linear stability analysis of parallel shear flows

Linear Stability Analysis of Parallel Shear Flows

A Thesis Submittedin Partial Fulfillment of the Requirements

for the Degree ofMaster of Technology

by

Amarjeet

to theDepartment of Aerospace EngineeringIndian Institute of Technology Kanpur

October, 2013

Certificate

It is certified that the work completed in the thesis entitled Linear Stability Analysis

of Parallel Shear Flows by Amarjeet has been carried out under my supervision and

this work has not been submitted elsewhere for a degree.

Prof. Sanjay Mittal

Department of Aerospace Engineering

Indian Institute of Technology Kanpur

Kanpur-208016.

October, 2013

Dedicated

to

my parents, sister and brother

Abstract

Linear Stability Analysis (LSA) of parallel shear flows has been a major area of research

during the last century and continues to be so in the 21st century. The famous Orr-

Sommerfeld (OS) equation, which is used for local LSA of parallel shear flows, does not

yield useful results for several situations. This is due to mismatch of critical Reynolds

number for the onset of instability, Recro, obtained from the OS analysis and critical

Reynolds number for the transition to turbulence, Recrt, obtained from experimental

analysis. In this work, the Cross-Stream Orr-Sommerfeld (CSOS) equation is derived

for local LSA of parallel shear flows to include disturbances moving with nonzero cross-

stream velocity component. The CSOS equation is applied to Plane Couette Flow (PCF)

and Plane Poiseuille Flow (PPF). A spectral collocation method based on Chebyshev

polynomials of the first kind has been used to solve the relevant equations. This gives

Recro equal to 360.63 and 916.85 for PCF and PPF respectively, which are in close

agreement with Recrt obtained from experimental analysis for the corresponding flows.

The aspects of global LSA of parallel shear flows for periodic boundary conditions

in streamwise and spanwise directions are also analysed. A stabilized finite element

method proposed by Mittal and Kumar (2007) has been utilized to do so. It is shown

that the local and global LSA give same result. Global LSA of PCF and PPF are done

for many cases (including near the onset of instability predicted by local LSA). Direct

Numerical Simulation (DNS) of the respective cases has been performed to crosscheck

the analysis and study the time evolution of the disturbances.

Acknowledgments

I want to take this opportunity to express my sincere gratitude towards my thesis su-

pervisor Prof. Sanjay Mittal. The present work might have no existence without him.

During my stay with Professor Mittal’s research group, I got the opportunity to work

on Linear Stability Analysis (LSA) of parallel shear flows. My interactions and associ-

ation with my thesis advisor and other members of the research group were conducive

in carrying out this research work. In particular, I acknowledge the interactions with

the research group and the guidance from my thesis advisor in the extension of the Orr-

Sommerfield (OS) analysis to include cross-flow moving perturbations, the form of the

perturbations, naming the analysis as Cross-Stream Orr-Sommerfeld (CSOS) analysis

and demonstrating its equivalence with global analysis for periodic boundary conditions.

I am grateful to Prof. Nirmalya Guha who always tries his best to have improve-

ment in every aspect of my life.

I am thankful to Ravi and Anubhav for their invaluable help during the start of my

thesis. I am also thankful to Prof. Bhaskar Kumar for his invaluable discussion towards

the end of my thesis. I feel lucky to have interacted with very humble, intellectual and

helpful person, Mr. V M Krushna Rao Kotteda, Ph.D. student, Dept. of Aerospace

Engineering, IIT Kanpur. I would also like to thank all my lab mates in the CFD

lab, especially Durgesh, Navrose, Vivek, Sai, Furqaan, Sambhav, Nishchal, Aekaansh

and Ajinkya, for their help and making the lab environment enjoyable. I express my

gratitude to all my friends both inside and outside IITK, who helped me directly or

indirectly for successful completion of my thesis.

Finally, I extend my heartfelt thanks to my parents, sister and brother for their

unflagging love, unparalleled support and silent prayers throughout.

Amarjeet

Contents

Certificate i

Dedication ii

Abstract iii

Acknowledgments iv

Contents v

List of Figures viii

List of Tables xi

Nomenclature xii

0 Thesis organization and conventions adopted 1

0.1 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

0.2 Conventions adopted . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1 Introduction and general results 3

1.1 Outline of LSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1.1 Steps involved in LSA . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Historical note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.1 PCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.1.1 Theoretical and numerical investigations of LSA of PCF 6

1.2.1.2 Experimental investigations of LSA of PCF . . . . . 7

1.2.2 PPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.2.1 Theoretical and numerical investigations of LSA of PPF 7

CONTENTS vi

1.2.2.2 Experimental investigations of LSA of PPF . . . . . 8

1.2.3 List of Recro and Recrt for PCF and PPF . . . . . . . . . . . . 8

1.3 Incompressible flow equations . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Nonlinear disturbance equations . . . . . . . . . . . . . . . . . . . . . 9

1.5 Linearized disturbance equations . . . . . . . . . . . . . . . . . . . . 10

1.6 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Local LSA 11

2.1 Formulation of the CSOS equation . . . . . . . . . . . . . . . . . . . 11

2.2 Comparison of the CSOS analysis with the OS analysis . . . . . . . . 15

2.3 Squire’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.1 Chebyshev polynomials of the first kind . . . . . . . . . . . . . 17

2.4.2 Spurious eigenvalues . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.3 Chebyshev discretization of the CSOS equation . . . . . . . . 18

2.5 2D LSA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.5.1 Construction of modes . . . . . . . . . . . . . . . . . . . . . . 20

2.6 Application of the CSOS equation to PCF . . . . . . . . . . . . . . . 22

2.6.1 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.2 General properties . . . . . . . . . . . . . . . . . . . . . . . . 22

2.6.3 General results . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.7 Application of the CSOS equation to PPF . . . . . . . . . . . . . . . 28

2.7.1 Convergence study . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.2 General properties . . . . . . . . . . . . . . . . . . . . . . . . 28

2.7.3 General results . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.8 Group velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 Global LSA 34

3.1 Global linear stability equations . . . . . . . . . . . . . . . . . . . . . 34

3.2 The finite element formulation . . . . . . . . . . . . . . . . . . . . . . 35

3.2.1 The incompressible flow equations . . . . . . . . . . . . . . . . 35

3.2.2 The nonlinear disturbance equations . . . . . . . . . . . . . . 36

3.2.3 The linearized disturbance equations . . . . . . . . . . . . . . 36

3.2.4 The global linear stability equations . . . . . . . . . . . . . . . 36

3.2.5 Stabilization terms . . . . . . . . . . . . . . . . . . . . . . . . 37

CONTENTS vii

3.3 Solution method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Steady and unsteady flow . . . . . . . . . . . . . . . . . . . . 38

3.3.2 The eigenvalue problem . . . . . . . . . . . . . . . . . . . . . 38

3.4 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4.1 Domain and boundary conditions . . . . . . . . . . . . . . . . 41

3.4.2 Mesh convergence study . . . . . . . . . . . . . . . . . . . . . 43

3.4.2.1 PCF . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.4.2.2 PPF . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.4.3 Effect of length of domain . . . . . . . . . . . . . . . . . . . . 45

3.4.3.1 PCF . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4.3.2 PPF . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4 Time evolution of modes 50

4.1 Evolution equation corresponding to local LSA . . . . . . . . . . . . . 50

4.2 Evolution equation corresponding to global LSA . . . . . . . . . . . . 51

4.3 DNS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.4 Growth rate at a general time . . . . . . . . . . . . . . . . . . . . . . 52

4.5 Comparison of DNS with EEL . . . . . . . . . . . . . . . . . . . . . . 53

4.6 DNS for crosschecking Recr . . . . . . . . . . . . . . . . . . . . . . . 53

4.6.1 PCF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.6.2 PPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5 Conclusions and future scope 58

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 Future scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Appendix 60

A Issues related to LSA 61

A.1 Linear combination of modes . . . . . . . . . . . . . . . . . . . . . . . 61

A.2 The two variants of PCF . . . . . . . . . . . . . . . . . . . . . . . . . 62

B 2D LSA Matlab code 63

B.1 Code I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

B.2 Code II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography 84

List of Figures

1.1 Schematic of (a) PCF; (b) PPF. . . . . . . . . . . . . . . . . . . . 5

2.1 v′− modes at Re = 500, α = 2 for (a) PCF; (b) PPF. The upper part

is for Vcy = 0.05 while the lower one is for Vcy = −0.05. In both

cases, the left part shows the real modes while the right one shows the

imaginary modes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2 Neutral surface for PCF. . . . . . . . . . . . . . . . . . . . . . . . 23

2.3 PCF at Vcy = 0. Contours of (a) constant λi; (b) constant growth

rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.4 PCF at Vcy = 0.11. Contours of (a) constant λi; (b) constant growth

rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Variation of Recr with Vcy for PCF. v′−modes of a few cases are also

shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.6 Variation of αcr with Vcy for PCF. . . . . . . . . . . . . . . . . . . 27

2.7 Variation of λicr with Vcy for PCF. . . . . . . . . . . . . . . . . . . 27

2.8 Neutral surface for PPF. . . . . . . . . . . . . . . . . . . . . . . . 29

2.9 PPF at Vcy = 0. Contours of (a) constant λi; (b) constant growth

rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.10 PPF at Vcy = 0.08. Contours of (a) constant λi; (b) constant growth

rate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.11 Variation of Recr with Vcy for PPF. v′−modes of a few cases are also

shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.12 Variation of αcr with Vcy for PPF. . . . . . . . . . . . . . . . . . . 33

2.13 Variation of λicr with Vcy for PPF. . . . . . . . . . . . . . . . . . . 33

LIST OF FIGURES ix

3.1 A schematic of the scatter of eigenvalues before (top) and after (bot-

tom) the onset of instability: λ-plane and µ-plane corresponds to the

eigenvalues and their inverse. A circle of unit radius is also shown in

the figure. [2] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Domain description and boundary conditions used for calculation of

basic state in case of (a) PCF; (b) PPF. . . . . . . . . . . . . . . . 41

3.3 Boundary conditions used for solving the disturbance equations and

LSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 A finite element mesh for L = H with Nelx = 25 and Nely = 300. 42

3.5 PCF at Re = 15000, L = H and Vcy = 0.08. The figure shows the v′−modes corresponding to (a) local LSA; (b) global LSA. . . . . . . 44

3.6 PPF at Re = 15000, L = H and Vcy = 0.07. The figure shows the v′−modes corresponding to (a) local LSA; (b) global LSA. . . . . . . 45

3.7 Variation of λr with respect to α for PCF at Re = 1000, Vcy = 0.16. 46

3.8 Variation of λr with respect to L for PCF at Re = 1000, Vcy = 0.16. 46

3.9 Variation of λr with respect to Ln for PCF at Re = 1000, Vcy = 0.16.

Expected values (computed from local LSA) are plotted with lines.

Solid dots represent computed values from global LSA. . . . . . . 47

3.10 v′− modes with respect to Lno for PCF at Re = 1000, Vcy = 0.16. . 47

3.11 Variation of λr with respect to α for PPF at Re = 3500, Vcy = 0.04. 48

3.12 Variation of λr with respect to L for PPF at Re = 3500, Vcy = 0.04. 48

3.13 Variation of λr with respect to Ln for PPF at Re = 3500, Vcy = 0.04.

Expected values (computed from local LSA) are plotted with lines.

Solid dots represent computed values from global LSA. . . . . . . 49

3.14 v′− modes with respect to Lno for PPF at Re = 3500, Vcy = 0.04. . 49

4.1 Time evolution of the normalized kinetic energy (density) of the distur-

bance (in Fourier space) for PCF at Re = 400, α = 0.85, Vcy = 0.11. 54

4.2 Time evolution of the normalized kinetic energy (density) of the distur-

bance (in Fourier space) for PPF at Re = 1000, α = 2.50, Vcy = 0.08. 54

4.3 Time evolution of the normalized kinetic energy of the disturbance for

PCF at Vcy = 0 and indicated combinations of Re and α. Also shown

is the time evolution of v′−mode for the combination Re = 10, α = 2. 55

LIST OF FIGURES x


PCF at Vcy = 0.11, α = 0.85 and indicated values of Re. Also shown

is the time evolution of v′− mode for the case of Re = 400. . . . . 56


PPF at Vcy = 0, α = 1.02 and indicated values of Re. Also shown is

the time evolution of v′− mode for the case of Re = 7000. . . . . . 57


PPF at Vcy = 0.08, α = 2.50 and indicated values of Re. Also shown

is the time evolution of v′− mode for the case of Re = 1000. . . . 57

A.1 PCF at Re = 10000, α = π, Vcy = 0. v′− modes from local LSA

corresponding to (a) positive Vtx; (b) negative Vtx. (c): v′− mode

from global LSA. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

A.2 The two variants of PCF: (a) PCF1; (b) PCF2. . . . . . . . . . . 62

List of Tables

1.1 List of universally accepted Recro and Recrt for PCF and PPF at

present time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1 Convergence study with respect to NP for PCF at Re = 15000, α =

π, Vcy = 0.08. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 Convergence study with respect to NP for PCF at Re = 20000, α =

1.38, Vcy = 0.2291. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Critical parameters for PCF. . . . . . . . . . . . . . . . . . . . . . 25

2.4 Convergence study with respect to NP for PPF at Re = 15000, α =

π, Vcy = 0.07. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.5 Convergence study with respect to NP for PPF at Re = 20000, α =

3.80, Vcy = 0.1348. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.6 Critical parameters for PPF. . . . . . . . . . . . . . . . . . . . . . 31

3.1 Mesh convergence study for PCF at Re = 15000, L = H, Vcy = 0.08. 43

3.2 Mesh convergence study for PPF at Re = 15000, L = H, Vcy = 0.07. 44

Nomenclature

UW Wall velocity

UC Center-plane velocity

H Channel height

L Length of domain in streamwise direction

Ls Length of domain in spanwise direction

Re Reynolds number

Recr Critical Reynolds number

Recro Critical Reynolds number for the onset of instability

Recrt Critical Reynolds number for the transition to turbulence

u Velocity

p Pressure

Vc Velocity of the moving frame with respect to the laboratory frame

Vt Propagation speed of the disturbance

Vg Group velocity

NOMENCLATURE xiii

Greek symbols

α Streamwise wave number

β Spanwise wave number

λ Eigenvalue

Acronyms

PCF Plane Couette Flow

PPF Plane Poiseuille Flow

2D Two-dimensional

3D Three-dimensional

LSA Linear Stability Analysis

DNS Direct Numerical Simulation

OS Orr-Sommerfeld

CSOS Cross-stream Orr-Sommerfeld

SUPG Streamline-Upwind/Petrov-Galerkin

PSPG Pressure-Stabilizing/Petrov-Galerkin

LSIC Least-Squares on Incompressibility Constraint

Chapter 0

Thesis organization and conventions

adopted

0.1 Thesis organization

The work has been presented in the following manner:

Chapter 1: This chapter contains the general idea related to Linear Stability Anal-

ysis (LSA). Literature review related to LSA of Plane Couette Flow (PCF) and

Plane Poiseuille Flow (PPF) is presented. Basic equations governing incompress-

ible fluid flow are also included in this chapter.

Chapter 2: In this chapter, derivation of the Cross-Stream Orr-Sommerfeld (CSOS)

equation and it’s solution method are presented. The chapter shows the equiva-

lence between the Orr-Sommerfeld (OS) equation and the CSOS equation for dis-

turbances moving with zero cross-stream velocity component. Application of the

CSOS equation to PCF and PPF is also presented in this chapter.

Chapter 3: This chapter presents the aspects of global LSA of parallel shear flows

for periodic boundary conditions in streamwise and spanwise directions. The rela-

tion between local and global LSA of parallel shear flows is also presented in this

chapter.

Chapter 4: This chapter aims to depict the time evolution of the disturbances

through Direct Numerical Simulation (DNS).

Chapter 5: In this chapter, conclusions and scope for future work are presented.

0.2 Conventions adopted 2

0.2 Conventions adopted

We present below the conventions adopted throughout the thesis:

All the equations and analyses are presented in absence of body force.

All the quantities are in dimensionless form. Following nondimensionalization

schemes have been adopted for the specific cases of PCF and PPF:

Channel half-height is the length scale. The corresponding velocity scales are

(a) half the velocity difference between walls velocities for PCF, (b) center-plane

velocity for PPF.

Unless specified, all the analyses will be corresponding to Cartesian coordinate

system.

In the images for modes and time evolution of modes, negative levels are shown by

dashed line while positive levels are shown by solid lines. Here magnitude of min-

imum and maximum values are equal (since only sinusoidal terms are present for

construction of modes and integral multiples of spatial period are shown). Darkest

colors in region of negative and positive levels show the minimum and maximum

values respectively. Values indicated by in-between colors can be found by linear

interpolation.

Due to property of periodicity in streamwise and spanwise directions, phase shifts

should be ignored while comparing the modes. Also, after studying subsequent

chapters, it can be observed that at a given condition, real and imaginary modes

will be same. Hence, only real modes have been shown unless specified.

Unless specified, the eigenvalues, modes and time evolution of modes correspond

to maximum growth rate.

Chapter 1

Introduction and general results

There is no general theory for complete analysis of fluid motion and there may never be.

The reason is that all kinds of laminar flows become turbulent as Re is increased. This

way there is a profound change in behaviour of the fluid flow. So, turbulence in fluid is

the culprit, as no complete understanding of turbulent flow exists. The complexity of

describing and studying turbulence in fluids is cleverly expressed by the British physicist

Horace Lamb in 1932 during an address to the British Association for the Advancement

of Science. He said, “I am an old man now, and when I die and go to heaven there are two

matters on which I hope for enlightenment. One is quantum electrodynamics, and the

other is the turbulent motion of fluids. And about the former I am rather optimistic.”

LSA of a laminar flow is the process of examining the response of the flow to

infinitesimal disturbances. Such disturbances may lead the flow to become unstable.

Unstable flows often evolve into turbulent flow. So, in the past, LSA has been used as a

tool to understand the mechanism for transition to turbulence.

1.1 Outline of LSA

In general, LSA can be done in two different ways. These are local LSA and global

LSA [1, 2]. A brief description of these approaches is presented in following paragraphs.

Local LSA: In local LSA, the flow field is either parallel or assumed to be locally

parallel at different streamwise locations. Then the method of parallel flow instability

is used to obtain an eigenvalue problem. The solution to the eigenvalue problem gives

growth rate, frequency and local modes. The advantage here is that it requires relatively

less computer memory and can be performed on measured velocity profiles. However, if

1.2 Historical note 4

the flow field is highly nonparallel, one must validate the results of local LSA via global

LSA.

Global LSA: In global LSA, a domain containing entire flow field is considered.

Here the method of nonparallel flow instability is applied to get an eigenvalue problem.

The solution of the problem gives growth rate, frequency and global modes. The ad-

vantage of using global LSA is that one does not need to repeat the computation for

different streamwise locations of the flow field. Compared to local LSA, global LSA

requires larger computer memory.

1.1.1 Steps involved in LSA

Following seven steps are adopted for both local and global LSA:

1. Selecting a basic solution Q of the flow problem.

2. Adding a disturbance variable q′ and substituting (Q+ q′) into the basic equations

governing the flow problem.

3. Subtracting the basic terms (satisfied by Q) from the equations resulting from step

2 to get the disturbance equation.

4. Linearizing the disturbance equation by assuming small disturbances.

5. Simplifying the linearized disturbance equation by assuming a special form for the

disturbances (if possible). For example, the multidimensional linearized distur-

bance equation can be simplified by assuming wavelike disturbances.

6. Solving for the eigenvalues.

7. Interpreting the stability conditions as well as showing charts for neutral surface(s),

growth rates and decay rates.

1.2 Historical note

The first scientific study of turbulence was done by Leonardo da Vinci (15th century).

He placed obstructions in water and analysed the result. Leonardo wrote, “Observe the

motion of the surface of the water, which resembles that of hair, which has two motions,

of which one is caused by the weight of the hair, the other by the direction of the curls;


thus the water has eddying motions, one part of which is due to the principal current,

the other to the random and reverse motion.”

Many researchers like Hagen, Helmholtz, Kelvin, Rayleigh and Reynolds studied

the stability of fluid motion appreciably in the 19th century. Among them, Reynolds

(1883) [3] is credited for providing a clear picture of turbulence due to his famous pipe

flow experiment. Through his experiments, Reynolds showed that the smooth flow (lam-

inar flow) breaks down when Re exceeds a fixed critical value of 13000. However, this

was very sensitive to disturbance in the water before entering the tube. Later experimen-

talists have introduced disturbances of finite amplitude at the intake or used roughened

pipes to find Recr equal to 2000.

The famous Orr-Sommerfeld equation for obtaining Recr in case of parallel shear

flows was derived independently by William McFadden Orr (1907) [4] and Arnold Som-

merfeld (1908) [5]. Sommerfeld, like many other mathematicians and physicists, was

not aware of Orr’s work. Orr’s contribution was ignored for some years even in Great

Britain. It should be noted that the OS equation is used for local LSA of parallel shear

flows where cross-stream velocity component of disturbances are not considered. How-

ever, Juniper (2007) [6] showed that the impulse response can grow upstream in some

directions with a cross-stream component in case of unconfined and confined inviscid

jet/wake flows.

In past, efforts have been made to understand the transition from laminar to tur-

bulent flow through transient growth analysis. The transient growth occurs due to

nonnormality of the OS operator [7]. Reddy and Henningson (1993) [8] showed that the

maximum transient growth behaves like O(Re2) in viscous channel flows.

Uc

−Uw

wU

x

y

x

y HH

(a) (b)

Figure 1.1: Schematic of (a) PCF; (b) PPF.


1.2.1 PCF

Plane Couette flow is named in honour of M. Couette who introduced it in 1890 to

measure viscosity. It is fluid flow between two infinite parallel plates/walls moving

relative to each other. The exact analytical solution of the Navier-Stokes equation for the

steady, fully developed flow gives a linear velocity profile. Owing to its simple base flow

profile, the stability of PCF has been studied extensively to understand the phenomenon

of transition to turbulence. A schematic of PCF is shown in Fig. 1.1(a).

1.2.1.1 Theoretical and numerical investigations of LSA of PCF

J. L. Synge (1938) [9] was amongst the first ones to investigate the linear stability of PCF

mathematically. He showed that below small value of Re, infinitesimal disturbances are

stable. Wasow (1953) [10] using the OS analysis of wavelike disturbances in streamwise

direction showed that PCF is stable for any given wave number, if the multiplication

of wave number and Re is sufficiently large. Lin (1955, p. 11) [11] concluded that all

existing investigations tend to show that the flow is stable but caution is advisable here

because not allRe and wave numbers have yet been considered fully. Deardorff (1963) [12]

used a trial-and-error numerical method for solving the OS equation and concluded that

the flow is definitely stable up to Re equal to 1430. This result further strengthened

the belief that PCF is stable to infinitesimal disturbances at all finite Re. Romanov

(1973) [13] using mathematical analysis of the OS equation conclusively showed that

PCF is linearly stable for all Re. Davis and Morris (1983) [14] used a Chebyshev/QR

numerical technique and confidently concluded that the flow is stable up to Re as large as

108. Herron (1991) [15] also showed that PCF is linearly stable for all Re. S. S. Vemuri

(2011) [16] found an unstable mode in PCF for the very first time using the method

developed by Mittal and Kumar (2007) [17]1 for global LSA of nonparallel flows. M. H.

Khan (2012) [22] and R. Kumar (2013) [23] used the same method and found the value

of Recro for PCF to be 372.5 in case of nonperiodic boundary conditions. R. Kumar also

found the value of Recro for PCF to be 361.3 in case of periodic boundary conditions

in streamwise and spanwise directions. S. S. Vemuri, M. H. Khan and R. Kumar also

performed DNS to crosscheck the LSA results.

1The method has been used in the past for global LSA of many fluid flows [18, 19, 20, 21].


1.2.1.2 Experimental investigations of LSA of PCF

Experimentally, PCF has been observed to be turbulent by Reichardt (1959) [24] at Re as

low as 375. Robertson (1959) [25] did hot-wire measurements and showed the flow to be

turbulent at a sufficiently large Re. Tillmark and Alfredsson (1992) [26] used an infinite-

belt type water channel with counter-moving walls. This was done for avoiding the less

satisfactory results that would otherwise be obtained as end disturbances propagating

into the channel inevitably contribute to transition. In order to visualize the flow pattern

in the streamwise-spanwise plane, they chose transparent belt and channel walls. For

finding the Re for transition, they introduced a large disturbance in the centre of the

channel, and investigated whether it developed into a turbulent region. The value of

Recrt was determined to be 360± 10.

1.2.2 PPF

Plane Poiseuille flow is named after the channel experiments by J. L. M. Poiseuille in

1840. It occurs when a fluid is forced between two stationary infinite parallel plates/walls

under constant pressure gradient. The exact analytical solution of the Navier-Stokes

equation for the steady, fully developed flow gives a parabolic velocity profile. PPF is

another parallel flow which has been studied extensively in terms of stability analysis to

understand the phenomenon of transition to turbulence. A schematic of PPF is shown

in Fig. 1.1(b).

1.2.2.1 Theoretical and numerical investigations of LSA of PPF

Heisenberg (1924) [27] was the first one to present theoretical results on the stability of

PPF. He calculated the part of the neutral stability curve using an asymptotic method.

Thomas (1953) [28] replaced the fourth order differential equation by a difference system

of the same order with a truncation error having eighth derivative. He solved resulting

linear algebraic equations by direct Gaussian elimination and reported the Recro to be

5780 for perturbation wavelength of 3.062 times the width of the channel. Lin (1955) [11]

proceeded further the asymptotic analysis and calculated the Recro to be approximately

5300 at nondimensional wave number of 1 (approximately). Iordanskii and Kulikovskii

(1965) [29] showed that no absolute instability exists in PPF for Re >> 1. Orszag

(1971) [30] solved the OS equation using expansions in Chebyshev polynomials and the

QR matrix eigenvalue algorithm. They also showed that results of great accuracy are

1.3 Incompressible flow equations 8

obtained very economically by this method. The Recro was found to be 5772.22.

1.2.2.2 Experimental investigations of LSA of PPF

Experimentally, PPF was found to be unstable at Re ≈ 1000 by Patel and Head

(1969) [31]. Experiment on transition to turbulence of PPF is very carefully done by

Nishioka, Iida and Ichikawa (1975) [32]. They took a long channel with large width-to-

height ratio. By reducing the background turbulence down to a level of 0.05%, they were

able to maintain the flow to be laminar at Re up to 8000. Similar kind of channel was

used by Kozlov and Ramazanov (1982) [33], but with a background turbulence level of

0.1%. This way they were able to keep the flow laminar at Re up to 7000. Klingmann

and Alfredsson (1990) [34] studied the development of turbulent spots in PPF at a Re of

1600. They found that the initial disturbance undergo a first stage of rapid expansion,

in which sharp internal shear layers form in the regions away from the symmetry plane

and precede the transition to turbulence.

1.2.3 List of Recro and Recrt for PCF and PPF

List of universally accepted Recro and Recrt [35] for PCF and PPF at present time is

shown in Table 1.1.

Table 1.1: List of universally accepted Recro and Recrt for PCF and PPF at presenttime.

Flow Recro (OS analysis) Recrt (experimental analysis)

PCF ∞ 360PPF 5772 1000

1.3 Incompressible flow equations

Let Ω ⊂ IRnsd and (0, T ) be the spatial and temporal domains, where nsd is the number

of space dimensions. Further suppose Γ denotes the boundary of Ω so that the set closure

Ω ≡ Ω ∪ Γ. The spatial and temporal coordinates are x ∈ Ω and t ∈ [0, T ] respectively.

1.4 Nonlinear disturbance equations 9

The Navier-Stokes equations governing incompressible fluid flow are

Momentum :∂u

∂t+ u ·∇u−∇ · σ = 0 on Ω× (0, T ), (1.3.1)

Continuity : ∇ · u = 0 on Ω× (0, T ), (1.3.2)

where u and σ are the velocity and stress tensor respectively. The stress tensor is equal

to the sum of its isotropic (−pI) and deviatoric (T) parts:

σ = −pI + T, T =2

Reε(u), ε(u) =

1

2((∇u) + (∇u)T ), (1.3.3)

where p and ε are the pressure and rate of strain tensor respectively. Eqs. (1.3.2) and

(1.3.3) give

∇ · σ = −∇p+1

Re∇2u. (1.3.4)

In general, both Dirichlet and Neumann type boundary conditions can be applied at

different segments of Γ:

u = g on Γg, (1.3.5)

n · σ = h on Γh, (1.3.6)

where Γg and Γh are complementary subsets of Γ.

1.4 Nonlinear disturbance equations

Let (U, P ) be the basic state solution to Eqs. (1.3.1) and (1.3.2), whose stability has

to be analysed. Further suppose (u′, p′) are the disturbance fields of the velocity and

pressure. Then the perturbed state solution (u, p) can be decomposed as:

(u, p) = (U, P ) + (u′, p′). (1.4.7)

Subtracting the equations for the basic state from the equations for the perturbed state,

one obtains following nonlinear disturbance equations:

∂u′

∂t+ u′ ·∇U + U ·∇u′ + u′ ·∇u′ −∇ · σ′ = 0 on Ω× (0, T ), (1.4.8)

∇ · u′ = 0 on Ω× (0, T ). (1.4.9)

1.5 Linearized disturbance equations 10

1.5 Linearized disturbance equations

For small u′ and p′, above set of equations can be linearized to obtain following linearized

disturbance equations:

∂u′

∂t+ u′ ·∇U + U ·∇u′ −∇ · σ′ = 0 on Ω× (0, T ), (1.5.10)

∇ · u′ = 0 on Ω× (0, T ). (1.5.11)

It should be noted that as the disturbance velocities grow above a few percent of

the base flow, nonlinear effects become important and the linear equations no longer

accurately predict the disturbance evolution [35].

1.6 Objectives

The objectives of present work are following:

To derive the CSOS equation for local LSA of parallel shear flows to include dis-

turbances moving with nonzero cross-stream velocity component.

To apply the CSOS equation to PCF and PPF for determination of respective

Recro.

To analyse the aspects of global LSA of parallel shear flows for periodic boundary

conditions in streamwise and spanwise directions.

To find out the relation between local LSA and global LSA of parallel shear flows.

To study the time evolution of the disturbances.

Chapter 2

Local LSA

2.1 Formulation of the CSOS equation

The base flow and disturbance field are:

(U, P ) = (

U(y)

V = 0

W = 0

, P (x)), (2.1.1)

(u′, p′) = (

u′(x, y, z, t))

v′(x, y, z, t))

w′(x, y, z, t))

, p′(x, y, z, t)), (2.1.2)

where x, y and z are streamwise, cross-stream and spanwise coordinate respectively.

Taking into account Eqs. (1.3.4), (2.1.1) and (2.1.2), the linearized momentum equa-

tion (1.5.10) can be written as:

∂u′

∂t+ U

∂u′

∂x+ v′

∂U

∂y+∂p′

∂x− 1

Re∇2u′ = 0 on Ω× (0, T ), (2.1.3)

∂v′

∂t+ U

∂v′

∂x+∂p′

∂y− 1

Re∇2v′ = 0 on Ω× (0, T ), (2.1.4)

∂w′

∂t+ U

∂w′

∂x+∂p′

∂z− 1

Re∇2w′ = 0 on Ω× (0, T ). (2.1.5)

2.1 Formulation of the CSOS equation 12

Similarly, the continuity equation (1.5.11) can be written as:

∂u′

∂x+∂v′

∂y+∂w′

∂z= 0 on Ω× (0, T ). (2.1.6)

Taking the divergence of the linearized momentum equations (2.1.3), (2.1.4), (2.1.5),

and using the continuity equation (2.1.6), one obtains following equation for the pertur-

bation pressure:

∇2p′ = −2∂U

∂y

∂v′

∂xon Ω× (0, T ). (2.1.7)

Using Eq. (2.1.7) with Eq. (2.1.4) to eliminate p′, one obtains following linearized dis-

turbance equation in v′:

[(∂

∂t+ U

∂

∂x)∇2 − ∂2U

∂y2

∂

∂x− 1

Re∇4]v′ = 0 on Ω× (0, T ). (2.1.8)

Before proceeding further, we will consider following point:

P2.1.1. Till now we have considered only the laboratory frame [x ≡ (x, y, z)].

However, our interest also lies in the moving frame [z ≡ (xm, ym, zm)]. The z-frame

is assumed to be moving with velocity Vc ≡(Vcx Vcy Vcz

)Twith respect to the

x-frame. The relations between the two frames are given by following equation [17]:

x = z + Vct, ∇x =∇z,∂

∂t

∣∣∣∣x

=∂

∂t

∣∣∣∣z

−Vc ·∇z. (2.1.9)

It should be noted that U(y) is computed in the x-frame. However, for all our

equations in the z-frame, it has to be interpreted as U(ym + Vcyt). Therefore in

the z-frame, the base flow varies with time if Vcy is nonzero. At t = 0, the base

flow is same in both x and z-frame. Hence, all the equations in the z-frame which

contain U or nonzero derivative of U , are valid for:t = 0 if Vcy 6= 0,

t ∈ [0, T ] if Vcy = 0.

Considering the limitation of the analysis, we define following notation for conve-


nience:

Ωm =

Ω× (t = 0+) if Vcy 6= 0,

Ω× (0, T ) if Vcy = 0.

Using Eq. (2.1.9), one can write Eqs. (2.1.7) and (2.1.8) in the z-frame as

∇2zp′ = −2

∂U

∂ym

∂v′

∂xmon Ωm (2.1.10)

and

[(∂

∂t

∣∣∣∣z

−Vc ·∇z + U∂

∂xm)∇2

z −∂2U

∂y2m

∂

∂xm− 1

Re∇4

z]v′ = 0 on Ωm (2.1.11)

respectively. We propose a disturbance field in the z-frame as following:

(

u′(xm, ym, zm, t)

v′(xm, ym, zm, t)

w′(xm, ym, zm, t)

, p′(xm, ym, zm, t)) = (

u(ym)

v(ym)

w(ym)

, p(ym))eiαxm+βzmeλt. (2.1.12)

Hence, the disturbance field in the x-frame is:

(

u′(x, y, z, t)

v′(x, y, z, t)

w′(x, y, z, t)

, p′(x, y, z, t))

= (

u(y − Vcyt)v(y − Vcyt)w(y − Vcyt)

, p(y − Vcyt))eiα(x−Vcxt)+β(z−Vczt)eλt

= (

u(y − Vcyt)v(y − Vcyt)w(y − Vcyt)

, p(y − Vcyt))eik·r−kVcxzteλt,

(2.1.13)

where k is the wavevector in the xz plane having magnitude |k| = k =√α2 + β2,

r = xx + zz and Vcxz =√V 2cx + V 2

cz. It should be noted that k and Vcxz are inclined at

the same angle with respect to x-axis in the xz plane. So, once three quantities among

α, β, Vcx and Vcz are given, the fourth quantity can easily be found. Substituting the

cross-stream velocity component of disturbance from Eq. (2.1.12) in Eq. (2.1.11) and

denoting the first derivative with respect to ym by the symbol Dm and subscript ym, we


get the CSOS equation:

[iαUymym − (αU − kVcxz)(D2m − k2)

+ Vcy(D3m − k2Dm) +

1

Re(D2

m − k2)2]v = λ[D2m − k2]v on Ωm. (2.1.14)

Since u′ = 0 in the free stream and at solid walls (no slip). Hence, from Eq. (2.1.12) or

Eq. (2.1.13), we see that each of u, v and w should be equal to zero in the free stream

and at solid walls. So, the proper boundary conditions on the CSOS equation are1:

v = vym = 0 in the free stream and at solid walls. (2.1.15)

It is important to note following points:

P2.1.2. Eq. (2.1.14) is an eigenvalue problem2 with λ = λr + iλi as the eigenvalue

and v = vr + ivi as the corresponding eigenfunction. “λr” gives growth rate while

“−λi” gives (circular) frequency in the moving frame.

P2.1.3. Let propagation velocity of the disturbance is Vt ≡(Vtx Vty Vtz

)T. It

is easy to check that the frequency in the laboratory frame is kVcxz − λi. Hence,

propagation speed of the disturbance in direction r is Vtxz = 1k(kVcxz−λi). Finally,

we have following relations:

Vtx = Vtxzα

k, Vty = Vcy, Vtz = Vtxz

β

k. (2.1.16)

P2.1.4. The solution (λ, v) to the eigenvalue problem (2.1.14) is in general func-

tion of (Re, α, Vcx, β, Vcz, Vcy). However, as the flow profile U(y) is invariant under

Galilean transformations with respect to Vcx and Vcz, the growth rate is indepen-

dent of Vcx and Vcz. The dependence of frequency on Vcx and Vcz is due to Doppler

effect. Hence, without loss of generality, one can take Vcx and Vcz to be zero

throughout.

P2.1.5. The growth rate at a given (Re, α, Vcx, β, Vcz, Vcy) is constant with respect

to time in the case of Vcy = 0. However, it varies with time in the case of Vcy 6= 0.3

1The condition vym = 0 coming from the continuity equation is introduced to make the problemcomplete.

2As the equation and its boundary conditions are homogeneous.3Mechanism for finding the growth rate at any given time is provided in Chapter 4.

2.2 Comparison of the CSOS analysis with the OS analysis 15

P2.1.6. At a given (Re, α, Vcx, β, Vcz, Vcy) many different values of λr are possible.

The condition on stability of the basic state is governed by the maximum value of

λr. The basic state will be unstable, stable or neutral according as the maximum

value of λr is positive, negative or zero.

Computation of u and w.— From Eqs. (2.1.4) and (2.1.5), one can obtain following:

(∂

∂t+ U

∂

∂x− 1

Re∇2)(

∂w′

∂y− ∂v′

∂z) +

∂U

∂y

∂w′

∂x= 0 on Ω× (0, T ). (2.1.17)

Further using Eqs. (2.1.9) and (2.1.12), Eq. (2.1.17) can be written as:

[λ− i(kVcxz + VcyDm − αU)− 1

Re(D2

m − k2)](wym − iβv) + iαUymw = 0

on Ωm. (2.1.18)

Again, from Eqs. (2.1.9), (2.1.12) and (2.1.13), the continuity equation (2.1.6) can be

written as:

iαu+ vym + iβw = 0 on Ω× (0, T ). (2.1.19)

Finally, by solving the CSOS equation, we can compute u and w from Eqs. (2.1.19) and

(2.1.18) respectively.

2.2 Comparison of the CSOS analysis with the OS

analysis

The disturbance field in the x-frame for the OS analysis is:

(

u′(x, y, z, t)

v′(x, y, z, t)

w′(x, y, z, t)

, p′(x, y, z, t)) = (

u(y)

v(y)

w(y)

, p(y))eik·r−ωt, (2.2.20)

where ω is the circular frequency. Comparison of Eq. (2.2.20) with Eq. (2.1.13) suggests

that the CSOS analysis degenerates to the OS analysis in case of Vcy = 0. For this case

one obtains:

v(y) = v(y), λr = ωi, λi = kVcxz − ωr. (2.2.21)

2.3 Squire’s theorem 16

2.3 Squire’s theorem

For β = 0, Eq. (2.1.14) reduces to:

[iUymym − (U − [kVcxz]2D

α2D

)(D2m − α2

2D)

+[Vcy]2D

α2D

(D3m − α2

2DDm) +1

α2DRe2D

(D2m − α2

2D)2]v =λ2D

α2D

[D2m − α2

2D]v on Ωm.

(2.3.22)

Above equation shows equivalence to Eq. (2.1.14) written in the form

[iUymym − (U − kVcxzα

)(D2m − k2)

+Vcyα

(D3m − k2Dm) +

1

αRe(D2

m − k2)2]v =λ

α[D2

m − k2]v on Ωm (2.3.23)

if to put

[kVcxz]2D

α2D

=kVcxzα

,

α2D = k,

[Vcy]2D

α2D

=Vcyα,

α2DRe2D = αRe,λ2D

α2D

=λ

α.

(2.3.24)

From Eq. (2.3.24) it follows that

Re2D = Reα

k< Re. (2.3.25)

Above analysis states that each 3D CSOS mode corresponds to a 2D CSOS mode at a

lower Re. This result justifies following:

Squire’s theorem. For a parallel shear flow, the Recro occurs for the case of a 2D

disturbance with β = 0.

2.4 Solution method 17

2.4 Solution method

In this section we present a spectral collocation method4 based on Chebyshev polynomials

of the first kind [35]. The method is highly accurate as well as easy to implement.

2.4.1 Chebyshev polynomials of the first kind

The Chebyshev polynomials can be defined in many ways. For numerical purposes, the

most practical definition is given in terms of trigonometric functions. The Chebyshev

polynomial of the first kind is defined as

Tn(y) = cos(n cos−1(y)) (2.4.26)

for all nonnegative integers n. Normally the range of the variable y, that is, the physical

domain is in the interval [−1, 1], or else the physical domain can be mapped into the

Chebyshev domain [−1, 1]. First few Chebyshev polynomials of the first kind are T0(y) =

1, T1(y) = y, T2(y) = 2y2 − 1, T3(y) = 4y3 − 3y. Following recurrence relation shows the

relation between Chebyshev polynomials of the first kind and their kth(k ≥ 1) derivatives:

T(k)0 (y) = 0,

T(k)1 (y) = T

(k−1)0 (y),

T(k)2 (y) = 4T

(k−1)1 (y),

T(k)n (y) = 2nT

(k−1)n−1 (y) +

n

n− 2T

(k)n−2(y) n = 3, 4, . . . .

(2.4.27)

The dependent variable is approximated by

f(y) =N∑n=0

anTn(y) (2.4.28)

and the discretization has been done by choosing Gauss-Lobatto collocation points:

yj = cos(jπ

N) j = 0, 1, 2, . . . , N. (2.4.29)

4In mathematics, a collocation method is a method for the numerical solution of ordinary differentialequations, partial differential equations and integral equations. The idea is to choose a finite-dimensionalspace of candidate solutions (usually, polynomials up to a certain degree) and a number of points in thedomain (called collocation points), and to select that solution which satisfies the given equation at thecollocation points.


At the Gauss-Lobatto collocation points, the Chebyshev polynomials of the first kind

have a simple form: Tn(yj) = cos(njπ/N).

2.4.2 Spurious eigenvalues

Following two types of spurious eigenvalues [36] can be present in the solution of hydro-

dynamic stability problems by spectral collocation method based on Chebyshev polyno-

mials:

1. Physically spurious eigenvalues. These are numerically computed eigenvalues

which are spurious because of misapplication of boundary conditions or some other

misrepresentation of the physics.

2. Numerically spurious eigenvalues. These are poor approximations to exact

eigenvalues because the mode is oscillating too rapidly to be resolved by taken

number of collocation points. Correct eigenvalues corresponding to numerically

spurious eigenvalues can be computed with great accuracy by choosing sufficiently

large number of collocation points.

Hence, utmost care should be taken for presentation of the results.

2.4.3 Chebyshev discretization of the CSOS equation

We will approximate the eigenfunction as

v(ym) =N∑n=0

anTn(ym). (2.4.30)

Hence, kth derivative of the eigenfunction is obtained by

Dkmv(ym) =

N∑n=0

anT(k)n (ym). (2.4.31)


Substituting Eqs. (2.4.30) and (2.4.31) into the CSOS equation (2.1.14), we get:

[iαUymymN∑n=0

anTn(ym)− (αU − kVcxz)(N∑n=0

anT(2)n (ym)− k2

N∑n=0

anTn(ym))

+ Vcy(N∑n=0

anT(3)n (ym)− k2

N∑n=0

anT(1)n (ym))

+1

Re(N∑n=0

anT(4)n (ym)− 2k2

N∑n=0

anT(2)n (ym) + k4

N∑n=0

anTn(ym))]

= λ[N∑n=0

anT(2)n (ym)− k2

N∑n=0

anTn(ym)] on Ωm. (2.4.32)

The boundary conditions (2.1.15) become

N∑n=0

anTn(1) =N∑n=0

anTn(−1) =N∑n=0

anT(1)n (1) =

N∑n=0

anT(1)n (−1) = 0. (2.4.33)

Eq. (2.4.32) in conjunction with Eq. (2.4.33) constitutes the following equation

Aa = λBa (2.4.34)

with the right-hand side equal to

λ

T0(1) T1(1) · · ·T

(1)0 (1) T

(1)1 (1) · · ·

T(2)0 (y2)− k2T0(y2) T

(2)1 (y2)− k2T1(y2) · · ·

......

. . .

T(2)0 (yN−2)− k2T0(yN−2) T

(2)1 (yN−2)− k2T1(yN−2) · · ·

T(1)0 (−1) T

(−1)1 (1) · · ·

T0(−1) T1(−1) · · ·

a0

a1

a2

...

aN−2

aN−1

aN

. (2.4.35)

The left-hand side of Eq. (2.4.34) can easily be written in a similar manner. The first,

second, 2nd last and last rows of B have been chosen to implement the four boundary

conditions. The same rows in A should be chosen as a complex multiple (mbc) of the

corresponding rows in B. A careful selection of mbc maps the spurious modes associated

with the boundary conditions to an arbitrary location in the complex plane.

2.5 2D LSA 20

Results are primarily obtained by using the open source package Chebfun that can

be included in MATLAB. Using this package, we got adaptive solution with machine

precision. However, this package was not able to provide solutions for many case. In

such cases, we used our own MATLAB code with unadaptive procedure. Before applying

the unadaptive procedure, a convergence study is done with respect to number of Gauss-

Lobatto collocation points (NP ≡ N + 1).

2.5 2D LSA

Due to Squire’s theorem, only 2D LSA (with β ≡ 0) is considered in the rest part of the

present chapter. In this case Eq. (2.1.14) simplifies to following:

[iαUymym − (U − Vcx)(D2m − α2)

+ Vcy(D3m − α2Dm) +

1

Re(D2

m − α2)2]v = λ[D2m − α2]v on Ωm. (2.5.36)

It is also important to note that in this case w′ has no significance and Eq. (2.1.19)

simplifies to following:

iαu+ vym = 0 on Ω× (0, T ). (2.5.37)

2.5.1 Construction of modes

The modes are constructed at t = 0 by running x from 0 to 2π/α and y from −H/2to H/2. In following paragraphs, a quantity with subscript M indicates that it is for

construction of mode.

v′−mode.— v′M = (vr + ivi)eiαx, where vr and vi are real (re) and imaginary (im)

part of v respectively. Expanding further, we getre(v′M) = vr cos(αx)− vi sin(αx),

im(v′M) = vr sin(αx) + vi cos(αx).(2.5.38)

2.5 2D LSA 21

u′−mode.— From Eq. (2.5.37) we see that u′M = (i/α)(Dmvr + iDmvi)eiαx. Ex-

panding further, we getre(u′M) = −1α

[Dmvr sin(αx) +Dmvi cos(αx)],

im(u′M) = 1α

[Dmvr cos(αx)−Dmvi sin(αx)].(2.5.39)

Another useful quantity is disturbance field of vorticity in spanwise direction given as

ζ ′z = ∂v′

∂x− ∂u′

∂y. Hence, ζ ′zM = ∂

∂x(veiαx) − ∂

∂y(ueiαx) = ∂

∂x(veiαx) − ∂

∂y( iαvyme

iαx) =

(i/α)(α2v −D2mv)eiαx. Expanding further, we getre(ζ ′zM) = −1

α[α2(vr sin(αx) + vi cos(αx))− (D2

mvr sin(αx) +D2mvi cos(αx))],

im(ζ ′zM) = 1α

[α2(vr cos(αx)− vi sin(αx))− (D2mvr cos(αx)−D2

mvi sin(αx))].

In the remaining part of the present chapter, results have been presented

with respect to Vcx = 0.

(a) (b)

Figure 2.1: v′− modes at Re = 500, α = 2 for (a) PCF; (b) PPF. The upper part is forVcy = 0.05 while the lower one is for Vcy = −0.05. In both cases, the left part shows thereal modes while the right one shows the imaginary modes.

2.6 Application of the CSOS equation to PCF 22

2.6 Application of the CSOS equation to PCF

2.6.1 Convergence study

Table 2.1 shows the convergence study with respect to NP for PCF at Re = 15000, α =

π, Vcy = 0.08 while Table 2.2 shows the convergence study with respect to NP for PCF

at Re = 20000, α = 1.38, Vcy = 0.2291. Observing the situation, we chose NP = 300

for all our computations upto Re = 15000. Similarly, NP = 600 is chosen for all the

computations between Re = 15000 and Re = 20000.

Table 2.1: Convergence study with respect to NP for PCF at Re = 15000, α = π, Vcy =0.08.

NP λr λi100 5.77192602E − 02 −1.46363408E + 00200 5.76191637E − 02 −1.46370572E + 00300 5.76191633E − 02 −1.46370572E + 00400 5.76191633E − 02 −1.46370572E + 00

Table 2.2: Convergence study with respect to NP for PCF at Re = 20000, α = 1.38, Vcy =0.2291.

NP λr λi200 9.42024749E − 03 4.97684022E − 01400 −8.68403329E − 05 5.00703404E − 01600 −8.68557277E − 05 5.00703409E − 01800 −8.68562027E − 05 5.00703408E − 01

2.6.2 General properties

Following properties are observed (which can also be extended to 3D analysis):

PCF Prop2.6.1. From the schematic of PCF (shown in Fig. 1.1(a)) we see that

the base velocity profile is anti-symmetric about center-line, that is, it shows the

origin symmetry. Hence, eigenvalues at a given (Re, α,±Vcy) will be λr ± iλi.

This implies that both (Vtx, Vty) and (−Vtx,−Vty) will be associated with the same


growth rate at a given (Re, α, Vcy). Fig. 2.1(a) shows the complete set of v′− modes

for a test case which is clearly justifying the origin symmetry.

PCF Prop2.6.2. Eigenvalues at a given (Re,±α, Vcy) will be λr ± iλi. This

implies that both α and −α will be associated with the same growth rate and

propagation speed of disturbance at a given (Re, α, Vcy).

PCF Prop2.6.3. The complex eigenvalues at a given (Re, α, Vcy = 0) will appear

as conjugate pairs. This implies that both Vtx and −Vtx will be associated with

the same growth rate at a given (Re, α, Vcy = 0).

PCF Prop2.6.4. The complex eigenvalues at a given (Re, α = 0, Vcy) will appear

as conjugate pairs. In particular, if Vcy = 0, then we get purely real eigenvalues.

From above properties, we can see that negative α and negative Vcy can be omitted for

further computations.

2.6.3 General results

Fig. 2.2 shows the neutral surface. It can be seen that the neutral surface does not touch

Vcy = 0 plane. This shows the consistency of the result with the OS analysis.

10

4.0 3.50.0

Vcy

α

0.00

0.10

0.05

0.15

0.25

0.51.01.52.02.53.0

0.20

86

42

0

Rex10−3

Figure 2.2: Neutral surface for PCF.


Fig. 2.3 should be considered as the regeneration of results corresponding to the

OS analysis for PCF [35]. Fig. 2.3(a) shows the contours of constant λi at Vcy = 0.

Similarly, Fig. 2.4(a) shows the contours of constant λi at Vcy = 0.11. Recalling PCF

Prop2.6.3 and PCF Prop2.6.4, only negative λi was chosen for creating the contours of

constant λi in case of conjugate pairs of complex eigenvalues. Fig. 2.4(b) clearly shows

the existence of unsatble modes in PCF for Vcy = 0.11.

Reα

−0.

50−0

.25

−0.2

0

−0.15

−0.10

−0.05

0 2000 4000 6000 8000 100000

1

2

3

4

Re

α

−3.0−2.5

−2.0−1.5

−1.0−0.5−0.05

0 2000 4000 6000 8000 100000

1

2

3

4

(b)(a)

Figure 2.3: PCF at Vcy = 0. Contours of (a) constant λi; (b) constant growth rate.

Re

α

−0.0

8

−0.040

0.04

0.060.08

0 2000 4000 6000 8000 100000

1

2

3

4

Re

α

−1.50−1.00

−0.50

00.25

0 2000 4000 6000 8000 100000

1

2

3

4

(b)(a)

Figure 2.4: PCF at Vcy = 0.11. Contours of (a) constant λi; (b) constant growth rate.

Table 2.3 shows the list of Recr with respect to Vcy. Corresponding αcr and λicr

have also been shown in the table. Following steps were applied for creating the table:

1. Vcy of interest was chosen.

2. α was varied in step of 0.01 and Re in step of 1.

3. We chose the two minimum Reynolds numbers, Re1 and Re2 for which the growth

rates have opposite sign.


4. Re1 and Re2 were interpolated to get Recr, which was corresponding to zero growth

rate.

Table 2.3: Critical parameters for PCF.

Vcy Recr αcr λicr0 ∞ − −

0.0012 19458.08 0.02 −2.159E − 030.0025 9152.45 0.03 1.021E − 030.0050 4582.34 0.06 2.033E − 030.0075 3061.70 0.09 3.029E − 03

0.01 2303.43 0.12 4.000E − 030.02 1174.22 0.22 1.585E − 020.03 806.69 0.32 2.684E − 020.04 629.07 0.41 4.089E − 020.05 527.19 0.49 5.787E − 020.06 463.26 0.56 7.773E − 020.07 421.24 0.63 9.671E − 020.08 393.29 0.70 1.150E − 010.09 375.19 0.76 1.363E − 010.10 364.61 0.82 1.571E − 010.11 360.63 0.85 1.886E − 010.12 361.58 0.92 2.051E − 010.13 368.55 0.97 2.287E − 010.14 381.71 1.02 2.521E − 010.15 402.30 1.07 2.756E − 010.16 432.55 1.11 3.028E − 010.17 476.49 1.15 3.302E − 010.18 541.52 1.19 3.577E − 010.19 642.68 1.23 3.855E − 010.20 815.18 1.27 4.137E − 010.21 1164.75 1.31 4.424E − 010.22 2214.57 1.34 4.750E − 01

0.2225 2909.97 1.35 4.824E − 010.2250 4287.44 1.36 4.898E − 010.2275 8304.12 1.36 5.007E − 010.2290 19386.76 1.37 5.038E − 01


0

4000

8000

12000

16000

20000

0.00 0.04 0.08 0.12 0.16 0.20 0.24

Re

cr

Vcy

360365370375380

0.09 0.10 0.11 0.12 0.13R

ecr

Vcy

crα =1.31

crα =0.85

α =1.37cr

Figure 2.5: Variation of Recr with Vcy for PCF. v′−modes of a few cases are also shown.

Fig. 2.5 shows the nonmonotonic variation of Recr with respect to Vcy, which sug-

gests Recro to be 360.63 at Vcy = 0.11 and α = 0.85. The corresponding λi = 1.886E−01

gives Vtx = 2.219E − 01. Fig. 2.6 shows the variation of αcr with respect to Vcy. It is

easy to observe that as we increase Vcy, value of αcr increases. Hence, the variation of αcr

with respect to Vcy is monotonic. It should be interesting to note the dip at Vcy = 0.11

where Recro exists. Fig. 2.7 shows the variation of λicr with respect to Vcy. Again it

is easy to check that as we increase Vcy, value of λicr increases. Hence, the variation

of λicr with respect to Vcy is monotonic. It should be interesting to note the bump at

Vcy = 0.11 where Recro exists. Overall, it can be said that variation of Recr, αcr and λicr

with respect to Vcy is nonlinear.

It should be noted that at the critical Vcy for the onset of instability, the plots of

Recr, αcr and λicr with respect to Vcy need not change their behaviour. Results (given

in next section) related to PPF justify the same.


0.00

0.25

0.50

0.75

1.00

1.25

1.50

0.00 0.04 0.08 0.12 0.16 0.20 0.24

α cr

Vcy

Figure 2.6: Variation of αcr with Vcy for PCF.

-0.10

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.00 0.04 0.08 0.12 0.16 0.20 0.24

λ icr

Vcy

Figure 2.7: Variation of λicr with Vcy for PCF.

2.7 Application of the CSOS equation to PPF 28

2.7 Application of the CSOS equation to PPF

2.7.1 Convergence study

Table 2.4 shows the convergence study with respect to NP for PPF at Re = 15000, α =

π, Vcy = 0.07 while Table 2.5 shows the convergence study with respect to NP for PPF

at Re = 20000, α = 3.80, Vcy = 0.1348. Observing the situation, we chose NP = 300

for all our computations upto Re = 15000. Similarly, NP = 600 is chosen for all the

computations between Re = 15000 and Re = 20000.

Table 2.4: Convergence study with respect to NP for PPF at Re = 15000, α = π, Vcy =0.07.

NP λr λi100 1.20517104E − 01 −1.95255455E + 00200 1.20481126E − 01 −1.95271135E + 00300 1.20481126E − 01 −1.95271135E + 00400 1.20481126E − 01 −1.95271135E + 00

Table 2.5: Convergence study with respect to NP for PPF at Re = 20000, α = 3.80, Vcy =0.1348.

NP λr λi200 −1.78015222E − 04 −2.90418614E + 00400 −1.84370827E − 04 −2.90418387E + 00600 −1.84370848E − 04 −2.90418387E + 00800 −1.84370656E − 04 −2.90418387E + 00

2.7.2 General properties

Following properties are observed (which can also be extended to 3D analysis):

PPF Prop2.7.1. From the schematic of PPF (shown in Fig. 1.1(b)) we see that

the base velocity profile is symmetric about center-line, that is, it shows the x-axis

symmetry. Hence, eigenvalues at a given (Re, α,±Vcy) will be same. This implies

that both (Vtx, Vty) and (Vtx,−Vty) will be associated with the same growth rate


at a given (Re, α, Vcy). Fig. 2.1(b) shows the complete set of v′− modes for a test

case which is clearly justifying the x-axis symmetry.

PPF Prop2.7.2. Eigenvalues at a given (Re,±α, Vcy) will be λr ± iλi. This

implies that both α and −α will be associated with the same growth rate and

propagation speed of disturbance at a given (Re, α, Vcy).

PPF Prop2.7.3. Unlike PCF, the complex eigenvalues at a given (Re, α, Vcy = 0)

will not appear as conjugate pairs in general. This implies that if both Vtx and

−Vtx exit at a given (Re, α, Vcy = 0), then they will not be associated with the

same growth rate in general.

PPF Prop2.7.4. The complex eigenvalues at a given (Re, α = 0, Vcy) will appear

as conjugate pairs. In particular, if Vcy = 0, then we get purely real eigenvalues.

From above properties, we can see that negative α and negative Vcy can be omitted for

further computations.

2.7.3 General results

Fig. 2.8 shows the neutral surface. Three modes are identified here. Mode C contains

the neutral curve obtained from the OS analysis.

Vcy

Mode A

α

4.0 3.50.00

0.03

0.09

0.12

0.15

0.06

3.0 2.5 2.0 1.5 1.0 0.5 0.0 02 4

68

10

Rex10−3

Mode C

Mode B

Figure 2.8: Neutral surface for PPF.


Fig. 2.9 should be considered as the regeneration of results corresponding to the

OS analysis for PPF [35]. Fig. 2.9(a) shows the contours of constant λi at Vcy = 0.

Here the thick line shown in blue colour represents a discontinuity in the λi contours.

Fig. 2.9(b) shows the contours of constant growth rate at Vcy = 0. It can be seen that

the contour of zero growth rate in this case belongs to Mode C. Fig. 2.10(a) suggests

that no discontinuity in the λi contours occurs in case of Vcy = 0.08. Fig. 2.10(b) shows

the contours of constant growth rate at Vcy = 0.08. Contours of zero growth rate in this

case belong to Mode A and Mode B.

Re

α

−0.

20−0

.15

−0.1

0

−0.07

−0.05

−0.02 0

0 2000 4000 6000 8000 100000

1

2

3

4

Re

α

−3.5−3.0

−2.5−2.0

−0.6−0.4

−0.2−0.025

0 2000 4000 6000 8000 100000

1

2

3

4

(b)(a)

Figure 2.9: PPF at Vcy = 0. Contours of (a) constant λi; (b) constant growth rate.

Re

α

−0.0

4

−0.02

0.03 0.06

0.09

00

0 2000 4000 6000 8000 100000

1

2

3

4

Re

α

−2.3−2.0

−1.7−1.1

−0.7−0.3

0.10 2000 4000 6000 8000 10000

0

1

2

3

4

(b)(a)

Figure 2.10: PPF at Vcy = 0.08. Contours of (a) constant λi; (b) constant growth rate.

Table 2.6 shows the list of Recr with respect to Vcy. Corresponding αcr and λicr

have also been shown in the table. Steps applied for creating this table are same as the

steps, which were applied for creating Table 2.3.


Table 2.6: Critical parameters for PPF.

Vcy Recr αcr λicr0 5772.26 1.02 −2.692E − 01

0.00110 8466.33 0.97 −2.356E − 010.00129 15310.11 1.03 −2.415E − 010.00130 15324.53 1.03 −2.416E − 010.00131 15336.80 1.03 −2.418E − 010.00132 15331.57 1.04 −2.455E − 010.00133 15323.66 1.04 −2.457E − 010.00250 10512.89 1.18 −3.269E − 010.00500 5830.69 1.34 −4.485E − 010.00750 4083.09 1.44 −5.392E − 01

0.01 3187.82 1.52 −6.163E − 010.02 1841.60 1.74 −8.526E − 010.03 1411.23 1.91 −1.043E + 000.04 1221.98 2.08 −1.222E + 000.05 1096.19 2.16 −1.354E + 00

0.0525 1076.81 2.23 −1.410E + 000.0550 1067.73 2.30 −1.465E + 000.0575 1067.45 2.37 −1.521E + 000.0600 1074.78 2.44 −1.576E + 000.0625 1088.88 2.51 −1.631E + 000.0635 1096.25 2.54 −1.654E + 000.0637 1097.85 2.55 −1.661E + 000.0639 1096.37 2.05 −1.417E + 000.0641 1090.46 2.05 −1.418E + 000.0650 1066.48 2.08 −1.441E + 000.0675 1011.12 2.15 −1.496E + 00

0.07 970.53 2.22 −1.552E + 000.08 916.85 2.50 −1.780E + 000.09 977.87 2.73 −1.974E + 000.10 1150.33 2.96 −2.170E + 000.11 1494.93 3.20 −2.376E + 000.12 2283.32 3.43 −2.579E + 00

0.1225 2658.57 3.49 −2.632E + 000.1250 3197.18 3.55 −2.685E + 000.1275 4036.06 3.62 −2.744E + 000.1300 5520.81 3.68 −2.798E + 000.1325 8860.57 3.74 −2.852E + 000.1347 19521.23 3.79 −2.898E + 00

2.8 Group velocity 32

α =2.50cr

α =3.43crα =1.02cr

α =3.79cr

0

4000

8000

12000

16000

20000

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

Re

cr

Vcy

900

1050

1200

0.07 0.08 0.09 0.10R

ecr

Vcy

Figure 2.11: Variation of Recr with Vcy for PPF. v′−modes of a few cases are alsoshown.

Fig. 2.11 shows the nonmonotonic variation of Recr with respect to Vcy, which

suggests Recro to be 916.85 at Vcy = 0.08 and α = 2.50. The corresponding λi =

−1.780E + 00 gives Vtx = 7.120E − 01. Fig. 2.12 shows the nonmonotonic variation

of αcr with respect to Vcy. It is interesting to note the discontinuity at Vcy = 0.0639.

Fig. 2.13 shows the nonmonotonic variation of λicr with respect to Vcy. Again, it should

be interesting to note the discontinuity at Vcy = 0.0639.

2.8 Group velocity

In our case, the streamwise wave number (α) is real while the complex (circular) fre-

quency is ω = iλ ≡ −λi + iλr. Hence, the group velocity in streamwise direction is

Vgx = ∂ω∂α

while the group velocity in cross-stream direction is Vgy = Vcy. It can further

be checked that near the onset of instability of PCF, Vgx = ±0.4 with Vgy = ±0.11.

Similarly, near the onset of instability of PPF, Vgx = 0.7 with Vgy = ±0.08.

2.8 Group velocity 33

0.00

1.00

2.00

3.00

4.00

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

α cr

Vcy

Figure 2.12: Variation of αcr with Vcy for PPF.

-3.00

-2.60

-2.20

-1.80

-1.40

-1.00

-0.60

-0.20

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

λ icr

Vcy

Figure 2.13: Variation of λicr with Vcy for PPF.

Chapter 3

Global LSA

Prof. Mittal advised me to apply the method of global analysis of convective instabilities

to parallel flows. The idea central to this application, including the form of perturbation,

was formulated by him. The idea of doing a global analysis with periodic boundary

conditions is also his.

This chapter concerns the global LSA of parallel shear flows for periodic boundary

conditions in streamwise and spanwise directions. The present chapter also compares

results of global LSA with local LSA. To study this chapter, one might have a look at

Eqs. (2.1.1), (2.1.2) and Point P2.1.1. Most of the contents in Sections 3.1, 3.2 and

3.3 are due to Kumar (2009) [2].

3.1 Global linear stability equations

Here we propose a disturbance field in the z-frame as following:

(u′(z, t), p′(z, t)) = (u(z), p(z))eλt. (3.1.1)

Hence, the disturbance field in the x-frame is:

(u′(x, t), p′(x, t)) = (u(x−Vct), p(x−Vct))eλt. (3.1.2)

3.2 The finite element formulation 35

Substituting Eq. (3.1.2) in Eqs. (1.5.10) and (1.5.11) and using Eq. (2.1.9) we have

following eigenvalue problem for global LSA in the z-frame:

λu + u ·∇zU + (U−Vc) ·∇zu−∇z · σ = 0 on Ωm, (3.1.3)

∇z · u = 0 on Ω× (0, T ). (3.1.4)

3.2 The finite element formulation

To solve the relevant set of equations we use a hybrid mix of stabilized finite element

formulation developed in the past by many researchers (Hughes and Brooks (1979) [37],

Brooks and Hughes (1982) [38], Hughes et al. (1986) [39], Tezduyar et al. (1992) [40]).

We consider a finite element discretization of Ω into subdomains Ωe, e = 1, 2, ..., nel,

where nel is the number of elements and define:

H1h = φh|φh ∈ C0(Ω), φh|Ωe ∈ P 1, e = 1, 2, ..., nel,

Shuuu = uh|uh ∈ [H1h(Ω)]2,uh = gh onΓg,

Vhuuu = uh|uh ∈ [H1h(Ω)]2,uh = 0 onΓg,

Shp = Vhp = q|q ∈ H1h(Ω),

where P 1 denotes the first order polynomials.

3.2.1 The incompressible flow equations

The application of the stabilized finite element method to the Navier-Stokes equa-

tions (1.3.1) and (1.3.2) results in the following problem: find uh ∈ Shuuu and ph ∈ Shpsuch that ∀wh ∈ Vhuuu , qh ∈ Vhp∫

Ω

wh ·(∂uh

∂t+ uh ·∇uh

)dΩ +

∫Ω

ε(wh) : σ(ph,uh)dΩ +

∫Ω

qh∇ · uhdΩ

+

nel∑e=1

∫Ωe

(τSUPGuh ·∇wh + τPSPG∇qh

).

[∂uh

∂t+ uh ·∇uh −∇ · σ(ph,uh)

]dΩe

+

nel∑e=1

∫Ωe

τLSIC∇ ·wh∇ · uhdΩe =

∫Γh

wh · hhdΓ.

(3.2.5)


3.2.2 The nonlinear disturbance equations

The application of the stabilized finite element method to the nonlinear disturbance

equations (1.4.8) and (1.4.9) results in the following problem: find u′h ∈ Vhuuu and p′h ∈ Vhpsuch that ∀wh ∈ Vhuuu and qh ∈ Vhp∫

Ω

wh ·(∂u′h

∂t+ u′h ·∇Uh + Uh ·∇u′h + u′h ·∇u′h

)dΩ +

∫Ω

ε(wh) : σ(p′h,u′h)dΩ

+

∫Ω

qh∇ · u′hdΩ +

nel∑e=1

∫Ωe

(τSUPGUh ·∇wh + τPSPG∇qh

).[

∂u′h

∂t+ u′h ·∇Uh + Uh ·∇u′h + u′h ·∇u′h −∇ · σ(p′h,u′h)

]dΩe

+

nel∑e=1

∫Ωe

τLSIC∇ ·wh∇ · u′hdΩe = 0.

(3.2.6)

3.2.3 The linearized disturbance equations

The application of the stabilized finite element method to the linearized disturbance

equations (1.5.10) and (1.5.11) results in the following problem: find u′h ∈ Vhuuu and

p′h ∈ Vhp such that ∀wh ∈ Vhuuu and qh ∈ Vhp

∫Ω

wh ·(∂u′h

∂t+ u′h ·∇Uh + Uh ·∇u′h

)dΩ +

∫Ω

ε(wh) : σ(p′h,u′h)dΩ

+

∫Ω

qh∇ · u′hdΩ +

nel∑e=1

∫Ωe

(τSUPGUh ·∇wh + τPSPG∇qh

).[

∂u′h

∂t+ u′h ·∇Uh + Uh ·∇u′h −∇ · σ(p′h,u′h)

]dΩe

+

nel∑e=1

∫Ωe

τLSIC∇ ·wh∇ · u′hdΩe = 0.

(3.2.7)

3.2.4 The global linear stability equations

The application of the stabilized finite element method to the global linear stability

equations (3.1.3) and (3.1.4) results in the following problem: find uh ∈ Vhuuu and ph ∈ Vhp


such that ∀wh ∈ Vhuuu and qh ∈ Vhp∫Ω

wh ·(λuh + uh ·∇Uh + (U−Vc)

h ·∇uh)dΩ +

∫Ω

ε(wh) : σ(ph, uh)dΩ

+

∫Ω

qh∇ · uhdΩ +

nel∑e=1

∫Ωe

(τSUPG(U−Vc)

h ·∇wh + τPSPG∇qh).[

λuh + uh ·∇Uh + (U−Vc)h ·∇uh −∇ · σ(ph, uh)

]dΩe

+

nel∑e=1

∫Ωe

τLSIC∇ · wh∇ · uhdΩe = 0. (3.2.8)

3.2.5 Stabilization terms

The first three terms equated to right-hand side in each of the variational formulations

given by Eqs. (3.2.5), (3.2.6), (3.2.7) and (3.2.8) constitute the Galerkin formulation

of the corresponding problem, which can possess numerical instability due to following

two sources:

1. Presence of the advection terms with high cell Re (based on the local velocity and

element length1).

2. Choosing an inappropriate combination of interpolation functions for velocity and

pressure.2

Hence, to give stability to the basic formulations, we use Streamline-Upwind/Petrov-

Galerkin (SUPG), Pressure-Stabilizing/Petrov-Galerkin (PSPG) and Least-Squares on

Incompressibility Constraint (LSIC) stabilization techniques with following coefficients

(based on its values for advection and diffusion limits):

τSUPG = τPSPG =

(1

τ 2ADV

+1

τ 2DIF

)− 12

, τLSIC =

(1

δ2ADV

+1

δ2DIF

)− 12

, (3.2.9)

where

τADV =he

2‖uh‖, τDIF =

(he)2

12ν, δADV =

he‖uh‖2

, δDIF =(he)2(‖uh‖)2

12ν. (3.2.10)

1Various definitions exist for element length. In this work, we take minimum edge length of anelement to be the element length.

2In other words, not following the Babuska-Brezzi condition.


Here he is the element length and ν is the kinematic viscosity. The SUPG and LSIC

stabilizations render the formulations stable in the presence of advection terms with

high cell Re. The PSPG stabilization allows us to use any combination of interpolation

functions for velocity and pressure.

3.3 Solution method

3.3.1 Steady and unsteady flow

Eqs. (3.2.5), (3.2.6), (3.2.7) and (3.2.8) are discretized by semi-discrete approach.

Quadrilateral element with bilinear interpolation functions for both velocity and pressure

is taken for spatial discretization while Crank-Nicholson scheme, which is 2nd order

accurate, is used for marching in time. The Newton-Raphson technique is used to solve

the nonlinear equations. To obtain the steady-state solution, we drop time dependent

terms from the relevant equations and apply direct solution method. This way, we are

able to get very accurate solution needed for the eigenvalue analysis. The unsteady-state

solution is obtained by applying the Generalized Minimal RESidual (GMRES) technique

in conjunction with diagonal preconditioners. This method is very economical compared

to the direct solution method.

3.3.2 The eigenvalue problem

Eq. (3.2.8) leads to a generalized nonsymmetric eigenvalue problem in the matrix form3

AGX− λBGX = 0, (3.3.11)

with λ = λr± i|λi| as the eigenvalues.4 “λr” gives growth rate while “−λit” gives (circu-

lar) frequency in the moving frame, where λit can be one among ±|λi| or both depending

upon the case. To get correct value(s) of λit (and hence corresponding eigenfunction(s)),

following concept can be used.

Let L and Ls be the respective length of domain in streamwise and spanwise direc-

tions. Due to application of periodic boundary conditions in the respective directions,

3The subscript G stands for matrices corresponding to global LSA.4Since AG and BG are real, the complex eigenvalues will appear as conjugate pairs. The corre-

sponding eigenfunctions will also be complex conjugate pairs (X and X).


we have:

α = 2π/L, β = 2π/Ls. (3.3.12)

Hence, Eqs. (3.1.1) and (3.1.2) can respectively be considered as equivalent to

(u′(z, t), p′(z, t)) = (

uH(ym)

vH(ym)

wH(ym)

, pH(ym))eiαxm+βzmeλt (3.3.13)

and

(u′(x, t), p′(x, t)) = (

uH(y − Vcyt)vH(y − Vcyt)wH(y − Vcyt)

, pH(y − Vcyt))eiα(x−Vcxt)+β(z−Vczt)eλt

= (

uH(y − Vcyt)vH(y − Vcyt)wH(y − Vcyt)

, pH(y − Vcyt))eik·r−kVcxzteλt. (3.3.14)

Here uH , vH , wH and pH are purely real, and meanings of k, r, k and Vcxz are same as

defined in Chapter 2. So, propagation speed of the disturbance in direction r is Vtxz =1k(kVcxz − λit). Now we calculate the propagation speed of the disturbance in direction

r from DNS5 and see which among 1k(kVcxz ± |λi|) is/are matching with the calculated

speed. Subsequently we can find the true value(s) of λit. Now it can easily be seen that

Points P2.1.3, P2.1.4 [after replacing the segment “The solution (λ, v) to the eigenvalue

problem (2.1.14)” with “The solution (λ,X) to the eigenvalue problem (3.3.11)”], P2.1.5

and P2.1.6 are completely true in the present case as well.

The present situation is complicated due to the fact that the continuity equation,

which is responsible for determining the pressure, causes the matrix BG to become

singular. Hence, to overcome this situation, the problem is transformed to

BGX − µAGX = 0, (3.3.15)

where µ = 1/λ. Even though λ = 1/µ, the λ-spectrum need not be reciprocal µ-

spectrum. It can be shown that just beyond the onset of instability, where only one of

the conjugate pair of eigenvalues have positive real parts, the rightmost of λ-spectrum

5In case of Vcy = 0, the propagation speed can also be calculated by multiplying the eigenfunctionsby ei−kVcxzteλt and then taking images at different time instants.


is reciprocal to rightmost of µ-spectrum (see Fig. 3.1).

λ−plane µ−plane

µ−planeλ−plane

. ..

. ..

..

. ..

. ..

. ...

.

.. .. ......

.

.

...

..

.. .. ......

.

.

... .

.

. ..

. ...

. ..

. .. ..

.

.

Figure 3.1: A schematic of the scatter of eigenvalues before (top) and after (bottom)the onset of instability: λ-plane and µ-plane corresponds to the eigenvalues and theirinverse. A circle of unit radius is also shown in the figure. [2]

Hence, the desired eigenvalue can be found by a shift of the origin to a proper loca-

tion. An appropriate value of shift of the origin is found by experience. The eigenvalue

problem is solved by subspace iteration method (Stewart (1975) [41]) in conjunction with

the shift-invert transformation.

3.4 Results and discussion 41

wU (2y/H )U=V=0

H

x

y

wU (2y/H )U=V=0

wUU=− ,V =0

,V =0U=

L

wU(a)

H

L

V=0U=Uc(1−(2y/H )2) x

y

V=0U=Uc(1−(2y/H )2)

U=0,V =0

U=0,V =0

(b)

Figure 3.2: Domain description and boundary conditions used for calculation of basicstate in case of (a) PCF; (b) PPF.

3.4 Results and discussion

Due to Squire’s theorem, only 2D LSA (with β ≡ 0) is considered in the rest part of the

present chapter. Also, in the remaining part of the present chapter, results have been

presented with respect to Vcx = 0. It can be noted that properties PCF Prop2.6.1 to

PCF Prop2.6.4 and PPF Prop2.7.1 to PPF Prop2.7.4 are completely true in the present

case as well.

3.4.1 Domain and boundary conditions

Fig. 3.2 shows schematic of domain along with boundary conditions used for calculation

of basic state. The boundary conditions used for solving the disturbance equations and

LSA are shown in Fig. 3.3.


u = 0/ , v = 0/

u = 0/ , v = 0/

L

x

y

Periodic boundaries

H

Figure 3.3: Boundary conditions used for solving the disturbance equations and LSA.

Figure 3.4: A finite element mesh for L = H with Nelx = 25 and Nely = 300.


3.4.2 Mesh convergence study

3.4.2.1 PCF

Table 3.1 shows the mesh convergence study with respect to Nelx (number of elements

in streamwise direction) and Nely (number of elements in cross-stream direction) at

Re = 15000, L = H,Vcy = 0.08. The upper and lower part of the table show the

respective effect of increment of Nelx and Nely on λ. The RelErλr on the upper part

of the table shows the relative error in λr with respect to λr corresponding to M5.

Similarly, the RelErλr on the lower part of the table shows the relative error in λr with

respect to λr corresponding to N4. It can be seen that every mesh gives reasonably good

solution. M4, which is equivalent to N3, has been considered as the mesh corresponding

to converged solution. This mesh is shown in Fig. 3.4. The mesh is kept dense near the

wall to capture the modes. For all our our further calculations, we chose Nelx = 25⌈LH

⌉and Nely = 300. Here d.e denotes the ceiling function6.

Table 3.1: Mesh convergence study for PCF at Re = 15000, L = H,Vcy = 0.08.

Mesh Nelx Nely λr λi RelErλrM1 10 300 5.632E − 02 −1.463E + 00 1.97%M2 15 300 5.701E − 02 −1.464E + 00 0.77%M3 20 300 5.727E − 02 −1.464E + 00 0.31%M4 25 300 5.738E − 02 −1.464E + 00 0.12%M5 30 300 5.745E − 02 −1.464E + 00 0.00%

N1 25 100 5.670E − 02 −1.465E + 00 1.25%N2 25 200 5.729E − 02 −1.464E + 00 0.22%N3 25 300 5.738E − 02 −1.464E + 00 0.07%N4 25 400 5.742E − 02 −1.464E + 00 0.00%

It can be noted that for the same given condition, that is, for Re = 15000, L = H

and Vcy = 0.08, local LSA gives λr = 5.762E − 02 and λi = −1.464E + 00. This value

of λ is quite close to the value of λ for the selected mesh. Fig. 3.5 shows the comparison

of v′− modes of the present condition for local and global LSA. The modes are quite

similar as expected.

6The ceiling function dxe gives the smallest integer ≥ x.


(a) (b)

Figure 3.5: PCF at Re = 15000, L = H and Vcy = 0.08. The figure shows the v′−modes corresponding to (a) local LSA; (b) global LSA.

3.4.2.2 PPF

Table 3.2 shows the mesh convergence study with respect to Nelx and Nely at Re =

15000, L = H,Vcy = 0.07. The structure of the table is similar to that of Table 3.1.

Like the case of PCF, here also every mesh gives reasonably good solution. M4, which is

equivalent to N3, has been considered as the mesh corresponding to converged solution.

Again the selected mesh (shown in Fig. 3.4) is kept dense near the wall to capture the

modes. For all our our further calculations, we chose Nelx = 25⌈LH

⌉and Nely = 300.

Table 3.2: Mesh convergence study for PPF at Re = 15000, L = H,Vcy = 0.07.

Mesh Nelx Nely λr λi RelErλrM1 10 300 1.200E − 01 −1.953E + 00 0.33%M2 15 300 1.202E − 01 −1.953E + 00 0.17%M3 20 300 1.203E − 01 −1.953E + 00 0.08%M4 25 300 1.204E − 01 −1.953E + 00 0.00%M5 30 300 1.204E − 01 −1.953E + 00 0.00%

N1 25 100 1.205E − 01 −1.953E + 00 0.08%N2 25 200 1.203E − 01 −1.953E + 00 0.08%N3 25 300 1.204E − 01 −1.953E + 00 0.00%N4 25 400 1.204E − 01 −1.953E + 00 0.00%

For the same given condition, that is, for Re = 15000, L = H and Vcy = 0.07, local

LSA gives λr = 1.205E − 01 and λi = −1.953E + 00. This value of λ is quite close to

the value of λ for the selected mesh. Fig. 3.6 shows the comparison of v′− modes of the

present condition for local and global LSA. The modes are quite similar as expected.


(a) (b)

Figure 3.6: PPF at Re = 15000, L = H and Vcy = 0.07. The figure shows the v′−modes corresponding to (a) local LSA; (b) global LSA.

3.4.3 Effect of length of domain

It is observed that in local LSA we get an αcr for a given Re and Vcy. Correspond-

ingly, we shall get a critical length, Lcr = 2π/αcr. In global LSA one extra property

is obtained. Here we can always find a length Ln = nL1, n = 1, 2, 3, . . . , which gives

same λ as L1 gives. It is also interesting to note that Ln will contain n spatial periods.

Correspondingly, we can obtain Lno as the optimal length of the domain with respect to

maximum growth rate containing n spatial periods. The growth rate becomes constant

as we increase the length to very large value. Examples related to above discussions are

presented for each PCF and PPF respectively in subsequent parts.

3.4.3.1 PCF

Fig. 3.7 shows the variation of λr with respect to α at Re = 1000, Vcy = 0.16. Com-

putations are done here from local LSA. Here we obtain αcr = 1.22. Fig. 3.8 shows

the corresponding variation of λr with respect to L. Lcr can be calculated to be 5.15.

Variation of λr with respect to Ln has been shown in Fig. 3.9. We have shown the v′−modes with respect to Lno in Fig. 3.10.

3.4.3.2 PPF

Fig. 3.11 shows the variation of λr (computed from local LSA) with respect to α at

Re = 3500, Vcy = 0.04. Here we obtain αcr = 2.27. Fig. 3.12 shows the corresponding

variation of λr with respect to L. Lcr can be calculated to be 2.77. Variation of λr with

respect to Ln has been shown in Fig. 3.13 while v′− modes with respect to Lno have

been shown in Fig. 3.14.


-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π

λ r

α

Figure 3.7: Variation of λr with respect to α for PCF at Re = 1000, Vcy = 0.16.

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

0 10 20 30 40 50 60 70 80 90 100

λ r

L

Figure 3.8: Variation of λr with respect to L for PCF at Re = 1000, Vcy = 0.16.


-0.18

-0.14

-0.10

-0.06

-0.02

0.02

0.06

0 3 6 9 12 15 18 21

λ r

L

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10 L11

Figure 3.9: Variation of λr with respect to Ln for PCF at Re = 1000, Vcy = 0.16.Expected values (computed from local LSA) are plotted with lines. Solid dots representcomputed values from global LSA.

=5.15L L 1o =

=L L 2o=10.30

=L L 3o=15.45

=L L 4o=20.60

Figure 3.10: v′− modes with respect to Lno for PCF at Re = 1000, Vcy = 0.16.


-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0 π/4 π/2 3π/4 π 5π/4 3π/2 7π/4 2π

λ r

α

Figure 3.11: Variation of λr with respect to α for PPF at Re = 3500, Vcy = 0.04.

-0.08

-0.06

-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0 10 20 30 40 50 60 70 80 90 100

λ r

L

Figure 3.12: Variation of λr with respect to L for PPF at Re = 3500, Vcy = 0.04.


-0.04

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

0 2 4 6 8 10 12

λ r

L

L1 L2 L3 L4 L5 L6 L7 L8 L9 L10

Figure 3.13: Variation of λr with respect to Ln for PPF at Re = 3500, Vcy = 0.04.Expected values (computed from local LSA) are plotted with lines. Solid dots representcomputed values from global LSA.

=2.77 1oL L =

=5.54 2oL L =

=8.31 3oL L =

=11.08 4o =L L

Figure 3.14: v′− modes with respect to Lno for PPF at Re = 3500, Vcy = 0.04.

Chapter 4

Time evolution of modes

In this chapter, we shall study the time evolution of modes [35]. All the analyses will be

done in the x-frame.

4.1 Evolution equation corresponding to local LSA

For obtaining the evolution equation, we assume the solutions of the form

(

u′(x, y, z, t)

v′(x, y, z, t)

w′(x, y, z, t)

, p′(x, y, z, t)) = (

u(y, t)

v(y, t)

w(y, t)

, p(y, t))eiαx+βz. (4.1.1)

Substituting the cross-stream velocity component of disturbance from Eq. (4.1.1) into

Eq. (2.1.8) and denoting the first derivative with respect to y by the symbol D, we get

following evolution equation in v:

[(∂

∂t+ iαU)(D2 − k2)− iαD2U − 1

Re(D2 − k2)2]v = 0 on Ω× (0, T ) (4.1.2)

with boundary conditions

v = Dv = 0 in the free stream and at solid walls (4.1.3)

and initial condition

v(y, 0) = vo. (4.1.4)

4.2 Evolution equation corresponding to global LSA 51

After solving the evolution equation in v, the horizontal velocities w and u can be

recovered with help of Eqs. (4.1.1), (2.1.17) and (2.1.6). Eq. (4.1.2) can further be

written as˙v = Aevv on Ω× (0, T ), (4.1.5)

where the evolution operator, Aev is given as

Aev = (D2 − k2)−1[−iαU(D2 − k2) + iαD2U +1

Re(D2 − k2)2]. (4.1.6)

The solution of Eq. (4.1.5) in conjunction with the initial condition (4.1.4) and boundary

conditions (4.1.3) can be expressed as

v(y, t) = eAevtvo. (4.1.7)

This represents the time evolution of an initial disturbance. We have solved the evolution

equation by spectral collocation method based on Chebyshev polynomials of the first

kind. It can be noted that here we do not need mbc explicitly for handling the boundary

conditions.

We shall use the term “EEL” for analysis related to evolution equation correspond-

ing to local LSA.

Disturbance measure.— A disturbance measure is required for quantifying the size

of the disturbance. We shall use kinetic energy density of the disturbance in Fourier

space [35] as the disturbance measure. It is given as

E =1

2

∫ H/2

−H/2(|u|2 + |v|2 + |w|2)dy =

1

2k2

∫ H/2

−H/2(|Du|2 + k2|v|2 + |η|2)dy, (4.1.8)

where η is corresponding to the cross-stream perturbation vorticity, η′ =∂u′

∂z− ∂w′

∂x.

4.2 Evolution equation corresponding to global

LSA

Eq. (1.5.10) in conjunction with Eq. (1.5.11) can be considered as evolution equation cor-

responding to global LSA for given initial condition (u′(x, t = 0), p′(x, t = 0)) = (u′o, p′o).

The technique for finding the solution to this equation has already been described in

Chapter 3.

4.3 DNS 52

The term “EEG” has been used for analysis related to evolution equation corre-

sponding to global LSA.

Disturbance measure.— As a disturbance measure, here we shall take kinetic energy

of the perturbation velocity u′ in the flow domain Ω:

E =1

2

∫Ω

u′ · u′dΩ. (4.2.9)

4.3 DNS

DNS is another option for studying the time evolution of modes. However, it can be

costlier compared to the EEL. We have performed DNS for the Eq. (1.4.8) in conjunction

with Eq. (1.4.9). Considered disturbance measure here is same as provided in Section 4.2.

Note.— Care should be taken here as the notation E has been used for disturbance

measure in cases of EEL, EEG and DNS.

4.4 Growth rate at a general time

We have seen the limitation of LSA by the methods proposed in Chapter 2 and Chapter 3.

We were not able to find the growth rate at t > 0 for the case of Vcy 6= 0. The discussion

involved in this section may be able to sort out the problem.

We conjecture that following equation is true in general:

(

u′(x, y, z, t+ ∆t)

v′(x, y, z, t+ ∆t)

w′(x, y, z, t+ ∆t)

, p′(x, y, z, t+ ∆t)) = (

u′(x, y, z, t)

v′(x, y, z, t)

w′(x, y, z, t)

, p′(x, y, z, t))eλ(t)t.

(4.4.10)

Here λ(t) = λr(t) + iλi(t) with λr(t) giving the growth rate and −λi(t) giving the

(circular) frequency at time t. To increase the accuracy of λ(t), we should decrease ∆t.

Considering Eq. (4.4.10), it is easy to check that

λr(t) =1

2∆tln

(E(t+ ∆t)

E(t)

). (4.4.11)

4.5 Comparison of DNS with EEL 53

It should be noted that Eq. (4.4.10) is surely true fort = 0 if Vcy 6= 0,

t ∈ [0, T ] if Vcy = 0.

Hence, Eq. (4.4.11) will have to be surely true fort = 0 if Vcy 6= 0,

t ∈ [0, T ] if Vcy = 0.

In the remaining part of the present chapter, results have been presented

with respect to Vcx = 0 corresponding to 2D LSA case at β ≡ 0. Also, DNS

has been done for modes obtained from global LSA.

4.5 Comparison of DNS with EEL

Fig. 4.1 shows the comparative results obtained from EEL and DNS for PCF at Re =

400, α = 0.85, Vcy = 0.11. Similarly, Fig. 4.2 shows the comparison of results obtained

from EEL and DNS for PPF at Re = 1000, α = 2.50, Vcy = 0.08. As expected, results

from EEL and DNS perfectly match for each case. We note that EEL, EEG and DNS

should give same results for evolution of modes if the magnitude of the initial disturbance

is small.

4.6 DNS for crosschecking Recr

To crosscheck the Recr with respect to Vcy, we have performed DNS for both PCF and

PPF for a few cases. At t = 0, the growth rate as well as the propagation speed of the

disturbance from DNS and LSA are in excellent agreement for each case. This increases

our confidence in the results obtained from LSA.

4.6 DNS for crosschecking Recr 54

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10

E(t

)/E

(0)

t

DNS EEL

Figure 4.1: Time evolution of the normalized kinetic energy (density) of the disturbance(in Fourier space) for PCF at Re = 400, α = 0.85, Vcy = 0.11.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10

E(t

)/E

(0)

t

DNS EEL

Figure 4.2: Time evolution of the normalized kinetic energy (density) of the disturbance(in Fourier space) for PPF at Re = 1000, α = 2.50, Vcy = 0.08.


4.6.1 PCF

To start with, firstly we present results for PCF at Vcy = 0. Here it is expected that there

should always be monotonic decrease in the disturbance energy for any combination of

Re and α. Fig. 4.3 clearly indicates the same for a few selected combinations of Re and

α.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10

E(t

)/E

(0)

t

Re=10000, α=πRe=1000, α=π

Re=10, α=2t=0

t

t

t

=0.5

=1.0

=2.0

Figure 4.3: Time evolution of the normalized kinetic energy of the disturbance for PCFat Vcy = 0 and indicated combinations of Re and α. Also shown is the time evolution ofv′− mode for the combination Re = 10, α = 2.

Fig. 4.4 shows the results for PCF at Vcy = 0.11, α = 0.85 and indicated values

of Re. It can be recalled that this Vcy and α is corresponding to the Recro for PCF.

The figure clearly shows the growth followed by a decay in the disturbance energy for

Re > Recro. This kind of growth might lead the flow to change its state from laminar to

turbulent. The figure also indicates that there might exist a critical Reynolds number,

Recrc(< Recro for PCF) with following properties:

For Recrc < Re < Recro, the disturbance energy will firstly decay. Then it will

increase and subsequently change its behaviour to have monotonic decay.

For Re < Recrc, the disturbance energy will decay monotonically.


0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 2 4 6 8 10

E(t

)/E

(0)

t

Re=400Re=361Re=300Re=250

-120

-80

-40

0

40

0.000 0.001 0.002

ln[E

(t)/

E(0

)]

t

×10-6

=0t

=2t

=4t

=6t

=8t

=10t

Figure 4.4: Time evolution of the normalized kinetic energy of the disturbance for PCFat Vcy = 0.11, α = 0.85 and indicated values of Re. Also shown is the time evolution ofv′− mode for the case of Re = 400.

4.6.2 PPF

To start with, here we present results for PPF at Vcy = 0, α = 1.02 and three chosen value

of Re(= 5000, 5773, 7000). As expected, Fig. 4.5 clearly shows the monotonic increase

in the disturbance energy for Re > 5772. The figure also shows the monotonic decrease

in the disturbance energy for Re < 5772.

Fig. 4.6 shows the results for PPF at Vcy = 0.08, α = 2.50 and three chosen value of

Re(= 840, 920, 1000). This Vcy and α is corresponding to the Recro for PPF. The figure

clearly shows the growth followed by a decay in the disturbance energy for Re > Recro,

which might lead the flow to change its state from laminar to turbulent. The figure also

suggests that, unlike PCF, here we have only monotonic decay in the disturbance energy

for Re < Recro.


0.94

0.96

0.98

1.00

1.02

1.04

0 1 2 3 4 5 6

E(t

)/E

(0)

t

Re=7000Re=5773Re=5000

=6t

=0t

=4t

=2t

Figure 4.5: Time evolution of the normalized kinetic energy of the disturbance for PPFat Vcy = 0, α = 1.02 and indicated values of Re. Also shown is the time evolution of v′−mode for the case of Re = 7000.

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 1 2 3 4 5 6

E(t

)/E

(0)

t

Re=1000Re=920Re=840

-30

-15

0

15

30

0.000 0.001 0.002

ln[E

(t)/

E(0

)]

t

×10-6

=0t

=2t

=4t

=6t

Figure 4.6: Time evolution of the normalized kinetic energy of the disturbance for PPFat Vcy = 0.08, α = 2.50 and indicated values of Re. Also shown is the time evolution ofv′− mode for the case of Re = 1000.

Chapter 5

Conclusions and future scope

5.1 Conclusions

The CSOS equation is derived for local LSA of parallel shear flows to include disturbances

moving with nonzero cross-stream velocity component. The equivalence of the CSOS

equation with the OS equation is shown for disturbances moving with zero cross-stream

velocity component. The critical Reynolds numbers for the onset of instability are found

to be 360.63 and 916.85 for PCF and PPF respectively. These numbers are in close

agreement with the critical Reynolds numbers for the transition to turbulence obtained

from experimental analyses of the corresponding flows.

It is shown that global LSA of parallel shear flows for periodic boundary conditions

in streamwise and spanwise directions is equivalent to the local LSA. The results from

LSA are in excellent agreement with DNS. The time evolution of modes suggests that

the growth rate varies with respect to time for the case of nonzero cross-stream velocity

component of disturbances.

5.2 Future scope

Following relevant studies are suggested:

The methods for LSA presented in the thesis should be applied/extended to other

parallel shear flows, e.g. Hagen-Poiseuille flow and weakly nonparallel shear flows,

e.g. Blasius boundary layer flow.

To get more idea of the relevant physical phenomena, the 3D global LSA cor-

5.2 Future scope 59

responding to nonperiodic boundary conditions should be done for parallel and

weakly nonparallel shear flows. Corresponding DNS will help further in linking the

group velocity with growth rate.

It should also be interesting to find the reasons for the observed behaviours of

evolution of modes.

More analysis can be done for the conjecture presented in Chapter 4.

The analysis should be explored to get more insight of turbulence.

Appendix

Appendix A

Issues related to LSA

A.1 Linear combination of modes

We know that if v and w are eigenvectors of a matrix A corresponding to the same

eigenvalue λ, so are v + w (provided v 6= w) and cv for any c 6= 0. The same is

true in our eigenvalue analysis as well. For a test case, we consider PCF at Re =

10000, α = π, Vcy = 0 (with Vcx = 0). The (least stable) eigenvalues for the present

case are λ = −1.182 × 10−1 ± 2.747i from local LSA and λ = −1.194 × 10−1 ± 2.748i

from global LSA. As expected, the values of λ obtained from local and global LSA are

reasonably same. Figs. A.1(a) and A.1(b) show the respective v′−modes corresponding

to λ = −1.182 × 10−1 − 2.747i and λ = −1.182 × 10−1 + 2.747i. Fig. A.1(c) shows the

v′−mode corresponding to one of the two mentioned eigenvalues obtained from global

LSA. The v′−mode corresponding to the other eigenvalue is of the same nature and

hence, not presented here. It can be seen that the v′−mode corresponding to global LSA

is linear combination of the two v′−modes corresponding to local LSA. It can further

be noted that Vtx corresponding to the v′−mode shown in Fig. A.1(a) is positive while

it is negative corresponding to the v′−mode shown in Fig. A.1(b). Hence, it is expected

that the portions near the upper and lower walls of Fig. A.1(c) would respectively be

travelling towards right and left with time. A DNS corresponding to the global LSA

clearly indicated the same.

A.2 The two variants of PCF 62

(a) (b) (c)

Figure A.1: PCF at Re = 10000, α = π, Vcy = 0. v′− modes from local LSA corre-sponding to (a) positive Vtx; (b) negative Vtx. (c): v′− mode from global LSA.

A.2 The two variants of PCF

Fig. A.2 shows the two variants of PCF, which are normally used in relevant analysis.

Following relations hold between the two variants:

Vcx|PCF2 = Vcx|PCF1 + UW , Vcy|PCF2 = Vcy|PCF1, Vcz|PCF2 = Vcz|PCF1,

Vtx|PCF2 = Vtx|PCF1 + UW , Vty|PCF2 = Vty|PCF1, Vtz|PCF2 = Vtz|PCF1. (A.2.1)

It should be noted that Eq. (A.2.1) stands due to dynamic similarity1.

−Uw

wU

x

y H

x

y HwU(y) U+

(a) (b)

U2 w

Figure A.2: The two variants of PCF: (a) PCF1; (b) PCF2.

1S. S. Vemuri (2011) [16] also tried to obtained results corresponding to the dynamic similarity incase of PCF.

Appendix B

2D LSA Matlab code

The following listed MATLAB codes are found by modification of relevant MATLAB rou-

tines (written by S. Reddy). The original MATLAB routines can be found in Appendix

A.6 of [35].

B.1 Code I

1. Main1.m

clc;

clear all;

%% =========================2−D LSA Matlab code========================

% This is 2−D LSA Matlab code, valid for time(t)=0 in general. The code

% uses Chebfun (a collection of algorithms and an open−source software

% system in object−oriented Matlab).

% The code is written for plane Couette flow and plane Poiseuille flow.

% The channel height is considered to be 2 for both flows. Velocity

% profile for plane Couette flow has been chosen to be U=y while that

% for plane Poiseuille flow has been chosen to be U=1−yˆ2. (y=0 being

% the center line for both profiles.)

% For simplicity, eigenfunctions have been shown without hat everywhere.

global iflow name Re alp Vcx Vcy neig see nthmode usc vsc zvorsc X Y

INPUT1();

[X,Y]=mesh1(alp);

[V,D]=solution1();

B.1 Code I 64

[Vsr,Vsi]=ploteig1(V,D);

mkmode1(Vsr,Vsi);

2. INPUT1.m

function []=INPUT1()


% Function to take the inputs.

iflow='poi'; % poi for plane Poiseuille flow, cou for plane Couette flow

Re=10000; % Reynolds number

alp=1.00; % Streamwise wave number

Vcx=0; % x−component of velocity of moving frame

Vcy=0; % y−component of velocity of moving frame

neig=4; % Number of eigenmode(s) (of largest growth rate)

% to be computed

see nthmode=1; % nth mode (starting from largest growth rate) to be seen

% For this case eigen function will also be plotted. (Set see nthmode=0,

% if don't want to see the mode.)

usc=1; % Scale by which we want to see the u'−mode (streamwise

% velocity component of perturbation)

vsc=1; % Scale by which we want to see the v'−mode (cross−stream% velocity component of perturbation)

zvorsc=1; % Scale by which we want to see the z−vorticity'−mode% (spanwise vorticity component of perturbation)

3. mesh1.m

function [X,Y]=mesh1(alp)

B.1 Code I 65

% This function creates mesh. This mesh has been used for making modes.

% Following is for making mesh in x−direction.

l1=0; l2=l1+2*pi*1/alp; % (l2−l1)=spatial period

X=linspace(l1,l2);

% Following is for making mesh in y−direction.

% Following is for lower half of the domain.

hw=0.1; % Height near the wall

hy1=0.00001; % First element width

nw=100; % No. of elements near the wall

nr=100; % No. of elements in rest part

ny=nw+nr; % Total no. of elements

r=ratio(nw,hw/hy1);

h(1)=hy1; % h(i)=height of ith element

for i=2:nw

h(i)=h(i−1)*r;end

Y(1)=−1;for i=2:nw+1

Y(i)=Y(i−1)+h(i−1);end

h(nw+1)=h(nw);

r=ratio(nw,(1−hw)/h(nw+1));for i=nw+2:ny

h(i)=h(i−1)*r;end

for i=nw+2:ny+1


% Following is for upper half of the domain. (Mirror image with respect

% to lower half of the domain.)

j=0;

for i=ny+2:2*ny+1

Y(i)=−Y(ny−j);j=j+1;

end

B.1 Code I 66

function [r]=ratio(n,s)

% Function to find the ratio (r) for an exponential mesh generation

% using Newton−Raphson method.

r=sˆ(1/(n−1));for i=1:50

res=−(rˆn−s*r+s−1);if abs(res)<10ˆ−10

break;

else

del r=res/(n*rˆ(n−1)−s);r=r+del r;

end

end

4. solution1.m

function [V,D]=solution1()


% Function to find the solution.

dom=[−1 1]; % Defines the domain we are working on.

% Assign the equation to two chebops A and B such that A(v)=lambda*B(v).

if strcmp(iflow,'poi')

A=chebop(@(ym,v) (diff(v,4)−2*alpˆ2*diff(v,2)+alpˆ4*v)/Re...−1i*alp*(diag(1−(ym).ˆ2−Vcx)*(diff(v,2)−alpˆ2*v)+2*v)...+Vcy*(diff(v,3)−alpˆ2*diff(v,1)),dom);

elseif strcmp(iflow,'cou')

A=chebop(@(ym,v) (diff(v,4)−2*alpˆ2*diff(v,2)+alpˆ4*v)/Re...−1i*alp*(diag(ym−Vcx)*(diff(v,2)−alpˆ2*v))...+Vcy*(diff(v,3)−alpˆ2*diff(v,1)),dom);

end

B=chebop(@(ym,v) diff(v,2)−alpˆ2*v,dom);

B.1 Code I 67

% Assign boundary conditions to the chebop.

A.bc=@(ym,v) [v(−1),feval(diff(v),−1),v(1),feval(diff(v),1)];

% Solve the eigenvalue problem.

[V,D]=eigs(A,B,neig,'lr');

D=diag(D);

disp('Eigen value(s) in decreasing order of growth rate:')

for i=1:1:neig

fprintf('%+3.3E %+3.3Ei \n',real(D(i)),imag(D(i)))end

5. ploteig1.m

function [Vsr,Vsi]=ploteig1(V,D)


% Function to plot the eigen values and eigen function.


name='plane Poiseuille flow';


name='plane Couette flow';

end

% Plotting of eigenvalues

figure(1)

plot(imag(D),real(D),'.r','markersize',14)

grid on

title(['Eigenvalue spectrum of ',name,' for Re=',num2str(Re,5)...

,', alp=',num2str(alp,4),', Vcx=',num2str(Vcx,3),', Vcy='...

,num2str(Vcy,3)])

xlabel('\lambda i'); ylabel('\lambda r');

% Plotting of eigenfunction

B.1 Code I 68

Vsr=real(V(:,see nthmode)); % Real part of eigen function of interest.

Vsi=imag(V(:,see nthmode)); % Imaginary part of eigen function of

% interest.

figure(2)

subplot(2,1,1)

plot(Vsr)

title('Real part of eigenfunction of the considered case');

grid on

xlabel('y'); ylabel('v r');

subplot(2,1,2)

plot(Vsi)

title('Imaginary part of eigenfunction of the considered case');

grid on

xlabel('y'); ylabel('v i');

6. mkmode1.m

function []=mkmode1(Vsr,Vsi)


% Function to create modes.

if see nthmode 6=0

% Value of eigen function

Vr=funval1(Vsr,Y);

Vi=funval1(Vsi,Y);

% Value of first derivative of eigen function

dVr=funval1(diff(Vsr),Y);

dVi=funval1(diff(Vsi),Y);

% Value of second derivative of eigen function

d2Vr=funval1(diff(Vsr,2),Y);

B.1 Code I 69

d2Vi=funval1(diff(Vsi,2),Y);

% Following is for finding the modes.

for i=1:1:length(Y)

for j=1:1:length(X)

umode(i,j)=usc*(−1/alp)*(dVr(i)*sin(alp*X(j))...+dVi(i)*cos(alp*X(j)));

vmode(i,j)=vsc*(Vr(i)*cos(alp*X(j))−Vi(i)*sin(alp*X(j)));zvormode(i,j)=zvorsc*(−1/alp)*(alpˆ2*(Vr(i)*sin(alp*X(j))...

+Vi(i)*cos(alp*X(j)))−(d2Vr(i)*sin(alp*X(j))...+d2Vi(i)*cos(alp*X(j))));

end

end

figure(3)

contourf(X,Y,umode,100,'LineStyle','none');

title('u''−mode'); xlabel('x'); ylabel('y');

colorbar;

figure(4)

contourf(X,Y,vmode,100,'LineStyle','none');

title('v''−mode'); xlabel('x'); ylabel('y');

colorbar;

figure(5)

contourf(X,Y,zvormode,100,'LineStyle','none');

title('z−vorticity''−mode'); xlabel('x'); ylabel('y');

colorbar;

end

7. funval1.m

function [val]=funval1(f,Y)

% Function to return the numerical value of given function(f) at

% specified points Y(i) defined in the array Y.

NP=length(f); % Number of collocation points

an rev=chebpoly(f); % Coefficients for Chebyshev poynomials T NP−1,% T NP−2,...,T 0

B.1 Code I 70

an=an rev(end:−1:1); % Coefficients for T 0,T 1,...,T NP−1for i=1:1:length(Y)

for j=1:1:NP

Tv(j,i)=cos((j−1)*acos(Y(i))); % Tv(j,i)=value of T j−1% at Y(i).

end

end

val=zeros(length(Y),1); % Initialize value of the function at Y(i)

for i=1:1:length(Y)

for j=1:1:NP

val(i,1)=val(i,1)+an(j)*Tv(j,i);

end

end

B.2 Code II 71

B.2 Code II

1. Main2.m

clc;

clear all;

%% =========================2−D LSA Matlab code========================

% This is 2−D LSA Matlab code, valid for time(t)=0 in general.

% The code is written for plane Couette flow and plane Poiseuille flow.

% The channel height is considered to be 2 for both flows. Velocity

% profile for plane Couette flow has been chosen to be U=y while that

% for plane Poiseuille flow has been chosen to be U=1−yˆ2. (y=0 being

% the center line for both profiles.)

% Through this code, evolution of the mode has also been presented.

% For simplicity, eigenfunctions have been shown without hat everywhere.

global iflow name Re alp Vcx Vcy neig see nthmode usc vsc zvorsc NP N ...

t X Y

INPUT2();

[X,Y]=mesh2(alp);

[V,D]=solution2();

[Vsr,Vsi]=ploteig2(V,D);

[upmax,vpmax,zvpmax]=mkmode2(Vsr,Vsi);

evolution2(Vsr,Vsi,upmax,vpmax,zvpmax);

2. INPUT2.m

function []=INPUT2()


t X Y

% Function to take the inputs.

iflow='poi'; % poi for plane Poiseuille flow, cou for plane Couette flow

Re=10000; % Reynolds number

alp=1.00; % Streamwise wave number

B.2 Code II 72

Vcx=0; % x−component of velocity of moving frame

Vcy=0; % y−component of velocity of moving frame

neig=4; % Number of eigenmode(s) (of largest growth rate)

% to be printed

see nthmode=1;% nth mode (starting from largest growth rate) to be seen

% For this case eigen function will also be plotted. (Set see nthmode=0,

% if don't want to see the mode.)

t=2; % Time at which we want to see the evolved mode.

usc=1; % Scale by which we want to see the u'−mode (streamwise

% velocity component of perturbation)

vsc=1; % Scale by which we want to see the v'−mode (cross−stream% velocity component of perturbation)

zvorsc=1; % Scale by which we want to see the z−vorticity'−mode% (spanwise vorticity component of perturbation)

NP=300; % Number of collocation points

N=NP−1;

3. mesh2.m

function [X,Y]=mesh2(alp)

% This function creates mesh. This mesh has been used for making modes,

% finding numerical integration in y−direction, etc.

% Following is for making mesh in x−direction.

l1=0; l2=l1+2*pi*1/alp; % (l2−l1)=spatial period

X=linspace(l1,l2);

% Following is for making mesh in y−direction.

% Following is for lower half of the domain.

B.2 Code II 73

hw=0.1; % Height near the wall

hy1=0.00001; % First element width

nw=100; % No. of elements near the wall

nr=100; % No. of elements in rest part

ny=nw+nr; % Total no. of elements

r=ratio(nw,hw/hy1);

h(1)=hy1; % h(i)=height of ith element

for i=2:nw

h(i)=h(i−1)*r;end

Y(1)=−1;for i=2:nw+1


h(nw+1)=h(nw);

r=ratio(nw,(1−hw)/h(nw+1));for i=nw+2:ny

h(i)=h(i−1)*r;end

for i=nw+2:ny+1


% Following is for upper half of the domain. (Mirror image with respect

% to lower half of the domain.)

j=0;

for i=ny+2:2*ny+1

Y(i)=−Y(ny−j);j=j+1;

end

function [r]=ratio(n,s)

% Function to find the ratio (r) for an exponential mesh generation

% using Newton−Raphson method.

r=sˆ(1/(n−1));for i=1:50

res=−(rˆn−s*r+s−1);if abs(res)<10ˆ−10

B.2 Code II 74

break;

else

del r=res/(n*rˆ(n−1)−s);r=r+del r;

end

end

4. Dmat2.m

function [D0,D1,D2,D3,D4]=Dmat2


t X Y

% Function to create Chebyshev differentiation matrices.

% D0, D1, D2, D3 and D4 are zero'th, first, second, third and fourth

% derivative matrices respectively.

% Create D0

D0=[]; % Assigns a 0−by−0 empty matrix to D0

vec=(0:1:N)';

for j=0:1:N

D0=[D0 cos(j*pi*vec/N)];

end

% Create higher derivative matrices

D1=[zeros(NP,1) D0(:,1) 4*D0(:,2)];

D2=[zeros(NP,1) zeros(NP,1) 4*D0(:,1)];

D3=[zeros(NP,1) zeros(NP,1) zeros(NP,1)];

D4=[zeros(NP,1) zeros(NP,1) zeros(NP,1)];

for n=3:1:N

D1=[D1 2*n*D0(:,n)+(n/(n−2))*D1(:,n−1)];D2=[D2 2*n*D1(:,n)+(n/(n−2))*D2(:,n−1)];D3=[D3 2*n*D2(:,n)+(n/(n−2))*D3(:,n−1)];D4=[D4 2*n*D3(:,n)+(n/(n−2))*D4(:,n−1)];

end

B.2 Code II 75

5. solution2.m

function [V,D]=solution2()


t X Y

% Function to find the solution.

[D0,D1,D2,D3,D4]=Dmat2; % Chebyshev differentiation matrices

vec=(0:1:N)';

ym=cos(pi*vec/N);

B=D2−alpˆ2*D0;if strcmp(iflow,'poi')

U=(ones(NP,1)−(ym).ˆ2);A=(D4−2*alpˆ2*D2+alpˆ4*D0)/Re...

−1i*alp*(((U−Vcx*ones(NP,1))*ones(1,NP)).*B+2*D0)...+Vcy*(D3−alpˆ2*D1);

mbc=−200−200i; % For handling the boundary conditions

A=[mbc*D0(1,:); mbc*D1(1,:); A(3:NP−2,:); mbc*D1(NP,:); mbc*D0(NP,:)];

B=[D0(1,:); D1(1,:); B(3:NP−2,:); D1(NP,:); D0(NP,:)];


U=ym;

A=(D4−2*alpˆ2*D2+alpˆ4*D0)/Re...−1i*alp*(((U−Vcx*ones(NP,1))*ones(1,NP)).*B)...+Vcy*(D3−alpˆ2*D1);

mbc=−200−200i; % For handling the boundary conditions

A=[mbc*D0(1,:); mbc*D1(1,:); A(3:NP−2,:); mbc*D1(NP,:); mbc*D0(NP,:)];

B=[D0(1,:); D1(1,:); B(3:NP−2,:); D1(NP,:); D0(NP,:)];

end

% Solve the eigenvalue problem.

[V,D]=eig(B\A);

% Following is for ordering of eigen values in descending order of real

% part and getting the corresponding eigen vectors.

D=diag(D);

B.2 Code II 76

[Dreal,IX]=sort(real(D),'descend');

V=V(:,IX);

D=D(IX); % Ordering related work ends here.

disp('Eigen value(s) in decreasing order of growth rate:')

for i=1:1:neig

fprintf('%+3.3E %+3.3Ei \n',real(D(i)),imag(D(i)))end

6. ploteig2.m

function [Vsr,Vsi]=ploteig2(V,D)


t X Y

% Function to plot the eigen values and eigen function.


name='plane Poiseuille flow';


name='plane Couette flow';

end

% Plotting of eigenvalues

figure(1)

plot(imag(D),real(D),'.r','markersize',14)

grid on

title(['Eigenvalue spectrum of ',name,' for Re=',num2str(Re,5)...

,', alp=',num2str(alp,4),', Vcx=',num2str(Vcx,3),', Vcy='...

,num2str(Vcy,3)])

xlabel('\lambda i'); ylabel('\lambda r');

% Plotting of eigenfunction

Vsr=real(V(:,see nthmode)); % Real part of eigen function of interest.

Vsi=imag(V(:,see nthmode)); % Imaginary part of eigen function of

% interest.

B.2 Code II 77

figure(2)

subplot(2,1,1)

plot(Y,funval2(Vsr,Y,NP,0))

title('Real part of eigenfunction of the considered case');

grid on


subplot(2,1,2)

plot(Y,funval2(Vsi,Y,NP,0))

title('Imaginary part of eigenfunction of the considered case');

grid on


7. mkmode2.m

function [upmax,vpmax,zvpmax]=mkmode2(Vsr,Vsi)


t X Y

% Function to create modes.

if see nthmode 6=0


Vr=funval2(Vsr,Y,NP,0);

Vi=funval2(Vsi,Y,NP,0);


dVr=funval2(Vsr,Y,NP,1);

dVi=funval2(Vsi,Y,NP,1);

% Value of second derivative of eigen function

d2Vr=funval2(Vsr,Y,NP,2);

d2Vi=funval2(Vsi,Y,NP,2);

% Following is for finding the modes.

B.2 Code II 78

for i=1:1:length(Y)

for j=1:1:length(X)

umode(i,j)=usc*(−1/alp)*(dVr(i)*sin(alp*X(j))...+dVi(i)*cos(alp*X(j)));

vmode(i,j)=vsc*(Vr(i)*cos(alp*X(j))−Vi(i)*sin(alp*X(j)));zvormode(i,j)=zvorsc*(−1/alp)*(alpˆ2*(Vr(i)*sin(alp*X(j))...

+Vi(i)*cos(alp*X(j)))−(d2Vr(i)*sin(alp*X(j))...+d2Vi(i)*cos(alp*X(j))));

end

end

upmax=max(max(umode));

vpmax=max(max(vmode));

zvpmax=max(max(zvormode));

figure(3)

contourf(X,Y,umode,100,'LineStyle','none');

title('u''−mode'); xlabel('x'); ylabel('y');

colorbar;

figure(4)

contourf(X,Y,vmode,100,'LineStyle','none');

title('v''−mode'); xlabel('x'); ylabel('y');

colorbar;

figure(5)

contourf(X,Y,zvormode,100,'LineStyle','none');

title('z−vorticity''−mode'); xlabel('x'); ylabel('y');

colorbar;

end

8. evolution2.m

function []=evolution2(Vsr,Vsi,upmax,vpmax,zvpmax)


t X Y

% Function to find the evolution of disturbance.


B.2 Code II 79

Vr=funval2(Vsr,Y,NP,0);

Vi=funval2(Vsi,Y,NP,0);


dVr=funval2(Vsr,Y,NP,1);

dVi=funval2(Vsi,Y,NP,1);

for i=1:1:length(Y)

uen0(i)=0.5*(abs((1i/alp)*(dVr(i)+1i*dVi(i))))ˆ2;

ven0(i)=0.5*(abs(Vr(i)+1i*Vi(i)))ˆ2;

en0(i)=uen0(i)+ven0(i);

end

enden0=trapz(Y,en0); % Kinetic energy density of the disturbance in

% Fourier space at initial time. (Linear interpolation is assumed. So

% trapezoidal method of numerical integration suffices.)

[D0,D1,D2,D3,D4]=Dmat2; % Chebyshev differentiation matrices

vec=(0:1:N)';

y=cos(pi*vec/N);

B=D2−alpˆ2*D0;if strcmp(iflow,'poi')

U=(ones(NP,1)−(y).ˆ2);A=(D4−2*alpˆ2*D2+alpˆ4*D0)/Re...

−1i*alp*((U*ones(1,NP)).*B+2*D0);A=[D0(1,:); D1(1,:); A(3:NP−2,:); D1(NP,:); D0(NP,:)];

B=[D0(1,:); D1(1,:); B(3:NP−2,:); D1(NP,:); D0(NP,:)];


U=y;

A=(D4−2*alpˆ2*D2+alpˆ4*D0)/Re...−1i*alp*((U*ones(1,NP)).*B);

A=[D0(1,:); D1(1,:); A(3:NP−2,:); D1(NP,:); D0(NP,:)];

B=[D0(1,:); D1(1,:); B(3:NP−2,:); D1(NP,:); D0(NP,:)];

end

C=(inv(B))*A;

Vt=expm(C*t)*(Vsr+Vsi*1i);

Vsrt=real(Vt);

Vsit=imag(Vt);

if see nthmode 6=0

B.2 Code II 80

% Value of evolved eigen function at time t

Vrt=funval2(Vsrt,Y,NP,0);

Vit=funval2(Vsit,Y,NP,0);

% Value of first derivative of evolved eigen function at time t

dVrt=funval2(Vsrt,Y,NP,1);

dVit=funval2(Vsit,Y,NP,1);

% Value of second derivative of evolved eigen function at time t

d2Vrt=funval2(Vsrt,Y,NP,2);

d2Vit=funval2(Vsit,Y,NP,2);

figure(6)

subplot(2,1,1)

plot(Y,Vrt)

title(['Evolved real part of eigenfunction of the considered case'...

,' at t=',num2str(t,5)]);

grid on


subplot(2,1,2)

plot(Y,Vit)

title(['Evolved imaginary part of eigenfunction of the considered'...

,' case at t=',num2str(t,5)]);

grid on


% Following is for finding the evolved modes at time t.

for i=1:1:length(Y)

for j=1:1:length(X)

umodet(i,j)=usc*(−1/alp)*(dVrt(i)*sin(alp*X(j))...+dVit(i)*cos(alp*X(j)));

vmodet(i,j)=vsc*(Vrt(i)*cos(alp*X(j))−Vit(i)*sin(alp*X(j)));zvormodet(i,j)=zvorsc*(−1/alp)*(alpˆ2*(Vrt(i)*sin(alp*X(j))...

+Vit(i)*cos(alp*X(j)))−(d2Vrt(i)*sin(alp*X(j))...+d2Vit(i)*cos(alp*X(j))));

end

end

figure(7)

B.2 Code II 81

contourf(X,Y,umodet,100,'LineStyle','none');

title(['Evolved u''−mode at t=',num2str(t,5)]); xlabel('x');

ylabel('y');

caxis([−1*upmax upmax]);

colorbar;

figure(8)

contourf(X,Y,vmodet,100,'LineStyle','none');

title(['Evolved v''−mode at t=',num2str(t,5)]); xlabel('x');

ylabel('y');

caxis([−1*vpmax vpmax]);

colorbar;

figure(9)

contourf(X,Y,zvormodet,100,'LineStyle','none');

title(['Evolved z−vorticity''−mode at t=',num2str(t,5)]); xlabel('x');

ylabel('y');

caxis([−1*zvpmax zvpmax]);

colorbar;

end

for i=1:1:length(Y)

uent(i)=0.5*(abs((1i/alp)*(dVrt(i)+1i*dVit(i))))ˆ2;

vent(i)=0.5*(abs(Vrt(i)+1i*Vit(i)))ˆ2;

ent(i)=uent(i)+vent(i);

end

endent=trapz(Y,ent); % Kinetic energy density of the disturbance in

% Fourier space at time t.(Again linear interpolation is assumed. So

% trapezoidal method of numerical integration suffices.)

disp(['Normalized kinetic energy of the disturbance at ...

t=',num2str(t,5),':'])

fprintf('%+12.12E \n',endent/enden0)

9. funval2.m

function [val]=funval2(an,Y,NP,k)

% Function to return the numerical value of a function (k=0 case) or

% it's 1st derivative (k=1 case) or it's 2nd derivative (k=2 case) at

% specified points Y(i) defined in the array Y, when 'an' is provided.

B.2 Code II 82

% Note: Here 'an' is a vector whose respective elements (from first to

% last) are coefficients for Chebyshev poynomials T 0,T 1,...,% T NP−1.

if k==0

for i=1:1:length(Y)

for j=1:1:NP

Tv0(j,i)=cos((j−1)*acos(Y(i))); % Tv0(j,i)=value of T j−1% at Y(i).

end

end

val=zeros(length(Y),1); % Initialize value at Y(i)

for i=1:1:length(Y)

for j=1:1:NP

val(i,1)=val(i,1)+an(j)*Tv0(j,i);

end

end

elseif k==1

for i=1:1:length(Y)

for j=1:1:NP

Tv0(j,i)=cos((j−1)*acos(Y(i)));if j==1

Tv1(j,i)=0; % Tv1(j,i)=value of 1st derivative of

% T j−1 at Y(i).

elseif j==2

Tv1(j,i)=Tv0(j−1,i);elseif j==3

Tv1(j,i)=4*Tv0(j−1,i);else

Tv1(j,i)=2*(j−1)*Tv0(j−1,i)+((j−1)/(j−3))*Tv1(j−2,i);end

end

end

val=zeros(length(Y),1);

for i=1:1:length(Y)

for j=1:1:NP


end

end

elseif k==2

B.2 Code II 83

for i=1:1:length(Y)

for j=1:1:NP

Tv0(j,i)=cos((j−1)*acos(Y(i)));if j==1

Tv1(j,i)=0;

Tv2(j,i)=0; % Tv2(j,i)=value of 2nd derivative of

% T j−1 at Y(i).

elseif j==2

Tv1(j,i)=Tv0(j−1,i);Tv2(j,i)=Tv1(j−1,i);

elseif j==3

Tv1(j,i)=4*Tv0(j−1,i);Tv2(j,i)=4*Tv1(j−1,i);

else

Tv1(j,i)=2*(j−1)*Tv0(j−1,i)+((j−1)/(j−3))*Tv1(j−2,i);Tv2(j,i)=2*(j−1)*Tv1(j−1,i)+((j−1)/(j−3))*Tv2(j−2,i);

end

end

end

val=zeros(length(Y),1);

for i=1:1:length(Y)

for j=1:1:NP


end

end

end

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