracking error obtained with GPRC for a frequency shift of 3 Hz decreases from 0.0259 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu m$</tex-math> </inline-formula> (CRC) and 0.2207 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu m$</tex-math> </inline-formula> (MRC) to 0.0101 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mu m$</tex-math> </inline-formula> at the nominal frequency of 1000 Hz, demonstrating the merits of the proposed GPRC. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —To enable automation systems, one of the crucial requirements is to track repetitive references with high precision. Although the normal repetitive control (RC) based schemes are successfully applied to improve the tracking accuracy of periodic trajectories, the existing RC schemes suffer from the problems of lower robustness against frequency shift and the unwanted gain amplification at the aperiodic frequencies due to Bode’s sensitivity integral. To overcome this problem, this paper proposes a novel general parametric repetitive control (GPRC) method via characterizing the loop properties quantitatively based on the IIR high pass filter design. Focusing on specific issues, the detailed variations to handle the errors at only the odd-and even-harmonics are also demonstrated. This framework leads to a flexible solution in practical implementations. The experimental validation on a piezo-actuated nanopositioning stage is comparatively presented in terms of tracking accuracy and rejection ability of the gain amplification at the aperiodic frequencies. With its flexibility and effectiveness, the proposed GPRC can be easily implemented in diverse applications.

Beyond optimal disturbances: a statistical framework for transient growth

The theory of transient growth describes how linear mechanisms can cause temporary amplification of disturbances even when the linearized system is asymptotically stable as defined by its eigenvalues. This growth is traditionally quantified by finding the initial disturbance that generates the maximum response at the peak time of its evolution. However, this can vastly overstate the growth of a real disturbance. In this paper, we introduce a statistical perspective on transient growth that models statistics of the energy amplification of the disturbances. We derive a formula for the mean energy amplification and spatial correlation of the growing disturbance in terms of the spatial correlation of the initial disturbance. The eigendecomposition of the correlation provides the most prevalent structures, which are the statistical analogue of the standard left singular vectors of the evolution operator. We also derive accurate confidence bounds on the growth by approximating the probability density function of the energy. Applying our analysis to Poiseuille flow yields a number of observations. First, the mean energy amplification is often drastically smaller than the maximum. In these cases, it is exceedingly unlikely to achieve near-optimal growth due to the exponential behaviour observed in the probability density function. Second, the characteristic length scale of the initial disturbances has a significant impact on the expected growth, with large-scale initial disturbances growing orders of magnitude more than small-scale ones. Finally, while the optimal growth scales quadratically with Reynolds number, the mean energy amplification scales only linearly for certain reasonable choices of the initial correlation.

Aeroacoustic investigation of airfoil at near-stall conditions

This paper presents a detailed aeroacoustic investigation of a controlled-diffusion airfoil at near-stall condition. The study aims at answering two research questions: identify the flow mechanism responsible for separation noise for an airfoil near-stall conditions and whether the noise is generated by a dipole for airfoil close to stall and can be quantified by Amiet's diffraction theory. The study uses synchronized particle image velocimetry, remote microphone probes, and far-field microphone measurements to perform experiments at two chord-based Reynolds numbers of about 150 000 and 250 000. The results show that when the airfoil is placed at a higher angle of attack, such as 15°, strong amplification of flow disturbance is seen, resulting in the rolling up of the shear layer in the aft region of the airfoil, forming large coherent structures. While these rollers play a central role in the increase in noise due to flow separation, the flapping of shear layer does not contribute to the separation noise. The present study conclusively shows that separation noise is dipolar in nature and that the quadrupolar contribution for low-speed airfoils at near-stall conditions can be neglected. However, the increase in flow disturbances measured close to the trailing edge of the airfoil implies that the assumption of small-amplitude disturbance is no longer valid, which is the central premise of the thin linearized airfoil theory. Outside the frequency range at which flow separation operates, Amiet's theory is able to predict the far-field noise even at high angles of attack.

The effect of leading-edge shape on separation-induced transition on the suction surface of a controlled-diffusion airfoil

To investigate the effect of leading-edge shape on separation-induced transition on suction surface, a large eddy simulation is performed on two compressor controlled-diffusion airfoils: a conventional one with an elliptical leading edge and an optimized one with a curvature-continuous design based on the B-spline description. The Reynolds number based on inflow velocity and chord length is 4.5×105. The critical angle of attack +4°, over which the aerodynamic loss rises sharply, is chosen for simulation. Two transitions are observed on the suction surface, one near the leading edge and the other at 40% chord length. The primary difference between the two airfoils lies in the leading-edge transition, which also leads to the distinction of fluctuating velocity amplitude and energy loss in the subsequent development of boundary layer flow. In order to provide an insight into the transition mechanism, the frequency spectrum analysis is conducted, and the results indicate that the amplification of disturbances during transition is dominated by Kelvin–Helmholtz instability. The mechanisms of energy transport and dissipation are analyzed, and the influence of leading-edge curvature on the initial state of boundary layer flow is elucidated from a dynamic perspective. The results show that continuous and large curvature distributions are more conducive to suppressing the formation of leading-edge separation bubble and delaying the onset of transition.

Data-driven analysis for disturbance amplification in car-following behavior of automated vehicles

This paper presents a data-driven framework to quantitatively analyze the disturbance amplification behavior of automated vehicles in car-following (CF). The data-driven framework can be applied to unknown CF controllers based on the concept of empirical frequency response function (FRF). Specifically, a well-known signal processing method, Welch's method, together with a short time Fourier transformation is developed to extract the empirical transfer functions from vehicle trajectories. The method is first developed assuming a generic linear controller with time-invariant CF control features (e.g., control gains) and later extended to capture time-variant features. The proposed methods are evaluated for estimation consistencies via synthetic data-based simulations. The evaluation includes the performances of the linear approximation accuracy for a linear time-invariant controller, a nonlinear controller, and a linear time-variant controller. Results indicate that our framework can provide reasonably consistent results as theoretical ones in terms of disturbance amplification. Further it can perform better than a linear theoretical analysis of disturbance amplification, particularly when nonlinearity in CF behavior is present. The methods are applied to existing field data collected from vehicles with adaptive cruise control (ACC) on the market. Findings reveal that all tested vehicles tend to amplify disturbances, particularly in low frequency (< 0.5 Hz). Further, the results demonstrate that these ACC vehicles exhibit time-varying features in terms of disturbance amplification ratio depending on the leading vehicle trajectories.

Synthesis of Output-Feedback Controllers for Mixed Traffic Systems in Presence of Disturbances and Uncertainties

In this paper, we study mixed traffic systems that move along a single-lane ring-road or open-road. The traffic flow forms a platoon, which includes a number of heterogeneous human-driven vehicles (HDVs) together with only one connected and automated vehicle (CAV) that receives information from a subset of neighbors. The dynamics of HDVs are assumed to follow a general continuous-time nonlinear car-following model, which is a function of their velocity, spacing, and the relative velocity. The acceleration of the single CAV is also directly controlled by a dynamical output-feedback controller. The ultimate goal of this work is to present a robust control strategy that can smoothen the traffic flow in the presence of undesired disturbances (e.g. abrupt deceleration) and parametric uncertainties. A prerequisite for synthesizing a dynamical output controller is the stabilizability and detectability of the underlying system. Accordingly, a theoretical analysis is presented first to prove the stabilizability and detectability of the mixed traffic flow system. Then, two <inline-formula> <tex-math notation="LaTeX">$H_\infty$</tex-math> </inline-formula> control strategies, with and without considering uncertainties in the system dynamics, are designed. The efficiency of the two control methods is subsequently illustrated through numerical simulations, and various experimental results are presented to demonstrate the effectiveness of the proposed controller to mitigate disturbance amplification and achieve platoon stability.

Linear stability analysis of a three-dimensional laminar separation bubble on a finite wing

Disturbance amplification in a laminar separation bubble forming on the suction surface of a cantilevered NACA0018 wing at an angle of attack of 6° and Reynolds number of 1.25 × 105 is studied using a combination of particle image velocimetry measurements and local and non-local linear stability analyses. The experimental measurements reveal the formation of largely two-dimensional velocity fluctuations leading to the formation of initially two-dimensional roll-up vortices in the separated shear layer at a nearly constant wavenumber and frequency, despite the substantial reduction in the effective angle of attack near the wing tip. The most amplified wavenumber and its growth rate are accurately predicted by linear stability theory outside of the regions of the separation bubble dominated by three-dimensional end effects, with negligible differences between local and non-local stability analyses. Near the wing root and tip, an increase in the magnitudes of spanwise velocities is observed within the laminar separation bubble, and linear stability calculations predict a relative increase in the amplification of spanwise wavenumbers with the same sign as the spanwise flow. However, the calculations show that the most amplified disturbance remains essentially two-dimensional across the entire wingspan.

Numerical Simulation of Two-Dimensional Shear Flow

Numerical experiments on finite difference solutions of time dependent two-dimensional Navier-Stokes equations are used to study the transition triggering off in a mixing layer of initial tanh(y) profile. The inflow is excited by sinusoidal waves resulting from the linear theory of hydrodynamic instability. Numerical realizations are compared, through stream wise growth of momentum thickness and vorticity plots, with Winant and Browand experiments.&#x0D; The transition control and triggering off are studied through the finite difference solution of the two-dimensional time dependant Navier-Stokes equations. In a shear flow, the transition results from hydrodynamic amplification of unstable disturbances stimulated by perturbations of diverse origin. In a unidirectional flow, the most unstable disturbances are two-dimensional transverse waves {2, 3} and experiments have shown the area where small disturbances grow exponentially {5,8,9,10}. Thus, a two dimensional numerical simulation is well suited to the study of transition triggering off.

A data-driven optimal disturbance procedure for free-stream turbulence induced transition

The investigation of free-stream turbulence induced transition by means of simple and effective numerical methods traditionally represents a major challenge in the aerodynamic field. In this work, a data-driven algorithm aimed at obtaining optimal forcing and response concerning free-stream turbulence induced boundary layer transition is introduced. The method, referred to as Data-driven Optimal Disturbance (DOD) in the following, relies on dynamic mode decomposition to compute the linear matrix responsible for disturbance evolution in the streamwise direction and opens the possibility for optimal disturbance analysis in an equation-free manner. The procedure has been applied to high-fidelity large-eddy simulation (LES) results concerning zero pressure gradient flows. Four different combinations of turbulence intensity Tu and integral length scale Lg have been adopted as boundary conditions to investigate the sensitivity of the transition route to the free-stream turbulence properties. Overall, DOD applied within the transitional region identifies highly energetic turbulent scales embedded in the free-stream as the optimal forcing inducing the formation of streaky structures within the boundary layer. Furthermore, streaky structures characterized by the same spanwise wavelength observed in the LES results are identified by DOD as the boundary layer response to the optimal forcing. Finally, the amplification of disturbances provided by DOD along the streamwise direction clearly resembles the well-established transient growth. Thus, DOD appears a useful tool to analyze the free-stream turbulence induced transition of boundary layers by a simple equation-free algorithm merely based on data analytics.

Effects of Upstream Wakes on the Boundary Layer Over a Low-Pressure Turbine Blade

Abstract In the present work, the evolution of the boundary layer over a low-pressure turbine blade is studied using direct numerical simulations, with the aim of investigating the unsteady flow field induced by the rotor-stator interaction. The freestream flow is characterized by the high level of freestream turbulence and periodically impinging wakes. As in the experiments, the wakes are shed by moving bars modeling the rotor blades and placed upstream of the turbine blades. To include the presence of the wake without employing an ad-hoc model, we simulate both the moving bars and the stationary blades in their respective frames of reference and the coupling of the two domains is done through appropriate boundary conditions. The presence of the wake mainly affects the development of the boundary layer on the suction side of the blade. In particular, the flow separation in the rear part of the blade is suppressed. Moreover, the presence of the wake introduces alternating regions in the streamwise direction of high- and low-velocity fluctuations inside the boundary layer. These fluctuations are responsible for significant variations of the shear stress. The analysis of the velocity fields allows the characterization of the streaky structures forced in the boundary layer by turbulence carried by upstream wakes. The breakdown events are observed once positive streamwise velocity fluctuations reach the end of the blade. Both the fluctuations induced by the migration of the wake in the blade passage and the presence of the streaks contribute to high values of the disturbance velocity inside the boundary layer with respect to a steady inflow case. The amplification of the boundary layer disturbances associated with different spanwise wavenumbers has been computed. It was found that the migration of the wake in the blade passage stands for the most part of the perturbations with zero spanwise wavenumber. The non-zero wavenumbers are found to be amplified in the rear part of the blade at the boundary between the low- and high-speed regions associated with the wakes.

Wing sweep effects on laminar separated flows

We reveal the effects of sweep on the wake dynamics around NACA 0015 wings at high angles of attack using direct numerical simulations and resolvent analysis. The influence of sweep on the wake dynamics is considered for sweep angles from $0^\circ$ to $45^\circ$ and angles of attack from $16^\circ$ to $30^\circ$ for a spanwise periodic wing at a chord-based Reynolds number of $400$ and a Mach number of $0.1$ . Wing sweep affects the wake dynamics, especially in terms of stability and spanwise fluctuations with implications on the development of three-dimensional (3-D) wakes. We observe that wing sweep attenuates spanwise fluctuations. Even as the sweep angle influences the wake, force and pressure coefficients can be collapsed for low angles of attack when examined in wall-normal and wingspan-normal independent flow components. Some small deviations at high sweep and incidence angles are attributed to vortical wake structures that impose secondary aerodynamic loads, revealed through the force element analysis. Furthermore, we conduct global resolvent analysis to uncover oblique modes with high disturbance amplification. The resolvent analysis also reveals the presence of wavemakers in the shear-dominated region associated with the emergence of 3-D wakes at high angles of attack. For flows at high sweep angles, the optimal convection speed of the response modes is shown to be faster than the optimal wavemakers speed suggesting a mechanism for the attenuation of perturbations. The present findings serve as a fundamental stepping stone to understanding separated flows at higher Reynolds numbers.

Instability and Transition of Hypersonic Boundary Layer on a Blunted Delta Wing

The instability and transition of hypersonic boundary layer on a blunted delta wing is performed by numerical simulation of experiments conducted in the quiet wind tunnel. Disturbances are calculated using linear stability theory to obtain the integrated disturbance growth rates and amplification factors from laminar steady base flows. Comparisons are made between the results of the stability analysis and the experimentally measured transition characteristics. The results show that the generalized inflections existing in velocity profiles reveal both the streamwise instability and crossflow instability of the boundary layer on the delta wing. In the parameter spaces of frequency and wave number, there is only one unstable region in which the first-mode and second-mode waves are integrated with each other at angle of attack, whereas the unstable regions of the first-mode and second-mode waves are separately distributed at angle of attack of . The numerically predicted factor distribution is in good agreement with the measured transition front, and the factor is about 3 when boundary-layer transition occurs. The low-frequency unstable waves with about 25–30 kHz play a key role in dominating the boundary-layer transition of the delta wing.

Oblique transition in hypersonic double-wedge flow

We utilize resolvent and weakly nonlinear analyses in combination with direct numerical simulations (DNS) to identify mechanisms for oblique transition in a Mach $5$ hypersonic flow over an adiabatic slender double wedge. Even though the laminar separated flow is globally stable, resolvent analysis demonstrates significant amplification of unsteady external disturbances to the linearized flow equations. These disturbances are introduced upstream of the separation zone and they lead to the appearance of oblique waves further downstream. We demonstrate that the large amplification of oblique waves arises from the growth of fluctuation shear stress due to streamline curvature of the laminar base flow in the separated shear layer. This is in contrast to the attached boundary layers, where no such mechanism exists. We also use a weakly nonlinear analysis to show that the resolvent operator associated with linearization around the laminar base flow governs the evolution of steady reattachment streaks that arise from quadratic interactions of unsteady oblique waves. These quadratic interactions generate vortical excitations in the reattaching shear layer which lead to the formation of streaks in the recirculation zone and their subsequent amplification, breakdown and transition to turbulence downstream. Our analysis of the energy budget shows that deceleration of the base flow near reattachment is primarily responsible for amplification of steady streaks. Finally, we employ DNS to examine latter stages of transition to turbulence and demonstrate the predictive power of a weakly nonlinear input–output framework in uncovering triggering mechanisms for oblique transition in separated high-speed boundary layer flows.

Boundary-Layer Instabilities over a Cone–Cylinder–Flare Model at Mach 6

Computations are performed to investigate the convective and global boundary-layer instabilities over a sharp cone–cylinder–flare model at zero-degree angle of attack. The model geometry and the flow conditions are selected to match the experiments conducted in the Boeing/AFOSR Mach 6 Quiet Tunnel at Purdue University. The geometry consists of a nominally sharp 5 deg half-angle cone, followed by a cylindrical segment, and then a 10 deg flare. Additionally, flare half angles equal to 8 and 12 deg and nosetip radii equal to 0.1, 1, and 5 mm are studied. An axisymmetric separation bubble is generated as a result of the laminar shock–boundary-layer interaction in the cylinder–flare region. The comparison of the laminar flow solution and the schlieren images shows a remarkable agreement between the respective locations of both the boundary-layer edge and the reattachment shock. The linear amplification of first and second Mack mode instabilities that begin to amplify in the cone region are computed with a combination of the parabolized stability equations and the harmonic linearized Navier–Stokes equations. The predicted frequency spectra of the surface pressure fluctuations associated with both planar and oblique instability waves capture the distinct lobes within the disturbance amplification spectra measured with PCB® and Kulite sensors. To our knowledge, this represents the first successful comparison between convective instability analysis and measured surface pressure fluctuations for a hypersonic configuration with a separation bubble. Finally, the global instability analysis shows that the laminar flow becomes supercritical for flare half angles larger than 8 deg.

Transition induced by wall-normal vibration in flow around a flat plate with roughness

The effects of wall-normal vibration on bypass transition in a boundary-layer flow over a flat plate with roughness elements in the form of circular cylinders are investigated using direct numerical simulations (DNS), linear Floquet analyses and dynamic mode decompositions (DMD). The vibration of the plate strengthens the streamwise vorticity, consequently enhancing the velocity streaks and reducing the critical Reynolds number for transition. A map is constructed to identify the coupling effect of the vibration amplitude and Reynolds number on transition. Among all investigated combinations of height and diameter of roughness, the critical Reynolds numbers at different vibration amplitude $A$ can be unified by a scaling function of $(1-10A)$ . Two instability modes are identified in the vibration-induced transition process: a wake mode immediately downstream of the roughness, related to the inviscid Kelvin–Helmholtz instability of the wall-normal shear in the wake; and a streak mode, in response to the spanwise shear of the streaky flow, occurring further downstream. The first one results in the generation of central hairpin vortices which then feed the second one. Further development of the streak instability leads to two arrays of hairpin vortices. Results from linear Floquet analyses and DMD further confirm the two modes of instability observed in DNS. A quantitative study suggests that the amplification of vibration-induced disturbance by the base shear dominates the production of streamwise vorticity and subsequently the hairpin vortices.

Optimal control of growth of instabilities in Taylor–Couette flow

The present work aims to achieve optimal control of instabilities in a standard Taylor–Couette flow. The motivation of the present study is to reduce the disturbance growth and delay the transition process to turbulence. We numerically employ control using a stability modifier, namely, wall transpiration. In the non-modal stability framework, we form a state-space model employing control actuation by means of periodic suction/blowing of fluid from the walls. The study is conducted for two cases of flow rotations: (i) counter-rotating cylinders and (ii) the stationary outer cylinder with inner cylinder rotating. The parametric study was performed with varying radii ratios, Reynolds numbers (Re), axial (α), and azimuthal (n) wavenumbers. The time evolution of governing equation is written in terms of perturbation velocities in radial (r) and azimuthal (θ) directions. The optimal feedback control is obtained using a linear quadratic regulator controller and feed backed to the system to reduce the maximum optimal growth of the instabilities in the flow. The perturbation kinetic energy is taken as the measure of the amplification of disturbances and used as the cost function to be minimized. We use Chebyshev spectral collocation method to discretize the equations and variational method to calculate the optimal growth. We studied four different parametric cases of radii ratios (η=r1r2= 0.1, 0.25, 0.5, 0.75), with angular velocity (Ω2Ω1) ratio fixed as μ=−1 and μ = 0. We choose the subcritical wavenumbers that led to a maximum transient energy growth corresponding to a Reynolds number ≈ 0.65 times the critical Reynolds number for the case of counter-rotation. For the case of the stationary outer cylinder, we showed the effect of the control in the modal analysis framework. The presented control technique resulted in a maximum of 72% reduction in the growth rate and the typical growth of perturbation energy.

Transition, intermittency and phase interference effects in airfoil secondary tones and acoustic feedback loop

A large eddy simulation is performed to study secondary tones generated by a NACA0012 airfoil at angle of attack of $\alpha = 3^{\circ}$ with freestream Mach number of $M_{\infty} = 0.3$ and Reynolds number of $Re = 5 \times 10^4$ . Laminar separation bubbles are observed over the suction side and near the trailing edge, on the pressure side. Vortex shedding occurs aft of the suction side separation bubble, and vortex interaction results in merging or bursting such that coherent structures or turbulent packets are advected towards the trailing edge. This mechanism modulates the amplitude of the incident pressure signal, leading to different levels of noise emission. Despite the intermittent occurrence of laminar–turbulent transition, the noise spectrum depicts a main tone with multiple equidistant secondary tones. To understand the role of flow instabilities on the tones, the linearised Navier–Stokes equations are examined in their operator form through biglobal stability and resolvent analyses, and by time evolution of disturbances using a matrix-free method. These linear global analyses reveal amplification of disturbances over the suction side separation bubble. Spanwise-averaged pressure fluctuations elucidate aspects of the acoustic feedback loop mechanism in the nonlinear solutions. This feedback process is self-sustained by acoustic waves radiated from the trailing edge, which reach the most sensitive flow location near the leading edge, as identified by the resolvent analysis. Flow disturbances arising from secondary diffraction and phase interference among the most unstable frequencies computed in the eigenspectrum are shown to have an important role in the feedback loop.