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Parallel Simulation, Reduced-Order Modeling, and Feedback Control of Vortex Shedding using Fluidic Actuators

Imran Akhtar

Dissertation submitted to the Faculty of the Engineering Science and Mechanics Department of Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of

Doctor of Philosophy in Engineering Mechanics

Ali H. Nayfeh, Chairman Saad A. Ragab Calvin J. Ribbens Muahmmad R. Hajj Mahendra P. Singh

April 28, 2008 Blacksburg, Virginia

Keywords: Vortex Shedding, Parallel Simulations, Reduced-Order Modeling, Feedback Control, Fluidic Actuation Copyright 2008, Imran Akhtar

Parallel Simulation, Reduced-Order Modeling, and Feedback Control of Vortex Shedding using Fluidic Actuators

Imran Akhtar

(ABSTRACT)

In most of the engineering and industrial flow applications, one encounters fluid-structure interaction. This interaction can lead to some undesirable forces acting on the structure, causing its damage or fatigue. The phenomenon, being complex in nature, requires thorough understanding of the flow physics. Analyzing canonical flows, such as the flow past a cylinder, provides fundamental concepts governing the fluid behavior. Despite a simpler geometry, studying such flows are a building block in an effort to comprehend, model, and control complicated flows. For the flow past a circular cylinder, we examine the phenomenon of vortex shedding observed in many bluff body wakes. We develop a parallel computational fluid dynamics (CFD) code to solve the incompressible Navier-Stokes equations on curvilinear coordinates to analyze vortex shedding. The algorithm is implemented on a distributed-memory, message-passing parallel computer, and a domain decomposition technique is employed to partition the grid into various processors. We validate and verify the numerical results with existing experimental and numerical studies. We analyse the performance of the parallel CFD solver by computing the speed-up and efficiency of the solver. We also show that the algorithm is scalable and can be efficiently employed to study other engineering problems requiring larger grid sizes and computational domains. Various other features of the solver, such as the turbulence model, moving boundary techniques, shear, and other canonical flows are also presented. Direct numerical simulations (DNS) are performed to simulate the flow past a circular cylinder to compute the velocity and pressure fields. Based on the flow realizations of the DNS data, we use the proper orthogonal decomposition (POD) tool to determine

the minimum degrees of freedom (or modes) required to represent the flow field. For the current nonlinear problem, the dominant POD modes are used in a Galerkin procedure to project the Navier-Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. We use long-time integration of the reduced-order model to calculate periodic solutions and alternatively use a shooting technique to home on the system limit cycles. We obtain the pressure-Poisson equation by taking the divergence of the Navier-Stokes equation and then project it onto the pressure POD modes. Then, we decompose the pressure into lift and drag components and compare the results with the CFD results. To reduce the fluctuating forces on the structure, we implement full-state feedback control on the low-dimensional model with suction applied aft of the separation point. The control algorithm is successfully simulated using the CFD code and suppression of vortex-shedding is achieved.

iii

Dedication To Mehral and Maham, my two wonderful daughters.

iv

Acknowledgments Well, what seemed to be a long endeavor four years ago is finally over. Looking back in time, there are many people who contributed directly or indirectly in this achievement and they all deserve my deepest gratitude. I would also like to thank the Government of Pakistan for the financial support during my graduate studies. My special thanks to Dr. Ali H. Nayfeh, my advisor, who guided me in my academic pursuit. It is an honour for me to work with him. I would also like to thank my other advisory committee members, Drs. Saad A. Ragab, Calvin J. Ribbens, Muhammam R. Hajj, and Mahendra P. Singh for their encouragement and assistance. I would also like to thank Dr. L. Glenn Kraige with whom I worked for a year and developed some interesting engineering softwares. I also want to thank Drs. John Burns and Jeff Borggaard in Interdisciplinary Center for Applied Mathematics in teaching me various mathematics tools which I used in my research. Thanks to all the faculty, staff, and students associated with the Nonlinear Dynamics Research Lab, namely former students (Drs. Waleed Faris, Pramod Malatkar, Muhammad Younis, Xiaopeng Zhao, Konda Chevva, Muhammad Daqaq, and Greg Vogl), former postdocs (Drs. Eihab Abdel-Rahman and Haider Arafat), and current students (Bashar Hammad, Osama Marzouk, A. Bharami, and M. Ghommem). Also, thanks to Mrs. Sally Shrader for keeping the Lab organized and assistance in official matters. My deepest thanks to my wife Nadia who supported me throughout my graduate studies. She was my wing woman whose presence was the source of my strength. Thanks to my daughters, Mehral and Maham, for keeping my life full of joy. Special thanks to my mom and dad, Mrs. Tabasum Rehan and Mr. Rehan Akhtar, whose love and prayers got me through. Thanks to my brothers, Mr. Abdul Rehman Akhtar and Mr. Muhammad Suleman v

Akhtar, and my sister, Mrs. Musammat K. Barlas, for their affection. I would also like to thank Mr. and Mrs. Sarfraz Iqbal, my parents in-laws, for their support. At the end, I pray from Almighty Allah, who bestowed all these blessings on me, to give me the strength and wisdom to use this knowlege the way He wants.

vi

Contents Abstract

ii

Dedication

iv

Acknowledgments

v

Contents

vii

List of Figures

xii

List of Tables

xviii

1 Introduction

1

1.1

Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

1.2

Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6

1.2.1

Modeling of Vortex-Induced Vibrations . . . . . . . . . . . . . . . . .

6

1.2.2

Parallel Computing and CFD . . . . . . . . . . . . . . . . . . . . . .

9

1.2.3

Flow Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

11

1.2.4

Flow Control using POD-based Reduced-Order Model . . . . . . . . .

14

1.3

Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

1.4

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

1.5

Summary and Outline of Dissertation . . . . . . . . . . . . . . . . . . . . . .

18

1.5.1

Part I - Parallel Simulations . . . . . . . . . . . . . . . . . . . . . . .

19

1.5.2

Part II - Reduced-Order Modeling . . . . . . . . . . . . . . . . . . . .

20

1.5.3

Part III - Active Flow Control . . . . . . . . . . . . . . . . . . . . . .

20

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1.5.4

I

Contents

Outline of the Dissertation . . . . . . . . . . . . . . . . . . . . . . . .

20

Parallel Simulations

24

2 Numerical Methodology

25

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

2.2

Governing Equations and Coordinate Transformation . . . . . . . . . . . . .

26

2.3

Spatial and Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . .

29

2.3.1

Spatial Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . .

29

2.3.2

Temporal Discretization . . . . . . . . . . . . . . . . . . . . . . . . .

29

Solution Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

2.4.1

Fractional-Step Method . . . . . . . . . . . . . . . . . . . . . . . . .

31

2.4.2

Pressure-Poisson Equation . . . . . . . . . . . . . . . . . . . . . . . .

32

2.4.3

Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . .

33

2.4.4

Sequence of Solution Procedure . . . . . . . . . . . . . . . . . . . . .

33

Parallel Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

2.5.2

Message-Passing Interface (MPI) . . . . . . . . . . . . . . . . . . . .

35

2.5.3

Implementation Technique . . . . . . . . . . . . . . . . . . . . . . . .

38

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

2.4

2.5

2.6

3 Validation, Verification, and Parallel Performance 3.1

43

Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.1.1

Three-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . .

44

3.1.2

Two-Dimensional Flows . . . . . . . . . . . . . . . . . . . . . . . . .

48

3.2

Flow Visualization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.3

Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3.1

Grid Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.3.2

Domain Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Parallel Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.4.1

56

3.4

Speed-Up and Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . viii

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3.4.2 3.5

Contents

Scalability Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

60

4 Features and Applications of the Parallel CFD Solver 4.1

II

62

DNS of Turbulent Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

63

4.1.1

Two-Dimensional Flow at ReD =1,000 . . . . . . . . . . . . . . . . . .

64

4.1.2

Three-Dimensional Flow at ReD =1,000 . . . . . . . . . . . . . . . . .

67

4.2

Shear Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71

4.3

Moving Boundary Methods . . . . . . . . . . . . . . . . . . . . . . . . . . .

75

4.3.1

Accelerated Reference Frame . . . . . . . . . . . . . . . . . . . . . . .

77

4.3.2

Forced Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

4.3.3

Crossflow Oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

4.4

Surface Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . .

98

4.5

Reynolds-Averaged Navier-Stokes (RANS) . . . . . . . . . . . . . . . . . . . 101

4.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

Reduced-Order Modeling

108

5 Oscillator Based Reduced-Order Model of Hydrodynamic Forces

109

5.1

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.2

ROM for Elliptic Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 5.2.1

Lift Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5.2.2

Drag Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.2.3

Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.3

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.4

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

6 Proper Orthogonal Decomposition based Galerkin Expansion

124

6.1

Mathematical Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

6.2

Flow Simulation Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3

POD Modes of Velocity Field . . . . . . . . . . . . . . . . . . . . . . . . . . 129 ix

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6.4

Contents

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

7 Reduced-Order Modeling of the Velocity Field

144

7.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7.2

Reduced-Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 7.2.1

Galerkin Projection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

7.2.2

Ordinary-Differential Equation Model . . . . . . . . . . . . . . . . . . 151

7.2.3

POD Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

7.3

Shooting Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

7.4

Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

7.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

8 Reduced-Order Modeling of the Pressure Field

167

8.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

8.2

Pressure Based Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

8.3

Pressure POD Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.4

Reduced-Order Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 8.4.1


8.4.2

Algebraic Equation Model . . . . . . . . . . . . . . . . . . . . . . . . 179

8.5

Lift and Drag Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

8.6

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

III

Active Flow Control

186

9 Feedback Control of Vortex Shedding

187

9.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

9.2

Control Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

9.3

9.2.1

Control Mechanism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

9.2.2

Location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Reduced-Order Model with Control . . . . . . . . . . . . . . . . . . . . . . . 193 9.3.1

Control Function Method . . . . . . . . . . . . . . . . . . . . . . . . 194 x

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Contents

9.3.2


9.3.3

Modified Reduced-Order Model . . . . . . . . . . . . . . . . . . . . . 196

9.3.4

Control Mode Design . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9.3.5

Control Law Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

9.3.6

Open-Loop Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

9.4

9.5

9.6

Full-State Feedback Control . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 9.4.1

System Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.4.2

Modified Feedback System . . . . . . . . . . . . . . . . . . . . . . . . 207

Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 9.5.1

Control of Direct Simulations . . . . . . . . . . . . . . . . . . . . . . 209

9.5.2

Limitations of Full-State Feedback Controller . . . . . . . . . . . . . 210

9.5.3

Controllability Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

10 Summary, Conclusions, and Recommendations for Future Work

214

10.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 10.1.1 Parallel CFD Solver . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 10.1.2 Reduced-Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . 216 10.1.3 Full-State Feedback Control . . . . . . . . . . . . . . . . . . . . . . . 216 10.2 Recommendations for Future Work . . . . . . . . . . . . . . . . . . . . . . . 217 Bibliography

219

xi

List of Figures 1.1

A typical von Kármán vortex street in the wake of a cylinder. . . . . . . . .

2

1.2

A block diagram depicting the reduced-order model procedure. . . . . . . . .

6

1.3

Illustration of the three main parts in the Dissertation: Vortex-Induced Vibration (VIV), Reduced-Order Modeling (ROM), and Active Feedback Control (AFC). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.1

19

Schematic of coordinate transformation from a physical domain to a computational domain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

2.2

Geometry in curvilinear coordinates. . . . . . . . . . . . . . . . . . . . . . .

27

2.3

A 2-D layout of an “O” grid over a circular cylinder, showing the inflow and outflow directions.

2.4

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Schematic of (a) shared-memory and (b) distributed-memory architecture. Note: Processor (P), Cache (C), Memory (M) . . . . . . . . . . . . . . . . .

2.5

36

(a) Processor group topology in a Cartesian structure and (b) communicator comprising 24 processors on 8 × 3 processor platform. . . . . . . . . . . . . .

2.6

28

40

(a) A 2-D layout of the 128×128×96 “O” grid distributed among 8 processors in the η-direction. The grid is plotted only for the region of processor “1”. (b) A 3-D layout of the complete domain and the grid is plotted for only one processor, indicating the load per processor in a 24 (8×3) processor platform. The grid is plotted only for the region of processor “1”. (Note: every 4th grid

3.1

point is plotted for clarity.) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

42

Time histories of (a) CL and (b) CD . . . . . . . . . . . . . . . . . . . . . . .

46

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List of Figures

3.2

Power spectrum of (a) CL and (b) CD . . . . . . . . . . . . . . . . . . . . . .

3.3

Mean surface CP distribution: present (solid) and Mittal and Balachandar (1995) (triangle). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.4

53

The instantaneous vorticity isosurfaces (level = 0.5, 1.0, 1.5) at ReD = 525. (Note: the plot is 50% transparent to view the inner structures.) . . . . . . .

3.7

52

The instantaneous streamwise vorticity isosurfaces (blue: ω x level at -0.5 and red: ω x level at +0.5) at ReD =525. . . . . . . . . . . . . . . . . . . . . . . .

3.6

49

The instantaneous spanwise vorticity isosurfaces (blue: ω z level at -0.5 and red: ω z level at +0.5) at ReD =525. . . . . . . . . . . . . . . . . . . . . . . .

3.5

47

54

Speed-up trends of the (a) 2-D and (b) 3-D CFD codes relative to two and four processors for different grid sizes. The dashed line represents the ideal speed-up. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3.8

Scalability study of the parallel code, the horizontal bars depict the spanwise length for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

58

60

Spanwise vorticity contours of one vortex shedding cycle over a stationary cylinder at ReD = 1, 000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

4.2

Time histories and power spectra of CL and CD at ReD = 1, 000. . . . . . . .

67

4.3

Geometry of the cylinder with arrows showing the flow direction and checkerboard pattern indicating 8 processors in the spanwise direction (left), isosurfaces of ωz at t = 0 (center) and t ≈ Ts /4 (right). . . . . . . . . . . . . . . .

69

4.4

Isosurfaces of ωz at t ≈ Ts /2 (left), t ≈ 3Ts /4 (center), and t ≈ Ts (right). . .

70

4.5

Time histories and power spectra of CL and CD at ReD = 1000. . . . . . . .

71

4.6

1-D energy spectrum at ReD =1,000. . . . . . . . . . . . . . . . . . . . . . . .

72

4.7

A schematic of some shear flows in spanwise directions. . . . . . . . . . . . .

74

4.8

Spanwise vorticity plotted on two planes along the span. . . . . . . . . . . .

76

4.9

Isosurfaces of the absolute vorticity (level=0.75, 1.5 and 3) colored by the spanwise vorticity (50% transparent). . . . . . . . . . . . . . . . . . . . . . .

77

4.10 A schematic of the possible stationary and moving boundary cases. . . . . .

81

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List of Figures

4.11 Frequency-amplitude plot: Lock-in (squares), quasi-periodic (triangles), and lock-in region obtained by Koopmann (1967)(circles). . . . . . . . . . . . . .

83

4.12 Time histories and power spectra of CL and CD for a stationary cylinder at ReD = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

84

4.13 Spanwise vorticity contours of one vortex shedding cycle over a stationary cylinder at ReD = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

85

4.14 Time histories and power spectra of CL and CD for a crossflow oscillating cylinder at ReD = 200 with fe /fs = 0.8 and Ay /D = 0.1. . . . . . . . . . . .

87


88


89

4.17 Spanwise vorticity contours of one oscillating cycle with fe /fs = 0.6 and Ay /D = 0.5 at ReD = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . .

90

4.18 Spanwise vorticity contours of one oscillating cycle with fe /fs = 1.2 and Ay /D = 0.5 at ReD = 200. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Spanwise vorticity contours of the flow past an oscillating cylinder.

91

. . . . .

93

4.20 Time histories of (a) CL and (b) CD . . . . . . . . . . . . . . . . . . . . . . .

95

4.21 Power spectrum of (a) CL and (b) CD . . . . . . . . . . . . . . . . . . . . . .

96

4.22 Cross-bispectrum of the lift and drag coefficient. . . . . . . . . . . . . . . . .

97

4.23 (a) A schematic of a synthetic jet actuation pair located on the cylinder surface and (b) velocity profiles for the jet: top hat (dashed), sinusoidal (solid), and sine square (dashdot). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 4.24 Time histories of CL and CD with control actuation. . . . . . . . . . . . . . . 101 4.25 Instantaneous spanwise vorticity isosurfaces (blue: ω z level at -1 and red: ω z level at +1) at ReD =3,900. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.26 (a) Time histories of the lift CL and drag CD coefficients obtained with unsteady RANS at ReD =3,900. . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.1

Instantaneous spanwise vorticity contours. . . . . . . . . . . . . . . . . . . . 115

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List of Figures

5.2

Time history of the lift (solid) and drag (dashed) coefficients. . . . . . . . . . 117

5.3

Spectral analysis parameters from the CFD simulations. . . . . . . . . . . . 119

5.4

Parameters of the van der Pol-Duffing oscillator model. . . . . . . . . . . . . 120

5.5

Comparison between the time histories obtained by the lift model with the CFD results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.6

Comparison between the time histories obtained by the drag model with the CFD results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

6.1

Average flow fields: (a) streamwise and (b) crossflow velocity components at ReD =100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6.2


6.3


6.4

Average flow fields: (a) streamwise, (b) crossflow, and (c) spanwise velocity components at ReD =525 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

6.5

Normalized eigenvalues. Square: Case I; triangle: Case II; circle; Case III, and diamond; Case IV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

6.6

The normalized cumulative energy. Square; Case I, triangle; Case II, circle; Case III, and diamond; Case IV. . . . . . . . . . . . . . . . . . . . . . . . . . 133

6.7

The streamwise velocity modes (φui , i = 1, 2, ..., 10) at ReD =100

6.8

The crossflow velocity modes (φvi , i = 1, 2, ..., 10) at ReD =100 . . . . . . . . . 135

6.9

The streamwise velocity modes (φui , i = 1, 2, ..., 10) at ReD =200

. . . . . . . 134

. . . . . . . 136

6.10 The crossflow velocity modes (φvi , i = 1, 2, ..., 10) at ReD =200 . . . . . . . . . 137 6.11 The streamwise velocity modes (φui , i = 1, 2, ..., 10) at ReD =525

. . . . . . . 138

6.12 The crossflow velocity modes (φvi , i = 1, 2, ..., 10) at ReD =525 . . . . . . . . . 139 6.13 Isosurfaces of the streamwise velocity modes (φui , i = 1, 2, ..., 10) at ReD =525 with 25% transparency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.14 Isosurfaces of the crossflow velocity modes (φvi , i = 1, 2, ..., 10) at ReD =525 with 25% transparency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

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List of Figures

6.15 Isosurfaces of the spanwise velocity modes (φw i , i = 1, 2, ..., 10) at ReD =525 with 25% transparency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 7.1

The velocity coefficients qi = 1, 2, ..., 8 at ReD =100. . . . . . . . . . . . . . . 154

7.2

A two-dimensional projection of the POD phase portrait on (q1 , q2 )-plane (solid) and snapshot portrait (triangle). . . . . . . . . . . . . . . . . . . . . . 155

7.3

Two-dimensional projections of the phase portraits of the velocity coefficients onto the plane (qi , a1 ) for qi = 3, 4, ..., 8 at ReD =100.. . . . . . . . . . . . . . 158

7.4

Divergence of the qi , i = 1, 2, ..., 4 to spurious limit cycles. . . . . . . . . . . . 159

7.5

Velocity coefficient qi = 1, 2, ..., 8 at ReD =100 . . . . . . . . . . . . . . . . . 160

7.6

Instantaneous u-velocity fields from (a) CFD and (b) POD. . . . . . . . . . . 162

7.7

Instantaneous u-velocity fields from (a) CFD and (b) POD. . . . . . . . . . . 163

8.1

Average pressure fields: (a) Case I, (b) Case II, (c) Case III, and (d) Case IV. 171

8.2

Pressure modes (Ψi , i = 1, 2, ..., 10) at ReD =100. . . . . . . . . . . . . . . . . 172

8.3

Pressure modes (Ψi , i = 1, 2, ..., 10) at ReD =200. . . . . . . . . . . . . . . . . 173

8.4

Pressure modes (Ψi , i = 1, 2, ..., 10) at ReD =525 . . . . . . . . . . . . . . . . 174

8.5

Isosurfaces of pressure modes (Ψi , i = 1, 2, ..., 10) at ReD =525 with 25% transparency. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

8.6

The pressure coefficients ai = 1, 2, ..., 8 at ReD =100. . . . . . . . . . . . . . . 180

8.7

A two-dimensional projection of the pressure coefficients onto the plane (a1 ,a2 ) at ReD =100. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

8.8

Two-dimensional projections of the phase portraits of the pressure coefficients onto the plane (ai , a1 ) for ai = 3, 4, ..., 8 at ReD =100. . . . . . . . . . . . . . 182

8.9

Comparison of (a) the mean pressure and (b) lift coefficients: POD (solid line) and CFD (triangle). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

9.1

Normalized eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

9.2

Cumulative effect of eigenvalues. . . . . . . . . . . . . . . . . . . . . . . . . . 190

9.3

Mean pressure POD mode at ReD = 200 . . . . . . . . . . . . . . . . . . . . 191

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9.4

List of Figures

The pressure POD modes on the cylinder surface at ReD = 200, (a) i = 1, 2, (b) i = 3, 4, (c) i = 5, 6, and (d) i = 7, 8: even (solid) and odd (dashed). . . . 192

9.5

Actuated flow at ReD =200: (a) streamwise velocity and (b) crossflow velocity. 198

9.6

Control mode at ReD =200: (a) streamwise velocity and (b) crossflow velocity. 198

9.7

Poles of open-loop system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

9.8

Schematic of the unstable poles moving to the left-half of the complex plane. 201

9.9

Pole locations: circle - OL, diamond - CL1, square - CL2, and delta - CL3. . 203

9.10 Time history of q1 with control actuation; (a) CL1, (b) CL2, and (c) CL3.

. 204

9.11 Time history of γ; (a) CL1, (b) CL2, and (c) CL3. . . . . . . . . . . . . . . . 206 9.12 System response q1 for CL1: solid - modified model and dashed - linear model. 208 9.13 Jet velocity associated with the control input in CL3. . . . . . . . . . . . . . 209 9.14 Time histories of CL and CD with feedback control (solid) and without feedback control (dashed). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 9.15 Spanwise vorticity contours with feedback controller from t=80-100 (Actuation begins at t=80). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

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List of Tables 3.1

Computed flow parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . .

45

3.2

Validation for the two cases. . . . . . . . . . . . . . . . . . . . . . . . . . . .

50

3.3

Grid dependence study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

3.4

Domain dependence study . . . . . . . . . . . . . . . . . . . . . . . . . . . .

56

3.5

Scalability study chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

4.1

Computed flow parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . .

68

4.2

Crossflow oscillations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

82

4.3

Computed flow parameters at Re=3900. . . . . . . . . . . . . . . . . . . . . 105

5.1

Spectral Analysis of the Lift and Drag Coefficients. . . . . . . . . . . . . . . 118

5.2

Parameters of the Reduced-Order Model. . . . . . . . . . . . . . . . . . . . . 118

6.1

POD configurations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

9.1

Pole Placement Chart. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

9.2

Feedback Gains of the Systems. . . . . . . . . . . . . . . . . . . . . . . . . . 205

9.3

System Response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

xviii

Chapter 1 Introduction The interactive fluid-structure phenomenon of flow separation and bluff body wakes has its fundamental significance in flow physics and its practical importance in aerodynamic and hydrodynamic applications. Since Roshko (1955) measured the vortex shedding period behind a bluff body, many researchers have investigated this phenomenon experimentally and numerically for a wide range of Reynolds numbers. The flow behind a circular cylinder has become the canonical problem for studying such external separated flows (Bishop and Hassan, 1963a; Karniadakis and Triantafyllou, 1992; Roshko, 1954; Tomboulides et al., 1989; Williamson, 1996; Wu et al., 1994).

1.1

Motivation

When flow passes over a bluff body at low Reynolds numbers, flow separation may take place over substantial parts of its surfaces but the flow around it remains steady. As the Reynolds number exceeds a critical value, instability in the separated shear layers develops, and nonlinear interaction of these layers with feedback from the wake leads to an organised and periodic motion of a regular array of concentrated vorticity, known as the von Kármán vortex street in the wake, as shown in Figure 1.1. This vortex shedding exerts oscillatory forces on the body, which are often decomposed into drag and lift components along the freestream and crossflow directions, respectively. If the body is capable of flexing or moving rigidly, these forces can cause it to oscillate and the classical vortex-induced vibration (VIV) 1

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Introduction

2

problem takes place. If the frequency of vortex shedding is close to a natural frequency of the body, the resulting resonance can generate large-amplitude oscillations, which may ultimately cause structural failure. Understanding this problem is of great interest in the design and maintenance of offshore structures, like mooring cables, risers, and spars, and of high-aspect ratio structures subject to air streams, like chimneys, high-rise buildings, bridges, and cable-suspension systems.

Figure 1.1: A typical von Kármán vortex street in the wake of a cylinder.

Despite the progress in research and technology, separating flows over complex shapes still remain a great challenge for its analytical, experimental, or numerical treatments. Due to the complicated nature of such flows, the scope of theoretical analysis has been in general very limited. Experimental techniques have become very sophisticated in recent years, but an extensive spatial and temporal interrogation of complex 3-D flow fields would quickly overwhelm available resources. Therefore, a fundamental analysis of the flow field around a canonical body would allow an understanding of the flow separation process. In terms of cost effectiveness, numerical simulations provide a promising approach, as compared to experimental investigation to investigating this problem. However, a thorough analysis of such flows would require a systematic sweep of the parameter space, including the Reynolds number, angle-of-attack, and thickness ratio. Moreover, for each simulation, a

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Introduction

3

wide range of variables would have to be computed with sufficient accuracy in both space and time, especially if turbulent structures are to be resolved adequately. The challenge for the computational fluid dynamicist would be to have an intelligent grid generation algorithm and an efficient solver capable of simulating the flow in the full parameter space and then to be able to analyze and interpret the gigabytes of data that would be generated from these runs. In order to model, predict, and control vortex shedding, full simulation of the coupled flow and structure equations is needed. This involves accounting for three-dimensional effects, turbulence structures, and elasticity, among other considerations. Advances in parallel computing technology in terms of speed and storage capacity has enabled the computational fluid dynamicists to solve more complex problems. Parallelism appears in various forms, such as lookahead, pipelining, vectorization, concurrency, simultaneity, data parallelism, partitioning, interleaving, overlapping, multiplicity, replication, time sharing, space sharing, multitasking, multiprogramming, multi threading, and distributed computing at different processing levels (Hwang, 1993). Exploitation of parallelism has created a new dimension in numerical solutions of significant scientific and engineering problems. However, even with these advances in hardware and software capabilities, direct numerical simulation (DNS) of flows with engineering relevance remains a challenging task. Hence, some type of modeling or approximation is introduced to simplify the flow computations and make them feasible on existing computer platforms. Usually, the approximation is introduced in the representation of the turbulent scales in the form of a turbulence model, which significantly reduces the spatial and temporal resolution requirements. Under these conditions, full simulation of canonical structures is overwhelming, even with todays computing power. So, simplifying assumptions about the flow and structure are often made and efforts to produce simple, accurate representations of the problem via reduced-order modeling are still on-going. Control of the flow field has been a subject of interest in recent years. It can lead to significant benefits in industrial applications. Few examples that illustrate the immediate benefits of flow control include increased aerodynamic efficiency by prevention/provocation of flow separation, reduced structural weight to reduce operating costs, enhanced mixing in combustion, and enhanced heat transfer in heat exchangers. Flow control methods can be

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classified as passive and active. Passive control devices, such as riblets, vortex generators, and boundary-layer trips, have been shown to be quite effective in delaying flow separation. Because passive devices can not adapt to flow changes, they might lose their efficiency during the process. On the other hand, active-control methods, such as acoustic excitation, vibrating ribbons or flaps, blowing and suction, couple the control input to the flow instabilities and can operate in a broad range of conditions. However, these active controllers require appropriate sensors and actuators. Because turbulent flows of interest in engineering applications have small time and length scales associated with the turbulent eddies, active controllers require large number of sensors and actuators with high-frequency responses. Another approach to actively control the flow over a body is to place it in the vicinity of another body. Gopalkrishnan et al. (1994) examined the interaction of a flapping foil with vortices shed by a bluff body. In their experiments, a hydrofoil was placed in the wake of a D-section cylinder, sufficiently far behind the cylinder so that it did not interfere with the vortex formation process. Their results indicated that the change in the efficiency of the foil is related to the spacing between the two structures. This control technique is inspired by the swimming patterns of aquatic animals (Drucker and Lauder, 2001; Fish and Lauder, 2006) and can be employed in Autonomous Underwater Vehicles (AUV) (Akhtar et al., 2007). Flow-induced vibrations are a direct consequence of the fluctuating hydrodynamic forces (lift and drag) on the cylinder. These fluctuations may have adverse effects on the performance and life of a structure. Control of the flow over a circular cylinder has received considerable attention because of its canonical nature and being a typical unstable flow. There are various control mechanisms, which have been employed to suppress vortex shedding on a circular cylinder. These methods include cylinder rotation, transverse motion, blowing/suction on the cylinder surface, and acoustic actuation. Significant progress has been made in developing various control strategies; however, the control of fluid flow remains an active field of research. Due to the inherent nonlinearity in the Navier-Stokes equations and complexity of infinite-dimensional flow dynamics, design and application of a control system still remains a challenging task. In order to reduce the complexity of the governing equation, low-dimensional models are often developed. Many reduced-order model techniques in fluid mechanics are derived from the proper orthogonal

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Introduction

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decomposition (POD)-Galerkin projection approach (Berkooz et al., 1993; Holmes et al., 1996). The POD provides a tool to formulate an optimal basis or minimum degrees of freedom (or modes) required to represent a dynamical system. The POD is also known as the Karhunen-Loeve expansion in statistics and principal component analysis or empirical orthogonal functions (EOF) in meteorology. The POD-Galerkin reduced-order models are constructed in two steps: 1. Computation of the POD eigenfunctions from the data ensemble of the flow field. 2. Galerkin projection of the Navier-Stokes equations onto a space spanned be a small number of POD eigenfunctions. In the first step, the variables from the high-fidelity simulation (typically CFD) are transferred to a finite number of basis functions or modes, which are relatively small in number as compared to the degrees of freedom involved in the simulation. The second step involves a translation of the full-system dynamics to the implied dynamics of these modes. The resulting dynamical system consists of a set of ordinary-differential equations (ODEs) in time in the modal amplitudes. Thus, reduced-order models obtained from this procedure can be used to apply a control strategy. This application of control design at the analytical level will be beneficial for the design of a controller in the real-time system. Some other applications of reduced-order models include shape optimization, aeroelastic stability analysis, and understanding of the nonlinear dynamics of the system. The real strength of reduced-order models lies in the predictive settings. The governing equations for most of physical dynamical systems (e.g., fluid flows) comprise partialdifferential equations (e.g., Navier-Stokes equations) which correspond to infinite degrees of freedom. Such systems are solved numerically using various CFD methods, thereby reducing the system to a finite degrees of freedom. However, for a time-varying, three-dimensional fluid flows, the degrees of freedom are of the order of millions. Applying a control strategy at this level remains an expensive endevour. Thus reduction of the dynamical system to few degrees of freedom may be viewed as an alternate discretization of the governing partial-differential equations. The key advantage of such models is that control theory can be applied for flow control design. A schematic of the hierarchy of flow models is represented

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6

Introduction

in Figure 1.2 where the degrees of freedom are systematically reduced while preserving the physics and dominant features of the flow.

Experiment

Flow Field

PDE

N-S

Discretize

CFD

POD modes Snapshots

ODE

Galerkin projection

Controller

Full-State Feedback

ROM

Actuation

Suction/ Blowing

Figure 1.2: A block diagram depicting the reduced-order model procedure.

1.2 1.2.1

Literature Review Modeling of Vortex-Induced Vibrations

Circular cylinders are extensively used in the study of bluff body fluid dynamics due to their geometric simplicity and common use in engineering applications. Bishop and Hassan (1963a) were among the earliest to suggest using a self-excited oscillator to represent the forces over a cylinder due to vortex shedding. Hartlen and Currie (1970) formulated a

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Introduction

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model for elastically restrained cylinders that are restricted to crossflow motions. They used a Rayleigh oscillator to describe the lift force and coupled it to the cylinder motion by a linear velocity term. Currie and Turnball (1987) proposed a similar model for the fluctuations in the drag. Skop and Griffin (1973) pointed out that the parameters in the model of Hartlen and Currie (1970) lacked clear connections to the physical parameters of the problem. They proposed a modified van der Pol oscillator to represent the lift coupled to a linear equation of motion for the structure. They also introduced the Skop-Griffin parameter, which is now commonly used in studying VIV problems. Iwan and Blevins (1974) considered the fluid mechanics of the vortex street and developed a model in terms of a fluid variable that captures the fluid-dynamics effects of the problem. Landl (1975) added a nonlinear aerodynamic damping term of fifth order to the van der Pol oscillator in his two-equation model, suggesting that this enables better capturing of some physical characteristics. However, the model involved many constants to be determined. Several attempts have been made to extend the wake-oscillator models to elastic structures, such as beams and cables. Iwan (1975) extended the model of Iwan and Blevins (1974) to predict the maximum VIV amplitude of taut strings and circular cylindrical beams with different end conditions. Iwan (1981) also derived an analytical model for the VIV of elastic structures under nonuniform flows. Skop and Griffin (1975) extended their rigid-cylinder model to elastic cylinders, with the goal of accurately capturing the asymptotic, self-limiting structural response near zero damping. Skop and Balasubramanian (1997) modified the model of Skop and Griffin (1975) by separating the fluctuating lift into two components: one satisfying a van der Pol oscillator that is driven by the transverse motion of the cylinder and the other linearly proportional to the transverse velocity of the cylinder (called the stall term). Krenk and Nielsen (1999) proposed a two-oscillator model in which the mutual forcing terms are developed based on the premise that energy flows directly between the fluid and structure. This means that the forcing terms correspond to the same flow of energy at all times. Kim and Perkins (2002) modeled the lift and drag by two nonlinearly coupled van der Pol equations in their study of resonant responses of suspended elastic cables. The coupling terms were introduced based on the fact that the main frequency of the drag component is

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Introduction

8

twice the main frequency of the lift component. Facchinetti et al. (2003) investigated the flow around a circular cylinder elastically restrained to move only in the crossflow direction. They developed a reduced-order model for the fluid-structure system by introducing a van der Pol model for the lift coefficient coupled with the cylinder equation of motion (a classical mass-spring-damper-system) through external forcing terms. They assumed the natural frequency and proposed three different coupling terms in the wake model: the cylinder position, velocity, and acceleration. They found that the acceleration coupling model matches the experimental data for a harmonically excited cylinder better than the velocity or position coupling models. Evangelinos et al. (2000) used direct numerical simulation (DNS) of the incompressible Navier-Stokes equations to solve, at Reynolds number of 1,000, for the fluid-structure problem of rigid and flexible cylinders allowed to move freely in the crossflow direction. The flexible cylinder (or beam) was represented by a simplified linear model with a forcing term proportional to the lift coefficient and no structural damping. The governing equations were transformed into the Fourier domain and solved with a spectral elements method that employs a hybrid grid in the x − y plane and Fourier complex modes in the z-direction. They considered short and long cylinders with length-to-diameter ratio of 4π and 378, respectively. They reported that the often-used empirical formula proposed by Skop and Balasubramanian (1997) overpredicts the drag coefficient. Norberg (2003) presented an overview of the fluctuating pressure and lift acting on a circular cylinder, taking special consideration of the influence of the Reynolds number and the relation between the fluctuating lift and flow features (e.g., laminar shedding, wake transition, and turbulent shedding) in the near-wake region. He compiled the then available data from the diverse experimental and numerical approaches, including two-dimensional and three-dimensional simulations, and reviewed the different measurement methods like the force-element method, total force method, electromagnetic method, and the family of momentum pressure methods. These results show significant changes in the fluctuating pressure distribution over the cylinder at different Reynolds numbers ranging from 47 to 200,000. A spanwise correlation length of about 30 cylinder diameters was observed for flows near the subcritical regime Re=30,000.

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Introduction

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Williamson and Govardhan (2004) gave a detailed review of the experimental and computational work on the VIV problem over the last two decades. The primary concern is with the free and forced oscillations of elastically mounted rigid cylinders and bodies having two degrees of freedom and the dynamics of cantilevers, pivoted cylinders, cables, and tethered bodies. They underline some of the fundamental questions in VIV and touched upon some debatable concepts, like the added mass and the effect of combined mass-damping parameter on the peak-amplitude data in the Griffin plot. Nayfeh et al. (2003) numerically simulated the two-dimensional flow past a stationary cylinder for a wide range of Reynolds numbers and calculated the lift and drag. They employed higher-order spectral analysis to determine the phase relations among the different spectral components of the lift and compared them with the phase information one obtains from closed-form approximate solutions of the van der Pol and Rayleigh oscillators. The model was later extended to include the transient flow over the cylinder (Marzouk et al., 2007). Their analytical model was based on two-dimensional simulations of flow over a stationary cylinder. Gabbai and Benaroya (2005) presented a comprehensive review of the studies (experimental, semi-experimental, and numerical) related to VIV problem. They covered a variety of issues related to the physics of the problem, like the dynamics of a vibrating cylinder in a flow, vortex-shedding modes, and the three-dimensionality effects. They also reviewed different models (e.g., wake models and force-decomposition models) and approaches (e.g., variational approach, vortex-in-cell approach (VIC), direct numerical simulation (DNS), and the finite-element method) used to simulate the problem of VIV of both rigid and elastic cylinders.

1.2.2

Parallel Computing and CFD

With the increase in the speed of micro-processors, the performance of sequential computers have become incredibly fast. However, it is insufficient for a large number of challenging applications in industry and research, which require much more performance. Parallel processing has emerged as a key enabling technology in modern computers, driven by the everincreasing demand for higher performance. Parallel computing provides fast and accurate

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Introduction

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solutions to scientific, engineering, business, social, and defense problems. One of the major applications of parallel processing is the field of CFD. Due to the presence of multiple scales, especially in the turbulent regime, large and small eddies need to be resolved for accurate prediction of the flow field. DNS of turbulent flows provides a detailed turbulence data, which are free from ambiguities, such as the effects of using Taylor’s hypothesis and one-dimensional surrogates. However, the number of degrees of freedom necessary for resolution in DNS increases rapidly with increasing Reynolds number. Therefore, the maximum resolution is limited by the available computing speed and memory. The first large-scale parallel computer was the ILLIAC IV at NASA Ames Research Center in 1965. It had 64 processors (called processing elements or PEs) typically running at the clock 12.5 MHz (Kim, 2003) with a total memory of 128 megabytes. However, over the past decade, parallel processing has emerged as a key enabling technology in modern computers, driven by the ever-increasing demand for higher performance, lower costs, and sustained productivity in real-life applications. Advances in computational power and parallel computing in the past two decades have revolutionized the field of CFD. DNS and large-eddy-simulation (LES), despite being limited to relatively simple geometries and moderate Reynolds number flows, have provided an insight into turbulent flow physics. Fundamental turbulent flows, such as turbulent boundary layers, free-shear layers, and wake flows, are better understood in terms of their physics and structure. The recently developed Earth Simulator in Japan has a peak performance and main memory of 40 TFlops and 10 TBytes, respectively. To date, DNSs of incompressible turbulence using spectral methods have been limited to a maximum of 40963 grid points in a periodic box (Kaneda et al., 2003). Simulating the three-dimensional flow over a simple geometry, such as a circular cylinder, at high Reynolds numbers still remains a formidable task. Three-dimensional structures become significant at high Reynolds numbers and affect the pressure distribution on the surface of the body. Dong and Karniadakis (2004) successfully simulated DNS of the flow past stationary and oscillating cylinders to analyse fluid-structure interaction at high Reynolds numbers with high resolution in the streamwise and crossflow directions and a large number of Fourier modes along the cylinder span. They employed an MPI/MPI two-level parallel algorithm

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on a Compaq Alpha cluster (1 GHz Alpha CPU) for the cylinder flow at Re=10,000 with a problem size of 300 million degrees of freedom. They used 256, 512, 1024, and 1536 processors to simulate the flow and computed the speed-up and efficiency for each case.

1.2.3

Flow Control

Flow separation over a bluff body may lead to vortex shedding, thereby producing significant fluctuating forces along with the mean drag. Much research is devoted to manipulate the flow to reduce the drag and suppress vortex shedding (Gopalkrishnan et al., 1994; Park et al., 1994; Roussopoulos, 1993; Tao et al., 1991; Tokumaro and Dimotakis, 1991). Using pneumatic devices as a means to modify the flow for desired purposes goes back to the 1930s, perhaps even earlier (Englar, 2000). Over a period of seventy years, various applications, methods, and strategies have been employed to control the flow field. Significant progress has been made by combining CFD, control methods, and sensor/actuator technologies. Successful control of turbulent flows, however, requires both a thorough understanding of the underlying physics of turbulent flows and an efficient control algorithm (Kim, 2003). Viscous flow past a circular cylinder has been extensively studied due to its simple geometry and because its behavior is representative of general bluff-body wakes. Various approaches for control of the flow around a circular cylinder have been employed to achieve reduction in the drag, noise, and vibration. Some passive ways of controlling vortex shedding include the use of end plates (Nishioka and Sato, 1974; Stansby, 1974), inhomogeneous inlet flows (Gaster, 1971; Gerich and Eckelmann, 1982), splitter plates (Apelt and West, 1975; Apelt et al., 1973; Cimbala and Garg, 1991; Gerrard, 1966; Kwon and Choi, 1996; Roshko, 1955; Unal and Rockwell, 1988), blockage (Shair et al., 1963), a second cylinder in the wake (Sakamoto and Haniu, 1994; Strykowski and Sreenivisan, 1990), base bleed (Bearman, 1967; Schumm et al., 1994; Wood, 1964), and periodic rotation of the cylinder (Homescu, 2002; Tokumaro and Dimotakis, 1991). Roussopoulos (1993) implemented feedback control using a speaker driven based on the velocity phase information measured at a point in the wake. He obtained suppression of the vortex shedding up to a Reynolds number of 60. In order to suppress vortex shedding, Gunzburger and Lee (1996) determined the amount of fluid to be injected or sucked on the

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Introduction

12

rear of the cylinder. The rate of blowing/suction was determined using a feedback laws of the pressure measurements at stations along the surface of the cylinder. Huang (1996) suppressed vortex shedding by feeding back sound. In an experimental study, Fujisawa and Nakabayashi (2002) evaluated the performance of active control of vortex shedding from a circular cylinder at Re=20,000 by a rotational feedback oscillation using neural networks. They successfully applied their neural network model, with a back propagation algorithm, to determine the control parameters, such as phase lag and feedback gain, that achieve optimum performance of the controller. They reported reduction in the drag and lift forces by 16% and 70%, respectively. Park et al. (1994) proposed a feedback approach using a pair of blowing/suction slots on the cylinder. A single feedback sensor was located in the wake. They showed complete suppression of vortex shedding at Re=60. Roussopoulos (1993) and Park et al. (1994) located sensors in the wake and constructed feedback controllers based on physical intuition through observation of the flow phenomenon. Their objective was to stabilize the wake. Active control can also be achieved by placing another structure (cylinder or airfoil) in the vicinity of the main cylinder (Gopalkrishnan et al., 1994; Sakamoto and Haniu, 1994). Min and Choi (1999) developed a systematic control method using a suboptimal control scheme to reduce drag in both laminar and turbulent flows. They also restricted the location of feedback sensors to the cylinder surface. They argued that placing sensors in the wake is not practical in many real applications. In their suboptimal feedback control procedure, they minimized two cost functionals (J1 , J2 ) and maximized a third functional (J3 ). The functional J1 is proportional to the pressure drag of the cylinder, J2 is proportional to the square of the difference between the target pressure (inviscid flow pressure) and the real flow pressure on the cylinder surface, and J3 is proportional to the square of the pressure gradient on the cylinder surface. The control input is blowing/suction on the cylinder surface. They numerically simulated several cases at Re=100 and Re=160 for each cost functional. They found that vortex shedding became weak or disappeared, and the mean drag and lift/drag fluctuations decreased significantly. They also observed that, for a given amount of blowing/suction, reducing J2 provides the largest drag reduction among the three cost functionals.

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Introduction

13

Fransson et al. (2004) experimentally investigated the flow around a porous cylinder subject to a moderate level of suction or blowing (around 5% of the oncoming streamwise velocity) for Reynolds numbers of the order of 104 . They observed that suction moves the separation line to the rear part of the cylinder as in case of turbulent boundary layers. They also found that the Strouhal number decreases with blowing, whereas suction has the opposite effect, and the suction/blowing rate can empirically be represented by an effective Reynolds number for the solid cylinder. In another experimental study, Zhang et al. (2004) investigated closed-loop control of vortex shedding from a spring-supported square cylinder in crossflow. The control action was implemented through perturbation of one cylinder surface through piezoelectric ceramic actuators controlled by a proportional-integral-derivative (PID) controller. Their control scheme based on the feedback of both of the flow and structural oscillation outperformed the open-loop approach and led to suppression of the Kármán vortex street. They observed reduction in the structural vibration, vortex shedding strength, and drag coefficient by 82%, 65%, and 35%, respectively. In a numerical investigation, Kim and Choi (2005) applied a distributed forcing (suction and blowing) to the flow over a circular cylinder (ReD =47-3,900) in order to reduce its drag. The forcing function, fixed in time, was varied in the spanwise direction, both in-phase and out-of-phase, from slots located at the upper and lower surfaces of the cylinder. The sinusoidal blowing/suction profile from each slot was defined as follows: φ1 (z) = φ2 (z) = φ0 sin(2πz/λ)

(1.1)

where φ1 and φ2 are the radial velocities at the upper and lower slots, respectively; z is the spanwise direction; φ0 is the forcing amplitude; and λ is the forcing wavelength. They observed that for all Reynolds numbers larger than 46, the distributed forcing suppresses vortex shedding, resulting in a decrease in the mean drag and the fluctuating forces. They observed in-phase distributed forcing to be more effective in reducing the drag than outof-phase forcing. They also explained the drag-reduction mechanism in terms of threedimensional vortical evolution through phase mismatch along the spanwise direction. A novel concept for active flow control is the use of micro-electro-mechanical systems

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Introduction

14

(MEMS) (Ho and Tai, 1998). MEMS technology plays an important role in producing arrays of large number of sensors and actuators at a reasonable cost. Several low-power MEMS, such as microflaps, surface heating elements, and synthetic jets, have been used for flow control. Synthetic jets (Glazer and Amitay, 2002) are active control devices with zero net mass flux and do not require internal fluid lines. Actuators based on synthetic-jet techniques have been shown to be very effective for several applications of aerodynamic flow control. Amitay et al. (1997) implemented experimentally synthetic jets on circular cylinders at Reynolds numbers of up to 1.3 × 105 . They found that the interaction between the jet and the main flow induces a local separation bubble, which acts as a virtual surface that displaces the streamlines outside the undisturbed boundary layer. Another innovative idea of nonzero time-mean mass-flow-rate synthetic jets, known as hybrid synthetic jets, have recently been introduced and seem to offer potential applications. Tesar (2007) discussed the basic configuration and topology of components in jet-type actuators used to generate hybrid-synthetic jets. This kind of fluidic actuators can effectively be used in a flow control problem where more suction is required rather than blowing and vice-versa. Choi et al. (2008) reviewed various control methods for flow over a bluff body and reviewed major achievements in this field. Some of the recent advances include three-dimensional forcing, active feedback control, control based on local and global instability, and control with a synthetic jet. They also classified the controls as boundary-layer controls and direct-wake modifications. In the boundary-layer control, the separation point is delayed by enhancing the near-wall streamwise momentum close to the separation point. On the other hand, the direct-wake control methods include splitter plates located at the base surface and base bleed, which delay the interaction of top and bottom shear layers in the wake. They also defined some quantitive measures of the control efficiency and discussed them for blowing/suction control.

1.2.4

Flow Control using POD-based Reduced-Order Model

POD has successfully been applied to many engineering and scientific applications, including low-dimensional dynamics modeling (Berkooz et al., 1993; Deane and Mavriplis, 1994;

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Introduction

15

Holmes et al., 1996; Ma and Karniadakis, 2002; Noack et al., 2003), image processing (Holmes et al., 1996), and pattern recognition (Sirovich and Kirby, 1987). POD was introduced in the field of turbulence by Lumley (Bakewell and Lumley, 1967) to identify coherent structures in the flow. Later, Sirovich (Sirovich, 1987) introduced the snapshot method to study the dynamics of some turbulent flows. In this method, a set of instantaneous flow field solutions, or snapshots, is obtained from an experimental data or a numerical simulation. This allows reduction of large data sets obtained from computational fluid dynamics (CFD) or particle image velocimetry (PIV), while still preserving the dominant features of the flow represented by POD eigenfunctions or modes. This technique has been successfully applied to grooved channel flows and laminar cylinder wakes (Deane et al., 1991; Ma and Karniadakis, 2002). Gillies (1998) represented the flow field by a finite set of coherent structures or modes, thereby allowing an efficient design of a closed-loop control algorithm. These modes were extracted by proper orthogonal decomposition (POD) from two dimensional numerical simulation data at Re=100. A neural network was used to provide an empirical prediction of the modal response of the wake to external control forcing. Graham et al. (1999a,b) developed a reduced-order model for the flow past a circular cylinder at Re = 100 using numerical simulations and the control action was achieved by cylinder rotation. They derived optimal control formulations for the penalty method and the control-function method using low-order models. In the control-function method, they added a spatial control mode alongwith a time-dependent input control in the Galerkin expansion of the velocity field. They found that the level of wake unsteadiness can be reduced, even when the low-order model is developed on the basis of limited flow field information. Tang and Aubry (2000) proposed a control technique inspired by the stability analysis of a low-dimensional vortex model. The instability of the twin vortices was controlled by introducing an additional pair of point vortices. The circulation of these controlling vortices was 10% or less than the circulation of the twin vortices, depending on the Reynolds number. This control had the effect of altering/suppressing vortex shedding by generating a reversed Kármán vortex street. They implemented their control scheme using a two-dimensional numerical simulation in the range of Re=100-1,000.

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Introduction

16

Singh et al. (2001) used two finite-dimensional reduced-order models for control system design. These models were based on POD of the two-dimensional incompressible, unsteady wake flow behind a circular cylinder at Re=100. Linear optimal control theory was used to develop the feedback gains. Their simulation results showed that, in a closed-loop system, asymptotic regulation of the mode amplitudes to the desired equilibrium state could be accomplished by appropriately rotating the cylinder. Also, they found that their linear controller was capable of suppressing large unsteady perturbations, despite the presence of nonlinearity in the flow dynamics of the two models. Cohen et al. (2004b) obtained flow field data for a cylinder at a Reynolds number around 100 from a Navier-Stokes simulation and a particle image velocimetry (PIV). A lowdimensional POD model was developed and linear stochastic estimation was used to map sensor readings of the velocity field on the POD mode coefficients. Their results showed that a five-sensor configuration can keep the root mean-square estimation error for the amplitudes of the first two modes to within 4% for the simulation data and with 10% for the PIV data. This procedure was developed to determine the number of sensors and their appropriate placement for feedback control suppression of the wake instability. Seigal et al. (2006) analyzed numerically the effect of feedback flow control on the wake of a circular cylinder at Re=100. They based their scheme on a low-dimensional model based on POD. They used linear proportional and differential feedback of the estimated first POD mode. They placed sensors in the wake and actuated the cylinder using its displacement normal to the flow. They reported a 15% reduction in the drag force and a 90% in the unsteady lift force. To enable real-time control of synthetic-jet actuated flows, Rediniotis et al. (2002) derived a low-dimensional model for the synthetic jet through a Galerkin projection of the NavierStokes equations on a POD basis. They discussed controllability of the reduced-order models. They also showed that certain choices of controls, observations, and states lead to well conceived controllable systems. It should also be noted that, in some cases of POD modes, while optimal in terms of energy representation of a data set, they may not be the best choice for reduced-order modeling (Kim and Bewley, 2007), especially at higher Reynolds numbers. In POD based models, low-

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Introduction

17

energy modes are generally truncated, therefore they do not account for observability and controlablity of these modes. Rowley (2005) demonstrated such cases in which POD based models are unable to capture the dynamics of the system. He proposed a balanced POD scheme in which the POD snapshot method was used to compute empirical Grammians.

1.3

Objectives

The objective of this research is to develop a technique to control the flow over a cylinder and hence reduce its lift and drag forces. The control strategy is implemented using a reducedorder model developed by projecting the Navier-Stokes equations onto the POD modes. It reduces the problem size from an infinite-dimensional space to a finite number of modes. These modes are calculated from the flow field data recorded from the numerical simulation of the flow past a cylinder. The objective is achieved in three phases and are enumerated as follows: 1. We developed a parallel three-dimensional CFD solver to simulate the flow past a circular cylinder. The solver is capable of simulating the flow over more general structures, such as elliptic cylinders and airfoils. Moreover, additional features, including moving boundaries, turbulence model, incoming shear flows, broaden application of the solver to other engineering and research problems. Based on the domain decomposition technique, parallel computing allows us to simulate complex problems efficiently. 2. We developed a reduced-order model for the velocity and pressure fields over a circular cylinder to reduce the problem size from a large-dimensional space to a low-dimensional space while capturing the flow physics. 3. We applied a control strategy using the reduced-order model through fluidic actuation on the cylinder surface to suppress vortex shedding. Full-state feedback control is designed using the modified reduced-order model with a control function. This closedloop control represents a formidable alternative to open-loop strategies and can be used for designing controllers to reduce the fluctuating forces.

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1.4

Introduction

18

Contributions

The key contributions are: 1. We developed a parallel three-dimensional CFD solver to simulate the flow past cylinders. Curvilinear coordinates are used to allow the code to solve for any arbitrary closed shape. Additional features are incorporated in the algorithm, which include turbulence modeling, moving boundaries, and modification in the surface boundary conditions. The turbulence model allows the code to simulate high Reynolds number flows. Motion of the structure can be simulated to study fluid-structure interaction. Modification in the boundary conditions can simulate blowing/suction on the surface or rotation of the structure and hence can be used to control the flow. Modifying the inflow boundary conditions can simulate shear flows in the spanwise direction. 2. We developed a low-dimensional model for the pressure field using the pressure POD modes. This reduced-order model predicts the lift and drag forces on the cylinder surface. 3. We used long-time integration to investigate whether the low-dimensional model can converge on the system physical limit cycle and a shooting method is applied to home on this limit cycle. 4. Using surface actuation as a control function, we designed a full-state feedback controller using the linear-time invariant (LTI) part of the reduced-order model. This closed-loop design can successfully be used to meet the requirements for controlling vortex shedding at design conditions. Moreover, it can also help in constructing openloop controllers in off-design conditions.

1.5

Summary and Outline of Dissertation

The Dissertation can be divided into three main parts. In the first part, we discuss the numerical methodology and parallel computing developed to simulate the flow and analyze

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19

Introduction

VIV problems. In the second part, we develop reduced-order models using both a selfexcited oscillator based approach and a POD-Galerkin expansion approach. In the last part, we modify the reduced-order model to include fluid actuation and apply full-state feedback control to suppress vortex shedding. The summary is illustrated in Figure 1.3. Next, we provide an overview of each part and present the outline of the Dissertation.

AFC

lel ra l ns Pa latio u Sim

Fu l Fee l-Stat dba e ck

VIV

Fluidic Actuation

ROM

Figure 1.3: Illustration of the three main parts in the Dissertation: Vortex-Induced Vibration (VIV), Reduced-Order Modeling (ROM), and Active Feedback Control (AFC).

1.5.1

Part I - Parallel Simulations

To implement and validate the control scheme, we develop an unsteady full 3D CFD code to simulate the flow field past a circular cylinder. The simulation captures the turbulence structures and spanwise interaction of the flow field. The code is parallelized by distributing the computing grid among various processors to reduce the workload per processor, thereby avoiding the immense requirement in terms of computational cost of serial computations. A two-dimensional (2-D) domain decomposition method is employed using message passing interface (MPI) libraries. The developed CFD tool is validated by comparing its results with existing experiments and other numerical studies. Moreover, the numerical results are verified by performing

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Introduction

20

grid and domain independent studies. The parallel performance of the solver is analyzed by computing its speed-up and efficiency for 2-D and 3-D simulations. A scalability study is also performed to ensure that the current solver scales well with problem size.

1.5.2

Part II - Reduced-Order Modeling

Using the CFD code, we perform direct simulations at various Reynolds numbers. The flow field is then ensembled in a matrix form at discrete time steps. Velocity and pressure POD modes are computed from this large data set. The dominant of these POD modes are used in a Galerkin procedure to project the Navier-Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finite-dimensional nonlinear dynamical system in time. The pressure POD modes are used in a similar manner for the pressure-Poisson equation to develop a reduced-order model for the pressure field. The pressure thus obtained is integrated over the cylinder surface to obtain the lift and drag forces acting on the structure.

1.5.3

Part III - Active Flow Control

The flow-control mechanism of interest in this study is through suction on the cylinder surface, which acts as an actuator for controlling the flow. The suction actuators are numerically modeled by imposing a velocity normal to the cylinder surface. Control of the flow past a cylinder is achieved using a reduced-order model approach. In this approach, the reduced-order model is modified to include an appropriate control function that incorporates suction/blowing on the cylinder surface. Full-state feedback control is applied to the modified model to reduce the fluctuations of the hydrodynamic forces.

1.5.4

Outline of the Dissertation

1. In Chapter 1, we introduce the phenomenon of vortex shedding and its importance in fluid-structure interactions. We present a detailed literature review of VIV, reducedorder modeling of VIV, parallel computing, and various flow control techniques to

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Introduction

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suppress vortex shedding. We also discuss the objective and key contributions of this research. 2. In Chapter 2, we define the computational domain, governing equations, and coordinate transformation for the current problem. We also discuss in detail the temporal and spatial discretization, the solution algorithm, and the boundary conditions. Then, we explain in detail implementation of parallel processing to the numerical methodology. It includes domain decomposition of the grid and the message passing routines among various processors. 3. In Chapter 3, we validate and verify the algorithm. We integrate the pressure over the cylinder surface to obtain the lift and drag coefficients and compute the dominant frequencies in their spectra. The numerical results obtained from the simulations are compared with existing experimental and numerical studies. We also compare the vortical structures in the wake. We perform domain and grid independence studies to establish accuracy of the results. We then analyze the parallel performance of the solver by simulating the flow with different number of processors and computing their speedup and efficiency. We also increase the grid size alongwith the number of processors to address the scalability aspect of the code. 4. In Chapter 4, we discuss additional features and applications of the parallel CFD code. We simulate the flow past elliptic cylinders with varying eccentricities. We also simulate 3-D shear flows where the incoming flow varies along the span of the cylinder. The solver is capable of simulating moving boundaries and uses the accelerated reference frame (ARF) method to simulate a vibrating cylinder. Fluidic actuation can also be simulated by appropriately modifying the surface boundary conditions. For high Reynolds number simulations, the Spalart-Allmaras turbulence model is used to compute the eddy viscosity. Each feature is presented with a test case and the results are validated. 5. In Chapter 5, we use the CFD solver to simulate the flow past elliptic cylinders, with varying eccentricities, and calculate the lift and drag coefficients for each case. Using

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Introduction

22

these numerical results as an input, we extend the previously developed van der PolDuffing oscillator model of lift and drag forces on a circular cylinder to elliptic cylinders. We provide an overview of the existing oscillator model and compute the parameters of the modified model for elliptic cylinders. 6. In Chapter 6, we present the procedure for data ensembling from the flow field. The data is decomposed into mean and fluctuating components. The fluctuations are expressed in a Galerkin expansion and the POD modes are computed. The velocity POD modes are computed for different Reynolds numbers. We also compare the eigenvalues which correspond to the energy contained in the corresponding mode. 7. In Chapter 7, we develop a reduced-order model of the velocity field. We project the Navier-Stokes equations onto the velocity POD modes to obtain a system of ordinarydifferential equations (ODE) in time. We solve the dynamical system to compute the velocity temporal coefficients. The POD simulations are compared with the CFD data as well as other POD models. Long-time integration of the reduced-order model is used to investigate whether it can home on the physical limit cycle. A shooting method is proposed to home on this limit cycle. 8. In Chapter 8, we develop a reduced-order model for the pressure field using the information from the reduced-order model of the velocity field. The pressure is expressed in a Galerkin procedure and the pressure-Poisson equation is projected onto the pressure POD modes and integrated over the domain to obtain an algebraic expression for the pressure time coefficients. The modeled pressure is then integrated over the surface to compute the lift and drag forces. 9. In Chapter 9, we apply a flow control strategy using the reduced-order model. We use fluidic actuators simulating suction on the cylinder surface as a means of controlling the flow. The model is modified to incorporate a control function mode, constructed from the actuated flow field. We follow a similar Galerkin projection of the Navier-stokes equations onto the POD modes. However, in the actuated case, the model is complex and contains extra terms originating from the input variable. Then, full-state feedback

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Introduction

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control is applied to the modified reduced-order model to suppress vortex shedding in the flow. 10. In Chapter 10, we summarize and conclude our results. The research performed in our work includes CFD, parallel computing, reduced-order modeling, and control theory, which provide the necessary building blocks and great potential for application in many engineering problems. We also present recommendation for future research.

Part I Parallel Simulations

24

Chapter 2 Numerical Methodology 2.1

Introduction

In this Chapter, we present the governing equations which describe the motion of an incompressible flow. The equations are derived from the fundamental principles of conservation of mass and momentum. We develop a parallel 3-D CFD code to compute the flow over a cylinder. The unsteady 3-D Navier-Stokes equations are generally considered to govern turbulent flows in the continuum regime. Resolving a turbulent flow by direct numerical simulation (DNS) requires resolution of all of the relevant length scales from the smallest eddies to scales on the order of the physical dimensions of the flow problem. The computation needs to be three-dimensional and the time steps needs to resolve small scale motions even if the time-mean aspects of the flow are two-dimensional and steady. The objective is to employ an efficient and accurate discretization and a solution method for unsteady flows in complex geometries. Moreover, the governing equations are solved on a parallel platform using the domain decomposition approach on a distributed-memory platform. Message passing interface (MPI) libraries are employed to partition the grid and to communicate among neighboring processors.

25

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2.2

26

Numerical Methodology

Governing Equations and Coordinate Transformation

The Navier-Stokes and continuity equations are the governing equations for the present problem. For incompressible flow, they can be represented as follows: Continuity Equation ∂uj =0 ∂xj

(2.1)

∂ 1 ∂p ∂ 2 ui ∂ui + (uj ui ) = − +µ , ∂t ∂xj ρ ∂xi ∂xj ∂xi

(2.2)

Momentum Equation

where i,j=1,2,3; the ui represent the Cartesian velocity components (u,v,w); p is the pressure; ρ is the fluid density; and µ is the fluid viscosity. Equations (2.1) and (2.2) are nondimensionalized using the diameter (D) of the cylinder as the length scale and the freestream velocity (U∞ ) as the velocity scale. Thus, the Reynolds number is given by ReD =

DU∞ , ν

where ν = µρ .

The other important geometric quantity, spanwise length of the body, is nondimensionalized using the cylinder diameter D and is denoted by Lz . In most of engineering and industrial applications, the fluid flows occur in the domains of complex geometry. To broaden the spectrum of applications, we employ body fitted curvilinear coordinates. Thus, the irregular physical domain is transformed into a regular computational space as shown in Figure 2.1. This approach provides two main advantages: 1. The boundary of the physical domain can be accurately represented. 2. The discretization of the governing equations and the application of boundary conditions are straightforward. In the present formulation, we transform the independent variables to the computational space keeping the Cartesian velocity components as the dependent variables. This allows the governing equations to be written in a strong-conservation-law form. We employ curvilinear coordinates (ξ, η, ζ) in an Eulerian reference frame, a planar view of which is shown in Figure 2.2. Here, X and Y indicate the Cartesian coordinates; ξ and

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Figure 2.1: Schematic of coordinate transformation from a physical domain to a computational domain. η represent a curvilinear coordinate system; and Lx and Ly are the major- and minor-axes, respectively. The thickness ratio of the ellipse is given by τ =

Ly . Lx

This choice of coordinate

system allows us to simulate the flow past any arbitrary closed shape in a body conformal “O” type grid, such as circular and elliptic cylinders and airfoils. For the specific case of a

Ș=ʌ/2

Y Ș Ly

U

ȟ Ș=0

Ș=ʌ

Lx

X

Ș=2ʌ

Ș=3ʌ/2

Figure 2.2: Geometry in curvilinear coordinates. circular cylinder, it is a polar grid, as shown in Figure 2.3. The grid is clustered near the body to resolve the flow adequately and capture the flow physics. The left half of the domain is the inflow, whereas the right half is the outflow. For convenience of writing the equations, we use tensor notation such that xi = (x, y, z) and ξi = (ξ, η, ζ), where i=1,2,3. Equations (2.1) and (2.2) are transformed into curvilinear

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Figure 2.3: A 2-D layout of an “O” grid over a circular cylinder, showing the inflow and outflow directions. coordinates in a strong conservative form as follows: ∂Um =0 ∂ξm ∂(J −1 ui ) ∂Fim + = 0, ∂t ∂ξm

(2.3) (2.4)

where the flux is defined as Fim = Um ui + J −1

1 ∂ui ∂ξm p− Gmn ∂xi ReD ∂ξn

(2.5)

Here, J −1 is the inverse of the Jacobian determinant or the volume of the cell; Um is the volume flux (contravariant velocity multiplied by J −1 ) normal to the surface of constant ξm ; and Gmn is the “mesh skewness tensor”. These quantities are defined as follows: ∂ξm Um = J −1 uj ∂xj ∂xi −1 J = det ∂ξj ∂ξm ∂ξn Gmn = J −1 ∂xj ∂xj

(2.6) (2.7) (2.8)

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2.3 2.3.1


29

Spatial and Temporal Discretization Spatial Discretization

A nonstaggered grid layout (Zang, 1993) is employed to solve the transformed Navier-Stokes equations. The Cartesian velocity components (u, v, w) and pressure (p) are defined at the center of the control volume in the computational space and the volume fluxes (U, V, W ) are defined at the mid points of its corresponding faces. All of the spatial derivatives are approximated with second-order accurate central differences except for the convective terms. Using the same central differencing for the convection terms may lead to spurious oscillations in the coarser regions of the grid, thereby leading to erroneous results. Control of the oscillations can be obtained either by applying a staggered formulation or by improving the central difference schemes by adding numerical viscosity. The former option is not applicable in the present case. However, the latter can be achieved by adding a nonlinear artificial viscosity term or using an upwinding formulation. To avoid this over-or-under shoots, various schemes have been suggested, such as the Godunov method (Gudonov, 1959), the total variation diminishing method (TVD) (Harten, 1983), the flux-corrected transport method (FCT) (Book et al., 1975), quadratic upwinding interpolation for convective kinematics (QUICK) (Leonard, 1979), the monotonic upstream scheme for conservation laws (MUSCL) method (Leer, 1979), and the essentially nonoscillatory (ENO) method (Harten and Osher, 1987). In the present formulation, we discretize the convective terms using a variation of QUICK; that is, we calculate the face values of the velocity variables (ui ) from the nodal values using a quadratic upwinding interpolation. The upwinding of QUICK is carried out by computing the positive and negative volume fluxes m| m| ) and ( Um −|U ), respectively, and using the generic stencil. ( Um +|U 2 2

2.3.2

Temporal Discretization

A semi-implicit scheme is employed to advance the solution in time. The diagonal viscous terms are advanced implicitly using the second-order accurate Crank-Nicolson method, whereas all of the other terms are advanced using the second-order accurate Adams-Bashforth

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method. The Adams-Bashforth scheme was chosen because of its computational efficiency when coupled with the fractional step method. The discretized equations are ∂Um =0 ∂ξm 3 1 uk+1 − uki = (Cik + DE (uki )) − (Cik−1 + DE (uik−1 )) J −1 i ∆t 2 2 1 + Ri (pk+1 ) + (DI (uk+1 + uki )), i 2 where the

∂ ∂ξm

(2.9)

(2.10)

represent the discrete finite-difference operators in the computational space;

the superscripts represent the time step; the Ci represent the convective terms; the Ri are the discrete operators for the pressure gradient terms; DI is the discrete operator representing the implicitly treated diagonal viscous terms; and DE is the discrete operator for the explicitly treated off-diagonal viscous terms. Mathematically, these terms are defined as follows: ∂ (Um ui ) ∂ξm ∂ξm ∂ (J −1 ) Ri = − ∂ξm ∂xi ∂ ∂ DI = (νGmn ) for m = n ∂ξm ∂ξn ∂ ∂ DE = (νGmn ) for m 6= n ∂ξm ∂ξn Ci = −

(2.11) (2.12) (2.13) (2.14)

It is important to note that, due to the orthogonality property for the specific case of a cylinder, the cross terms for the mesh skewness tensor Gmn are zero; that is, when m 6= n. Therefore, the terms in Equation (2.14) are identically zero and the stencil for the problem is a seven-point stencil.

2.4

Solution Algorithm

Commonly used numerical methods to solve the incompressible Navier-Stokes equations include the artificial compressibility method (Chorin, 1968) and the fractional-step or projection method (Chorin, 1967). The key feature of the artificial compressibility method is that it allows the use of efficient compressible flow algorithms for computing incompressible flows. However, for a time-dependent problem, the system of equations may become highly

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stiff (Steger and Kutler, 1977). On the other hand, a fractional-step method relies on the idea of operator splitting to uncouple the pressure computation from that of the velocity field, it is efficient for solving the incompressible Navier-Stokes equations. The theory for this method is quite well developed (Chorin, 1997) and it has been successfully used in a variety of flow configurations (Mittal and Balachandar, 1995; Orszag and Kells, 1985; Street and Hussaini, 1991). The fractional-step method splits the momentum equation into (a) an advection-diffusion equation - momentum equation solved without the pressure term, (b) a pressure-Poisson equation - constructed by implicit coupling between the continuity equation and the pressure in the momentum equation, thus satisfying the constraint of mass conservation.

2.4.1

Fractional-Step Method

In the present formulation, a fractional-step method is applied to advance the solution in time. Numerical solutions of the incompressible Navier-Stokes equations have been obtained in Cartesian (Kim and Moin, 1985) as well as in curvilinear coordinate systems (Zang et al., 1994). In the present study, the governing equations are solved using a methodology similar to that employed by Zang (1993). However, the algorithm is extended to parallel computing platforms and the 2-D domain decomposition technique is employed to distribute the problem among different processors. Implementation of the parallel processing is discussed in detail in Section 2.5. This time-splitting method leads to the three-step predictor-corrector solution. In the first step, a predicted velocity field is computed, which is not constrained by the continuity equation. Secondly, the pressure field is determined by solving the pressure-Poisson equation (details later). In the final step, the predicted velocity field is corrected with the pressure to obtain the true velocity at the end of the time step. The predictor equation (advectiondiffusion equation) is defined as follows:

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∆t 3 k ∆t ∗ k I − −1 DI (ui − ui ) = −1 (Ci + DE (uki )) 2J J 2 1 k−1 k−1 k − (Ci + DE (ui )) + DI (ui ) 2

in Ω

(2.15)

where Ω refers to the interior of the computational domain and I is the identity matrix. The variable u∗i is called the “intermediate velocity”, which is not constrained by continuity. Unlike the solution procedure of approximate factorization adopted by Zang (1993) to solve Equation (2.15), the line successive over relaxation (LSOR) scheme is employed to solve the advection-diffusion equation with a relaxation factor of 1.5. The corrector equation is represented as follows: ∆t k+1 ∗ k+1 ui − ui = −1 Ri (φ ) in Ω J ∇ · uk+1 = 0 in Ω i where the function φ is related to the pressure p by Ri (φ) ∆t −1 DI Ri (p) = J − 2 J −1

2.4.2

(2.16) (2.17)

(2.18)

Pressure-Poisson Equation

The correction step in Equation (2.16) can be rewritten in terms of volume flux as follows: k+1 ∆t k+1 ∗ mn ∂φ Um = Um − −1 G (2.19) J ∂ξn ∗ m where Um = J −1 ∂ξ (u∗j )f ace is known as the intermediate volume flux. In the present work, ∂xj

QUICK is used for interpolating the u∗j on the respective cell faces. Substituting Equation (2.19) into Equation (2.9) yields the following pressure-Poisson equation for φn+1 : k+1 ∗ 1 Um ∂ mn ∂φ G = ∂m ∂ξn ∆t ∂ξm

(2.20)

It can be noted that, in the pressure correction step, the coefficients consist only of the mesh skewness tensor Gmn . This elliptic equation is solved using the bi-conjugate gradient stabilized method (BiCGStab).

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2.4.3


33

Boundary Conditions

Application of suitable and well-posed boundary conditions is crucial for any simulation. Boundary conditions for the intermediate velocity fields in fractional-step (time-splitting) methods are generally a source of ambiguity. At each complete time-step, only the boundary conditions for the velocity field are given and those of the intermediate velocity are unknown. Appropriate boundary conditions are applied on different sections of the domain. For the inflow boundary condition, we use a Dirichlet boundary condition. This follows the reasonable assumption that, far upstream of the body, the displacement effect of the body is negligible and thus the flow is sufficiently close to potential flow. This inflow boundary condition has been used in most of the simulations of flows over bodies, varying in shape from streamlined to bluff, except for the cases when the effects of incoming turbulence are being analyzed. For the outflow boundary condition, we use a Neumann boundary condition. Noslip and no-penetration boundary conditions are applied on the cylinder surface except for the cases where suction/blowing is applied on the cylinder surface. An appropriate modification of the boundary condition to simulate suction/blowing is discussed in Chapter 4. Moreover, periodic boundary conditions are applied in the η and ζ directions. The boundary conditions for the intermediate velocity are u∗i = uki −

∂u∗i =0 ∂ξ

∆t Ri (φk ) J −1

on ∂ΩSb

(2.21)

on ∂ΩSo

(2.22)

where ∂ΩSb and ∂ΩSo refer to the body surface and outer boundary, respectively. For the pressure, we apply Neumann boundary conditions for solving the pressure-Poisson equation.

2.4.4

Sequence of Solution Procedure

The sequence followed to solve the governing equations is as follows: 1. Generate the grid over the cylinder. 2. Solve the advection-diffusion equation for the intermediate velocity u∗i at the cell centers.

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34

3. Interpolate the u∗i onto the cell faces to compute the contravariant velocity Ui∗ and compute the RHS of Equation (2.20). 4. Solve Equation (2.20) to compute the φk+1 . 5. Compute the uk+1 from Equation (2.16) and the Uik+1 from Equation (2.19) to complete i one time step. 6. Repeat the procedure (step 2-5) to march in time. Implicit treatment of the viscous terms removes the viscous stability limit; however, the stability of the numerical scheme is restricted by the CFL condition. The local CFL number is defined as |u1 | |u2 | |u3 | + + ∆t ∆x ∆y ∆z ∆t = (|U1 | + |U2 | + |U3 |) −1 J

CF L =

(2.23)

where ∆x, ∆y, and ∆z are the grid spacing in Cartesian coordinates. According to the stability criterion of the numerical scheme, the maximum CFL number within the computational domain should be less than unity.

2.5 2.5.1

Parallel Implementation Introduction

A multi-processor system is a computer with more than one processor. A common way to classify parallel machines is to distinguish them by the manner processors access the system’s main memory. From an architectural point of view, two categories of parallel computers are modeled. These physical models are distinguished by having either a shared common memory or unshared distributed memories. A shared-memory architecture accomplishes interprocessor coordination by providing a global shared memory, which each processor can address as shown in Figure 2.4(a). All memory locations can be accessed through usual load and store operations. Access to a remote location results in a copy of the appropriate

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35

cache line in the processor’s cache. This allows programming of the system to be simple and easy. However, shared-memory systems can only be scaled to moderate number of processors, typically around 128. Distributed-memory architecture consist of multiple computers, often called nodes. They are interconnected processing nodes, each of which consists of an autonomous processor and its local memory with a processor-to-processor interconnection network as shown in Figure 2.4(b). Nodes share data by explicitly passing messages through this interconnection network. The message-passing network provides point-to-point static connections among the nodes. All local memories are private and are accessible only by local processors. Internode communication is performed by passing messages through the static connection network. The major advantage of distributed-memory platform is their ability to scale to a very large number of nodes. In comparison to the shared-memory systems, distributed-memory systems are harder to program. It is the challenge for the program developer to assign data to each processor such that most of the data accessed during the computation is already in the local memory of that processor. Shared-memory computers do not have some of the problems encountered by distributed-memory architectures, such as message sending latency as data is queued and forwarded by intermediate nodes. However, other problems, such as data access synchronization and cache coherency, must be solved.

2.5.2

Message-Passing Interface (MPI)

The current code uses MPI routines for message passing among different nodes. The messagepassing interface (MPI) standard is a library that allows the user to do problems in parallel using message passing for communicating between processes (processors). It is not a language (like Fortran 90, C++ or HPF ) or even an extension to a language. Instead, it is a library, which the local, standard, serial compiler (f77, f90, cc etc) uses. It includes routines for pointto-point communication, one-sided communication, collective communication, and parallel IO. MPI consists of more than 300 functions, however, most of the programs developed using MPI can be efficiently developed using less than 20 functions. There are some fundamental functions that are required to initialize and set up the parallel computing platform, which are defined as follows:

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P0

P1

PN

C0

C1

CN

36

Shared-Memory (a)

P0

P1

PN

C0

C1

CN

M0

M1

Mn

Message-passing interconnection

(b)

Figure 2.4: Schematic of (a) shared-memory and (b) distributed-memory architecture. Note: Processor (P), Cache (C), Memory (M)

• MPI INIT initializes the MPI library. It is called at the beginning of a parallel operation before any other MPI routines are executed. • MPI COMM SIZE determines the number of processors executing the parallel program. Typically, the user specifies the number of processors at the beginning of the program.

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• MPI COMM RANK returns the unique processor identifier. During the execution and message passing of the parallel program, it serves as the “name” of the processor. • MPI FINALIZE frees any resources used by the library and is called at the end of the program. The message passing routines depend on the concept of a communicator to transfer data from one node to the other. A communicator consists of a process group and a communication context. MPI COMM RANK assigns numbers to the processors from 0 to Np − 1, where Np is the total number of processors in that group. The communication context of the communicator identifies the message through an integer known as tag, which is matched during the corresponding receive operation. This aspect of message passing helps in building parallel libraries and does not allow interference of the messages outside the communicator. Typically, the default communicator is MPI COMM WORLD, which includes all processors in the application. The processor group topology is a Cartesian structure. The MPI routines used to define the communicators in the code are as follows: • MPI SENDRECV passes message within communicators. The send-receive operations combine in one call sending of a message to one destination and receiving of another message from another process. • MPI CART CREATE returns a handle to a new communicator to which the Cartesian topology information is attached. • MPI CART GET returns the Cartesian topology information that was associated with a communicator by MPI-CART-CREATE. • MPI CART SHIFT is called for a Cartesian process group. It provides the calling process with the above identifiers, which then can be utilized during message sending or receiving operations. The user specifies the coordinate direction and the size of the step (positive or negative).

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• MPI CART SUB is used to partition the communicator group into subgroups that form lower-dimensional Cartesian subgrids and to build for each subgroup a communicator with the associated subgrid Cartesian topology. The initial step in designing a parallel program is to break down the “big” problem into discrete “small” problems so that it can be distributed to multiple tasks. This step is known as decomposition or partitioning. There are two fundamental ways to partition computational work among parallel tasks, functional decomposition and domain decomposition. In the former approach, the focus is on the computation that is to be performed rather than on the data manipulated by the computation. The problem is decomposed according to the work that must be done. Each task then performs a portion of the overall work. Solving the pressure-Poisson equation on the whole grid using multiple processors would be an example of functional decomposition. In the domain decomposition method, data associated with a problem is decomposed and each processor then works on a portion of the data. The domain decomposition approach is used in the present code and its implementation is discussed in the next section.

2.5.3

Implementation Technique

To communicate data among processors, we define one communicator for the 1-D domain decomposition where the grid is not partitioned in the ζ-direction, as shown in Figure 2.6(a). The processor group, known as COMM-1D, involves all of the processors distributed in the η-direction. Using MPI routines, each processor knows its neighborhood (say, the East and West directions). In the case shown in Figure 2.6(b), where the grid is decomposed into the η- and ζ- directions (i.e., 2-D domain decomposition), we define three communicators for message passing. The first processor group, known as COMM-2D, consists of all the processors where each processor knows its neighbors in the East, West, North, and South directions, as shown in Figure 2.5(a). This allows passing the primitive variables to the neighboring processors, as and when required. This group is also responsible for MPI ALL REDUCE operations. For example, using any iterative solver, we require the global tolerance to satisfy certain tolerance

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criteria. We use MPI ALL REDUCE (...,MPI MAX,...), where each processor identifies the local maximum tolerance and based on the global maximum tolerance continues the iterative process till it satisfies the tolerance criteria. The second group COMM-1DY groups the processors in the η-direction as in the case of 1-D domain decomposition; that is East and West. It allows message passing where the communication is required only in the η-direction. For example, while calculating sectional lift and drag forces on the cylinder, each processor integrates the pressure in its domain and the result is summed using MPI ALL REDUCE (..., MPI SUM,...). In the third group COMM-1DZ, the processors in the ζ-direction are grouped together for explicit communication in the ζ-direction; that is North and South. As an example, we require this communication for computing quantities averaged over the length (span) of the cylinder. In Figure 2.5(b), we show a communicator topology comprising 24 processors in a Cartesian structure with 8 processors in the η-direction and 3 processors in the ζ-direction. Periodicity in the processor layout is defined using MPI CART CREATE. An “O” type grid is employed to simulate the flow over a cylinder. For the specific case of a circular cylinder, the generalized coordinates (ξ, η, ζ) can be represented by polar coordinates (r, θ, z). However, to maintain curvilinear coordinates, we use the former notation. To implement a message-passing interface (MPI), we use a two-dimensional domain decomposition technique such that each processor gets a “slice” of the grid, as shown in Figure 2.6(a). In this figure, a two-dimensional view of the grid is shown and is divided among 8 processors. The spanwise dimension is also divided into domains such that each “slice” has some “depth” as depicted in Figure 2.6(b). For example, if a 128 × 128 × 96 grid in (r, θ, z) is decomposed into 8 × 3 processors in the θ and z-directions, then each processor will have a load of 128 × 16 × 32 grid points. This technique allows for a simple way to implement boundary conditions and keeps an equal load distribution among the processors. Moreover, the data points are exchanged only in two directions, the η- and ζ-directions, thereby reducing the inter-processor communication cost.

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ȗ PN Pw

Pi

PE

PS

Ș (a)

ȗ 16

17

18

19

20

21

22

23

8

9

10

11

12

13

14

15

0

1

2

3

4

5

6

7

Ș (b)

Figure 2.5: (a) Processor group topology in a Cartesian structure and (b) communicator comprising 24 processors on 8 × 3 processor platform.

2.6

Summary

In this Chapter, we presented the numerical methodology of the parallel CFD solver developed during the research to analyse VIV. The governing equations are written in curvilinear coordinates and solved using a predictor-corrector method known as the fractional-step method. In the first step, the advection-diffusion equation is solved without the pressure term to compute the intermediate velocity field. The pressure-Poisson equation is then solved to compute the pressure field and the velocity is updated in the correction step. The time stepping is restricted by the viscous stability criterion of the CFL number, which should be

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kept less than unity. To reduce the computation time, we employed the domain decomposition technique to distribute the computational domain equally among various processors, each solving a portion of the grid. Communication among different processors is performed using MPI libraries. We also discussed the MPI routines used in the grid distribution and other computing operations during the simulation.

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42


(a) Y

Z

X

z-partition =1 z-partition =2 z-partition =3

(b)

Figure 2.6: (a) A 2-D layout of the 128 × 128 × 96 “O” grid distributed among 8 processors in the η-direction. The grid is plotted only for the region of processor “1”. (b) A 3-D layout of the complete domain and the grid is plotted for only one processor, indicating the load per processor in a 24 (8×3) processor platform. The grid is plotted only for the region of processor “1”. (Note: every 4th grid point is plotted for clarity.)

Chapter 3 Validation, Verification, and Parallel Performance Starting from a steady wake, the flow over bluff bodies undergoes a sequence of transitions as the Reynolds number is increased. In the case of a circular cylinder, the wake transitions from a steady to an unsteady state at a Reynolds number (based on the diameter D) of about 40 and the well known Kármán vortex street is observed. The second bifurcation, known as Mode A instability, occurs around ReD ≈180 at which there is a transition from a two-dimensional to a three-dimensional wake. According to Williamson (1996), the threedimensionality at these Reynolds numbers manifests itself in the form of vortex loops and pairs of streamwise vortices, which have a characteristic spanwise wavelength of about 4D. As the Reynolds number is increased beyond 260, a gradual transition occurs where the vortex loops give way to fine-scale streamwise vortices with a spanwise spacing of about D. This transition is called Mode B instability. A further transition occurs at around ReD ≈ 105 (critical Reynolds number range), where the boundary layer becomes turbulent before separation. The aim of this Chapter is to demonstrate that the solver is capable of capturing the important flow physics in a satisfactory manner by comparing the results of the present simulations with available experimental and numerical data. We simulated some test cases to validate the current CFD solver.

43

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3.1 3.1.1

Validation, Verification & Performance

44

Validation Three-Dimensional Flows

We performed a DNS of the flow over a circular cylinder at ReD =525 on a 128 × 192 × 32 grid. In the simulation, the outer domain is 30D with a nondimensional spanwise length (Lz ) of π. A CFL (Courant-Friedrich-Levy) number based on the convection term in curvilinear coordinates is used as a guideline in choosing the time step. The simulations show that a stable time stepping is achieved for a CFL ≈ 0.2, which for the grid used corresponds to a nondimensional time step size ∆t = 2 × 10−3 .

The fluid force on the cylinder is the manifestation of the pressure and shear stresses acting on the surface of the cylinder. The net force can be decomposed into two components, namely lift and drag forces. These forces are nondimensionalized with respect to the dynamic pressure. The coefficients of lift and drag can thus be written in terms of the pressure and shear stresses as follows: 1 CL = − Lz CD = −

1 Lz

ZLz Z2π 0 0 ZLz Z2π 0

0

p sin θ −

1 ωz cos θ dθdz ReD

(3.1)

p cos θ +

1 ωz sin θ dθdz ReD

(3.2)

where ωz is the spanwise vorticity component on the cylinder surface. The results obtained from this simulation are presented in Table 3.1. The time histories of the lift and drag coefficients are plotted in Figures 3.1(a) and 3.1(b); CD represents the mean drag and the Strouhal number (St) is the nondimensionalized frequency defined as St = f D/U∞ .

The

power spectra of the lift and drag coefficients are plotted, respectively, in Figures 3.2(a) and 3.2(b). In order to reduce the statistical variance, the data is divided into 8 realizations, each consisting of 512 data points. The data sampling frequency is 10 and the frequency resolution is 0.0098. In Figure 3.2(a), the lift power spectrum shows a large peak corresponding to the shedding frequency (fs ) of 0.20 and smaller peaks at the odd harmonics 3fs and 5fs . The corresponding time history shows the lift coefficient fluctuating periodically about a zero mean. Therefore, it can be inferred that the lift coefficient oscillates at the shedding

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Table 3.1: Computed flow parameters. Data from

CD

St

1.15 − 1.2

−

−

0.21

3-D DNS (Mittal and Balachandar, 1996)

1.24

0.220

3-D DNS (present)

1.25

0.20

Experiment (Weiselsberger, 1922) Experiment (Roshko, 1954)

frequency and that its behavior is influenced by cubic, and, to a lower extent, higher-order nonlinearities. On the other hand, in Figure 3.2(b), the drag power spectrum shows a large peak at twice the shedding frequency (2fs ) and smaller peaks at the even harmonics 4fs and 6fs . The time history in Figure 3.1(b) shows the drag fluctuating periodically about a nonzero mean. The fluctuating term mainly varies quadratically with the lift coefficient, as the influence of higher-order even nonlinearities is quite small.

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1.5

1

CL

0.5

0

-0.5

-1

-1.5 450

460

470

480

490

500

480

490

500

Time

(a)

1.4

CD

1.3

1.2

1.1 450

460

470

Time

(b)

Figure 3.1: Time histories of (a) CL and (b) CD .

46

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0

10

−1

10

CL

−2

10

−3

10

−4

10

−1

0

10

10

Frequency

(a)

0

10

−1

10

CD

−2

10

−3

10

−4

10

−1

0

10

10

Frequency

(b)

Figure 3.2: Power spectrum of (a) CL and (b) CD .

47

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3.1.2


48

Two-Dimensional Flows

An “O” type grid allows us to simulate the flow past other closed surfaces. A systematic approach would be to start by studying a configuration that is more general than a circular cylinder and can characterize typical engineering flow configurations. Elliptic cylinders seem to provide such a configuration and changes in the eccentricity allow for shapes ranging from a circular cylinder to a flat plate. To validate the two-dimensional CFD results, we simulate the flow past elliptic cylinders. We compare the peak-to-valley lift coefficient CLp−v , the mean drag coefficient CD , the Strouhal number St, and mean surface pressure coefficient CP on the cylinder for the following two cases: • Case I - τ = 0.5 (Elliptic Cylinder). • Case II - τ = 1.0 (Circular Cylinder). Case I We simulate the flow past an elliptic cylinder with τ = 0.5 and a Reynolds number of 525 based on the projected length (Ly ) of the elliptic cylinder. The simulation results are compared with the numerical study of Mittal and Balachandar (1995). We perform the two-dimensional simulation on a 192 × 256 grid and the numerical results are presented in Table 3.2. We observe a good agreement in CLp−v , CD , and St. In Figure 3.3(a), we

plot the mean surface pressure coefficient CP over the surface of the ellipse. The pressure distribution is normalized such that CP = 1.0 at the stagnation point; that is, η = 180◦ . The CP distribution compares well with the results of Mittal and Balachandar (1995). Case II We also simulate the flow past a circular cylinder at Re=525 on a 192 × 256 grid. In the simulation, the two-dimensional domain is 30D. For the specific case of a circular cylinder Lx = Ly = D. Similar to Case I, we compute the physical parameters associated with the flow and compare the results in Table 3.2. We also compare the CP distribution over the cylinder surface and observe a good agreement between the two numerical studies.

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49

1

CP

0.5

0

-0.5

-1 0

90

(a)

180

η

270

360

270

360

τ = 0.5

1 0.5

CP

0 -0.5 -1 -1.5 0

90

(b)

180

η

τ = 1.0

Figure 3.3: Mean surface CP distribution: present (solid) and Mittal and Balachandar (1995) (triangle).

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Case

Table 3.2: Validation for the two cases. Study CLp−v CD

St 0.2∗

I

Mittal and Balachandar (1995)

1.21

0.78

I

Present Study

1.05

0.766 0.216

II

Mittal and Balachandar (1995)

2.42

1.44

II

Present Study

2.366 1.415 0.225

0.22

*Re=1000

3.2

Flow Visualization

The difference between the two-dimensional and three-dimensional simulations is a manifestation of the vortical structures developed due to spanwise flow. For the three-dimensional DNS case, we simulated the flow past a cylinder at ReD =525 with Lz = 2π. The increase in the spanwise length allows the formation of three-dimensional structures with a lesser constraint of the periodic boundary condition. We observe three-dimensional structures in the wake after the onset of Mode B instability. As the Reynolds number is increased above 500, the wake becomes turbulent and exhibits vortex instability in all directions. The instantaneous streamwise vorticity ωx and spanwise vorticity ωz isosurfaces are plotted in Figures 3.4 and 3.5, respectively. The role of spanwise rollers in the generation of streamwise structures can be visualized by plotting the isosurfaces of the absolute vorticity in the near wake of a circular cylinder. These results are in line with the experimental visualization of Williamson (1991) and numerical study of Mittal and Balachandar (1995). The absolute vorticity is defined as |ω| =

q

ωx2 + ωy2 + ωz2

(3.3)

The isosurfaces of the absolute vorticity |ω| are plotted for three different levels of 0.5, 1.0, and 1.5. The inclined, top, and side views are plotted in Figure 3.6. These figures show a snapshot when the bottom shear layer is in an early stage of rollup into a counter-clockwise rotating roller (CCR1 ) and the attached clockwise rotating roller (CR1 ) in the process of separating from the shear layer, as shown in Figure 3.6(c). A separated counter-clockwise rotating roller (CCR2 ) and a clockwise rotating roller (CR2 ) can also be observed at a

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downstream locations in the wake. As seen in Figure 3.6(b), these downstream rotating rollers (CCR2 &CR2 ) of opposite directions are connected to finger-like structures known as ribs (R1). Similarly, the next pair of rotating rollers of opposite directions are also connected through these ribs (R2). The arrows on ribs (R2) in Figure 3.6(b) denote the direction of these streamwise structures. Hairpin structures are formed due to clockwise and counterclockwise rotating cores (Mittal, 1995) and are shown in Figure 3.6(a). We observe that they are displaced by half a wavelength along the span.

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(a) Inclined view

(b) Top view

(c) Side view

Figure 3.4: The instantaneous spanwise vorticity isosurfaces (blue: ω z level at -0.5 and red: ω z level at +0.5) at ReD =525.

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(a) Inclined view

(b) Top view

(c) Side view

Figure 3.5: The instantaneous streamwise vorticity isosurfaces (blue: ω x level at -0.5 and red: ω x level at +0.5) at ReD =525.

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(a) Inclined view

(b) Top view

(c) Side view

Figure 3.6: The instantaneous vorticity isosurfaces (level = 0.5, 1.0, 1.5) at ReD = 525. (Note: the plot is 50% transparent to view the inner structures.)

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3.3

55


Verification

Grid and domain independence studies are critical in order to verify the accuracy of the computational results. Therefore, we perform a study to verify the grid/domain independence of our two-dimensional and three-dimensional simulation results at ReD =525. We compare some physical parameters, such as mean drag coefficient, and Strouhal number, to establish grid/domain independence.

3.3.1

Grid Dependence

We simulate the flow past a cylinder over a computational domain of 30D with 192×256 grid points. In order to check the dependence on the grid, the grid size is increased to 288 × 384; that is, 50% increase in grid points in both of the ξ- and η- directions. These simulations are run on 8 processors. For three-dimensional simulations, the computational domain is 30D and the spanwise length is Lz = π with 128 × 192 × 32 grid points. For our analysis, the grid size is increased to 192 × 256 × 48 distributed over 16 processors with an 8 × 2 configuration. The results from these simulations are given in Table 3.3. It is observed that the percentage difference for the mean drag coefficeint and St is less than 5%, which clearly indicates that the flow in the vicinity of the structure is virtually grid independent. Table 3.3: Grid dependence study Dim

3.3.2

Grid size (ξ × η × ζ) Domain Size CPU (η × ζ)

CD

St

2

192 × 256

30D

8×1

1.415 0.2197

2

288 × 384

30D

8×1

1.429 0.2246

3

128 × 192 × 32

30D

8×2

1.25

0.20

3

192 × 256 × 48

30D

8×2

1.261

0.204

Domain Dependence

To establish domain independence, the domain is increased to 50D; that is, a 60% increase in domain size, with 216 × 256 grid points. The slight increase in the grid points in the ξ-

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direction on the larger domain is chosen to maintain comparable grid spacing in the two grids. The flow is simulated on these grids using 8 processors. Similarly, for three-dimensional simulations, the computational domain is increased to 50D, keeping the spanwise length constant (Lz = π). Increasing the domain in the spanwise direction changes the geometry of the structure and can not be considered a similar flow. Again, the grid points in the larger domain are adjusted to keep comparable grid spacing in the ξ-direction. Therfore, the grid size is 144 × 192 × 32 distributed over 16 processors with an 8 × 2 configuration. The results from these simulations are given in Table 3.4 and indicate that the domain size of 30D is quite adequate for the current study. Table 3.4: Domain dependence study Dim

3.4

Grid size (ξ × η × ζ) Domain Size CPU (η × ζ)

CD

St

2

192 × 256

30D

8×1

1.415 0.2197

2

216 × 256

50D

8×1

1.381

0.228

3

128 × 192 × 32

30D

8×2

1.25

0.20

3

144 × 192 × 32

50D

8×2

1.267

0.197

Parallel Performance

The performance of a computer system depends on a large number of factors. We define few parallel performance parameters and study speed-up, efficiency, and scalability of the incompressible CFD solver.

3.4.1

Speed-Up and Efficiency

The speed-up and efficiency are typically defined as follows: Simulation time on 1 processor Simulation time on Np processors Achieved speed-up Efficiency = Ideal speed-up

Speed-Up =

(3.4) (3.5)

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Several simulations were performed to check the speed-up of the parallel code. For twodimensional cases, the grid is decomposed only in the η-direction. Figure 3.7(a) shows the speed-up curves for two different grid sizes at ReD =200 along with the linear (ideal) speed-up curve. We note that the speed-up curves are with respect to 2 processors. It is observed that the 256 × 256 grid shows better performance than the 128 × 128 grid because the ratio of computation to communication (per processor) is greater for the large problem. The 256 × 256 case yields a parallel efficiency of 73% on 16 processors. For the three-dimensional case, the speed-up is calculated with respect to 4 processors since the organization of the code is inherently parallel and the boundary conditions are defined assuming at least four processors, two in each of the η- and ζ-directions. The speedup curve for the 128 × 192 × 48 grid at ReD =525 is plotted in Figure 3.7(b). These results indicate significant savings in the computation time as the number of processors increases from 4 to 64. We observe a parallel efficiency of 90% for this 3-D case on 64 processors.

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(a)

16 (16 x 4)

Speed-Up (WRT to 4 Processors)

14 12 (16 x 3)

10 8 (16 x 2)

6 4

(8 x 2) (4 x 3)

2

(4 x 2) (2 x 2)

0

0

8

16

24 32 40 Number of Processors

48

56

64

(b)

Figure 3.7: Speed-up trends of the (a) 2-D and (b) 3-D CFD codes relative to two and four processors for different grid sizes. The dashed line represents the ideal speed-up.

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3.4.2


59

Scalability Study

The simplest definition of scalability is that the performance of a computer system increases linearly with respect to the number of processors used for a given application. In other words, the processing time for the parallel algorithm remains the same with the increase in the problem size (workload) and number of processors, Np . The performance parameter affects the scalability of the computer architecture and the application program involved and must be conducted for a given application program/algorithm. A good understanding of scalability will help evaluate the performance of parallel computer architectures for largescale applications. In our scalability analysis, the flow is simulated at ReD =525 with a computational domain of 30D × Lz , as given in Table 3.5. In Case 1, the domain is decomposed into 16 processors such that the load on each machine is 128 × 24 × 32. In Cases 2-4, Lz along with the grid points are doubled each time to increase the problem size (workload). However, the number of processors in the ζ-direction are also doubled to keep the workload per machine constant. All of the cases are simulated up to 100 time units, long enough to avoid transient effects. In Figure 3.8, we plot the physical time in hours it took to complete the simulation for each case. The horizontal bars along the x axis are plotted for comparison among different Lz cases. The study shows that the time remains fairly constant with the increase in the problem size and the number of processors, indicating that the parallel algorithm is scalable. Table 3.5: Scalability study chart. Case

Lz

Grid size (ξ × η × ζ)

1

6.28(2π)

128 × 192 × 64

2

12.56(4π)

128 × 192 × 128

3

25.13(8π)

128 × 192 × 256

4

50.26(16π)

128 × 192 × 512

Grid points Processors (η × ζ) 1.57 × 106 3.14 × 106 6.29 × 106

12.58 × 106

16(8 × 2) 32(8 × 4) 64(8 × 8) 128(8 × 16)

In order to model and predict the VIV problem, full simulation of the coupled flow and structure equations is required. This fluid-structure interaction is complex and involves three-dimensional effects, turbulence structures, and elasticity, among other considerations.

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Figure 3.8: Scalability study of the parallel code, the horizontal bars depict the spanwise length for comparison.

However, in general, many of these structures are quite long, reaching several thousand feet, and flows around them may reach very high Reynolds numbers. This capability of the parallel CFD solver to simulate long cylinders is extremely important to study VIV of offshore structures.

3.5

Summary

In this Chapter, we validated the parallel CFD solver by simulating benchmark cases for twoand three-dimensional flows. We simulated the flow past a circular cylinder and compared the mean pressure distribution, the mean drag coefficient, and the Strouhal number with existing experimental and numerical results. We also performed grid and domain dependence studies and verified that the numerical results obtained from the simulations are independent of the grid and domain sizes. In addition to validation and verification, we also investigated the parallel performance of the CFD solver. Since the motivation of this research is the

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analysis of VIV in offshore structures, which are long pipes and cables, it is important that the CFD solver be capable of simulating these long structures. Therefore, we computed the performance parameters, such as speed-up and efficiency, of the parallel solver and obtained close to linear increase in the speed-up as the number of processors is increased. Moreover, we increased the problem size (i.e., span of the cylinder,) and the number of processors and observed fairly constant computation time, indicating that the parallel algorithm is scalable.

Chapter 4 Features and Applications of the Parallel CFD Solver The parallel solver developed during this research is used, so far, to solve the flow past a circular cylinder. Many engineering and industrial applications, on the other hand, involve flows over complex bodies and structures, like aircraft, rotor blades, and bridges, which can hardly be modeled as a flow over a circular cylinder. Parameters, such as thickness ratio and angle-of-attack, can greatly influence the nature of separation in such flows. Therefore, richer separation phenomena, ranging from steady to unsteady two-dimensional and threedimensional phenomena, are exhibited in flows over complex bodies. Moreover, in many fluid-structure interactions, the flow is turbulent and requires fine grid resolution to resolve the small scales present in the flow. At times, due to computational constraint, these small scales or eddies can not be captured everywhere in the domain especially at high Reynolds numbers. Therefore, we require some mathematical model to represent the small scales in a turbulent flow. As mentioned earlier, the governing equations for the flow problem are written in curvilinear coordinates and, therefore, are capable of solving other complex flows of engineering relevance. In this Chapter, various features of the algorithm are discussed along with their application. We perform DNS at relatively higher Reynolds numbers and observe turbulent structures in the flow field. We also simulate the flow past a cylinder under the influence of sheared freestream in the spanwise direction. Later, we modify the CFD solver so that 62

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it can simulate moving boundaries. We perform various forced-oscillation configurations to validate the numerical results. The solver can also simulate a suction/blowing jet on the cylinder surface by appropriately modifying the surface boundary conditions. This feature is employed effectively to control the flow field in the wake and subsequently suppress vortex shedding to reduce the fluctuating forces on the cylinder. We introduce control parameters of the jet and perform a test case. In order to simulate turbulent flows at high Reynolds numbers, we use the Spalart-Allmaras model to capture the effect of fine scales. We discuss implementation of the model in detail and simulate unsteady flow at ReD =3,900 using the Reynolds-Averaged Navier-Stokes equations.

4.1

DNS of Turbulent Flows

Despite a simple flow geometry, the flow past a cylinder undergoes various transitions as the Reynolds number is increased. These transitions are the manifestation of instabilities of the flow. Starting from the laminar flow at Re ≈ 5, the wake consists of a steady recirculation region of two symmetrically placed vortices on each side of the wake whose length grows as the Reynolds number increases. As the Reynolds number exceeds 40, the downstream end of the recirculation region starts to develop instabilities, which is the manifestation of a Hopf bifurcation. These instabilities grow in amplitude with increasing Reynolds number, resulting in an increase in the Reynolds stresses in the near wake region, a decrease in the formation length, and increase in the base suction pressure. The onset of this instability results in the laminar von Kármán vortex street. The next transition regime, the wake-transition regime, is associated with two discontinuous changes in the wake formation. At the first discontinuity, Mode A instability at ReD ≈ 180, we observe the inception of vortex loops and the flow no longer remains twodimensional. We observe deformation of the primary vortices at a wavelength of approximately 4 diameters and formation of streamwise vortex pairs. The next transition, Mode B instability, occurs around ReD ≈ 260. This mode consists of finer scale streamwise vortices with a spanwise length scale of approximately one diameter. The base suction pressure CP b is maximum during this transition. According to Williamson (1996), it is associated with a

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peak in the Reynolds stresses and a relatively ordered 3-D streamwise vortex structure in the near wake. As the Reynolds number is increased, the effects of third-dimensionality become important and alter the physics of the flow, and we start to observe disorder in the fine scales in the wake, resulting in a reduction in the Reynolds stresses and base suction pressure and increase in the formation length (Mittal and Balachandar, 1995). As the Reynolds number is increased to 500, the wake start to become turbulent and becomes fully turbulent around Re=1,000. In the shear-layer transition regime extending to approximately Re=200,000, the base suction pressure and the 2-D Reynolds stresses increase, while the formation length and Strouhal number decrease. The boundary layer then becomes turbulent and a sharp decrease in the base suction pressure and drag is observed. This critical transition is associated with a separation-reattachment bubble, causing the boundary layer to separate further downstream. Williamson (1996) provided a detailed and comprehensive review of different transition regimes for the flow past a circular cylinder. We performed a DNS of a turbulent wake for two- and three-dimensional flows past a circular cylinder at ReD =1,000. The purpose of these simulations is to ensure that, given enough grid resolution, the parallel CFD solver is capable of capturing not only the dominant structures, but also the fine-scale features present in turbulent flows. We compared our numerical results with the experiment (Norberg, 1994) and the numerical study (Evangelinos and Karniadakis, 1999). Evangelinos and Karniadakis (1999) investigated VIV for stationary, rigid, and flexible cylinders at ReD =1,000. In their study, the computational domain for the x − y plane extended 69D downstream and 22D upstream, top and bottom of the cylinder. For three-dimensional flows, they performed DNS over a circular cylinder with a nondimensionalized spanwise length Lz = 4π. They employed a spectral element method on an unstructured hybrid grids with hp-refinement. For the stationary case, they used 16 independent Fourier modes in the periodic spanwise direction.

4.1.1

Two-Dimensional Flow at ReD =1,000

We performed a two-dimensional flow past a circular cylinder on a 192 × 256 grid with a computational domain of 30D. The grid was distributed over 8 processors such that each processor has a work load of 192×32 grid points. The simulation was performed long enough

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to reach steady state. We plot the spanwise vorticity over one vortex shedding cycle in Figure 4.1. Each snapshot was taken after a Ts /10 interval, where Ts is the vortex shedding time period. We observe that the shear layer rolls up to form discrete vortices of opposite signs during a shedding cycle. We computed the lift an drag coefficients on the cylinder and plotted them in Figures 4.2(a) and (b). We observe an increase in the lift and drag coefficients as the Reynolds number is increased from ReD =525 to 1,000. We then performed the fast Fourier transform of the time histories of the lift and drag coefficients to compute the dominant frequencies. In Figure 4.2(c), the lift spectrum shows a dominant peak at fs = 0.235, which corresponds to the vortex shedding frequency or the Strouhal number. We also observe odd harmonics at 3fs and 5fs . Similarly, in the drag spectrum, we observe even harmonics of the vortex shedding frequency at 2fs and 4fs . The mean drag CD is 1.486 and compares well the numerical results obtained by Evangelinos and Karniadakis (1999).

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Figure 4.1: Spanwise vorticity contours of one vortex shedding cycle over a stationary cylinder at ReD = 1, 000.

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1.5

1.7

1

1.6

CD

CL

0.5 0

1.5 1.4

−0.5 1.3

−1

1.2

−1.5 400

420

440

T ime

460

480

500

400

420

440

(a) CL

460

480

500

(b) CD

0

0

10

10

−1

−1

10

Spectrum

10

Spectrum

T ime

−2

10

−3

10

−4

−2

10

−3

10

−4

10

10

0.2

0.4

0.6

0.8

1

1.2

F requency

1.4

1.6

1.8

0.2

0.4

0.6

(c) CL

0.8

1

1.2

F requency

1.4

1.6

1.8

(d) CD

Figure 4.2: Time histories and power spectra of CL and CD at ReD = 1, 000.

4.1.2

Three-Dimensional Flow at ReD =1,000

We performed a DNS on a 192 × 256 × 192 grid distributed over 64 (8 × 8) processors with a computational domain of 30 × 4π. The load per processor was 192 × 32 × 24 grid points. The spanwise length Lz of the cylinder was kept 4π to match the computational domain in the numerical study of Evangelinos and Karniadakis (1999), as shown in Figure 4.3(left). The grid spacing in the spanwise direction ∆z = 0.06545, which is fine enough to capture the fine length scales generated along the cylinder span. The grid spacing was motivated from an LES study by Kravchenko and Moin (2000) at ReD = 3, 900. For flow visualization,

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we plot the isosurfaces of spanwise vorticity ωz in Figures 4.3 and 4.4. The vortex shedding cycle is distributed in five snapshots at t ≈ 0, Ts /4, Ts /2, 3Ts /4, and Ts . We observe that the wake is no longer ordered and is fully turbulent. However, we can identify some dominant features such as ribs and hairpin vortices in the wake. We compute the lift and drag coefficients on the cylinder surface and plot the time histories in Figure 4.5(a) and (b). From the Figures, we observe the effect of the thirddimensionality in the flow on the hydrodynamic forces. In the two-dimensional case, we observe that the amplitudes of the lift and drag coefficients remain constant, while in the three-dimensional case, we observed that the amplitudes vary due to the presence of high frequencies in the flow. We discuss the pointwise statistics focusing on the structure as well as the wake. In Figure 4.6, the 1-D energy spectrum at (x/D, y/D) = (1.0, 0.5) shows a peak at the shedding frequency 0.20. Moreover, it exhibits a − 53 law in the inertial range, which extends about half a decade in wave number. We also perform a two-dimensional simulation on a 128 × 192 grid using 8 processors. A comparison of the mean drag coefficient and the Strouhal number (St) is given in Table 4.1. We compute the mean drag coefficient and the Strouhal number for the two- and threedimensional flows at ReD =1,000. In Table 4.1, these quantities are compared with those obtained by Norberg (1994) and Evangelinos and Karniadakis (1999). Table 4.1: Computed flow parameters. Data from

CD

St

Experiment (Norberg, 1994)

1.0

0.21

2-D (Evangelinos and Karniadakis, 1999)

1.54

0.238

2-D (present)

1.486 0.235

3-D DNS (Evangelinos and Karniadakis, 1999)

1.02

0.202

3-D DNS (present)

1.11

0.205

Imran Akhtar Features and Applications Figure 4.3: Geometry of the cylinder with arrows showing the flow direction and checkerboard pattern indicating 8 processors in the spanwise direction (left), isosurfaces of ωz at t = 0 (center) and t ≈ Ts /4 (right).

69

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Figure 4.4: Isosurfaces of ωz at t ≈ Ts /2 (left), t ≈ 3Ts /4 (center), and t ≈ Ts (right).

70

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0.6 1.25 0.4 1.2

CD

CL

0.2 0

1.15 1.1

−0.2 1.05 −0.4 1 −0.6 150

200

250

300

350

T ime

400

450

500

150

200

250

(a) CL

350

T ime

400

450

500

(b) CD

0

0

10

10

−1

−1

10

Spectrum

10

Spectrum

300

−2

10

−3

10

−4

−2

10

−3

10

−4

10

10

0.2

0.4

0.6

0.8

1

1.2

F requency

(c) CL

1.4

1.6

1.8

0.2

0.4

0.6

0.8

1

1.2

F requency

1.4

1.6

1.8

(d) CD

Figure 4.5: Time histories and power spectra of CL and CD at ReD = 1000.

4.2

Shear Flows

Offshore platforms and marine risers are subjected to VIV and these vibrations are difficult to predict even when the current is uniform and unidirectional. These risers are hundreds of feet long and bring oil and gas from the sea floor. Among other loading factors, spars and risers are subjected to shear currents due to a velocity that varies with depth. The problem becomes more challenging when the current varies along the axis of the cylinder and changes direction as well. Cunff et al. (2002) discussed several approaches to provide a reliable method to compute the fatigue life of risers subjected to currents. They reviewed analytical, numerical, and experimental efforts in this area of research. Experimental investigation of

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0

10

−2

10

slope = − 5/3

−4

Energy spectrum

10

−6

10

−8

10

−10

10

−2

10

−1

0

10

10

1

10

Frequency

Figure 4.6: 1-D energy spectrum at ReD =1,000.

such problems is limited by cost, however, numerical simulations provide a useful tool for the analysis of fluid-structure interaction under varying flow conditions. When the current has a spanwise shear, the flow past the cylinder is influenced by the secondary flow generated by the leading edge stagnation pressure gradient. Variation in the incoming flow causes significant changes in the flow pattern in the wake. The influence on the vortex shedding is a function of the Reynolds number and the shear parameter, defined as β=

D ∂u , avg U∞ ∂z max

(4.1)

avg where ( ∂u ) is the maximum slope of the profile and U∞ is the freestream velocity deter∂z max

mined by the average of the velocities over the cylinder span. There have been experimental as well as numerical studies for analyzing the effect of freestream shear on the structure. In one of the pioneer works, Maull and Young (1973) noted that a spanwise variation of the base pressure across the cellular wake structures in a linear shear flow past an airfoil shaped body with a flat trailing edge. Mair and Stansby (1975) performed a similar study on slender cylinders and found the base pressure to be a function of Reynolds number and the shear parameter. Later, Peltzer and Rooney (1981) conducted experiments with 0 < β < 0.026 and observed spanwise variation of the base

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pressure. However, they did not provide its relationship with the cellular pattern in the wake. In another experimental study, Vandiver et al. (1996) analyzed the occurrence of lock-in under highly sheared conditions. They employed two dimensionless parameters to predict lock-in under such conditions. The first parameter is Ns , the number of potentially responding modes with the vortex shedding frequency bandwidth generated by the shear flow. Secondly, ∆V /Vavr (shear parameter) is the ratio of the change in velocity of the flow over the length of the cylinder to the spatially averaged flow velocity over the length of the cylinder, as defined in Equation (4.1). They observed that in the presence of highly sheared conditions, lock-in may occur because the power available to a single mode of vortex shedding is dominant over other modes. Mukhopadhyay et al. (1999) performed a DNS study of a linear shear flow past a circular cylinder at a mean Reynolds number of 131.5 and β = 0.2. In their numerical approach, they modeled the boundary conditions in the spanwise directions as end plates so that no flow crossed the computational boundaries along the span. They observed the same cellular shedding pattern as observed in the experiments at much higher Reynolds numbers. In another numerical investigation of shear flows, Xu and Dalton (2001) conducted an LES of the flow past a circular cylinder at the Reynolds numbers 1000, 10000, and 20000 with β = 0.3093. They employed a 3-D spectral/finite-difference approximation alongwith the Smagorinsky model to solve the space-filtered Navier-Stokes equations. Due to the spectral method in the spanwise direction, the freestream velocity is modeled as U (z) = U0 − Up sin(2πz/L),

(4.2)

where U0 is the mean velocity, Up is the maximum velocity in the sinusoidal wave, and L is the wavelength of the freestream. The modeling ensures the application of periodic boundary conditions in the spanwise directions described as follows: u(ξ, η, 0) = u(ξ, η, L), p(ξ, η, 0) = p(ξ, η, L).

(4.3)

The periodic boundary conditions in the spanwise direction impose a restriction on the application of linear shear. However, they focused on the flow regime from L/4 − 3L/4 and

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analyzed the forces acting in that shear flow region. They observed that the drag coefficient in the shear flow has a slight axial dependence. They also noted a phase difference in the length-averaged lift coefficient between a uniform and a shear flow, especially at higher Reynolds numbers. In the current study, the parallel CFD solver is capable of simulating shear flows in the spanwise directions. A schematic of few examples of shear flows are shown in Figure 4.7. The three different spanwise variations plotted include (a) sinusoidal, (b) linear, and (c) exponential variations. It is important to note that a sinusoidal spanwise variation requires periodic boundary conditions in the ζ-direction, while, in the two other cases, we need other condition (e.g., symmetric boundary conditions) are needed. Most of the codes available for CFD simulations employ spectral methods in the spanwise direction and are limited to periodic boundary conditions. However, we use a finite-difference approach in all of the directions, thus enabling simulation of other shear flows.

U

U(z) X Y

Z

+

(a)

(b)

(c)

Figure 4.7: A schematic of some shear flows in spanwise directions.

We simulated the flow past a cylinder with an incoming shear flow of type (a). The computational domain for the flow is 30 × 10 with 128 × 192 × 96 grid points. A spanwise sinusoidal distribution is added to the incoming flow with an amplitude of 0.1, and the

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Reynolds number based on the midspan velocity is 525. Based on the velocity variation, the Reynolds number ranges approximately from 470 to 580 and is three-dimensional in the wake. In Figure 4.8, the spanwise vorticity is plotted at two different planes along the span. We also plot the isosurfaces of the instantaneous absolute vorticity in Figure 4.9 and observe the effect of the incoming flow variation along the span. Clockwise rotating and counterclockwise rotating vortices (or rollers) form an S -shaped curve due to the sinusoidal velocity variation. As seen in the figure, these downstream rotating rollers of opposite directions are connected to finger-like structures, known as ribs, and are dominant in the high Reynolds number region.

4.3

Moving Boundary Methods

In a more realistic fluid-structure interaction, a structure may be elastic or excited externally. The behavior of a structure undergoing VIV depends mainly on the natural frequencies of the body and vortex shedding from the body. If the frequency of vortex the shedding is close to a natural frequency of the body, it might oscillate with high amplitudes, eventually leading to its damage or complete structural failure. In order to simulate the motion of the structure, three basic methods can be identified for the numerical discretization of the flow problem. In the immersed boundary (IB) methods, the flow is simulated with immersed boundaries on grids that do not conform to the shapes of these boundaries. The advantage of this method appears in the grid generation, which does not require coordinate transformation at every increment of body motion. However, applying appropriate boundary conditions is not straightforward and resolving the boundary layer at high Reynolds numbers increases the grid-size requirement faster than a corresponding bodyconformal grid. For detailed description and comparison of this method, readers are referred to Mittal and Iaccarino (2005). In another approach known as arbitrary Lagrangian-Eulerian (ALE) method, the computational mesh local to the structure is distorted continuously in time as the structure moves. The boundary conditions on the body and in the far field are usually fixed in time. In terms of computational cost, the disadvantage of this approach is the re-meshing and the

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Figure 4.8: Spanwise vorticity plotted on two planes along the span.

temporal changes of the mesh interpolation functions. An alternative to this method is the accelerating reference frame (ARF) method. Unlike ALE, the mesh is fixed to the structure and the momentum equations and corresponding boundary conditions are modified to allow for the motion. In this way, the computational overhead associated with the coordinate transformation at every time step can be avoided, however, the method is limited in its application.

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Figure 4.9: Isosurfaces of the absolute vorticity (level=0.75, 1.5 and 3) colored by the spanwise vorticity (50% transparent).

4.3.1

Accelerated Reference Frame

In our problem of the flow past an oscillating cylinder, the ARF method is a suitable candidate for simulating moving boundaries. The momentum equations can be directly coupled with the cylinder motion by adding a frame acceleration term and modifying the boundary conditions to accommodate the moving reference frame (Blackburn and Henderson, 1999). Thus, the governing equations describing the relative motion of incompressible fluid in the reference frame attached to the structure are

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Continuity Equation ∂uj =0 ∂xj

(4.4)

∂ui ∂ 1 ∂p ∂ 2 ui + (uj ui ) = − +µ − xï , ∂t ∂xj ρ ∂xi ∂xj ∂xi

(4.5)

Momentum Equation

where i,j=1,2,3. In an inertial reference frame, the structure moves in response to the force per unit length F exerted on it by the fluid according to x¨ + 2ςωn x˙ + ωn2 x = F/m

(4.6)

where m is the mass per unit length, ωn is the natural frequency, and ς is dimensionless structural damping and x˙ = v is the structure velocity. At the domain boundary, the velocity boundary condition is modified to include the effect of moving body, such that u = uD − v

(4.7)

where uD is the velocity in the inertial reference frame. On the structure surface, the velocity boundary condition is typically ui = 0. Pressure boundary conditions are obtained by dotting the domain unit outward normal n into Equation (4.5), and using the vector identity ∇2 u = ∇(∇ · u − ∇ × ∇ × u). The result is ∂ui ∂p = ρn · (u · ∇)u − ν∇ × ∇ × u − xï − . ∂n ∂t In the far field boundary,

4.3.2

∂ui ∂t

= −xï , while on the surface

∂ui ∂t

(4.8)

= 0 and ui = 0.

Forced Oscillations

Many experimental and numerical studies have been performed to understand, model, and predict the phenomenon of VIV for fixed, excited, and elastic cylinders. The problem of a vibrating cylinder due to exerted forces goes back to the work of Strouhal (1878) in the area of aeroacoustic and to the work of Rayleigh (1879) on the oscillations of violin strings subject to the incoming wind. The first significant contribution to this problem is credited to

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Bishop and Hassan (1963b) who experimentally studied the flow past an externally-oscillated cylinder over a Reynolds number range of 5,850 to 10,800 within the lock-in frequency range. The lock-in frequency range is the bandwidth where the lift frequency is entrained by the oscillating frequency of the cylinder. The cylinder motion was adjusted via a Scotch-yoke mechanism, and the fluid forces were computed from the measured forces in running and static water. They reported that this bandwidth is a function of the Reynolds number and the amplitude of motion. Williamson and Roshko (1988) performed an experimental study of the flow over a cylinder undergoing simple harmonic motion in a low Reynolds number range, 300 to 1,000. They used aluminum particles on the fluid surface for visualization. They observed various vortex patterns and constructed a map of vortex synchronization regions using as coordinates as a function of the nondimensional wavelength of cylinder motion (λ/D). They denoted certain vortex patterns by symbols, such as “P+S”, “S”, and “2P”. Here “S” denotes a single vortex, and “P” signifies a pair of vortices of opposite signs. Thus, the vortex pattern “P+S” is one in which a pair and a single vortex are shed during each oscillation cycle. Similarly, two vortex pairs are shed per cycle in a “2P” wake pattern, while two singlets are shed in a “2S” wake pattern. Krishnamoorthy et al. (2001) investigated the wake patterns of an oscillating cylinder at ReD =1,500. They varied the frequency of oscillation from one-third to three times the natural shedding frequency with a nondimensional amplitude of 0.22. They observed abrupt switching between some vortex patterns and a phase jump in the vortex shedding relative to cylinder displacement. In a recent experimental study, Krishnamoorthy et al. (2001) investigated the wake patterns of an oscillating cylinder at ReD =1,500. They varied the frequency of oscillation from one-third to three times the natural shedding frequency with a nondimensional amplitude of 0.22. They observed abrupt switching between some vortex patterns and a phase jump in the vortex shedding relative to the cylinder displacement. Recent numerical studies on the flow around stationary and oscillating cylinders include the direct numerical simulations by Dong and Karniadakis (2004) at Re=10,000. The cylinder is excited at a nondimensional motion amplitude of 0.3 with Lz = π. They used the spectral element method with polynomial orders ranging from 5 to 8 and 300 million degrees

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of freedom to resolve the flow at this high Reynolds number. The computational domain employed for the simulations extended 20 diameters upstream and 50 diameters downstream, with 40 diameters in the crossflow directions. The problem was solved in a coordinate system that is attached to the cylinder. Al-Jamal and Dalton (2005) performed a two-dimensional LES to compare the flow past a forced and an elastically vibrating cylinder within the lock-in region at Reynolds number of 8,000. They used a vorticity formulation instead of primitive-variable formulation used in the current CFD solver. The problem was solved in an absolute coordinate system over an O-grid with a radius of five cylinder diameters and used a second-order central difference scheme for all of the spatial derivatives. They employed the Smagorinsky model to simulate the subgrid scales. For the elastically mounted cylinder, spectral and complex demodulation analyses show that the phase angle between the lift and motion is strongly modulated. This is remarkably different from what is typically observed in the externally excited motion. In an experimental study, Kim and Williams (2006) investigated the nonlinear coupling of the fluctuating drag and lift forces on cylinders undergoing forced oscillations at Re=15,200. They measured the surface pressure fluctuations for both inline and crossflow oscillations. They reported quadratic nonlinear interaction between the von Kármán vortex-shedding modes and the forcing modes. Marzouk and Nayfeh (2007) performed two-dimensional unsteady RANS simulations of the flow around stationary and crossflow oscillating cylinders with different frequencies and amplitudes in the lock-in regime at Re=500. The problem was solved on a curvilinear, O-grid with a radius of 25 cylinder diameters. They employed the artificial compressibility method to solve the governing equations with second-order accurate schemes in both time and space: central differences for the diffusion term and an upwind scheme for the convective terms. They used the one equation Spalart-Almaras model to represent the unresolved scales in the flow. They observed discontinuities between the frequency response and phase angle of the lift relative to the displacement. They identified two distinctive modes, namely, a low-lift mode and a high-lift mode. In the former case, the lift and displacement are out-of-phase while in the latter case, they are in-phase. They also observed that the motion amplitude must exceed a certain threshold for these discontinuities to take place, in agreement with

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other studies. For a comprehensive review of the VIV phenomenon, the reader is referred to Williamson and Govardhan (2004). In the current methodology, the equation governing the nondimensional displacement (forced-oscillation) of the cylinder is modeled by a simple harmonic motion in the in-line and crossflow directions as follows: (x, y) = (Ax sin(2πfe t), Ay sin(2πfe t))

(4.9)

where Ax and Ay are the nondimensional amplitudes in the inline and crossflow directions, respectively, and fe is the nondimensional excitation frequency. Therefore, the CFD flow solver is capable of simulating single- and two-degrees-of-freedom motions, as shown in Figure 4.10.

U

U

U

Case I

Case II

Case III

Figure 4.10: A schematic of the possible stationary and moving boundary cases.

4.3.3

Crossflow Oscillations

Most of the experimental and numerical studies have ignored the streamwise component of the motion since the crossflow in VIV problems is much more influential, with the exception of cases where the density of the cylinder is small relative to the fluid density (Jauvtis and Williamson, 2004). Moreover, this assumption reduces the parameters in the simulations and enables better understanding of the effect of this component alone. To validate this feature of the parallel CFD solver, we simulate Case I with different configurations of amplitudes, frequencies, and Reynolds numbers. We then compare our

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results with other experimental and numerical studies. In Table 4.2, we present details of the simulations performed and discuss the hydrodynamic forces on the cylinder as well as the wake configuration. It is important to note that the only objective of this Section is to establish that the CFD solver is capable of simulating moving boundaries and capture the physics in such flows. Table 4.2: Crossflow oscillations. fe /fs Ay /D Comments

Case

ReD

A

200

−

−

B

200

0.80

0.10

Forced response

C

200

0.97

0.10

Forced response

D

200

1.20

0.10

Forced response

E

200

0.60

0.50

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M

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Bispectrum analysis

Stationary cylinder

We simulated different cases of the flow past a circular cylinder undergoing crossflow oscillations with varying frequencies and amplitudes. We computed the lift and drag coefficients and their spectra to identify the dominant frequencies. We summarize all the cases simulated at ReD = 200 and 525 and validate the numerical results. In Figure 4.11, we plot the lock-in region (area enclosed) in the amplitude-frequency plane obtained by Koopmann (1967) at ReD = 200. We also indicate the points (triangles and squares) where the numerical simulations were performed. We observe that Cases C, F, G, and H are within the lock-in region obtained in the experimental study. On the other hand, Cases B, D, and E lie outside this region and we observe that the vortex shedding frequency is quasi-periodic.

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We note that the key objective of this exercise of simulating the flow at various locations in the frequency-amplitude plane is only to validate the solver’s capability to simulate moving boundaries and not to investigate the flow physics of oscillating cylinders. However, the CFD solver can be effectively used to study forced-oscillating cylinders. This feature of the CFD code can be further enhanced and may be employed in analyzing the VIV problem for elastic and flexible cylinders

Ay / D

0.6

0.4

0.2

0

0.6

0.8

1

1.2

fe / fs Figure 4.11: Frequency-amplitude plot: Lock-in (squares), quasi-periodic (triangles), and lock-in region obtained by Koopmann (1967)(circles).

Case A We simulate crossflow oscillations at ReD = 200 and compare the results with the numerical study of Meneghini and Bearman (1995). They used the vorticity and stream function form of the governing equations and solved them using the operator-splitting technique in which the convection and diffusion of the vorticity are treated separately. They simulated twodimensional flow at Re=200 on a 170×128 polar grid undergoing forced crossflow oscillations for a range of amplitude ratios (Ay /D) and frequency ratios (fe /fs ). In the stationary cylinder case, we simulate a two-dimensional flow and calculate the lift and drag coefficients. In Figure 4.12, we plot the time histories of the lift and drag

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coefficients and calculate the corresponding frequency spectra. The Strouhal number in the lift spectrum is the shedding frequency fs . We observe odd harmonics in the lift spectrum and even harmonics in the drag spectrum. In Figure 4.13, we plot the instantaneous spanwise vorticity over the stationary cylinder. Each snapshot is taken over a Ts /10 interval and a vortex pair of opposite sign is shed during the shedding cycle.

0.8 1.4

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Figure 4.12: Time histories and power spectra of CL and CD for a stationary cylinder at ReD = 200.

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Figure 4.13: Spanwise vorticity contours of one vortex shedding cycle over a stationary cylinder at ReD = 200.

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Cases B-D We simulate the flow around an oscillating cylinder with an amplitude of Ay /D = 0.1 at different frequency ratios fe /fs . The objective is to identify the frequency response and compute the dominant frequencies in the lift and drag coefficients. In Case B, we plot the time histories of the lift and drag coefficients in Figures 4.14(a) and (b). We observe a quasi-periodic response in the hydrodynamic forces acting on the cylinder surface. We perform spectral analysis of the lift coefficients and plot them in Figure 4.14(c). We observe two distinctive peaks corresponding to the excitation frequency and the vortex shedding frequency. We observe even harmonics of the dominant frequencies in the spectrum of the drag coefficients in Figure 4.14(d). We also observe a peak at fs − fe , which is the difference between the vortex shedding frequency and the excitation frequency. In Case C, we set fe /fs = 0.97 and observe the lock-in phenomenon and that vortex shedding occurs at the excitation frequency. In Figures 4.15(a) and (b), we observe that the lift and drag coefficients are periodic with a decrease in the lift coefficient amplitude and an increase in the mean drag coefficient. The spectral analysis, in Figures 4.15(c) and (d) identifies only dominant peaks at fe and 2fe , respectively, alongwith their harmonics. We then increased the frequency ratio fe /fs to 1.2 in Case D. In Figures 4.16(a) and (b), we observe a quasi-periodic behavior in the lift and drag coefficients with a significant increase in their amplitudes. We also observe additional frequency components in the frequency domain as shown in Figures 4.16(c) and (d). The dominant frequencies in the lift coefficient spectrum remain to be the vortex shedding frequency and the excitation frequency. However, the nonlinear interaction between various frequency components gives rise to multiple frequencies in the lift and drag spectra. Cases E-H In the next four Cases E-H, we increase the amplitude of oscillation to Ay /D = 0.5 and vary the frequency ratio from 0.6 to 1.2 with an increment of approximately 0.2. For Case E, we plot the instantaneous spanwise vorticity ωz over one oscillation period in Figure 4.17. Each snapshot is taken for Te /10 intervals where Te is the time period of one oscillation.

Imran Akhtar

1.5

1.6

1

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CL

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(d) CD spectrum

Figure 4.14: Time histories and power spectra of CL and CD for a crossflow oscillating cylinder at ReD = 200 with fe /fs = 0.8 and Ay /D = 0.1.

The vortex patterns indicate the nonsynchronous region at this excitation frequency and amplitude. From Figure 4.11, this frequency-amplitude configuration lies within the lock-in region. In Figure 4.18, we plot the instantaneous spanwise vorticity ωz over one oscillation period and observe a nonconventional vortex shedding pattern in this synchronous region.

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0.8 1.5

0.6

1.45

0.4

1.4

CD

CL

0.2 0

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Figure 4.17: Spanwise vorticity contours of one oscillating cycle with fe /fs = 0.6 and Ay /D = 0.5 at ReD = 200.

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Figure 4.18: Spanwise vorticity contours of one oscillating cycle with fe /fs = 1.2 and Ay /D = 0.5 at ReD = 200.

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Case J In the last four cases, we increase the Reynolds number to 525 for the crossflow oscillations. The two-dimensional flow around the cylinder has been discussed in detail in Chapter 3. Cases K-L In the last two cases, we increased the Reynolds number to 525 and observed the shedding pattern for Ay /D=0.05 and 0.25 at the frequency ratio of fe /fs =0.82. The response in Case K is quasi-periodic and lies in the nonsynchronous region. The amplitude of oscillation is then increased to Ay =0.25. For this amplitude of oscillation, the lock-in range of the lift and drag extends from fe /fs =0.78 to 0.99 (Marzouk and Nayfeh, 2007), which is consistent with our findings. In Figure 4.19, we plot spanwise vorticity contours for the two nondimensional amplitudes. We observe the difference in the vortex patterns in the lock-in and lock-out cases. Case M The dynamical system under study is a nonlinear system. An important property of any nonlinear system is that different frequency components in the system can interact with each other and generate new frequencies. These new components typically appear as the sum and the difference of frequencies. The phase of the frequency provides the information if it is a result of a nonlinear interaction of other frequencies. In order to analyze the quadratic coupling between the excitation and aerodynamics forces, we compute the cross-bispectrum between the motion and the lift coefficient. The cross-bispectrum can be expressed as 1 Bxxy (fi , fj ) = limT →∞ E X ∗ (fi )X ∗ (fj )Y (fi + fj ) T

(4.10)

where X(f ) = f f t(x(t)), Y (f ) = f f t(y(t)), and E[...] denotes the time-average. Also, X ∗ is the complex conjugate of X. In the above formulation, x(t) is the time series of motion of the cylinder and y(t) is the lift coefficient time history. The cross-bispectrum is often normalized to yield the cross-bicoherence. The cross-bicoherence is defined as (Powers and Im, 1995) |Byxx (f1 , f2 )|2 1 T →∞ T E[|YT (f1 + f2 )|2 ]E[|XT (f1 )|2 ]E[|XT (f2 )|2 ]

b2yxx (f1 , f2 ) = lim

(4.11)

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(a) Ay=0.25 (Lock-in)

(b) Ay=0.05 (Lock-out)

Figure 4.19: Spanwise vorticity contours of the flow past an oscillating cylinder.

Based on Schwarz’s inequality, the value of the cross-bicoherence is bounded between [0, 1]. A value of cross-bicoherence close to one indicates strong quadratic phase coupling and a value near zero indicates the absence of coupling. The cross-bicoherence has several symmetries and the principal domains defined by f1 are 0 ≤ f1 and 0 ≤ f1 + f2 ≤ fN . In an experimental study, Kim and Williams (2006) investigated the nonlinear coupling of the fluctuating drag and lift forces on cylinders undergoing forced oscillations at Re=15,200.

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They measured the surface pressure fluctuations for both inline and crossflow oscillations. They reported quadratic nonlinear interaction between the von Kármán vortex-shedding modes and the forcing modes. In all of the cases, power spectra are computed for the lift and drag forces on the cylinder surface. In order to reduce statistical variance, the data was divided into 10 realizations, each consisting of 512 data points. The data sampling frequency is 10 and frequency resolution is 0.0098. For higher order spectral moments, the number of realizations is 46, each with 1024 data points. In this case, we analyze nonlinear interaction between the lift and drag using higherorder spectral analysis. Figure 4.20 and 4.21 shows the time histories and power spectra of the lift and drag coefficients, respectively. It is observed that the motion of the cylinder induces a significant change in the fluctuating forces acting on the cylinder surface, where two dominant frequencies in the lift spectrum are observed. In addition to the fundamental and forcing frequencies, we also observe multiple frequencies in the lift and drag spectra. In order to identify the type of coupling between the different frequencies, we perform bispectrum analysis on the lift and drag data.

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1

CL

0.5

0

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−1 350

400

450

Time

(a)

1.5

1.4

1.3

CD

1.2

1.1

1

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0.8 350

400

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(b)

Figure 4.20: Time histories of (a) CL and (b) CD

450

Imran Akhtar


fs fe

2fs-fe 3fs

5fs

(a)

2fs fs-fe

2fe 4fs

(b)

Figure 4.21: Power spectrum of (a) CL and (b) CD

96

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Figure 4.22 shows cross-bispectrum of lift and drag (LLD); we observe that the drag is quadratically coupled with the lift on the cylinder. The value of the bispectrum in the frequency plane determines the degree of quadratic coupling in the system. Near zero value indicates independent random phases between X(fi ), X(fj ) and Y (fi + fj ) and shows that the data is uncorrelated. Conversely, higher contour levels in the frequency plane identifies quadratic coupling between the two frequencies. We note that the symmetric properties are used in the bispectrum plane to reduce the computational cost. The results show that the new frequencies are generated at the sum and difference of the shedding and oscillating frequencies. Also, we infer a quadratic nature of the nonlinearity from the fact that peaks are observed at twice the excitation frequency. The magnitudes of the peaks appearing in the

(fs,, fs)

(fs,, -fe)

(3fs,, -fs)

(fs,, -(2fs-fe))

Figure 4.22: Cross-bispectrum of the lift and drag coefficient.

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cross-bicoherence are summarized below, along with the coupling relations between peaks in the spectra. • fs + fs → 2fs , b2LLD = 0.99 • 3fs − fs → 2fs , b2LLD = 0.97 • fs − fe → 0.05, b2LLD = 0.945 Elyyan and Akhtar (2008) performed detailed bispectrum analysis of the crossflow oscillating cylinder at this Reynolds number.

4.4

Surface Boundary Conditions

To this end, we employ no-slip and no-penetration boundary conditions on the body surface. However, the boundary conditions can be modified to simulate other flow configurations, such as the case of a rotating cylinder along its axial direction. We can also enforce fixed or time-varying suction/blowing on the surface to perturb the flow field in the near vicinity of the cylinder for control purposes. Time-periodic suction and blowing is a special case representing zero-net-mass-flux actuation, also known as a synthetic jet. Suction and blowing can be used as a controlling mechanism for the flow field. For illustration purposes, a pair of actuators is located symmetrically from the stagnation point, as shown in Figure 4.23(a). The actuator is modeled by imposing a fixed or time-varying velocity normal to the cylinder surface. The defining parameters for the actuator are the peak velocity (VjA ), operating frequency (fj ), location (θj ), and width (Sθ ) of the jet orifice; that is, Vj = g(θj )VjA sin 2πfj

U∞ t D

(4.12)

The frequency of the actuator is chosen to be proportional to the natural shedding frequency, fj = cfs , where c is a constant of proportionality. In the actuator model, g(θj ) defines the profile of the jet velocity as a function of Sθ , such as a top hat function, a sine function, and a sine square function, as shown in Figure 4.23(b) (Rizzetta et al., 1998). Typically, the top hat function may represent blowing while the sine square function is suitable for

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modeling suction on the surface. In our simulations, we use the sine function to represent the jet profile on the cylinder surface. Another important parameter in the dynamics of the actuated flow is the momentum coefficient, which is defined as Cµ =

Vj2 (t)Sθ 2 D U∞

(4.13)

It is a measure of the momentum transferred by the jet to the main flow. It is important to note that the jet profiles, shown in Figure 4.23(b), correspond to different momentum coefficients and are plotted for comparison only. For the same Cµ , the jet velocity for the sine profile should be 1.57 times the jet velocity of the top hat profile, while the jet velocity for the sine square profile should be twice the jet velocity of the top hat profile. From experience, the sine profile converges better than the top hat function during the simulation. We simulated the flow past a circular cylinder at ReD =200 with and without flow actuation for comparison purposes. We place a pair of actuators at θj = 75◦ and θj = 285◦ with a sine wave profile. For this test case, we apply suction control through these actuators with VjA being approximately 30% of the free-stream velocity and Sθ being about 19◦ , such that Cµ = 0.00687 for each actuator. During the simulation, the control is switched on at 80 time units, long enough to reach steady state. In Figure 4.24, we plot the lift and drag coefficients to demonstrate the effect of control. This control technique is used in Chapter 8 to construct the control function for the reduced-order model. We can also simulate periodic suction and blowing by choosing Vj to be an oscillatory function. It corresponds to a zero-net-mass-flux actuator or a synthetic jet. Various combinations of jet location, velocity, and operating frequency can be used to span the variable space in order to find an optimum combination for the desired requirement. This direct search approach with a multivariable domain search is tedious. However, with better search methods (Kolda et al., 2003) and combined with the parallel CFD solver, the optimization problem can be solved efficiently.

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Vj

U

Tj

ST

(a)

(b)

Figure 4.23: (a) A schematic of a synthetic jet actuation pair located on the cylinder surface and (b) velocity profiles for the jet: top hat (dashed), sinusoidal (solid), and sine square (dashdot).

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1

CL , CD

Control ON 0.5

0

-0.5 60

80

100

120

Time Figure 4.24: Time histories of CL and CD with control actuation.

4.5

Reynolds-Averaged Navier-Stokes (RANS)

Turbulent flow is irregular, random, and chaotic. The flow consists of a spectrum of different scales (eddies) distributed over a wide range of scales, where the largest eddies are of the order of the flow geometry, such as the diameter of the cylinder. The smallest scales are dissipated into internal energy by viscous forces. The largest eddies extract their energy from the mean flow. This process of energy transfer from the large eddies to the smallest eddies is called cascade process. The parallel CFD solver developed during the current research performs DNS for a moderate range of Reynolds numbers. As the Reynolds number increases, the grid requirement for DNS increases as Re9/4 . Therefore, some type of modeling or approximation is needed to simplify the flow computations and make them feasible on existing computer platforms. An alternative to DNS is the LES approach, which resolves the energy-containing turbulent scales while modeling the subgrid scales (SGS). LES provides information about a wide range of spatial and temporal scales in the flow at a cost that is significantly lower than the cost of DNS. However, LES computations of high Reynolds number flows with complex geometries remain a formidable task. In the conventional RANS modeling, which is used extensively

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in the engineering community, turbulence models are used to represent the effect of all of the turbulent fluctuations. For this purpose, the unsteady 3-D Reynolds-Averaged NavierStokes (RANS) equations are also solved to simulate the flow at high Reynolds numbers. In the present code, the Spalart-Allamras (S-A) model is used to simulate the flow at high Reynolds numbers. For the turbulent flow, we decompose the primitive variables into a mean component and a fluctuating component; that is, ui = u¯i + u′i P = P¯ + p′

(4.14)

Substituting Equation (4.14) into Equations (2.1) and (2.2), time average the continuity equation and Navier-Stokes equations and obtain ∂ u¯j =0 ∂xj ∂ u¯i ∂ 1 ∂p ∂ ∂ u¯i ∂ u¯j + (¯ uj u¯i ) = − + + ) − u i uj ν( ∂t ∂xj ρ ∂xi ∂xj ∂xj ∂xi

(4.15)

where ui uj is called the Reynolds stress tensor. It is a symmetric tensor and represents correlation between fluctuating velocities. In order to close Equation (4.15), we need to model ui uj , which is called the closure problem. The S-A model is a one-equation turbulence model in which a transport equation is constructed for a working variable based on several physical and experimental facts (Spalart and Allamras, 1992). It provides robustness and accuracy for complicated turbulent flows and takes into account the transition location. Instead of using a transport equation for the eddy viscosity, it defines a working variable that is related to the eddy viscosity by algebraic equations. This indirect calculation has some advantages. Contrary to other transport equations for the eddy viscosity variables, the S-A transport equation does not require a grid finer than that needed to resolve the mean-velocity field accurately. For some eddy viscosity variables (e.g., ǫ in the k − ǫ model), there is ambiguity in the boundary conditions. However, boundary conditions for the working variable are well defined and the S-A transport equation is numerically robust in that it allows small nonzero reference values without degrading the accuracy.

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We solve the RANS equations and a transport equation for the turbulence model. The Reynolds stresses are given by −ui uj = 2νt Sij

(4.16)

where νt is the eddy viscosity given by νt = νefv1 ,

χ3 , χ3 + c3v1 νe χ≡ . ν

fv1 =

The eddy viscosity νt is obtained by solving the following transport equation for the auxiliary variable νe:

2 1 De ν νe 2 e + ∇ · (ν + νe)∇νe ν) = cb1 Se ν − cw1 fw ν + cb2 (∇e | {z } e Dt σ | {z } | {z } d | {z } P D V

(4.17)

ǫ

The left-hand side contains the unsteady and convective terms. The first term on the righthand side is the production term (P ), the second term is a destruction term (ǫ), the last term is the diffusion term (V + D). We note that the S-A model in the present form is without the trip function. The trip function terms provide control over the laminar regions of the shear layers and locate transition where desired. However, in the current application, we do not require such transition control, thus ignoring these terms. The other quantities in Equation (4.17) are as follows: ¯ + νe fv2 , Se = |S| κ2 de2 χ fv2 = 1 − , 1 + χfv1 1 + c6 , fw = g 6 w3 g + c6w3 g = r + cw2 (r6 − r), r≡

νe

e 2 de2 Sκ

,

¯ = (2S¯ij S¯ij ) 12 is the magnitude of the vorticity and de is the distance to the closest where |S|

wall. For large r, fw approaches a constant, so large values of r can be truncated to 10 or

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so. The constants are cb1 = 0.1355, cb2 = 0.622, 2 σ= , 3 κ = 0.41, cw1 =

cb1 (1 + cb2 ) + , κ2 σ

cw2 = 0.3, cw3 = 2.0, cv1 = 7.1 We use a semi-implicit time-advancement using the Crank-Nicolson method for production, destruction, and wall-normal diffusion, and the Adams-Bashforth method for the other terms. Thus, we obtain n+1/2

νen+1 = νen + ∆t P n+1/2 − ǫn+1/2 + Dξ

n+1/2

+ Vξ

3 + ∆t Dηn + Dζn + Vηn + Vζn 2 1 − ∆t Dηn−1 + Dζn−1 + Vηn−1 + Vζn−1 2

(4.18)

The model is discretized using a cell-centered finite difference scheme. However, the convection term is discretized using the first-order accurate upwinding scheme. The eddy viscosity is computed at every time step with the wall boundary condition νe = 0. In the freestream

νe = 0, however, due to numerical errors, νe may be negative near the edge of boundary layer. Since the exact values can not be negative, νe/10 is acceptable.

We performed a 3-D unsteady RANS simulation at ReD =3,900 on a 128×192×48 grid on

a domain of 30D × π. Figure 4.26 shows the time histories of the lift and drag coefficients. It can be observed that, due to the effects of turbulence, higher frequencies are dominant. The spanwise vorticity is plotted in Figure 4.25 and the shear-layer instability can be observed over the cylinder. We compare our numerical results with the experimental results of Ong and Wallace (1996) and the numerical study of Kravchenko and Moin (2000). In their numerical study,

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Figure 4.25: Instantaneous spanwise vorticity isosurfaces (blue: ω z level at -1 and red: ω z level at +1) at ReD =3,900.

Kravchenko and Moin (2000) performed numerical simulations of the flow over a circular cylinder with Lz = π at Re=3,900. They used 48 grid points to resolve the domain in the spanwise directions. They employed LES with the dynamic subgrid scale model (Germano et al., 1991) to account for the turbulent scales that were not resolved by the grid. The mean-drag coefficients and the Strouhal number obtained from the simulations are in good agreement with the published data at this Reynolds number, as shown in Table 4.3. Table 4.3: Computed flow parameters at Re=3900. C¯D Data from St Experiment (Ong and Wallace, 1996) 0.99 ± 0.05

0.215 ± 0.005

LES (Kravchenko and Moin, 2000)

1.04

0.210

3-D URANS (present)

1.06

0.218

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Figure 4.26: (a) Time histories of the lift CL and drag CD coefficients obtained with unsteady RANS at ReD =3,900.

4.6

Summary

The parallel CFD solver is capable of simulating complex flows for closed shaped bodies. We performed various test cases to show different features and applications of the solver. We performed DNS at a moderate Reynolds number of 1,000. We computed the flow parameters and compared the numerical results with existing studies at this Reynolds number. Moreover, we observed that the energy spectrum follows the -5/3 cascade law in the inertial subrange. We then showed that the solver can be employed to analyze shear flows in the spanwise direction. This feature is of special interest for the fluid dynamics of offshore structures, which are hundreds and thousands of feet long and are subjected to nonuniform ocean currents. The numerical scheme used in the algorithm allows simulation of linear and exponential shear flows and flows with the periodic boundary conditions often required in spectral methods. Because the research focuses on the uniform flow around a cylinder, we did not emphasize numerical simulations of shear flows. However, we recommend the use of this CFD tool to perform detailed research in shear flows.

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Next, we elaborated in detail the capability of the CFD solver to simulate moving boundaries. We used ARF technique to study single- and two-degrees-of-freedom. We simulated moving cylinders crossflow oscillations of a circular cylinder at the Reynolds numbers 200 and 525, both in the synchronous and nonsynchronous regions. The numerical results are compared and validated with existing experimental and numerical results. In order to simulate turbulent flows at high Reynolds numbers, we incorporated the Spalart-Allmaras turbulence model to account for the effect of small eddies, which can not be resolved due to computational constraint. We then tested and validated the turbulence model at a Reynolds number of 3,900 and compared the numerical results with existing results in the literature. In addition, we modified the boundary conditions on the cylinder surface to simulate suction, blowing, and synthetic jets as a means to control the flow field. This feature is used in Chapter 8 to develop a full-state feedback controller to suppress vortex shedding.

Part II Reduced-Order Modeling

108

Chapter 5 Oscillator Based Reduced-Order Model of Hydrodynamic Forces As discussed in the preceding Chapters, the parallel CFD solver is based on curvilinear coordinates and is capable of simulating the flow past elliptic cylinders. We performed flow simulations past elliptic cylinders with different eccentricities and computed the lift and drag forces on them. These numerical results are employed to extend our van der Pol-Duffing oscillator model for the circular cylinder lift and drag coefficients (Marzouk et al., 2007). We briefly discuss the background of the van der Pol-Duffing model and present a more general lift and drag model for elliptic cylinders (Akhtar et al., 2008a). The CFD results of the lift and drag coefficients serve as an input to construct the analytical model.

5.1

Background

Nayfeh et al. (2003) investigated two wake-oscillator models for the lift, namely the van der Pol and Rayleigh oscillators. Using a higher-order spectral moments analysis, they found that the phase angle φ13 between the lift components at fs and 3fs is around 90◦ . Based on this finding, they concluded that the van der Pol oscillator is more suitable for modeling the steady-state lift coefficient; that is, C¨L + ω 2 CL = µC˙ L − αCL2 C˙ L 109

(5.1)

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Oscillator Based ROM

110

The angular frequency ω in Equation (5.1) is related (but not equal) to the angular shedding frequency ωs = 2πfs and the parameters µ and α represent the linear and nonlinear damping coefficients, respectively. The values of µ and α are taken to be positive, so that the linear damping is destabilizing while the nonlinear damping is stabilizing. As a consequence, small disturbances grow and large ones decay, both eventually approaching a stable limit cycle. The values of the parameters in Equation (5.1) depend on the Reynolds number. Using the method of multiple scales (Nayfeh, 1973, 1981) and assuming that the oscillator is weakly damped (i.e., µ = O(ǫ) and α = O(ǫ) where ǫ ≪ 1 is a small bookkeeping parameter), the following second-order approximate solution was obtained: CL (t) = a(t) sin[ωt + θ(t) + η(t)] −

α a(t)3 sin[3ωt + 3θ(t)] + · · · 32ω

≡ a1 (t) sin[ωt + θ(t) + η(t)] − a3 (t) sin[3ωt + 3θ(t)] + · · · (5.2) h i 16ω where η(t) = tan−1 αa(t) and the amplitude a(t) and phase θ(t) are governed by the 2 −4µ modulation equations

1 da = (4µa − αa3 ) dt 8 1 11 2 4 dθ 3 2 2 =− µ − αµa + α a dt 8ω 2 32

(5.3) (5.4)

Setting a˙ = 0 in Equation (5.3), one obtains the steady-state value a as a solution of a(4µ − αa2 ) = 0. There are two possibilities: the trivial solution a = 0 or the nontrivial p solution a = 2 µ/α. For the nontrivial solution, it follows from Equation (5.2) that r r µ µ µ and a3 = (5.5) a1 = 2 α 4ω α Moreover, the angle η =

π 2

and, from Equation (5.4), the corresponding expression for

θ˙ = −µ2 /16ω. Consequently, the angular shedding frequency is given by ωs = ω + θ˙ = ω −

µ2 16ω

(5.6)

Equation (5.6) shows that the angular frequency ω of the van der Pol oscillator is not exactly equal to the angular shedding frequency ωs , as one would predict from a first-order expansion (Nayfeh et al., 2003). Hence, an improved second-order approximate expression for the steady-state lift coefficient becomes CL (t) ≈ a1 cos(ωs t) + a3 cos(3ωs t + π2 )

(5.7)

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The methodology used to identify the system parameters for a given Reynolds number is as follows: 1. The CFD solver is used to calculate the time history of the lift coefficient. 2. Spectral analysis is performed on the steady-state part of the CFD data to extract the values of a1 , a3 , and fs (or ωs ). 3. Equations (5.5) and (5.6) are then solved for the nonlinear and linear damping coefficients α and µ and the angular frequency ω. 4. With all of the parameters identified, Equation (5.1) is numerically integrated using a Runge-Kutta routine and the results are compared with the CFD results. Nayfeh et al. (2005) later examined the possibility of using Equation (5.1) to model the lift coefficient in the transient region as well. They simulated the lift model with different initial conditions in the transient region and observed the response of their van der Pol oscillator model. They observed that the closer the initial conditions of the model to the steady-state are, the better the matching between the CFD and the lift model is. Their results showed that the “closeness” of the initial conditions to the steady state depends on the Reynolds number. At low Reynolds numbers, the transient due to the numerical simulation (e.g., impulse or zero initial conditions) decays much slower and the physical transient behavior of the dynamical system dominates later. However, at higher Reynolds numbers, the numerical transients decay much faster and the van der Pol model predicts the transient and steady-state responses accurately. As an example, the nondimensional time for the numerical transients to decay is approximately 50 and 16 for the Reynolds numbers of 200 and 100,000, respectively. Nayfeh et al. (2003) examined the phase relation between the major peaks in the drag and lift and found that it is near 270◦ . Hence, they inferred that the periodic component of the drag is proportional to −CL C˙ L and proposed the following drag model: CD (t) = hCD i − 2

a2 CL (t)C˙ L (t) ωs a21

(5.8)

where a2 is the amplitude of the drag component at 2fs and hCD i is the mean value. In the steady state, the mean component of the drag hCD i = hCD iss is constant; while in the

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transient state, hCD i is a monotonically increasing function of time. The constant value hCD iss is determined from the CFD steady-state time history of the drag and the value of a2 is determined from its spectral analysis. Qin (2004) proposed that the quadratic term in the drag model should be of the form CL2 instead of CL C˙ L . He also found linear coherence between the drag and lift components at fs and 3fs and introduced a linear lift term in the drag model to account for it. In the above models, it was assumed that the phase φ13 = 90◦ and the phase φ12 = 270◦ . However, Marzouk et al. (2007) observed that these phases depend on the Reynolds number and the phases φ13 and φ12 varied up to ±25◦ and ±85◦ , respectively. In order to match the phases, they added a Duffing-type cubic term to the van der Pol oscillator model for the lift coefficient.

5.2 5.2.1

ROM for Elliptic Cylinders Lift Model

In order to accurately predict the phase φ13 , Marzouk et al. (2007) added a Duffing-type nonlinearity (with γ as its coefficient) in Equation (5.1). The coefficient γ allows for matching the phases of the model prediction and the CFD results. Hence, the modified model is given as C¨L + ω 2 CL = µC˙ L − αCL2 C˙ L − γCL3

(5.9)

The solution of Equation (5.9) is written in the following form: CL (t) = c1 cos(ωs t) + c2 sin(ωs t) + c3 cos(3ωs t) + c4 sin(3ωs t)

(5.10)

Substituting Equation (5.10) into Equation (5.9) and separating the terms multiplying the different sine and cosine functions yields the fourth-order linear system Ay = b

(5.11)

where y = {ω 2 , µ, α, γ}, b = {c1 , c2 , 9c3 , 9c4 }ωs2 , and A = A(ci , ωs ). The ci are determined by matching Equation (5.10) with the steady-state CFD solution, which is expressed as CL (t) = a1 cos(ωs t) + a3 cos(3ωs t + φ13 )

(5.12)

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113

Drag Model

The lift and drag forces acting on the cylinder have a common source; that is, the pressure distribution on the surface. In view of the two-to-one frequency relationship between the dominant peaks in the lift and drag, the drag is quadratically related to the lift. However, the actual value of the phase may vary from one Reynolds number to another by up to 85◦ , which is quite significant. Hence, Marzouk et al. (2007) proposed a new model in which the drag is proportional to both CL C˙ L and CL2 in the following manner: CD (t) = hCD i + 2k1

a2 a2 CL2 − CL2 + 2k2 CL C˙ L 2 a1 ωs a21

(5.13)

The first expression in equation (5.13) represents the mean component of the drag, which reaches a constant value hCD iss in the steady-state region. The term − hCL2 i in the second

expression in Equation (5.13) negates the DC component introduced by CL2 .

To determine the contributions of both quadratic expressions to the overall behavior of the drag, Marzouk et al. (2007) matched the amplitude a2 and phase φ12 obtained from the model with the CFD results in the steady state. Substituting Equation (5.12) into Equation (5.13) yields CD (t) = hCD iss + a2 [k1 cos(2ωs t) − k2 sin(2ωs t)] + · · ·

(5.14)

Equation (5.14) is then compared with the CFD result CD (t) = hCD i + a2 cos(2ωs t + φ12 ) + · · ·

(5.15)

to obtain k1 = cos φ12 and k2 = sin φ12 .

5.2.3

Numerical Simulations

We numerically simulate the flow past elliptic cylinders with different thickness ratios on a 192 × 256 grid. The Reynolds number for the current study is 525 and is based on the major-axis of the ellipse. We analyse five elliptic cylinders: τ = 0.6 (Case 1) to τ = 1.0 (Case 5), with an increment of 0.1. All cases are simulated using eight processors and are run long enough to reach steady state. Grid and domain independence studies are critical for verifying the accuracy of the computational results. Detailed analyses of the grid and domain independence study at this Reynolds number are discussed in Chapter 3.

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In Figure 5.1, we plot the instantaneous spanwise vorticity for all cases. We observe a similar vortex shedding pattern, however the shedding frequency is different in each case. Moreover, the width of the wake increases with increasing τ , thereby increasing the drag. In other words, the size of the vortices being shed is of the order of Ly and the vorticity increases with τ . This is also evident from the fact that the projected area of the cylinders, as “seen” by the flow, increases with increasing τ .

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(a)

τ = 0.6

(b)

τ = 0.7

(c)

τ = 0.8

(d)

τ = 0.9

(e)

τ = 1.0

Figure 5.1: Instantaneous spanwise vorticity contours.

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116

Results and Discussion

We compute the time history of the pressure distribution on the surface. Using Equation (3.2), we compute the lift and drag coefficients for each case. We observe an increase in the amplitudes of the fluctuating forces as τ increases, as shown in Figures 5.2(a-e). Using spectral analysis, we compute the dominant frequencies and the phases φ12 and φ13 . The results are given in Table 5.1. In Figure 5.3, we plot the shedding frequencies and the amplitudes a1 , a2 , and a3 of the dominant frequencies fs , 2fs , and 3fs , respectively. We note that odd harmonics appear in the lift spectrum while even harmonics appear in the drag spectrum. In Figure 5.3(a), we observe that the shedding frequency fs decreases with increasing τ , while the corresponding amplitude a1 increases as τ increases, as shown in Figure 5.3(b). We observe approximately a five fold increase in a1 as τ increases from 0.6 to 1.0. Likewise, there is a similar trend in a2 and a3 , however the harmonics are much stronger in Case 5 than for Case 1 when compared with the respective a1 . For example, in Case 1, a2 and a3 are 3.4% and 0.23% of a1 , respectively, whereas in Case 5, they are 10.9% and 3%, respectively. Thus, the power distribution in the frequency spectrum is related to the projected area of the cylinder. Moreover, we also note a steady increase in the mean drag, a direct consequence of the increase in the projected length of the cylinder, as shown in Figure 5.3(c). We compute the phases between the fundamental frequency and its harmonics and show the results in Figure 5.3(d). We observe that φ12 varies between −30◦ and −19◦ and φ13

ranges from 71◦ to 97◦ . We note that the phase information is critical in establishing the

accuracy of the reduced-order model. We also compute the angular shedding frequency (ωs = 2πfs ) and list the result in Table 5.1. We use the CFD data and compute the hydrodynamic forces on elliptic cylinders. We perform the spectral analysis on the time histories of the lift and drag coefficients to obtain the dominant frequencies and their magnitudes. This data serves as an input to determine the parameters of the reduced-order model in Equation (5.9). We list the model parameters for the five cases of elliptic cylinders in Table 5.2. As mentioned earlier, the angular frequency ω of the van der Pol-Duffing oscillator model is

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0.5

0.6

0.4 0.4

CL , CD

CL , CD

0.3 0.2 0.1

0.2 0

0 −0.2

−0.1 −0.2

−0.4

300

305

310

315

(a)

320

T ime

325

330

335

340

300

305

310

τ = 0.6

315

(b)

320

T ime

325

330

335

340

330

335

340

τ = 0.7

1 0.8

1

0.6 0.5

CL , CD

0.2 0

0

−0.2 −0.5

−0.4 −0.6 300

305

310

315

(c)

320

T ime

325

330

335

340

300

305

310

τ = 0.8

315

(d)

320

T ime

325

τ = 0.9

1.5 1

CL , CD

CL , CD

0.4

0.5 0 −0.5 −1 300

305

310

315

(e)

320

T ime

325

330

335

340

τ = 1.0

Figure 5.2: Time history of the lift (solid) and drag (dashed) coefficients.

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Table 5.1: Spectral Analysis of the Lift and Drag Coefficients. Case 1

Case 2 Case 3 Case 4 Case 5

fs

0.318

0.273

0.248

0.231

0.225

a1

0.238

0.425

0.643

0.891

1.178

a2

0.0081

0.0202

0.0437

0.0836

0.129

a3

0.00055

0.0028

0.0087

0.0182

0.036

0.51

0.66

0.85

1.08

1.42

−30◦

−19◦

−21◦

−20◦

φ13

81◦

−25◦

ωs

1.998

1.715

1.558

1.451

1.414

hCD iss φ12

71◦

86◦

98◦

97◦

close (but not equal) to the angular shedding frequency ωs . In Figure 5.4(a), we plot the liftTable 5.2: Parameters of the Reduced-Order Model. Case 1

Case 2

Case 3

Case 4

Case 5

ω

1.992

1.671

1.528

1.484

1.436

µ

0.037

0.084

0.169

0.236

0.35

α

2.582

1.831

1.623

1.198

1.012

γ

0.855

1.146

0.279

k1

0.866

0.906

0.946

k2

−0.5

−0.423 −0.326

−0.137 −0.042 0.934

0.94

−0.358 −0.342

model coefficients µ, α, and γ as a function of τ . The values µ and α are positive, indicating the presence of a limit cycle. From the values of α and γ, it is clear that the influence of the Duffing term, though lesser in magnitude, cannot be neglected in the modeling. Likewise, we compute the coefficients k1 and k2 in the drag model and plot them in Figure 5.4(b). The values of k1 and k2 suggest that both of the terms CL2 and CL C˙L contribute significantly in the reduced-order modeling of the drag. Using the parameters in the reduced-order model, we compute the time histories of the lift and drag coefficients and compare them with the

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0.4

1.2 1

a1, a2, a3

fs

0.3

0.2

0.1

0.8 0.6 0.4 0.2 0

0

0.6

0.7

0.8

τ

0.9

1

(a) Fundamental frequency fs

0.6

0.7

0.8

τ

0.9

1

(b) a1 (square), a2 (diamond), and a3 (delta). 180

90

φ12 , φ13

CD

1.5

1

0.5

0

0

-90

0.6

0.7

0.8

τ

0.9

(c) Mean drag coefficient

1

-180

0.6

0.7

0.8

τ

0.9

1

(d) φ12 (delta) and φ13 (square).

Figure 5.3: Spectral analysis parameters from the CFD simulations.

numerical results obtained from the CFD. In Figure 5.5, we plot the time histories of the lift coefficient obtained for all of the cases. We observe a good agreement between the results of the model and the CFD results. Similarly, using the solution for CL and the parameters (k1 ,k2 ), we compute the drag coefficients from Equation (5.13). We plot the time histories of the drag coefficients obtained from the model for all cases and compare them with the CFD

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2.5

µ,α,γ

2 1.5 1 0.5 0 0.6

0.7

0.8

τ

0.9

1

(a) µ(square), α(delta), and γ(diamond).

1

k1 , k2

0.5 0

-0.5 -1 0.6

0.7

0.8

τ

0.9

1

(b) k1 (square) and k2 (delta).

Figure 5.4: Parameters of the van der Pol-Duffing oscillator model.

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results. Figure 5.6 shows that the model results compare well with the CFD data. Hence, the van der Pol-Duffing oscillator predicts the CFD simulation results both in amplitude and phase. Moreover, the reduced-order model is not only valid for a circular cylinder but also for elliptic cylinders with varying thickness ratios.

5.4

Summary

The lift and drag coefficients for two-dimensional flows over elliptic cylinders were examined and modeled as a preliminary step to understanding the complex problem of vortex shedding. We extended our circular cylinder van der Pol-Duffing oscillator for the lift and drag to elliptic cylinders. We performed numerical simulations over elliptic cylinders with varying eccentricities using the parallel CFD code. We studied five cases of elliptic cylinders and calculated the hydrodynamic forces acting on each cylinder. We performed spectral analysis of the time histories of the lift and drag coefficients and computed the dominant frequencies. Using this data for different elliptic cylinders, we developed a reduced-order model to represent the lift and drag coefficients acting on them. The model results compare well with the CFD data, indicating generality of the reduced-order model. Moreover, we can also perform more simulations on elliptic cylinders with the incoming flow at different angles-of-attack. It will lead to a more general reduced-order model of lift and drag forces on a bluff body.

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0.8 Model CFD

Model CFD

0.4

CL

CL

0.2

0

−0.2

−0.4 200

122


0

−0.4

202

204

206

208

210

212

214

216

218

−0.8 200

220

202

204

206

208

Time

(a)

210

212

214

216

218

220

214

216

218

220

Time

τ = 0.6

(b)

τ = 0.7

1.5 1

Model CFD

Model CFD 1

0.5

CL

CL

0.5

0

0

−0.5 −0.5

−1 −1

200

202

204

206

208

210

212

214

216

218

−1.5 200

220

202

204

206

208

Time

(c)

210

212

Time

τ = 0.8

(d)

τ = 0.9

Model CFD 1.2

CL

0.2

−0.8

−1.8 200

202

204

206

208

210

212

214

216

218

220

Time

(e)

τ = 1.0

Figure 5.5: Comparison between the time histories obtained by the lift model with the CFD results.

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Model CFD

Model CFD 0.69

0.52

CD

CD

0.67

0.512

0.504 200

0.65

202

204

206

208

210

212

214

216

218

0.63 200

220

202

204

206

208

Time

(a)

210

212

214

216

218

220

214

216

218

220

Time

τ = 0.6

(b)

0.94

τ = 0.7

1.25 Model CFD

Model CFD 1.2

0.9

CD

CD

1.15

0.86

1.1

1.05 0.82 1

0.78 200

202

204

206

208

210

212

214

216

218

0.95 200

220

202

204

206

208

Time

(c)

210

212

Time

τ = 0.8

(d)

τ = 0.9

Model CFD

CD

1.6

1.4

1.2 200

202

204

206

208

210

212

214

216

218

220

Time

(e)

τ = 1.0

Figure 5.6: Comparison between the time histories obtained by the drag model with the CFD results.

Chapter 6 Proper Orthogonal Decomposition based Galerkin Expansion Low-dimensional models successfully describe the qualitative properties of the cylinder wake (Deane and Mavriplis, 1994; Ma and Karniadakis, 2002; Noack et al., 2003). The Galerkin expansion is the natural candidate to describe globally synchronized dynamics, such as the von Kármán vortex street (Rempfer et al., 2003). However, there are a number of options available for the reduced basis, such as the Langrange, Hermite, Taylor, proper orthogonal decomposition (POD), and centroidal Voronoi tessellations (CVT) basis (Burkardt et al., 2002). The choice of the basis in the Galerkin expansion would affect its dimension and properties. Noack et al. (2003) classified these modes in terms of mathematical, physical, and empirical approaches, but we restrict our discussion to the empirical approach. The reduced basis cannot be determined a priori as in finite-element methods, rather the reduced basis is determined a posteriori, using experimental results (particle image velocimetry) or numerical simulation data previously obtained for a given flow configuration. For reducedorder modeling, a promising choice of such basis is the POD basis. In this Chapter, the POD eigenfunctions of the flow past a circular cylinder are discussed. These eigenfunctions are used to develop a reduced-order dynamical system. The data used to generate the POD basis has been generated from the DNS of the flow past a cylinder at different Reynolds numbers.

124

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POD-Galerkin Expansion

125

Mathematical Formulation

The POD provides a tool to formulate an optimal basis or minimum degrees of freedom (or modes) required to represent a dynamical system. POD is also known as the Karhunen-Loeve expansion in statistics and principal component analysis or empirical orthogonal functions (EOF) in meteorology. It has successfully been applied to many engineering and scientific systems, including low-dimensional dynamics modeling (Berkooz et al., 1993; Deane and Mavriplis, 1994; Ma and Karniadakis, 2002; Noack et al., 2003), image processing (Holmes et al., 1996), and pattern recognition (Sirovich and Kirby, 1987). POD has been widely used to identify the coherent structures (Bakewell and Lumley, 1967) in turbulent flows and examine their stability (Holmes et al., 1996). The flow field data (u, v, w) is generated from an experiment or a numerical simulation and is assembled in a matrix W3N ×S , as shown in Equation (6.1); each column represents one time instant or a snapshot and S is the total number of snapshots for N grid points in the domain. The vorticity field can also be used for POD, however, in the case of the velocity field, the eigenvalues are a direct measure of the kinetic energy  (1) (2) u1 ... u  1 ..  ..  . .   (1)  uN u(2) ... N   (1) (2)  v1 v1 ...  ..  .. W= . .   (1) (2)  vN vN ...   (1)  w1 w1(2) . . .   .. ..  . .  (1) (2) wN wN . . .

in each mode.  (S) u1  ..  .   (S)  uN   (S)  v1   ..  .   (S)  vN   (S)  w1   ..  .   (S) wN

Mathematically, we compute Φ for which the following quantity is maximum:

|u, Φ|2 , kΦk2

(6.1)

(6.2)

where h.i denotes the ensemble average. Applying variational calculus, one can show that

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Equation (6.2) is equivalent to a Fredholm integral eigenvalue problem represented as Z Rij (x, x′ )Φj (x′ ) dx = λΦj (x), (6.3) Ω

where i, j are the number of velocity components and R(x, x′ ) is the two-point space-time correlation tensor. In a tutorial, Smith et al. (2005) discuss various methods to compute the POD functions and their application in low-dimensional modeling of turbulence. In the classical POD or direct method, originally introduced by Lumley (Bakewell and Lumley, 1967), Rij is a two-point spatial-correlation tensor and the eigenfunctions are the POD modes. In this approach, the average operator is estimated in time. On the other hand, if the average operator is evaluated as a space average over the domain of interest, the method is known as the method of snapshot (Sirovich, 1987). In this approach, we formulate a temporal-correlation function from the snapshots and transform it into an eigenvalue problem as follows:

where (a, b) =

R

Cij = (ui , uj ) ,

(6.4)

a.b dΩ represents the inner product between a and b. The POD modes are

Ω

then computed by solving the eigenvalue problem CQ = Qλ

(6.5)

where Q and λ are the eigenvectors and eigenvalues, respectively. Since C is nonnegative Hermitian, Q is orthogonal by definition. The POD modes are computed as follows: 1 Φi = √ WQi λi

(6.6)

An important characteristic of these modes is orthogonality; that is, Φi .Φj = δij , where δij is the kronecker delta. The optimality of the POD modes lies in capturing the greatest possible fraction of the total kinetic energy for a projection onto the given set of modes. In another approach, we compute the singular value decomposition (SVD) of the data ensemble, W = UΣV T , where U represents the POD basis. This method has a limited application, especially when the grid size (N ) is large. Beattie et al. (2006) proposed an algorithm, inspired from domain decomposition ideas, for a scalable parallel efficient computation of the POD basis vector with low communication overhead. An SVD of a (spatial) subdomain time

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history is calculated locally on each processor followed by the exchange of a small number of dominant Vp (right singular vector on the pth processor) with other processors. An iterative application of this step showed promising results for complex fluid flows, gravity currents, and two-dimensional flow past a square cylinder. We have discussed three methods to compute the POD eigenfunctions, which can be summarised as follows: 1. WW T ∈ RN ×N : WW T Ui = σi2 Ui (if N ≪ S) 1 2. W T W ∈ RS×S : W T WVi = σi2 Vi and Ui = WVi (if S ≪ N) σi N ×S 3. W ∈ R : WVi = σi Ui where



(u)

φ  1,1  ..  .   (u)  φ1,N   (v)  φ1,1   . U =  ..   (v)  φ1,N   (w)  φ1,1   ..  .  (w) φ1,N

(u)

φ2,1 .. . (u)

φ2,N (v)

φ2,1 .. . (v)

φ2,N (w)

φ2,1 .. .

(w)

φ2,N

 (u) . . . φS,1  ..  .   (u)  . . . φS,N   (v)  . . . φS,1   ..  .  and  (v)  . . . φS,N   (w)  . . . φS,1   ..  .   (w) . . . φS,N



   Σ = diag    

σ1 σ2 .. . σS

(6.7) (6.8) (6.9)

       

(6.10)

In Equation (6.10), Σ contains the singular values of W. The corresponding singular values σi are real and positive arranged in Σ in descending order. The σi are related to the eigenvalues λi obtained in the method of snapshots by λi = σi2 . They are used for ordering the POD eigenfunctions for the dynamical system and represent energy contained in each eigenfunction.

6.2

Flow Simulation Data

Among other factors, both of the number of snapshots and the number of POD modes used in the simulation affect the accuracy of the reduced-order model. Increasing the number

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of modes increases the accuracy of the simulation, but it also increases the dimension of the solution space spanned by the modes. Similarly, increasing the number of snapshots increases the flow dynamics information, however, it limits the computational power and available computer memory. Deane et al. (1991) observed that 20 snapshots are sufficient for the construction of the first eight eigenfunctions. Numerical studies (Noack et al., 2005) suggest that the first M POD modes, where M is even, resolve the first M/2 temporal harmonics and require 2M number of snapshots for convergence. In the current study, we simulate the flow past a circular cylinder at different Reynolds numbers. Details of each simulation are given in Table 6.1. For ReD =100 and 200, we took 40 snapshots over a period of one vortex-shedding cycle. For a relatively higher Reynolds number of 525, 80 snapshots were recorded over two vortex-shedding cycles in order to capture more information of the flow field. We note the increase in the domain size; it is intentionally done to minimize the effect of the pressure term on the outflow boundary (details are discussed later in Chapter 6). Table 6.1: POD configurations. Case

ReD

Domain Lz

Grid size

Snapshots (Period)

I

100

50D

−

192 × 256

40(1)

II

200

50D

−

192 × 256

40(1)

III

525

50D

−

192 × 256

80(2)

IV

525

40D

π

128 × 192 × 32

80(2)

The POD eigenfunctions are used as a basis for a Galerkin projection of the incompressible Navier-Stokes equations. These POD eigenfunctions are orthogonal, divergence-free, and satisfy the boundary conditions. These properties are independent of the number of snapshots. Details of the reduced-order model of the velocity and pressure are discussed in Chapters 7 and 8, respectively. We write the velocity field as the sum of the mean flow (¯ u) ¯ = hui, where hi is the time average of and the velocity fluctuations (u′ ). The mean flow u the assembled data, is subtracted from W. Then, the fluctuations are expanded in terms of

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the POD eigenfunctions (Φi ) as follows: ¯ (x) + u(x, t) ≈ u

M X

qi (t)Φi (x),

(6.11)

i=1

where M is the number of POD modes used in the projection. The average flow fields are plotted in Figures 6.1-6.4. We observe an increase in the maximum streamwise velocity component with an increasing Reynolds number. Similarly, the region of negative streamwise velocity decreases for higher Reynolds numbers. For the 3-D flow, the average fields have spanwise variations due to flow instability in the third dimension.

(a)

(b)

Figure 6.1: Average flow fields: (a) streamwise and (b) crossflow velocity components at ReD =100.

6.3

POD Modes of Velocity Field

In our parallel CFD simulations, each processor records its own flow field data. We used SVD to compute the eigenfunctions and corresponding eigenvalues. These eigenvalues quantify the energy contained in the ith mode of the expansion; that is, λi = h(Φi , Φi )2Ω i = hqi2 i,

(6.12)

where λi /2 can be considered as the kinetic energy of Φi . This property of the POD modes makes the Galerkin approximation efficient for the current study.

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(a)

(b)


(a)

(b)


In Figure 6.5, we plot the first 30 eigenvalues obtained from different cases. Each eigenP value is normalized as λi / Si=1 λi . We observe that the two modes of each pair have values

of the same order and they decrease from one pair to the next approximately in a geometric progression. In Case I, the energy decays much faster as compared to other Reynolds numbers, indicating that fewer modes required to represent the flow field. Similarly, Case II shows a similar trend but with a less steeper decay in energy than Case I. In the other two

cases III and IV, we simulate 2-D and 3-D flows at ReD =525 and compute the POD modes.

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(a)

(b)

(c)

Figure 6.4: Average flow fields: (a) streamwise, (b) crossflow, and (c) spanwise velocity components at ReD =525

The energy decay is more for the 2-D flow than that for the corresponding 3-D flow. In order to find the contribution of each mode to the total energy of the system, we define P the cumulative energy as Ei = Si=1 λi . In Figure 6.6, we plot variation of the cumulative

energy of the flow with the number of modes. We observe that most of the energy is contained

in the first few modes. The first two eigenfunctions in each case clearly contain most of the energy of the system. In Case I, the first ten modes contain more than 99.9% of the total flow energy. Similary, Case II shows a similar trend but requires more modes to capture the same amount of energy as in Case I. For the other two cases, we observe more than 30 modes are required to capture 99.9% of the energy. The energy content spanned by the first 30

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100 10

-1

λ

10-2 10

-3

10-4 10

-5

10-6

5

10

15

20

25

30

Mode Figure 6.5: Normalized eigenvalues. Square: Case I; triangle: Case II; circle; Case III, and diamond; Case IV.

modes is more in a 2-D flow than in a 3-D flow. In Figures 6.7-6.15, we plot the sreamwise, crossflow, and spanwise velocity modes for all of the cases (I-IV). In the two-dimensional cases, we plot contours of the first ten most energetic POD modes of the streamwise and crossflow velocities for all of the cases, whereas in Case IV, we plot isosurfaces of the spanwise POD modes. Like the eigenvalue pairs, we observe a similar pattern in the POD modes. The pattern is either symmetric or anti-symmetric with respect to the vertical axis. From the corresponding eigenvalues, we observe that most of the energy is contained in the first two modes, and the corresponding POD modes show large structures in the near-wake region of the cylinder. Looking at the first eight POD modes of the streamwise and crossflow velocity components in Figures 6.7 and 6.8, respectively, we observe that the POD modes φui (i = 1, 2, 5, 6) of the streamwise component are antisymmetric with respect to the x -axis; that is, φui (x, −y) = −φui (x, y),

(6.13a)

while the φui (i = 3, 4, 7, 8) are symmetric; that is, φui (x, −y) = φui (x, y).

(6.13b)

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Cumulative Energy

1 0.8 0.6 0.4 0.2 0

2

4

6

8

10

Mode Figure 6.6: The normalized cumulative energy. Square; Case I, triangle; Case II, circle; Case III, and diamond; Case IV.

On the other hand, the POD modes φvi of the crossflow component are symmetric for i = 1, 2, 5, 6 with respect to the x -axis; that is, φvi (x, −y) = φvi (x, y),

(6.14a)

and antisymmetric for i = 3, 4, 7, 8; that is, φvi (x, −y) = −φvi (x, y). We observe a similar trend for other Reynolds numbers.

(6.14b)

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Figure 6.7: The streamwise velocity modes (φui , i = 1, 2, ..., 10) at ReD =100

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Figure 6.8: The crossflow velocity modes (φvi , i = 1, 2, ..., 10) at ReD =100

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Figure 6.13: Isosurfaces of the streamwise velocity modes (φui , i = 1, 2, ..., 10) at ReD =525 with 25% transparency.

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Figure 6.14: Isosurfaces of the crossflow velocity modes (φvi , i = 1, 2, ..., 10) at ReD =525 with 25% transparency.

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Figure 6.15: Isosurfaces of the spanwise velocity modes (φw i , i = 1, 2, ..., 10) at ReD =525 with 25% transparency.

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Summary

In this Chapter, we simulated the flow past a circular cylinder at various Reynolds numbers and recorded the velocity field (snapshots) at equal intervals. The velocity was then resolved into mean and fluctuating components and expanded in a Galerkin approach. The spatial term was represented by POD modes that optimally capture the energy content in the flow field. Moreover, the POD modes are orthogonal, divergence-free, and satisfy the boundary conditions. We computed the POD modes for different Reynolds numbers and observed symmetry within these modes. The temporal coefficients are computed in Chapter 7 by projecting the Navier-Stokes equations onto these POD modes.

Chapter 7 Reduced-Order Modeling of the Velocity Field 7.1

Introduction

Reduced-order modeling of incompressible flows plays an important role in academic and industrial research. These models allow an analytical insight into the physical phenomena and enable application of dynamical systems theory and control methods. The POD approach was first introduced in the field of turbulence by Lumley (Bakewell and Lumley, 1967) to identify the coherent structures in the flow and examine their stability (Holmes et al., 1996). Later, Sirovich Sirovich (1987) introduced the snapshot method to study the dynamics of some turbulent flows. In this method, a set of instantaneous flow field solutions, or snapshots, is obtained from either experimental data or a numerical simulation. This allows reduction of large data sets obtained from computational fluid dynamics (CFD) or particle image velocimetry (PIV), while still preserving the dominant features of the flow represented by the POD eigenfunctions or modes. The optimal basis functions capture the dominant energy content from numerical and experimental data and make the POD eigenfunctions a suitable candidate for the reduced-order model. Low-dimensional dynamical models consist of a set of nonlinear ordinary-differential equations (ODE) of the form q˙ = F(q). 144

(7.1)

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This low-dimensional model can be used to effectively apply various strategies for flow control. The POD approach has been implemented successfully in the low-dimensional modeling of laminar cylinder wakes. Deane et al. (1991) modeled the dynamics of the flow past a cylinder with an eight-dimensional Galerkin model. Ma and Karniadakis (2002) generalized the POD Galerkin models to predict the three-dimensional transition initiated by the Mode A instability (Williamson, 1996) at Re ≈ 180. The results of the POD based reduced-order model, obtained in these studies, are in agreement with the numerical simulations, at least for short-time integration. However, longtime integration of the reduced-order model might not produce the limit cycles obtained with the CFD code. The solution can drift to some erroneous state even if it is initialized with the correct periodic state. This instability is associated with the presence of multiple spurious limit cycles. Foias et al. (1991) investigated the existence of multiple spurious steady states in the Galerkin expansion of the Kuramoto-Sivashinsky equation. For a similar equation, Aubry et al. (1993) also found spurious states in their POD model that captured 99.99% of the system’s energy. They observed that the predicted solution is not the right limit cycle. In the POD model of the flow past a cylinder, Sirisup and Karniadakis (2004) showed that the onset of divergence from the correct limit cycle depends on the number of modes in the Galerkin expansion, the Reynolds number, and the flow geometry. They used a spectral vanishing viscosity (SVV) method (Tadmor, 1989), which adds a small amount of modedependent dissipation satisfying the entropy condition while retaining the spectral accuracy. Thus, Equation (7.1) is modified as q˙ = F(q) − H(q; ǫ, Qǫ ),

(7.2)

where ǫ → 0 is a viscosity amplitude and Qǫ is a viscosity convolution kernel. The SVV is typically applied to the higher-order modes and the numerical value of ǫ depends on the number of modes for which the SVV is activated. The parameters for the SVV model are found by an empirical method and a bifurcation analysis. However, their exact values are not known a priori and depend on the flow geometry and the number of POD modes. In the current study, we discuss a shooting method for homing on the correct limit cycle.

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In this Chapter, we develop a reduced-order model for the velocity field of the flow past a circular cylinder. We simulate a two-dimensional flow past a stationary cylinder at different Reynolds numbers, as given in Table 6.1. As an example, we develop a reduced-oder model for ReD =100 using the first ten dominant modes. Then, the flow field is reconstructed using the ODE model and compared with the CFD results. An important aspect regarding the long-time integration of the POD based Galerkin models is discussed in detail. We applied a shooting method to compute the period and appropriate initial conditions to accurately predict the limit cycles of the dynamical system.

7.2 7.2.1

Reduced-Order Model Galerkin Projection

The POD eigenfunctions are used as a basis in a Galerkin projection of the incompressible Navier-Stokes equations. The projection is performed by the inner product of the POD modes with Equation (2.2) as ∂u 1 2 Φk , + (u.∇)u + ∇p − ∇ u = 0, (7.3) ∂t ReD R where (a, b) = Ω a · b dΩ represents the inner product between a and b and k = 1, 2, ..., M .

We substitute Equation (6.11) into Equation (7.3) and perform the inner product of each term. We collect them to obtain a reduced-order model comprising a set of M ordinary-

differential equations. The inner product reduces the two (three) momentum equations, depending upon two (three) dimensional flow, to only one equation. In other words, the system is reduced from 2M (3M ) equations to M equations. To elaborate various terms in the reduced-order model, we expand each term individually. Time-derivative term From the Galerkin expansion in Equation (6.11), we note that the average velocity is only a function of space and is independent of time. Thus, the temporal operator acts on the

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velocity coefficients and the partial derivative term is converted to full differential; that is ∂¯ u(x) ∂u′ (x, t) ∂u + = Φk , Φk , ∂t ∂t ∂t PM ∂ m=1 qm (t)Φm (x) = Φk , 0 + ∂t M X dqm = Φk , Φm (7.4) dt m=1 From the definition of the POD modes, the eigenfunctions Φi (φui , φvi , φw i ) are orthogonal by construction; that is,

Φk ,

M X

m=1

Φm

= Φk , Φk = (φuk , φuk ) + (φvk , φvk ) + (φvk , φvk ) = σk

(7.5)

Substituting Equation (7.5) into Equation (7.4), we obtain dqk ∂u = σk Φk , ∂t dt

(7.6)

Convection term We substitute Equation (6.11) into the second term of Equation (7.3) and separate the terms, ′ ′ Φk , u.∇u = Φk , (¯ u + u ) · ∇(¯ u+u) ′ ′ ′ ′ = Φk , u ¯ · ∇¯ u+u ¯ · ∇u + u · ∇¯ u + u · ∇u =

Φk , u ¯ · ∇¯ u+u ¯·∇ +

=

M X

m=1

qm Φm · ∇

Φk , u ¯ · ∇¯ u+u ¯· +

M X M X

m=1 n=1

M X

m=1 M X

qm Φm + qn Φn

n=1

M X

m=1

∇Φm qm +

Φm · ∇Φn qm qn

M X

qm Φm · ∇¯ u

M X

Φm · ∇¯ uq m

m=1

m=1

(7.7)

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We expand the inner product of each term on the right-hand side of Equation (7.7). The first term is Φk , u ¯ · ∇¯ u = (φuk , u ¯ · ∇u) + (φvk , u ¯ · ∇v) + (φw ¯ · ∇w) k,u ∂ u¯ ∂ u¯ ∂ u¯ = φuk u¯ + v¯ + w¯ ∂x ∂y ∂z ∂¯ v ∂¯ v ∂¯ v + v¯ + w¯ + φvk u¯ ∂x ∂y ∂z ∂ w ¯ ∂ w ¯ ∂ w¯ + φw ¯ + v¯ + w¯ k u ∂x ∂y ∂z

(7.8)

The second term is Φk , u ¯·

M X

m=1

∇Φm qm = + +

M X

m=1 M X

m=1 M X m=1

φuk u¯ φvk u¯

∂φu ∂φu ∂φum + v¯ m + w¯ m qm ∂x ∂y ∂z

∂φv ∂φv ∂φvm + v¯ m + w¯ m qm ∂x ∂y ∂z

φw ¯ k u

∂φw ∂φw ∂φw m + v¯ m + w¯ m qm ∂x ∂y ∂z

∂φu ∂φu ∂φu = φuk u¯ m + v¯ m + w¯ m ∂x ∂y ∂z v v ∂φ ∂φv ∂φ + φvk u¯ m + v¯ m + w¯ m ∂x ∂y ∂z w w ∂φm ∂φm ∂φw m w + φk u¯ ∗q + v¯ + w¯ ∂x ∂y ∂z

(7.9)

The third term is M X

M X ∂ u¯ ∂ u¯ ∂ u¯ qm + φvm + φw Φk , qm Φm · ∇¯ u = φuk φum m ∂x ∂y ∂z m=1 m=1

+ +

M X

m=1 M X

m=1

φvk φum

∂¯ v ∂¯ v ∂¯ v + φvm + φw qm m ∂x ∂y ∂z

u φw k φm

∂ w¯ ∂ w¯ ∂ w¯ + φvm + φw qm m ∂x ∂y ∂z

∂ u¯ ∂ u¯ ∂ u¯ = φuk φum + φvm + φw m ∂x ∂y ∂z ∂¯ v ∂¯ v ∂¯ v + φvm + φw + φvk φum m ∂x ∂y ∂z ¯ ¯ ¯ v ∂w w ∂w w u ∂w ∗q + φm + φm + φk φm ∂x ∂y ∂z

(7.10)

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The fourth term is Φk ,

M X

m=1

qm Φm · ∇

M X

qn Φn =

n=1

φuk ,

+ φvk , + φw k,

M X M X

m=1 n=1 M X M X

m=1 n=1 M X M X m=1 n=1

qm qn Φn · ∇φum qm qn Φn · ∇φvm

qm qn Φn · ∇φw m

(7.11)

Equation (7.11) is further split into sub-terms for convenience as follows: φuk ,

M X M X

M X M X ∂φu ∂φum ∂φu ∗ ) qm qn Φn · ∇φum = φuk , qm qn (φun ∗ m + φvn ∗ m + φw n ∂x ∂y ∂z m=1 n=1 m=1 n=1 ∂φum ∂φum ∂φum ′ u u v w = q ∗ φk (φn ∗ + φn ∗ + φn ∗ ) ∗ q (7.12) ∂x ∂y ∂z

Similarly, the other sub-terms can be expanded as φvk ,

M X M X

m=1 n=1

qm qn Φn ·

∇φvm

qm qn Φn ·

∇φw m

M X M X

∂φv ∂φvm ∂φvm + φvn ∗ m + φw ∗ ) n ∂x ∂y ∂z m=1 n=1 ∂φvm ∂φvm ∂φvm ′ v u v w = q ∗ φk (φn ∗ + φn ∗ + φn ∗ ) ∗ q (7.13) ∂x ∂y ∂z φuk ,

=

qm qn (φun ∗

and φw k,

M X M X

m=1 n=1

M X M X

∂φw ∂φw ∂φw m m + φvn ∗ m + φw ∗ ) n ∂x ∂y ∂z m=1 n=1 ∂φw ∂φw ∂φw m m m u v w = q′ ∗ φw (φ ∗ + φ ∗ + φ ∗ ) ∗ q (7.14) k n n n ∂x ∂y ∂z

=

φuk ,

qm qn (φun ∗

Combining Equations (7.12), (7.13), and (7.14), we obtain Φk ,

M X

m=1

qm Φm · ∇

M X n=1

qn Φn

∂φu ∂φu ∂φum = q ∗ φuk (φun ∗ m + φvn ∗ m + φw ∗ ) n ∂x ∂y ∂z ′

∂φvm ∂φv ∂φvm + φvn ∗ m + φw ) n ∗ ∂x ∂y ∂z ∂φw ∂φw ∂φw m m m v w w u + φn ∗ + φn ∗ ) ∗q + φk (φn ∗ ∂x ∂y ∂z + φvk (φun ∗

Equation (7.15) represents the quadratic term in the model.

(7.15)

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Pressure term The pressure term in the model is also projected onto the POD modes as follows: Z Φk , ∇p = Φk · ∇p Z Z = − (∇ · Φk )p + p(n · Φk )

(7.16)

Ωs

We note that using Green’s theorem and the divergence-free property, the pressure term drops out from Equation (7.21) for the case of p=0 on the outerflow boundary Ωso (Ma and Karniadakis, 2002). The POD eigenfunctions are identically zero on the inflow boundary because the average flow is subtracted from the total flow. However, in case of Neumann boundary conditions on Ωso , the contribution of the pressure term is not exactly zero for the cylinder wake. The outer domain is intentionally kept at 25D from the cylinder to minimize the pressure effects. Hence, the pressure is neglected on the outflow boundary so the pressure term vanishes in the reduced-order model (Noack et al., 2005). Thus, Φk , ∇p = 0

(7.17)

Diffusion term We substitute Equation (6.11) into the diffusion term to obtain 1 1 2 ′ 2 Φk , Φk , ∇ (¯ u+u) ∇ u =− ReD ReD M X 1 2 2 =− Φk , ∇ u ¯+ qm ∇ Φm ReD m=1

(7.18)

We expand each term on the right-hand side of Equation (7.18) into its components. The term containing the mean velocity term is ∂ 2 u¯ ∂ 2 u¯ ∂ 2 u¯ 2 Φk , ∇ u ¯ = φuk ( 2 + 2 + 2 ) ∂x ∂y ∂z 2 2 ∂ v¯ ∂ v¯ ∂ 2 v¯ + φvk ( 2 + 2 + 2 ) ∂x ∂y ∂z 2 2 ∂ w¯ ∂ w¯ ∂ 2 w¯ + + ). + φw ( k ∂x2 ∂y 2 ∂z 2

(7.19)

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Likewise, the term containing the POD mode is expanded as follows: X M M X ∂ 2 φum ∂ 2 φum ∂ 2 φum 2 Φk , qm ∇ Φm = qm φuk ( + + ) 2 2 2 ∂x ∂y ∂z m=1 m=1 + +

M X

m=1 M X

qm φvk (

∂ 2 φvm ∂ 2 φvm ∂ 2 φvm + + ) ∂x2 ∂y 2 ∂z 2

qm φw k(

m=1

∂ 2 φw ∂ 2 φw ∂ 2 φw m m m + + ) 2 2 2 ∂x ∂y ∂z

∂ 2 φum ∂ 2 φum ∂ 2 φum = φuk ( + + ) ∂x2 ∂y 2 ∂z 2 ∂ 2 φvm ∂ 2 φvm ∂ 2 φvm + φvk ( + + ) ∂x2 ∂y 2 ∂z 2 2 w ∂ 2 φw ∂ 2 φw m m w ∂ φm + + ) ∗q + φk ( ∂x2 ∂y 2 ∂z 2

7.2.2

(7.20)

Ordinary-Differential Equation Model

We combine Equations (7.6), (7.15), (7.17), (7.19), and (7.20) and form a set of M ordinarydifferential equations as follows: q˙k (t) = Ak +

M X

m=1

Bkm qm (t) +

M X M X

m=1 n=1

Ckmn qn (t)qm (t),

(7.21)

where Ak =

1 ¯ ) − (Φk , u ¯ .∇¯ u), (Φk , ∇2 u ReD

¯ .∇Φm ) − (Φk , φm .∇¯ Bkm = −(Φk , u u) +

1 (Φk , ∇2 φm ), ReD

Ckmn = −(Φk , φm .∇Φn ), In Equation (7.21), A is an M × 1 vector resulting from the average flow field, C is a tensor that represents the quadratic nonlinearity, and B is the linear part of the dynamical system. It is an M × M matrix comprising three terms: the first two terms are the byproduct of the nonlinearity in the Navier-Stokes equations (convection term) and originate from the interaction of the average field with the eigenfunctions, whereas the third term results from the linear dissipative operator and is a function of ReD . Thus, we can write B as the sum of

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a linear component and a component originating from the nonlinearity as follows: B = BN L +

1 L B . ReD

(7.22)

From linear stability analysis, we can compute the eigenvalues of B and hence ascertain the stability of the trivial solution. Moreover, the model is normalised by σj which is like a mass matrix in the system. In Equation (7.21), we transform the spatial derivatives to curvilinear coordinates and compute A, B, and C using a spatial discretization schemes similar to that in the CFD solver. We note that the projection procedure is performed using the same domain decomposition as that used in the CFD simulation. Each processor computes the inner product of the part assigned to it and the results are summed using MPI commands.

7.2.3

POD Simulation

We choose the first ten POD modes to develop a reduced-order model for the current flow configuration. We perform the Galerkin projection and compute Ak , Bkm , and Ckmn , where k, m, n = 1, 2, ..., 10 in Equation (7.21). Thus, the CFD problem with 192 × 256 degrees of freedom is reduced to a ten-dimensional dynamical system. The eigenvalues of B consist of five complex conjugate pairs. For small Re, all eigenvalues are in the left half of the complex plane. As the Reynolds number increases past the critical value Rec ≈ 40, a pair of these eigenvalues cross transversely the imaginary axis; that is the system undergoes a Hopf bifurcation. Consequently, vortex shedding is initiated leading to a periodic solution. It would be interesting to determine the contributions of the linear and nonlinear components to the eigenvalues of B. The eigenvalues of BL /ReD are real and located in the left half of the complex plane. Thus, the instability arises from the terms generated due to the nonlinear interaction of the average flow and the POD eigenfunctions from the Navier-Stokes projection. The instability due to the linear part of the dynamical system would lead to the growth of the velocity with time. However, the nonlinear component Cijk in the reduced-order model plays a stabilizing role and limits the growth, giving rise to a limit cycle. We integrate Equation (7.21) simultaneously using the ode45 function in Matlab and

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compute the qk (t). The temporal evolution of the first eight velocity coefficients represent an oscillatory solution, as shown in Figure 7.1. The pairs (q1 , q2 ), (q3 , q4 ), (q5 , q6 ), and (q7 , q8 ) have a frequency of ω, 2ω, 3ω, and 4ω, respectively, with a 90◦ phase difference within each pair. We also project each snapshot onto the POD modes to compute the velocity coefficient q˜i (t) as follows: q˜i (t) = (u′ (x, ti ), Φ(x))Ω .

(7.23)

For the ten-dimensional Galerkin model, we perform a two-dimensional projection of the phase portrait on the plane (q1 , q2 ) and compare it with the projection obtained on the (˜ q1 , q˜2 )-plane. We observe that the two projections compare well as shown in Figure 7.2. We also plot projection of the limit cycle onto the (q1 , qi )-plane, where i = 3, 4, ..., 10. We then integrate the dynamical system for long time to study the stability of the limit cycle. We observe that the states qi diverges from the physical limit cycle to a spurious limit cycle. In Figure 7.4, we plot the first eight states and observe that the divergence is gradual for some of the states, whereas it is abrupt and develops an offset in the other states. This long-time integration divergence occurs after approximately 400 shedding cycles. However, it is found to depend on the number of modes being used in the reduced-order model. Increasing the number of modes, which correspond to approximately 100% of the energy, we found that the solution of the reduced-order model still drifts to a nonphysical limit cycle. Because experiments and CFD simulations yield only one limit cycle corresponding to the correct solution of the physical problem, the appearence of spurious limit cycles is due to the Galerkin projection. To investigate convergence of the solution of the reduced-order system, we use a direct approach in the time domain. We apply a shooting method (Nayfeh and Balachandran, 1995) to adjust the initial conditions so that the system does not drift to a spurious limit cycle. We discuss this method in detail in the next section.

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2

q3,4

q1,2

0.1

0

0

-0.1 -2

360

370

380

360

Time

(a) q1 (solid) and q2 (dashed)

380

(b) q3 (solid) and q4 (dashed)

0.05

q7,8

0.1

q5,6

370

Time

0

-0.1

0

-0.05

360

370

380

Time

(c) q5 (solid) and q6 (dashed)

360

370

380

Time

(d) q7 (solid) and q8 (dashed)

Figure 7.1: The velocity coefficients qi = 1, 2, ..., 8 at ReD =100.

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q2

2

0

-2

-2

0

q1

2

Figure 7.2: A two-dimensional projection of the POD phase portrait on (q1 , q2 )-plane (solid) and snapshot portrait (triangle).

7.3

Shooting Method

In this method, the initial-value problem (7.1) is converted into a two-point boundary-value problem as follows: q˙ = F(q)

(7.24a)

q(0) = α

(7.24b)

q(T ) = α,

(7.24c)

where α is the initial guess and T is the period of the limit cycle. In other words, we seek an initial condition q(0) = α and a solution q(t; α) with a minimal period T such that q(T, α) = α.

(7.25)

Hence, starting from a point on the limit cycle (i.e., q(0) = α) and integrating Equation (7.24a) over the period T of the limit cycle should bring back the trajectory to its initial condition. Because we do not know α and T exactly, we start with a guess of α0 and T0 and use a Newton-Raphson scheme to correct this guess; that is, we calculate δα = α − α0

(7.26a)

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δT = T − T0 ,

(7.26b)

such that Equation (7.25) is satisfied within a tolerance ε; that is, q(T0 + δT, α0 + δα) − (α0 + δα) ≤ ε.

(7.27)

We expand Equation (7.27) in a Taylor series about (T0 , α0 ) and obtain ∂q ∂q (T0 , α0 ) − I δα − (T0 , α0 )δT = α0 − q(T0 , α0 ), ∂α ∂T where ∂∂q α is an M × M matrix, I is an M × M identity matrix, and Moreover, it follows from Equation (7.1) that

∂q ∂T

(7.28) is an M × 1 vector.

∂q (T0 , α0 ) = F(α0 ). ∂t

(7.29)

In order to compute ∂∂q α , we differentiate Equation (7.1) and the initial conditions with respect to α, and obtain ∂q d ∂q ( ) = Dq F(q) (7.30a) dt ∂α ∂α ∂q (0) = I, (7.30b) ∂α where Dq F(q) is the Jacobian of F(q). We solve Equations (7.24a), (7.24b), (7.30a), and (7.30b) to obtain q(t, α0 ) and ∂∂q α at (T0 , α0 ) simultaneously. Equation (7.28) constitutes a system of M with M + 1 unknowns. Since the phase is arbitrary for a periodic solution of an autonomous system, we impose a phase condition to obtain an extra equation. There are several ways to specify the phase condition and are as follows: 1. Fix a component αk in the initial vector α0 . 2. Set the k th component Fk (x) of the vector field F(x) equal to zero. 3. Require the corrections δα to be normal to the vector field F; that is, FT δα = 0. This condition is called the orthogonality condition. 4. Use an integrated form of the orthogonality condition.

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In the present case, we use the orthogonality condition and obtain the following system:      ∂q α − q(T0 , α0 ) (T , α ) − I F(α0 ) δα  ∂α 0 0 .  = 0 (7.31) T 0 F (α0 ) 0 δT For the reduced-order model of the flow past a cylinder, developed in Equation (7.21), the Jacobian is computed as M M X X ∂ q˙k = Bkl + Cklm qm + Ckln qn . ∂ql m=1 n=1

(7.32)

We then differentiate Equation (7.21) with respect to the αi (t) to obtain the matrix ∂∂q α as follows: M M M M X M X X ∂qk X X ∂qn ∂qm ∂ q˙k = + qm + . (7.33) Bkm Ckmn Ckmn qn ∂αi m=1 ∂αi m=1 n=1 ∂αi ∂α i m=1 n=1 Thus, we solve Equation (7.33), which is a set of M 2 equations and the M equations in the original dynamical system. Therefore, in terms of computational cost, we solve M 2 + M equations in the shooting method to compute the new initial conditions and time period T of the limit cycle. We note that the shooting method is sensitive to the initial time period T0 and requires a “good” guess for convergence. In the present study, the first snapshot data provides a reasonable initial guess α0 for α. We solve 110 equations to compute the new initial conditions. We then integrate Equation (7.21) with the modified initial conditions. In Figure 7.5, we compute the time response of the first eight states for approximately 4000 vortex-shedding cycles and observe a stable solution. The figure shows the envelope of the oscillating response of the states for 20000 time units. The reduced-order model thus obtained is stable and does not require any additional term to stabilize the dynamical system. It is important to note that this iterative procedure of correcting the initial conditions works well when almost 100% of the energy is contained in the modes of the Galerkin expansion. However, it is in no way a substitute of modeling higher frequency scales due to the truncated modes and should not be confused with viscosity models employed at higher Reynolds numbers.

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0.2

q4

q3

0.2

0

-0.2

0

-0.2

-2

0

2

-2

q1

0

-0.2

0

-0.2

-2

0

2

-2

q1

0

2

q1

0.1

q8

0.1

q7

2

0.2

q6

q5

0.2

0

-0.1

0

-0.1

-2

0

2

-2

q1

0

2

q1

0.1

q10

0.1

q9

0

q1

0

-0.1

0

-0.1

-2

0

2

q1

-2

0

2

q1

Figure 7.3: Two-dimensional projections of the phase portraits of the velocity coefficients onto the plane (qi , a1 ) for qi = 3, 4, ..., 8 at ReD =100..

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Figure 7.4: Divergence of the qi , i = 1, 2, ..., 4 to spurious limit cycles.

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Figure 7.5: Velocity coefficient qi = 1, 2, ..., 8 at ReD =100

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Results and Discussion

The Galerkin approximation of the system can be validated by reconstructing the velocity fields from Equation (6.11). In Figures 7.6 and 7.7, we compare the instantaneous u and v velocity fields, respectively, with the CFD results. The two velocity fields constructed using the POD simulation compare well with the CFD snapshot. Depending on the problem, the number and variety of snapshots used to generate the POD modes can have a strong influence on the general applicability of the reduced-order model. Despite the accuracy of the model at this specific Reynolds number, the model lacks robustness away from the reference simulation. Deane et al. (1991) note that the accuracy of the model predictions rapidly deteriorate as we move away from the decomposition value. For the flow around a circular cylinder, the wavelengths of the wake structures are strongly affected by the Reynolds number. Thus, the POD basis generated from an experiment or a numerical simulation are only useful within a narrow bandwidth of parameters close to that reference. However, for the grooved channel flow, the horizontal wave number is primarily determined by the channel periodicity, and thus active control simulations can produce reasonable results over a wide variety of Reynolds numbers. In the lid-driven-cavity problem, Arian et al. (2000) observed large errors in the solution when the control input is away from the reference on which the model is based. A quick solution is to generate a wide range of reference simulations to generate new modes. Graham et al. (1999a,b) used an optimal control scheme through cylinder rotation in their reduced-order model for the flow past a circular cylinder at Re = 100. They encountered a similar problem in their control methodology. When the POD modes were generated from the snapshot at a given oscillation frequency, the active control model worked only near that particular frequency. Therefore, they incorporated multiple frequency information in their snapshot data set through varying frequency sinusoid, or chirp, defined as follows: γ = γ0 sin

2πt 2πt + Aπ sin T1 T2

(7.34)

Using appropriate numerical values of γ0 , T1 , T2 , and A, they recorded 75 snapshots taken at time intervals of 0.64. They found the chirp excitation approach to produce the desired broadening of the basis function set for two different initial conditions: locked-on and un-

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(a)

(b)

Figure 7.6: Instantaneous u-velocity fields from (a) CFD and (b) POD.

forced steady vortex shedding. The POD modes, thus obtained, are capable of simulating a wider range of control inputs. They tested two cases; halting the cylinder rotation when the

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(a)

(b)

Figure 7.7: Instantaneous u-velocity fields from (a) CFD and (b) POD.

flow is in a locked-on state and starting a sinusoid cylinder rotation from the unforced state. They observed that their models followed the transient for a certain length of time, before

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the prediction error grew to a significant level. They related this error to the projection error, which represents the extent to which the POD basis functions are unable to capture the flow. Another limitation of this method is the inability to identify bifurcation points, if any, in the vicinity of control parameter(s). This is obvious from the fact that the snapshots data, from which the reduced-order model is derived, does not include any bifurcation information. In order to capture any qualitative changes in the dynamics, the snapshot data must include the relevant flow field information to construct appropriate eigenfunctions. For a typical three-dimensional incompressible flow past a circular cylinder, the flow undergoes various transitions with increasing Reynolds number. Near the Reynolds number 185, a second bifurcation, also known as Mode A instability, occurs resulting in transition from a twodimensional to a three-dimensional wake (Williamson, 1996). Ma and Karniadakis (2002) investigated the stability and dynamics of three dimensional limit-cycle states for the flow past a cylinder using low-dimensional modeling. They employed spectral DNS to obtain snapshots of the flow field and used the POD to extract the most energetic modes. For their simulation, they used 412 triangular elements of spectral order seven, with eight Fourier collocation points along the span, Lz = 4π. They investigated the Reynolds number range of 180 ± 20 to capture the Mode A instability. From their simulations, they observed the Mode A instability at Re≈185. They analyzed two systems. The first (system A) was constructed based on 40 POD modes at Re=185; the second (system B) was constructed by combining 20 modes extracted at Re=185 and 20 modes extracted at Re=182. There are two procedures to construct this hybrid system, either by concatenating the two sets of snapshots at Re=185 and Re=182 or by extracting the POD modes at each state and orthonormalizing the entire set. For both systems A and B, they developed reduced-order models for M =6, 10, 20, and 40. For M =20 in system A, they obtained a stable limit cycle for at least 500 convective time units, corresponding to 100 shedding cycles, which matched well with their DNS data. The simulation with M =40 was similar to that of M =20 for the first 15 modes. However, they noted a difference in the amplitudes q(t) for higher modes. In addition to correctly exhibiting a stable and convergent response at Re=185, the hybrid system B correctly predicted the jump in the Strouhal

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number versus Reynolds number curve. However, both systems A and B at Re=189 show a stable response only up to 5 shedding cycles and then diverge. The eigenfunctions, thus obtained, contained the bifurcation information along the spanwise direction. Their model predicted the Mode B instability and the discontinuity in the Strouhal number discovered experimentally. Therefore, the reference data must contain sufficient information about the flow field to predict any qualitative change within that flow regime. The snapshot data is typically recorded in the steady state and does not include any transient behavior of the dynamical system. Noack et al. (2003) suggested a generalization for POD based Galerkin models to include the transient behavior. In their Galerkin approximation, they included an additional vector and termed it a shift-mode. They obtained the steady solution us with a Newton iteration, employing the discretized steady NavierStokes equations. The shift-mode was constructed in the following Gram-Schimdt procedure ¯ − us : starting from the mean-field correction u ¯ − us ua∆ := u ub∆

:=

u∆ :=

ua∆

−

M X i=1

ub∆ ||ub∆ || Ω

ua∆ , ui

Ω

ui (7.35)

where u∆ is the M + 1th mode. It represents the shift of the short term averaged flow away from the POD space and is effectively a normalized mean-field correction. Inclusion of the shift-mode keeps the system orthonormal; that is, (ui , uj )Ω = δij for all i, j = 1, 2, ..., M + 1. The authors compared the two reduced-order models: System A (without the shift-mode) and System B (with the shift-mode). They discussed the two models with reference to the periodic solution, energy flow analysis, transient solution, and Reynolds number dependence. For predicting the periodic solution, the accuracy of Systems A and B was comparable since the shift-mode did not play any role. They also compared the transient behavior of the two systems with a global linear stability analysis of the steady solution and with a DNS. They observed that the transient time in System A was more than one hundred shedding periods and thus was significantly over predicted. However, System B was in closer agreement with the numerical solution. In the Reynolds number dependence study, System A predicted the

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critical Reynolds number to be 80, far larger than the correct value of approximately 47. With a similar model, Deane et al. (1991) also reported the onset of supercritical Hopf bifurcation at a Reynolds number around 80. System B, on the other hand, accurately predicted the bifurcation point and that the Strouhal number increases with Reynolds number. However, the Strouhal-Reynolds number relationship was only qualitatively correct. Comparing the two systems, they observed System B to be more robust and predict the flow physics much better than that predicted by System A. The approaches of adding shift-modes and mixing the snapshots to capture the flow physics in a wide range are encouraging and expand the application of POD based reducedorder models. The varying flow features are reflected in the additional or modified eigenfunctions, and the control parameter can be varied over a wider range to predict the flow dynamics. However, keeping these limitations in mind, one might argue the benefits of lowdimensional models. In Chapter 9, we introduce a control function approach and incorporate an input variable in the reduced-order model. We apply full-state feedback control to the modified reduced-order model and compute appropriate gains to reduce vortex shedding.

7.5

Summary

In this Chapter, we used the dominant of the POD modes (a small number of eigenfunctions or modes) in a Galerkin procedure to project the Navier-Stokes equations onto a low-dimensional space, thereby reducing the distributed-parameter problem into a finitedimensional nonlinear dynamical system in time. We integrated the model using a RungeKutta scheme to compute the temporal coefficients for long time and observed that the limit cycle drifts to a spurious limit cycle. We investigated the stability of the model and presented a shooting method to compute initial conditions on the limit cycle and its period. Although the POD based models lack robustness away from the reference simulation, they provide an analytical insight into the physical phenomena. We also discussed in detail various remedies to generalize the applicability of the reduced-order models to a wide range of Reynolds numbers. Moreover, these models, being a set of ODE, enable application of dynamical systems theory and control methods.

Chapter 8 Reduced-Order Modeling of the Pressure Field 8.1

Introduction

In general, the POD based approach models the velocity field in the flow. However, in most of the engineering and industrial applications of fluid-structure interaction, the pressure distribution over the surface is essential for computing the hydrodynamic forces over the structure. These forces may lead the dynamical system to undesired vibrations, thus causing damage or even failure of the structure. It is therefore important to model the pressure field in addition to the velocity field (Akhtar et al., 2008b). We present a pressure model for incompressible flows based on the Galerkin projection of the pressure-Poisson equation onto the POD modes. In incompressible flows, the pressurePoisson equation is obtained by taking the divergence of the Navier-Stokes equations in vector form and applying the continuity constraint. The model requires snapshots of the pressure field, in addition to snapshots of the velocity field, for computing the pressure POD modes. We project the pressure-Poisson equation onto the pressure POD modes and develop a reduced-order model for the pressure field. The pressure is then integrated over the cylinder to obtain the hydrodynamic forces acting on the cylinder. In the preceding chapter, we developed a reduced-order model for the velocity field and compared the POD results with those of the CFD code by reconstructing the velocity fields. 167

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The flow field around a cylinder affects the pressure distribution over its surface. The pressure force, when decomposed into crossflow and inflow directions, produces lift and drag forces, respectively. These physical quantities provide a measure of accuracy of the lowdimensional models. Therefore, it would be interesting to extend the model and compute the hydrodynamic forces on the structure in order to appreciate application of reduced-order analyses.

8.2

Pressure Based Models

In the Galerkin projection of Navier-Stokes equations onto the POD modes and application of Green’s theorem, the pressure term drops out from the model. The divergence-free property of the POD modes is essential for this step. In incompressible flows, the pressure-Poisson equation ∂ui ∂uj ∂ 2p =− , 2 ∂xi ∂xj ∂xi

(8.1)

where i and j refer to the Cartesian components of the vector, is the governing equation for the pressure. Noack et al. (2005) discussed contribution of the pressure term in the Galerkin models. They developed a low-dimensional Galerkin model for spatially evolving laminar and transitional shear layers based on 2D and 3D Navier-Stokes simulations. They observed that the effect of the pressure term is significant for a two-dimensional mixing layer, is less pronounced for the three-dimensional analogue, and is small in an absolutely unstable wake flow. Substituting the Galerkin approximation in Equation (6.11) into Equation (8.1) yields ∆p = −

3 X 3 X ∂Φm ∂Φnj i

i=1 j=1

∂xj ∂xi

qm qn ,

(8.2)

where m and n denote the expansion modes. They expanded the solution of Equation (8.2) as a function of the velocity field in the form p=

N X N X

pmn qm (t)qn (t),

(8.3)

m=0 n=0

where N is the number of POD modes and the pmn represent the partial pressures, which satisfy the Neumann condition (i.e., n · ∇p = 0). However, the objective of their work was

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to model the pressure gradient term as a function of the qm (t). Therefore, they let X N X N ∇pjk qj (t)qk (t) (Φi , −∇p)Ω = − Φi

Ω

j=0 k=0

=

N X N X j=0 k=0

′ Cijk qj (t)qk (t),

(8.4)

′ where Cijk = (Φi , −∇p)Ω for i = 1, 2, ..., N . Thus, this pressure model leads to an additional

quadratic term in the Galerkin system in Equation (7.21). In another empirical pressure model, they neglected the quadratic terms qj qk and kept only the linear terms. We note that q0 q0 = 1 and q0 qi = qi constitute the linear terms in the Galerkin expansion. They recorded the pressure snapshot data pm and computed the corresponding qm (t) from the snapshot data. The pressure gradient term thus obtained is given by (Φi , −∇p)Ω =

N X j=0

Bij′ qj (t),

(8.5)

where the coefficients of Bij′ are determined from linear regression of the pm and qm (t). In another numerical study, Cohen et al. (2004a) investigated feedback flow control of the wake of a “D” shaped cylinder using the POD approach. They obtained the steady-state streamwise velocity u and pressure p data from 100 equally-spaced snapshots over approximately 15 vortex-shedding cycles at Re=300. The flow field (u,p) is decomposed as the sum of an average component and a fluctuation component using the temporal coefficients, given in Equation (6.11). Using the inner product, they computed the empirical correlation matrix and solved the eigenvalue problem to obtain the Φ(x). The q(t) are obtained by projecting the snapshots onto the POD modes. For their closed-loop control design, the qi (t) could not be measured directly and therefore they designed an estimator to compute the states of the system using the Linear Stochastic Estimator (LSE) approach. The q1−3 are mapped onto P the extracted sensor signals from the pressure signals ps as qm (t) = ns=1 Csm ps (t), where

n is the number of sensors and the Csm represent the coefficients of the linear mapping. This approach of measuring the pressure field is sensitive to the number of sensors and their configurations and depends on the accuracy of the estimator.

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Pressure POD Eigenfunctions

We use the reduced-order model of the velocity field to develop a model for the pressure field. Knowledge of the velocity field enables us to construct a pressure model from the pressure-Poisson equation. A naive approach would be to reconstruct the velocity field from the reduced-order model at every time step and solve the pressure-Poisson equation over the entire domain to compute the pressure at every instant. However, the computational cost of this approach would be of the same order of magnitude as solving the complete Navier-Stokes equations in CFD. An alternate approach is to compute the POD eigenfunctions of the pressure field and use a similar Galerkin procedure for the pressure-Poisson equation. The pressure field is written in a matrix PN ×S , each column representing one time instant as follows:   (1) (2) (S) p p1 . . . p1   1 ..  ..  .. P= . .  .   (1) (2) (S) pN pN . . . pN

(8.6)

Similar to the velocity field, we write the pressure field as a sum of a mean component (P¯ ) and a fluctuations component (p′ ). The average pressure P¯ = hpi is subtracted from P. We perform the SVD of the resulting P and compute the pressure POD modes (Ψi ). Next, the pressure is expanded in terms of the Ψi (x) as follows: p(x, t) ≈ P¯ (x) +

M X

am (t)Ψm (x).

(8.7)

m=1

We note that, in the Galerkin expansion in Equation (8.7), the temporal coefficients ai (t) are different from the qi (t). We also note that integrating the average pressure over the cylinder surface yields zero lift and the mean drag corresponding to its Reynolds number. The velocity POD modes Φ(x) are known a priori and the pressure modes Ψ(x) are computed from the respective data ensemble in a similar manner. The average pressure for all of the cases given in Table 6.1 are shown in Figure 8.1. The POD eigenfunctions are calculated for all of the cases mentioned in Table 6.1. These POD eigenfunctions are plotted in Figures 8.2, 8.3, 8.4, and 8.5.

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(a)

(b)

(c)

(d)

Figure 8.1: Average pressure fields: (a) Case I, (b) Case II, (c) Case III, and (d) Case IV.

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Figure 8.2: Pressure modes (Ψi , i = 1, 2, ..., 10) at ReD =100.

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Figure 8.3: Pressure modes (Ψi , i = 1, 2, ..., 10) at ReD =200.

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Figure 8.4: Pressure modes (Ψi , i = 1, 2, ..., 10) at ReD =525

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Figure 8.5: Isosurfaces of pressure modes (Ψi , i = 1, 2, ..., 10) at ReD =525 with 25% transparency.

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Reduced-Order Model

8.4.1

Galerkin Projection

We substitute Equations (6.11) and (8.7) into Equation (8.1) and obtain M M ∂Φ(m)i ∂ 2 P¯ X ∂ 2 Ψm ∂ u¯i X + =− + am (t) qm (t) ∂xi 2 m=1 ∂xi 2 ∂xj m=1 ∂xj M ∂Φ(n)j ∂ u¯j X . . + qn (t) ∂xi n=1 ∂xi

(8.8)

We then project Equation (8.8) onto the pressure POD modes and perform the inner product as follows:

M M ∂Φ(n)j ∂ 2 P¯ X ∂ u¯i X ∂ u¯i ∂ u¯j ∂ 2 Ψm Ψk , q (t) + = − + a (t) n m ∂xi 2 m=1 ∂xi 2 ∂xj ∂xi ∂xj n=1 ∂xi M ∂Φ(m)i ∂ u¯j X qm (t) + ∂xi m=1 ∂xj

M ∂Φ(m)i X ∂Φ(n)j + qm (t) qn (t) . ∂xj n=1 ∂xi m=1 M X

(8.9)

Term I The first term on the left-hand side of Equation (8.9)represents a vector of M components and is the manifestation of the average pressure field projected onto the pressure POD modes; that is,

∂ 2 P¯ Ψk , ∂xi 2

= Ψk

∂ P¯ ∂ P¯ ∂ P¯ + + ∂x2 ∂y 2 ∂z 2

(8.10)

Term II We obtain a matrix of order M × M by projecting the fluctuating component of the pressure field onto the pressure POD modes as X M M X ∂ 2 Ψm ∂ 2 Ψm ∂ 2 Ψm ∂ 2 Ψm = a Ψ + + Ψk , am (t) m k 2 2 2 ∂x ∂x ∂y ∂z 2 i m=1 m=1 ∂ 2 Ψm ∂ 2 Ψm ∂ 2 Ψm = Ψk + + ∗a ∂x2 ∂y 2 ∂z 2 This term contains the unknown temporal coefficient vector a(t).

(8.11)

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Term III The first term on the right-hand side is the projection of the quantity associated with the average velocity field and yields a vector of order M. We expand the index notation into the velocity components and obtain ∂ u¯i ∂ u¯j ∂ u¯ ∂ u¯ ∂ u¯ ∂¯ v ∂ u¯ ∂ w¯ + + + Ψk , = Ψk ∂xj ∂xi ∂x ∂x ∂y ∂x ∂z ∂x v ∂¯ v ∂ w¯ ∂¯ v ∂ u¯ ∂¯ v ∂¯ + + + ∂x ∂y ∂y ∂y ∂z ∂y ∂ w¯ ∂ u¯ ∂ w¯ ∂¯ v ∂ w¯ ∂ w¯ + + ∂x ∂z ∂y ∂z ∂z ∂z

(8.12)

Term IV This term is originated from the quantities involving average and fluctuating velocity components. The product is projected onto the pressure POD modes, resulting in M ×M matrix multiplying the vector q(t) as

M M X ∂Φ(n)j ∂ u¯ ∂φun ∂ u¯ ∂φvn ∂ u¯ ∂φw ∂ u¯i X n qn = Ψk qn (t) + + + Ψk , ∂xj n=1 ∂xi ∂x ∂x ∂y ∂x ∂z ∂x n=1 v ∂φvn ∂¯ v ∂φw ∂¯ v ∂φun ∂¯ n + + + ∂x ∂y ∂y ∂y ∂z ∂y ∂ w¯ ∂φun ∂ w¯ ∂φvn ∂ w¯ ∂φw n + + ∂x ∂z ∂y ∂z ∂z ∂z

(8.13)

It represents a linear term in q(t) in the reduced-order model and can be rewritten as follows:

M ∂Φ(n)j ∂ u¯i X ∂ u¯ ∂φun ∂ u¯ ∂φvn ∂ u¯ ∂φw n Ψk , = Ψk qn (t) + + + ∂xj n=1 ∂xi ∂x ∂x ∂y ∂x ∂z ∂x v ∂φvn ∂¯ v ∂φw ∂¯ v ∂φun ∂¯ n + + + ∂x ∂y ∂y ∂y ∂z ∂y ¯ ∂φw ∂ w¯ ∂φun ∂ w¯ ∂φvn ∂ w n ∗q + + ∂x ∂z ∂y ∂z ∂z ∂z

(8.14)

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Term V Similar to Term IV, we expand this term and obtain M M X ∂Φ(n)i ∂ u¯j X ∂ u¯ ∂φum ∂ u¯ ∂φvm ∂ u¯ ∂φw m Ψk , = Ψk qn + + + qm (t) ∂xi m=1 ∂xj ∂x ∂x ∂y ∂x ∂z ∂x n=1 v ∂φvm ∂¯ v ∂φw ∂¯ v ∂φum ∂¯ m + + + ∂x ∂y ∂y ∂y ∂z ∂y ∂ w¯ ∂φum ∂ w¯ ∂φvm ∂ w¯ ∂φw m + + ∂x ∂z ∂y ∂z ∂z ∂z

(8.15)

M ∂Φ(n)i ∂ u¯ ∂φum ∂ u¯ ∂φvm ∂ u¯ ∂φw ∂ u¯j X m = Ψk + + + qm (t) Ψk , ∂xi m=1 ∂xj ∂x ∂x ∂y ∂x ∂z ∂x v ∂φvm ∂¯ v ∂φw ∂¯ v ∂φum ∂¯ m + + + ∂x ∂y ∂y ∂y ∂z ∂y ∂ w¯ ∂φum ∂ w¯ ∂φvm ∂ w¯ ∂φw m ∗q + + ∂x ∂z ∂y ∂z ∂z ∂z

(8.16)

Term VI This is a nonlinear term and represents quadratic nonlinearity in q(t). It is given by

X M M X M ∂Φ(n)j ∂Φ(m)i X = Ψk , qn (t) qm (t) Ψk qn qm ∂x ∂x j i n=1 m=1 n=1 m=1 M X

∂φum ∂φun ∂φum ∂φvn ∂φum ∂φw n + + + ∂x ∂x ∂y ∂x ∂z ∂x ∂φvm ∂φun ∂φvm ∂φvn ∂φvm ∂φw n + + + ∂x ∂y ∂y ∂y ∂z ∂y u ∂φw ∂φw ∂φv ∂φw ∂φw m ∂φn + m n+ m n ∂x ∂z ∂y ∂z ∂z ∂z

(8.17)

We perform the inner product and obtain a tensor of M × M × M order in the form

M ∂Φ(m)i X ∂Φ(n)j ∂φum ∂φun ∂φum ∂φvn ∂φum ∂φw n ′ = q ∗ Ψk qn + + + Ψk , qm ∂x ∂x ∂x ∂x ∂y ∂x ∂z ∂x j i n=1 m=1 M X

∂φvm ∂φun ∂φvm ∂φvn ∂φvm ∂φw n + + + ∂x ∂y ∂y ∂y ∂z ∂y u v w ∂φw ∂φw ∂φw m ∂φn m ∂φn m ∂φn ∗q + + ∂x ∂z ∂y ∂z ∂z ∂z (8.18)

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Algebraic Equation Model

Equation (8.8) is then transformed into curvilinear coordinates using a transformation similar to that used in the CFD code where the Φi (x) and the qi (t) are known a priori. The reducedorder model for the pressure field thus obtained is Dkm ak (t) = Ek +

M X

m=1

Fkm qm (t) +

M X M X

m=1 n=1

Gkmn qn (t)qm (t),

(8.19)

where Dkm = (Ψk , ∇2 Ψi ) Ek = −(Ψk , ∇2 P¯ ) − (Ψk , ∇¯ u : ∇¯ u) Fkm = −(Φk , ∇¯ u : ∇Ψm ) − (Φk , ∇Ψm : ∇¯ u) Gkmn = −(Φk , ∇Ψm : ∇Ψn ). Equations (8.19) constitute a set of algebraic equations quadratic in terms of the qi . We compute the qi by integrating the dynamical system in Equation (7.21). Using these coefficients, we compute the ai from Equation (8.19). Similar to the qi , we plot the pairs (a1 , a2 ), (a3 , a4 ), (a5 , a6 ), and (a7 , a8 ) in Figure 8.6 with a 90◦ phase difference within each pair. We note that the fundamental frequency in a1 and a2 corresponds to the Strouhal number in the CFD simulation. Thus, we recreate the pressure field using Equation (8.7). The time coefficient a2 is plotted against a1 in Figure 8.7 The time coefficients are plotted against a1 in Figure 8.8. We observe that the ai phase portraits have a similar structure when compared to the corresponding qi . However, there is a shift in the limit cycles, which is obvious from the fact that there is a quadratic relationship between qi and ai .

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0.5

a3,4

a1,2

0.1

0

0

-0.1 -0.5

360

370

380

360

Time

(a) a1 (solid) and a2 (dashed)

380

(b) a3 (solid) and a4 (dashed)

0.01

a7,8

0.02

a5,6

370

Time

0

-0.02

0

-0.01

360

370

380

Time

(c) a5 (solid) and a6 (dashed)

360

370

380

Time

(d) a7 (solid) and a8 (dashed)

Figure 8.6: The pressure coefficients ai = 1, 2, ..., 8 at ReD =100.

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a2

0.4

0

-0.4 -0.4

0

0.4

a1 Figure 8.7: A two-dimensional projection of the pressure coefficients onto the plane (a1 ,a2 ) at ReD =100.

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a4

a3

0.2

0

-0.2

0

-0.2 -0.4

0

0.4

-0.4

a1

0.4

0.04

a6

a5

0

a1

0.04

0

-0.04

0

-0.04 -0.4

0

0.4

-0.4

a1

0

0.4

a1

0.02

a8

0.02

a7

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0

-0.02

0

-0.02 -0.4

0

0.4

a1

-0.4

0

0.4

a1

Figure 8.8: Two-dimensional projections of the phase portraits of the pressure coefficients onto the plane (ai , a1 ) for ai = 3, 4, ..., 8 at ReD =100.

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Lift and Drag Model

To compute the lift and drag coefficients, we substitute Equations (6.11) and (8.7) into Equation (3.2) to obtain 1 CL = − Lz

ZLz Z2π X M 0

1 − ReD 1 CD = − Lz

0 M X

m=0

∂φvm ∂φum qm (t)( − ) cos θ dθdz ∂x ∂y m=0

ZLz Z2π X M 0

1 + ReD

am (t)Ψm (x) sin θ

0 M X

(8.20)

am (t)Ψm (x) cos θ

m=0

∂φvm ∂φum qm (t)( − ) sin θ dθdz ∂x ∂y m=0

(8.21)

where q0 and a0 correspond to the temporal coefficients of the mean velocity and mean pressure and equal to unity. We compute the mean-pressure distribution over the cylinder surface from the POD simulation and compare it with the CFD simulation in Figure 8.9(a). We observe a good agreement between the two results. The pressure is then integrated over the surface to compute the lift and drag forces on the cylinder using Equation (3.2). It should be noted that Equation (3.2) contains both of the pressure and shear components. The shear component is computed from the modeled velocity field. In Figure 8.9(b), we plot the lift coefficient computed from the POD approximation along with lift calculated from the CFD solver. We also observe good agreement in the mean and fluctuating drag components.

8.6

Summary

We developed a reduced-order model for the lift and drag forces on a cylinder using the POD approach. In addition to the velocity field, we also recorded the pressure data from the DNS. Similar to the velocity field, we decomposed the pressure field into mean and fluctuating components. However, in incompressible flows, the pressure is governed by the pressurePoisson equation. We computed the pressure POD modes and projected the pressure-Poisson

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equation onto the dominant of these modes. In this projection procedure, the velocity POD modes and its temporal coefficients are known a priori. We reconstructed the pressure field using the pressure model and integrated it over the surface to obtain the lift and drag forces.

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(a)

(b)

Figure 8.9: Comparison of (a) the mean pressure and (b) lift coefficients: POD (solid line) and CFD (triangle).

Part III Active Flow Control

186

Chapter 9 Feedback Control of Vortex Shedding 9.1

Introduction

Feedback control of fluid-structure interaction is of practical importance from the perspective of wake modification and VIV reduction. Therefore, the development of design models and computational schemes for controlling vortex production and its shedding pattern has received considerable interest in research as well as in industry. Despite recent progress in CFD and parallel computing, flow control through a computational approach still remains a formidable task. Application of control on systems described by PDEs is a challenging problem. A typical example is the control of fluid dynamical systems in which the Navier-Stokes equations are the state equations. To apply a control technique, one often reduces the PDEs to ODEs to simplify the complexity of the dynamical system. For control design purposes, the POD approach enables modeling the Navier-Stokes equations as a set of ODEs. We use the CFD data of the flow past a circular cylinder and compute the POD eigenfunctions from the velocity and pressure fields. The Navier-Stokes equations are then projected onto a low-dimensional space to develop a reduced-order model. The focus of this work is to develop a reduced-order model that can potentially be used in an active control of vortex shedding in the wake. The POD based reduced-order models are limited to low Reynolds number flows. Due to the presence of small scales in the turbulence, a large data set of modes is required to capture enough energy to model the dynamical system. However, development of an 187

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efficient control algorithm for low-dimensional models of canonical flows can be beneficial in understanding and application of real-time controllers. In this chapter, we develop a reduced-order model of the flow past a cylinder at ReD = 200. The nonlinear model is, then, modified to incorporate fluidic actuation on the cylinder surface. Using feedback linearization, we reduce the dynamical system to a linear-time-invariant (LTI) system. We design a full-state feedback controller to stabilize the linear dynamical model. Based on the controller design, we compute the jet velocity of the fluidic actuator and perform direct simulations.

9.2

Control Strategy

Different types of flow control strategies have been developed and successfully implemented for the flow past a circular cylinder, such as optimal control techniques, full-state feedback control, neural networks, and proportional closed-loop feedback control (Homescu, 2002; Kim and Bewley, 2007; Li et al., 2003; Min and Choi, 1999). In this section, we restrict our discussion to control techniques for reduced-order models. Optimal control and PID control are widely used techniques to control vortex shedding.

9.2.1

Control Mechanism

For an active flow control of the cylinder wake, various actuation mechanisms are employed, such as cylinder rotation, crossflow motion, and fluidic actuation on the cylinder surface. Using an optimal control strategy, Graham et al. (1999a) achieved the control action through cylinder rotation. Similarly, Singh et al. (2001) used a linear controller to suppress vortex shedding by appropriately rotating the cylinder. In another optimal control approach, Bergmann et al. (2005) used the angular velocity of the rotating cylinder as their control function. Seigal et al. (2006) used linear proportional and differential feedback of the estimated first POD mode and actuated the cylinder in the crossflow direction to reduce the lift and drag forces. In our analysis, we apply suction on the cylinder surface as a mechanism to control the flow. Suction actuators are modeled by modifying the boundary conditions on the cylinder

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surface as discussed in Chapter 4. We use a pair of actuators placed symmetrically about the x -axis, as shown in Figure 4.23(a). In general, if appropriately applied, suction delays separation of the boundary layer, thus decreasing the width of the von Kármán vortex street. It reduces the fluctuating forces and reduces the drag on the cylinder, whereas blowing has the opposite effect.

9.2.2

Location

Placement of fluidic actuators on the cylinder surface plays an important role in the effectiveness of flow control and therefore requires special attention. In a direct search method, we can simulate flows with different actuator locations and find the optimum position that produces maximum reduction in the lift and drag coefficients. However, this is not an efficient approach and has a huge computational cost. Moreover, any change in the actuator properties, such as jet velocity, frequency, etc., might require more simulations to find the optimum location. In case of canonical flows, we can place the fluidic actuators based on intuition and knowledge of such flows. For example, in the flow past a cylinder, we can place the actuators near or aft of the separation point to effectively perturb and control the flow. Likewise, in case of airfoils, placing the actuators near the leading edge increases the lift (Ausseur and Pinier, 2003). However, for complex flows, a more general approach with some physical insight is required to find the appropriate location of the actuators. In an attempt to obtain such a criterion, we analyse the pressure POD modes and their eigenvalues computed from the flow field data. Unlike the velocity POD modes, the lower eigenvalues of the pressure do not occur in pairs while the higher eigenvalues occur in pairs, as shown in Figure 9.1. From Figure 9.2, we note that the first pressure POD mode is the most dominant and contains more than 75% of the cumulative effect.

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10

0

10-2 -4

λ

p

10

10-6 10 10

-8

-10

5

10

15

20

25

30

Mode Figure 9.1: Normalized eigenvalues.

Cumulative λp

1 0.8 0.6 0.4 0.2 0

4

8

12

Mode Figure 9.2: Cumulative effect of eigenvalues.

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The objective of our work is to reduce the hydrodynamic forces, which are a direct consequence of the pressure distribution on the cylinder. Therefore, it is more appropriate to analyze the pressure POD modes on the cylinder surface. In Figure 9.3, we plot the average pressure P¯ over the cylinder surface obtained from Equation (8.7). We note that θ = 0◦ corresponds to the base point and θ = 180◦ corresponds to the stagnation point on the cylinder. We observe that the average pressure represents a typical mean-pressure distribution curve over the cylinder surface except for its magnitude. The remainder of the pressure contribution comes from the pressure POD modes alongwith their corresponding coefficients.

Figure 9.3: Mean pressure POD mode at ReD = 200

Figure 9.4 shows the first eight pressure POD modes on the cylinder surface. From Figure 9.4(a), we observe that the absolute maxima of the first POD mode occur at θ ≈ 80◦

and θ ≈ 280◦ . For the higher modes, the absolute maxima occur close to the base point. However, based on the eigenvalues, the contributions of the higher modes are much less than that of the first mode. In other words, the orientation of the first mode can effectively be used in choosing the actuator placement. Based on this criterion, we place a pair of actuators

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at θj = 75◦ and θj = 285◦ . The fluidic actuators modeled and used in the current analysis have the same parameters as discussed in Chapter 4.

(a)

(b)

(c)

(d)

Figure 9.4: The pressure POD modes on the cylinder surface at ReD = 200, (a) i = 1, 2, (b) i = 3, 4, (c) i = 5, 6, and (d) i = 7, 8: even (solid) and odd (dashed).

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193

Reduced-Order Model with Control

For the flow control, we use suction on the cylinder surface through fluidic actuators. We simulate the actuated flow at ReD =200 and record snapshot data . We perform the POD analysis of the actuated field and compute the POD modes. We observe that the eigenvalue spectrum is still concentrated around the first few modes and the first twelve eigenvalues contain more than 99% energy of the system. However, in the actuated flow, the velocity expansion in Equation 6.11 is no longer a candidate for u¯ in the Galerkin projection, as it does not satisfy the boundary conditions for varying suction velocities required in the control action. Graham et al. (1999a) suggested two approaches to incorporate variable input control in the reduced-order model. In the first approach, known as the “control function method”, a suitable control function is included in the velocity expansion to account for the inhomogeneous boundary conditions on the cylinder surface. The POD modes, used in the modified expansion, retain the homogeneous boundary conditions. This method is discussed in detail in Section 9.3.1. In the second approach, the “penalty method”, the velocity expansion remains the same as for the unactuated flow and the essential boundary condition is enforced in an integral “weak” fashion. Thus, to simulate the actuator on the cylinder surface, we express the normal velocity as follows: ˆξ − ǫ uξ = γVjA e

∂u ∂ξ

(9.1)

ˆξ is the unit normal vector. where γ is the variable input control, ǫ is a small parameter, and e We note that the negative value of VjA corresponds to suction. For control purposes, we apply the control function method to develop a reduced-order model. There are three main advantages of using the control function approach over the penalty method. First, it explicitly enforces the homogeneous boundary conditions on the POD modes. Second, it ensures that the pressure term drops out in the Galerkin projection. On the other hand, the control function formulation leads to a complex system of equations in comparison to the penalty method. However, there is no dependence on the parameter ǫ, whose value cannot be determined a priori and requires numerical experimentation.

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Control Function Method

In the control function approach, we expand the velocity field as ¯ (x) + u(x, t) ≈ u

M X

qi (t)Φi (x) +

i=1

Mc X

γi (t)Γi (x),

(9.2)

i=1

where Mc is the total number of control modes, the Γi (x) are a suitable divergence-free control functions that satisfy the inhomogeneous boundary condition due to fluidic actuators, and γ(t) is the variable control input. The POD eigenfunctions are computed after subtracting the control function from each snapshot. Mathematically, we can redefine the snapshot matrix W as

k

W = u(x, t ) −

Mc X

γi (tk )Γi (x)

(9.3)

i=1

In other words, the snapshot data is modified such that the POD modes satisfy homogeneous boundary conditions on the cylinder surface. Moreover, the basis functions in Equation (9.2) retain the divergence-free property.

9.3.2

Galerkin Projection

The reduced-order model developed here is more general and corresponds to M POD modes and Mc control modes. We substitute Equation (9.2) into Equation (2.2) and expand each term to understand its contribution in the modified reduced-order model. Time-derivative term We note that Equation (9.2) contains two time dependent states; qi and γi . Therefore, we obtain two time derivative terms in the expansion as follows: Mc M ∂ X ∂u ∂¯ u ∂ X = + γ n Γn + qm Ωn ∂t ∂t ∂t n=1 ∂t m=1

=

Mc X n=1

γ˙ n Γn +

M X

m=1

q˙m Φm

(9.4)

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Convection term The convection term is more complex and generates linear and nonlinear terms in the model. The quadratic terms obtained in the expansion are in q, γ, and their combinations; that is, Mc M Mc M X X X X u · ∇u = u ¯+ γ n Γn + qm Φm · ∇ u ¯+ γ n Γn + qm Φm n=1

m=1

=u ¯ · ∇¯ u+u ¯· + +

Mc X

n=1 M X

m=1

Mc X n=1

γn Γn · ∇¯ u+

n=1

γn ∇Γn + u ¯· Mc X

γ n Γn ·

n=1 M X

qm Φm · ∇¯ u+

m=1

M X

m=1

qm ∇Φm

Mc X

γm ∇Γm m=1 Mc X

qm Φm ·

n=1

m=1

+

Mc X

γ n Γn ·

n=1 M X

γn ∇Γn +

m=1

M X

qm ∇Φm m=1 M X

qm Φm ·

n=1

qn ∇Φn

(9.5)

Rearranging the terms in Equation (9.5) leads to u · ∇u = u ¯ · ∇¯ u+ + +

Mc X n=1

M X u ¯ · ∇Γn + Γn · ∇¯ u γn + u ¯ · ∇Φm + Φm · ∇¯ u qm m=1

Mc X Mc X

Γn · ∇Γm γm γn +

m=1 n=1

Φm · ∇Γn + Γn · ∇Φm γn qm

n=1 m=1 M X Mc X

M X M X

m=1 n=1

Φm · ∇Φn qm qn (9.6)

or in matrix form as u · ∇u = u ¯ · ∇¯ u + u ¯ · ∇Γ + Γ · ∇¯ u ∗γ+ u ¯ · ∇Φ + Φ · ∇¯ u ∗q + γ ′ ∗ Γ · ∇Γ ∗ γ + q′ ∗ Φ · ∇Φ ∗ q + γ ′ ∗ Γ · ∇Φ + Φ · ∇Γ ∗ q

(9.7)

where q′ and γ ′ denote the transpose of the vectors. Diffusion term The diffusion term is also expanded as follows: ∇2 u = ∇2 u ¯+ = ∇2 u ¯+

Mc X

n=1 M c X n=1

γ n Γn +

M X

qm Φm

m=1 M X

γn ∇Γn +

m=1

qm ∇Φm

(9.8)

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Modified Reduced-Order Model

We substitute Equation (9.2) into the Navier-Stokes equations, project these equations along the Φk , and obtain q˙k (t) = Ak + + +

Mc X

m=1 Mc X

m=1

M X

m=1

Bkm qm (t) +

Hkm γ˙ m (t) + Kkm γm (t) +

M X M X

m=1 n=1

M X Mc X

m=1 n=1 Mc X Mc X

m=1 n=1

Ckmn qn (t)qm (t)

Jkmn qm (t)γn (t) Lkmn γm (t)γn (t),

(9.9)

where Ak =

1 ¯ ) − (Φk , u ¯ .∇¯ u), (Φk , ∇2 u ReD 1 (Φk , ∇2 Φm ), ReD

¯ .∇Φm ) − (Φk , Φm .∇¯ Bkm = −(Φk , u u) + Ckmn = −(Φk , Φm .∇Φn ), Hkm = −(Φk , Γm ), Jkmn = −(Φk , Γn .∇Φm ) − (Φk , Φm .∇Γn ), ¯ .∇Γm ) − (Φk , Γm .∇¯ Kkm = −(Φk , u u) +

1 (Φk , ∇2 Γm ), ReD

Lkmn = −(Φk , Γm .∇Γn ). Mathematically, there is no restriction on the total number of control functions, however, it is only constrained by physical realizations. In general, one control function has successfully been implemented to control the flow past a cylinder. A commonly used control function is the rotation of the cylinder (Graham et al., 1999a,b; Singh et al., 2001). Similarly, Rediniotis et al. (2002) used two control functions for a synthetic jet actuator. In a reduced-order model of a heaving airfoil, Lewin and Haj-Hariri (2005) developed three control modes to simulate heaving, pitching, and lagging. Using their reduced-order model, they simulated heaving motions that are both similar to and different from the motion(s) used to generate the basis functions. In the current study, we use one control function to simulate a pair of fluidic actuators on the cylinder.

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Control Mode Design

There is no specific way to construct the control mode; however, it should satisfy the divergence-free inhomogeneous boundary conditions. A convenient and simple way is to simulate suction with a suitable jet velocity on the cylinder surface in a stationary flow. A snapshot of the flow field is a good choice for the control mode that corresponds to γ = 1. The control mode, thus obtained, is divergence-free and satisfies the inhomogeneous boundary conditions for the actuators. However, the effect of the control mode is local due to the absence of the freestream flow in the simulation. An alternate way to generate a control mode is as follows. We simulate the flow past a cylinder with suction control on the cylinder surface. We call this flow as the actuated flow. We take S snapshots of the flow field and compute the average actuated flow field, as shown in Figure 9.5. Similarly, we simulate the flow without any fluidic actuation (i.e., the unactuated flow) and compute the unactuated flow field from the ensembled snapshot data. Hence, we can compute the control function as the difference between the two average flow fields as follows: ¯ (x)Actuated − u ¯ (x)U nactuated Γ(x) = u

(9.10)

The key advantage of this method is that the flow perturbations in the control mode are not restricted to the local regions close to the actuators. Rather, the effect of the difference between the two average fields is also present in the wake, as shown in Figure 9.6. Appropriate application of the control input, using this function, can favorably alter the flow field over a wider part of the domain. In other words, the global aspect of this method makes it a better candidate for the construction of the control mode. In our analysis, we compute the control mode using this method. Since all of the boundary conditions on the domain are satisfied in the expansion, there is no contribution of the pressure term in the reduced-order model.

9.3.5

Control Law Design

In the reduced-order model, developed in Equation (9.9), the time derivative of the control signal γ is present. This indicates that the control system designed is of an integral nature. The integral control technique is usually used when the closed-loop system is to reject a

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(a)

(b)

Figure 9.5: Actuated flow at ReD =200: (a) streamwise velocity and (b) crossflow velocity.

(a)

(b)

Figure 9.6: Control mode at ReD =200: (a) streamwise velocity and (b) crossflow velocity.

constant disturbance. Such behavior is observed further in this chapter, and an appropriate discussion along with physical insights is provided. To this end, we cast the system of Equation (9.9) into the standard form of state space equations as follows:        H B K q q˙  γc   +   = 1 0 0 γ γ˙

(9.11)

such that Hγc = Hγ + (A + qT Cq + J γq + Lγ 2 )

(9.12)

Here, γc is a new control input to the system. This new control input is constructed in a way to cancel the nonlinearities in the system and further implement a full-state feedback

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scheme. In other words, we are using the feedback linearization technique and then solving an LTI full-state feedback control problem. Hence, with one control input, the new dynamical system has M + 1 dimensions. In the above formulation, A, H, K, and L are vectors of order M × 1, B and J are matrices of order M × M , and C is a second-order tensor of order M . Thus, we reduced our nonlinear model to an augmented LTI control problem of the following form: x˙ = |{z}

(M +1)×1

where 

A=

9.3.6

B K 0

0

x + |{z} B |{z}

A |{z}

(M +1)×(M +1)





 ,B=

H 1

(M +1)×1





 ,x=

(M +1)×1

q γ

u |{z}

(9.13)

1×1



 and u = γc

Open-Loop Control

In an open-loop control, a system is controlled directly and only by an input signal. Any change or perturbation in the system does not affect its control input. In our case, the actuated flow is an example of an open-loop control where a constant suction on the cylinder surface is applied through fluidic actuators. Similarly, the reduced-order dynamical system, developed in Equation (9.11), represents an open-loop control system where the control input γ is not dependent on the states of the system. Therefore, the controller cannot sense any change in the states. However, the eigenvalues of the state matrix A, commonly referred to as poles, provide a means to analyze the dynamical behavior and stability of the system. The pole locations of a system in the complex plane can also assist in designing a feedback controller and favorably alter the system dynamics. We compute the open-loop poles for the LTI system, derived in Equation (9.11), and plot them in Figure 9.7. We observe six complex conjugate pairs (µi ± iωi ), where i = 1, 2, ..., 6, and a pole at the origin. A single pair of poles is in the right-half of the complex plane, thus rendering the behavior of the system unstable. It is important to note that the imaginary part of the unstable pair µ1 ± ω1 i = 0.0016 ± 1.16i corresponds to the vortex-shedding frequency at ReD =200. Since ω = 2πf , therefore f = 0.185 is the Strouhal number at this Reynolds number. The physical interpretation of the poles is significant and emphasizes the

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benefits of analytical tools, such as reduced-order modeling. Despite the simplification of a complex flow problem, physical aspects and important flow features are captured in the model. In order to stabilize the above system, we need to move the unstable poles to the

5

Imag

2.5

0

-2.5

-5 -0.3

-0.2

-0.1

0

0.1

Real Figure 9.7: Poles of open-loop system.

left-half of the complex plane, as shown in Figure 9.8. The new locations of these poles alter the dynamical behavior and controls the response of the system.

9.4

Full-State Feedback Control

The state transition matrix, eigenvalues, eigenvectors, and therefore the system are dependent on the matrix A. Thus, if we manipulate A, we can change (control) the dynamics of the system. We wish to design a full-state feedback controller for this system. Hence, we

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1

Imag

0.5 0 -0.5 -1

-0.002

0.000

0.002

Real Figure 9.8: Schematic of the unstable poles moving to the left-half of the complex plane.

choose γc to be a linear function of the states of the system as follows:    q  γc = −G  γ 

(9.14)

where G1×(M +1) is a matrix of static (i.e., constant) gains multiplied by the state vector. If Equation (9.14) is substituted into Equation (9.12), then the closed-loop system becomes         H B K q˙ q  G   −   =  (9.15) 1 0 0 γ˙ γ | {z } Ac

As the purpose of the control law is to stabilize the system, it is desired to have the poles of the system in the left-half of the complex plane. In other words, we would like the eigenvalues of Ac to have negative real parts. This will guarantee exponential stability of the system.

The pole placement technique is one method to obtain a feedback gain that favorably alters the dynamics of the system. Other techniques include the Linear Quadratic Regulator

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technique, H 2 control, and H ∞ control. Placement of the poles may be driven by design specifications, such as settling time, rise time, overshoot, etc. In the current analysis, we study three cases of pole placement as shown in Table 9.1. The choice of the new pole locations is arbitrary and is not meant for any specific design requirements. In CL1, we Table 9.1: Pole Placement Chart. Case Unstable Poles New Pole Location OL

0.0016 ± 1.16i

-

CL1

0.0016 ± 1.16i

−0.0016 ± 1.16i

CL2

0.0016 ± 1.16i

−0.0160 ± 1.16i

CL3

0.0016 ± 1.16i

−0.0320 ± 1.16i

placed the poles by taking a mirror image on the complex plane. Similarly, in CL2 and CL3, the magnitudes of the real parts of the poles are increased by 10 and 20 times, respectively, and placed in the left-half of the complex plane. In all these cases, the imaginary part is unchanged. The aim of this exercise is to broaden the spectrum of our control application, as regards to pole placement. However, |µ1 | is intentionally kept lower than the other stable poles of OL to minimize their effect on the response of the system. Figure 9.9 shows a graphical representation of these cases.

9.4.1

System Response

The key advantage of reducing the problem to an LTI form is that we can utilize the MATLAB Control Toolbox. We use the “place” command to compute respective gains for each case and construct the augmented dynamical system, as given in Equation (9.15). We then integrate the system to compute its response in each case. In Figure 9.10, we plot the response of the first state q1 for each case. In Figure 9.10(a), we observe that the system response of CL1 is too slow and the settling time is over 1000 time units. It is consistent with the fact that the poles, despite being in the left-half of the complex plane, are close to the imaginary axis. However, the response of CL2 is much faster and the settling time is approximately 300 time units, as shown in Figure 9.10(b). In CL3,

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1

Imag

0.5 0 -0.5 -1

-0.04

-0.02

0.00

Real Figure 9.9: Pole locations: circle - OL, diamond - CL1, square - CL2, and delta - CL3.

the poles are placed further to the left resulting in a decay in q1 . From Figure 9.10(c), we observe the settling time for CL3 is about 150 time units.

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4 3 2

q1 (t)

1 0 −1 −2 −3 −4 0

50

100

150

200

100

150

200

150

200

T ime

(a) 4 3 2

q1 (t)

1 0 −1 −2 −3 −4 0

50

T ime

(b) 4 3 2

q1 (t)

1 0 −1 −2 −3 −4 0

50

100

T ime

(c)

Figure 9.10: Time history of q1 with control actuation; (a) CL1, (b) CL2, and (c) CL3.

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Depending upon the design requirements, we can place the poles to meet those criteria. Increasing the pole frequency at constant damping ratio increases the bandwidth and decreases the time constant. Poles close to the imaginary axis have a damped oscillatory response and a longer time constant as observed in CL1. On the other hand, poles far from the imaginary axis have a shorter time constant as in CL3. In all cases, the response is oscillatory due to the imaginary part of the poles. Appropriate gains computed for the closed-loop systems are given in Table 9.2. We observe larger gains for systems having shorter time constants. Table 9.2: Feedback Gains of the Systems. State

Gains of CL1 ×10−3



q1

20.332

112.669

215.379

q2

-1.988

-8.951

-15.386

q3

0.071

0.388

0.738

q4

0.0136

0.0811

0.159

q5

-1.405

-7.666

-14.551

q6

-0.587

-3.563

-7.065

q7

-0.002

-0.008

-0.0116

q8

-0.061

-0.335

-0.636

q9

0.237

1.358

2.637

q10

0.328

1.765

3.326

q11

0.018

0.102

-0.197

q12

-0.008

-0.046

-0.0872

γ

6.331

34.725

66.080

In Figure 9.11, we plot the input variable γ for all three case. As observed in the state response, the input control γ shows a similar trend in all cases. However, it is interesting to note that the peak bounds (γmin and γmax ) and steady-state γss show clear differences in these cases, as shown in Table 9.3.

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0.02 0

γ

−0.02 −0.04 −0.06 −0.08 −0.1 0

50

100

150

200

100

150

200

150

200

T ime

(a) 0.1 0 −0.1

γ

−0.2 −0.3 −0.4 −0.5

0

50

T ime

(b) 0.2 0

γ

−0.2 −0.4 −0.6 −0.8 −1 0

50

100

T ime

(c)

Figure 9.11: Time history of γ; (a) CL1, (b) CL2, and (c) CL3.

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As discussed above, the settling time of the state decreases by moving the poles further away from the imaginary axis in the left-half of the complex plane. However, in order to achieve quick response, we need to increase the input to the system. The steady-state input in CL1 is -0.04, which is approximately 6 and 11 times lower than those in CL2 and CL3, respectively. Thus, the control design is a compromise between the system response and available resources. Table 9.3: System Response. Case

Pole Location

Settling time for q1

γmin

γmax

γss

OL

0.0016 ± 1.16i

-

-

-

-

CL1

−0.0016 ± 1.16i

1000

-0.095

0.01

-0.04

CL2

−0.0160 ± 1.16i

300

-0.520

0.05

-0.24

CL3

−0.0320 ± 1.16i

150

-0.960

0.10

-0.45

In most of engineering systems, it is desired to suppress the oscillations or perturbations. Although, it seems to be a simple mathematical problem with a straightforward solution, it can be physically constrained. The limitation comes from the input or feasible actuation associated with the controller. The fluidic actuators are limited in their capability of providing suction/blowing and can at times be inefficient in terms of the energy requirement to attain desired objectives.

9.4.2

Modified Feedback System

We design our controller by linearizing the dynamical system using the control law defined in Equation (9.12). We compute the gains corresponding to the desired pole locations and construct a closed-loop feedback system, as shown in Equation (9.15). The dynamical system, thus obtained, is stable and is integrated to compute the system response. In the linear open-loop system, the presence of a complex conjugate pair in the right-half of the complex plane corresponds to linear instability of the system. However, in the physical system, we observe a stable periodic solution (limit cycle) in the form of vortex shedding as a result of a Hopf bifurcation. The nonlinearity entraps the unstable linear part and

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stabilizes the system. Also, from the normal form (Nayfeh, 2000) of Hopf bifurcation, the positive damping coefficient in the nonlinearity plays a stabilizing role in the dynamics of the system. Thus, including the nonlinear term in our dynamical system adds damping to the system at no additional cost. Therefore, we modify the closed-loop feedback model and incorporate in it the nonlinear term as follows: x˙ k =

M +1 X

Ackm xm +

m=1

M +1 M +1 X X m=1 n=1

 

Ckmn 0 0

0



 xn xm

(9.16)

In order to verify our argument, we simulated CL1 for the two models; that is, with and without the nonlinear term. In Figure 9.12, we observe that the modified nonlinear dynamical system exhibits more damping as compared to the linear system and shows stabilizing behavior. However, the relative increase in damping due to nonlinearity as compared to the damping in the linear system decreases for CL2 and CL3. In other words, the positive linear damping in CL2 and CL3 is already large enough due to pole locations so that the additional damping from nonlinearity is not substantial. Anyways, addition of a nonlinear term, already computed in our reduced-order model, does not require any extra effort. Hence, we improve our control model by adding the nonlinear term. 4 3 2

q1 (t)

1 0 −1 −2 −3 −4 150

160

170

T ime

180

190

200

Figure 9.12: System response q1 for CL1: solid - modified model and dashed - linear model.

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Discussion Control of Direct Simulations

The control strategy, developed so far, is the control of a mathematical model. In other words, we apply full-state feedback control to the reduced-order model of the flow past a cylinder and not on the real flow. For an active control design, it is important to compute the input for our fluidic actuators on the cylinder surface. In this section, we use the system response of CL3 to calculate the control input associated with the actuator using Equation (9.12). We transform the control input into a jet velocity (Vj ) of the actuator using the control mode information. The jet velocity Vj (t) of the fluidic actuator is plotted in Figure 9.13. A negative velocity indicates suction and an oscillatory actuation is due to the control design. From Figure 9.13, we observe a steady-state VjA =-0.66. In Figure 4.24, we show the effect of −0.45 −0.5 −0.55

Vj

−0.6 −0.65 −0.7 −0.75 −0.8 −0.85 0

50

100

T ime

150

200

Figure 9.13: Jet velocity associated with the control input in CL3.

suction on the lift and drag coefficient. The flow control, actuated with VjA =-0.31, achieves a 50% suppression in the lift fluctuations and a 25% reduction in the mean-drag coefficient. Using the results obtained in the CL3 simulation, we design an open-loop controller and perform a CFD simulation. In Figure 9.14, we plot the time histories of CL and CD for two cases. In the first case, we use VjA = −0.31, based on which we design our control mode. In the second case, we simulate the flow control using the results of our full-state

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CL , CD

1

Control ON

0.5

0

-0.5 60

80

100

120

Time Figure 9.14: Time histories of CL and CD with feedback control (solid) and without feedback control (dashed).

feedback control; that is, VjA = −0.66. We observe no fluctuations in the force response, which indicates complete suppression of vortex shedding. Similarly, the mean drag is further reduced to 0.7; that is almost 45% decrease from the uncontrolled flow. In Figure 9.15, we plot ten frames of the spanwise vorticity equally distributed over 20 nondimensional time units (t = 80 − 100). We actuate the feedback controller at t = 80 and observe gradual suppression of vortex shedding, resulting in the reduction of fluctuating forces.

9.5.2

Limitations of Full-State Feedback Controller

Although the control law presented in the preceding section guarantees asymptotic stability of the closed-loop system, the full-state feedback technique lacks the experimental implementation aspect. In real-life dynamic systems, often few measurements are made compared to the number of the states that dominate its dynamics. These limitations are sometimes

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Figure 9.15: Spanwise vorticity contours with feedback controller from t=80-100 (Actuation begins at t=80).

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the result of difficulties associated with sensor placement and maintenance. Other times, the states are not measurable all together, which is the case for the system discussed in this work. In this case, an estimator is mathematically constructed to extract the instantaneous values of the system states from measurement data. In such a case, the control problem branches into a control problem and an estimation problem. Fortunately enough, for linear systems, these two problems are separable. The estimation treatment brings in a host of challenges of its own. The control engineer has to survey the structure under study for the possible nature and placement of a sensor. Moreover, a theoretical analysis should be carried out to judge whether the sensor output is rich enough to construct back the system states. On the other hand, sensors have their limitations as well; sensing is often a stochastic process. In this spirit, the estimation problem is a true challenge to many control engineers.

9.5.3

Controllability Issues

The developments in the preceding section demonstrated that the system is controllable as a feedback gain could be obtained to stabilize the linear system. However, further analysis is needed to examine the controllability of the overall nonlinear system. Possible tools to be used in carrying such an analysis are Lie Algebra and Lie Brackets, which are usually used to study controllability of nonlinear systems. The controller design in this Dissertation is a feedback linearization scheme augmented with a full-state feedback controller. For more advanced control methods, such as adaptive control, one requires a reference system. The closed-loop system developed in this work can serve as a reference model for an adaptive controller that stabilizes the overall nonlinear system.

9.6

Summary

We modified the reduced-order model developed in Chapter 6 using a control function method. Suction is used as the mechanism to control the flow field around a cylinder. We placed the suction actuators on the cylinder surface at the optimum locations obtained

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from the dominant pressure POD mode. Using the DNS of the controlled flow field, we developed a suitable control function that satisfies the inhomogeneous boundary condition on the cylinder surface. We substituted the modified velocity expansion into the Navier-Stokes equations and projected it onto the velocity POD modes. We used feedback linearization to reduce the model to an LTI system. We employed full-state feedback control to obtain suitable gains that stabilize the linear problem. We discussed different cases of moving the poles to the left-half of the complex plane and computed the system response for each case. We later translated the input variable gain into the suction velocity and simulated the flow. We observed reduction in the mean drag coefficient and complete suppression of the fluctuating components in both the lift and drag coefficients.

Chapter 10 Summary, Conclusions, and Recommendations for Future Work The research performed here is motivated by fluid-structure interactions resulting in VIV. If the frequency of these vibrations is close to a natural frequency of the body, the resulting resonance can generate large-amplitude oscillations, which may ultimately cause structural failure. The salient features of this Dissertation include the development of a three-dimensional parallel CFD solver to simulate the flow past a circular cylinder and compute the hydrodynamic forces on the structure. We also developed a reduced-order model of the flow field and the fluctuating forces based on the CFD results. Then, we applied feedback control to reduce these forces by using fluidic actuators. In this chapter, we summarize these results and present concluding remarks and recommendations for future work.

10.1

Summary and Conclusions

Numerical simulations, reduced-order modeling, and feedback control of fluid flows constitute fundamental ingredients in many engineering and industrial applications. We summarize the results in three categories and highlight the contributions and achievements made in each category.

214

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10.1.1

Conclusions

215

Parallel CFD Solver

With increasing parallel computing technology, the numerical approach provides a means to simulate three-dimensional complex flows and analyze its flow physics. We developed a parallel numerical methodology to study the fluid-structure interaction using MPI. We used a fractional-step solution algorithm to solve the incompressible Navier-Stokes equations. The CFD solver is based on curvilinear coordinates, which enable simulation of the flow past arbitrary closed shapes, such as circular and elliptic cylinders and airfoils. We employed the domain decomposition technique, which divides the computational domain into multiple processors, so that the work load is evenly distributed among processors and the computation to communication ratio is efficient. A detailed validation and verification of the solver was performed to establish confidence in the accuracy of the numerical results. We computed the speed-up and efficiency of the parallel computations to indicate the parallel performance of the solver. Moreover, the scalability study showed that the solver can be efficiently used to perform fluid-structure analysis with larger computational and grid sizes. In addition to basic flow configuration of the flow past a stationary structure, the CFD solver is capable of simulating moving boundaries using ARF in two degrees of freedom. We performed several simulations of a crossflow oscillating cylinder in the synchronous and nonsynchronous regions; good agreement was found between simulations and experimental results. The surface boundary conditions can be modified to simulate fluidic actuation on the cylinder for flow control application. Due to computational constraints to resolve small structures (eddies) at high Reynolds numbers, we incorporated a turbulent model in the CFD solver. The Spalart-Allmaras model was used to simulate the flow past a cylinder, as a test case at Re=3,900; the results are promising. In short, we developed an efficient computational tool capable of simulating complex fluid flows. Various features of the solver enables its application to many fluid-structureinteraction problems.

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10.1.2

Conclusions

216

Reduced-Order Models

Even with present day technology of parallel computing, it would take hours to simulate different VIV configurations. The degrees of freedom required to simulate a full 3-D NavierStokes equation simulation is of the order of millions. Moreover, it is difficult to apply flow control strategies directly to the Navier-Stokes equations. Therefore, we developed a reducedorder model to reduce the degrees of freedom in the current problem. Using the parallel CFD solver, we extended the van der Pol-Duffing oscillator model for the hydrodynamic forces on a circular cylinder to elliptic cylinders with varying eccentricities. We employed a POD-Galerkin expansion to develop reduced-order models for the velocity and pressure fields of the flow around a cylinder. Using the flow field snapshots from the CFD simulations, we computed the eigenfunctions for the velocity and pressure by using the POD approach. The model is obtained from a Galerkin projection of the Navier-Stokes equations onto the velocity modes. Although the POD based models lack robustness away from the reference simulation, they provide an analytical insight into the physical phenomena. Moreover, these models, being a set of ODE, enable application of dynamical systems theory and control methods. We also investigated the stability of the model and presented a shooting method to compute initial conditions on the limit cycle and its period. A pressure model was then developed from a Galerkin projection of the pressure-Poisson equation onto the pressure modes. The lift and drag coefficients were then computed by integrating the pressure and shear forces on the cylinder surface. The spectrum of applications of the reduced-order models can be broadened by combining the snapshot data in different flow regimes and computing the POD eigenfunctions. The modes thus obtained may be used to model the flow characteristics over a wider range of Reynolds numbers. Moreover, control techniques can effectively be applied to the reducedorder model to suppress vortex shedding.

10.1.3

Full-State Feedback Control

The key objective of the research is to reduce or suppress vortex shedding. For flow control, we applied suction on the cylinder surface through fluid actuators. We modified the reduced-

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217

order model to incorporate a control function mode, constructed from the actuated flow field. A similar Galerkin projection of the Navier-stokes equations onto the POD modes lead to a more complex model, containing extra terms originating from the input variable. In the linear open-loop system of the reduced-order model, we observed a single complex-conjugate pair in right-half of the complex plane, indicating a Hopf bifurcation in the dynamical system. We applied a full-state feedback controller and computed appropriate gains to move the poles to the left-half of the complex plane. This stabilizes the linear system and suppresses vortex shedding, thereby reducing the lift and drag forces on the cylinder.

10.2

Recommendations for Future Work

Combination of three subjects covered in this research has a lot of potential in many engineering applications. The parallel CFD solver could be used to study forced-oscillations in single- and two-degrees-of-freedom motion not only for cylinders, but also airfoils. Shear flows is another aspect of research in vortex-induced vibrations. The capability of the CFD solver could also be enhanced to simulate fluid-structure vibrations by coupling the equations of motion of the structure and fluid. For reduced-order modeling, some other estimation procedures for computing the temporal coefficients, such as the Linear Stochastic Estimator (LSE) can be used. Viscosity models should also be introduced in the reduced-order models high-Reynolds-numbers applications. The full-state feedback control law developed to suppress vortex shedding has its application limitations. Real dynamical systems require sensors and estimators for feedback control. In this work, the feedback control implemented utilized all the states of the system. In real systems, this is not usually the case, as a limited number of sensors is usually available/realizable. Moreover, the states of the model considered in this work are not physical states, and hence cannot be sensed to begin with. Consequently, a state estimator is a necessity for this system. Also, the locations of these sensors might be critical for implementation, as this will directly affect the estimated states, which in turn will be used to control the system. Moreover, the controllability of the system should be considered in a more thorough manner. Performance of the closed-loop system will be directly influenced by the locations of the actuators. Later, suction actuators will be

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Conclusions

replaced by synthetic jets to improve the efficiency of the control mechanism.

218

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