flow control using the parabolized stability equations

Thesis presented to the Instituto Tecnológico de Aeronáutica, in partial fulfillment of the requirements for the degree of Doctor of Science in the Program of Aeronautical Engineering, Field of Aerodynamics, Propulsion and Energy.

Kenzo Sasaki

FLOW CONTROL USING THE PARABOLIZED STABILITY EQUATIONS

Thesis approved in its final version by signatories below:

Prof. Dr. André Valdetaro Gomes Cavalieri Advisor

Prof. Dr. Luiz Carlos Sandoval Góes Prorector of Graduate Studies and Research

Campo Montenegro São José dos Campos, SP - Brazil 2016

Cataloging-in Publication Data Documentation and Information Division Sasaki, Kenzo Flow control using the parabolized stability equations / Kenzo Sasaki. S˜ ao José dos Campos, 2016. 116f. Thesis of Doctor of Science – Course of Aeronautical Engineering. Area of Aerodynamics, Propulsion and Energy – Instituto Tecnológico de Aeronáutica, 2016. Advisor: Prof. Dr. André Valdetaro Gomes Cavalieri. 1. Aeroacoustics. 2. Instability Waves. 3. Jet Noise. I. Instituto Tecnológico de Aeronáutica. II. Title.

BIBLIOGRAPHIC REFERENCE SASAKI, Kenzo. Flow control using the parabolized stability equations. 2016. 116f. Thesis of Doctor of Science – Instituto Tecnológico de Aeronáutica, São José dos Campos.

CESSION OF RIGHTS AUTHOR’S NAME: Kenzo Sasaki PUBLICATION TITLE: Flow control using the PUBLICATION KIND/YEAR: Thesis / 2016

parabolized stability equations.

It is granted to Instituto Tecnológico de Aeronáutica permission to reproduce copies of this thesis and to only loan or to sell copies for academic and scientific purposes. The author reserves other publication rights and no part of this thesis can be reproduced without the authorization of the author.

Kenzo Sasaki Av. Cidade Jardim, 679 12.233-066 – São José dos Campos–SP

FLOW CONTROL USING THE PARABOLIZED STABILITY EQUATIONS

Kenzo Sasaki

Thesis Committee Composition:

Prof. Prof. Prof. Prof.

Dr. Dr. Dr. Dr.

André Valdetaro Gomes Cavalieri André Valdetaro Gomes Cavalieri Márcio Teixeira Mendon¸ca Paulo Afonso de Oliveira Soviero

ITA

President Advisor Jury Jury

-

ITA ITA IAE ITA

To my future wife, Luciana and to my parents, Jorge and Inês.

“Because if you’re willing to go through all the battling you got to go through to get where you want to get, who’s got the right to stop you?” — Rocky Balboa

Abstract We develop a framework for the closed-loop control of fluctuations along flow systems, which is based in reduced-order models. Such models were obtained from several techniques, which were either identification or PSE-based (where PSE stands for Parabolized Stability Equations) and were validated for two convective problems, a shear-layer and a high-Reynolds turbulent jet. One of the main results of this work is a further investigation of the applicability of these techniques, particularly for the PSE-based transfer function, which supplies further evidence on the importance of wavepacket structures for the real-time behaviour of flows and its link to linear control. Furthermore, we have posed the problem in terms of control methodologies, introducing several of the concepts and possible strategies for obtaining real-time alterations of a fluid system, some of which were already tested and validated on a non-linear simulation. Understanding the behaviour of the different control-laws that were already implemented and modelling of each of the individual necessary components for the current closed-loop control is also one of the main results obtained so far. Regarding the further work to be done until the end of the PhD, the foundations related to theoretical aspects which are necessary to proceed are also being addressed over this document.

List of Figures

FIGURE 1.1 – Qualitative description about the processes involved in jet noise generation, as presented on (GU MUNDSSON, 2009). . . . . . . . . . . . 19 FIGURE 1.2 – Mean flow profile for the axial velocity, for a Mach 0.9 turbulent jet.

20

FIGURE 1.3 – Mean axial velocity flow profile at constant axial slices, for a Mach 0.9 turbulent jet. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 FIGURE 1.4 – Visualization of a turbulent jet (MOORE, 1977). . . . . . . . . . . . 23 FIGURE 1.5 – Schematic representation of boundary-layer transition, reproduced from (KACHANOV, 1994). . . . . . . . . . . . . . . . . . . . . . . . . 26 FIGURE 2.1 – Comparison between the total PSE solution q e

Rx 0

α(ξ) dξ

with the cor-

responding Fourier component of the DNS. . . . . . . . . . . . . . . 32 FIGURE 2.2 – Center-line power spectral densities of the axial velocity - comparison between the implemented PSE method (squares) and data presented in (CAVALIERI et al., 2013): earlier PSE calculation (solid line) and experimental values obtained from hot-wire measurements (circles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 FIGURE 2.3 – β parameter plot for a Mach 0.9 turbulent jet. Dark line represents the region where PSE saturates, predicting a stable behaviour for the fluctuations along the flow. . . . . . . . . . . . . . . . . . . . . . 36

LIST OF FIGURES

viii

FIGURE 2.4 – Frequency and time-domain transfer functions between three inputoutput combinations (top: xi /D, xo /D) = (0.05, 1.7); middle: (1.3, 2.5); bottom: (2.1, 3.3)) for pressure fluctuations on the 8◦ conical surface. Blue dots: experimental data; solid red lines: PSE. . . . . . . . . . . 48 FIGURE 2.5 – Comparison between measured non-dimensional pressure fluctuations, PSE transfer function, empirical transfer function and ARMAX predictions, with x/D = 2.1 and 3.3 as the input and output positions, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . 49 FIGURE 2.6 – Comparison between measured non-dimensional pressure fluctuations (solid line) and PSE transfer function prediction (dashed line) for x/D = 0.5, 1.3 and 2.1, in the first, second and third lines, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 FIGURE 2.7 – Peak correlation between PSE predicted signal and the measured data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 FIGURE 2.8 – Peak correlation between the predicted signal with the different methods and the measured data considering the input at xo /D = 2.1. 52 FIGURE 2.9 – Comparison between measured non-dimensional pressure fluctuations POD filtered (in blue) and PSE transfer function prediction (in red) for xi /D = 0.5, 1.3 and 2.1, in the first, second and third lines, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 FIGURE 2.10 –Peak correlation values between the PSE transfer function prediction and measured data, considering POD filtered values. . . . . . . . . . 54 FIGURE 2.11 –PSE transfer function pressure field prediction for several time slices obtained from measurements along the x/D = 0.5 ring. . . . . . . . 54 FIGURE 2.12 –PSE transfer function axial velocity field prediction for several time slices, obtained from measurements along the x/D = 0.5 ring. . . . . 55

LIST OF FIGURES

ix

FIGURE 2.13 –Comparison between PSE transfer function prediction and results from the non-linear simulation, for input in (x, y) = (75, 0) and output in (100, 0). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 FIGURE 2.14 –Resulting correlation between prediction and simulation for several input and output combinations. . . . . . . . . . . . . . . . . . . . . 56 FIGURE 2.15 –DNS of the bidimensional shear-layer in comparison against PSE transfer functions predicion from a single measurement. . . . . . . . 57 FIGURE 3.1 – Chevron geometry applied to a turbofan engine, as presented in (SAIYED et al., 2003). . . . . . . . . . . . . . . . . . . . . . . . . . . 60 FIGURE 3.2 – Example of open-loop active strategy for the control of noise from a turbulent round jet (KŒNIG, 2011). . . . . . . . . . . . . . . . . . . 61 FIGURE 3.3 – Classification of actuator types (CATTAFESTA; SHEPLAK, 2011). . . 62 FIGURE 3.4 – Actuation behaviour along time and space. . . . . . . . . . . . . . . 63 FIGURE 3.5 – Chosen contour path of integration, followed anti-clockwise, a semi circle of infinite radius joined to a curve of area tending to zero, which is made to capture the Kelvin-Helmholtz mode. . . . . . . . . 66 FIGURE 3.6 – Frequency domain comparison of the theoretical and empirical transfer functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 FIGURE 3.7 – Comparison between prediction and simulation for a given perturbation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 FIGURE 3.8 – Correlation between prediction and simulation at several input-output combinations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 FIGURE 3.9 – Basic block diagram for feedforward control. . . . . . . . . . . . . . 72 FIGURE 3.10 –Basic block diagram for feedback control. . . . . . . . . . . . . . . . 72 FIGURE 3.11 –Basic block diagram for feedback control when sensor and objective are not at the same position. . . . . . . . . . . . . . . . . . . . . . . 72

LIST OF FIGURES

x

FIGURE 3.12 –Turbulent jet and bidimensional shear-layer impulse responses obtained via PSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 FIGURE 3.13 –Resulting frequency domain transfer functions between estimation, actuation and output and computed gain for the compensator. . . . 86 FIGURE 3.14 –Resulting gain obtained from the frequency inversion method, in the time-domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 FIGURE 3.15 –Open-loop prediction obtained from the PSE transfer function and the controlled case via system inversion, considering the linear control framework, with the objective of minimizing the axial-velocity component.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

FIGURE 3.16 –Open-loop prediction obtained from the PSE transfer function and the controlled case via proportional law, considering the linear control framework, with the axial velocity fluctuation as the control objective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 FIGURE 3.17 –Open-loop prediction obtained from the PSE transfer function and the controlled case via proportional integral law, considering the linear control framework, with the axial velocity fluctuation as the control objective. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 FIGURE 3.18 –Behaviour of the transfer functions of equation 3.44 in the frequency domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 FIGURE 3.19 –Comparison of the uncontrolled simulation against the controlled case, using system inversion, for the axial and transverse velocity fluctuations, with the time to the fluctuation to reach the actuator (a), transient (b), buffer (c) and time to reach the actuation (d), highlighted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 FIGURE 3.20 –Turbulent kinetic energy for the uncontrolled and controlled simulations via single body force, using system inversion. . . . . . . . . . 92

LIST OF FIGURES

xi

FIGURE 3.21 –Turbulent kinetic energy for the uncontrolled and controlled simulations via single body force, comparison of the three control methodologies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 FIGURE 3.22 –Comparison of the vorticity fluctuations for the uncontrolled (a, c and e) and controlled (b, d and f) cases. The delay in the vortex pairing becomes apparent. . . . . . . . . . . . . . . . . . . . . . . . 93 FIGURE 3.23 –Block diagram of the aligned actuators problem, for this case there are sensors in positions 1 and 4, actuators in 2 and 5 and the objectives are at 3 and 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 FIGURE 3.24 –Results for the controlled and uncontrolled transverse velocity component for a single and two aligned actuators at position of X = 175. 95 FIGURE 3.25 –Comparison of the vorticity fluctuations for the uncontrolled (a, c and e) and controlled (b, d and f) with two aligned actuators (right) cases. The delay in the vortex pairing becomes apparent. . . . . . . 96 FIGURE 4.1 – Comparison between the system inversion and On/Off control schemes, for the transverse velocity component, at the position of the objective. 99 FIGURE 4.2 – Illustration of the minimum buffer size and minimum period for sampling, for a resulting given gain k(t). . . . . . . . . . . . . . . . 99 FIGURE 4.3 – Robustness verification for amplitude modification of the inflow perturbations. RMS of the streamwise and transverse velocity components of the controlled case divided by the corresponding open-loop scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 FIGURE 4.4 – Robustness verification for Reynolds modification in relation to the corresponding design Reynolds number. RMS of the streamwise and transverse velocity components of the controlled case divided by the corresponding open-loop scenario. . . . . . . . . . . . . . . . . . . . 102

LIST OF FIGURES

xii

FIGURE 4.5 – Robustness verification for perturbations between input and objective. RMS of the streamwise and transverse velocity components of the controlled case divided by the corresponding open-loop scenario. 103 FIGURE 4.6 – Mean axial and transverse velocity components on a Blasius boundarylayer without pressure gradients. . . . . . . . . . . . . . . . . . . . . 105 FIGURE A.1 – Wind-tunnel facility ‘Bruit & Vent’. . . . . . . . . . . . . . . . . . . 114 FIGURE A.2 – Experimental setup: near field microphone cage array in the 7-ring configuration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 FIGURE B.1 – Instantaneous vorticity field of the mixing layer DNS. . . . . . . . . 117

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2

Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.1 1.3

Structure of the document . . . . . . . . . . . . . . . . . . . . . . . . 17

Jet Aeroacoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.1

Characteristics of the sound field of turbulent jets . . . . . . . . . . . 19

1.3.2

Lighthill’s Analogy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.3.3

Large scale coherent structures and noise generation . . . . . . . . . . 22

1.4

Common features of the flows under study . . . . . . . . . . . . . . 24

1.4.1

Absolute and convective instabilities

1.4.2

Transition of boundary layers to turbulence . . . . . . . . . . . . . . . 26

1.4.3

Link with the aeroacoustics of turbulent jets . . . . . . . . . . . . . . 27

1.5

2

16

. . . . . . . . . . . . . . . . . . 25

Partial Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1

28

Frequency domain models . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.1.1

Parabolized Stability Equations . . . . . . . . . . . . . . . . . . . . . 29

2.1.2

Empirical Frequency Response Functions . . . . . . . . . . . . . . . . 34

CONTENTS 2.2

Real-time estimations

. . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2.1

Single-Input-Single-Output Systems . . . . . . . . . . . . . . . . . . . 38

2.2.2

Identification models - ARMAX . . . . . . . . . . . . . . . . . . . . . 41

2.2.3

Single-Input-Multiple-Outputs Systems . . . . . . . . . . . . . . . . . 43

2.2.4

Multiple-Inputs-Single-Input Systems . . . . . . . . . . . . . . . . . . 43

2.2.5

Multiple-Inputs-Multiple-Inputs Systems . . . . . . . . . . . . . . . . 46

2.3

3

xiv

Results for real-time estimations . . . . . . . . . . . . . . . . . . . . . 46

2.3.1

Time-domain predictions for the turbulent jet . . . . . . . . . . . . . 47

2.3.2

The shear-layer problem . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.3.3

Partial conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

The control problem . . . . . . . . . . . . . . . . . . . . . . . . . 3.1

59

Active and passive, opened and closed-loop control of flows . . . . 59

3.1.1

Modelling of actuators . . . . . . . . . . . . . . . . . . . . . . . . . . 61

3.2

Feedback and Feedforward control strategies . . . . . . . . . . . . . 69

3.3

Flow control applications . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3.1

Model construction for control . . . . . . . . . . . . . . . . . . . . . . 74

3.3.2

Model-based control

3.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . 77

Feedforward: Frequency Inversion . . . . . . . . . . . . . . . . . . . . 78

3.4.1

Feedforward: Proportional-Integral Controllers . . . . . . . . . . . . . 79

3.4.2

Feedback control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.4.3

State-Space formulation for control . . . . . . . . . . . . . . . . . . . 83

3.4.4

Adaptative control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.5

Results closed-loop control of a free shear-layer . . . . . . . . . . . 85

CONTENTS

4

xv

3.5.1

Linearized system results . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.5.2

DNS Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.5.3

Dealing with disturbances - Aligned Actuators . . . . . . . . . . . . . 94

Conclusions and future work . . . . . . . . . . . . . . . . . . 4.1

97

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.1.1

Experimental implementation for jet noise control . . . . . . . . . . . 97

4.1.2

Robustness issues - Is feedback more efficient than feedforward? . . . 100

4.1.3

Blasius Boundary Layer - Attempting to delay transition . . . . . . . 104

4.2

Summary of results and conclusions

. . . . . . . . . . . . . . . . . . 104

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 Appendix A – Jet experimental measurements . . . . . . . . 113 Appendix B – DNS of the bidimensional shear-layer . . . 116

1 Introduction

1.1

Motivation

Altering the behaviour of a flow, either passively or actively, in open or closed-loop may lead to several sorts of industrial and academic applications which are related to modification of the growth of the unsteady fluctuations along a fluidic system. Applications are related both to the increase and decrease of growth rates along a system; Increasing the growth of fluctuations and therefore shortening the route to turbulence could lead to an augmentation of combustion or mixing efficiencies(KIM; BEWLEY, 2007). On the other side, a reduction in drag could save millions of dollars in shipping worlwide(CORBETT; KOEHLER, 2003), as the primary expenses of an airline are related to fuel consumptions. Another issue which is related to the unsteadiness fluctuations are the aeroacoustic emissions coming from a turbulent system, such as a jet, boundary-layer or the separated flow over a wing flap. To civil aviation, a reduction of 10 decibels is set as the objective by 2020 in Europe (BOWES et al., 2009). All such problems could be tackled via flow control strategies which will be the object of study of this work. The problem is posed over the framework of reduced-order models, which provide physical insight into the dynamics of the evaluated systems and may be regarded as one of the main results of this study.

CHAPTER 1. INTRODUCTION

1.2

17

Objective

This work aims at establishing a framework for the active, closed-loop control of instability waves in flow systems. In order to obtain such objective, an actuation must be added to the system, which the time behaviour is based in a limited set of measurements (inputs) to alter the behaviour of the unsteady fluctuations at a given position (output). This is accomplished via estimations of the flow fluctuations, which are made via reduced-order models. Such models for flows and actuation, which follow a physical approach based on the principles of linear stability theory and the Parabolized Stability Equations, supply not only an estimation of the flow dynamics, but also physical insight into the physical phenomena involved and therefore, are also the object of the following study. The similarity of the applicable tools both for aeroacoustics and transition to turbulence make both of these problems promising candidates for the use of the prediction and control techniques presented in this thesis. Therefore, the remaining parts of this chapter will be dedicated firstly to the aeroacoustics of jets and then to a brief revision on the routes to turbulence on flow systems, focusing on boundary layers. In what refers to aeroacoustic applications, we are not directly interested in calculating the resulting sound field, as the methods we propose, at least for subsonic applications, are unable to educe it without a further step which is related to the coupling of acoustic analogies.

1.2.1

Structure of the document

The remaining chapters of this qualification are organized as follows; Chapter 2 is dedicated to establishing the theoretical background behind the reducedorder models proposed for the modelling of fluctuations on convective systems. A summary on the parabolized stability equations (PSE) and frequency-response functions is proposed along with a recollection of results for time-domain estimations of turbulent


18

fluctuations. For this purpose, some of the results found in (SASAKI, 2014) will be revised and addressed in more detail along with other time-domain modelling techniques that will be used along the subsequent chapters. Chapter 3 presents a control framework applied to fluidic systems. For this purpose, some aspects of control theory will be revised. Results for active control of a free-shear layer, both in the linear and non-linear frameworks. Finally, the conclusions and foreseeable work until the end of this thesis is presented in Chapter 4.

1.3

Jet Aeroacoustics

The high level of acoustic intensities irradiated from jet engines motivated several studies aiming at determining the mechanisms of wave generation. Such researches culminated on the works of M.J. Lighthill (LIGHTHILL, 1952; LIGHTHILL, 1954) which gave birth to a new field, related to the study of acoustic waves generated aerodynamically. Lighthill performed a Reynolds decomposition over the Navier-Stokes equations in order to formulate the problem as as inhomogeneous wave equation, with the remaining terms acting as a sources. Such concepts will be presented in more detail in 1.3.2. The most relevant sources in jets, at least to what concerns the low frequency acoustic emissions at shallow angles, are large-scale coherent structures, corresponding to a typical spatio-temporal behaviour, which are denominated wavepackets (CAVALIERI et al., 2012; CAVALIERI et al.,

2012; CAVALIERI et al., 2013; JORDAN; COLONIUS, 2013).

The discovery of such structures by Mollo-Christensen (MOLLO-CHRISTENSEN, 1963; MOLLO-CHRISTENSEN,

1967) led to a better understanding of the link between turbulence

and the far acoustic field emanating from a jet. The simplicity and richness of information provided by such methods which provide insight into the dynamics of a turbulent field, without prohibitive computational costs, allow for the construction of real-time theoretical models for turbulent systems. Obtaining


19

such reduced order models and understanding the role they may play in flow control is one of the central ideas of this thesis.

1.3.1

Characteristics of the sound field of turbulent jets

The acoustic field radiated from a turbulent jet, is usually referred to as jet noise. Figure 1.1 presents a qualitative description of the processes which are involved with the noise generation.

FIGURE 1.1 – Qualitative description about the processes involved in jet noise generation, as presented on (GU MUNDSSON, 2009).

The boundary layer inside the nozzle suffers a series of instabilities that will eventually transition to turbulence, and the far region from the boundary layer presents hot spots and vorticity coming from the engine combustion and different turbine stages. Considering regions close to a jet nozzle in static conditions, the mean velocity decays sharply with increasing radius, going to zero outside of the jet. This will lead to inflections along the mean profile which subject the jet to the Kelvin-Helmholtz instability(CRIMINALE et al., 2003). Such instability will cause fluctuations to grow, and in case the jet is initially laminar, they will force the transition to turbulence. For the profiles under study over this document, which the profile was obtained either experimentally or via a Large-Eddy-Simulation (LES), the profiles are turbulent ever since the nozzle boundary layer. For such cases, the Kelvin-Helmholtz instability is still present, however it acts as a model for the evolution of coherent structures over the flow. The central region of the jet may be regarded as an irrotational flow, which is therefore


20

1 0.8

R/D

0.5

0

Mixing Layer

0.6

Potential core 0.4

−0.5

−1 0

0.2

0.5

1

1.5

2

2.5 X/D

3

3.5

4

4.5

5

0

FIGURE 1.2 – Mean flow profile for the axial velocity, for a Mach 0.9 turbulent jet.

FIGURE 1.3 – Mean axial velocity flow profile at constant axial slices, for a Mach 0.9 turbulent jet. called potential core. This region extends until the mixing layer collapses, as depicted in figure 1.2. Downstream of such region, the velocity profile becomes bell-shaped 1.3, causing the flow to be stable to inviscid perturbations, as the Kelvin-Helmholtz instabilities.

1.3.2

Lighthill’s Analogy

The works by Lighthill (LIGHTHILL, 1952; LIGHTHILL, 1954) which led to the theory of aeroacoustics may be considered one of the pioneering studies which allowed for the prediction and understanding of some of the characteristics of jet noise. Based on the continuity, Navier-Stokes and energy equations, Lighthill was able to


21

obtain an inhomogeneous wave equation, where the forcing terms have an aerodynamic origin, and act as source terms for the equation. This idea is frequently referred to as Lighthill’s analogy, as it seeks to replace the complex phenomena of sound generation by an equivalent source term. Some of the concepts and consequences of this theory will be briefly summarized over this section. Considering an observer which detects the acoustic field at a given position x and time t, in a fluid at rest, where the ambient the speed of sound is c0 , the fluctuation terms of the fluidic variables will follow a linear wave equation, which is presented below, for the density;

∂ 2 ρ0 − c20 ∇2 ρ0 = 0 ∂t02

(1.1)

Within Lighthill’s analogy, the fluid is assumed to be at rest at the position of the observer, and extends throughout space, such that refractions and reflections are not taken into account. For such case, propagation and generation of sound may be studied separately. By considering a Reynolds decomposition of the flow variables (mean plus fluctuation), and without eliminating the second order non-linear terms, Lighthill was able to obtain the following equation for the density:

∂ 2 ρ0 ∂ 2 Tij 2 2 0 − c ∇ ρ = 0 ∂t02 ∂xi ∂xj

(1.2)

with Tij being defined as Lighthill’s tensor;

Tij = ρ0 ui uj − τij + (p + ρc20 )δij

(1.3)

where p and ui are the pressure and velocity along the i direction, δij is Kroenecker’s Delta function and τij is the viscous stress tensor. On this equation, three terms act as sources: non-linear stresses originated from


22

Reynolds tensor (ρ0 ui uj ), viscous stresses (τij ) and the deviation of a uniform isentropic behaviour ((p + ρc20 )δij ). Equation 1.3 is exact and is therefore as difficult to be solved as the original equations. However, the strength of Lighthill’s equation appears as some simplifing hypothesis are assumed for the flow. The tensor Tij will only be non-null within the area of the source, in the near field of the flow under study. Therefore, a Green’s function formalism may be used. Furthermore, by not considering surface effects nor transients of the solution (only free turbulence is considered), simplifing to the inviscid, isentropic case, which is a valid approach for a high Reynolds, cold jet; Lighthill was able to obtain the acoustic intensity of the jet as being 8 and its directivity as proportional to (1 − cos(θ))−5 . proportional to Ujet

These power-laws for the directivity and radiated intensity are considered reasonable approximations, but only for very clean jets. As mentioned on 1.3.1, the lip line of jets is unstable and subject to the Kelvin-Helmholtz instability, which will lead to growing structures radiating differently than a compact source composed of free turbulence, as Lighthill hypothesised. On the following section some of this characteristics which ultimately led to the modelling of wavepacket structure within the velocity field of jets will be summarized.

1.3.3

Large scale coherent structures and noise generation

When the pressure and velocity fluctuations are measured within the near-field of jets, they reveal the existence of large scale, coherent structures, which are depicted on figure 1.4. Perhaps the earliest observation of such structures was made by MollChristensen (MOLLO-CHRISTENSEN, 1963; MOLLO-CHRISTENSEN, 1967), who verified a spatial correlation between pressure measurement which extended over one jet diameter and presented an ondulatory behaviour. The experiments of Crow and Champagne (CROW, 1972), which consisted of flow visualizations allowed for a better understanding of such fluctuations, confirmed the existence


23

of such structures, with a repeating wave-like spatial structure.

FIGURE 1.4 – Visualization of a turbulent jet (MOORE, 1977).

Such structures exist on the near field of jets and are a prediction of the linear theory; The low amplitude of the fluctuations on the near field, for structures with low azimuthal number, allow a linear approximation, and the inflection of the mean velocity profile lead to Kelvin-Helmholtz instability waves which model the behaviour of large scale structures Such instability waves will behave on a coherent manner on a structure denominated wavepacket (SASAKI, 2014; JORDAN; COLONIUS, 2013; CAVALIERI et al., 2012). These wavepackets are characterized by a low azimuthal wave-number, nearly constant convection speed and coherence along the axial and radial directions, which causes them to be acoustically efficient, in spite of presenting a small contribution to the total turbulent kinetic energy. More recently, models which consider this inherent behaviour of jets, by using the wavepacket Ansatz in the linear solution, have seen a relative success in predicting and offering physical insight into some of the characteristics of the acoustic field of turbulent jets; Cavalieri et al. (CAVALIERI et al., 2013) have shown the existence of a correlation between the near and far acoustic fields of jets, for the axisymmetric mode of the velocity fluctuations. It is also shown how a method that explores the wavepacket structure of the solution is able to predict in the frequency domain the velocity fluctuations within the


24

near-field for the axisymmetric and first helical modes. In the articles (CAVALIERI et al., 2011; CAVALIERI et al., 2012) it is shown how a simple wavepacket model, acting as the source term for Lighthill’s equation is able to reproduce the observed experimental directivity, indicating that a jet may not be regarded as a compact source, but as an extended one, for which the interference between between its different parts plays an important role to the total sound field. The linear stability formalism which supports wavepacket models, such as the Parabolized Stability Equations, is the basis for the Reduced Order Models we propose over this thesis, for the control of the maximum amplitude of the fluctuations along convectively unstable systems. By doing so, we expect to obtain reductions in the resulting acoustic radiations or a postponement of the transition to turbulence and therefore a reduction of drag.

1.4

Common features of the flows under study

There are several routes that will lead to a turbulent behaviour of a flow, several of which share the same characteristics of those related to noise generation on a turbulent free jet, and consequently these problems share some of the available tools, both for prediction and control. The one that will be mostly explored over this thesis is the linearization of the equations over a mean turbulent or laminar flow and the construction of a generalized eigenvalue problem. The multiple scales of the behaviour at hand (CRIGHTON; GASTER, 1976), and the slow divergence of the mean-flow are also common characteristics of these problems, which allow their study via the Parabolized Stability Equations (PSE). This work aims at setting a paradigm for the control of convective systems (e.g. cold jets, boundary-layers, shear-layers etc.), by using such method as the prediction tool.


1.4.1

25

Absolute and convective instabilities

In the classical linear theory, one is interested in the behaviour of infinitesimal wavelike perturbations around the mean flow, of the type φ(y)exp(i(kx − ωt)), such structures are defined as instability waves. Replacing this Ansatz on the linearized equations and enforcing appropriate boundary conditions leads to a generalized eigenvalue problem. Solutions of such problem will only exist when the wavenumber and frequency obey a dispersion relation of the form D[k, ω, R] = 0, on which R is a given parameter related to the stability of the flow. There are two strategies to solve this problem; imposing a given real k and obtaining ω as a complex quantity - resulting in temporal stability analysis; imposing a real ω and obtaining the correspondent complex k - resulting in spatial stability analysis. The choice of method is intrinsically related to the nature of the flow under under study. When realising a stability analysis of these types, two sorts of unstable behaviour will be possible, absolute and convective instabilities; For the first case, the fluctuations grow both in time and space, leading an unstable behaviour throughout the flow. Such behaviour is normally related to a feedback mechanism which leads to a resonant condition of the equations, and a saddle point on the resulting eigenvalues of the linearized problem. Absolute instabilities are found in problems such as hot jets and Von-Karman vortex streaks. Convective instabilities, on the other had, grow in time while being carried along the flow with the mean speed. For this case, the downstream behaviour is directly related to the fluctuations upstream and no feedback mechanism is expected to occur. Cold jets, boundary and shear-layers are examples of such problems, and will be the object of study of this thesis. For a more complete explanation on convective and absolute instabilities, the reader is referred to the works of (HUERRE; MONKEWITZ, 1985; HUERRE; MONKEWITZ, 1990; HUERRE et al.,

2000).


1.4.2

26

Transition of boundary layers to turbulence

There are many ways through which a boundary-layer may transition to turbulence, the first one is related to the convective instability mechanism which was explained in the previous section 1.4.1; For such case, the flow is unstable to perturbations which occur after a given critical Reynolds number which will grow exponentially along the spanwise direction, these structures are called Tollmien-Schlichting (TS) waves. The amplification will occur until the amplitude of the perturbation is such that the linearization hypothesis is no longer valid, resulting on nonlinear interactions, after which turbulent behaviour results. This process is explained in more detail in (KACHANOV, 1994), and schematically in figure 1.5.

FIGURE 1.5 – Schematic representation of boundary-layer transition, reproduced from (KACHANOV, 1994). This mechanism for the initial growth is described by the Orr-Sommerfeld equation which leads to the Tollmien-Schlichting waves. Evaluation of the Orr-Sommerfeld equation results in a critical Reynolds number for the instability of the resulting waves to be around Recrit ≈ 1500, based on the boundary layer thickness as the characteristic length. Such critical number has been confirmed experimentally, but only for flows with very low disturbance levels. As the background turbulence is increased, a mechanism related to the non-normality of the eigenmodes may occur and the flow may experience a large transient growth of the fluctuations (HENNINGSON et al., 1993). If this growth is large enough, the linear predictions may be bypassed and the transition to turbulence occurs prior to the linear-theory predictions. This is referred to as a bypass transition. This work will focus on the low-disturbance scenario, and transient growth results will


27

not be initially considered.

1.4.3

Link with the aeroacoustics of turbulent jets

The jets which are the object of study of this work present a turbulent behaviour ever since the boundary layer inside the nozzle, which is forced to transition via a trip. For such cases, when the jet is cold, it is susceptible to the Kelvin-Helmholts convective instability which will lead to the appearance of large-scale coherent structures called wavepackets. Differently from the boundary layer problem on which these structures (the TollmienSchilichting waves) will eventually grow to amplitudes that will force the transition to turbulence, the wavepackets within the near-field of the jet will work as sound sources, as presented on the previous sections. Targeting on reducing such structures, may therefore lead to a delay on the transition to turbulence of a laminar flow, or to a reduction of the sound field coming from a turbulent system, such as the jets presented over this qualification.

1.5

Partial Conclusions

By this time, the reader is hopefully convinced about the importance of wavepacket models for the prediction and generation of acoustic waves and their relation to the transition to turbulence of flows. The communality of the tools used for both of these fields allows us to tackle them with the same strategies and similar reduced order models. The use of such models in the time domain, both as real-time prediction and control is the central idea of this work, as to set a prototypical framework which may be adapted to other systems, with applications expected to all sorts of convectively unstable systems. With such ideas at hand, over the next chapter the methods that explore the linear wavepacket nature of the solution will be more formally presented. Their capabilities as frequency and time-domain prediction tools will be shown.

2 Methods This chapter is dedicated to establishing the importance of the linear formalism to the construction of reduced-order models for control oriented applications. The frequencydomain models to be considered are the Parabolized Stability Equations and empirical impulse responses. We then proceed with the complementing time-domain behaviour, to which we add identification methods. The applicability of such strategies to fluid systems relies on the hypothesis that linearized models are able to capture the relations between the inputs and outputs of a system and the most important dynamic process of the turbulent field. On that sense, one of the main difficulties in applying linear control theory to fluidic systems related to the high dimensionality of most flows and their intrinsic non-linear behaviour, in particular for the turbulent cases, on which a high discretization may be required to resolve the system accurately (KIM; BEWLEY, 2007), and each flow variable and grid point may be regarded as a state-variable to be resolved. In order to cope with difficulties, a possible strategy is to deal with a reduced-order model of the system, for which the linear control laws are derived and tested a posteriori on a non-linear simulation, or an experiment; examples are the the works of (BELSON et al.,

2013; FABBIANE et al., 2015; SEMERARO et al., 2013a). The complication is now how

to obtain a reduced order model that is both computationally inexpensive and accurate in capturing the said relationship between inputs and outputs of the system. In what follows, models based on the wavepacket background stablished by the PSE method will be derived, both in frequency, and time-domain.

CHAPTER 2. METHODS

2.1 2.1.1

29

Frequency domain models Parabolized Stability Equations

The parabolized stability equations (PSE) represents an extension of the parallel-flow linear stability problem, by considering the slow streamwise variation of the flow within its modal decomposition. The assumption of a slowly varying base-flow allows the use of the multiple-scales method (CRIGHTON; GASTER, 1976) to decompose the perturbation associated with a given frequency ω into slowly and rapidly wave-like behaviours (HERBERT, 1997). Over the next section, this decomposition will be applied to two mean flows, a shearlayer and a round jet.

2.1.1.1

PSE Applied to a Bidimensional Incompressible Flow

We will follow the same Ansatz as (HERBERT, 1997), for a bidimensional system, which includes terms to account both for the slow variation of the fluctuations and their wave-like behaviour, and reads as 2.1

q 0 (x, y, t) = q(x, y) ei(

Rx 0

α(ξ) dξ−ωt)

.

on which q(x, y) = (u, v, p)T is the slowly varying part and Γ = ei(

(2.1) Rx 0

α(ξ) dξ−ωt)

is the

wavelike part. The decomposition (2.1) is introduced into the Navier-Stokes and continuity equations. Non-linear terms are neglected by assuming small perturbations, and moreover the first axial derivatives of α and the second axial derivatives of q may be eliminated by assuming a slow variation of these quantities, which will lead to a parabolization of the resulting equations.

CHAPTER 2. METHODS

30

After simplifying by Γ, we obtain ∂u ∂U ∂U ∂u ∂p 1 ∂u ∂ 2u 2 −iωu + iαU u + U +u +v +V = −iαp − + 2iα −α u+ 2 ∂x ∂x ∂y ∂y ∂x Re ∂x ∂y 2 ∂v ∂V ∂V ∂v ∂p 1 ∂v ∂ v −iωv + iαU v + U +u +v +V =− + 2iα − α2 v + 2 ∂x ∂x ∂y ∂y ∂y Re ∂x ∂y ∂u ∂v + iαu + = 0, ∂x ∂y (2.2)

The system (2.2) can be written in matrix form, as in (LI; MALIK, 1996):

(E + αF )

∂q + (A + αB + α2 C)q = 0, ∂x

(2.3)

with 



U 0  E= 0 U  1 0  −iω +   A=  

1  0 ,  0





2i − Re

0

 F =  0  0

∂U + V ∂· − ∂x ∂y ∂V ∂x

1 Re

2i − Re

0

∂ 2· ∂y 2

0 

iU 0 i    , B= 0 iU 0     i 0 0



 0  0 ,  0

1

 Re  C= 0  0



∂U ∂y −iω + ∂V + V ∂· − ∂y ∂y ∂· ∂y  0 0  . 1 0  Re  0 0

1 Re

∂ 2· ∂y 2

0  ∂·  , ∂y   0

(2.4)

Decomposition (2.1) is a priori not unique because of the presence of the x variable in q(x, y) as well in α(x) and because no evolution equation is given for α. In order to overcome this ambiguity, a normalization constraint is added (HERBERT, 1997) such that the exponential dependence (real and imaginary) is absorbed by the wavelike term ei

Rx 0

α(ξ) dξ

:

∂q q, ∂x

Z

∞

= y

q. −∞

∂q dy = 0. ∂x

(2.5)

CHAPTER 2. METHODS

31

A uniform discretization in the stream-wise direction and a grid of Chebyshev collocation points in the transversal direction are used for the numerical procedure. Starting with (q0 , α0 ), q(x, y) can be obtained by integrating equation (2.3) in space using an implicit Euler scheme, with iterations (index (n)) in α such that the constraint (2.5) is respected (SASAKI, 2014)  2 (n) (n) (n)     E + αj+1 F + ∆x A + αj+1 B + αj+1 C

(n) qj+1 = Ej+1 + αj+1 F qj j+1

 i (qj+1 , qj+1 − qj )y (n+1) (n)   .  αj+1 = αj+1 − ∆x (qj+1 , qj+1 )y

.

(2.6) The PSE has been validated using the DNS results of the bi-dimensional mixing layer with a hyperbolic tangent mean profile, with the base-flow chosen as the mean-flow determined by the average of the DNS results and the Kelvin-Helmholtz mode from spatial linear stability used as inflow condition. This DNS code has been supplied by Damien Biau, and it has also been used in (BIAU, 2016). The space step is fixed at dx = 10, and we use N y = 501 Chebyshev collocation points, mapped to [−yinf : yinf ] with yinf = 80. Rx

Figure 2.1 compares the total PSE solution q e

0

α(ξ) dξ

with the corresponding Fourier

component of the DNS for two frequencies ω = 0.11 and ω = 0.22. The results are qualitatively consistent from figure 2.1(a) to figure 2.1(d). The PSD of the axial component at the centerline |u(y = 0)|2 is shown figure 2.1(e) and 2.1(f). The agreement is good until x ≈ 300 and we obtain results comparable with (CHEUNG; LELE, 2009a).

2.1.1.2

PSE Applied to a Compressible Jet

The application of the PSE to the jet problem follows the same idea as the one for the bidimensional problem. Therefore, only the main differences between these two strategies will be highlighted. The same methodology was used in other works by this author, where further details may be found (SASAKI, 2014; SASAKI et al., 2015). Cylindrical coordinates are used and therefore the system presents five variables, and the fluctuation term becomes q = [ux , ur , uθ , ρ, T ]T , axial, radial and azimuthal compo-

CHAPTER 2. METHODS

32

100

150

100

DNS

10 DNS

100

PSE

-50

0

0

y

0

0

y

5

50

50

PSE -5

-50

-100 -150

-100 0

100

200

300

-10

-100

400

0

150

200

250

300

x

(a) Axial component u ; ω = 0.11

(b) Axial component u ; ω = 0.22 100

100

DNS

10 DNS

100

5

50

0

0

y

50

y

100

x

150

PSE

-50

0

0 PSE

-5

-50

-100 -150

-100 0

100

200

300

-10

-100

400

0

50

100

150

200

250

300

x

x

(c) Tranversal component v ; ω = 0.11

(d) Tranversal component v ; ω = 0.22

105

PSE DNS

104 103 102

|uc |2

|uc |2

50

101 100 10−1 10−2

0

100

200

300

400

500

600

105 104 103 102 101 100 10−1 10−2 10−3 10−4

PSE DNS

0

100

200

x

(e) PSD of the centerline streamwise velocity component |u(y = 0)|2 ; ω = 0.11

300

400

500

600

x

(f) PSD of the centerline streamwise velocity component |u(y = 0)|2 ; ω = 0.22 Rx

FIGURE 2.1 – Comparison between the total PSE solution q e sponding Fourier component of the DNS.

0

α(ξ) dξ

with the corre-

CHAPTER 2. METHODS

33

nents of velocity, density and temperature, respectively. The flow is homogeneous along the azimuthal and time directions, and normal modes may be considered along those directions;

XX q ˜ x, r eimθ e−iωt , q0 x, t = ω

(2.7)

m

with q ˜ representing the complex amplitude for a given frequency ω and azimuthal wavenumber m. Once again the axial direction is simplified with the assumption of a slowly varying direction.

Rx 0 0 ˆ (x, r)ei α(x )dx eimθ e−iωt , q0 x, t = q

(2.8)

where α = αr + iαi is the complex axial wavenumber, whose imaginary part will define growth or decay of disturbances along the axial direction. By replacing the wavepacket Anzatz (2.8) into the linearised, compressible Euler, Energy and Continuity equations, a matrix system is obtained, ∂ˆ ∂ˆ q q A q + B q, α, ω q ˆ+C q +D q = 0, ∂x ∂r

(2.9)

where the high Reynolds number of the flows considered justifies the neglect of the viscous terms, which is not the case for the shear-layer problem. Linearisation allows the independent calculation of each ω − m combination. The resulting matrices may be found in (SASAKI, 2014), and the other steps of the solution, normalization, first calculation considering the parallel flow assumption and numerical methods follow analogously to what was presented previously in 2.1.1.1. A validation of the PSE for the jet problem is given in figure 2.2, in the frequencydomain. As observed in previous works, there is a good agreement against experimental data in growth region of the flow, which corresponds to the potential core of the jet. There

CHAPTER 2. METHODS

34

are, however, mismatches as more downstream regions are considered. A further advantage of the PSE modelling which was shown recently on the work of (CAVALIERI et al., 2016) is its ability to model frequencies as high as Strouhal number of 4.0, which lie on the region of greater sensitivity of the human ear and therefore are of greater industrial interest to be controlled. Figure 2.3 presents a plot of the β parameter to the Mach 0.9 turbulent jet presented in (CAVALIERI et al., 2016), as defined in equation 2.10. This parameter is a measurement of the similarity between two flows and indicates that close to the jet nozzle, where PSE predicts and unstable behaviour for the eigenfunctions of the flow, it is able to reproduce correctly the behaviour of the fluctuations, for all the evaluated Strouhal numbers.

β(x, St) =

2.1.2

hˆ uP SE (x, r, St)ˆ uP OD (x, r, St)i kˆ uP SE (x, r, St)kkˆ uP OD (x, r, St)k

(2.10)

Empirical Frequency Response Functions

As a consequence of the formalism established in sections 2.1.1.1 and 2.1.1.2 is that the PSE may be understood as a frequency domain model, on which each frequency is resolved separately and no interaction between them is expected to occur. An alternative way to obtain such an idea is to perform a linear system identification, via the available unsteady measurements. An analogous procedure has been used by (DAHAN et al., 2012) to characterize the response of the a flow to an actuation and obtain a control law. Based on the measured pressure data, with the hypothesis that a linear relation holds between the fluctuation field at two different axial locations along the jet, the optimal frequency response, in the least squares sense, may be defined from the auto and cross-

CHAPTER 2. METHODS

35

St=0.2

0

ux.ux*/Uj²

ux.ux*/Uj²

10

−5

10

−10

10

2

4

6

8

10

10

12

x/D St=0.4

0

ux.ux*/Uj²

ux.ux*/Uj²

−10

2

4

6

8

10

6

8

10

12

8

10

12

8

10

12

8

10

12

x/D St=0.5

−5

10

10

12

x/D St=0.6

0

0

2

4

6 x/D St=0.7

0

10 ux.ux*/Uj²

ux.ux*/Uj²

4

−10

0

10

−5

10

−10

−5

10

−10

0

2

4

6

8

10

10

12

x/D St=0.8

0

0

2

4

6 x/D St=0.9

0

10 ux.ux*/Uj²

10 ux.ux*/Uj²

2

0

−5

−5

10

−10

10

0

10

10

10

−5

10

−10

0

10

10

St=0.3

0

10

−5

10

−10

0

2

4

6 x/D

8

10

12

10

0

2

4

6 x/D

FIGURE 2.2 – Center-line power spectral densities of the axial velocity - comparison between the implemented PSE method (squares) and data presented in (CAVALIERI et al., 2013): earlier PSE calculation (solid line) and experimental values obtained from hot-wire measurements (circles)

CHAPTER 2. METHODS

36

β for m=0 0.9

4

0.8

3.5

0.7 3

0.6

St

2.5

0.5

2

0.4

1.5

0.3 0.2

1

0.1

0.5 1

2 X/D

3

4

FIGURE 2.3 – β parameter plot for a Mach 0.9 turbulent jet. Dark line represents the region where PSE saturates, predicting a stable behaviour for the fluctuations along the flow.

CHAPTER 2. METHODS

37

spectra of the input and output signals (BENDAT; PIERSOL, 2011) as

G(ω) =

Sry (ω) , Srr (ω)

(2.11)

where Srr and Sry are, respectively, the auto and cross-spectra of the input and output signals.

2.2

Real-time estimations

The results of sections 2.1.1.1, 2.1.1.2 and 2.1.2 present how to build frequency domain models for linearized systems. From such frequency domain models, the time behaviour of the systems may be obtained via an inverse Fourier transform, so that Input/Output relations may be derived. This idea is related to the proposition of model-based control, on which a the description of the system is obtained a priori either via identification or theoretical techniques. Several examples of such may be found in the literature; (FABBIANE et al., 2015) compares the performance of model and adaptive-based control techniques on a experimental ˜ preparation to the control of a Blasius boundary-layer, (INIGO et al., 2016) apply system

identification to obtain time domain models for amplifier flows and educe a feed-forward control scheme, on a similar topology to the one proposed on this work. (GAUTIER; AIDER, 2014; HERVE´ et al., 2012) apply ARMAX (Auto-Regressive-MovingAverage-Exogenous) to model the relationship between perturbations and their result downstream, also on a convectively unstable flow and apply a feedforward control scheme to reduce the fluctuations downstream. A similar approach to what this thesis proposes is presented in (LI; GASTER, 2006), on which a model based on the linear stability theory is obtained, over which the control law is built. The testing of the results follows afterwards, on a non-linear scenario. The objective of this section is to present time-domain relations that may be obtained

CHAPTER 2. METHODS

38

from frequency models, as these are one of the necessary ingredients to obtain a closedloop control law for a system. Several of the results which will presented over this chapter may be found in (SASAKI et al., 2015).

2.2.1

Single-Input-Single-Output Systems

Given that the PSE does not allow for interactions between the different frequencies, it may be used to build a frequency domain model which, when inverse Fourier transformed will grant a SISO time-domain modelling of the ststem. Such model is readily available when taking the inverse Fourier of equation 2.11, which is consistent with the theoretical background established by PSE. In what follows R(ω) denotes a quantity in the frequency domain, and r(t) is the same quantity in the time domain; the two are related by the direct and inverse Fourier transform pair. Defining R(ω) and Y (ω) as the complex quantities at two separated positions along the system and assuming a linear relation between these, a frequency domain transfer function can be obtained as,

G(ω) =

Y (ω) . R(ω)

(2.12)

A time-domain transfer function, g(t), is then available following an inverse Fourier transform,

1 g(t) = 2π

Z

∞

G(ω)e−iωt dω.

(2.13)

−∞

Once g(t) is obtained, the output y(t) of the system can be predicted by convolution with the input r(t) and a SISO system is defined as: Z

∞

g(τ )r(t − τ )dτ .

y(t) =

(2.14)

0

However, only the behaviour associated with linear PSE (i.e. linear wavepackets) is

CHAPTER 2. METHODS

39

being predicted; as shown in earlier works (CAVALIERI et al., 2013), downstream of the potential core, the PSE model presents an early saturation when compared with actual experimental data and thus a worsening of the prediction performance is expected in that region. Implicit in this procedure is that a single mode dominates the dynamics between the positions of input and output, such that a single upstream sensor allows an estimation of the amplitudes and phases of this mode. This will be the case for jets, shear and boundarylayers, which will thus allow the estimation of the output from a single convolution. In this sense, the PSE transfer function can be seen as a propagator of this mode between the two positions. Extensions to cases when more modes are present in the system, may be calculated using the resolvent operator; such idea may be found in (BENEDDINE et al., 2016). It is also important to keep in mind that such model is not expected to capture the exact behaviour of the flow fluctuations at the output position. The basic input/output relations are expected to be well captured and a phase difference of no more than 90 ◦

between prediction and measurement must be maintained so that the control system

remains stable, and no positive feedback occurs between the actuation and original openloop behaviour of the system.1 Once the time-domain model has been obtained, the convolution in eq. 2.14 is an operation with low computational cost. Considering that g(t) is previously determined, the convolution in eq. 2.14 involves nt multiplications and sums, where nt is the number of time steps used to discretise 2.14. This reduced number of operations makes such an approach a candidate for implementation in experimental settings and non-linear simulations. Equation 2.14 is only useful for real-time prediction of the flow behaviour when there is a causal relationship between input and output signals, which implies g(t < 0) = 1

Such instabilitization of the system when phase margin diminishes is only possible in feedback control schemes, which alter the positions of the poles of the system. For the feedforward case, as the phase error starts to occur, the performance of the controller is expected to worsen, but without causing the system to be unstable.

CHAPTER 2. METHODS

40

0. This is the case for convection dominated flows, such as the cold jets evaluated on this work, if the output y(t) is downstream of the input r(t). In this case structures detected at a given position will evolve convectively, affecting downstream stations at later times. Indeed, the Kelvin-Helmholtz, or Tollmien-Schlichting modes considered here have a positive generalised group velocities ((DAVIS et al., 2000)), which should lead to causal transfer functions if one attempts to estimate the downstream behaviour from an upstream input. PSE may be thus used to calculate the evolution of the fluctuations for such case; The computation is repeated for many Strouhal numbers (the low computational cost of PSE makes this feasible), and a change of variables is necessary in order to perform the inverse Fourier transform. The following presents the mathematical details of this derivation for the jet problem, the formalism is very much similar when applied to different systems, such as the shear-layer.

dω = 2π

Ujet dSt D

(2.15)

where Ujet is the jet velocity. Using this definition the inverse transform is,

1 Ujet g(t) = 2π 2π D

Z

∞

G

−∞

Ujet Ujet St e−i D St.t dSt. D

(2.16)

Or, by defining F −1 [G(ω)] as the Inverse Fourier Transform of G(ω): Ujet −1 Ujet g(t) = 2π F G St . D D

(2.17)

Use of the scaling property of the Fourier Transform gives,

Ujet g t 2π = F −1 [G(St)] . D

(2.18)

CHAPTER 2. METHODS

41

Defining the non-dimensional time, t∗ = t

Ujet , D

we have,

g(2πt∗ ) = F −1 [G(St)] .

(2.19)

The frequency domain transfer function G(St) is obtained via the calculation for each Strouhal number separately and is performed until St ≈ 2.0, in intervals of 0.02. To ensure an appropriate time range, these frequency results are projected into a finer grid of frequencies. This is realized by interpolating the amplitudes and phases of the PSE solution separately. The PSE system has the particularity that the minimum marching step size (δx) for convergence depends on the Strouhal number considered (HERBERT, 1997). The lower the frequency, the greater the value of δx for convergence. This property may lead to difficulties for the low frequencies (St ≤ 0.2) and also to a mismatch between the axial grids used for each Strouhal calculation. These difficulties are dealt with by solving each frequency with its appropriate axial step and interpolating them to a finer axial grid. Such interpolation is also made by considering the amplitude and phase of the solution separately.

2.2.2

Identification models - ARMAX

Another approach for time-domain system-identification-based models is to use ARMAX. The following describes briefly its use to fluidic systems. The same approach of (PIANTANIDA et al., 2014) is used. The objective is again to build a SISO model. The identification is performed using an ARMAX, Auto-Regressive-Moving-Average-eXogenous model (the Matlab routine armax was used to perform the calculations), whose Ansatz takes the form:

y(t) +

na X k=1

ak y(t − k) =

nX b +nd k=nd

bk r(t − k) + E(t)

(2.20)

CHAPTER 2. METHODS

42

E(t) =

nc X

ck e(t − k) + e(t).

(2.21)

k=1

where y(t) is the simulated output value at time t, given by an expression that includes an autoregressive part, y(t − k), an exogeneous part represented by r(t − k), with r the measured input to the system (sensor). A moving-average term, E(t), accounts for phenomena that impact the measurement, y, but are not observed by the sensor. The identification works by means of a regression that seeks the coefficients ak , bk and ck that lead to a minimal-variance noise e(t). The model requires the parameters na , nd , nb , nc , to be set and these can be related to certain properties of the flow, such as, the convection velocity of wavepackets and their characteristic length and time scales (PIANTANIDA et al., 2014). The coefficient na represents the number of previous output data on which the current output depends. It is set by computing the autocorrelation function of the output signal. The parameter has been set as the temporal shift approximately corresponding to half of the first oscillation period of the correlation function. The nd parameter indicates the temporal delay between the input and output signal; when different from 0, it implies that the effect of the input on the output is not instantaneous, as is the case for compressible systems. nb represents the number of past input samples on which the current output depends. These two parameters may be obtained by considering the cross-correlation between s and y. For the precise values of the parameters na , nd , nb , nc a sensitivity evaluation must be performed prior to the optimization of the parameters ak , bk and ck , which is out of the scope of this qualification. The results for the SISO model predictions will be shown in 2.3 for the jet and a shear-layer.

CHAPTER 2. METHODS

2.2.3

43

Single-Input-Multiple-Outputs Systems

Analogously to the case of the SISO modelling, obtaining systems with multiple-inputs and/or multiple-outputs may be done either from a theoretical or identified manner. The advantage of such systems is their versatility when compared to single case, on which multiple sensors may be used to increase the representativity of the model and multiple outputs will allow for a more robust reduction of the fluctuations on a greater region of the domain. Examples of such systems may be found on the works of (LI; GASTER, 2006; BAGHERI; HENNINGSON,

2011) both of which apply these methods to an amplifier flow case.

Auto and cross correlations may be used both for SIMO and MIMO systems, with the first one corresponding to a particular case of the second (BENDAT; PIERSOL, 2011). Therefore, only the MIMO case will be presented in detail. Also, for systems with a single mode, it is not expected that adding input positions will increase the performance of the model, in comparison to the best prediction with a single input, as no information is being added to prediction, i.e. only the Kelvin-Helmholtz mode is being observed, either to the jet or shear-layer problem. On the other hand, the PSE approach permits a fine application of the SIMO case on which measurements on a single location is used to predict the entire flow field downstream of the input position. The transfer function may be obtained independently for the single input and any other number of outputs, in the same way presented in 2.2.1. And by doing so, a SIMO model may be obtained between any pair of fluid variables and positions (as long as these respect the causality principle). Results for these predictions will also be shown in 2.3.

2.2.4

Multiple-Inputs-Single-Input Systems

Obtaining systems with multiple inputs relies on the hypothesis that there is no absolute redundancy between the chosen input positions (BENDAT; PIERSOL, 2011), i.e.

CHAPTER 2. METHODS

44

the normalized coherence between inputs must not be equal to unity, such that the new measurements will add information to the prediction.

2

For that case, the output in the frequency domain is written in terms of the q considered inputs as in equation 2.22, where Hj are the MISO transfer functions for each output and N (ω) is the error which is uncorrelated to each of the inputs.

Y (ω) =

q X

Hj (ω)Xj (ω) + N (ω)

(2.22)

j=1

By multiplying both sides of equation 2.22 by Xi∗ , for i = 1, 2...q the expected values become:

E[Xi∗ Y

(ω)] = E[

q X

Hj (ω)Xi∗ Xj (ω)] + E[Xi∗ N (ω)]

(2.23)

j=1

Given that the error is associated to uncorrelated data to the inputs, E[Xi∗ N (ω)] = 0 the problem reduces to equation 2.24

Giy =

q X

Hj (ω)Gij (ω)

(2.24)

j=1

where Giy is the cross-correlation between inputs and output and Gij the cross-correlation, if i 6= j or the auto-correlation, if i = j between inputs. Equation 2.24 is a square system where the MISO transfer functions, Hj (ω), are the unknowns. For the particular case of two inputs, application of Cramer s rule to equation 2.24 yields;

G1y 1 − H1 =

G12 G2y G22 G1y

2 G11 (1 − γ12 )

(2.25)

2 For a system with a single mode, as those studied over this thesis, multiple-inputs-multiple-outputs systems may not have a direct application. Nevertheless, methods will be presented in order to complete the scenario of prediction techniques one may consider.

CHAPTER 2. METHODS

45 G2y 1 − H2 =

G21 G1y G11 G2y

2 ) G21 (1 − γ12

(2.26)

2 With γ12 the coherence function between inputs 1 and 2 (which is why this value could

not be equal to one, on the beginning). The PSE MISO transfer function may also derived from equation 2.24 which, for q inputs is written as:     G1y = H1 G11 + H2 G12 + . . . + Hq G1q        G2y = H1 G21 + H2 G22 + . . . + Hq G2q  ..   .        Gqy = H1 Gq1 + H2 Gq2 + . . . + Hq Gqq

(2.27)

Dividing each equation by Gjj the problem may be regarded in terms of PSE transfer functions;     Gp1y = H1 + H2 Gp12 + . . . + Hq Gp1q        Gp2y = H1 Gp21 + H2 + . . . + Hq Gp2q  ..   .        Gpqy = H1 Gpq1 + Gp2 Gq2 + . . . + Hq

(2.28)

where Gpij represents the PSE transfer function between positions i and j. Solution of the system of equations in 2.28 will grant the Hj transfer functions obtained theoretically via PSE.

CHAPTER 2. METHODS

2.2.5

46

Multiple-Inputs-Multiple-Inputs Systems

One way to obtain a MIMO system is to consider a series of multiple-input-singleoutput systems, such that:   P P   Y1 (ω) = qj=1 H1j (ω)Xj (ω) 7−→ Giy1 = qj=1 H1j (ω)Gy1  ij (ω)      Y (ω) = Pq H (ω)X (ω) 7−→ G = Pq H (ω)Gy2 (ω)  2 2j j iy2 2j ij j=1 j=1  ..   .       P P  Yq (ω) = qj=1 Hqj (ω)Xj (ω) 7−→ Giyq = qj=1 Hqj (ω)Gyq ij (ω)

(2.29)

And so, by proceeding analogously to the previous sections, each output may be solved separately as a MISO system. As on the previous section, dividing each equation by Gyii will make the PSE transfer functions appear for each output. Solving q linear systems of equations will grant the MIMO transfer functions H and the matrix system of inputs and outputs becomes: 





Y1 (ω) H11 (ω) H12 (ω) . . .    Y2 (ω) H21 (ω) H22 (ω) . . .     . = . .. ...  ..   .. .       Yq (ω) Hq1 (ω) Hq2 (ω) . . .

  H1q (ω) X1 (ω)   X2 (ω) H2q (ω)    .  ..    .   ..     Hqq (ω) Xq (ω)

(2.30)

Obtaining the set of equations when the number of inputs and outputs is different is straightforward.

2.3

Results for real-time estimations

Throughout this section, some results for the real-time modelling of the fluctuations will be shown, for two problems; The jet case in 2.3.1, for which experimental measurements were available (BREAKEY et al., 2013); The bidimensional shear-layer, for which a DNS provided unsteady data for testing. This corresponds to a summary of the results

CHAPTER 2. METHODS

47

presented in the Master’s thesis, (SASAKI, 2014), two conference papers, (SASAKI et al., 2015; SASAKI et al., 2016) and a submitted Journal of Fluid Mechanics article.

2.3.1

Time-domain predictions for the turbulent jet

Time-domain predictions for a turbulent jet results used for validation and comparison with the real-time predictions of this section are based on measurements of (BREAKEY et al.,

2013). A detailed description of the experimental setup may be found in (BREAKEY et

al.,

2013; CAVALIERI et al., 2012; PIANTANIDA et al., 2014) and is briefly presented in the

appendix A. Figure 2.4 presents frequency and time-domain comparisons of the transfer functions obtained by PSE with those obtained experimentally (see the previous subsection for description of how the experimental transfer functions are computed). Three input-ouput combinations are shown for pressure measurements at radial positions on a 8◦ conical surface (cf. figure A.2).The frequency domain transfer functions present a peak around St = 0.3, where the comparison with experiment is most convincing ((CAVALIERI et al., 2013)). As the input is moved further downstream, the predicted output decreases in magnitude. The time-domain model presents causal behaviour, as expected, for the input and output positions considered. From figure 2.4 an equivalence between the two frequency and time-domain models is observed, with PSE being able to reproduce the impulse response of the system. Comparisons of the time prediction using PSE, Empirical frequency impulse responses and ARMAX are shown is figure 2.5. The quantity which is shown is the non-dimensional pressure fluctuation p0 = p/ρ0 c20 , with ρ0 and c0 as the ambient density and sound velocity and the input and output positions are (xi /D, xo /D), along the conical surface, as presented in the Appendix. Figure 2.6 shows how the prediction deteriorates as the input and output pair become further appart. Phases of fluctuations are nonetheless well reproduced.

CHAPTER 2. METHODS

48

6

0.25 0.2

5

0.15 0.1 g(t*)

G(St)

4 3

0.05 0

2 −0.05 1 0 −2

−0.1 −1

0 St

1

−0.15 0

2

2

4

6

8

10

6

8

10

6

8

10

t*

2.5 0.1 2

0.08 0.06

G(St)

g(t*)

1.5

1

0.04 0.02 0 −0.02

0.5

−0.04 0 −2

−1

0 St

1

−0.06 0

2

2

4 t*

1.6 0.06

1.4 1.2

0.04 g(t*)

G(St)

1 0.8

0.02

0.6 0.4

0

0.2 0 −2

−1

0 St

1

2

−0.02 0

2

4 *

t

FIGURE 2.4 – Frequency and time-domain transfer functions between three input-output combinations (top: xi /D, xo /D) = (0.05, 1.7); middle: (1.3, 2.5); bottom: (2.1, 3.3)) for pressure fluctuations on the 8◦ conical surface. Blue dots: experimental data; solid red lines: PSE.

CHAPTER 2. METHODS

49

Input

p′(t*) − input

−3

1

x 10

0 −1 0

10

20

30

p′(t*) − output

x 10

p′(t*) − output

60

70

80

90

100

Measured Output PSE TF

0 −1 0

10

20

30

1

40

50 t*

60

70

x 10

80

90

100

Measured Output ARMAX

Output x ARMAX

−3

0 −1 0

10

20

30

1

40

50 t*

60

70

x 10

80

90

100

Measured Output Experimental TF

Output x Experimental TF

−3

p′(t*) − output

50 t*

Output x PSE TF

−3

1

40

0 −1 0

10

20

30

40

50 t*

60

70

80

90

100

FIGURE 2.5 – Comparison between measured non-dimensional pressure fluctuations, PSE transfer function, empirical transfer function and ARMAX predictions, with x/D = 2.1 and 3.3 as the input and output positions, respectively.

CHAPTER 2. METHODS

−3

xo/D=1.7

−3

1

−1 0 −3

x 10

20 t* xo/D=2.5

x 10

20 t* xo/D=3.7

40

p′(t*)

p′(t*)

−1 0

20 t*

0

−3

x 10

xo/D=3.3

x 10

40

p′(t*)

0

20 t*

40

−3

x 10

xo/D=5.7

0.5

0

−1 0

−1 0

1

0 −0.5

−0.5

−0.5

40

0

xo/D=4.5

0.5

0.5

−1 0

20 t* −3

1

20 t*

x 10

−0.5

−1 0

40

−1 0

0.5

−0.5

−0.5

40

−3

20 t* xo/D=4.9

0

1

0.5

0

xo/D=4.1

−0.5

−3

1

0.5 p′(t*)

0

−1 0

40

−3

x 10

0.5

−0.5

−0.5

p′(t*)

1

p′(t*)

p′(t*)

p′(t*)

0

1

xo/D=2.9

0.5

0.5

1

x 10

p′(t*)

1

x 10

50

20 t*

40

−1 0

20 t*

40

FIGURE 2.6 – Comparison between measured non-dimensional pressure fluctuations (solid line) and PSE transfer function prediction (dashed line) for x/D = 0.5, 1.3 and 2.1, in the first, second and third lines, respectively.

CHAPTER 2. METHODS

51 Input = 0.5 Input = 1.3 Input = 2.1

1 0.9 0.8 0.7

C

0.6 0.5 0.4 0.3 0.2 0.1 0 1

2

3

4 xo/D

5

6

7

FIGURE 2.7 – Peak correlation between PSE predicted signal and the measured data. As a performance metric we use the normalized peak correlation coefficient between the estimation y(t) and measurement f (t), C = max(Cf y (τ )), where Cf y (τ ) is defined in equation 2.31 R∞

f (t)y(t + τ )dt qR Cf y (τ ) = qR −∞ ∞ ∞ 2 (t)dt f y 2 (t)dt −∞ −∞

(2.31)

at the downstream position, shown in figure 2.7 for PSE. Correlations of up to 0.8 are observed for small axial separations between input and output. The correlation decreases as the output station is moved downstream away from the potential core of the jet, a consequence of the limitations of the linear model. Another interesting characteristic is the fact that the correlation increases when the input is moved from 0.5D to 2.1D. This is consistent with the observations of (CAVALIERI et al.,

2013) that PSE predictions differ from experimental data in the first axial region,

untill x/D = 2, due to a transition of the fluctuations from the boundary layer exiting nozzle to the Kelvin-Helmholtz mode.

CHAPTER 2. METHODS

52

1 PSE Transfer Function Experimental Transfer Function ARMAX

0.9 0.8 0.7

C

0.6 0.5 0.4 0.3 0.2 0.1 0 3

3.5

4

4.5

5 xo/D

5.5

6

6.5

7

FIGURE 2.8 – Peak correlation between the predicted signal with the different methods and the measured data considering the input at xo /D = 2.1. The trend is similar for the empirical transfer function and ARMAX estimations, as shown in Figure 2.8 with a comparison of the peak correlation values, for the three methods considering an input position at x/D = 2.1. The results presented for the time predictions considered the axissymmetric (m = 0) mode of the pressure fluctuation. A way to improve the comparisons is to use POD filtered data (for a review on POD filtering techniques, refer to(KERSCHEN et al., 2005) and POD applied to wavepacket eduction (SUZUKI; COLONIUS, 2006; PIANTANIDA et al., 2014)). Figure 2.9 presents the comparison of the PSE time-domain transfer function convolved with POD experimental data and its comparison to the actual POD filtered measured data, with the same configurations of input-output pairs of figure 2.6. The peak correlation values are shown in figure 2.10. Although not presented, a similar trend is observed when ARMAX or the experimental transfer function are used to perform the estimation. The performance of the three methods is considerably improved when the POD filtering is applied to the original input-output near-field data. This supports the idea that the POD filtering retains the salient correlated wavepacket dynamics, which is well predicted

CHAPTER 2. METHODS −3

x 10

xo/D=1.7

−3

1

−0.5 20 t* −3

x 10

xo/D=2.5

1

20 * t

0

−3

x 10

xo/D=3.3

20 * t −3

1

0.5 p′(t*)

−0.5

x 10

xo/D=4.5

−3

1

40

0

−1 0

x 10

xo/D=5.7

0.5

−0.5 20 t*

40

−1 0

40

0.5

0

20 * t

0 −0.5

−1 0

40

x 10

0.5

−0.5

−1 0

40

−3

20 t* xo/D=4.9

o

0

−1 0

40

p′(t*)

p′(t*)

p′(t*)

−0.5

p′(t*)

x 10

20 * t xo/D=3.7

0.5

0

x /D=4.1

−0.5

−3

1

0.5

−1 0

0

−1 0

40

−3

x 10

0.5

−0.5

−1 0

1

1

p′(t*)

0

1

xo/D=2.9

0.5 p′(t*)

p′(t*)

0.5

x 10

p′(t*)

1

53

0 −0.5

20 * t

40

−1 0

20 * t

40

FIGURE 2.9 – Comparison between measured non-dimensional pressure fluctuations POD filtered (in blue) and PSE transfer function prediction (in red) for xi /D = 0.5, 1.3 and 2.1, in the first, second and third lines, respectively. by the transfer function. A deterioration nonetheless persists as the output position is moved downstream, away from the potential core of the jet. However, the correlation between prediction and POD-filtered pressure is at least 0.6 for the cases studied here. As mentioned on previous sections, PSE may be used to obtain SIMO models which really on unsteady measurements of a single position to predict the entire flow field and fluidic variables downstream of it, which makes it a stronger technique than the more experimentally dependent ones. Figures 2.11 and 2.12 present such prediction, using the unsteady pressure fluctuation as the input variable and pressure or axial velocity as the output variables.

CHAPTER 2. METHODS

54

1 Input = 0.5 Input = 1.3 Input = 2.1

0.9 0.8 0.7

C

0.6 0.5 0.4 0.3 0.2 0.1 0 1

2

3

4 xo/D

5

6

7

FIGURE 2.10 – Peak correlation values between the PSE transfer function prediction and measured data, considering POD filtered values.

t*=81.6

t*=122.4

2

2

0.01

0.01

1.5 0

1

R/D

R/D

1.5

0.5

0

1 0.5

1

2

3

4 X/D

5

6

7

−0.01

1

2

3

t*=163.2

5

6

7

−0.01

t*=204

2

0.01

2

1.5

0.01

1.5 0

1 0.5

R/D

R/D

4 X/D

0

1 0.5

1

2

3

4 X/D

5

6

7

−0.01

1

2

3

4 X/D

5

6

7

−0.01

FIGURE 2.11 – PSE transfer function pressure field prediction for several time slices obtained from measurements along the x/D = 0.5 ring.

CHAPTER 2. METHODS

55

t*=81.6

t*=122.4

2

2

20

20

1.5 0

1

R/D

R/D

1.5

0.5

0

1 0.5

1

2

3

4 X/D

5

6

7

−20

1

2

3

t*=163.2

5

6

7

−20

t*=204

2

2

20

20

1.5 0

1 0.5

R/D

1.5 R/D

4 X/D

0

1 0.5

1

2

3

4 X/D

5

6

7

−20

1

2

3

4 X/D

5

6

7

−20

FIGURE 2.12 – PSE transfer function axial velocity field prediction for several time slices, obtained from measurements along the x/D = 0.5 ring.

2.3.2

The shear-layer problem

Obtaining the PSE and Empirical transfer functions of the 2D shear-layer follows the same approach outlined in section 2.2.1 and performed for the turbulent jet in 2.3.1, therefore only the validation figures will be presented over this section. The unsteady results for the velocity and pressure fluctuation were obtained from a Direct Numerical Simulation of the flow, as presented in B. Figure 2.13 presents the PSE prediction for the transverse velocity component in comparison against the non-linear DNS. Figure 2.14 shows the resulting correlation between prediction and simulation, for several input and output combinations. Figure 2.15 shows the SIMO prediction for the shear-layer problem in comparison against the DNS results, for the axial and transverse velocity components. The input position was chosen at X = 75 and the PSE transfer functions were used to obtain timedomain predictions throughout the field. The agreement between the reduced-order model and the non-linear DNS is compelling. Analogously to what was shown for the jet problem, similar results were obtained for the empirical transfer function predictions. The results of this section validate the use of PSE for the prediction of unsteady fluctuations on a 2D shear-layer, which will be the test bench for the closed-loop control schemes presented in the following section.

CHAPTER 2. METHODS

56

DNS PSE TF Prediction

0.2 0.15

v(t)

0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 400

600

800

1000 t

1200

1400

1600

FIGURE 2.13 – Comparison between PSE transfer function prediction and results from the non-linear simulation, for input in (x, y) = (75, 0) and output in (100, 0).

1 0.9 0.8 0.7

Corr

0.6

Input X=75 Input X=50 Input X=150

0.5 0.4 0.3 0.2 0.1 0 50

100

150 X

200

250

FIGURE 2.14 – Resulting correlation between prediction and simulation for several input and output combinations.

CHAPTER 2. METHODS

57

Y

PSETF v, T*=800 5

0.2

0

0

−5 100

110

120

130

140

150

−0.2

X

Y

DNS v,T*=800 5

0.2

0

0

−5 100

110

120

130

140

150

−0.2

Y

X PSETF u, T*=800 5

0.2

0

0

−5 100

110

120

130

140

150

−0.2

X

Y

DNS u,T*=800 5

0.2

0

0

−5 100

110

120

130

140

150

−0.2

X

FIGURE 2.15 – DNS of the bidimensional shear-layer in comparison against PSE transfer functions predicion from a single measurement.

CHAPTER 2. METHODS

2.3.3

58

Partial conclusions

This section presents the strength of reduced order models in predicting the timebehaviour of fluctuations. Although varying in terms of the necessary experimental level for performing the prediction, they show a similar level of agreement against experimental or non-linear simulations. Perhaps the most interesting technique is the PSE based transfer function, it presents a low-computational cost both for the construction and application of the model, allowing for the definition of a SIMO system for the prediction of any fluidic variable at any downstream location of the measurement. Over the next chapter the strength of this technique will be shown for closed-loop control of fluctuations. As a final conclusion of this chapter one may consider the importance of wavepacket models, not only as statical entities for the prediction of power spectral densities, but also the instantaneous fluctuations, which is a further demonstration of such models for large-scale structures.

3 The control problem The purpose of this Chapter is to present some the concepts involved in flow control techniques. To that matter, some of the aspects of systems control will be revised, such as: opened and closed-loop control, feedback and feedforward strategies and adaptive control. Some of the specifics of flow control applications such as model construction via reduction techniques will also be dealt with. Communalities between hydrodynamics stability theory and dynamic systems control theory will also be highlighted during the remainder of this chapter. Several of such ideas were already applied in this work to the active control of a shearlayer and the results of it will be shown in 3.5; the remaining are related to expected addition to this thesis which have not yet been applied.

3.1

Active and passive, opened and closed-loop control of flows

Perhaps the first distinction on what relates to control strategies to fluids should be made between passive and active control. For the first case, the design of the plant is made so as to minimize a certain given quantity, without the necessity to continuously inject energy or mass into the system. Examples of such, for the turbulent jet may be regarded as the use of chevrons (GU MUNDSSON,

´ et al., 2015) in order to attenuate large 2009; BRIDGES; BROWN, 2004; LAJUS

scale structures within the lip-line. For such case, the design is made at a given operating

CHAPTER 3. THE CONTROL PROBLEM

60

FIGURE 3.1 – Chevron geometry applied to a turbofan engine, as presented in (SAIYED et al., 2003). condition and no change can be made as the state of the flow changes. Figure 3.1 presents an example of a serrated nozzle to form a chevron. Delaying the transition to turbulence via a passive strategy is also possible and is referred to as natural laminar flow designs (KACHANOV, 1994). Another aerodynamic application is the use of vortex generator to the inhibition of detached flow (GENTRY; JACOBI,

1997).

For the active case, on the other hand, there is a continuous acting over the controlled system. For such case a further distinction is needed; If the actuator acts in accordance with the measurement of incoming perturbations or based on a continuous observation of the behaviour of the objective, the resulting strategy is a closed loop; in case the acting is made without such knowledge, the strategy is said to be an open loop 1 . An example of an active opened-loop strategy to controlling a jet may be found in (KŒNIG, 2011), on a which a fluidic actuator continuously inject mass in the centerline of the jet 3.2. Examples of active closed-loop control for fluids are numerous (BELSON et al., 2013; FABBIANE et al., 1

2014) and will be dealt with in greater depth over the next sections, as

some authors (BELSON et al., 2013; FABBIANE et al., 2014) prefer the use of reactive and non-reactive control, as the closed and open-loop nomenclature may present a certain ambiguity. This idea will explained in greater depth in section 3.2


61

FIGURE 3.2 – Example of open-loop active strategy for the control of noise from a turbulent round jet (KŒNIG, 2011). these correspond to the one of the central objectives of this thesis. Another necessary ingredient of active control schemes is the actuator, therefore its types and some modelling particularities will be shown in the next section, along with preliminary results for the modelling of a body force on the shear-layer DNS.

3.1.1

Modelling of actuators

The active, closed-loop control we apply over this work implies the use of an actuator on the flow. The amplitude is expected to be sufficient to alter the amplitude of the unsteady fluctuations but not high enough that it would alter the mean flow structure. The work of (CATTAFESTA; SHEPLAK, 2011) proposes the classification presented in figure 3.3 for the actuating types along a flow, considering its function. The most common type are the fluidic actuators which work by injection or suction along the flow. Another class includes a moving body inside the flow, which shall alter


62

FIGURE 3.3 – Classification of actuator types (CATTAFESTA; SHEPLAK, 2011). the boundary conditions of the problem. The final class are the plasma actuators, which allow for a very fast time response and without moving parts which has made them gain popularity in flow control applications. Plasma actuators function by breaking the dielectric barrier of the flow thus allowing for the introduction of a body force locally. This is therefore the type of actuation that is imposed for the construction of the proposed control laws and implementation in 3.5. For this work, it is assumed a plasma actuator will impose a body force at a given position of the flow with a fixed spatial structure and frequency which is consistent with those of the fluctuating motions of the velocity and pressure terms. In this manner, the actuator is supposed to be given and the design issues of this component will not be addressed. In this way, the modelling that will be performed already assumes a given frequency and spatial support for the actuation. For an extensive review on plasma actuators and their modelling, the reader is referred to the works of (CORKE et al., 2010; MOREAU, 2007). For the bidimensional shear-layer in B, the actuation is given by equation 3.1, where (x2 , y2 ) are the axial and transverse positions where the actuation is inserted and L will determine its spatial decay, and a(t) is supposed to be able to act in frequencies which


63

a(t)

0.05 0 −0.05 −0.1 0

2000

4000

6000

8000

10000

t

Y

10 0.8 0.6 0.4 0.2

0

−10 90

95

100 X

105

110

FIGURE 3.4 – Actuation behaviour along time and space. are comparable to those of the fluctuations

f (x, y, t) = a(t)e−

(x−x2 )2 L2

e−

(y−y2 )2 L2

(3.1)

and its time and spatial behaviours are show in figure 3.4, with a(t) set to be a broadband time signal. There are now two ways to model the effect of the actuation on the flow. We can either to take the empirical transfer function between a(t) and a chosen output position, or to project this body force into the Kelvin-Helmholtz mode, to see how it will alter the flow locally, using PSE a posteriori to see its effect downstream. The first one is based on the System’s theory of the frequency response of the system (FABBIANE et al., 2014), and the second comes from the signalling problem defined in the stability theory of flows (HUERRE; MONKEWITZ, 1990; HUERRE; MONKEWITZ, 1985; HUERRE et al., 2000). Empirical transfer function of the actuation With the actuation defined as equation 3.1, the same approach outlined in section 2.1.2 may be used to obtain a relation in between a(t) and the corresponding output, and


64

for this case the input will be considered a(t). In order obtain such relation, the DNS is simulated without any perturbations, other than the body force at a fixed position and the auto and cross correlations are computed between a(t) and itself and between a(t) and the defined output, respectively. Such strategy is based on the System’s theory for obtaining frequency response of dynamical systems (BENDAT; PIERSOL, 2011). Kelvin-Helmholtz Projection To obtain the Kelvin-Helmholtz projection, we start by considering the flow locally parallel close to the perturbation so one may write a forced LST problem which, for the forced bidimensional case, becomes;       ˆ uˆ fx  uˆ            L vˆ − αF vˆ = fˆy        0 pˆ pˆ

(3.2)

where the forcing terms, are written in terms of their Fourier transforms.

fˆx (α, y) =

Z

∞

fx (x, y)e−iαx dx

(3.3)

fy (x, y)e−iαx dx

(3.4)

−∞

fˆy (α, y) =

Z

∞

−∞

It should also be noted that there is a time dependence of the time exp(−iωt) in all the terms of this section, as the equations are posed in terms of their time and space Fourier transforms. Writing the velocity and pressure fluctuations as a linear combination of the eigen-


65

functions defined from the eigenvalue problem L{φi } = αi Fφi ,   u    X v  = ai φi     i p

(3.5)

So that equation 3.2 becomes;

X

[ai αi F φi − ai αF φi ] = fˆ

(3.6)

i

Projecting equation 3.6 into the eigenfunctions of the adjoint problem ψi ;

X

[ai αi hψi , F φi i − ai αhψi , F φi i] = hψi , fî

(3.7)

i

Given that the eigenfunctions of the direct and adjoint problems are biorthogonal, i.e. hψi , F φj i = δij , equation 3.7 simplifies to;

aj (αj − α) = hψj , fî

(3.8)

and

aj =

hψj , fî αj − α

(3.9)

The velocity fluctuations in equation 3.5 become, if one considers that the KelvinHelmholtz mode will dominate the dynamics of the system, as is the case for the bidimensional shear layer of B and axisymmetric jet;   uˆ   X hψj , fî hψKH , fî vˆ = φ ≈ φKH i   α − α α − α j KH   j pˆ

(3.10)


66

FIGURE 3.5 – Chosen contour path of integration, followed anti-clockwise, a semi circle of infinite radius joined to a curve of area tending to zero, which is made to capture the Kelvin-Helmholtz mode. The inverse Fourier transform of equation 3.10 will grant the fluctuation in the (x, y) domain;   uˆ Z ∞   hψK−H , fî vˆ = 1 φK−H eiαx dα   2π −∞ αK−H − α   pˆ

(3.11)

The integral in equation 3.11 is solved in the complex plane, using residue theory, a semi-circular path of integration along the αi −αr is chosen such that the Kelvin-Helmholtz mode is captured. Figure 3.5 presents the scheme for this integral. Given the integral in equation 3.11 will tend to zero if =(α) > 0, its value will be null along the semi-circle. Therefore it may be approximated by Cauchy’s principal value which will be equal to the residue calculated at αKH . The velocity fluctuations become, for a given force f (x, y) as given in equation 3.12.   u    v  = ihΨK−H (y), fˆ(αK−h , y)ieiαK−H x φK−H (y)     p

(3.12)


67

With the LST prediction of the velocity and pressure fluctuations, they may be evolved using the PSE strategy, as for the estimation case and the transfer function is calculated using the formalism derived in chapter 2. For the particular case where the forcing term is a Dirac delta function, f (x, y) = δ(y)δ(x), we have fˆ = δ(y), given the definition of the Fourier transform and the filtering property of the delta. For such case, the amplitude of the projection into the KelvinHelmholtz mode is given exclusively by the adjoint mode at y = 0, as per equation 3.13.

hΨK−H (y), fˆ(αK−h , y)i = ΨK−H (0)

(3.13)

This study, which corresponds to finding the impulse response of the system may be used to determine sensitivity characteristics of the flow, leading to the optimal positioning of actuators along axial and transverse directions. A comparison of the resulting transfer functions in the frequency domain obtained via the empirical and theoretical strategies is shown in figure 3.6 and proves the equivalence of them. Once the transfer function has been obtained, the prediction of the output is given by the convolution with the time-domain input, as established in the formalism of chapter 2. The resulting prediction, in time in comparison with the simulated output is shown, for the Kelvin-Helmholtz projection strategy in figure 3.7, for the actuation at (X, Y ) = 100 and output at (X, Y ) = 125. An equivalent result is obtained when the empirically obtained transfer function is used. Finally, the correlations between prediction and non-linear simulation, for several input-output combinations are shown in figure 3.8. When the closed-loop system is considered, such correlation will be useful to define the most convenient locations for sensoractuator-objective positions, in terms of the most effective predictions of the reduced order models. A final comment regarding this theoretical procedure for obtaining the response of


68

6 Empirical TF PSE TF

5

|G|

4 3 2 1 0

−0.2

−0.1

0 St

0.1

0.2

0.3

FIGURE 3.6 – Frequency domain comparison of the theoretical and empirical transfer functions.

DNS PSE TF Prediction

0.2 0.15

v(t)

0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 400

600

800

1000 t

1200

1400

1600

FIGURE 3.7 – Comparison between prediction and simulation for a given perturbation.


69

1

0.8

Corr

0.6

0.4 Input − X=50 Input − X=100 Input − X=130 Input − X=155

0.2

0 60

80

100

120

140

160

180

200

220

240

X

FIGURE 3.8 – Correlation between prediction and simulation at several input-output combinations. the system to a perturbation is that it may be regarded as the resolvent operator of the problem, as it allows, via a convolution, to obtain the resulting behaviour of the flow in response to a inhomogeneity given, in this case, by a body force. It is expected, that by such, it may be possible to obtain either via an optimization or via and SVD decomposition, the positions of greatest sensitivity to disturbances, which are a valuable information when dealing with control strategies. For further information on the resolvent operator, the reader is referred to the article (BENEDDINE et al., 2016).

3.2

Feedback and Feedforward control strategies

One of the most important classifications when dealing with active flow control regards the separation between feedback and feedforward control strategies. Although hardly ever used alone i.e. without a feedback loop, for system’s applications, feedforward has seen several uses in flow control, particularly in convectively unstable flows, such as the one that will be dealt with in section 3.5. The bibliography on the subject is vast, dealing with both strategies to a flow control mind set. To feedback control, the reader is referred to (DADFAR et al., 2015; BAGHERI et

CHAPTER 3. THE CONTROL PROBLEM al.,

70

2009a; BAGHERI et al., 2009b; SEMERARO et al., 2013b; SEMERARO et al., 2011) all of

which deal with different control-laws for the reduction of Tollmien-Schlichting waves in boundary layers, both in bi-dimensional and three-dimensional cases. For the feedforward control, (LI; GASTER, 2006; FABBIANE et al., 2015) use wave cancellation techniques to control boundary-layer instabilities and (GAUTIER; AIDER, 2014) use an ARMAX-modelbased control for a backward-facing step problem. Most of the review of this section is based on the works of (BELSON et al., 2013; FABBIANE et al.,

2014) which provide a nice insight into the differences between these two

strategies applied to flow control. The main difference between feedforward and feedback, in a flow with convective nature, is related to the position of the actuators and sensors; For the first case, the actuator is downstream of the input sensor, in a way that it acts by opposite phase cancellation of the incoming disturbance. Sometimes this law is said to be non-reactive as although the control is made on a closed-loop way, there is no way to reject disturbances that are introduced between sensor and actuator, and there is no way to correct the control action considering the measured output behaviour. If the positions of sensor and actuators are reversed, a feedback strategy is posed, where the resulting behaviour at the position of the input or objective is fed back and used in the definition of the actuating signal. For strongly convective flows, feedback only leads to significant performances if the positions of the sensor and actuator are close to one another (BELSON et al., 2013; FABBIANE et al.,

2014), otherwise the actuation will be unable to alter the flow accordingly.

An evaluation of the Bode diagram of the closed-loop control and the corresponding phase margins, along with the sensitivity transfer functions, will supply the necessary information on whether or not the chosen position is appropriate. Care must be taken as the feedback control will change the position of the poles of the closed-loop system, and it is therefore capable of de-stabilizing the plant 2 ; feedforward, on the other hand, is unable 2

from a control point of view, convective unstable flows may be regarded as stable plants, as at the position of the objective the amplitude of the fluctuations will not grow indefinitely, when in an openedloop condition


71

to do so. Nevertheless, the main disadvantage of feedforward control is that it is unable to deal with perturbations downstream of the sensor position making its use in practical applications difficult. Differences between model and real system are also not seen on a purely feedforward loop, and although it will not make the plant unstable, its effectiveness may be reduced rapidly in such cases. Such differences, when known a priori may be quantified and will allow for the definition on whether or not feedforward alone will work (DEVASIA, 2002). From this moment and on, the following nomenclature will be used to denote the positions and transfer functions along the controlled system:

• Indexes 1, 2 and 3 will refer to positions downstream of each other, and will be used to denote transfer functions and locations of the elements involved into the control-law. • GIij refers to an estimation transfer function between positions i and j. • GAij refers to an actuation transfer function (how the actuator affects the behaviour of the chosen flow variable) between positions i, where it is located and j.

Using such nomenclature, the canonical feedforward block diagram is shown in figure 3.9.


72

FIGURE 3.9 – Basic block diagram for feedforward control.

FIGURE 3.10 – Basic block diagram for feedback control.

FIGURE 3.11 – Basic block diagram for feedback control when sensor and objective are not at the same position.


73

For the feedback case, there are two possible block diagrams: the classical case, where the objective is sensed so that differences between the expected state and real one are immediately brought back into the actuation definition. The other possibility is to feed back a given input position, for this case it is unnecessary to observe the output position. These two strategies are show in figures 3.10 and 3.11, respectively. The previous block diagrams do not consider the presence of exogenous disturbances, which could be added anywhere on the actuator-sensor-objective branch. The resulting closed-loop relations between input and output signals are shown in equations 3.14 to 3.16.

O(ω) = [GI13 (ω) + KGA23 (ω)]I(ω)(F eedf orward)

O(ω) =

GI13 (ω) I(ω)(ClassicF eedback) 1 − KGA23 (ω)GI13 (ω)

O(ω) = [GI23 (ω) +

KGA23 (ω) ]I(ω)(M odif iedF eedback) 1 − KGI21 (ω)

(3.14)

(3.15)

(3.16)

With the block diagrams assembled, the control strategy consists into minimizing the transfer function between in and output positions, such that the fluctuations at the position of the objective are attenuated. There are several ways to do so, some of which were already applied and will be shown in 3.5. They are either based into a good representation of the system (model-based control) or into a black-box representation of it, which seeks to represent the observed behaviour of the flow. These two strategies will be presented on the following sections.


3.3 3.3.1

74

Flow control applications Model construction for control

There are two manners to describe a dynamic system via a linear model; The first one is to obtain the state space of the system, i.e. the input/output interaction is given from a set of ordinary differential equations.

State-space representation As the interest is on the dynamics of small disturbances over the plant, it is convenient to write the fluidic varibles using a Reynolds decomposition, into a mean plus a fluctuating quantity. For a bi-dimensional system, this reads

u(x, t) = U(x) + u0 (x, t)p(x, t) = P (x) + p0 (x, t)

(3.17)

Replacing this decomposition into the Navier-Stokes and Continuity equation will lead to, after eliminating the terms of order higher than 2 for the fluctuations:

1 2 0 ∂u0 = −U.∇u0 − u0 .∇U − ∇p0 + ∇ u +f ∂t Re

(3.18)

∇u0 = 0

(3.19)

With the term f (x, t) defining the forcing of the system, and it may be further decomposed into a disturbance and an actuation term;

f (x, t) = bd (x)d(t) + bu (x)u(t)

(3.20)

Where bd and bu will define the coordinates and spatial support of the considered disturbances and actuation, respectively. The measurements of the input, i(t), and output, o(t)


75

are defined from:

Z i(t) =

Z

0

cy (x)u (x, t)dΩ + η(t)o(t) =

cz (x)u0 (x, t)dΩ

(3.21)

where η(t) represents a disturbance, and the matrices cy (x) and cz (x) define the sensors positions and the length of the area they are able to observe. This set of partial differential equations in 3.18 and 3.19 may be transformed into ordinary differential equations in time, such that a Linear-Time-Invariant system results:

˙ = Aq(t) + Bd d(t) + Bu u(t) q(t)

(3.22)

y(t) = Cy q(t) + η(t)

(3.23)

z(t) = Cz q(t)

(3.24)

where q(t) is the state vector, composed from the velocity and pressure components. Obtaining the matrices A, Bd and Bu are the objective of identification and theoretical methods for control, defined via state-space representation and several strategies may be found in the literature; A theoretical transformation of the partial differential equations into and ordinary set may accomplished via a Galerkin projection (QUARTERONI, 2010). In the realm of system identification techniques, one may find the Eigensystem Realization Algorithm (ERA) (see (SCHOEN et al., 2006) for an explanation on the technique for model reduction purposes), which is based on the linear impulse response of the system, obtained from experimental data or an simulation; or balanced truncation, which may be obtained from proper orthogonal decomposition (see (MA et al., 2011; KIM; BEWLEY, 2007) for a review on the technique applied for model reduction, with a linear control mindset). The control techniques that are evaluated over this work, however, rely in a into classi-


76

cal control theory, which is based in time and frequency domain transfer functions, which correspond to a Finite Impulse Response (FIR) of the system (see, for example (OGATA; YANG,

1970) for a precise definition between classical and modern control theories, with

a Systems application point of view). Such transfer functions were obtained in Chapter 2 using theoretical (PSE-based) and empirical (frequency responses obtained via a simulation and ARMAX identification) strategies. In spite of this, such theoretical impulse response functions may be used to obtain state-space-models via ERA. Another possibility is the conversion from the frequency domain to state-space by using a polynomial to fit the transfer function and proceed with the analytically inverse Laplace transform. Such idea also allows for obtaining other important characteristics of the system, from a control perspective, such as the number of poles and zeros or its eigenvalue/vector pairs. Finite-Impulse-Response representation The FIR representation is based on a convolution to define the input/output relationship, equation 3.25, as presented in 2. The strategies proposed into this work consider the PSE method to obtain g(τ ), which as far as the author is concerned had not yet been done 3 .

Z

∞

g(τ )r(t − τ )dτ .

y(t) =

(3.25)

0

The previous expression may be written in a discretized form, and since the system is stable, from a control application perspective, equation 3.25 may be truncated, as g(τ ) will have to tend to zero for large values of τ . And the Finite Impulse Response of the system is defined from:

y(n) =

N X

g(i)r(n − i)∆(τ )

(3.26)

i=0 3

It is also important to notice that PSE provides the theoretical impulse response of the system, which may be used via the Eigensytem Realization Algorithm to lead to a state-space representation of the system.


77

0.06 0.05 0.04 g(τ)

0.03 0.02 0.01 0 −0.01 −0.02 0

5

10

τ

15

20

25

0.8 0.6

g(τ)

0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 0

50

100

150

200

250 τ

300

350

400

450

500

FIGURE 3.12 – Turbulent jet and bidimensional shear-layer impulse responses obtained via PSE Another strategy to obtain the finite number of coefficients that describe the inputoutput relationship along the flow is to proceed via an identification, such as the leastmean-squares-algorithm (FABBIANE et al., 2014); this type of model is, for the moment, out of the scope of this work. Figure 3.12 presents the Finite-Impulse-Response of the turbulent jet and the bidimensional shear-layer which were already shown in the frequency domain in chapter 2.

3.3.2

Model-based control

This section aims at formulating the control laws to the free-shear layer, as described in section 3.5. So far only frequency domain techniques applied to a feedforward strategy have been evaluated, however other strategies will be briefly presented. The objective of flow control strategies is usually to attenuate the output of a system, such that the growth of the fluctuations is diminished and so are the phenomena related to


78

this, such as the transition to turbulence or the resulting radiation coming from a turbulent jet. The rationale consists in obtaining the closed-loop transfer function between input and output signals (i(t) and o(t)) and minimize it, assuming a given shape to the gains that link actuation and output signals. Optimal control ideas, which account for the power of the actuating signal may also be regarded.

3.4

Feedforward: Frequency Inversion

The first technique we evaluate is based on a precise inversion of the plant to cancel the incoming input; in the Systems theory it is called Frequency Inversion (DEVASIA, 2002). This idea has been applied successfully in aerospace systems and flexible structures (TOMLIN et al., 1995; MARTIN et al., 1994; CLAYTON et al., 2008) and more recently for flow control via wave-cancellation of incoming structures in a boundary-layer (LI; GASTER, 2006). By starting with the closed-loop relation for the feedforward problem, equation 3.27;

O(ω) = [GI13 (ω) + KGA23 (ω)]I(ω)

(3.27)

it is noticeable that if K presents a frequency dependence set by equation 3.28, it is possible to precisely cancel the incoming disturbance related to GI13 .

K(ω) = −

GI13 (ω) GA23 (ω)

(3.28)

And the actuation signal, u(t), is obtained by a convolution with the input,

Z

T

k(τ )i(t − τ )dτ

u(t) =

(3.29)

−T

where K(τ ) is the inverse Fourier transform of K(ω). If the system is causal, the integral in equation 3.29 is taken only for the positive values of τ .


79

Such inversion-based controllers are strongly dependent on the fidelity of the reducedorder model in comparison against the actual non-linear system. (DEVASIA, 2002) stablishes a limit for the maximum uncertainty of the reduced-order model (∆(ω)) to be used for this strategy in terms of the amplitude of the actual non-linear system at a given frequency, G0 (ω), divided by its condition number, as defined in (GOLUB, 1989). This condition implies that frequencies where there is not a good representation of the plant, or where the system is not controllable, for which the estimation (GI13 (ω)) and actuation (GA23 (ω)) transfer functions tend to zero, should be filtered-out or removed from the problem. If such condition is met, this type of control-law is expected to act according to its specification. Another important characteristic of this feedforward-based inversion is that it does not change the poles of the closed-loop system, therefore the stability of the plant is maintained. This is an important feature of the controller, when dealing with convectively unstable flows, as such systems are globally stable at a fixed position, which makes such strategy very interesting for this application. For globally unstable flows, on the other other hand, feedback loops should be used to improve the performance of the controller.

3.4.1

Feedforward: Proportional-Integral Controllers

The name Proportional (P) and Proportional-Integral (PI) controllers is related to the type of actuation that is considered and how it is related to a given measured or estimated signal. Minimization of the output may be performed either in the frequency or timedomains. In this work, the consideration is that the actuation is proportional/integral to the estimation of the signal at the positions nearby the actuator. A further distinction that can be made is whether the gains are scalar or present a frequency/time dependence. Whereas the second case is commonly found in the literature for flow control systems, the second one, which is inspired in system’s applications which present a much lower number of degrees of freedom, presents some advantages in terms of implementation of the actuation.


80

For this thesis, the scalar case has been studied so far. For a single estimative, the minimization is performed on time using a gradient-based searching algorithm. The problem that is minimized is given in equation 3.30, for the proportional case (the integral is analogous, however a second gain, proportional to the integral of the actuation is also included).

Z

T

Z i(t − τ )gi3 (t)dτ + kp

o(t) = 0

T

i(t − τ )gi2 (t)dτ ? ga (t)

(3.30)

0

Or, in the frequency domain, the output may be written as,

O(ω) = I(ω)GI13 (ω) + [Kp I(ω)GA23 (ω)]

(3.31)

and Kp is found by solving: 4 :

Kp = −

GI13 GI12 GA23

(3.32)

For two estimatives, the gain equation becomes,

Kp1 G1I2 + Kp2 G2I2 = −

GI3 (ω) GA (ω)

(3.33)

which may be solved either via an optimization problem or via a pseudo-inverse matrix. Generalizing for N estimatives, N X

Kpi GiI2 = GI3 (ω)

(3.34)

i=1

And equation 3.34 will have an exact solution for the proportional gains if the number 4

it is important to notice that equation 3.32 is not solved exactly, if the gain is a scalar, as the transfer function present a frequency dependence, and an optimization problem must be considered, either on the frequency or time-domain, equation 3.30)


81

of gains is equal to the number of unstable frequencies, in the discretized problem.

[Kp ] = [G−1 I2 ][GI3 ]

(3.35)

The method outlined above allows for the direct definition of the number of degrees of freedom of the controller by determining the number of estimatives that are chosen, this enables the definition of the complexity of the controller. Adding a term to penalize for the actuation or the amplitude of the input and exogenous perturbations is also possible, and result in optimal and robust controllers, respectively For an optimal controller, an usual functional for minimization includes the actuation amplitude u(t), which is penalized via a value R, Z J=

T

(o(t)2 + Ra(t)2 )dt

(3.36)

0

for an H2 controller, the function to be minimized includes the expected value of the output, input and disturbances,

J=

E(|o(t)|2 ) E(i(t)2 + w(t)2 )

(3.37)

where w(t) is the disturbance, and it is normally considered to be white noise. Another possibility is to deal with a frequency-dependent gain, which is more commonly found in the literature (BELSON et al., 2013; BAGHERI et al., 2009b; GAUTIER; AIDER, 2014). For such cases, the optimization is performed for the same equations, but allow the gains to be frequency/time-dependent. By doing so, the technique is expected to be less sensitive to perturbations than the resulting frequency inversion controller. H2 and H∞ norms for frequency minimization As it was explained over this section, the use of scalar gains does not lead to a null output, as the transfer function that compose the equation for the resulting gain (3.32)


82

have different frequency dependences. Therefore, an optimization procedure may be used to find the optimal gains via search algorithm. In the time-domain, the equation to be minimized is given in 3.30, which may be generalized to any number of estimations and gains. In the frequency domain, a useful strategy corresponds to minimizing a given norm of the output signal, via the closed-loop gains. Two useful norms for this purpose are the H2 and H∞ norms, as presented in equations 3.38 and 3.39, for SISO systems. Analogously to the time-domain minimization, the value of the gains may be found via a gradient-based search algorithm.

kOk2 =

1 2π

Z

∞ 2

|O(ω) |dω

(3.38)

−∞

kOk∞ = max(O(ω))

3.4.2

1/2

(3.39)

Feedback control

The feedback control also relies on a minimization of the output via a gain, that is either frequency dependent or a scalar. The closed-loop systems are given by equations 3.15 and 3.16 and the optimization is made by considering the PI, H2 or optimal strategies, on the same way as defined in the previous section. Nevertheless there are two peculiarities that should be observed when dealing with applications to strongly convective systems:

• The causality will cause the resulting transfer function between sensor and actuator to be non-zero only for a brief non-zero period of time, and when these positions are very close to one another. Such behaviour has been pointed out in the works of (BELSON et al., 2013); • Feedback controllers are able to alter the location of the poles of the system and


83

therefore change its stability properties. It is useful to observe the Bode diagrams between the different positions to ensure proper phase margins, particularly for convective problems, and prevent the de-stabilization. The stability transfer function, equation 3.40, allows to a quantification of the robustness for a computation of its infinity norm. It is advisable to use gains and positions for the sensor-actuator pair such that S(ω) ≤ 2 (SKOGESTAD; POSTLETHWAITE, 2007).

S(ω) =

1 1 − GI12 (ω)K(ω)

(3.40)

Feedback controllers have not yet been evaluated over this work. A more detailed study comparing these different strategies, particularly on what concerns robustness issues, is expected to be added to the final thesis.

3.4.3

State-Space formulation for control

This strategy is based on the definition of the control gains from the formulation defined in equations 3.22 to 3.24. The purpose of this formulation is to compute a gain K(t) such that the norm of the output is minimized. The signal u(t) = K(t)i(t)

(3.41)

which relates a given state to the actuation, is fedback (or fedforward, but for this case the input is considered) into the system. For the feedforward control case, the LQG regulator is often the choice, as it mimics the system’s behaviour from a measurement or observation (a Kalman filter may be used as an observer, for example (FABBIANE et al., 2014)) of the plant to act in a way to cancel the fluctuations downstream of the actuator. It is also necessary to define a controller, which usually considers an optimization problem in order to balance output and control action; the Linear Quadratic Regulator (LQR) is sometimes used for this purpose (FABBIANE et al., 2015).


84

For the moment, state-space formulations have not yet been explored over this work, for further aspects of these techniques the reader is referred to the work of (KIM; BEWLEY, 2007; BEWLEY; LIU, 1998)

3.4.4

Adaptative control

On the opposite side to the model-based control and the reduction techniques presented in 3.3.1 and 3.3.2, there is the possibility of dealing with adaptive control laws. For this case, the gains of the compensator are adjusted in real-time in order to account for changes and uncertainties on the plant, external disturbances. This strategy also has its roots in the Systems theory, and has been more recently addressed for flow control. A nice example is the work by (FABBIANE et al., 2014) on which an adaptive feedforward control scheme is compared to a model based strategy, and the first case is seen to be more robust to the evaluated uncertainties. The basic concept behind adaptive control compensators is to monitor on-line the performance in terms of a given metric and update the gains such that a correction is computed. Describing the control law in terms of the finite-impulse-response of the system, in a discretized manner, leads to equation 3.42.

u(n) =

N X

K(i)y(n − i)

(3.42)

i=1

The difference between equation 3.42 and that for the model-based techniques is that K is updated at each time-step, with the objective of minimizing a given norm. For the least-mean-squares algorithm, the norm is the square of the output 3.43,

minK(i) (o(t)2 )

(3.43)

and an optimization strategy, such as the gradient or the steepest-descent methods may be chosen to update K 5 . 5

for a stationary condition, the gain K(i) will no longer change and therefore an adaptive scheme is


85

Nevertheless, for this thesis, adaptive control is out of the foreseeable scope, as the intention is to deal exclusively with model-based control, and to work with PSE as a time and frequency domain model, both to the open-loop and actuated cases.

3.5

Results closed-loop control of a free shear-layer

The purpose of this section is to present the results of the application of some of the closed-loop control techniques shown over this chapter. The system to be dealt with is the bidimensional mixing layer described in Appendix B and the reduced order models correspond to those developed and validated in 3. The control structure that was chosen is the feedforward diagram of figure 3.9 and equation 3.14 for the closed loop behaviour of the output. Frequency inversion and PI compensators were the initial choice of controllers, whereas the PI case consisted of a scalar gain obtained via a time-domain optimization, rather than a frequency-domain minimization of a given norm. Implementation of the positions to be tested for sensor, actuator and objective was made considering figures 2.14 and 3.8, which present the correlations between prediction and the results of the non-linear simulation. Evaluation of such cases shows that the region 75 ≤ X ≤ 150 corresponds to where the PSE based models are most representative of the actual dynamics of the flow. The chosen positions for testing were X = 75, 100 and 125 for sensor, actuator and output, respectively and the objective aimed at reducing the transverse velocity component, using a transverse force. Although it would be desirable that the vortex pairing mechanism be postponed, as this is the main sound source mechanism on shear-layer (COLONIUS et al., 1997; WEI; FREUND, 2006; CHEUNG; LELE, 2009b), no restrictions over this variable were imposed. Minimization of the axial velocity component also proved to work, but a lower efficiency of the controller was observed. not necessary, as the gain could be computed only once, from an identification method, for example. The strength of the adaptive controller, however relies on situations for which disturbances and alterations of the plant are present and are not well captured by a theoretical model


86

|GI13|

2 1 0

−0.1

−0.05

0 St

0.05

0.1

0.15

−0.1

−0.05

0 St

0.05

0.1

0.15

−0.1

−0.05

0 St

0.05

0.1

0.15

|GA23|

4 2 0

ℜ(K)

2 0 −2

FIGURE 3.13 – Resulting frequency domain transfer functions between estimation, actuation and output and computed gain for the compensator. Figure 3.13 shows the estimation (GI13 ) and actuation GA23 transfer functions and the resulting gain obtained from the inversion procedure. As proposed in (DEVASIA, 2002) and explained in 3.4, it is necessary to eliminate uncontrollable frequencies and areas where the reduced order model is not representative, where the actuation and estimation TFs tend to zero. This is made by hand, forcing the resulting gain to go to zero smoothly on such regions. Figure 3.14 presents the inverse Fourier transform of the gain, which will grant the time-behaviour of the actuation.


87

0.2

k(t)

0.1

0

−0.1

−0.2 0

50

100

150

T

ad

FIGURE 3.14 – Resulting gain obtained from the frequency inversion method, in the time-domain.

3.5.1

Linearized system results

Prior to the implementation into the DNS, the gains obtained from the frequency inversion, P and PI controllers were tested in the linearized system, three estimation positions were considered for the P and PI definition of the gains. The results for the three methodologies are shown in figures 3.15 to 3.17. Understanding the differences - Degrees of freedom of the controllers The difference in performance between the three controllers is noticeable, even in the linearized descriptions, particularly the incapacity of the PI compensators in forcing the output to zero. This trend may be understood in terms of the number of degrees of freedom of the three controllers. By recalling equation 3.34 it may be observed that the output would only go to zero, even in the linear framework, if the number of estimation positions was equal to the number of evaluated unstable frequencies. The previous plots were obtained via estimations at three positions only, therefore the output signal does not go to zero. Such behaviour may be further studied by breaking down each of the components that form the frequency inverted gain, which leads to a null output signal. This is shown in equation 3.44, where GA22 (ω) represents the transfer function between the body force at


88

0.15 Uncontrolled Prediction Controlled Prediction 0.1

0.05

v(t)

0

−0.05

−0.1

−0.15

−0.2 0

500

1000

1500

t

FIGURE 3.15 – Open-loop prediction obtained from the PSE transfer function and the controlled case via system inversion, considering the linear control framework, with the objective of minimizing the axial-velocity component.

0.4 Uncontrolled Controlled − Proportional 0.3

0.2

v(t)

0.1

0

−0.1

−0.2

−0.3

−0.4 0

500

1000

1500

t

FIGURE 3.16 – Open-loop prediction obtained from the PSE transfer function and the controlled case via proportional law, considering the linear control framework, with the axial velocity fluctuation as the control objective.


0.4

89

Uncontrolled Controlled − Proportional Integral

0.3

0.2

v(t)

0.1

0

−0.1

−0.2

−0.3

−0.4 0

500

1000

1500

t

FIGURE 3.17 – Open-loop prediction obtained from the PSE transfer function and the controlled case via proportional integral law, considering the linear control framework, with the axial velocity fluctuation as the control objective. the position of the actuator and the corresponding resulting velocity field at the same position.

K(ω) = −

1 GI13 GI13 =− GA23 GA22 GI23

(3.44)

Figure 3.18 presents the frequency domain behaviour of these three transfer functions and also the fraction GI13 /GI23 . The conclusion is that there is no constant of proportionality between these transfer functions, and therefore it is not possible to obtain zero output signal with a single scalar gain.

3.5.2

DNS Results

The control laws derived and tested on the linearized system were then implemented on the DNS which contains non-linear interactions absent in the linearised models, and hence constitutes on a more challenging problem for the derived control law. The implementation


90 G

I13

5

G

I23

4.5

GI13/GI23 GA22

4 3.5

|G|

3 2.5 2 1.5 1 0.5 −0.04

−0.03

−0.02

−0.01

0 St

0.01

0.02

0.03

0.04

FIGURE 3.18 – Behaviour of the transfer functions of equation 3.44 in the frequency domain. considered a buffer size of two hundred timesteps and a discretization in time for the control gains k(τ ) which was fine enough to capture the most unstable frequencies. Figure 3.19 presents the results of the control action via system inversion, via a body force positioned at the transverse direction, aiming at reductions of transverse velocity components, respectively. The position of the objective was (X, Y ) = (125, 0). Although for each control scheme the objective was at either axial or transverse velocity components separately, the actuation presents the beneficial effect of reducing both components. Figure 3.20 presents the comparison of the turbulent kinetic energy obtained from application of either transverse or axial body forces, and shows that the reductions remain throughout the flow, downstream of the actuation. Furthermore, using a body force on the y-axis appears to be more efficient, at least at the position of the objective. The Kelvin-Helmholtz instability of the mixing layer leads to significant growth rates, which are not completely cancelled by the actuation; however, significant decreases of amplitude are obtained in downstream stations, with TKE reductions of up to 50%.


0.2

(a)

(b)

(c)

91

Uncontrolled Controlled fx

(d)

Controlled fy

u(t)

0.15 0.1 0.05 0 −0.05 −0.1 0

500

1000 t

1500

2000

500

1000 t

1500

2000

0.1 0.05

v(t)

0 −0.05 −0.1 −0.15 0

FIGURE 3.19 – Comparison of the uncontrolled simulation against the controlled case, using system inversion, for the axial and transverse velocity fluctuations, with the time to the fluctuation to reach the actuator (a), transient (b), buffer (c) and time to reach the actuation (d), highlighted. Figure 3.21 presents the resulting turbulent kinetic energy of the flow when it is controlled with a proportional or proportional-integral control law and their comparison against the system inversion method. We observe that as the number of degrees of freedom of the control law increases (from proportional, proportional-integral to system inversion), the consequent reduction of the amplitude of the fluctuations becomes more


92

0

10

Uncontrolled Controlled fy Controlled f

x

−1

10

−2

TKE

10

−3

10

−4

10

−5

10

0

50

100

150

200 x

250

300

350

400

FIGURE 3.20 – Turbulent kinetic energy for the uncontrolled and controlled simulations via single body force, using system inversion. significant, which is reflected on the turbulent kinetic energy of the flow. 0

10

−1

Uncontrolled System Inversion Proportional Proportional Integral

10

−2

TKE

10

−3

10

−4

10

−5

10

0

50

100

150

200 x

250

300

350

400

FIGURE 3.21 – Turbulent kinetic energy for the uncontrolled and controlled simulations via single body force, comparison of the three control methodologies.

Finally, the vorticity fluctuations of the uncontrolled and system inversion controlled cases are presented in figure 3.22. As the control action takes place, vortex roll-up and pairing pairing is delayed. This supplies evidence that for the analogous compressible mixing layer problem (COLONIUS et al., 1997; WEI; FREUND, 2006; CHEUNG; LELE, 2009b) or low Reynolds number jet (MITCHELL et al., 1999; VIOLATO; SCARANO, 2013) the control action would result in lower sound radiation.


93

y

t =400 − Uncontrolled 10 0 −10 0

50

100

150

200 x

250

300

350

400

300

350

400

300

350

400

300

350

400

300

350

400

300

350

400

(a)

y

t =400 − Controlled (transient and buffer time) 10 0 −10 0

50

100

150

200 x

250

(b)

y

t =1200 − Uncontrolled 10 0 −10 0

50

100

150

200 x

250

(c)

y

t =1200 − Controlled 10 0 −10 0

50

100

150

200 x

250

(d)

y

t =2000 − Uncontrolled 10 0 −10 0

50

100

150

200 x

250

(e)

y

t =2000 − Controlled 10 0 −10 0

50

100

150

200 x

250

(f)

FIGURE 3.22 – Comparison of the vorticity fluctuations for the uncontrolled (a, c and e) and controlled (b, d and f) cases. The delay in the vortex pairing becomes apparent.


3.5.3

94

Dealing with disturbances - Aligned Actuators

As it is seen on the work of (FABBIANE et al., 2014), for a model-based feedforward applied to flow control, this type of strategy is very sensitive to disturbances that occur within the sensor-actuation branch, which is also confirmed for this system, as it will be shown in 4. A way to deal with this matter, and also to increase the performance of the controller even without the presence of disturbances, is to align actuators, and design each control set independently. The linearity of the models and the strong convectivity of the shear-layer allows for such a thing. The resulting block diagram of the implementation is shown in figure 3.23, from which it becomes evident that disturbances introduced between positions 1 and 2 will be sensed on the second feedforward branch, increasing the overall robustness of the law.

FIGURE 3.23 – Block diagram of the aligned actuators problem, for this case there are sensors in positions 1 and 4, actuators in 2 and 5 and the objectives are at 3 and 6.

This implementation was tested on the DNS, using the frequency inversion technique, with the objective of minimizing the transverse velocity component. The chosen positions were X = 75, 100, 125, 125, 150 and 175, for positions 1 to 6, as defined in 3.23. Figure 3.24 shows a comparison of the transverse velocity component at X = 175 for the single and aligned actuation cases, proving the performance of the second is considerably increased; similar results were also obtained for the axial velocity component. Figure 3.25 confirms the reductions postpone the vortex roll-up and pairing even further when more actuators are considered. It is expected that the greatest advantages of

CHAPTER 3. THE CONTROL PROBLEM 0.3 0.2

95

Uncontrolled Controlled X = 125 Controlled Aligned Actuation

v(t)

0.1 0 −0.1 −0.2 −0.3 −0.4 0

200

400

600

800

1000 t

1200

1400

1600

1800

2000

FIGURE 3.24 – Results for the controlled and uncontrolled transverse velocity component for a single and two aligned actuators at position of X = 175. such implementation would be proved when exogenous disturbances are inserted into the system.


96

y

t =400 − Uncontrolled 10 0 −10 0

50

100

150

200 x

250

300

350

400

300

350

400

300

350

400

300

350

400

300

350

400

300

350

400

(a)

y

t =400 − Controlled (transient and buffer time) 10 0 −10 0

50

100

150

200 x

250

(b)

y

t =1200 − Uncontrolled 10 0 −10 0

50

100

150

200 x

250

(c)

y

t =1200 − Controlled 10 0 −10 0

50

100

150

200 x

250

(d)

y

t =2000 − Uncontrolled 10 0 −10 0

50

100

150

200 x

250

(e)

y

t =2000 − Controlled 10 0 −10 0

50

100

150

200 x

250

(f)

FIGURE 3.25 – Comparison of the vorticity fluctuations for the uncontrolled (a, c and e) and controlled (b, d and f) with two aligned actuators (right) cases. The delay in the vortex pairing becomes apparent.

4 Conclusions and future work On the final chapter of this qualification document, three future works that are directly related to the current results are foreseen, some of which are already on-going activities. The summary of results and conclusions of the work until this moment will also be presented.

4.1

Future Work

4.1.1

Experimental implementation for jet noise control

The first idea relates to the experimental counterpart of this work, with the implementation of the methodologies presented herein for closed-loop control of the turbulent jet presented in chapter 2, for which the prediction tools have been extensively tested and validated. The major difficulties resulting from an experimental implementation relate to following: • Uncertainties of the model and noise on the sensor-actuator-objective path, which are known flaws of the feedforward control scheme, were not considered in this first design, and could possibly lead to degradation of the performance when applying these methods experimentally; • Although it we are able to determine how an actuation will affect the heviour of the fluctuations downstream, the modelling of a real system will involve a series of

CHAPTER 4. CONCLUSIONS AND FUTURE WORK

98

particularities (valves, delays in the activation of components etc.) that make it more practical to use experimentally obtained frequency responses; • The high convective velocities of the fluctuations of practical jets will require fast actuating valves to be able to act on the most unstable Strouhal numbers and a fast sampling rate for reading the data, in order to prevent aliasing phenomena.

A first attempt to apply these methods to a jet experiment occurred in November 2015, and the DNS of the shear-layer has been able to provide insight on some aspects of the control. Although corresponding to a complete different system, it was possible to test the effects of buffer size and discretization for the sampling rate. Simpler control strategies, such as On/Off actuation, on which a fixed amplitude is kept for the actuating system was also evaluated. For this scenario, the control-law signal is given by a square wave of fixed amplitude, as equation 4.1 proposes.

aOn−Of f (t) = sign(a(t))RM S(a(t))

(4.1)

Figure 4.1 shows a comparison of the On/Off actuation compared to original system inversion case, the same set of positions as those evaluated in 3, with the tranvers velocity component as the objective variable were considered. The conclusion of these preliminary studies is that the considered buffer size to actuation has to be in accordance with the peak occurring the time behaviour of the gain, which is related to the convective time to the actuation to reach the objective position. Time discretization has to be fine enough in order to capture the most unstable frequencies, as given by the Nyquist theorem. Figure 4.2 illustrates these two characteristics. Finally, it is possible to act with an On/Off signal, where the amplitude of the actuation is set at a constant value and a square-shaped signal is considered to act. To do so, the amplitude of the actuation is chosen as the root-mean-square of the predicted signal, via the linearized problem.


99

0.1

Uncontrolled On/off control System Inversion control

0.05

v(t)

0

−0.05

−0.1

−0.15 0

200

400

600

800

1000 t

1200

1400

1600

1800

2000

FIGURE 4.1 – Comparison between the system inversion and On/Off control schemes, for the transverse velocity component, at the position of the objective.

0.3 0.2

2.Tmin

Buffer

0.1

k(t)

0 −0.1 −0.2 −0.3 −0.4 −0.5 0

50

100 t

150

200

FIGURE 4.2 – Illustration of the minimum buffer size and minimum period for sampling, for a resulting given gain k(t).


4.1.2

100

Robustness issues - Is feedback more efficient than feedforward?

As mentioned on chapter 3, feedforward control schemes are in general not robust for plant uncertainties and unmodelled disturbances. On the other hand, feedback control laws were only seen to be effective on such strongly convective systems when the distances between actuator and objective were quite small (BELSON et al., 2013). This may cause difficulties both due to experimental and practical limitations and due to the step size limitations that the PSE method is constrained to. An evaluation of the associated feedback+feedforward loop, as it is commonly seen in system’s applications, may be able to lead to both robustness and performance in terms of amplitude minimization of the fluctuations. Figure 4.3, 4.4 and 4.5 present preliminary robustness studies for the feedforward scheme applied to the shear-layer of chapter 3, aiming at the transverse velocity components. Although not so sensitive to Reynolds number and mean flow velocity variations, when disturbances are introduced between input and objective, the effectiveness of the control law rapidly degrades. Although not leading to a destabilization of the plant, depending on the amplitude of the disturbances, the open and closed-loop cases become completely equivalent. What is seen on system’s and flight mechanics’ applications is the combination of a feedforward to a feedback strategy to grant both speed of response and robustness. The derivation and testing of such strategies is also within the scope of this thesis, as a foreseeable future work, along with the testing of control designs based on the state-space of the system, for which the current plan is to proceed with on of the following:

• Use the Eigensystem Realization Algorithm to obtain from the impulse response of the system, taken either from PSE or an identification method, to the state-space matrices of the open-loop system and its response to actuation. • It is also possible to fit a polynomial to the frequency domain models and inverse


101

0.9

RMSu

0.8

RMS(Ctrl)/RMS(Uncntrl)

0.7 0.6 0.5 0

0.5

1

1.5

2 2.5 Amp/Amp

3

3.5

4

4.5

0

0.45

RMSv

0.4


0.35 0.3 0.25 0

0.5

1

1.5

2 2.5 Amp/Amp

3

3.5

4

4.5

0

FIGURE 4.3 – Robustness verification for amplitude modification of the inflow perturbations. RMS of the streamwise and transverse velocity components of the controlled case divided by the corresponding open-loop scenario.


102

1

RMSu

0.9


0.8 0.7 50

100 Re

150

0.45 RMS(Ctrl)/RMS(Uncntrl)

RMSv

0.4 0.35 0.3 0.25 50

100 Re

150

FIGURE 4.4 – Robustness verification for Reynolds modification in relation to the corresponding design Reynolds number. RMS of the streamwise and transverse velocity components of the controlled case divided by the corresponding open-loop scenario.


103

RMSu

1.06 1.04 1.02 1 0.05


0.06

0.07

0.08 0.09 Amplitude

0.1

0.11

0.12

0.95

RMSv

0.9 0.85 0.8 0.75 0.05

RMS(Ctrl)/RMS(Uncntrl) 0.06

0.07

0.08 0.09 Amplitude

0.1

0.11

0.12

FIGURE 4.5 – Robustness verification for perturbations between input and objective. RMS of the streamwise and transverse velocity components of the controlled case divided by the corresponding open-loop scenario.


104

Fourier transform this system considering the relation between the derivatives in the time-domain and the resulting degree of the polynomial in the frequency/Laplace domain.

4.1.3

Blasius Boundary Layer - Attempting to delay transition

Finally, one of the future works of this thesis the application of the control framework established herein to the real-time control of Tollmien-Schlichting waves in a Blasius boundary-layer. Figure 4.6 shows the mean flow for such case. The strategy is to use PSE based transfer function and set the control in order to postpone the transition to turbulence. The main practical application of this problem is the reduction of drag via a hybrid laminar flow control, as the friction drag over a flat plate is of an order of magnitude greater for a turbulent boundary layer, in comparison against the laminar case (FABBIANE et al., 2014). This problem will require adaptations of the pseudo-spectral method mapping in order to concentrate more points close to the wall and also a change on boundary conditions. The PSE code presented in chapter 3 will be the starting point for these.

4.2

Summary of results and conclusions

Throughout this qualification document, a framework for real-time model-based control of fluctuations in convectively unstable systems has been presented. We have developed reduced order models which were founded on the physics of the flows under study via linearization around a given mean behaviour. In that sense, the work initially developed in (SASAKI, 2014) has been extended to other flows and used to build different time models. This result acts as a further proof of the importance of wavepacket methodologies beyond statistical quantities, but as entities able to educe real-time properties of the flows under study, and is one of the main results of this qualification document. Furthermore, a general feedforward control law which may be easily adapted to other


105

U 1 20 15 y

0.5

10 5 0

300

400

500 x V

600

700

0 −5

x 10 4

15

3

10

2

5

1

y

20

0

300

400

500 x

600

700

0

FIGURE 4.6 – Mean axial and transverse velocity components on a Blasius boundarylayer without pressure gradients. flow systems, along with the transfer functions formalism via PSE has been tested and validated both in the linear and non-linear simulations of a convective unstable flow. Applications to other systems aiming at noise or drag reduction may be readily obtained after the adaptation of the corresponding transfer functions. Techniques based on system inversion, where the gain is frequency dependent to lead to wave cancellation, and proportional and proportional integral, where the gains are scalar values, were tested with their particularities, difficulties and advantages highlighted. For the case without unmodelled disturbances, the frequency inversion feedforward gain was able to lead to diminishments on the turbulent kinetic energy of up to 50 percent, at the position of the objective. As a consequence of that, a delay of the vortex roll-up and pairing phenomena, in the shear-layer system was also evident, within the controlled system. The description and understanding of each component (flow and actuation model) via several descriptions (PSE, Empirical Frequency Impulse Response, Kelvin-Helmholtz projection, ARMAX, MIMO system etc.) is one of the main contributions of this work,


106

as it may allow for other control designs just by rearranging the scheme and changing the ”block diagram” of the proposed control law. As for the proposed future projects, it is believed that the necessary ingredients to proceed with each one of them is readily available from the current qualification work developed so far.

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Appendix A - Jet experimental measurements The experiments were carried out at the ‘Bruit & Vent’ (figure A.1) anechoic jetnoise facility of the Institut Pprime, Poitiers, France. An unforced, Mach M = 0.6 and M = 0.4 jets, issuing from a 50mm diameter nozzle, previously used in a number of other studies (CAVALIERI et al., 2013; BREAKEY et al., 2013), was considered. The corresponding Reynolds number, based on the nozzle diameter, and the air density, viscosity and velocity at the nozzle exit, was 5.7 x 105 . Pressure measurements were made in the jet near field (see figure A.2) using a 42microphone cage array comprising seven 6-microphone rings that partially cover a conical surface of inclination angle 8◦ following the jet divergence. The spacing between successive rings was 0.75D. This measurement configuration permits azimuthal Fourier-series decomposition of the pressure field, providing real-time sensing of the space time behaviour of azimuthal modes m = 0, m = ±1 & m = ±2 (as per equation A.1) over an extended axial region. The present work focus on azimuthal number m = 0, due to its importance for sound radiation in turbulent jets (CAVALIERI et al., 2013). Near the nozzle lip an additional ring array of 14 microphones was installed in order to investigate the relationship between the near-lip fluctuations of the flow and the downstream evolution. A second campaign obtained data over a wider range of axial positions, ranging from 0.5D to 8.9D, with a finer resolution of 0.4D. An array of four 6-microphone rings, each independently displaceable in the axial direction, was used. While the first campaign

APPENDIX A. JET EXPERIMENTAL MEASUREMENTS

114

FIGURE A.1 – Wind-tunnel facility ‘Bruit & Vent’. provided simultaneous measurement of the near-field pressure field, this experiment was designed to obtain fine-resolution measurement of the statistical properties of the jet fluctuations via a two-point correlation analysis. The two-point Cross-Spectral Matrix (CSM) can be build and used in the application of Proper Orthogonal Decomposition.

Z p˜m (x, t) = p˜(x, m, t) =

p0 (x, φ, t)eimφ dφ.

(A.1)

APPENDIX A. JET EXPERIMENTAL MEASUREMENTS

115

FIGURE A.2 – Experimental setup: near field microphone cage array in the 7-ring configuration

Appendix B - DNS of the bidimensional shear-layer The direct numerical simulation (DNS) solves the incompressible bi-dimensional NavierStokes equations using a spectral method. The streamwise direction is discretized using Nx = 512 Fourier modes and the transversal direction using Ny = 70 Chebyshev polynomials. The domain is x = [0 : Lx ], y = [−yinf : yinf ] with Lx = 600, and yinf = 200. The simulation non-dimensional time step is dt = 0.02, where solutions are stored every dtsave = 0.4 for a time T = 2000. The inflow condition is specified as

Ub (y) =

y 1 1 + tanh , 2 2

(B.1)

plus some unsteady perturbations defined by random variables for the control. The inflow (B.1) imposes a unitary velocity difference across the shear ∆U and a unitary vorticity and momentum thickness. The Reynolds number used is Re =

∆U δω ν

= 100, where ν is the kinematic viscosity

and δω is the vorticity thickness. Because we are using a Fourier decomposition in the non-homogeneous streamwise direction, a sponge zone is applied at the end of the domain, so that the solution q reached the inflow state qi n. It consists of adding a forcing term f = −λ(q − qin ) with

APPENDIX B. DNS OF THE BIDIMENSIONAL SHEAR-LAYER 0.2

30

y

117

0

0.1

-30

0 0

100

200

300

400

x FIGURE B.1 – Instantaneous vorticity field of the mixing layer DNS. λ =

20 2

1 + tanh

x−0.85 Lx 10

. Figure B.1 shows an instantaneous vorticity field of the

mixing layer simulation used for the PSE validation.

flow control using the parabolized stability equations

flow control using the parabolized stability equations

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