Sensitivity analysis and passive control of the secondary instability in the wake of a cylinder

2019 ◽  
Vol 864 ◽  
pp. 45-72 ◽  
Author(s):  
F. Giannetti ◽  
S. Camarri ◽  
V. Citro
Keyword(s):  
Passive Control ◽  
Stokes Equations ◽  
Classical Problem ◽  
Base Flow ◽  
Small Object ◽  
Stability Properties ◽  
Floquet Exponent ◽  
Flow Configurations ◽  
The Stability ◽  
Time Periodic

The stability properties of selected flow configurations, usually denoted as base flows, can be significantly altered by small modifications of the flow, which can be caused, for instance, by a non-intrusive passive control. This aspect is amply demonstrated in the literature by ad hoc sensitivity studies which, however, focus on configurations characterised by a steady base flow. Nevertheless, several flow configurations of interest are characterised by a time-periodic base flow. To this purpose, we propose here an original theoretical framework suitable to quantify the effects of base-flow variations in the stability properties of saturated time-periodic limit cycles. In particular, starting from a Floquet analysis of the linearised Navier–Stokes equations and using adjoint methods, it is possible to estimate the variation of a selected Floquet exponent caused by a generic structural perturbation of the base-flow equations. This link is expressed concisely using the adjoint operators coming from the analysis, and the final result, when applied to spatially localised disturbances, is used to build spatial sensitivity and control maps. These maps identify the regions of the flow where the placement of a infinitesimal small object produces the largest effect on the Floquet exponent and may also provide a quantification of this effect. Such analysis brings useful insights both for passive control strategies and for further characterising the investigated instability. As an example of application, the proposed analysis is applied here to the three-dimensional flow instabilities in the wake past a circular cylinder. This is a classical problem which has been widely studied in the literature. Nevertheless, by applying the proposed analysis we derive original results comprising a further characterisation of the instability and related control maps. We finally show that the control maps obtained here are in very good agreement with control experiments documented in the literature.

scholarly journals The acoustic impedance of a laminar viscous jet through a thin circular aperture

2019 ◽  
Vol 864 ◽  
pp. 5-44 ◽  
Author(s):  
David Fabre ◽  
Raffaele Longobardi ◽  
Paul Bonnefis ◽  
Paolo Luchini
Keyword(s):  
Reynolds Number ◽  
Stokes Equations ◽  
Axisymmetric Flow ◽  
Classical Problem ◽  
Base Flow ◽  
Large Reynolds Number ◽  
Linear Perturbation ◽  
Circular Aperture ◽  
Vena Contracta ◽  
Numerical Resolution

The unsteady axisymmetric flow through a circular aperture in a thin plate subjected to harmonic forcing (for instance under the effect of an incident acoustic wave) is a classical problem first considered by Howe (Proc. R. Soc. Lond. A, vol. 366, 1979, pp. 205–223), using an inviscid model. The purpose of this work is to reconsider this problem through a numerical resolution of the incompressible linearized Navier–Stokes equations (LNSE) in the laminar regime, corresponding to $Re=[500,5000]$. We first compute a steady base flow which allows us to describe the vena contracta phenomenon in agreement with experiments. We then solve a linear problem allowing us to characterize both the spatial amplification of the perturbations and the impedance (or equivalently the Rayleigh conductivity), which is a key quantity to investigate the response of the jet to acoustic forcing. Since the linear perturbation is characterized by a strong spatial amplification, the numerical resolution requires the use of a complex mapping of the axial coordinate in order to enlarge the range of Reynolds number investigated. The results show that the impedances computed with $Re\gtrsim 1500$ collapse onto a single curve, indicating that a large Reynolds number asymptotic regime is effectively reached. However, expressing the results in terms of conductivity leads to substantial deviation with respect to Howe’s model. Finally, we investigate the case of finite-amplitude perturbations through direct numerical simulations (DNS). We show that the impedance predicted by the linear approach remains valid for amplitudes up to order $10^{-1}$, despite the fact that the spatial evolution of the perturbations in the jet is strongly nonlinear.

scholarly journals Direct and adjoint global modes of a recirculation bubble: lift-up and convective non-normalities

2009 ◽  
Vol 622 ◽  
pp. 1-21 ◽  
Author(s):  
OLIVIER MARQUET ◽  
MATTEO LOMBARDI ◽  
JEAN-MARC CHOMAZ ◽  
DENIS SIPP ◽  
LAURENT JACQUIN
Keyword(s):  
High Speed ◽  
Passive Control ◽  
Three Dimensional ◽  
Adjoint Problem ◽  
Base Flow ◽  
Mode Analysis ◽  
Global Mode ◽  
Recirculation Bubble ◽  
Global Modes ◽  
The Stability

The stability of the recirculation bubble behind a smoothed backward-facing step is numerically computed. Destabilization occurs first through a stationary three-dimensional mode. Analysis of the direct global mode shows that the instability corresponds to a deformation of the recirculation bubble in which streamwise vortices induce low- and high-speed streaks as in the classical lift-up mechanism. Formulation of the adjoint problem and computation of the adjoint global mode show that both the lift-up mechanism associated with the transport of the base flow by the perturbation and the convective non-normality associated with the transport of the perturbation by the base flow explain the properties of the flow. The lift-up non-normality differentiates the direct and adjoint modes by their component: the direct is dominated by the streamwise component and the adjoint by the cross-stream component. The convective non-normality results in a different localization of the direct and adjoint global modes, respectively downstream and upstream. The implications of these properties for the control problem are considered. Passive control, to be most efficient, should modify the flow inside the recirculation bubble where direct and adjoint global modes overlap, whereas active control, by for example blowing and suction at the wall, should be placed just upstream of the separation point where the pressure of the adjoint global mode is maximum.

scholarly journals The Navier–Stokes equations on the plane with time-dependent external forces

2021 ◽  
Vol 2 (4) ◽  
Author(s):  
Masao Yamazaki
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Almost Periodic ◽  
Existence Of A Solution ◽  
Navier Stokes Equations ◽  
Time Periodic Solutions ◽  
The Stability ◽  
External Forces ◽  
Stationary Navier Stokes Equations ◽  
Time Periodic

AbstractWe are concerned with the non-stationary Navier–Stokes equations on the whole plane with external forces which are non-decaying in time, and give a sufficient condition on the external forces for the existence of a solution which exists for whole time. Typical examples are time-periodic solutions and solutions almost periodic in time. The stability under perturbation is also verified.

Stability of the wakes of cylinders with triangular cross-sections

2018 ◽  
Vol 844 ◽  
pp. 721-745 ◽  
Author(s):  
Zhi Y. Ng ◽  
Tony Vo ◽  
Gregory J. Sheard
Keyword(s):  
Stability Analysis ◽  
Cross Sections ◽  
Reynolds Numbers ◽  
Base Flow ◽  
Half Period ◽  
Two Dimensional ◽  
Instability Modes ◽  
The Stability ◽  
Time Periodic ◽  
Mode A

The stability of the wakes of cylinders with triangular cross-sections at incidence is investigated using Floquet stability analysis to elucidate the effects of cylinder inclination on the dominant flow instability. The upper limit of the Reynolds numbers (scaled by the height projected by the cylinder in this study) at which the wake of the two-dimensional base flow is time periodic is$Re\approx 140$for most cylinder inclinations, exceeding which the flow becomes aperiodic, restricting the range of Reynolds numbers permitted for the stability analysis. Two different instability modes are predicted to manifest as the first-occurring mode at various cylinder inclinations – a regular mode possessing perturbation structures consistent with mode A dominates the wakes of cylinders at inclinations$\unicode[STIX]{x1D6FC}\lesssim 34.6^{\circ }$and$\unicode[STIX]{x1D6FC}\gtrsim 55.4^{\circ }$, with a subharmonic mode consistent with mode C emerging as the primary mode in the wakes of the cylinder at the intermediate range of inclinations. For all inclinations, the mode B branch is not detected within the range of Reynolds numbers examined. The peak instability growth rates corresponding to mode A for all cylinder inclinations describe a linear variation with$(Re-Re_{A})/Re_{A}$, where$Re_{A}$is the mode A transition Reynolds number, while those corresponding to mode C vary only approximately linearly. The generalized trend most pertinently shows mode C to develop more rapidly than mode A at inclinations which permit it. Examination of the near wake of the two-dimensional time-periodic base flow demonstrates the dependence of the development and intensity of mode C on imbalances in the flow solution over each shedding period, directly implying that the two-dimensional base flow solutions deviate from the half-period-flip map as the cylinder inclination is increased. The degree of asymmetry of the two-dimensional base flow relative to the ideal half-period-flip map is quantified using several measures. The results show distinctly different trends in these asymmetry measures between inclinations where modes A or C are dominant, agreeing with results from the stability analysis. The nature of the predicted instability modes at transition are also investigated by applying the Stuart–Landau equation, showing the transitions to be supercritical for all cylinder inclinations, with mode C being consistently more strongly supercritical than mode A.

Sensitivity analysis and passive control of cylinder flow

2008 ◽  
Vol 615 ◽  
pp. 221-252 ◽  
Author(s):  
OLIVIER MARQUET ◽  
DENIS SIPP ◽  
LAURENT JACQUIN
Keyword(s):  
Sensitivity Analysis ◽  
Vortex Shedding ◽  
Passive Control ◽  
Control Device ◽  
Base Flow ◽  
Linear Stability Theory ◽  
Sensitivity Analyses ◽  
Global Modes ◽  
The Stability ◽  
Steady Force

A general theoretical formalism is developed to assess how base-flow modifications may alter the stability properties of flows studied in a global approach of linear stability theory. It also comprises a systematic approach to the passive control of globally unstable flows by the use of small control devices. This formalism is based on a sensitivity analysis of any global eigenvalue to base-flow modifications. The base-flow modifications investigated are either arbitrary or specific ones induced by a steady force. This leads to a definition of the so-called sensitivity to base-flow modifications and sensitivity to a steady force. These sensitivity analyses are applied to the unstable global modes responsible for the onset of vortex shedding in the wake of a cylinder for Reynolds numbers in the range 47≤Re≤80. First, it is demonstrated how the sensitivity to arbitrary base-flow modifications may be used to identify regions and properties of the base flow that contribute to the onset of vortex shedding. Secondly, the sensitivity to a steady force determines the regions of the flow where a steady force acting on the base flow stabilizes the unstable global modes. Upon modelling the presence of a control device by a steady force acting on the base flow, these predictions are then extensively compared with the experimental results of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, p. 71). A physical interpretation of the suppression of vortex shedding by use of a control cylinder is proposed in the light of the sensitivity analysis.

Global stability result for parabolic Cauchy problems

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Mourad Choulli ◽  
Masahiro Yamamoto
Keyword(s):  
New York ◽  
Partial Differential Equations ◽  
Global Stability ◽  
Classical Problem ◽  
Stability Result ◽  
Cauchy Problems ◽  
Smooth Solutions ◽  
Linear Partial Differential Equations ◽  
Ill Posed ◽  
The Stability

AbstractUniqueness of parabolic Cauchy problems is nowadays a classical problem and since Hadamard [Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Dover, New York, 1953], these kind of problems are known to be ill-posed and even severely ill-posed. Until now, there are only few partial results concerning the quantification of the stability of parabolic Cauchy problems. We bring in the present work an answer to this issue for smooth solutions under the minimal condition that the domain is Lipschitz.

scholarly journals Time periodic Navier–Stokes equations in a thin tube structure

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
R. Juodagalvytė ◽  
G. Panasenko ◽  
K. Pileckas
Keyword(s):  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Thin Tube ◽  
Time Periodic ◽  
Tube Structure

scholarly journals Velocity and acceleration level inverse kinematic calculation alternatives for redundant manipulators

Meccanica ◽  
2021 ◽  
Author(s):  
Dóra Patkó ◽  
Ambrus Zelei
Keyword(s):  
Degrees Of Freedom ◽  
Direct Method ◽  
Tracking Error ◽  
Inverse Kinematic ◽  
Linear Feedback ◽  
Joint Position ◽  
Stability Properties ◽  
Acceleration Level ◽  
Error Elimination ◽  
The Stability

AbstractFor both non-redundant and redundant systems, the inverse kinematics (IK) calculation is a fundamental step in the control algorithm of fully actuated serial manipulators. The tool-center-point (TCP) position is given and the joint coordinates are determined by the IK. Depending on the task, robotic manipulators can be kinematically redundant. That is when the desired task possesses lower dimensions than the degrees-of-freedom of a redundant manipulator. The IK calculation can be implemented numerically in several alternative ways not only in case of the redundant but also in the non-redundant case. We study the stability properties and the feasibility of a tracking error feedback and a direct tracking error elimination approach of the numerical implementation of IK calculation both on velocity and acceleration levels. The feedback approach expresses the joint position increment stepwise based on the local velocity or acceleration of the desired TCP trajectory and linear feedback terms. In the direct error elimination concept, the increment of the joint position is directly given by the approximate error between the desired and the realized TCP position, by assuming constant TCP velocity or acceleration. We investigate the possibility of the implementation of the direct method on acceleration level. The investigated IK methods are unified in a framework that utilizes the idea of the auxiliary input. Our closed form results and numerical case study examples show the stability properties, benefits and disadvantages of the assessed IK implementations.

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